LMFDB - Embedded newform 39.5.g.a.31.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,5,Mod(31,39)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("39.31");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	39.g (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.03142856027$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 $ x^20 + 5446x^16 - 1452x^15 + 106320x^13 + 8376897x^12 - 1643220x^11 + 1054152x^10 + 232107192x^9 + 3337033804x^8 + 3105930144x^7 + 2297014848x^6 + 21034904256x^5 + 158658808624x^4 + 235144077504x^3 + 177846480000x^2 + 27353289600*x + 2103506496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			31.2
Root			$4.32919 - 4.32919i$ of defining polynomial
Character	$\chi$	$=$	39.31
Dual form			39.5.g.a.34.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.32919 + 4.32919i) q^{2} +5.19615 q^{3} -21.4837i q^{4} +(-26.0804 + 26.0804i) q^{5} +(-22.4951 + 22.4951i) q^{6} +(-16.9579 - 16.9579i) q^{7} +(23.7399 + 23.7399i) q^{8} +27.0000 q^{9} -225.814i q^{10} +(-69.5293 - 69.5293i) q^{11} -111.633i q^{12} +(-93.5895 - 140.720i) q^{13} +146.828 q^{14} +(-135.518 + 135.518i) q^{15} +138.190 q^{16} +166.635i q^{17} +(-116.888 + 116.888i) q^{18} +(-468.820 + 468.820i) q^{19} +(560.303 + 560.303i) q^{20} +(-88.1158 - 88.1158i) q^{21} +602.011 q^{22} +388.867i q^{23} +(123.356 + 123.356i) q^{24} -735.373i q^{25} +(1014.37 + 204.035i) q^{26} +140.296 q^{27} +(-364.318 + 364.318i) q^{28} -767.891 q^{29} -1173.36i q^{30} +(-66.8738 + 66.8738i) q^{31} +(-978.089 + 978.089i) q^{32} +(-361.285 - 361.285i) q^{33} +(-721.395 - 721.395i) q^{34} +884.537 q^{35} -580.060i q^{36} +(1532.33 + 1532.33i) q^{37} -4059.22i q^{38} +(-486.305 - 731.200i) q^{39} -1238.29 q^{40} +(802.986 - 802.986i) q^{41} +762.940 q^{42} +2798.62i q^{43} +(-1493.75 + 1493.75i) q^{44} +(-704.170 + 704.170i) q^{45} +(-1683.48 - 1683.48i) q^{46} +(33.8918 + 33.8918i) q^{47} +718.056 q^{48} -1825.86i q^{49} +(3183.56 + 3183.56i) q^{50} +865.862i q^{51} +(-3023.18 + 2010.65i) q^{52} -3139.54 q^{53} +(-607.368 + 607.368i) q^{54} +3626.70 q^{55} -805.159i q^{56} +(-2436.06 + 2436.06i) q^{57} +(3324.34 - 3324.34i) q^{58} +(-3959.16 - 3959.16i) q^{59} +(2911.42 + 2911.42i) q^{60} +504.230 q^{61} -579.018i q^{62} +(-457.863 - 457.863i) q^{63} -6257.62i q^{64} +(6110.87 + 1229.17i) q^{65} +3128.14 q^{66} +(4218.53 - 4218.53i) q^{67} +3579.94 q^{68} +2020.61i q^{69} +(-3829.33 + 3829.33i) q^{70} +(-3025.86 + 3025.86i) q^{71} +(640.978 + 640.978i) q^{72} +(3088.98 + 3088.98i) q^{73} -13267.5 q^{74} -3821.11i q^{75} +(10072.0 + 10072.0i) q^{76} +2358.14i q^{77} +(5270.81 + 1060.20i) q^{78} -9376.55 q^{79} +(-3604.05 + 3604.05i) q^{80} +729.000 q^{81} +6952.55i q^{82} +(6229.50 - 6229.50i) q^{83} +(-1893.05 + 1893.05i) q^{84} +(-4345.91 - 4345.91i) q^{85} +(-12115.7 - 12115.7i) q^{86} -3990.08 q^{87} -3301.24i q^{88} +(8558.42 + 8558.42i) q^{89} -6096.97i q^{90} +(-799.227 + 3973.39i) q^{91} +8354.30 q^{92} +(-347.486 + 347.486i) q^{93} -293.448 q^{94} -24454.0i q^{95} +(-5082.30 + 5082.30i) q^{96} +(2882.26 - 2882.26i) q^{97} +(7904.48 + 7904.48i) q^{98} +(-1877.29 - 1877.29i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 24 q^{5} - 24 q^{7} + 540 q^{9} + 372 q^{11} - 224 q^{13} + 480 q^{14} - 252 q^{15} - 2328 q^{16} - 840 q^{19} + 228 q^{20} + 936 q^{21} + 3536 q^{22} - 1404 q^{24} - 828 q^{26} - 1984 q^{28} - 5064 q^{29}+ \cdots + 10044 q^{99}+O(q^{100}) $ 20 * q + 24 * q^5 - 24 * q^7 + 540 * q^9 + 372 * q^11 - 224 * q^13 + 480 * q^14 - 252 * q^15 - 2328 * q^16 - 840 * q^19 + 228 * q^20 + 936 * q^21 + 3536 * q^22 - 1404 * q^24 - 828 * q^26 - 1984 * q^28 - 5064 * q^29 + 1712 * q^31 - 7260 * q^32 - 540 * q^33 + 8040 * q^34 + 5976 * q^35 - 868 * q^37 - 2952 * q^39 + 6336 * q^40 - 6000 * q^41 + 648 * q^42 + 20868 * q^44 + 648 * q^45 - 1992 * q^46 + 1692 * q^47 + 2592 * q^48 + 8952 * q^50 - 12416 * q^52 - 18816 * q^53 - 3280 * q^55 - 9360 * q^57 + 1256 * q^58 - 6588 * q^59 + 9036 * q^60 - 5512 * q^61 - 648 * q^63 + 33048 * q^65 + 13464 * q^66 - 4704 * q^67 - 28296 * q^68 - 31096 * q^70 + 9756 * q^71 + 13692 * q^73 - 28296 * q^74 - 17072 * q^76 - 23868 * q^78 - 5056 * q^79 + 42228 * q^80 + 14580 * q^81 + 24708 * q^83 + 26496 * q^84 - 4752 * q^85 - 77424 * q^86 + 5256 * q^87 + 36000 * q^89 + 19104 * q^91 + 12912 * q^92 - 18432 * q^93 + 112832 * q^94 - 52344 * q^96 + 50012 * q^97 + 11544 * q^98 + 10044 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$.

$n$	$14$	$28$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−4.32919	+	4.32919i	−1.08230	+	1.08230i	−0.0860014	+	0.996295i	$0.527409\pi$
$2$	−4.32919	+	4.32919i	−1.08230	+	1.08230i	−0.996295	+	0.0860014i	$0.972591\pi$
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$		−	21.4837i		−	1.34273i
$5$	−26.0804	+	26.0804i	−1.04322	+	1.04322i	−0.0441922	+	0.999023i	$0.514071\pi$
$5$	−26.0804	+	26.0804i	−1.04322	+	1.04322i	−0.999023	+	0.0441922i	$0.985929\pi$
$6$	−22.4951	+	22.4951i	−0.624864	+	0.624864i
$7$	−16.9579	−	16.9579i	−0.346080	−	0.346080i	0.512567	−	0.858647i	$-0.328694\pi$
$7$	−16.9579	−	16.9579i	−0.346080	−	0.346080i	−0.858647	+	0.512567i	$0.828694\pi$
$8$	23.7399	+	23.7399i	0.370937	+	0.370937i
$9$	27.0000			0.333333
$10$		−	225.814i		−	2.25814i
$11$	−69.5293	−	69.5293i	−0.574623	−	0.574623i	0.358794	−	0.933417i	$-0.383188\pi$
$11$	−69.5293	−	69.5293i	−0.574623	−	0.574623i	−0.933417	+	0.358794i	$0.883188\pi$
$12$		−	111.633i		−	0.775226i
$13$	−93.5895	−	140.720i	−0.553784	−	0.832660i
$13$	−93.5895	−	140.720i	−0.553784	−	0.832660i
$14$	146.828			0.749121
$15$	−135.518	+	135.518i	−0.602301	+	0.602301i
$16$	138.190			0.539804
$17$			166.635i			0.576593i	0.957541	+	0.288296i	$0.0930888\pi$
$17$			166.635i			0.576593i	−0.957541	+	0.288296i	$0.906911\pi$
$18$	−116.888	+	116.888i	−0.360765	+	0.360765i
$19$	−468.820	+	468.820i	−1.29867	+	1.29867i	−0.369401	+	0.929270i	$0.620437\pi$
$19$	−468.820	+	468.820i	−1.29867	+	1.29867i	−0.929270	+	0.369401i	$0.879563\pi$
$20$	560.303	+	560.303i	1.40076	+	1.40076i
$21$	−88.1158	−	88.1158i	−0.199809	−	0.199809i
$22$	602.011			1.24382
$23$			388.867i			0.735098i	0.930004	+	0.367549i	$0.119803\pi$
$23$			388.867i			0.735098i	−0.930004	+	0.367549i	$0.880197\pi$
$24$	123.356	+	123.356i	0.214160	+	0.214160i
$25$		−	735.373i		−	1.17660i
$26$	1014.37	+	204.035i	1.50054	+	0.301827i
$27$	140.296			0.192450
$28$	−364.318	+	364.318i	−0.464692	+	0.464692i
$29$	−767.891			−0.913069			−0.456534	−	0.889706i	$-0.650909\pi$
$29$	−767.891			−0.913069			−0.456534	+	0.889706i	$0.650909\pi$
$30$		−	1173.36i		−	1.30374i
$31$	−66.8738	+	66.8738i	−0.0695877	+	0.0695877i	−0.741044	−	0.671456i	$-0.765669\pi$
$31$	−66.8738	+	66.8738i	−0.0695877	+	0.0695877i	0.671456	+	0.741044i	$0.265669\pi$
$32$	−978.089	+	978.089i	−0.955165	+	0.955165i
$33$	−361.285	−	361.285i	−0.331758	−	0.331758i
$34$	−721.395	−	721.395i	−0.624044	−	0.624044i
$35$	884.537			0.722071
$36$		−	580.060i		−	0.447577i
$37$	1532.33	+	1532.33i	1.11931	+	1.11931i	0.991843	+	0.127465i	$0.0406841\pi$
$37$	1532.33	+	1532.33i	1.11931	+	1.11931i	0.127465	+	0.991843i	$0.459316\pi$
$38$		−	4059.22i		−	2.81109i
$39$	−486.305	−	731.200i	−0.319727	−	0.480737i
$40$	−1238.29			−0.773933
$41$	802.986	−	802.986i	0.477684	−	0.477684i	−0.426707	−	0.904390i	$-0.640326\pi$
$41$	802.986	−	802.986i	0.477684	−	0.477684i	0.904390	+	0.426707i	$0.140326\pi$
$42$	762.940			0.432505
$43$			2798.62i			1.51358i	0.653655	+	0.756792i	$0.273235\pi$
$43$			2798.62i			1.51358i	−0.653655	+	0.756792i	$0.726765\pi$
$44$	−1493.75	+	1493.75i	−0.771564	+	0.771564i
$45$	−704.170	+	704.170i	−0.347738	+	0.347738i
$46$	−1683.48	−	1683.48i	−0.795594	−	0.795594i
$47$	33.8918	+	33.8918i	0.0153426	+	0.0153426i	0.714736	−	0.699394i	$-0.246547\pi$
$47$	33.8918	+	33.8918i	0.0153426	+	0.0153426i	−0.699394	+	0.714736i	$0.746547\pi$
$48$	718.056			0.311656
$49$		−	1825.86i		−	0.760458i
$50$	3183.56	+	3183.56i	1.27343	+	1.27343i
$51$			865.862i			0.332896i
$52$	−3023.18	+	2010.65i	−1.11804	+	0.743583i
$53$	−3139.54			−1.11767			−0.558836	−	0.829278i	$-0.688752\pi$
$53$	−3139.54			−1.11767			−0.558836	+	0.829278i	$0.688752\pi$
$54$	−607.368	+	607.368i	−0.208288	+	0.208288i
$55$	3626.70			1.19891
$56$		−	805.159i		−	0.256747i
$57$	−2436.06	+	2436.06i	−0.749788	+	0.749788i
$58$	3324.34	−	3324.34i	0.988211	−	0.988211i
$59$	−3959.16	−	3959.16i	−1.13736	−	1.13736i	−0.988921	−	0.148440i	$-0.952575\pi$
$59$	−3959.16	−	3959.16i	−1.13736	−	1.13736i	−0.148440	−	0.988921i	$-0.547425\pi$
$60$	2911.42	+	2911.42i	0.808728	+	0.808728i
$61$	504.230			0.135509			0.0677546	−	0.997702i	$-0.478417\pi$
$61$	504.230			0.135509			0.0677546	+	0.997702i	$0.478417\pi$
$62$		−	579.018i		−	0.150629i
$63$	−457.863	−	457.863i	−0.115360	−	0.115360i
$64$		−	6257.62i		−	1.52774i
$65$	6110.87	+	1229.17i	1.44636	+	0.290928i
$66$	3128.14			0.718122
$67$	4218.53	−	4218.53i	0.939748	−	0.939748i	−0.0585372	−	0.998285i	$-0.518644\pi$
$67$	4218.53	−	4218.53i	0.939748	−	0.939748i	0.998285	+	0.0585372i	$0.0186436\pi$
$68$	3579.94			0.774209
$69$			2020.61i			0.424409i
$70$	−3829.33	+	3829.33i	−0.781495	+	0.781495i
$71$	−3025.86	+	3025.86i	−0.600249	+	0.600249i	−0.940379	−	0.340130i	$-0.889529\pi$
$71$	−3025.86	+	3025.86i	−0.600249	+	0.600249i	0.340130	+	0.940379i	$0.389529\pi$
$72$	640.978	+	640.978i	0.123646	+	0.123646i
$73$	3088.98	+	3088.98i	0.579654	+	0.579654i	0.934808	−	0.355154i	$-0.115571\pi$
$73$	3088.98	+	3088.98i	0.579654	+	0.579654i	−0.355154	+	0.934808i	$0.615571\pi$
$74$	−13267.5			−2.42285
$75$		−	3821.11i		−	0.679308i
$76$	10072.0	+	10072.0i	1.74377	+	1.74377i
$77$			2358.14i			0.397730i
$78$	5270.81	+	1060.20i	0.866339	+	0.174260i
$79$	−9376.55			−1.50241			−0.751205	−	0.660069i	$-0.770527\pi$
$79$	−9376.55			−1.50241			−0.751205	+	0.660069i	$0.770527\pi$
$80$	−3604.05	+	3604.05i	−0.563132	+	0.563132i
$81$	729.000			0.111111
$82$			6952.55i			1.03399i
$83$	6229.50	−	6229.50i	0.904267	−	0.904267i	−0.0915344	−	0.995802i	$-0.529177\pi$
$83$	6229.50	−	6229.50i	0.904267	−	0.904267i	0.995802	+	0.0915344i	$0.0291772\pi$
$84$	−1893.05	+	1893.05i	−0.268290	+	0.268290i
$85$	−4345.91	−	4345.91i	−0.601510	−	0.601510i
$86$	−12115.7	−	12115.7i	−1.63815	−	1.63815i
$87$	−3990.08			−0.527160
$88$		−	3301.24i		−	0.426297i
$89$	8558.42	+	8558.42i	1.08047	+	1.08047i	0.996465	+	0.0840075i	$0.0267720\pi$
$89$	8558.42	+	8558.42i	1.08047	+	1.08047i	0.0840075	+	0.996465i	$0.473228\pi$
$90$		−	6096.97i		−	0.752712i
$91$	−799.227	+	3973.39i	−0.0965133	+	0.479820i
$92$	8354.30			0.987039
$93$	−347.486	+	347.486i	−0.0401765	+	0.0401765i
$94$	−293.448			−0.0332105
$95$		−	24454.0i		−	2.70959i
$96$	−5082.30	+	5082.30i	−0.551465	+	0.551465i
$97$	2882.26	−	2882.26i	0.306330	−	0.306330i	−0.537154	−	0.843484i	$-0.680501\pi$
$97$	2882.26	−	2882.26i	0.306330	−	0.306330i	0.843484	+	0.537154i	$0.180501\pi$
$98$	7904.48	+	7904.48i	0.823041	+	0.823041i
$99$	−1877.29	−	1877.29i	−0.191541	−	0.191541i
$100$	−15798.5			−1.57985
$101$		−	5594.65i		−	0.548441i	−0.961667	−	0.274220i	$-0.911580\pi$
$101$		−	5594.65i		−	0.548441i	0.961667	−	0.274220i	$-0.0884198\pi$
$102$	−3748.48	−	3748.48i	−0.360292	−	0.360292i
$103$			6792.74i			0.640281i	0.947370	+	0.320140i	$0.103730\pi$
$103$			6792.74i			0.640281i	−0.947370	+	0.320140i	$0.896270\pi$
$104$	1118.86	−	5562.48i	0.103445	−	0.514283i
$105$	4596.19			0.416888
$106$	13591.7	−	13591.7i	1.20965	−	1.20965i
$107$	−8462.24			−0.739125			−0.369563	−	0.929206i	$-0.620492\pi$
$107$	−8462.24			−0.739125			−0.369563	+	0.929206i	$0.620492\pi$
$108$		−	3014.08i		−	0.258409i
$109$	−12077.9	+	12077.9i	−1.01657	+	1.01657i	−0.0167121	+	0.999860i	$0.505320\pi$
$109$	−12077.9	+	12077.9i	−1.01657	+	1.01657i	−0.999860	+	0.0167121i	$0.994680\pi$
$110$	−15700.7	+	15700.7i	−1.29758	+	1.29758i
$111$	7962.24	+	7962.24i	0.646233	+	0.646233i
$112$	−2343.41	−	2343.41i	−0.186815	−	0.186815i
$113$	14864.2			1.16409			0.582043	−	0.813158i	$-0.302254\pi$
$113$	14864.2			1.16409			0.582043	+	0.813158i	$0.302254\pi$
$114$		−	21092.3i		−	1.62299i
$115$	−10141.8	−	10141.8i	−0.766866	−	0.766866i
$116$			16497.1i			1.22601i
$117$	−2526.92	−	3799.43i	−0.184595	−	0.277553i
$118$	34279.8			2.46192
$119$	2825.78	−	2825.78i	0.199547	−	0.199547i
$120$	−6434.36			−0.446831
$121$		−	4972.34i		−	0.339618i
$122$	−2182.90	+	2182.90i	−0.146661	+	0.146661i
$123$	4172.44	−	4172.44i	0.275791	−	0.275791i
$124$	1436.70	+	1436.70i	0.0934376	+	0.0934376i
$125$	2878.56	+	2878.56i	0.184228	+	0.184228i
$126$	3964.35			0.249707
$127$			10995.6i			0.681728i	0.940113	+	0.340864i	$0.110720\pi$
$127$			10995.6i			0.681728i	−0.940113	+	0.340864i	$0.889280\pi$
$128$	11441.0	+	11441.0i	0.698301	+	0.698301i
$129$			14542.0i			0.873868i
$130$	−31776.4	+	21133.8i	−1.88026	+	1.25052i
$131$	14023.3			0.817160			0.408580	−	0.912722i	$-0.366024\pi$
$131$	14023.3			0.817160			0.408580	+	0.912722i	$0.366024\pi$
$132$	−7761.74	+	7761.74i	−0.445462	+	0.445462i
$133$	15900.4			0.898887
$134$			36525.6i			2.03417i
$135$	−3658.98	+	3658.98i	−0.200767	+	0.200767i
$136$	−3955.91	+	3955.91i	−0.213879	+	0.213879i
$137$	20682.4	+	20682.4i	1.10194	+	1.10194i	0.994176	+	0.107769i	$0.0343706\pi$
$137$	20682.4	+	20682.4i	1.10194	+	1.10194i	0.107769	+	0.994176i	$0.465629\pi$
$138$	−8747.60	−	8747.60i	−0.459336	−	0.459336i
$139$	−12584.1			−0.651317			−0.325658	−	0.945487i	$-0.605586\pi$
$139$	−12584.1			−0.651317			−0.325658	+	0.945487i	$0.605586\pi$
$140$		−	19003.1i		−	0.969547i
$141$	176.107	+	176.107i	0.00885807	+	0.00885807i
$142$		−	26199.0i		−	1.29929i
$143$	−3276.92	+	16291.4i	−0.160248	+	0.796682i
$144$	3731.13			0.179935
$145$	20026.9	−	20026.9i	0.952527	−	0.952527i
$146$	−26745.5			−1.25472
$147$		−	9487.44i		−	0.439051i
$148$	32920.2	−	32920.2i	1.50293	−	1.50293i
$149$	−1317.22	+	1317.22i	−0.0593315	+	0.0593315i	−0.736150	−	0.676818i	$-0.763358\pi$
$149$	−1317.22	+	1317.22i	−0.0593315	+	0.0593315i	0.676818	+	0.736150i	$0.263358\pi$
$150$	16542.3	+	16542.3i	0.735213	+	0.735213i
$151$	−6003.08	−	6003.08i	−0.263282	−	0.263282i	0.563104	−	0.826386i	$-0.309607\pi$
$151$	−6003.08	−	6003.08i	−0.263282	−	0.263282i	−0.826386	+	0.563104i	$0.809607\pi$
$152$	−22259.5			−0.963449
$153$			4499.15i			0.192198i
$154$	−10208.8	−	10208.8i	−0.430462	−	0.430462i
$155$		−	3488.19i		−	0.145190i
$156$	−15708.9	+	10447.6i	−0.645500	+	0.429308i
$157$	−24232.6			−0.983107			−0.491553	−	0.870847i	$-0.663571\pi$
$157$	−24232.6			−0.983107			−0.491553	+	0.870847i	$0.663571\pi$
$158$	40592.8	−	40592.8i	1.62605	−	1.62605i
$159$	−16313.5			−0.645288
$160$		−	51017.9i		−	1.99289i
$161$	6594.37	−	6594.37i	0.254403	−	0.254403i
$162$	−3155.98	+	3155.98i	−0.120255	+	0.120255i
$163$	−24239.5	−	24239.5i	−0.912322	−	0.912322i	0.0841325	−	0.996455i	$-0.473188\pi$
$163$	−24239.5	−	24239.5i	−0.912322	−	0.912322i	−0.996455	+	0.0841325i	$0.973188\pi$
$164$	−17251.1	−	17251.1i	−0.641401	−	0.641401i
$165$	18844.9			0.692191
$166$			53937.3i			1.95737i
$167$	−17352.4	−	17352.4i	−0.622195	−	0.622195i	0.323897	−	0.946092i	$-0.395007\pi$
$167$	−17352.4	−	17352.4i	−0.622195	−	0.622195i	−0.946092	+	0.323897i	$0.895007\pi$
$168$		−	4183.73i		−	0.148233i
$169$	−11043.0	+	26339.8i	−0.386646	+	0.922228i
$170$	37628.5			1.30202
$171$	−12658.1	+	12658.1i	−0.432890	+	0.432890i
$172$	60124.7			2.03234
$173$			2780.29i			0.0928961i	0.998921	+	0.0464480i	$0.0147902\pi$
$173$			2780.29i			0.0928961i	−0.998921	+	0.0464480i	$0.985210\pi$
$174$	17273.8	−	17273.8i	0.570544	−	0.570544i
$175$	−12470.4	+	12470.4i	−0.407196	+	0.407196i
$176$	−9608.25	−	9608.25i	−0.310184	−	0.310184i
$177$	−20572.4	−	20572.4i	−0.656656	−	0.656656i
$178$	−74102.0			−2.33878
$179$		−	2972.36i		−	0.0927674i	−0.998924	−	0.0463837i	$-0.985230\pi$
$179$		−	2972.36i		−	0.0927674i	0.998924	−	0.0463837i	$-0.0147697\pi$
$180$	15128.2	+	15128.2i	0.466919	+	0.466919i
$181$			44034.0i			1.34410i	0.740507	+	0.672049i	$0.234586\pi$
$181$			44034.0i			1.34410i	−0.740507	+	0.672049i	$0.765414\pi$
$182$	−13741.5	−	20661.5i	−0.414852	−	0.623764i
$183$	2620.05			0.0782362
$184$	−9231.68	+	9231.68i	−0.272675	+	0.272675i
$185$	−79927.7			−2.33536
$186$		−	3008.67i		−	0.0869657i
$187$	11586.0	−	11586.0i	0.331323	−	0.331323i
$188$	728.122	−	728.122i	0.0206010	−	0.0206010i
$189$	−2379.13	−	2379.13i	−0.0666031	−	0.0666031i
$190$	105866.	+	105866.i	2.93258	+	2.93258i
$191$	−3688.96			−0.101120			−0.0505601	−	0.998721i	$-0.516101\pi$
$191$	−3688.96			−0.101120			−0.0505601	+	0.998721i	$0.516101\pi$
$192$		−	32515.5i		−	0.882040i
$193$	19216.9	+	19216.9i	0.515904	+	0.515904i	0.916329	−	0.400426i	$-0.131138\pi$
$193$	19216.9	+	19216.9i	0.515904	+	0.515904i	−0.400426	+	0.916329i	$0.631138\pi$
$194$			24955.7i			0.663079i
$195$	31753.0	+	6386.95i	0.835056	+	0.167967i
$196$	−39226.2			−1.02109
$197$	−15917.0	+	15917.0i	−0.410136	+	0.410136i	−0.881786	−	0.471650i	$-0.843659\pi$
$197$	−15917.0	+	15917.0i	−0.410136	+	0.410136i	0.471650	+	0.881786i	$0.343659\pi$
$198$	16254.3			0.414608
$199$		−	23311.0i		−	0.588646i	−0.955706	−	0.294323i	$-0.904906\pi$
$199$		−	23311.0i		−	0.588646i	0.955706	−	0.294323i	$-0.0950941\pi$
$200$	17457.7	−	17457.7i	0.436443	−	0.436443i
$201$	21920.1	−	21920.1i	0.542564	−	0.542564i
$202$	24220.3	+	24220.3i	0.593576	+	0.593576i
$203$	13021.8	+	13021.8i	0.315995	+	0.315995i
$204$	18601.9			0.446990
$205$			41884.4i			0.996654i
$206$	−29407.0	−	29407.0i	−0.692974	−	0.692974i
$207$			10499.4i			0.245033i
$208$	−12933.1	−	19446.0i	−0.298935	−	0.449474i
$209$	65193.5			1.49249
$210$	−19897.8	+	19897.8i	−0.451196	+	0.451196i
$211$	10045.5			0.225636			0.112818	−	0.993616i	$-0.464012\pi$
$211$	10045.5			0.225636			0.112818	+	0.993616i	$0.464012\pi$
$212$			67448.9i			1.50073i
$213$	−15722.8	+	15722.8i	−0.346554	+	0.346554i
$214$	36634.6	−	36634.6i	0.799953	−	0.799953i
$215$	−72989.0	−	72989.0i	−1.57899	−	1.57899i
$216$	3330.62	+	3330.62i	0.0713868	+	0.0713868i
$217$	2268.08			0.0481658
$218$		−	104575.i		−	2.20047i
$219$	16050.8	+	16050.8i	0.334664	+	0.334664i
$220$		−	77915.0i		−	1.60981i
$221$	23448.8	−	15595.3i	0.480106	−	0.319308i
$222$	−68940.0			−1.39883
$223$	24945.0	−	24945.0i	0.501620	−	0.501620i	−0.410321	−	0.911941i	$-0.634583\pi$
$223$	24945.0	−	24945.0i	0.501620	−	0.501620i	0.911941	+	0.410321i	$0.134583\pi$
$224$	33172.7			0.661126
$225$		−	19855.1i		−	0.392199i
$226$	−64349.9	+	64349.9i	−1.25989	+	1.25989i
$227$	15576.9	−	15576.9i	0.302294	−	0.302294i	−0.539617	−	0.841911i	$-0.681431\pi$
$227$	15576.9	−	15576.9i	0.302294	−	0.302294i	0.841911	+	0.539617i	$0.181431\pi$
$228$	52335.6	+	52335.6i	1.00676	+	1.00676i
$229$	−24930.7	−	24930.7i	−0.475405	−	0.475405i	0.428254	−	0.903658i	$-0.359129\pi$
$229$	−24930.7	−	24930.7i	−0.475405	−	0.475405i	−0.903658	+	0.428254i	$0.859129\pi$
$230$	87811.5			1.65995
$231$			12253.3i			0.229630i
$232$	−18229.7	−	18229.7i	−0.338691	−	0.338691i
$233$			22434.3i			0.413239i	0.978421	+	0.206619i	$0.0662462\pi$
$233$			22434.3i			0.413239i	−0.978421	+	0.206619i	$0.933754\pi$
$234$	27387.9	+	5508.94i	0.500181	+	0.100609i
$235$	−1767.82			−0.0320113
$236$	−85057.3	+	85057.3i	−1.52717	+	1.52717i
$237$	−48722.0			−0.867417
$238$			24466.7i			0.431938i
$239$	−66829.7	+	66829.7i	−1.16997	+	1.16997i	−0.187751	+	0.982217i	$0.560120\pi$
$239$	−66829.7	+	66829.7i	−1.16997	+	1.16997i	−0.982217	+	0.187751i	$0.939880\pi$
$240$	−18727.2	+	18727.2i	−0.325124	+	0.325124i
$241$	−34827.1	−	34827.1i	−0.599629	−	0.599629i	0.340585	−	0.940214i	$-0.389375\pi$
$241$	−34827.1	−	34827.1i	−0.599629	−	0.599629i	−0.940214	+	0.340585i	$0.889375\pi$
$242$	21526.2	+	21526.2i	0.367567	+	0.367567i
$243$	3788.00			0.0641500
$244$		−	10832.7i		−	0.181952i
$245$	47619.1	+	47619.1i	0.793321	+	0.793321i
$246$			36126.5i			0.596975i
$247$	109849.	+	22095.5i	1.80054	+	0.362168i
$248$	−3175.16			−0.0516253
$249$	32369.4	−	32369.4i	0.522079	−	0.522079i
$250$	−24923.6			−0.398778
$251$		−	4768.38i		−	0.0756873i	−0.999284	−	0.0378437i	$-0.987951\pi$
$251$		−	4768.38i		−	0.0756873i	0.999284	−	0.0378437i	$-0.0120489\pi$
$252$	−9836.60	+	9836.60i	−0.154897	+	0.154897i
$253$	27037.7	−	27037.7i	0.422404	−	0.422404i
$254$	−47601.9	−	47601.9i	−0.737832	−	0.737832i
$255$	−22582.0	−	22582.0i	−0.347282	−	0.347282i
$256$	1061.74			0.0162008
$257$			62743.6i			0.949956i	0.879998	+	0.474978i	$0.157544\pi$
$257$			62743.6i			0.949956i	−0.879998	+	0.474978i	$0.842456\pi$
$258$	−62955.2	−	62955.2i	−0.945785	−	0.945785i
$259$		−	51970.3i		−	0.774740i
$260$	26407.1	−	131284.i	0.390638	−	1.94207i
$261$	−20733.1			−0.304356
$262$	−60709.4	+	60709.4i	−0.884409	+	0.884409i
$263$	45421.0			0.656667			0.328333	−	0.944562i	$-0.393513\pi$
$263$	45421.0			0.656667			0.328333	+	0.944562i	$0.393513\pi$
$264$		−	17153.8i		−	0.246123i
$265$	81880.4	−	81880.4i	1.16597	−	1.16597i
$266$	−68835.8	+	68835.8i	−0.972862	+	0.972862i
$267$	44470.9	+	44470.9i	0.623811	+	0.623811i
$268$	−90629.6	−	90629.6i	−1.26183	−	1.26183i
$269$	79205.8			1.09459			0.547296	−	0.836939i	$-0.315657\pi$
$269$	79205.8			1.09459			0.547296	+	0.836939i	$0.315657\pi$
$270$		−	31680.8i		−	0.434579i
$271$	−16180.5	−	16180.5i	−0.220320	−	0.220320i	0.588313	−	0.808633i	$-0.299792\pi$
$271$	−16180.5	−	16180.5i	−0.220320	−	0.220320i	−0.808633	+	0.588313i	$0.799792\pi$
$272$			23027.3i			0.311247i
$273$	−4152.90	+	20646.3i	−0.0557220	+	0.277024i
$274$	−179076.			−2.38526
$275$	−51130.0	+	51130.0i	−0.676099	+	0.676099i
$276$	43410.2			0.569867
$277$		−	2884.24i		−	0.0375899i	−0.999823	−	0.0187950i	$-0.994017\pi$
$277$		−	2884.24i		−	0.0375899i	0.999823	−	0.0187950i	$-0.00598298\pi$
$278$	54478.9	−	54478.9i	0.704918	−	0.704918i
$279$	−1805.59	+	1805.59i	−0.0231959	+	0.0231959i
$280$	20998.9	+	20998.9i	0.267843	+	0.267843i
$281$	−103231.	−	103231.i	−1.30736	−	1.30736i	−0.923311	−	0.384052i	$-0.874528\pi$
$281$	−103231.	−	103231.i	−1.30736	−	1.30736i	−0.384052	−	0.923311i	$-0.625472\pi$
$282$	−1524.80			−0.0191741
$283$			78581.6i			0.981178i	0.871391	+	0.490589i	$0.163218\pi$
$283$			78581.6i			0.981178i	−0.871391	+	0.490589i	$0.836782\pi$
$284$	65006.6	+	65006.6i	0.805973	+	0.805973i
$285$		−	127067.i		−	1.56438i
$286$	−56341.9	−	84714.7i	−0.688810	−	1.03568i
$287$	−27233.9			−0.330633
$288$	−26408.4	+	26408.4i	−0.318388	+	0.318388i
$289$	55753.7			0.667541
$290$			173400.i			2.06183i
$291$	14976.6	−	14976.6i	0.176860	−	0.176860i
$292$	66362.7	−	66362.7i	0.778320	−	0.778320i
$293$	−5490.45	−	5490.45i	−0.0639547	−	0.0639547i	0.674406	−	0.738361i	$-0.264400\pi$
$293$	−5490.45	−	5490.45i	−0.0639547	−	0.0639547i	−0.738361	+	0.674406i	$0.764400\pi$
$294$	41072.9	+	41072.9i	0.475183	+	0.475183i
$295$	206513.			2.37303
$296$			72755.0i			0.830385i
$297$	−9754.70	−	9754.70i	−0.110586	−	0.110586i
$298$		−	11405.0i		−	0.128429i
$299$	54721.2	−	36393.9i	0.612087	−	0.407086i
$300$	−82091.5			−0.912128
$301$	47458.7	−	47458.7i	0.523821	−	0.523821i
$302$	51976.9			0.569897
$303$		−	29070.6i		−	0.316642i
$304$	−64786.2	+	64786.2i	−0.701028	+	0.701028i
$305$	−13150.5	+	13150.5i	−0.141365	+	0.141365i
$306$	−19477.7	−	19477.7i	−0.208015	−	0.208015i
$307$	36642.5	+	36642.5i	0.388784	+	0.388784i	0.874254	−	0.485469i	$-0.161351\pi$
$307$	36642.5	+	36642.5i	0.388784	+	0.388784i	−0.485469	+	0.874254i	$0.661351\pi$
$308$	50661.6			0.534045
$309$			35296.1i			0.369666i
$310$	15101.0	+	15101.0i	0.157139	+	0.157139i
$311$		−	57004.6i		−	0.589371i	−0.955594	−	0.294686i	$-0.904785\pi$
$311$		−	57004.6i		−	0.589371i	0.955594	−	0.294686i	$-0.0952149\pi$
$312$	5813.79	−	28903.5i	0.0597242	−	0.296921i
$313$	−168911.			−1.72413			−0.862064	−	0.506799i	$-0.830829\pi$
$313$	−168911.			−1.72413			−0.862064	+	0.506799i	$0.830829\pi$
$314$	104907.	−	104907.i	1.06401	−	1.06401i
$315$	23882.5			0.240690
$316$			201443.i			2.01733i
$317$	−57585.7	+	57585.7i	−0.573055	+	0.573055i	−0.932981	−	0.359926i	$-0.882802\pi$
$317$	−57585.7	+	57585.7i	−0.573055	+	0.573055i	0.359926	+	0.932981i	$0.382802\pi$
$318$	70624.3	−	70624.3i	0.698393	−	0.698393i
$319$	53390.9	+	53390.9i	0.524670	+	0.524670i
$320$	163201.	+	163201.i	1.59376	+	1.59376i
$321$	−43971.1			−0.426734
$322$			57096.5i			0.550678i
$323$	−78122.0	−	78122.0i	−0.748804	−	0.748804i
$324$		−	15661.6i		−	0.149192i
$325$	−103481.	+	68823.2i	−0.979705	+	0.651580i
$326$	209874.			1.97481
$327$	−62758.6	+	62758.6i	−0.586918	+	0.586918i
$328$	38125.7			0.354381
$329$		−	1149.47i		−	0.0106195i
$330$	−81583.1	+	81583.1i	−0.749156	+	0.749156i
$331$	−134787.	+	134787.i	−1.23025	+	1.23025i	−0.266382	+	0.963868i	$0.585828\pi$
$331$	−134787.	+	134787.i	−1.23025	+	1.23025i	−0.963868	+	0.266382i	$0.914172\pi$
$332$	−133833.	−	133833.i	−1.21419	−	1.21419i
$333$	41373.0	+	41373.0i	0.373103	+	0.373103i
$334$	150243.			1.34680
$335$			220042.i			1.96072i
$336$	−12176.7	−	12176.7i	−0.107858	−	0.107858i
$337$			56729.9i			0.499520i	0.968308	+	0.249760i	$0.0803517\pi$
$337$			56729.9i			0.499520i	−0.968308	+	0.249760i	$0.919648\pi$
$338$	−66222.5	−	161837.i	−0.579659	−	1.41659i
$339$	77236.7			0.672085
$340$	−93366.3	+	93366.3i	−0.807667	+	0.807667i
$341$	9299.38			0.0799734
$342$		−	109599.i		−	0.937031i
$343$	−71678.7	+	71678.7i	−0.609259	+	0.609259i
$344$	−66439.0	+	66439.0i	−0.561444	+	0.561444i
$345$	−52698.3	−	52698.3i	−0.442750	−	0.442750i
$346$	−12036.4	−	12036.4i	−0.100541	−	0.100541i
$347$	−183864.			−1.52699			−0.763497	−	0.645811i	$-0.776519\pi$
$347$	−183864.			−1.52699			−0.763497	+	0.645811i	$0.776519\pi$
$348$			85721.6i			0.707835i
$349$	−72786.9	−	72786.9i	−0.597588	−	0.597588i	0.342082	−	0.939670i	$-0.388868\pi$
$349$	−72786.9	−	72786.9i	−0.597588	−	0.597588i	−0.939670	+	0.342082i	$0.888868\pi$
$350$		−	107973.i		−	0.881413i
$351$	−13130.2	−	19742.4i	−0.106576	−	0.160246i
$352$	136012.			1.09772
$353$	106752.	−	106752.i	0.856695	−	0.856695i	−0.134252	−	0.990947i	$-0.542863\pi$
$353$	106752.	−	106752.i	0.856695	−	0.856695i	0.990947	+	0.134252i	$0.0428632\pi$
$354$	178123.			1.42139
$355$		−	157831.i		−	1.25238i
$356$	183867.	−	183867.i	1.45078	−	1.45078i
$357$	14683.2	−	14683.2i	0.115209	−	0.115209i
$358$	12867.9	+	12867.9i	0.100402	+	0.100402i
$359$	−26893.9	−	26893.9i	−0.208672	−	0.208672i	0.595031	−	0.803703i	$-0.297140\pi$
$359$	−26893.9	−	26893.9i	−0.208672	−	0.208672i	−0.803703	+	0.595031i	$0.797140\pi$
$360$	−33433.9			−0.257978
$361$		−	309264.i		−	2.37309i
$362$	−190631.	−	190631.i	−1.45471	−	1.45471i
$363$		−	25837.1i		−	0.196078i
$364$	85363.1	+	17170.3i	0.644269	+	0.129591i
$365$	−161123.			−1.20941
$366$	−11342.7	+	11342.7i	−0.0846748	+	0.0846748i
$367$	6492.32			0.0482023			0.0241012	−	0.999710i	$-0.492328\pi$
$367$	6492.32			0.0482023			0.0241012	+	0.999710i	$0.492328\pi$
$368$			53737.5i			0.396809i
$369$	21680.6	−	21680.6i	0.159228	−	0.159228i
$370$	346022.	−	346022.i	2.52755	−	2.52755i
$371$	53240.0	+	53240.0i	0.386804	+	0.386804i
$372$	7465.29	+	7465.29i	0.0539462	+	0.0539462i
$373$	−53286.9			−0.383004			−0.191502	−	0.981492i	$-0.561336\pi$
$373$	−53286.9			−0.383004			−0.191502	+	0.981492i	$0.561336\pi$
$374$			100316.i			0.717180i
$375$	14957.4	+	14957.4i	0.106364	+	0.106364i
$376$			1609.18i			0.0113823i
$377$	71866.5	+	108057.i	0.505643	+	0.760276i
$378$	20599.4			0.144168
$379$	−92136.8	+	92136.8i	−0.641438	+	0.641438i	−0.950909	−	0.309471i	$-0.899848\pi$
$379$	−92136.8	+	92136.8i	−0.641438	+	0.641438i	0.309471	+	0.950909i	$0.399848\pi$
$380$	−525363.			−3.63825
$381$			57134.8i			0.393596i
$382$	15970.2	−	15970.2i	0.109442	−	0.109442i
$383$	−15911.5	+	15911.5i	−0.108471	+	0.108471i	−0.759259	−	0.650788i	$-0.774439\pi$
$383$	−15911.5	+	15911.5i	−0.108471	+	0.108471i	0.650788	+	0.759259i	$0.274439\pi$
$384$	59449.0	+	59449.0i	0.403164	+	0.403164i
$385$	−61501.3	−	61501.3i	−0.414918	−	0.414918i
$386$	−166387.			−1.11672
$387$			75562.7i			0.504528i
$388$	−61921.5	−	61921.5i	−0.411319	−	0.411319i
$389$			215552.i			1.42447i	0.701941	+	0.712236i	$0.252317\pi$
$389$			215552.i			1.42447i	−0.701941	+	0.712236i	$0.747683\pi$
$390$	−165115.	+	109814.i	−1.08557	+	0.721988i
$391$	−64799.0			−0.423852
$392$	43345.8	−	43345.8i	0.282082	−	0.282082i
$393$	72867.1			0.471788
$394$		−	137815.i		−	0.887778i
$395$	244544.	−	244544.i	1.56734	−	1.56734i
$396$	−40331.2	+	40331.2i	−0.257188	+	0.257188i
$397$	81274.7	+	81274.7i	0.515673	+	0.515673i	0.916259	−	0.400586i	$-0.131194\pi$
$397$	81274.7	+	81274.7i	0.515673	+	0.515673i	−0.400586	+	0.916259i	$0.631194\pi$
$398$	100917.	+	100917.i	0.637089	+	0.637089i
$399$	82621.0			0.518973
$400$		−	101621.i		−	0.635132i
$401$	170564.	+	170564.i	1.06072	+	1.06072i	0.998034	+	0.0626817i	$0.0199653\pi$
$401$	170564.	+	170564.i	1.06072	+	1.06072i	0.0626817	+	0.998034i	$0.480035\pi$
$402$			189793.i			1.17443i
$403$	15669.1	+	3151.77i	0.0964795	+	0.0194063i
$404$	−120194.			−0.736409
$405$	−19012.6	+	19012.6i	−0.115913	+	0.115913i
$406$	−112748.			−0.683999
$407$		−	213084.i		−	1.28636i
$408$	−20555.5	+	20555.5i	−0.123483	+	0.123483i
$409$	−92288.1	+	92288.1i	−0.551695	+	0.551695i	−0.926930	−	0.375235i	$-0.877562\pi$
$409$	−92288.1	+	92288.1i	−0.551695	+	0.551695i	0.375235	+	0.926930i	$0.377562\pi$
$410$	−181325.	−	181325.i	−1.07867	−	1.07867i
$411$	107469.	+	107469.i	0.636208	+	0.636208i
$412$	145933.			0.859725
$413$			134278.i			0.787235i
$414$	−45453.9	−	45453.9i	−0.265198	−	0.265198i
$415$			324935.i			1.88669i
$416$	229175.	+	46097.4i	1.32428	+	0.266373i
$417$	−65388.9			−0.376038
$418$	−282235.	+	282235.i	−1.61532	+	1.61532i
$419$	−20127.4			−0.114646			−0.0573232	−	0.998356i	$-0.518257\pi$
$419$	−20127.4			−0.114646			−0.0573232	+	0.998356i	$0.518257\pi$
$420$		−	98743.1i		−	0.559768i
$421$	171209.	−	171209.i	0.965965	−	0.965965i	−0.0334748	−	0.999440i	$-0.510657\pi$
$421$	171209.	−	171209.i	0.965965	−	0.965965i	0.999440	+	0.0334748i	$0.0106573\pi$
$422$	−43489.0	+	43489.0i	−0.244205	+	0.244205i
$423$	915.080	+	915.080i	0.00511421	+	0.00511421i
$424$	−74532.5	−	74532.5i	−0.414586	−	0.414586i
$425$	122539.			0.678417
$426$		−	136134.i		−	0.750148i
$427$	−8550.68	−	8550.68i	−0.0468970	−	0.0468970i
$428$			181800.i			0.992446i
$429$	−17027.4	+	84652.4i	−0.0925195	+	0.459965i
$430$	631966.			3.41788
$431$	127549.	−	127549.i	0.686630	−	0.686630i	−0.274856	−	0.961485i	$-0.588630\pi$
$431$	127549.	−	127549.i	0.686630	−	0.686630i	0.961485	+	0.274856i	$0.0886301\pi$
$432$	19387.5			0.103885
$433$			68957.0i			0.367792i	0.982946	+	0.183896i	$0.0588710\pi$
$433$			68957.0i			0.367792i	−0.982946	+	0.183896i	$0.941129\pi$
$434$	−9818.93	+	9818.93i	−0.0521297	+	0.0521297i
$435$	104063.	−	104063.i	0.549942	−	0.549942i
$436$	259478.	+	259478.i	1.36498	+	1.36498i
$437$	−182309.	−	182309.i	−0.954651	−	0.954651i
$438$	−138974.			−0.724410
$439$		−	113397.i		−	0.588400i	−0.955744	−	0.294200i	$-0.904947\pi$
$439$		−	113397.i		−	0.588400i	0.955744	−	0.294200i	$-0.0950531\pi$
$440$	86097.7	+	86097.7i	0.444720	+	0.444720i
$441$		−	49298.2i		−	0.253486i
$442$	−33999.4	+	169029.i	−0.174031	+	0.865203i
$443$	92267.8			0.470157			0.235079	−	0.971976i	$-0.424465\pi$
$443$	92267.8			0.470157			0.235079	+	0.971976i	$0.424465\pi$
$444$	171058.	−	171058.i	0.867717	−	0.867717i
$445$	−446414.			−2.25433
$446$			215984.i			1.08580i
$447$	−6844.47	+	6844.47i	−0.0342551	+	0.0342551i
$448$	−106116.	+	106116.i	−0.528719	+	0.528719i
$449$	−17542.3	−	17542.3i	−0.0870148	−	0.0870148i	0.662260	−	0.749274i	$-0.269598\pi$
$449$	−17542.3	−	17542.3i	−0.0870148	−	0.0870148i	−0.749274	+	0.662260i	$0.769598\pi$
$450$	85956.2	+	85956.2i	0.424475	+	0.424475i
$451$	−111662.			−0.548976
$452$		−	319338.i		−	1.56305i
$453$	−31192.9	−	31192.9i	−0.152006	−	0.152006i
$454$			134871.i			0.654343i
$455$	−82783.4	−	124472.i	−0.399872	−	0.601240i
$456$	−115664.			−0.556248
$457$	104731.	−	104731.i	0.501466	−	0.501466i	−0.410428	−	0.911893i	$-0.634621\pi$
$457$	104731.	−	104731.i	0.501466	−	0.501466i	0.911893	+	0.410428i	$0.134621\pi$
$458$	215859.			1.02906
$459$			23378.3i			0.110965i
$460$	−217883.	+	217883.i	−1.02969	+	1.02969i
$461$	−12382.8	+	12382.8i	−0.0582663	+	0.0582663i	−0.735640	−	0.677373i	$-0.763118\pi$
$461$	−12382.8	+	12382.8i	−0.0582663	+	0.0582663i	0.677373	+	0.735640i	$0.263118\pi$
$462$	−53046.7	−	53046.7i	−0.248527	−	0.248527i
$463$	156318.	+	156318.i	0.729199	+	0.729199i	0.970460	−	0.241261i	$-0.0775611\pi$
$463$	156318.	+	156318.i	0.729199	+	0.729199i	−0.241261	+	0.970460i	$0.577561\pi$
$464$	−106115.			−0.492878
$465$		−	18125.2i		−	0.0838255i
$466$	−97122.3	−	97122.3i	−0.447247	−	0.447247i
$467$			3880.03i			0.0177911i	0.999960	+	0.00889553i	$0.00283157\pi$
$467$			3880.03i			0.0177911i	−0.999960	+	0.00889553i	$0.997168\pi$
$468$	−81625.8	+	54287.5i	−0.372680	+	0.247861i
$469$	−143075.			−0.650455
$470$	7653.24	−	7653.24i	0.0346457	−	0.0346457i
$471$	−125916.			−0.567597
$472$		−	187980.i		−	0.843778i
$473$	194586.	−	194586.i	0.869740	−	0.869740i
$474$	210926.	−	210926.i	0.938803	−	0.938803i
$475$	344758.	+	344758.i	1.52801	+	1.52801i
$476$	−60708.3	−	60708.3i	−0.267938	−	0.267938i
$477$	−84767.6			−0.372557
$478$		−	578637.i		−	2.53250i
$479$	53446.0	+	53446.0i	0.232940	+	0.232940i	0.813919	−	0.580979i	$-0.197330\pi$
$479$	53446.0	+	53446.0i	0.232940	+	0.232940i	−0.580979	+	0.813919i	$0.697330\pi$
$480$		−	265097.i		−	1.15059i
$481$	72218.9	−	359040.i	0.312148	−	1.55186i
$482$	301546.			1.29795
$483$	34265.3	−	34265.3i	0.146879	−	0.146879i
$484$	−106824.			−0.456015
$485$			150341.i			0.639136i
$486$	−16398.9	+	16398.9i	−0.0694293	+	0.0694293i
$487$	−39739.5	+	39739.5i	−0.167558	+	0.167558i	−0.785905	−	0.618347i	$-0.787803\pi$
$487$	−39739.5	+	39739.5i	−0.167558	+	0.167558i	0.618347	+	0.785905i	$0.287803\pi$
$488$	11970.4	+	11970.4i	0.0502653	+	0.0502653i
$489$	−125952.	−	125952.i	−0.526729	−	0.526729i
$490$	−412304.			−1.71722
$491$			183063.i			0.759342i	0.925122	+	0.379671i	$0.123963\pi$
$491$			183063.i			0.759342i	−0.925122	+	0.379671i	$0.876037\pi$
$492$	−89639.4	−	89639.4i	−0.370313	−	0.370313i
$493$		−	127958.i		−	0.526469i
$494$	−571212.	+	379900.i	−2.34069	+	1.55674i
$495$	97921.0			0.399637
$496$	−9241.28	+	9241.28i	−0.0375638	+	0.0375638i
$497$	102624.			0.415468
$498$			280266.i			1.13009i
$499$	−57389.5	+	57389.5i	−0.230479	+	0.230479i	−0.812893	−	0.582414i	$-0.802108\pi$
$499$	−57389.5	+	57389.5i	−0.230479	+	0.230479i	0.582414	+	0.812893i	$0.302108\pi$
$500$	61842.1	−	61842.1i	0.247368	−	0.247368i
$501$	−90165.7	−	90165.7i	−0.359224	−	0.359224i
$502$	20643.2	+	20643.2i	0.0819161	+	0.0819161i
$503$	−192555.			−0.761061			−0.380530	−	0.924768i	$-0.624259\pi$
$503$	−192555.			−0.761061			−0.380530	+	0.924768i	$0.624259\pi$
$504$		−	21739.3i		−	0.0855824i
$505$	145910.	+	145910.i	0.572142	+	0.572142i
$506$			234102.i			0.914333i
$507$	−57381.1	+	136865.i	−0.223230	+	0.532449i
$508$	236226.			0.915377
$509$	−84764.7	+	84764.7i	−0.327175	+	0.327175i	−0.851511	−	0.524336i	$-0.824313\pi$
$509$	−84764.7	+	84764.7i	−0.327175	+	0.327175i	0.524336	+	0.851511i	$0.324313\pi$
$510$	195524.			0.751724
$511$		−	104765.i		−	0.401213i
$512$	−187652.	+	187652.i	−0.715835	+	0.715835i
$513$	−65773.7	+	65773.7i	−0.249929	+	0.249929i
$514$	−271629.	−	271629.i	−1.02813	−	1.02813i
$515$	−177157.	−	177157.i	−0.667951	−	0.667951i
$516$	312417.			1.17337
$517$		−	4712.95i		−	0.0176324i
$518$	224989.	+	224989.i	0.838498	+	0.838498i
$519$			14446.8i			0.0536336i
$520$	115891.	+	174252.i	0.428592	+	0.644424i
$521$	314076.			1.15707			0.578534	−	0.815658i	$-0.303625\pi$
$521$	314076.			1.15707			0.578534	+	0.815658i	$0.303625\pi$
$522$	89757.2	−	89757.2i	0.329404	−	0.329404i
$523$	−37022.8			−0.135352			−0.0676762	−	0.997707i	$-0.521558\pi$
$523$	−37022.8			−0.135352			−0.0676762	+	0.997707i	$0.521558\pi$
$524$		−	301272.i		−	1.09723i
$525$	−64798.0	+	64798.0i	−0.235095	+	0.235095i
$526$	−196636.	+	196636.i	−0.710708	+	0.710708i
$527$	−11143.5	−	11143.5i	−0.0401238	−	0.0401238i
$528$	−49925.9	−	49925.9i	−0.179085	−	0.179085i
$529$	128624.			0.459631
$530$			708951.i			2.52386i
$531$	−106897.	−	106897.i	−0.379121	−	0.379121i
$532$		−	341600.i		−	1.20696i
$533$	−188147.	−	37844.8i	−0.662282	−	0.133214i
$534$	−385045.			−1.35030
$535$	220699.	−	220699.i	0.771067	−	0.771067i
$536$	200295.			0.697174
$537$		−	15444.8i		−	0.0535593i
$538$	−342897.	+	342897.i	−1.18467	+	1.18467i
$539$	−126951.	+	126951.i	−0.436976	+	0.436976i
$540$	78608.3	+	78608.3i	0.269576	+	0.269576i
$541$	−38918.2	−	38918.2i	−0.132971	−	0.132971i	0.637489	−	0.770460i	$-0.279973\pi$
$541$	−38918.2	−	38918.2i	−0.132971	−	0.132971i	−0.770460	+	0.637489i	$0.779973\pi$
$542$	140097.			0.476903
$543$			228807.i			0.776015i
$544$	−162984.	−	162984.i	−0.550741	−	0.550741i
$545$		−	629992.i		−	2.12101i
$546$	−71403.2	−	107361.i	−0.239515	−	0.360130i
$547$	−383774.			−1.28263			−0.641314	−	0.767279i	$-0.721610\pi$
$547$	−383774.			−1.28263			−0.641314	+	0.767279i	$0.721610\pi$
$548$	444334.	−	444334.i	1.47962	−	1.47962i
$549$	13614.2			0.0451697
$550$		−	442702.i		−	1.46348i
$551$	360003.	−	360003.i	1.18578	−	1.18578i
$552$	−47969.2	+	47969.2i	−0.157429	+	0.157429i
$553$	159007.	+	159007.i	0.519954	+	0.519954i
$554$	12486.4	+	12486.4i	0.0406835	+	0.0406835i
$555$	−415316.			−1.34832
$556$			270353.i			0.874543i
$557$	280053.	+	280053.i	0.902672	+	0.902672i	0.995667	−	0.0929944i	$-0.0296439\pi$
$557$	280053.	+	280053.i	0.902672	+	0.902672i	−0.0929944	+	0.995667i	$0.529644\pi$
$558$		−	15633.5i		−	0.0502097i
$559$	393820.	−	261921.i	1.26030	−	0.838199i
$560$	122234.			0.389777
$561$	60202.8	−	60202.8i	0.191290	−	0.191290i
$562$	893810.			2.82991
$563$			50493.5i			0.159301i	0.996823	+	0.0796505i	$0.0253804\pi$
$563$			50493.5i			0.159301i	−0.996823	+	0.0796505i	$0.974620\pi$
$564$	3783.43	−	3783.43i	0.0118940	−	0.0118940i
$565$	−387664.	+	387664.i	−1.21439	+	1.21439i
$566$	−340194.	−	340194.i	−1.06193	−	1.06193i
$567$	−12362.3	−	12362.3i	−0.0384533	−	0.0384533i
$568$	−143667.			−0.445309
$569$		−	301875.i		−	0.932400i	−0.884679	−	0.466200i	$-0.845623\pi$
$569$		−	301875.i		−	0.932400i	0.884679	−	0.466200i	$-0.154377\pi$
$570$	550096.	+	550096.i	1.69312	+	1.69312i
$571$		−	290046.i		−	0.889601i	−0.895630	−	0.444800i	$-0.853275\pi$
$571$		−	290046.i		−	0.889601i	0.895630	−	0.444800i	$-0.146725\pi$
$572$	349999.	+	70400.4i	1.06973	+	0.215171i
$573$	−19168.4			−0.0583817
$574$	117901.	−	117901.i	0.357843	−	0.357843i
$575$	285962.			0.864914
$576$		−	168956.i		−	0.509246i
$577$	127765.	−	127765.i	0.383760	−	0.383760i	−0.488695	−	0.872455i	$-0.662527\pi$
$577$	127765.	−	127765.i	0.383760	−	0.383760i	0.872455	+	0.488695i	$0.162527\pi$
$578$	−241368.	+	241368.i	−0.722477	+	0.722477i
$579$	99853.9	+	99853.9i	0.297857	+	0.297857i
$580$	−430252.	−	430252.i	−1.27899	−	1.27899i
$581$	−211278.			−0.625897
$582$			129673.i			0.382829i
$583$	218290.	+	218290.i	0.642240	+	0.642240i
$584$			146664.i			0.430030i
$585$	164994.	+	33187.6i	0.482120	+	0.0969759i
$586$	47538.4			0.138436
$587$	−432681.	+	432681.i	−1.25572	+	1.25572i	−0.302598	+	0.953118i	$0.597854\pi$
$587$	−432681.	+	432681.i	−1.25572	+	1.25572i	−0.953118	+	0.302598i	$0.902146\pi$
$588$	−203825.			−0.589527
$589$		−	62703.6i		−	0.180743i
$590$	−894031.	+	894031.i	−2.56832	+	2.56832i
$591$	−82707.0	+	82707.0i	−0.236792	+	0.236792i
$592$	211753.	+	211753.i	0.604207	+	0.604207i
$593$	351836.	+	351836.i	1.00053	+	1.00053i	1.00000		0.000531251i	$0.000169102\pi$
$593$	351836.	+	351836.i	1.00053	+	1.00053i	0.000531251		1.00000i	$0.499831\pi$
$594$	84459.8			0.239374
$595$			147395.i			0.416341i
$596$	28298.7	+	28298.7i	0.0796663	+	0.0796663i
$597$		−	121127.i		−	0.339855i
$598$	−79342.4	+	394454.i	−0.221872	+	1.10305i
$599$	−484810.			−1.35119			−0.675597	−	0.737271i	$-0.736114\pi$
$599$	−484810.			−1.35119			−0.675597	+	0.737271i	$0.736114\pi$
$600$	90712.9	−	90712.9i	0.251980	−	0.251980i
$601$	214570.			0.594046			0.297023	−	0.954870i	$-0.404006\pi$
$601$	214570.			0.594046			0.297023	+	0.954870i	$0.404006\pi$
$602$			410915.i			1.13386i
$603$	113900.	−	113900.i	0.313249	−	0.313249i
$604$	−128968.	+	128968.i	−0.353516	+	0.353516i
$605$	129681.	+	129681.i	0.354294	+	0.354294i
$606$	125852.	+	125852.i	0.342701	+	0.342701i
$607$	−2800.08			−0.00759964			−0.00379982	−	0.999993i	$-0.501210\pi$
$607$	−2800.08			−0.00759964			−0.00379982	+	0.999993i	$0.501210\pi$
$608$		−	917096.i		−	2.48089i
$609$	67663.3	+	67663.3i	0.182440	+	0.182440i
$610$		−	113862.i		−	0.305998i
$611$	1597.32	−	7941.17i	0.00427869	−	0.0212717i
$612$	96658.4			0.258070
$613$	−92722.6	+	92722.6i	−0.246754	+	0.246754i	−0.819637	−	0.572883i	$-0.805825\pi$
$613$	−92722.6	+	92722.6i	−0.246754	+	0.246754i	0.572883	+	0.819637i	$0.305825\pi$
$614$	−317265.			−0.841560
$615$			217638.i			0.575418i
$616$	−55982.2	+	55982.2i	−0.147533	+	0.147533i
$617$	260927.	−	260927.i	0.685407	−	0.685407i	−0.275806	−	0.961213i	$-0.588945\pi$
$617$	260927.	−	260927.i	0.685407	−	0.685407i	0.961213	+	0.275806i	$0.0889447\pi$
$618$	−152803.	−	152803.i	−0.400089	−	0.400089i
$619$	103595.	+	103595.i	0.270369	+	0.270369i	0.829249	−	0.558880i	$-0.188769\pi$
$619$	103595.	+	103595.i	0.270369	+	0.270369i	−0.558880	+	0.829249i	$0.688769\pi$
$620$	−74939.2			−0.194951
$621$			54556.5i			0.141470i
$622$	246783.	+	246783.i	0.637874	+	0.637874i
$623$		−	290266.i		−	0.747859i
$624$	−67202.5	−	101045.i	−0.172590	−	0.259504i
$625$	309460.			0.792218
$626$	731248.	−	731248.i	1.86602	−	1.86602i
$627$	338755.			0.861690
$628$			520606.i			1.32005i
$629$	−255341.	+	255341.i	−0.645385	+	0.645385i
$630$	−103392.	+	103392.i	−0.260498	+	0.260498i
$631$	−300176.	−	300176.i	−0.753907	−	0.753907i	0.221299	−	0.975206i	$-0.428970\pi$
$631$	−300176.	−	300176.i	−0.753907	−	0.753907i	−0.975206	+	0.221299i	$0.928970\pi$
$632$	−222599.	−	222599.i	−0.557299	−	0.557299i
$633$	52198.2			0.130271
$634$		−	498599.i		−	1.24043i
$635$	−286769.	−	286769.i	−0.711189	−	0.711189i
$636$			350475.i			0.866449i
$637$	−256934.	+	170881.i	−0.633203	+	0.421130i
$638$	−462279.			−1.13570
$639$	−81698.1	+	81698.1i	−0.200083	+	0.200083i
$640$	−596770.			−1.45696
$641$			466494.i			1.13535i	0.823253	+	0.567675i	$0.192157\pi$
$641$			466494.i			1.13535i	−0.823253	+	0.567675i	$0.807843\pi$
$642$	190359.	−	190359.i	0.461853	−	0.461853i
$643$	439463.	−	439463.i	1.06292	−	1.06292i	0.0650372	−	0.997883i	$-0.479283\pi$
$643$	439463.	−	439463.i	1.06292	−	1.06292i	0.997883	−	0.0650372i	$-0.0207166\pi$
$644$	−141671.	−	141671.i	−0.341594	−	0.341594i
$645$	−379262.	−	379262.i	−0.911633	−	0.911633i
$646$	676409.			1.62086
$647$		−	562437.i		−	1.34358i	−0.740740	−	0.671792i	$-0.765525\pi$
$647$		−	562437.i		−	1.34358i	0.740740	−	0.671792i	$-0.234475\pi$
$648$	17306.4	+	17306.4i	0.0412152	+	0.0412152i
$649$			550555.i			1.30711i
$650$	150042.	−	745938.i	0.355128	−	1.76553i
$651$	11785.3			0.0278085
$652$	−520754.	+	520754.i	−1.22500	+	1.22500i
$653$	37629.2			0.0882467			0.0441234	−	0.999026i	$-0.485951\pi$
$653$	37629.2			0.0882467			0.0441234	+	0.999026i	$0.485951\pi$
$654$		−	543387.i		−	1.27044i
$655$	−365733.	+	365733.i	−0.852474	+	0.852474i
$656$	110965.	−	110965.i	0.257856	−	0.257856i
$657$	83402.4	+	83402.4i	0.193218	+	0.193218i
$658$	4976.27	+	4976.27i	0.0114935	+	0.0114935i
$659$	18033.4			0.0415248			0.0207624	−	0.999784i	$-0.493391\pi$
$659$	18033.4			0.0415248			0.0207624	+	0.999784i	$0.493391\pi$
$660$		−	404858.i		−	0.929426i
$661$	536607.	+	536607.i	1.22816	+	1.22816i	0.964661	+	0.263495i	$0.0848754\pi$
$661$	536607.	+	536607.i	1.22816	+	1.22816i	0.263495	+	0.964661i	$0.415125\pi$
$662$		−	1.16704e6i		−	2.66299i
$663$	121844.	−	81035.7i	0.277189	−	0.184353i
$664$	295776.			0.670852
$665$	−414689.	+	414689.i	−0.937733	+	0.937733i
$666$	−358223.			−0.807616
$667$		−	298607.i		−	0.671195i
$668$	−372794.	+	372794.i	−0.835440	+	0.835440i
$669$	129618.	−	129618.i	0.289610	−	0.289610i
$670$	−952601.	−	952601.i	−2.12208	−	2.12208i
$671$	−35058.7	−	35058.7i	−0.0778666	−	0.0778666i
$672$	172370.			0.381701
$673$			75975.6i			0.167743i	0.996477	+	0.0838715i	$0.0267285\pi$
$673$			75975.6i			0.167743i	−0.996477	+	0.0838715i	$0.973271\pi$
$674$	−245594.	−	245594.i	−0.540628	−	0.540628i
$675$		−	103170.i		−	0.226436i
$676$	565875.	+	237244.i	1.23830	+	0.519162i
$677$	438381.			0.956478			0.478239	−	0.878230i	$-0.341275\pi$
$677$	438381.			0.956478			0.478239	+	0.878230i	$0.341275\pi$
$678$	−334372.	+	334372.i	−0.727395	+	0.727395i
$679$	−97754.1			−0.212029
$680$		−	206343.i		−	0.446244i
$681$	80940.0	−	80940.0i	0.174530	−	0.174530i
$682$	−40258.7	+	40258.7i	−0.0865549	+	0.0865549i
$683$	−14323.9	−	14323.9i	−0.0307057	−	0.0307057i	0.691587	−	0.722293i	$-0.256912\pi$
$683$	−14323.9	−	14323.9i	−0.0307057	−	0.0307057i	−0.722293	+	0.691587i	$0.756912\pi$
$684$	271944.	+	271944.i	0.581255	+	0.581255i
$685$	−1.07881e6			−2.29913
$686$		−	620620.i		−	1.31880i
$687$	−129544.	−	129544.i	−0.274475	−	0.274475i
$688$			386741.i			0.817039i
$689$	293828.	+	441795.i	0.618949	+	0.930641i
$690$	456282.			0.958374
$691$	64315.1	−	64315.1i	0.134697	−	0.134697i	−0.636544	−	0.771241i	$-0.719637\pi$
$691$	64315.1	−	64315.1i	0.134697	−	0.134697i	0.771241	+	0.636544i	$0.219637\pi$
$692$	59730.8			0.124734
$693$			63669.9i			0.132577i
$694$	795981.	−	795981.i	1.65266	−	1.65266i
$695$	328198.	−	328198.i	0.679464	−	0.679464i
$696$	−94724.2	−	94724.2i	−0.195543	−	0.195543i
$697$	133806.	+	133806.i	0.275429	+	0.275429i
$698$	630216.			1.29354
$699$			116572.i			0.238583i
$700$	267910.	+	267910.i	0.546755	+	0.546755i
$701$			77204.1i			0.157110i	0.996910	+	0.0785551i	$0.0250307\pi$
$701$			77204.1i			0.157110i	−0.996910	+	0.0785551i	$0.974969\pi$
$702$	142312.	+	28625.3i	0.288780	+	0.0580865i
$703$	−1.43678e6			−2.90723
$704$	−435088.	+	435088.i	−0.877873	+	0.877873i
$705$	−9185.89			−0.0184817
$706$			924298.i			1.85440i
$707$	−94873.4	+	94873.4i	−0.189804	+	0.189804i
$708$	−441971.	+	441971.i	−0.881712	+	0.881712i
$709$	561910.	+	561910.i	1.11783	+	1.11783i	0.992060	+	0.125766i	$0.0401389\pi$
$709$	561910.	+	561910.i	1.11783	+	1.11783i	0.125766	+	0.992060i	$0.459861\pi$
$710$	683279.	+	683279.i	1.35544	+	1.35544i
$711$	−253167.			−0.500804
$712$			406353.i			0.801574i
$713$	−26005.0	−	26005.0i	−0.0511538	−	0.0511538i
$714$			127133.i			0.249380i
$715$	−339421.	−	510348.i	−0.663937	−	0.998285i
$716$	−63857.3			−0.124562
$717$	−347258.	+	347258.i	−0.675481	+	0.675481i
$718$	232857.			0.451690
$719$		−	324777.i		−	0.628243i	−0.949383	−	0.314121i	$-0.898290\pi$
$719$		−	324777.i		−	0.628243i	0.949383	−	0.314121i	$-0.101710\pi$
$720$	−97309.2	+	97309.2i	−0.187711	+	0.187711i
$721$	115191.	−	115191.i	0.221588	−	0.221588i
$722$	1.33886e6	+	1.33886e6i	2.56839	+	2.56839i
$723$	−180967.	−	180967.i	−0.346196	−	0.346196i
$724$	946013.			1.80476
$725$			564686.i			1.07431i
$726$	111853.	+	111853.i	0.212215	+	0.212215i
$727$		−	922269.i		−	1.74497i	−0.488638	−	0.872487i	$-0.662506\pi$
$727$		−	922269.i		−	1.74497i	0.488638	−	0.872487i	$-0.337494\pi$
$728$	−113302.	+	75354.5i	−0.213783	+	0.142183i
$729$	19683.0			0.0370370
$730$	697533.	−	697533.i	1.30894	−	1.30894i
$731$	−466348.			−0.872722
$732$		−	56288.4i		−	0.105050i
$733$	295621.	−	295621.i	0.550209	−	0.550209i	−0.376292	−	0.926501i	$-0.622801\pi$
$733$	295621.	−	295621.i	0.550209	−	0.550209i	0.926501	+	0.376292i	$0.122801\pi$
$734$	−28106.5	+	28106.5i	−0.0521692	+	0.0521692i
$735$	247436.	+	247436.i	0.458024	+	0.458024i
$736$	−380346.	−	380346.i	−0.702140	−	0.702140i
$737$	−586623.			−1.08000
$738$			187719.i			0.344663i
$739$	369186.	+	369186.i	0.676015	+	0.676015i	0.959096	−	0.283081i	$-0.0913566\pi$
$739$	369186.	+	369186.i	0.676015	+	0.676015i	−0.283081	+	0.959096i	$0.591357\pi$
$740$			1.71714e6i			3.13576i
$741$	570791.	+	114812.i	1.03954	+	0.209098i
$742$	−460972.			−0.837272
$743$	−671107.	+	671107.i	−1.21567	+	1.21567i	−0.246530	+	0.969135i	$0.579290\pi$
$743$	−671107.	+	671107.i	−1.21567	+	1.21567i	−0.969135	+	0.246530i	$0.920710\pi$
$744$	−16498.6			−0.0298059
$745$		−	68707.2i		−	0.123791i
$746$	230689.	−	230689.i	0.414523	−	0.414523i
$747$	168196.	−	168196.i	0.301422	−	0.301422i
$748$	−248911.	−	248911.i	−0.444878	−	0.444878i
$749$	143502.	+	143502.i	0.255796	+	0.255796i
$750$	−129507.			−0.230235
$751$			167627.i			0.297210i	0.988897	+	0.148605i	$0.0474784\pi$
$751$			167627.i			0.297210i	−0.988897	+	0.148605i	$0.952522\pi$
$752$	4683.51	+	4683.51i	0.00828201	+	0.00828201i
$753$		−	24777.2i		−	0.0436981i
$754$	−778924.	−	156676.i	−1.37010	−	0.275588i
$755$	313125.			0.549319
$756$	−51112.5	+	51112.5i	−0.0894300	+	0.0894300i
$757$	−578076.			−1.00877			−0.504386	−	0.863478i	$-0.668281\pi$
$757$	−578076.			−1.00877			−0.504386	+	0.863478i	$0.668281\pi$
$758$		−	797754.i		−	1.38845i
$759$	140492.	−	140492.i	0.243875	−	0.243875i
$760$	580537.	−	580537.i	1.00508	−	1.00508i
$761$	−762616.	−	762616.i	−1.31685	−	1.31685i	−0.916254	−	0.400597i	$-0.868803\pi$
$761$	−762616.	−	762616.i	−1.31685	−	1.31685i	−0.400597	−	0.916254i	$-0.631197\pi$
$762$	−247347.	−	247347.i	−0.425987	−	0.425987i
$763$	409632.			0.703630
$764$			79252.6i			0.135777i
$765$	−117340.	−	117340.i	−0.200503	−	0.200503i
$766$		−	137768.i		−	0.234795i
$767$	−186595.	+	927666.i	−0.317183	+	1.57689i
$768$	5516.94			0.00935354
$769$	698760.	−	698760.i	1.18161	−	1.18161i	0.202287	−	0.979326i	$-0.435163\pi$
$769$	698760.	−	698760.i	1.18161	−	1.18161i	0.979326	−	0.202287i	$-0.0648374\pi$
$770$	532501.			0.898129
$771$			326026.i			0.548457i
$772$	412850.	−	412850.i	0.692720	−	0.692720i
$773$	−311294.	+	311294.i	−0.520968	+	0.520968i	−0.917864	−	0.396895i	$-0.870088\pi$
$773$	−311294.	+	311294.i	−0.520968	+	0.520968i	0.396895	+	0.917864i	$0.370088\pi$
$774$	−327125.	−	327125.i	−0.546049	−	0.546049i
$775$	49177.2	+	49177.2i	0.0818766	+	0.0818766i
$776$	136849.			0.227258
$777$		−	270046.i		−	0.447296i
$778$	−933166.	−	933166.i	−1.54170	−	1.54170i
$779$			752912.i			1.24071i
$780$	137215.	−	682172.i	0.225535	−	1.12126i
$781$	420771.			0.689833
$782$	280527.	−	280527.i	0.458734	−	0.458734i
$783$	−107732.			−0.175720
$784$		−	252315.i		−	0.410498i
$785$	631996.	−	631996.i	1.02559	−	1.02559i
$786$	−315455.	+	315455.i	−0.510614	+	0.510614i
$787$	−528223.	−	528223.i	−0.852841	−	0.852841i	0.137642	−	0.990482i	$-0.456048\pi$
$787$	−528223.	−	528223.i	−0.852841	−	0.852841i	−0.990482	+	0.137642i	$0.956048\pi$
$788$	341955.	+	341955.i	0.550702	+	0.550702i
$789$	236014.			0.379127
$790$			2.11735e6i			3.39265i
$791$	−252066.	−	252066.i	−0.402866	−	0.402866i
$792$		−	89133.6i		−	0.142099i
$793$	−47190.6	−	70955.0i	−0.0750428	−	0.112833i
$794$	−703706.			−1.11622
$795$	425463.	−	425463.i	0.673175	−	0.673175i
$796$	−500805.			−0.790393
$797$		−	550495.i		−	0.866636i	−0.901241	−	0.433318i	$-0.857343\pi$
$797$		−	550495.i		−	0.866636i	0.901241	−	0.433318i	$-0.142657\pi$
$798$	−357682.	+	357682.i	−0.561682	+	0.561682i
$799$	−5647.58	+	5647.58i	−0.00884644	+	0.00884644i
$800$	719260.	+	719260.i	1.12384	+	1.12384i
$801$	231077.	+	231077.i	0.360158	+	0.360158i
$802$	−1.47681e6			−2.29602
$803$		−	429549.i		−	0.666165i
$804$	−470925.	−	470925.i	−0.728517	−	0.728517i
$805$			343967.i			0.530793i
$806$	−81479.2	+	54190.0i	−0.125423	+	0.0834160i
$807$	411565.			0.631963
$808$	132817.	−	132817.i	0.203437	−	0.203437i
$809$	705446.			1.07787			0.538936	−	0.842347i	$-0.318827\pi$
$809$	705446.			1.07787			0.538936	+	0.842347i	$0.318827\pi$
$810$		−	164618.i		−	0.250904i
$811$	476089.	−	476089.i	0.723846	−	0.723846i	−0.245540	−	0.969386i	$-0.578965\pi$
$811$	476089.	−	476089.i	0.723846	−	0.723846i	0.969386	+	0.245540i	$0.0789654\pi$
$812$	279757.	−	279757.i	0.424296	−	0.424296i
$813$	−84076.5	−	84076.5i	−0.127202	−	0.127202i
$814$	922481.	+	922481.i	1.39222	+	1.39222i
$815$	1.26435e6			1.90350
$816$			119653.i			0.179699i
$817$	−1.31205e6	−	1.31205e6i	−1.96565	−	1.96565i
$818$		−	799064.i		−	1.19419i
$819$	−21579.1	+	107282.i	−0.0321711	+	0.159940i
$820$	899831.			1.33824
$821$	162224.	−	162224.i	0.240675	−	0.240675i	−0.576455	−	0.817129i	$-0.695564\pi$
$821$	162224.	−	162224.i	0.240675	−	0.240675i	0.817129	+	0.576455i	$0.195564\pi$
$822$	−930506.			−1.37713
$823$			1.34448e6i			1.98497i	0.122378	+	0.992484i	$0.460948\pi$
$823$			1.34448e6i			1.98497i	−0.122378	+	0.992484i	$0.539052\pi$
$824$	−161259.	+	161259.i	−0.237504	+	0.237504i
$825$	−265679.	+	265679.i	−0.390346	+	0.390346i
$826$	−581314.	−	581314.i	−0.852022	−	0.852022i
$827$	−542193.	−	542193.i	−0.792762	−	0.792762i	0.189180	−	0.981942i	$-0.439417\pi$
$827$	−542193.	−	542193.i	−0.792762	−	0.792762i	−0.981942	+	0.189180i	$0.939417\pi$
$828$	225566.			0.329013
$829$			485612.i			0.706611i	0.935508	+	0.353305i	$0.114942\pi$
$829$			485612.i			0.706611i	−0.935508	+	0.353305i	$0.885058\pi$
$830$	−1.40671e6	−	1.40671e6i	−2.04196	−	2.04196i
$831$		−	14986.9i		−	0.0217026i
$832$	−880569.	+	585648.i	−1.27209	+	0.846038i
$833$	304253.			0.438474
$834$	283081.	−	283081.i	0.406985	−	0.406985i
$835$	905114.			1.29817
$836$		−	1.40060e6i		−	2.00401i
$837$	−9382.13	+	9382.13i	−0.0133922	+	0.0133922i
$838$	87135.4	−	87135.4i	0.124081	−	0.124081i
$839$	702797.	+	702797.i	0.998404	+	0.998404i	0.999999	−	0.00159522i	$-0.000507774\pi$
$839$	702797.	+	702797.i	0.998404	+	0.998404i	−0.00159522	+	0.999999i	$0.500508\pi$
$840$	109113.	+	109113.i	0.154639	+	0.154639i
$841$	−117625.			−0.166305
$842$			1.48239e6i			2.09092i
$843$	−536403.	−	536403.i	−0.754807	−	0.754807i
$844$		−	215815.i		−	0.302969i
$845$	−398945.	−	974957.i	−0.558727	−	1.36544i
$846$	−7923.10			−0.0110702
$847$	−84320.5	+	84320.5i	−0.117535	+	0.117535i
$848$	−433853.			−0.603324
$849$			408322.i			0.566484i
$850$	−530494.	+	530494.i	−0.734248	+	0.734248i
$851$	−595874.	+	595874.i	−0.822801	+	0.822801i
$852$	337784.	+	337784.i	0.465329	+	0.465329i
$853$	528758.	+	528758.i	0.726706	+	0.726706i	0.969962	−	0.243256i	$-0.0782154\pi$
$853$	528758.	+	528758.i	0.726706	+	0.726706i	−0.243256	+	0.969962i	$0.578215\pi$
$854$	74034.9			0.101513
$855$		−	660259.i		−	0.903196i
$856$	−200893.	−	200893.i	−0.274169	−	0.274169i
$857$			427769.i			0.582435i	0.956657	+	0.291217i	$0.0940603\pi$
$857$			427769.i			0.582435i	−0.956657	+	0.291217i	$0.905940\pi$
$858$	−292761.	−	440191.i	−0.397685	−	0.597952i
$859$	−110526.			−0.149788			−0.0748941	−	0.997191i	$-0.523862\pi$
$859$	−110526.			−0.149788			−0.0748941	+	0.997191i	$0.523862\pi$
$860$	−1.56807e6	+	1.56807e6i	−2.12016	+	2.12016i
$861$	−141512.			−0.190891
$862$			1.10437e6i			1.48627i
$863$	115225.	−	115225.i	0.154713	−	0.154713i	−0.625506	−	0.780219i	$-0.715108\pi$
$863$	115225.	−	115225.i	0.154713	−	0.154713i	0.780219	+	0.625506i	$0.215108\pi$
$864$	−137222.	+	137222.i	−0.183822	+	0.183822i
$865$	−72510.9	−	72510.9i	−0.0969106	−	0.0969106i
$866$	−298528.	−	298528.i	−0.398060	−	0.398060i
$867$	289705.			0.385405
$868$		−	48726.7i		−	0.0646737i
$869$	651945.	+	651945.i	0.863319	+	0.863319i
$870$			901014.i			1.19040i
$871$	−988440.	−	198819.i	−1.30291	−	0.262073i
$872$	−573457.			−0.754168
$873$	77820.9	−	77820.9i	0.102110	−	0.102110i
$874$	1.57850e6			2.06643
$875$		−	97628.6i		−	0.127515i
$876$	344831.	−	344831.i	0.449363	−	0.449363i
$877$	352595.	−	352595.i	0.458435	−	0.458435i	−0.439707	−	0.898141i	$-0.644918\pi$
$877$	352595.	−	352595.i	0.458435	−	0.458435i	0.898141	+	0.439707i	$0.144918\pi$
$878$	490917.	+	490917.i	0.636823	+	0.636823i
$879$	−28529.2	−	28529.2i	−0.0369243	−	0.0369243i
$880$	501174.			0.647177
$881$			776892.i			1.00094i	0.865753	+	0.500471i	$0.166840\pi$
$881$			776892.i			1.00094i	−0.865753	+	0.500471i	$0.833160\pi$
$882$	213421.	+	213421.i	0.274347	+	0.274347i
$883$			990585.i			1.27049i	0.772312	+	0.635244i	$0.219100\pi$
$883$			990585.i			1.27049i	−0.772312	+	0.635244i	$0.780900\pi$
$884$	−335045.	−	503768.i	−0.428745	−	0.644653i
$885$	1.07307e6			1.37007
$886$	−399445.	+	399445.i	−0.508849	+	0.508849i
$887$	951592.			1.20949			0.604747	−	0.796418i	$-0.293274\pi$
$887$	951592.			1.20949			0.604747	+	0.796418i	$0.293274\pi$
$888$			378046.i			0.479423i
$889$	186462.	−	186462.i	0.235932	−	0.235932i
$890$	1.93261e6	−	1.93261e6i	2.43985	−	2.43985i
$891$	−50686.9	−	50686.9i	−0.0638470	−	0.0638470i
$892$	−535912.	−	535912.i	−0.673540	−	0.673540i
$893$	−31778.4			−0.0398500
$894$		−	59262.0i		−	0.0741483i
$895$	77520.3	+	77520.3i	0.0967764	+	0.0967764i
$896$		−	388030.i		−	0.483336i
$897$	284340.	−	189108.i	0.353389	−	0.235031i
$898$	151887.			0.188352
$899$	51351.8	−	51351.8i	0.0635384	−	0.0635384i
$900$	−426560.			−0.526617
$901$		−	523158.i		−	0.644442i
$902$	483406.	−	483406.i	0.594154	−	0.594154i
$903$	246603.	−	246603.i	0.302428	−	0.302428i
$904$	352875.	+	352875.i	0.431802	+	0.431802i
$905$	−1.14842e6	−	1.14842e6i	−1.40218	−	1.40218i
$906$	270080.			0.329030
$907$		−	122762.i		−	0.149228i	−0.997212	−	0.0746141i	$-0.976227\pi$
$907$		−	122762.i		−	0.149228i	0.997212	−	0.0746141i	$-0.0237725\pi$
$908$	−334650.	−	334650.i	−0.405900	−	0.405900i
$909$		−	151055.i		−	0.182814i
$910$	897246.	+	180476.i	1.08350	+	0.217940i
$911$	93939.2			0.113191			0.0565953	−	0.998397i	$-0.481976\pi$
$911$	93939.2			0.113191			0.0565953	+	0.998397i	$0.481976\pi$
$912$	−336639.	+	336639.i	−0.404739	+	0.404739i
$913$	−866266.			−1.03923
$914$			906796.i			1.08547i
$915$	−68332.0	+	68332.0i	−0.0816172	+	0.0816172i
$916$	−535604.	+	535604.i	−0.638341	+	0.638341i
$917$	−237805.	−	237805.i	−0.282802	−	0.282802i
$918$	−101209.	−	101209.i	−0.120097	−	0.120097i
$919$	217922.			0.258030			0.129015	−	0.991643i	$-0.458818\pi$
$919$	217922.			0.258030			0.129015	+	0.991643i	$0.458818\pi$
$920$		−	481531.i		−	0.568917i
$921$	190400.	+	190400.i	0.224465	+	0.224465i
$922$		−	107215.i		−	0.126123i
$923$	708986.	+	142609.i	0.832212	+	0.167395i
$924$	263246.			0.308331
$925$	1.12684e6	−	1.12684e6i	1.31697	−	1.31697i
$926$	−1.35346e6			−1.57842
$927$			183404.i			0.213427i
$928$	751065.	−	751065.i	0.872131	−	0.872131i
$929$	−489424.	+	489424.i	−0.567092	+	0.567092i	−0.931313	−	0.364221i	$-0.881335\pi$
$929$	−489424.	+	489424.i	−0.567092	+	0.567092i	0.364221	+	0.931313i	$0.381335\pi$
$930$	78467.2	+	78467.2i	0.0907240	+	0.0907240i
$931$	856000.	+	856000.i	0.987584	+	0.987584i
$932$	481972.			0.554868
$933$		−	296204.i		−	0.340274i
$934$	−16797.4	−	16797.4i	−0.0192552	−	0.0192552i
$935$			604337.i			0.691283i
$936$	30209.3	−	150187.i	0.0344818	−	0.171428i
$937$	74994.8			0.0854185			0.0427093	−	0.999088i	$-0.486401\pi$
$937$	74994.8			0.0854185			0.0427093	+	0.999088i	$0.486401\pi$
$938$	619397.	−	619397.i	0.703985	−	0.703985i
$939$	−877688.			−0.995426
$940$			37979.4i			0.0429826i
$941$	−746527.	+	746527.i	−0.843075	+	0.843075i	−0.989258	−	0.146183i	$-0.953301\pi$
$941$	−746527.	+	746527.i	−0.843075	+	0.843075i	0.146183	+	0.989258i	$0.453301\pi$
$942$	545115.	−	545115.i	0.614308	−	0.614308i
$943$	312255.	+	312255.i	0.351144	+	0.351144i
$944$	−547115.	−	547115.i	−0.613953	−	0.613953i
$945$	124097.			0.138963
$946$			1.68480e6i			1.88263i
$947$	−392710.	−	392710.i	−0.437897	−	0.437897i	0.453407	−	0.891304i	$-0.350208\pi$
$947$	−392710.	−	392710.i	−0.437897	−	0.437897i	−0.891304	+	0.453407i	$0.850208\pi$
$948$			1.04673e6i			1.16471i
$949$	145584.	−	723776.i	0.161652	−	0.803659i
$950$	−2.98504e6			−3.30752
$951$	−299224.	+	299224.i	−0.330854	+	0.330854i
$952$	134168.			0.148039
$953$		−	920738.i		−	1.01380i	−0.862006	−	0.506898i	$-0.830792\pi$
$953$		−	920738.i		−	1.01380i	0.862006	−	0.506898i	$-0.169208\pi$
$954$	366975.	−	366975.i	0.403218	−	0.403218i
$955$	96209.5	−	96209.5i	0.105490	−	0.105490i
$956$	1.43575e6	+	1.43575e6i	1.57095	+	1.57095i
$957$	277427.	+	277427.i	0.302918	+	0.302918i
$958$	−462756.			−0.504221
$959$		−	701460.i		−	0.762721i
$960$	848018.	+	848018.i	0.920158	+	0.920158i
$961$			914577.i			0.990315i
$962$	1.24170e6	+	1.86700e6i	1.34173	+	2.01741i
$963$	−228481.			−0.246375
$964$	−748214.	+	748214.i	−0.805141	+	0.805141i
$965$	−1.00237e6			−1.07640
$966$			296682.i			0.317934i
$967$	−683545.	+	683545.i	−0.730994	+	0.730994i	−0.970817	−	0.239823i	$-0.922911\pi$
$967$	−683545.	+	683545.i	−0.730994	+	0.730994i	0.239823	+	0.970817i	$0.422911\pi$
$968$	118043.	−	118043.i	0.125977	−	0.125977i
$969$	−405934.	−	405934.i	−0.432322	−	0.432322i
$970$	−650853.	−	650853.i	−0.691734	−	0.691734i
$971$	−107226.			−0.113726			−0.0568632	−	0.998382i	$-0.518110\pi$
$971$	−107226.			−0.113726			−0.0568632	+	0.998382i	$0.518110\pi$
$972$		−	81380.1i		−	0.0861362i
$973$	213400.	+	213400.i	0.225407	+	0.225407i
$974$		−	344080.i		−	0.362695i
$975$	−537705.	+	357616.i	−0.565633	+	0.376190i
$976$	69679.4			0.0731484
$977$	−730243.	+	730243.i	−0.765030	+	0.765030i	−0.977227	−	0.212197i	$-0.931938\pi$
$977$	−730243.	+	730243.i	−0.765030	+	0.765030i	0.212197	+	0.977227i	$0.431938\pi$
$978$	1.09054e6			1.14015
$979$		−	1.19012e6i		−	1.24173i
$980$	1.02303e6	−	1.02303e6i	1.06522	−	1.06522i
$981$	−326103.	+	326103.i	−0.338857	+	0.338857i
$982$	−792514.	−	792514.i	−0.821833	−	0.821833i
$983$	180212.	+	180212.i	0.186500	+	0.186500i	0.794181	−	0.607681i	$-0.207900\pi$
$983$	180212.	+	180212.i	0.186500	+	0.186500i	−0.607681	+	0.794181i	$0.707900\pi$
$984$	198107.			0.204602
$985$		−	830241.i		−	0.855720i
$986$	553953.	+	553953.i	0.569795	+	0.569795i
$987$		−	5972.82i		−	0.00613119i
$988$	474693.	−	2.35996e6i	0.486294	−	2.41763i
$989$	−1.08829e6			−1.11263
$990$	−423918.	+	423918.i	−0.432525	+	0.432525i
$991$	−662719.			−0.674811			−0.337405	−	0.941359i	$-0.609549\pi$
$991$	−662719.			−0.674811			−0.337405	+	0.941359i	$0.609549\pi$
$992$		−	130817.i		−	0.132936i
$993$	−700376.	+	700376.i	−0.710285	+	0.710285i
$994$	−444280.	+	444280.i	−0.449659	+	0.449659i
$995$	607959.	+	607959.i	0.614084	+	0.614084i
$996$	−695415.	−	695415.i	−0.701012	−	0.701012i
$997$	464482.			0.467281			0.233641	−	0.972323i	$-0.424936\pi$
$997$	464482.			0.467281			0.233641	+	0.972323i	$0.424936\pi$
$998$		−	496899.i		−	0.498893i
$999$	214980.	+	214980.i	0.215411	+	0.215411i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.5.g.a.31.2	✓	20
3.2	odd	2		117.5.j.b.109.9		20
13.8	odd	4	inner	39.5.g.a.34.2	yes	20
39.8	even	4		117.5.j.b.73.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.2	✓	20	1.1	even	1	trivial
39.5.g.a.34.2	yes	20	13.8	odd	4	inner
117.5.j.b.73.9		20	39.8	even	4
117.5.j.b.109.9		20	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	39.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.32919 + 4.32919i) q^{2} +5.19615 q^{3} -21.4837i q^{4} +(-26.0804 + 26.0804i) q^{5} +(-22.4951 + 22.4951i) q^{6} +(-16.9579 - 16.9579i) q^{7} +(23.7399 + 23.7399i) q^{8} +27.0000 q^{9} -225.814i q^{10} +(-69.5293 - 69.5293i) q^{11} -111.633i q^{12} +(-93.5895 - 140.720i) q^{13} +146.828 q^{14} +(-135.518 + 135.518i) q^{15} +138.190 q^{16} +166.635i q^{17} +(-116.888 + 116.888i) q^{18} +(-468.820 + 468.820i) q^{19} +(560.303 + 560.303i) q^{20} +(-88.1158 - 88.1158i) q^{21} +602.011 q^{22} +388.867i q^{23} +(123.356 + 123.356i) q^{24} -735.373i q^{25} +(1014.37 + 204.035i) q^{26} +140.296 q^{27} +(-364.318 + 364.318i) q^{28} -767.891 q^{29} -1173.36i q^{30} +(-66.8738 + 66.8738i) q^{31} +(-978.089 + 978.089i) q^{32} +(-361.285 - 361.285i) q^{33} +(-721.395 - 721.395i) q^{34} +884.537 q^{35} -580.060i q^{36} +(1532.33 + 1532.33i) q^{37} -4059.22i q^{38} +(-486.305 - 731.200i) q^{39} -1238.29 q^{40} +(802.986 - 802.986i) q^{41} +762.940 q^{42} +2798.62i q^{43} +(-1493.75 + 1493.75i) q^{44} +(-704.170 + 704.170i) q^{45} +(-1683.48 - 1683.48i) q^{46} +(33.8918 + 33.8918i) q^{47} +718.056 q^{48} -1825.86i q^{49} +(3183.56 + 3183.56i) q^{50} +865.862i q^{51} +(-3023.18 + 2010.65i) q^{52} -3139.54 q^{53} +(-607.368 + 607.368i) q^{54} +3626.70 q^{55} -805.159i q^{56} +(-2436.06 + 2436.06i) q^{57} +(3324.34 - 3324.34i) q^{58} +(-3959.16 - 3959.16i) q^{59} +(2911.42 + 2911.42i) q^{60} +504.230 q^{61} -579.018i q^{62} +(-457.863 - 457.863i) q^{63} -6257.62i q^{64} +(6110.87 + 1229.17i) q^{65} +3128.14 q^{66} +(4218.53 - 4218.53i) q^{67} +3579.94 q^{68} +2020.61i q^{69} +(-3829.33 + 3829.33i) q^{70} +(-3025.86 + 3025.86i) q^{71} +(640.978 + 640.978i) q^{72} +(3088.98 + 3088.98i) q^{73} -13267.5 q^{74} -3821.11i q^{75} +(10072.0 + 10072.0i) q^{76} +2358.14i q^{77} +(5270.81 + 1060.20i) q^{78} -9376.55 q^{79} +(-3604.05 + 3604.05i) q^{80} +729.000 q^{81} +6952.55i q^{82} +(6229.50 - 6229.50i) q^{83} +(-1893.05 + 1893.05i) q^{84} +(-4345.91 - 4345.91i) q^{85} +(-12115.7 - 12115.7i) q^{86} -3990.08 q^{87} -3301.24i q^{88} +(8558.42 + 8558.42i) q^{89} -6096.97i q^{90} +(-799.227 + 3973.39i) q^{91} +8354.30 q^{92} +(-347.486 + 347.486i) q^{93} -293.448 q^{94} -24454.0i q^{95} +(-5082.30 + 5082.30i) q^{96} +(2882.26 - 2882.26i) q^{97} +(7904.48 + 7904.48i) q^{98} +(-1877.29 - 1877.29i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 24 q^{5} - 24 q^{7} + 540 q^{9} + 372 q^{11} - 224 q^{13} + 480 q^{14} - 252 q^{15} - 2328 q^{16} - 840 q^{19} + 228 q^{20} + 936 q^{21} + 3536 q^{22} - 1404 q^{24} - 828 q^{26} - 1984 q^{28} - 5064 q^{29}+ \cdots + 10044 q^{99}+O(q^{100}) \) 20 * q + 24 * q^5 - 24 * q^7 + 540 * q^9 + 372 * q^11 - 224 * q^13 + 480 * q^14 - 252 * q^15 - 2328 * q^16 - 840 * q^19 + 228 * q^20 + 936 * q^21 + 3536 * q^22 - 1404 * q^24 - 828 * q^26 - 1984 * q^28 - 5064 * q^29 + 1712 * q^31 - 7260 * q^32 - 540 * q^33 + 8040 * q^34 + 5976 * q^35 - 868 * q^37 - 2952 * q^39 + 6336 * q^40 - 6000 * q^41 + 648 * q^42 + 20868 * q^44 + 648 * q^45 - 1992 * q^46 + 1692 * q^47 + 2592 * q^48 + 8952 * q^50 - 12416 * q^52 - 18816 * q^53 - 3280 * q^55 - 9360 * q^57 + 1256 * q^58 - 6588 * q^59 + 9036 * q^60 - 5512 * q^61 - 648 * q^63 + 33048 * q^65 + 13464 * q^66 - 4704 * q^67 - 28296 * q^68 - 31096 * q^70 + 9756 * q^71 + 13692 * q^73 - 28296 * q^74 - 17072 * q^76 - 23868 * q^78 - 5056 * q^79 + 42228 * q^80 + 14580 * q^81 + 24708 * q^83 + 26496 * q^84 - 4752 * q^85 - 77424 * q^86 + 5256 * q^87 + 36000 * q^89 + 19104 * q^91 + 12912 * q^92 - 18432 * q^93 + 112832 * q^94 - 52344 * q^96 + 50012 * q^97 + 11544 * q^98 + 10044 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more