LMFDB - Embedded newform 117.5.j.b.109.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,5,Mod(73,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("117.73");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	117.j (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:	$12.0942856808$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			109.9
Root			$4.32919 - 4.32919i$ of defining polynomial
Character	$\chi$	$=$	117.109
Dual form			117.5.j.b.73.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.32919 - 4.32919i) q^{2} -21.4837i q^{4} +(26.0804 - 26.0804i) q^{5} +(-16.9579 - 16.9579i) q^{7} +(-23.7399 - 23.7399i) q^{8} -225.814i q^{10} +(69.5293 + 69.5293i) q^{11} +(-93.5895 - 140.720i) q^{13} -146.828 q^{14} +138.190 q^{16} -166.635i q^{17} +(-468.820 + 468.820i) q^{19} +(-560.303 - 560.303i) q^{20} +602.011 q^{22} -388.867i q^{23} -735.373i q^{25} +(-1014.37 - 204.035i) q^{26} +(-364.318 + 364.318i) q^{28} +767.891 q^{29} +(-66.8738 + 66.8738i) q^{31} +(978.089 - 978.089i) q^{32} +(-721.395 - 721.395i) q^{34} -884.537 q^{35} +(1532.33 + 1532.33i) q^{37} +4059.22i q^{38} -1238.29 q^{40} +(-802.986 + 802.986i) q^{41} +2798.62i q^{43} +(1493.75 - 1493.75i) q^{44} +(-1683.48 - 1683.48i) q^{46} +(-33.8918 - 33.8918i) q^{47} -1825.86i q^{49} +(-3183.56 - 3183.56i) q^{50} +(-3023.18 + 2010.65i) q^{52} +3139.54 q^{53} +3626.70 q^{55} +805.159i q^{56} +(3324.34 - 3324.34i) q^{58} +(3959.16 + 3959.16i) q^{59} +504.230 q^{61} +579.018i q^{62} -6257.62i q^{64} +(-6110.87 - 1229.17i) q^{65} +(4218.53 - 4218.53i) q^{67} -3579.94 q^{68} +(-3829.33 + 3829.33i) q^{70} +(3025.86 - 3025.86i) q^{71} +(3088.98 + 3088.98i) q^{73} +13267.5 q^{74} +(10072.0 + 10072.0i) q^{76} -2358.14i q^{77} -9376.55 q^{79} +(3604.05 - 3604.05i) q^{80} +6952.55i q^{82} +(-6229.50 + 6229.50i) q^{83} +(-4345.91 - 4345.91i) q^{85} +(12115.7 + 12115.7i) q^{86} -3301.24i q^{88} +(-8558.42 - 8558.42i) q^{89} +(-799.227 + 3973.39i) q^{91} -8354.30 q^{92} -293.448 q^{94} +24454.0i q^{95} +(2882.26 - 2882.26i) q^{97} +(-7904.48 - 7904.48i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) $ 20 * q - 24 * q^5 - 24 * q^7 - 372 * q^11 - 224 * q^13 - 480 * q^14 - 2328 * q^16 - 840 * q^19 - 228 * q^20 + 3536 * q^22 + 828 * q^26 - 1984 * q^28 + 5064 * q^29 + 1712 * q^31 + 7260 * q^32 + 8040 * q^34 - 5976 * q^35 - 868 * q^37 + 6336 * q^40 + 6000 * q^41 - 20868 * q^44 - 1992 * q^46 - 1692 * q^47 - 8952 * q^50 - 12416 * q^52 + 18816 * q^53 - 3280 * q^55 + 1256 * q^58 + 6588 * q^59 - 5512 * q^61 - 33048 * q^65 - 4704 * q^67 + 28296 * q^68 - 31096 * q^70 - 9756 * q^71 + 13692 * q^73 + 28296 * q^74 - 17072 * q^76 - 5056 * q^79 - 42228 * q^80 - 24708 * q^83 - 4752 * q^85 + 77424 * q^86 - 36000 * q^89 + 19104 * q^91 - 12912 * q^92 + 112832 * q^94 + 50012 * q^97 - 11544 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$

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	$-0.472591\pi$
$2$	4.32919	−	4.32919i	1.08230	−	1.08230i	0.996295	−	0.0860014i	$-0.0274090\pi$
$3$		0			0
$3$		0			0
$4$		−	21.4837i		−	1.34273i
$5$	26.0804	−	26.0804i	1.04322	−	1.04322i	0.0441922	−	0.999023i	$-0.485929\pi$
$5$	26.0804	−	26.0804i	1.04322	−	1.04322i	0.999023	−	0.0441922i	$-0.0140714\pi$
$6$		0			0
$7$	−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
$9$		0			0
$10$		−	225.814i		−	2.25814i
$11$	69.5293	+	69.5293i	0.574623	+	0.574623i	0.933417	−	0.358794i	$-0.116812\pi$
$11$	69.5293	+	69.5293i	0.574623	+	0.574623i	−0.358794	+	0.933417i	$0.616812\pi$
$12$		0			0
$13$	−93.5895	−	140.720i	−0.553784	−	0.832660i
$13$	−93.5895	−	140.720i	−0.553784	−	0.832660i
$14$	−146.828			−0.749121
$15$		0			0
$16$	138.190			0.539804
$17$		−	166.635i		−	0.576593i	−0.957541	−	0.288296i	$-0.906911\pi$
$17$		−	166.635i		−	0.576593i	0.957541	−	0.288296i	$-0.0930888\pi$
$18$		0			0
$19$	−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
$21$		0			0
$22$	602.011			1.24382
$23$		−	388.867i		−	0.735098i	−0.930004	−	0.367549i	$-0.880197\pi$
$23$		−	388.867i		−	0.735098i	0.930004	−	0.367549i	$-0.119803\pi$
$24$		0			0
$25$		−	735.373i		−	1.17660i
$26$	−1014.37	−	204.035i	−1.50054	−	0.301827i
$27$		0			0
$28$	−364.318	+	364.318i	−0.464692	+	0.464692i
$29$	767.891			0.913069			0.456534	−	0.889706i	$-0.349091\pi$
$29$	767.891			0.913069			0.456534	+	0.889706i	$0.349091\pi$
$30$		0			0
$31$	−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
$33$		0			0
$34$	−721.395	−	721.395i	−0.624044	−	0.624044i
$35$	−884.537			−0.722071
$36$		0			0
$37$	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
$39$		0			0
$40$	−1238.29			−0.773933
$41$	−802.986	+	802.986i	−0.477684	+	0.477684i	−0.904390	−	0.426707i	$-0.859674\pi$
$41$	−802.986	+	802.986i	−0.477684	+	0.477684i	0.426707	+	0.904390i	$0.359674\pi$
$42$		0			0
$43$			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
$45$		0			0
$46$	−1683.48	−	1683.48i	−0.795594	−	0.795594i
$47$	−33.8918	−	33.8918i	−0.0153426	−	0.0153426i	0.699394	−	0.714736i	$-0.253453\pi$
$47$	−33.8918	−	33.8918i	−0.0153426	−	0.0153426i	−0.714736	+	0.699394i	$0.753453\pi$
$48$		0			0
$49$		−	1825.86i		−	0.760458i
$50$	−3183.56	−	3183.56i	−1.27343	−	1.27343i
$51$		0			0
$52$	−3023.18	+	2010.65i	−1.11804	+	0.743583i
$53$	3139.54			1.11767			0.558836	−	0.829278i	$-0.311248\pi$
$53$	3139.54			1.11767			0.558836	+	0.829278i	$0.311248\pi$
$54$		0			0
$55$	3626.70			1.19891
$56$			805.159i			0.256747i
$57$		0			0
$58$	3324.34	−	3324.34i	0.988211	−	0.988211i
$59$	3959.16	+	3959.16i	1.13736	+	1.13736i	0.988921	+	0.148440i	$0.0474253\pi$
$59$	3959.16	+	3959.16i	1.13736	+	1.13736i	0.148440	+	0.988921i	$0.452575\pi$
$60$		0			0
$61$	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
$63$		0			0
$64$		−	6257.62i		−	1.52774i
$65$	−6110.87	−	1229.17i	−1.44636	−	0.290928i
$66$		0			0
$67$	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
$69$		0			0
$70$	−3829.33	+	3829.33i	−0.781495	+	0.781495i
$71$	3025.86	−	3025.86i	0.600249	−	0.600249i	−0.340130	−	0.940379i	$-0.610471\pi$
$71$	3025.86	−	3025.86i	0.600249	−	0.600249i	0.940379	+	0.340130i	$0.110471\pi$
$72$		0			0
$73$	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
$75$		0			0
$76$	10072.0	+	10072.0i	1.74377	+	1.74377i
$77$		−	2358.14i		−	0.397730i
$78$		0			0
$79$	−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
$81$		0			0
$82$			6952.55i			1.03399i
$83$	−6229.50	+	6229.50i	−0.904267	+	0.904267i	−0.995802	−	0.0915344i	$-0.970823\pi$
$83$	−6229.50	+	6229.50i	−0.904267	+	0.904267i	0.0915344	+	0.995802i	$0.470823\pi$
$84$		0			0
$85$	−4345.91	−	4345.91i	−0.601510	−	0.601510i
$86$	12115.7	+	12115.7i	1.63815	+	1.63815i
$87$		0			0
$88$		−	3301.24i		−	0.426297i
$89$	−8558.42	−	8558.42i	−1.08047	−	1.08047i	−0.996465	−	0.0840075i	$-0.973228\pi$
$89$	−8558.42	−	8558.42i	−1.08047	−	1.08047i	−0.0840075	−	0.996465i	$-0.526772\pi$
$90$		0			0
$91$	−799.227	+	3973.39i	−0.0965133	+	0.479820i
$92$	−8354.30			−0.987039
$93$		0			0
$94$	−293.448			−0.0332105
$95$			24454.0i			2.70959i
$96$		0			0
$97$	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
$99$		0			0
$100$	−15798.5			−1.57985
$101$			5594.65i			0.548441i	0.961667	+	0.274220i	$0.0884198\pi$
$101$			5594.65i			0.548441i	−0.961667	+	0.274220i	$0.911580\pi$
$102$		0			0
$103$			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
$105$		0			0
$106$	13591.7	−	13591.7i	1.20965	−	1.20965i
$107$	8462.24			0.739125			0.369563	−	0.929206i	$-0.379508\pi$
$107$	8462.24			0.739125			0.369563	+	0.929206i	$0.379508\pi$
$108$		0			0
$109$	−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
$111$		0			0
$112$	−2343.41	−	2343.41i	−0.186815	−	0.186815i
$113$	−14864.2			−1.16409			−0.582043	−	0.813158i	$-0.697746\pi$
$113$	−14864.2			−1.16409			−0.582043	+	0.813158i	$0.697746\pi$
$114$		0			0
$115$	−10141.8	−	10141.8i	−0.766866	−	0.766866i
$116$		−	16497.1i		−	1.22601i
$117$		0			0
$118$	34279.8			2.46192
$119$	−2825.78	+	2825.78i	−0.199547	+	0.199547i
$120$		0			0
$121$		−	4972.34i		−	0.339618i
$122$	2182.90	−	2182.90i	0.146661	−	0.146661i
$123$		0			0
$124$	1436.70	+	1436.70i	0.0934376	+	0.0934376i
$125$	−2878.56	−	2878.56i	−0.184228	−	0.184228i
$126$		0			0
$127$			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
$129$		0			0
$130$	−31776.4	+	21133.8i	−1.88026	+	1.25052i
$131$	−14023.3			−0.817160			−0.408580	−	0.912722i	$-0.633976\pi$
$131$	−14023.3			−0.817160			−0.408580	+	0.912722i	$0.633976\pi$
$132$		0			0
$133$	15900.4			0.898887
$134$		−	36525.6i		−	2.03417i
$135$		0			0
$136$	−3955.91	+	3955.91i	−0.213879	+	0.213879i
$137$	−20682.4	−	20682.4i	−1.10194	−	1.10194i	−0.994176	−	0.107769i	$-0.965629\pi$
$137$	−20682.4	−	20682.4i	−1.10194	−	1.10194i	−0.107769	−	0.994176i	$-0.534371\pi$
$138$		0			0
$139$	−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
$141$		0			0
$142$		−	26199.0i		−	1.29929i
$143$	3276.92	−	16291.4i	0.160248	−	0.796682i
$144$		0			0
$145$	20026.9	−	20026.9i	0.952527	−	0.952527i
$146$	26745.5			1.25472
$147$		0			0
$148$	32920.2	−	32920.2i	1.50293	−	1.50293i
$149$	1317.22	−	1317.22i	0.0593315	−	0.0593315i	−0.676818	−	0.736150i	$-0.736642\pi$
$149$	1317.22	−	1317.22i	0.0593315	−	0.0593315i	0.736150	+	0.676818i	$0.236642\pi$
$150$		0			0
$151$	−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
$153$		0			0
$154$	−10208.8	−	10208.8i	−0.430462	−	0.430462i
$155$			3488.19i			0.145190i
$156$		0			0
$157$	−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
$159$		0			0
$160$		−	51017.9i		−	1.99289i
$161$	−6594.37	+	6594.37i	−0.254403	+	0.254403i
$162$		0			0
$163$	−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
$165$		0			0
$166$			53937.3i			1.95737i
$167$	17352.4	+	17352.4i	0.622195	+	0.622195i	0.946092	−	0.323897i	$-0.104993\pi$
$167$	17352.4	+	17352.4i	0.622195	+	0.622195i	−0.323897	+	0.946092i	$0.604993\pi$
$168$		0			0
$169$	−11043.0	+	26339.8i	−0.386646	+	0.922228i
$170$	−37628.5			−1.30202
$171$		0			0
$172$	60124.7			2.03234
$173$		−	2780.29i		−	0.0928961i	−0.998921	−	0.0464480i	$-0.985210\pi$
$173$		−	2780.29i		−	0.0928961i	0.998921	−	0.0464480i	$-0.0147902\pi$
$174$		0			0
$175$	−12470.4	+	12470.4i	−0.407196	+	0.407196i
$176$	9608.25	+	9608.25i	0.310184	+	0.310184i
$177$		0			0
$178$	−74102.0			−2.33878
$179$			2972.36i			0.0927674i	0.998924	+	0.0463837i	$0.0147697\pi$
$179$			2972.36i			0.0927674i	−0.998924	+	0.0463837i	$0.985230\pi$
$180$		0			0
$181$			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
$183$		0			0
$184$	−9231.68	+	9231.68i	−0.272675	+	0.272675i
$185$	79927.7			2.33536
$186$		0			0
$187$	11586.0	−	11586.0i	0.331323	−	0.331323i
$188$	−728.122	+	728.122i	−0.0206010	+	0.0206010i
$189$		0			0
$190$	105866.	+	105866.i	2.93258	+	2.93258i
$191$	3688.96			0.101120			0.0505601	−	0.998721i	$-0.483899\pi$
$191$	3688.96			0.101120			0.0505601	+	0.998721i	$0.483899\pi$
$192$		0			0
$193$	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
$195$		0			0
$196$	−39226.2			−1.02109
$197$	15917.0	−	15917.0i	0.410136	−	0.410136i	−0.471650	−	0.881786i	$-0.656341\pi$
$197$	15917.0	−	15917.0i	0.410136	−	0.410136i	0.881786	+	0.471650i	$0.156341\pi$
$198$		0			0
$199$		−	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
$201$		0			0
$202$	24220.3	+	24220.3i	0.593576	+	0.593576i
$203$	−13021.8	−	13021.8i	−0.315995	−	0.315995i
$204$		0			0
$205$			41884.4i			0.996654i
$206$	29407.0	+	29407.0i	0.692974	+	0.692974i
$207$		0			0
$208$	−12933.1	−	19446.0i	−0.298935	−	0.449474i
$209$	−65193.5			−1.49249
$210$		0			0
$211$	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
$213$		0			0
$214$	36634.6	−	36634.6i	0.799953	−	0.799953i
$215$	72989.0	+	72989.0i	1.57899	+	1.57899i
$216$		0			0
$217$	2268.08			0.0481658
$218$			104575.i			2.20047i
$219$		0			0
$220$		−	77915.0i		−	1.60981i
$221$	−23448.8	+	15595.3i	−0.480106	+	0.319308i
$222$		0			0
$223$	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
$225$		0			0
$226$	−64349.9	+	64349.9i	−1.25989	+	1.25989i
$227$	−15576.9	+	15576.9i	−0.302294	+	0.302294i	−0.841911	−	0.539617i	$-0.818569\pi$
$227$	−15576.9	+	15576.9i	−0.302294	+	0.302294i	0.539617	+	0.841911i	$0.318569\pi$
$228$		0			0
$229$	−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
$231$		0			0
$232$	−18229.7	−	18229.7i	−0.338691	−	0.338691i
$233$		−	22434.3i		−	0.413239i	−0.978421	−	0.206619i	$-0.933754\pi$
$233$		−	22434.3i		−	0.413239i	0.978421	−	0.206619i	$-0.0662462\pi$
$234$		0			0
$235$	−1767.82			−0.0320113
$236$	85057.3	−	85057.3i	1.52717	−	1.52717i
$237$		0			0
$238$			24466.7i			0.431938i
$239$	66829.7	−	66829.7i	1.16997	−	1.16997i	0.187751	−	0.982217i	$-0.439880\pi$
$239$	66829.7	−	66829.7i	1.16997	−	1.16997i	0.982217	−	0.187751i	$-0.0601199\pi$
$240$		0			0
$241$	−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
$243$		0			0
$244$		−	10832.7i		−	0.181952i
$245$	−47619.1	−	47619.1i	−0.793321	−	0.793321i
$246$		0			0
$247$	109849.	+	22095.5i	1.80054	+	0.362168i
$248$	3175.16			0.0516253
$249$		0			0
$250$	−24923.6			−0.398778
$251$			4768.38i			0.0756873i	0.999284	+	0.0378437i	$0.0120489\pi$
$251$			4768.38i			0.0756873i	−0.999284	+	0.0378437i	$0.987951\pi$
$252$		0			0
$253$	27037.7	−	27037.7i	0.422404	−	0.422404i
$254$	47601.9	+	47601.9i	0.737832	+	0.737832i
$255$		0			0
$256$	1061.74			0.0162008
$257$		−	62743.6i		−	0.949956i	−0.879998	−	0.474978i	$-0.842456\pi$
$257$		−	62743.6i		−	0.949956i	0.879998	−	0.474978i	$-0.157544\pi$
$258$		0			0
$259$		−	51970.3i		−	0.774740i
$260$	−26407.1	+	131284.i	−0.390638	+	1.94207i
$261$		0			0
$262$	−60709.4	+	60709.4i	−0.884409	+	0.884409i
$263$	−45421.0			−0.656667			−0.328333	−	0.944562i	$-0.606487\pi$
$263$	−45421.0			−0.656667			−0.328333	+	0.944562i	$0.606487\pi$
$264$		0			0
$265$	81880.4	−	81880.4i	1.16597	−	1.16597i
$266$	68835.8	−	68835.8i	0.972862	−	0.972862i
$267$		0			0
$268$	−90629.6	−	90629.6i	−1.26183	−	1.26183i
$269$	−79205.8			−1.09459			−0.547296	−	0.836939i	$-0.684343\pi$
$269$	−79205.8			−1.09459			−0.547296	+	0.836939i	$0.684343\pi$
$270$		0			0
$271$	−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
$273$		0			0
$274$	−179076.			−2.38526
$275$	51130.0	−	51130.0i	0.676099	−	0.676099i
$276$		0			0
$277$		−	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
$279$		0			0
$280$	20998.9	+	20998.9i	0.267843	+	0.267843i
$281$	103231.	+	103231.i	1.30736	+	1.30736i	0.923311	+	0.384052i	$0.125472\pi$
$281$	103231.	+	103231.i	1.30736	+	1.30736i	0.384052	+	0.923311i	$0.374528\pi$
$282$		0			0
$283$			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
$285$		0			0
$286$	−56341.9	−	84714.7i	−0.688810	−	1.03568i
$287$	27233.9			0.330633
$288$		0			0
$289$	55753.7			0.667541
$290$		−	173400.i		−	2.06183i
$291$		0			0
$292$	66362.7	−	66362.7i	0.778320	−	0.778320i
$293$	5490.45	+	5490.45i	0.0639547	+	0.0639547i	0.738361	−	0.674406i	$-0.235600\pi$
$293$	5490.45	+	5490.45i	0.0639547	+	0.0639547i	−0.674406	+	0.738361i	$0.735600\pi$
$294$		0			0
$295$	206513.			2.37303
$296$		−	72755.0i		−	0.830385i
$297$		0			0
$298$		−	11405.0i		−	0.128429i
$299$	−54721.2	+	36393.9i	−0.612087	+	0.407086i
$300$		0			0
$301$	47458.7	−	47458.7i	0.523821	−	0.523821i
$302$	−51976.9			−0.569897
$303$		0			0
$304$	−64786.2	+	64786.2i	−0.701028	+	0.701028i
$305$	13150.5	−	13150.5i	0.141365	−	0.141365i
$306$		0			0
$307$	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
$309$		0			0
$310$	15101.0	+	15101.0i	0.157139	+	0.157139i
$311$			57004.6i			0.589371i	0.955594	+	0.294686i	$0.0952149\pi$
$311$			57004.6i			0.589371i	−0.955594	+	0.294686i	$0.904785\pi$
$312$		0			0
$313$	−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
$315$		0			0
$316$			201443.i			2.01733i
$317$	57585.7	−	57585.7i	0.573055	−	0.573055i	−0.359926	−	0.932981i	$-0.617198\pi$
$317$	57585.7	−	57585.7i	0.573055	−	0.573055i	0.932981	+	0.359926i	$0.117198\pi$
$318$		0			0
$319$	53390.9	+	53390.9i	0.524670	+	0.524670i
$320$	−163201.	−	163201.i	−1.59376	−	1.59376i
$321$		0			0
$322$			57096.5i			0.550678i
$323$	78122.0	+	78122.0i	0.748804	+	0.748804i
$324$		0			0
$325$	−103481.	+	68823.2i	−0.979705	+	0.651580i
$326$	−209874.			−1.97481
$327$		0			0
$328$	38125.7			0.354381
$329$			1149.47i			0.0106195i
$330$		0			0
$331$	−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
$333$		0			0
$334$	150243.			1.34680
$335$		−	220042.i		−	1.96072i
$336$		0			0
$337$			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
$339$		0			0
$340$	−93366.3	+	93366.3i	−0.807667	+	0.807667i
$341$	−9299.38			−0.0799734
$342$		0			0
$343$	−71678.7	+	71678.7i	−0.609259	+	0.609259i
$344$	66439.0	−	66439.0i	0.561444	−	0.561444i
$345$		0			0
$346$	−12036.4	−	12036.4i	−0.100541	−	0.100541i
$347$	183864.			1.52699			0.763497	−	0.645811i	$-0.223481\pi$
$347$	183864.			1.52699			0.763497	+	0.645811i	$0.223481\pi$
$348$		0			0
$349$	−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
$351$		0			0
$352$	136012.			1.09772
$353$	−106752.	+	106752.i	−0.856695	+	0.856695i	−0.990947	−	0.134252i	$-0.957137\pi$
$353$	−106752.	+	106752.i	−0.856695	+	0.856695i	0.134252	+	0.990947i	$0.457137\pi$
$354$		0			0
$355$		−	157831.i		−	1.25238i
$356$	−183867.	+	183867.i	−1.45078	+	1.45078i
$357$		0			0
$358$	12867.9	+	12867.9i	0.100402	+	0.100402i
$359$	26893.9	+	26893.9i	0.208672	+	0.208672i	0.803703	−	0.595031i	$-0.202860\pi$
$359$	26893.9	+	26893.9i	0.208672	+	0.208672i	−0.595031	+	0.803703i	$0.702860\pi$
$360$		0			0
$361$		−	309264.i		−	2.37309i
$362$	190631.	+	190631.i	1.45471	+	1.45471i
$363$		0			0
$364$	85363.1	+	17170.3i	0.644269	+	0.129591i
$365$	161123.			1.20941
$366$		0			0
$367$	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
$369$		0			0
$370$	346022.	−	346022.i	2.52755	−	2.52755i
$371$	−53240.0	−	53240.0i	−0.386804	−	0.386804i
$372$		0			0
$373$	−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
$375$		0			0
$376$			1609.18i			0.0113823i
$377$	−71866.5	−	108057.i	−0.505643	−	0.760276i
$378$		0			0
$379$	−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
$381$		0			0
$382$	15970.2	−	15970.2i	0.109442	−	0.109442i
$383$	15911.5	−	15911.5i	0.108471	−	0.108471i	−0.650788	−	0.759259i	$-0.725561\pi$
$383$	15911.5	−	15911.5i	0.108471	−	0.108471i	0.759259	+	0.650788i	$0.225561\pi$
$384$		0			0
$385$	−61501.3	−	61501.3i	−0.414918	−	0.414918i
$386$	166387.			1.11672
$387$		0			0
$388$	−61921.5	−	61921.5i	−0.411319	−	0.411319i
$389$		−	215552.i		−	1.42447i	−0.701941	−	0.712236i	$-0.747683\pi$
$389$		−	215552.i		−	1.42447i	0.701941	−	0.712236i	$-0.252317\pi$
$390$		0			0
$391$	−64799.0			−0.423852
$392$	−43345.8	+	43345.8i	−0.282082	+	0.282082i
$393$		0			0
$394$		−	137815.i		−	0.887778i
$395$	−244544.	+	244544.i	−1.56734	+	1.56734i
$396$		0			0
$397$	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
$399$		0			0
$400$		−	101621.i		−	0.635132i
$401$	−170564.	−	170564.i	−1.06072	−	1.06072i	−0.998034	−	0.0626817i	$-0.980035\pi$
$401$	−170564.	−	170564.i	−1.06072	−	1.06072i	−0.0626817	−	0.998034i	$-0.519965\pi$
$402$		0			0
$403$	15669.1	+	3151.77i	0.0964795	+	0.0194063i
$404$	120194.			0.736409
$405$		0			0
$406$	−112748.			−0.683999
$407$			213084.i			1.28636i
$408$		0			0
$409$	−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
$411$		0			0
$412$	145933.			0.859725
$413$		−	134278.i		−	0.787235i
$414$		0			0
$415$			324935.i			1.88669i
$416$	−229175.	−	46097.4i	−1.32428	−	0.266373i
$417$		0			0
$418$	−282235.	+	282235.i	−1.61532	+	1.61532i
$419$	20127.4			0.114646			0.0573232	−	0.998356i	$-0.481743\pi$
$419$	20127.4			0.114646			0.0573232	+	0.998356i	$0.481743\pi$
$420$		0			0
$421$	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
$423$		0			0
$424$	−74532.5	−	74532.5i	−0.414586	−	0.414586i
$425$	−122539.			−0.678417
$426$		0			0
$427$	−8550.68	−	8550.68i	−0.0468970	−	0.0468970i
$428$		−	181800.i		−	0.992446i
$429$		0			0
$430$	631966.			3.41788
$431$	−127549.	+	127549.i	−0.686630	+	0.686630i	−0.961485	−	0.274856i	$-0.911370\pi$
$431$	−127549.	+	127549.i	−0.686630	+	0.686630i	0.274856	+	0.961485i	$0.411370\pi$
$432$		0			0
$433$			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
$435$		0			0
$436$	259478.	+	259478.i	1.36498	+	1.36498i
$437$	182309.	+	182309.i	0.954651	+	0.954651i
$438$		0			0
$439$		−	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
$441$		0			0
$442$	−33999.4	+	169029.i	−0.174031	+	0.865203i
$443$	−92267.8			−0.470157			−0.235079	−	0.971976i	$-0.575535\pi$
$443$	−92267.8			−0.470157			−0.235079	+	0.971976i	$0.575535\pi$
$444$		0			0
$445$	−446414.			−2.25433
$446$		−	215984.i		−	1.08580i
$447$		0			0
$448$	−106116.	+	106116.i	−0.528719	+	0.528719i
$449$	17542.3	+	17542.3i	0.0870148	+	0.0870148i	0.749274	−	0.662260i	$-0.230402\pi$
$449$	17542.3	+	17542.3i	0.0870148	+	0.0870148i	−0.662260	+	0.749274i	$0.730402\pi$
$450$		0			0
$451$	−111662.			−0.548976
$452$			319338.i			1.56305i
$453$		0			0
$454$			134871.i			0.654343i
$455$	82783.4	+	124472.i	0.399872	+	0.601240i
$456$		0			0
$457$	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
$459$		0			0
$460$	−217883.	+	217883.i	−1.02969	+	1.02969i
$461$	12382.8	−	12382.8i	0.0582663	−	0.0582663i	−0.677373	−	0.735640i	$-0.736882\pi$
$461$	12382.8	−	12382.8i	0.0582663	−	0.0582663i	0.735640	+	0.677373i	$0.236882\pi$
$462$		0			0
$463$	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
$465$		0			0
$466$	−97122.3	−	97122.3i	−0.447247	−	0.447247i
$467$		−	3880.03i		−	0.0177911i	−0.999960	−	0.00889553i	$-0.997168\pi$
$467$		−	3880.03i		−	0.0177911i	0.999960	−	0.00889553i	$-0.00283157\pi$
$468$		0			0
$469$	−143075.			−0.650455
$470$	−7653.24	+	7653.24i	−0.0346457	+	0.0346457i
$471$		0			0
$472$		−	187980.i		−	0.843778i
$473$	−194586.	+	194586.i	−0.869740	+	0.869740i
$474$		0			0
$475$	344758.	+	344758.i	1.52801	+	1.52801i
$476$	60708.3	+	60708.3i	0.267938	+	0.267938i
$477$		0			0
$478$		−	578637.i		−	2.53250i
$479$	−53446.0	−	53446.0i	−0.232940	−	0.232940i	0.580979	−	0.813919i	$-0.302670\pi$
$479$	−53446.0	−	53446.0i	−0.232940	−	0.232940i	−0.813919	+	0.580979i	$0.802670\pi$
$480$		0			0
$481$	72218.9	−	359040.i	0.312148	−	1.55186i
$482$	−301546.			−1.29795
$483$		0			0
$484$	−106824.			−0.456015
$485$		−	150341.i		−	0.639136i
$486$		0			0
$487$	−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
$489$		0			0
$490$	−412304.			−1.71722
$491$		−	183063.i		−	0.759342i	−0.925122	−	0.379671i	$-0.876037\pi$
$491$		−	183063.i		−	0.759342i	0.925122	−	0.379671i	$-0.123963\pi$
$492$		0			0
$493$		−	127958.i		−	0.526469i
$494$	571212.	−	379900.i	2.34069	−	1.55674i
$495$		0			0
$496$	−9241.28	+	9241.28i	−0.0375638	+	0.0375638i
$497$	−102624.			−0.415468
$498$		0			0
$499$	−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
$501$		0			0
$502$	20643.2	+	20643.2i	0.0819161	+	0.0819161i
$503$	192555.			0.761061			0.380530	−	0.924768i	$-0.375741\pi$
$503$	192555.			0.761061			0.380530	+	0.924768i	$0.375741\pi$
$504$		0			0
$505$	145910.	+	145910.i	0.572142	+	0.572142i
$506$		−	234102.i		−	0.914333i
$507$		0			0
$508$	236226.			0.915377
$509$	84764.7	−	84764.7i	0.327175	−	0.327175i	−0.524336	−	0.851511i	$-0.675687\pi$
$509$	84764.7	−	84764.7i	0.327175	−	0.327175i	0.851511	+	0.524336i	$0.175687\pi$
$510$		0			0
$511$		−	104765.i		−	0.401213i
$512$	187652.	−	187652.i	0.715835	−	0.715835i
$513$		0			0
$514$	−271629.	−	271629.i	−1.02813	−	1.02813i
$515$	177157.	+	177157.i	0.667951	+	0.667951i
$516$		0			0
$517$		−	4712.95i		−	0.0176324i
$518$	−224989.	−	224989.i	−0.838498	−	0.838498i
$519$		0			0
$520$	115891.	+	174252.i	0.428592	+	0.644424i
$521$	−314076.			−1.15707			−0.578534	−	0.815658i	$-0.696375\pi$
$521$	−314076.			−1.15707			−0.578534	+	0.815658i	$0.696375\pi$
$522$		0			0
$523$	−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
$525$		0			0
$526$	−196636.	+	196636.i	−0.710708	+	0.710708i
$527$	11143.5	+	11143.5i	0.0401238	+	0.0401238i
$528$		0			0
$529$	128624.			0.459631
$530$		−	708951.i		−	2.52386i
$531$		0			0
$532$		−	341600.i		−	1.20696i
$533$	188147.	+	37844.8i	0.662282	+	0.133214i
$534$		0			0
$535$	220699.	−	220699.i	0.771067	−	0.771067i
$536$	−200295.			−0.697174
$537$		0			0
$538$	−342897.	+	342897.i	−1.18467	+	1.18467i
$539$	126951.	−	126951.i	0.436976	−	0.436976i
$540$		0			0
$541$	−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
$543$		0			0
$544$	−162984.	−	162984.i	−0.550741	−	0.550741i
$545$			629992.i			2.12101i
$546$		0			0
$547$	−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
$549$		0			0
$550$		−	442702.i		−	1.46348i
$551$	−360003.	+	360003.i	−1.18578	+	1.18578i
$552$		0			0
$553$	159007.	+	159007.i	0.519954	+	0.519954i
$554$	−12486.4	−	12486.4i	−0.0406835	−	0.0406835i
$555$		0			0
$556$			270353.i			0.874543i
$557$	−280053.	−	280053.i	−0.902672	−	0.902672i	0.0929944	−	0.995667i	$-0.470356\pi$
$557$	−280053.	−	280053.i	−0.902672	−	0.902672i	−0.995667	+	0.0929944i	$0.970356\pi$
$558$		0			0
$559$	393820.	−	261921.i	1.26030	−	0.838199i
$560$	−122234.			−0.389777
$561$		0			0
$562$	893810.			2.82991
$563$		−	50493.5i		−	0.159301i	−0.996823	−	0.0796505i	$-0.974620\pi$
$563$		−	50493.5i		−	0.159301i	0.996823	−	0.0796505i	$-0.0253804\pi$
$564$		0			0
$565$	−387664.	+	387664.i	−1.21439	+	1.21439i
$566$	340194.	+	340194.i	1.06193	+	1.06193i
$567$		0			0
$568$	−143667.			−0.445309
$569$			301875.i			0.932400i	0.884679	+	0.466200i	$0.154377\pi$
$569$			301875.i			0.932400i	−0.884679	+	0.466200i	$0.845623\pi$
$570$		0			0
$571$		−	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
$573$		0			0
$574$	117901.	−	117901.i	0.357843	−	0.357843i
$575$	−285962.			−0.864914
$576$		0			0
$577$	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
$579$		0			0
$580$	−430252.	−	430252.i	−1.27899	−	1.27899i
$581$	211278.			0.625897
$582$		0			0
$583$	218290.	+	218290.i	0.642240	+	0.642240i
$584$		−	146664.i		−	0.430030i
$585$		0			0
$586$	47538.4			0.138436
$587$	432681.	−	432681.i	1.25572	−	1.25572i	0.302598	−	0.953118i	$-0.402146\pi$
$587$	432681.	−	432681.i	1.25572	−	1.25572i	0.953118	−	0.302598i	$-0.0978541\pi$
$588$		0			0
$589$		−	62703.6i		−	0.180743i
$590$	894031.	−	894031.i	2.56832	−	2.56832i
$591$		0			0
$592$	211753.	+	211753.i	0.604207	+	0.604207i
$593$	−351836.	−	351836.i	−1.00053	−	1.00053i	−1.00000		0.000531251i	$-0.999831\pi$
$593$	−351836.	−	351836.i	−1.00053	−	1.00053i	−0.000531251	−	1.00000i	$-0.500169\pi$
$594$		0			0
$595$			147395.i			0.416341i
$596$	−28298.7	−	28298.7i	−0.0796663	−	0.0796663i
$597$		0			0
$598$	−79342.4	+	394454.i	−0.221872	+	1.10305i
$599$	484810.			1.35119			0.675597	−	0.737271i	$-0.263886\pi$
$599$	484810.			1.35119			0.675597	+	0.737271i	$0.263886\pi$
$600$		0			0
$601$	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
$603$		0			0
$604$	−128968.	+	128968.i	−0.353516	+	0.353516i
$605$	−129681.	−	129681.i	−0.354294	−	0.354294i
$606$		0			0
$607$	−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
$609$		0			0
$610$		−	113862.i		−	0.305998i
$611$	−1597.32	+	7941.17i	−0.00427869	+	0.0212717i
$612$		0			0
$613$	−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
$615$		0			0
$616$	−55982.2	+	55982.2i	−0.147533	+	0.147533i
$617$	−260927.	+	260927.i	−0.685407	+	0.685407i	−0.961213	−	0.275806i	$-0.911055\pi$
$617$	−260927.	+	260927.i	−0.685407	+	0.685407i	0.275806	+	0.961213i	$0.411055\pi$
$618$		0			0
$619$	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
$621$		0			0
$622$	246783.	+	246783.i	0.637874	+	0.637874i
$623$			290266.i			0.747859i
$624$		0			0
$625$	309460.			0.792218
$626$	−731248.	+	731248.i	−1.86602	+	1.86602i
$627$		0			0
$628$			520606.i			1.32005i
$629$	255341.	−	255341.i	0.645385	−	0.645385i
$630$		0			0
$631$	−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
$633$		0			0
$634$		−	498599.i		−	1.24043i
$635$	286769.	+	286769.i	0.711189	+	0.711189i
$636$		0			0
$637$	−256934.	+	170881.i	−0.633203	+	0.421130i
$638$	462279.			1.13570
$639$		0			0
$640$	−596770.			−1.45696
$641$		−	466494.i		−	1.13535i	−0.823253	−	0.567675i	$-0.807843\pi$
$641$		−	466494.i		−	1.13535i	0.823253	−	0.567675i	$-0.192157\pi$
$642$		0			0
$643$	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
$645$		0			0
$646$	676409.			1.62086
$647$			562437.i			1.34358i	0.740740	+	0.671792i	$0.234475\pi$
$647$			562437.i			1.34358i	−0.740740	+	0.671792i	$0.765525\pi$
$648$		0			0
$649$			550555.i			1.30711i
$650$	−150042.	+	745938.i	−0.355128	+	1.76553i
$651$		0			0
$652$	−520754.	+	520754.i	−1.22500	+	1.22500i
$653$	−37629.2			−0.0882467			−0.0441234	−	0.999026i	$-0.514049\pi$
$653$	−37629.2			−0.0882467			−0.0441234	+	0.999026i	$0.514049\pi$
$654$		0			0
$655$	−365733.	+	365733.i	−0.852474	+	0.852474i
$656$	−110965.	+	110965.i	−0.257856	+	0.257856i
$657$		0			0
$658$	4976.27	+	4976.27i	0.0114935	+	0.0114935i
$659$	−18033.4			−0.0415248			−0.0207624	−	0.999784i	$-0.506609\pi$
$659$	−18033.4			−0.0415248			−0.0207624	+	0.999784i	$0.506609\pi$
$660$		0			0
$661$	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
$663$		0			0
$664$	295776.			0.670852
$665$	414689.	−	414689.i	0.937733	−	0.937733i
$666$		0			0
$667$		−	298607.i		−	0.671195i
$668$	372794.	−	372794.i	0.835440	−	0.835440i
$669$		0			0
$670$	−952601.	−	952601.i	−2.12208	−	2.12208i
$671$	35058.7	+	35058.7i	0.0778666	+	0.0778666i
$672$		0			0
$673$			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
$675$		0			0
$676$	565875.	+	237244.i	1.23830	+	0.519162i
$677$	−438381.			−0.956478			−0.478239	−	0.878230i	$-0.658725\pi$
$677$	−438381.			−0.956478			−0.478239	+	0.878230i	$0.658725\pi$
$678$		0			0
$679$	−97754.1			−0.212029
$680$			206343.i			0.446244i
$681$		0			0
$682$	−40258.7	+	40258.7i	−0.0865549	+	0.0865549i
$683$	14323.9	+	14323.9i	0.0307057	+	0.0307057i	0.722293	−	0.691587i	$-0.243088\pi$
$683$	14323.9	+	14323.9i	0.0307057	+	0.0307057i	−0.691587	+	0.722293i	$0.743088\pi$
$684$		0			0
$685$	−1.07881e6			−2.29913
$686$			620620.i			1.31880i
$687$		0			0
$688$			386741.i			0.817039i
$689$	−293828.	−	441795.i	−0.618949	−	0.930641i
$690$		0			0
$691$	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
$693$		0			0
$694$	795981.	−	795981.i	1.65266	−	1.65266i
$695$	−328198.	+	328198.i	−0.679464	+	0.679464i
$696$		0			0
$697$	133806.	+	133806.i	0.275429	+	0.275429i
$698$	−630216.			−1.29354
$699$		0			0
$700$	267910.	+	267910.i	0.546755	+	0.546755i
$701$		−	77204.1i		−	0.157110i	−0.996910	−	0.0785551i	$-0.974969\pi$
$701$		−	77204.1i		−	0.157110i	0.996910	−	0.0785551i	$-0.0250307\pi$
$702$		0			0
$703$	−1.43678e6			−2.90723
$704$	435088.	−	435088.i	0.877873	−	0.877873i
$705$		0			0
$706$			924298.i			1.85440i
$707$	94873.4	−	94873.4i	0.189804	−	0.189804i
$708$		0			0
$709$	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
$711$		0			0
$712$			406353.i			0.801574i
$713$	26005.0	+	26005.0i	0.0511538	+	0.0511538i
$714$		0			0
$715$	−339421.	−	510348.i	−0.663937	−	0.998285i
$716$	63857.3			0.124562
$717$		0			0
$718$	232857.			0.451690
$719$			324777.i			0.628243i	0.949383	+	0.314121i	$0.101710\pi$
$719$			324777.i			0.628243i	−0.949383	+	0.314121i	$0.898290\pi$
$720$		0			0
$721$	115191.	−	115191.i	0.221588	−	0.221588i
$722$	−1.33886e6	−	1.33886e6i	−2.56839	−	2.56839i
$723$		0			0
$724$	946013.			1.80476
$725$		−	564686.i		−	1.07431i
$726$		0			0
$727$		−	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
$729$		0			0
$730$	697533.	−	697533.i	1.30894	−	1.30894i
$731$	466348.			0.872722
$732$		0			0
$733$	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
$735$		0			0
$736$	−380346.	−	380346.i	−0.702140	−	0.702140i
$737$	586623.			1.08000
$738$		0			0
$739$	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
$741$		0			0
$742$	−460972.			−0.837272
$743$	671107.	−	671107.i	1.21567	−	1.21567i	0.246530	−	0.969135i	$-0.420710\pi$
$743$	671107.	−	671107.i	1.21567	−	1.21567i	0.969135	−	0.246530i	$-0.0792905\pi$
$744$		0			0
$745$		−	68707.2i		−	0.123791i
$746$	−230689.	+	230689.i	−0.414523	+	0.414523i
$747$		0			0
$748$	−248911.	−	248911.i	−0.444878	−	0.444878i
$749$	−143502.	−	143502.i	−0.255796	−	0.255796i
$750$		0			0
$751$			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
$753$		0			0
$754$	−778924.	−	156676.i	−1.37010	−	0.275588i
$755$	−313125.			−0.549319
$756$		0			0
$757$	−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
$759$		0			0
$760$	580537.	−	580537.i	1.00508	−	1.00508i
$761$	762616.	+	762616.i	1.31685	+	1.31685i	0.916254	+	0.400597i	$0.131197\pi$
$761$	762616.	+	762616.i	1.31685	+	1.31685i	0.400597	+	0.916254i	$0.368803\pi$
$762$		0			0
$763$	409632.			0.703630
$764$		−	79252.6i		−	0.135777i
$765$		0			0
$766$		−	137768.i		−	0.234795i
$767$	186595.	−	927666.i	0.317183	−	1.57689i
$768$		0			0
$769$	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
$771$		0			0
$772$	412850.	−	412850.i	0.692720	−	0.692720i
$773$	311294.	−	311294.i	0.520968	−	0.520968i	−0.396895	−	0.917864i	$-0.629912\pi$
$773$	311294.	−	311294.i	0.520968	−	0.520968i	0.917864	+	0.396895i	$0.129912\pi$
$774$		0			0
$775$	49177.2	+	49177.2i	0.0818766	+	0.0818766i
$776$	−136849.			−0.227258
$777$		0			0
$778$	−933166.	−	933166.i	−1.54170	−	1.54170i
$779$		−	752912.i		−	1.24071i
$780$		0			0
$781$	420771.			0.689833
$782$	−280527.	+	280527.i	−0.458734	+	0.458734i
$783$		0			0
$784$		−	252315.i		−	0.410498i
$785$	−631996.	+	631996.i	−1.02559	+	1.02559i
$786$		0			0
$787$	−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
$789$		0			0
$790$			2.11735e6i			3.39265i
$791$	252066.	+	252066.i	0.402866	+	0.402866i
$792$		0			0
$793$	−47190.6	−	70955.0i	−0.0750428	−	0.112833i
$794$	703706.			1.11622
$795$		0			0
$796$	−500805.			−0.790393
$797$			550495.i			0.866636i	0.901241	+	0.433318i	$0.142657\pi$
$797$			550495.i			0.866636i	−0.901241	+	0.433318i	$0.857343\pi$
$798$		0			0
$799$	−5647.58	+	5647.58i	−0.00884644	+	0.00884644i
$800$	−719260.	−	719260.i	−1.12384	−	1.12384i
$801$		0			0
$802$	−1.47681e6			−2.29602
$803$			429549.i			0.666165i
$804$		0			0
$805$			343967.i			0.530793i
$806$	81479.2	−	54190.0i	0.125423	−	0.0834160i
$807$		0			0
$808$	132817.	−	132817.i	0.203437	−	0.203437i
$809$	−705446.			−1.07787			−0.538936	−	0.842347i	$-0.681173\pi$
$809$	−705446.			−1.07787			−0.538936	+	0.842347i	$0.681173\pi$
$810$		0			0
$811$	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
$813$		0			0
$814$	922481.	+	922481.i	1.39222	+	1.39222i
$815$	−1.26435e6			−1.90350
$816$		0			0
$817$	−1.31205e6	−	1.31205e6i	−1.96565	−	1.96565i
$818$			799064.i			1.19419i
$819$		0			0
$820$	899831.			1.33824
$821$	−162224.	+	162224.i	−0.240675	+	0.240675i	−0.817129	−	0.576455i	$-0.804436\pi$
$821$	−162224.	+	162224.i	−0.240675	+	0.240675i	0.576455	+	0.817129i	$0.304436\pi$
$822$		0			0
$823$			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
$825$		0			0
$826$	−581314.	−	581314.i	−0.852022	−	0.852022i
$827$	542193.	+	542193.i	0.792762	+	0.792762i	0.981942	−	0.189180i	$-0.0605830\pi$
$827$	542193.	+	542193.i	0.792762	+	0.792762i	−0.189180	+	0.981942i	$0.560583\pi$
$828$		0			0
$829$			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
$831$		0			0
$832$	−880569.	+	585648.i	−1.27209	+	0.846038i
$833$	−304253.			−0.438474
$834$		0			0
$835$	905114.			1.29817
$836$			1.40060e6i			2.00401i
$837$		0			0
$838$	87135.4	−	87135.4i	0.124081	−	0.124081i
$839$	−702797.	−	702797.i	−0.998404	−	0.998404i	0.00159522	−	0.999999i	$-0.499492\pi$
$839$	−702797.	−	702797.i	−0.998404	−	0.998404i	−0.999999	+	0.00159522i	$0.999492\pi$
$840$		0			0
$841$	−117625.			−0.166305
$842$		−	1.48239e6i		−	2.09092i
$843$		0			0
$844$		−	215815.i		−	0.302969i
$845$	398945.	+	974957.i	0.558727	+	1.36544i
$846$		0			0
$847$	−84320.5	+	84320.5i	−0.117535	+	0.117535i
$848$	433853.			0.603324
$849$		0			0
$850$	−530494.	+	530494.i	−0.734248	+	0.734248i
$851$	595874.	−	595874.i	0.822801	−	0.822801i
$852$		0			0
$853$	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
$855$		0			0
$856$	−200893.	−	200893.i	−0.274169	−	0.274169i
$857$		−	427769.i		−	0.582435i	−0.956657	−	0.291217i	$-0.905940\pi$
$857$		−	427769.i		−	0.582435i	0.956657	−	0.291217i	$-0.0940603\pi$
$858$		0			0
$859$	−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
$861$		0			0
$862$			1.10437e6i			1.48627i
$863$	−115225.	+	115225.i	−0.154713	+	0.154713i	−0.780219	−	0.625506i	$-0.784892\pi$
$863$	−115225.	+	115225.i	−0.154713	+	0.154713i	0.625506	+	0.780219i	$0.284892\pi$
$864$		0			0
$865$	−72510.9	−	72510.9i	−0.0969106	−	0.0969106i
$866$	298528.	+	298528.i	0.398060	+	0.398060i
$867$		0			0
$868$		−	48726.7i		−	0.0646737i
$869$	−651945.	−	651945.i	−0.863319	−	0.863319i
$870$		0			0
$871$	−988440.	−	198819.i	−1.30291	−	0.262073i
$872$	573457.			0.754168
$873$		0			0
$874$	1.57850e6			2.06643
$875$			97628.6i			0.127515i
$876$		0			0
$877$	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
$879$		0			0
$880$	501174.			0.647177
$881$		−	776892.i		−	1.00094i	−0.865753	−	0.500471i	$-0.833160\pi$
$881$		−	776892.i		−	1.00094i	0.865753	−	0.500471i	$-0.166840\pi$
$882$		0			0
$883$			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
$885$		0			0
$886$	−399445.	+	399445.i	−0.508849	+	0.508849i
$887$	−951592.			−1.20949			−0.604747	−	0.796418i	$-0.706726\pi$
$887$	−951592.			−1.20949			−0.604747	+	0.796418i	$0.706726\pi$
$888$		0			0
$889$	186462.	−	186462.i	0.235932	−	0.235932i
$890$	−1.93261e6	+	1.93261e6i	−2.43985	+	2.43985i
$891$		0			0
$892$	−535912.	−	535912.i	−0.673540	−	0.673540i
$893$	31778.4			0.0398500
$894$		0			0
$895$	77520.3	+	77520.3i	0.0967764	+	0.0967764i
$896$			388030.i			0.483336i
$897$		0			0
$898$	151887.			0.188352
$899$	−51351.8	+	51351.8i	−0.0635384	+	0.0635384i
$900$		0			0
$901$		−	523158.i		−	0.644442i
$902$	−483406.	+	483406.i	−0.594154	+	0.594154i
$903$		0			0
$904$	352875.	+	352875.i	0.431802	+	0.431802i
$905$	1.14842e6	+	1.14842e6i	1.40218	+	1.40218i
$906$		0			0
$907$		−	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
$909$		0			0
$910$	897246.	+	180476.i	1.08350	+	0.217940i
$911$	−93939.2			−0.113191			−0.0565953	−	0.998397i	$-0.518024\pi$
$911$	−93939.2			−0.113191			−0.0565953	+	0.998397i	$0.518024\pi$
$912$		0			0
$913$	−866266.			−1.03923
$914$		−	906796.i		−	1.08547i
$915$		0			0
$916$	−535604.	+	535604.i	−0.638341	+	0.638341i
$917$	237805.	+	237805.i	0.282802	+	0.282802i
$918$		0			0
$919$	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
$921$		0			0
$922$		−	107215.i		−	0.126123i
$923$	−708986.	−	142609.i	−0.832212	−	0.167395i
$924$		0			0
$925$	1.12684e6	−	1.12684e6i	1.31697	−	1.31697i
$926$	1.35346e6			1.57842
$927$		0			0
$928$	751065.	−	751065.i	0.872131	−	0.872131i
$929$	489424.	−	489424.i	0.567092	−	0.567092i	−0.364221	−	0.931313i	$-0.618665\pi$
$929$	489424.	−	489424.i	0.567092	−	0.567092i	0.931313	+	0.364221i	$0.118665\pi$
$930$		0			0
$931$	856000.	+	856000.i	0.987584	+	0.987584i
$932$	−481972.			−0.554868
$933$		0			0
$934$	−16797.4	−	16797.4i	−0.0192552	−	0.0192552i
$935$		−	604337.i		−	0.691283i
$936$		0			0
$937$	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
$939$		0			0
$940$			37979.4i			0.0429826i
$941$	746527.	−	746527.i	0.843075	−	0.843075i	−0.146183	−	0.989258i	$-0.546699\pi$
$941$	746527.	−	746527.i	0.843075	−	0.843075i	0.989258	+	0.146183i	$0.0466988\pi$
$942$		0			0
$943$	312255.	+	312255.i	0.351144	+	0.351144i
$944$	547115.	+	547115.i	0.613953	+	0.613953i
$945$		0			0
$946$			1.68480e6i			1.88263i
$947$	392710.	+	392710.i	0.437897	+	0.437897i	0.891304	−	0.453407i	$-0.149792\pi$
$947$	392710.	+	392710.i	0.437897	+	0.437897i	−0.453407	+	0.891304i	$0.649792\pi$
$948$		0			0
$949$	145584.	−	723776.i	0.161652	−	0.803659i
$950$	2.98504e6			3.30752
$951$		0			0
$952$	134168.			0.148039
$953$			920738.i			1.01380i	0.862006	+	0.506898i	$0.169208\pi$
$953$			920738.i			1.01380i	−0.862006	+	0.506898i	$0.830792\pi$
$954$		0			0
$955$	96209.5	−	96209.5i	0.105490	−	0.105490i
$956$	−1.43575e6	−	1.43575e6i	−1.57095	−	1.57095i
$957$		0			0
$958$	−462756.			−0.504221
$959$			701460.i			0.762721i
$960$		0			0
$961$			914577.i			0.990315i
$962$	−1.24170e6	−	1.86700e6i	−1.34173	−	2.01741i
$963$		0			0
$964$	−748214.	+	748214.i	−0.805141	+	0.805141i
$965$	1.00237e6			1.07640
$966$		0			0
$967$	−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
$969$		0			0
$970$	−650853.	−	650853.i	−0.691734	−	0.691734i
$971$	107226.			0.113726			0.0568632	−	0.998382i	$-0.481890\pi$
$971$	107226.			0.113726			0.0568632	+	0.998382i	$0.481890\pi$
$972$		0			0
$973$	213400.	+	213400.i	0.225407	+	0.225407i
$974$			344080.i			0.362695i
$975$		0			0
$976$	69679.4			0.0731484
$977$	730243.	−	730243.i	0.765030	−	0.765030i	−0.212197	−	0.977227i	$-0.568062\pi$
$977$	730243.	−	730243.i	0.765030	−	0.765030i	0.977227	+	0.212197i	$0.0680619\pi$
$978$		0			0
$979$		−	1.19012e6i		−	1.24173i
$980$	−1.02303e6	+	1.02303e6i	−1.06522	+	1.06522i
$981$		0			0
$982$	−792514.	−	792514.i	−0.821833	−	0.821833i
$983$	−180212.	−	180212.i	−0.186500	−	0.186500i	0.607681	−	0.794181i	$-0.292100\pi$
$983$	−180212.	−	180212.i	−0.186500	−	0.186500i	−0.794181	+	0.607681i	$0.792100\pi$
$984$		0			0
$985$		−	830241.i		−	0.855720i
$986$	−553953.	−	553953.i	−0.569795	−	0.569795i
$987$		0			0
$988$	474693.	−	2.35996e6i	0.486294	−	2.41763i
$989$	1.08829e6			1.11263
$990$		0			0
$991$	−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
$993$		0			0
$994$	−444280.	+	444280.i	−0.449659	+	0.449659i
$995$	−607959.	−	607959.i	−0.614084	−	0.614084i
$996$		0			0
$997$	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
$999$		0			0

Twists

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

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

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	117.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.32919 - 4.32919i) q^{2} -21.4837i q^{4} +(26.0804 - 26.0804i) q^{5} +(-16.9579 - 16.9579i) q^{7} +(-23.7399 - 23.7399i) q^{8} -225.814i q^{10} +(69.5293 + 69.5293i) q^{11} +(-93.5895 - 140.720i) q^{13} -146.828 q^{14} +138.190 q^{16} -166.635i q^{17} +(-468.820 + 468.820i) q^{19} +(-560.303 - 560.303i) q^{20} +602.011 q^{22} -388.867i q^{23} -735.373i q^{25} +(-1014.37 - 204.035i) q^{26} +(-364.318 + 364.318i) q^{28} +767.891 q^{29} +(-66.8738 + 66.8738i) q^{31} +(978.089 - 978.089i) q^{32} +(-721.395 - 721.395i) q^{34} -884.537 q^{35} +(1532.33 + 1532.33i) q^{37} +4059.22i q^{38} -1238.29 q^{40} +(-802.986 + 802.986i) q^{41} +2798.62i q^{43} +(1493.75 - 1493.75i) q^{44} +(-1683.48 - 1683.48i) q^{46} +(-33.8918 - 33.8918i) q^{47} -1825.86i q^{49} +(-3183.56 - 3183.56i) q^{50} +(-3023.18 + 2010.65i) q^{52} +3139.54 q^{53} +3626.70 q^{55} +805.159i q^{56} +(3324.34 - 3324.34i) q^{58} +(3959.16 + 3959.16i) q^{59} +504.230 q^{61} +579.018i q^{62} -6257.62i q^{64} +(-6110.87 - 1229.17i) q^{65} +(4218.53 - 4218.53i) q^{67} -3579.94 q^{68} +(-3829.33 + 3829.33i) q^{70} +(3025.86 - 3025.86i) q^{71} +(3088.98 + 3088.98i) q^{73} +13267.5 q^{74} +(10072.0 + 10072.0i) q^{76} -2358.14i q^{77} -9376.55 q^{79} +(3604.05 - 3604.05i) q^{80} +6952.55i q^{82} +(-6229.50 + 6229.50i) q^{83} +(-4345.91 - 4345.91i) q^{85} +(12115.7 + 12115.7i) q^{86} -3301.24i q^{88} +(-8558.42 - 8558.42i) q^{89} +(-799.227 + 3973.39i) q^{91} -8354.30 q^{92} -293.448 q^{94} +24454.0i q^{95} +(2882.26 - 2882.26i) q^{97} +(-7904.48 - 7904.48i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) 20 * q - 24 * q^5 - 24 * q^7 - 372 * q^11 - 224 * q^13 - 480 * q^14 - 2328 * q^16 - 840 * q^19 - 228 * q^20 + 3536 * q^22 + 828 * q^26 - 1984 * q^28 + 5064 * q^29 + 1712 * q^31 + 7260 * q^32 + 8040 * q^34 - 5976 * q^35 - 868 * q^37 + 6336 * q^40 + 6000 * q^41 - 20868 * q^44 - 1992 * q^46 - 1692 * q^47 - 8952 * q^50 - 12416 * q^52 + 18816 * q^53 - 3280 * q^55 + 1256 * q^58 + 6588 * q^59 - 5512 * q^61 - 33048 * q^65 - 4704 * q^67 + 28296 * q^68 - 31096 * q^70 - 9756 * q^71 + 13692 * q^73 + 28296 * q^74 - 17072 * q^76 - 5056 * q^79 - 42228 * q^80 - 24708 * q^83 - 4752 * q^85 + 77424 * q^86 - 36000 * q^89 + 19104 * q^91 - 12912 * q^92 + 112832 * q^94 + 50012 * q^97 - 11544 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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