LMFDB - Embedded newform 69.5.d.a.22.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,5,Mod(22,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

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("69.22");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	69.d (of order $2$, degree $1$, 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:	$7.13252745279$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 5598 x^{14} + 11369517 x^{12} + 11272666128 x^{10} + 5958872960073 x^{8} + \cdots + 13\!\cdots\!52 $ x^16 + 5598x^14 + 11369517x^12 + 11272666128x^10 + 5958872960073x^8 + 1661250441139200x^6 + 215215956313867368x^4 + 8236967436040727664*x^2 + 13861812915684254052
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.15
Root			$-19.1905i$ of defining polynomial
Character	$\chi$	$=$	69.22
Dual form			69.5.d.a.22.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.44175 q^{2} -5.19615 q^{3} +39.3796 q^{4} -49.1084i q^{5} -38.6685 q^{6} +11.1740i q^{7} +173.985 q^{8} +27.0000 q^{9} +O(q^{10})$	\(q+7.44175 q^{2} -5.19615 q^{3} +39.3796 q^{4} -49.1084i q^{5} -38.6685 q^{6} +11.1740i q^{7} +173.985 q^{8} +27.0000 q^{9} -365.452i q^{10} +158.229i q^{11} -204.623 q^{12} +3.48595 q^{13} +83.1543i q^{14} +255.174i q^{15} +664.681 q^{16} +273.535i q^{17} +200.927 q^{18} -195.603i q^{19} -1933.87i q^{20} -58.0619i q^{21} +1177.50i q^{22} +(415.728 + 327.126i) q^{23} -904.054 q^{24} -1786.63 q^{25} +25.9416 q^{26} -140.296 q^{27} +440.029i q^{28} -1097.57 q^{29} +1898.94i q^{30} +362.469 q^{31} +2162.62 q^{32} -822.183i q^{33} +2035.58i q^{34} +548.738 q^{35} +1063.25 q^{36} -1385.65i q^{37} -1455.63i q^{38} -18.1135 q^{39} -8544.13i q^{40} +1118.33 q^{41} -432.082i q^{42} +2501.85i q^{43} +6231.01i q^{44} -1325.93i q^{45} +(3093.74 + 2434.39i) q^{46} -420.680 q^{47} -3453.78 q^{48} +2276.14 q^{49} -13295.7 q^{50} -1421.33i q^{51} +137.276 q^{52} +2758.71i q^{53} -1044.05 q^{54} +7770.38 q^{55} +1944.12i q^{56} +1016.39i q^{57} -8167.86 q^{58} -3196.59 q^{59} +10048.7i q^{60} -3444.94i q^{61} +2697.40 q^{62} +301.699i q^{63} +5458.80 q^{64} -171.189i q^{65} -6118.48i q^{66} -5967.36i q^{67} +10771.7i q^{68} +(-2160.18 - 1699.80i) q^{69} +4083.57 q^{70} +3422.34 q^{71} +4697.60 q^{72} -4879.75 q^{73} -10311.7i q^{74} +9283.60 q^{75} -7702.79i q^{76} -1768.06 q^{77} -134.796 q^{78} -968.983i q^{79} -32641.4i q^{80} +729.000 q^{81} +8322.30 q^{82} -6564.33i q^{83} -2286.46i q^{84} +13432.8 q^{85} +18618.2i q^{86} +5703.15 q^{87} +27529.6i q^{88} +8072.73i q^{89} -9867.20i q^{90} +38.9521i q^{91} +(16371.2 + 12882.1i) q^{92} -1883.44 q^{93} -3130.60 q^{94} -9605.76 q^{95} -11237.3 q^{96} -9573.09i q^{97} +16938.5 q^{98} +4272.19i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{2} + 144 q^{4} - 36 q^{6} + 372 q^{8} + 432 q^{9}+O(q^{10}) $ 16 * q + 12 * q^2 + 144 * q^4 - 36 * q^6 + 372 * q^8 + 432 * q^9	$ 16 q + 12 q^{2} + 144 q^{4} - 36 q^{6} + 372 q^{8} + 432 q^{9} + 104 q^{13} + 680 q^{16} + 324 q^{18} - 732 q^{23} - 1764 q^{24} - 2984 q^{25} + 1800 q^{26} - 3528 q^{29} - 400 q^{31} + 5244 q^{32} + 912 q^{35} + 3888 q^{36} + 2016 q^{39} + 1008 q^{41} - 1168 q^{46} - 8664 q^{47} - 2016 q^{48} + 7240 q^{49} - 18852 q^{50} - 20952 q^{52} - 972 q^{54} + 6816 q^{55} - 13352 q^{58} + 20112 q^{59} + 4248 q^{62} - 896 q^{64} - 10044 q^{69} - 10680 q^{70} + 40368 q^{71} + 10044 q^{72} - 9568 q^{73} + 7560 q^{75} + 2952 q^{77} - 6912 q^{78} + 11664 q^{81} + 71800 q^{82} + 42744 q^{85} + 8352 q^{87} - 9876 q^{92} - 10008 q^{93} + 73720 q^{94} + 33312 q^{95} - 24948 q^{96} - 59052 q^{98}+O(q^{100}) $ 16 * q + 12 * q^2 + 144 * q^4 - 36 * q^6 + 372 * q^8 + 432 * q^9 + 104 * q^13 + 680 * q^16 + 324 * q^18 - 732 * q^23 - 1764 * q^24 - 2984 * q^25 + 1800 * q^26 - 3528 * q^29 - 400 * q^31 + 5244 * q^32 + 912 * q^35 + 3888 * q^36 + 2016 * q^39 + 1008 * q^41 - 1168 * q^46 - 8664 * q^47 - 2016 * q^48 + 7240 * q^49 - 18852 * q^50 - 20952 * q^52 - 972 * q^54 + 6816 * q^55 - 13352 * q^58 + 20112 * q^59 + 4248 * q^62 - 896 * q^64 - 10044 * q^69 - 10680 * q^70 + 40368 * q^71 + 10044 * q^72 - 9568 * q^73 + 7560 * q^75 + 2952 * q^77 - 6912 * q^78 + 11664 * q^81 + 71800 * q^82 + 42744 * q^85 + 8352 * q^87 - 9876 * q^92 - 10008 * q^93 + 73720 * q^94 + 33312 * q^95 - 24948 * q^96 - 59052 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$1$

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 $

$2$	7.44175			1.86044			0.930219	−	0.367006i	$-0.119617\pi$
$2$	7.44175			1.86044			0.930219	+	0.367006i	$0.119617\pi$
$3$	−5.19615			−0.577350
$3$	−5.19615			−0.577350
$4$	39.3796			2.46123
$5$		−	49.1084i		−	1.96433i	−0.188010	−	0.982167i	$-0.560204\pi$
$5$		−	49.1084i		−	1.96433i	0.188010	−	0.982167i	$-0.439796\pi$
$6$	−38.6685			−1.07412
$7$			11.1740i			0.228041i	0.993478	+	0.114021i	$0.0363730\pi$
$7$			11.1740i			0.228041i	−0.993478	+	0.114021i	$0.963627\pi$
$8$	173.985			2.71852
$9$	27.0000			0.333333
$10$		−	365.452i		−	3.65452i
$11$			158.229i			1.30768i	0.756633	+	0.653840i	$0.226843\pi$
$11$			158.229i			1.30768i	−0.756633	+	0.653840i	$0.773157\pi$
$12$	−204.623			−1.42099
$13$	3.48595			0.0206269			0.0103135	−	0.999947i	$-0.496717\pi$
$13$	3.48595			0.0206269			0.0103135	+	0.999947i	$0.496717\pi$
$14$			83.1543i			0.424257i
$15$			255.174i			1.13411i
$16$	664.681			2.59641
$17$			273.535i			0.946487i	0.880932	+	0.473244i	$0.156917\pi$
$17$			273.535i			0.946487i	−0.880932	+	0.473244i	$0.843083\pi$
$18$	200.927			0.620146
$19$		−	195.603i		−	0.541838i	−0.962602	−	0.270919i	$-0.912672\pi$
$19$		−	195.603i		−	0.541838i	0.962602	−	0.270919i	$-0.0873275\pi$
$20$		−	1933.87i		−	4.83467i
$21$		−	58.0619i		−	0.131660i
$22$			1177.50i			2.43286i
$23$	415.728	+	327.126i	0.785874	+	0.618386i
$23$	415.728	+	327.126i	0.785874	+	0.618386i
$24$	−904.054			−1.56954
$25$	−1786.63			−2.85861
$26$	25.9416			0.0383751
$27$	−140.296			−0.192450
$28$			440.029i			0.561261i
$29$	−1097.57			−1.30508			−0.652540	−	0.757754i	$-0.726297\pi$
$29$	−1097.57			−1.30508			−0.652540	+	0.757754i	$0.726297\pi$
$30$			1898.94i			2.10994i
$31$	362.469			0.377179			0.188589	−	0.982056i	$-0.439608\pi$
$31$	362.469			0.377179			0.188589	+	0.982056i	$0.439608\pi$
$32$	2162.62			2.11194
$33$		−	822.183i		−	0.754989i
$34$			2035.58i			1.76088i
$35$	548.738			0.447949
$36$	1063.25			0.820409
$37$		−	1385.65i		−	1.01217i	−0.862485	−	0.506083i	$-0.831093\pi$
$37$		−	1385.65i		−	1.01217i	0.862485	−	0.506083i	$-0.168907\pi$
$38$		−	1455.63i		−	1.00806i
$39$	−18.1135			−0.0119090
$40$		−	8544.13i		−	5.34008i
$41$	1118.33			0.665274			0.332637	−	0.943055i	$-0.392062\pi$
$41$	1118.33			0.665274			0.332637	+	0.943055i	$0.392062\pi$
$42$		−	432.082i		−	0.244945i
$43$			2501.85i			1.35308i	0.736404	+	0.676542i	$0.236522\pi$
$43$			2501.85i			1.35308i	−0.736404	+	0.676542i	$0.763478\pi$
$44$			6231.01i			3.21850i
$45$		−	1325.93i		−	0.654778i
$46$	3093.74	+	2434.39i	1.46207	+	1.15047i
$47$	−420.680			−0.190439			−0.0952196	−	0.995456i	$-0.530355\pi$
$47$	−420.680			−0.190439			−0.0952196	+	0.995456i	$0.530355\pi$
$48$	−3453.78			−1.49904
$49$	2276.14			0.947997
$50$	−13295.7			−5.31826
$51$		−	1421.33i		−	0.546455i
$52$	137.276			0.0507676
$53$			2758.71i			0.982097i	0.871132	+	0.491049i	$0.163386\pi$
$53$			2758.71i			0.982097i	−0.871132	+	0.491049i	$0.836614\pi$
$54$	−1044.05			−0.358041
$55$	7770.38			2.56872
$56$			1944.12i			0.619935i
$57$			1016.39i			0.312830i
$58$	−8167.86			−2.42802
$59$	−3196.59			−0.918298			−0.459149	−	0.888359i	$-0.651846\pi$
$59$	−3196.59			−0.918298			−0.459149	+	0.888359i	$0.651846\pi$
$60$			10048.7i			2.79130i
$61$		−	3444.94i		−	0.925809i	−0.886408	−	0.462905i	$-0.846807\pi$
$61$		−	3444.94i		−	0.925809i	0.886408	−	0.462905i	$-0.153193\pi$
$62$	2697.40			0.701718
$63$			301.699i			0.0760138i
$64$	5458.80			1.33271
$65$		−	171.189i		−	0.0405182i
$66$		−	6118.48i		−	1.40461i
$67$		−	5967.36i		−	1.32933i	−0.747141	−	0.664665i	$-0.768574\pi$
$67$		−	5967.36i		−	1.32933i	0.747141	−	0.664665i	$-0.231426\pi$
$68$			10771.7i			2.32952i
$69$	−2160.18	−	1699.80i	−0.453725	−	0.357025i
$70$	4083.57			0.833382
$71$	3422.34			0.678900			0.339450	−	0.940624i	$-0.389759\pi$
$71$	3422.34			0.678900			0.339450	+	0.940624i	$0.389759\pi$
$72$	4697.60			0.906173
$73$	−4879.75			−0.915697			−0.457849	−	0.889030i	$-0.651380\pi$
$73$	−4879.75			−0.915697			−0.457849	+	0.889030i	$0.651380\pi$
$74$		−	10311.7i		−	1.88307i
$75$	9283.60			1.65042
$76$		−	7702.79i		−	1.33359i
$77$	−1768.06			−0.298205
$78$	−134.796			−0.0221559
$79$		−	968.983i		−	0.155261i	−0.996982	−	0.0776304i	$-0.975265\pi$
$79$		−	968.983i		−	0.155261i	0.996982	−	0.0776304i	$-0.0247354\pi$
$80$		−	32641.4i		−	5.10021i
$81$	729.000			0.111111
$82$	8322.30			1.23770
$83$		−	6564.33i		−	0.952872i	−0.879209	−	0.476436i	$-0.841928\pi$
$83$		−	6564.33i		−	0.952872i	0.879209	−	0.476436i	$-0.158072\pi$
$84$		−	2286.46i		−	0.324044i
$85$	13432.8			1.85922
$86$			18618.2i			2.51733i
$87$	5703.15			0.753488
$88$			27529.6i			3.55495i
$89$			8072.73i			1.01916i	0.860425	+	0.509578i	$0.170198\pi$
$89$			8072.73i			1.01916i	−0.860425	+	0.509578i	$0.829802\pi$
$90$		−	9867.20i		−	1.21817i
$91$			38.9521i			0.00470380i
$92$	16371.2	+	12882.1i	1.93421	+	1.52199i
$93$	−1883.44			−0.217764
$94$	−3130.60			−0.354300
$95$	−9605.76			−1.06435
$96$	−11237.3			−1.21933
$97$		−	9573.09i		−	1.01744i	−0.860932	−	0.508720i	$-0.830119\pi$
$97$		−	9573.09i		−	1.01744i	0.860932	−	0.508720i	$-0.169881\pi$
$98$	16938.5			1.76369
$99$			4272.19i			0.435893i
$100$	−70356.8			−7.03568
$101$	−7144.87			−0.700409			−0.350204	−	0.936673i	$-0.613888\pi$
$101$	−7144.87			−0.700409			−0.350204	+	0.936673i	$0.613888\pi$
$102$		−	10577.2i		−	1.01664i
$103$			7771.56i			0.732544i	0.930508	+	0.366272i	$0.119366\pi$
$103$			7771.56i			0.732544i	−0.930508	+	0.366272i	$0.880634\pi$
$104$	606.504			0.0560747
$105$	−2851.33			−0.258624
$106$			20529.6i			1.82713i
$107$		−	16682.6i		−	1.45713i	−0.684979	−	0.728563i	$-0.740189\pi$
$107$		−	16682.6i		−	1.45713i	0.684979	−	0.728563i	$-0.259811\pi$
$108$	−5524.81			−0.473663
$109$		−	2073.27i		−	0.174503i	−0.996186	−	0.0872517i	$-0.972192\pi$
$109$		−	2073.27i		−	0.174503i	0.996186	−	0.0872517i	$-0.0278084\pi$
$110$	57825.2			4.77894
$111$			7200.07i			0.584374i
$112$			7427.16i			0.592089i
$113$		−	10792.9i		−	0.845239i	−0.906307	−	0.422619i	$-0.861111\pi$
$113$		−	10792.9i		−	0.845239i	0.906307	−	0.422619i	$-0.138889\pi$
$114$			7563.68i			0.582001i
$115$	16064.6	−	20415.7i	1.21472	−	1.54372i
$116$	−43222.0			−3.21210
$117$	94.1207			0.00687565
$118$	−23788.2			−1.70843
$119$	−3056.49			−0.215838
$120$			44396.6i			3.08310i
$121$	−10395.5			−0.710027
$122$		−	25636.3i		−	1.72241i
$123$	−5810.99			−0.384096
$124$	14273.9			0.928323
$125$			57045.7i			3.65093i
$126$			2245.17i			0.141419i
$127$	8526.40			0.528638			0.264319	−	0.964435i	$-0.414853\pi$
$127$	8526.40			0.528638			0.264319	+	0.964435i	$0.414853\pi$
$128$	6021.06			0.367497
$129$		−	13000.0i		−	0.781204i
$130$		−	1273.95i		−	0.0753816i
$131$	−28518.1			−1.66180			−0.830900	−	0.556422i	$-0.812174\pi$
$131$	−28518.1			−1.66180			−0.830900	+	0.556422i	$0.812174\pi$
$132$		−	32377.3i		−	1.85820i
$133$	2185.68			0.123561
$134$		−	44407.6i		−	2.47314i
$135$			6889.71i			0.378036i
$136$			47591.0i			2.57304i
$137$			6175.76i			0.329041i	0.986374	+	0.164520i	$0.0526076\pi$
$137$			6175.76i			0.329041i	−0.986374	+	0.164520i	$0.947392\pi$
$138$	−16075.5	−	12649.5i	−0.844126	−	0.664223i
$139$	15175.2			0.785423			0.392711	−	0.919662i	$-0.371537\pi$
$139$	15175.2			0.785423			0.392711	+	0.919662i	$0.371537\pi$
$140$	21609.1			1.10250
$141$	2185.92			0.109950
$142$	25468.2			1.26305
$143$			551.580i			0.0269734i
$144$	17946.4			0.865470
$145$			53900.0i			2.56361i
$146$	−36313.9			−1.70360
$147$	−11827.2			−0.547326
$148$		−	54566.6i		−	2.49117i
$149$			21863.0i			0.984777i	0.870376	+	0.492389i	$0.163876\pi$
$149$			21863.0i			0.984777i	−0.870376	+	0.492389i	$0.836124\pi$
$150$	69086.2			3.07050
$151$	−7673.52			−0.336543			−0.168272	−	0.985741i	$-0.553819\pi$
$151$	−7673.52			−0.336543			−0.168272	+	0.985741i	$0.553819\pi$
$152$		−	34032.1i		−	1.47300i
$153$			7385.44i			0.315496i
$154$	−13157.4			−0.554792
$155$		−	17800.3i		−	0.740905i
$156$	−713.304			−0.0293107
$157$			3908.72i			0.158575i	0.996852	+	0.0792876i	$0.0252646\pi$
$157$			3908.72i			0.158575i	−0.996852	+	0.0792876i	$0.974735\pi$
$158$		−	7210.93i		−	0.288853i
$159$		−	14334.7i		−	0.567014i
$160$		−	106203.i		−	4.14855i
$161$	−3655.32	+	4645.35i	−0.141018	+	0.179212i
$162$	5425.03			0.206715
$163$	−91.9597			−0.00346117			−0.00173058	−	0.999999i	$-0.500551\pi$
$163$	−91.9597			−0.00346117			−0.00173058	+	0.999999i	$0.500551\pi$
$164$	44039.3			1.63739
$165$	−40376.1			−1.48305
$166$		−	48850.1i		−	1.77276i
$167$	3823.49			0.137097			0.0685484	−	0.997648i	$-0.478163\pi$
$167$	3823.49			0.137097			0.0685484	+	0.997648i	$0.478163\pi$
$168$		−	10101.9i		−	0.357920i
$169$	−28548.8			−0.999575
$170$	99963.8			3.45896
$171$		−	5281.29i		−	0.180613i
$172$			98522.0i			3.33025i
$173$	9580.01			0.320091			0.160046	−	0.987110i	$-0.448836\pi$
$173$	9580.01			0.320091			0.160046	+	0.987110i	$0.448836\pi$
$174$	42441.4			1.40182
$175$		−	19963.9i		−	0.651881i
$176$			105172.i			3.39527i
$177$	16610.0			0.530179
$178$			60075.2i			1.89608i
$179$	−27250.4			−0.850484			−0.425242	−	0.905080i	$-0.639811\pi$
$179$	−27250.4			−0.850484			−0.425242	+	0.905080i	$0.639811\pi$
$180$		−	52214.4i		−	1.61156i
$181$		−	60533.0i		−	1.84772i	−0.382736	−	0.923858i	$-0.625018\pi$
$181$		−	60533.0i		−	1.84772i	0.382736	−	0.923858i	$-0.374982\pi$
$182$			289.872i			0.00875112i
$183$			17900.4i			0.534516i
$184$	72330.5	+	56915.2i	2.13642	+	1.68110i
$185$	−68047.2			−1.98823
$186$	−14016.1			−0.405137
$187$	−43281.2			−1.23770
$188$	−16566.2			−0.468714
$189$		−	1567.67i		−	0.0438866i
$190$	−71483.7			−1.98016
$191$			35942.8i			0.985246i	0.870243	+	0.492623i	$0.163962\pi$
$191$			35942.8i			0.985246i	−0.870243	+	0.492623i	$0.836038\pi$
$192$	−28364.8			−0.769443
$193$	8323.05			0.223443			0.111722	−	0.993740i	$-0.464363\pi$
$193$	8323.05			0.223443			0.111722	+	0.993740i	$0.464363\pi$
$194$		−	71240.6i		−	1.89288i
$195$			889.526i			0.0233932i
$196$	89633.6			2.33324
$197$	40003.9			1.03079			0.515395	−	0.856953i	$-0.327645\pi$
$197$	40003.9			1.03079			0.515395	+	0.856953i	$0.327645\pi$
$198$			31792.6i			0.810952i
$199$			22691.3i			0.572998i	0.958081	+	0.286499i	$0.0924914\pi$
$199$			22691.3i			0.572998i	−0.958081	+	0.286499i	$0.907509\pi$
$200$	−310847.			−7.77118
$201$			31007.3i			0.767489i
$202$	−53170.3			−1.30307
$203$		−	12264.3i		−	0.297612i
$204$		−	55971.4i		−	1.34495i
$205$		−	54919.2i		−	1.30682i
$206$			57834.0i			1.36285i
$207$	11224.6	+	8832.41i	0.261958	+	0.206129i
$208$	2317.05			0.0535560
$209$	30950.2			0.708550
$210$	−21218.9			−0.481153
$211$	41396.3			0.929816			0.464908	−	0.885359i	$-0.346087\pi$
$211$	41396.3			0.929816			0.464908	+	0.885359i	$0.346087\pi$
$212$			108637.i			2.41716i
$213$	−17783.0			−0.391963
$214$		−	124148.i		−	2.71089i
$215$	122862.			2.65791
$216$	−24409.5			−0.523179
$217$			4050.24i			0.0860124i
$218$		−	15428.8i		−	0.324653i
$219$	25355.9			0.528678
$220$	305995.			6.32220
$221$			953.529i			0.0195231i
$222$			53581.1i			1.08719i
$223$	30876.5			0.620895			0.310448	−	0.950590i	$-0.399521\pi$
$223$	30876.5			0.620895			0.310448	+	0.950590i	$0.399521\pi$
$224$			24165.2i			0.481609i
$225$	−48239.0			−0.952869
$226$		−	80317.7i		−	1.57251i
$227$		−	17872.9i		−	0.346852i	−0.984847	−	0.173426i	$-0.944516\pi$
$227$		−	17872.9i		−	0.346852i	0.984847	−	0.173426i	$-0.0554838\pi$
$228$			40024.9i			0.769946i
$229$			22033.5i			0.420158i	0.977684	+	0.210079i	$0.0673721\pi$
$229$			22033.5i			0.420158i	−0.977684	+	0.210079i	$0.932628\pi$
$230$	119549.	−	151928.i	2.25990	−	2.87199i
$231$	9187.10			0.172169
$232$	−190961.			−3.54789
$233$	−38266.3			−0.704862			−0.352431	−	0.935838i	$-0.614645\pi$
$233$	−38266.3			−0.704862			−0.352431	+	0.935838i	$0.614645\pi$
$234$	700.423			0.0127917
$235$			20658.9i			0.374086i
$236$	−125881.			−2.26014
$237$			5034.98i			0.0896399i
$238$	−22745.6			−0.401553
$239$	82469.8			1.44377			0.721887	−	0.692011i	$-0.243275\pi$
$239$	82469.8			1.44377			0.721887	+	0.692011i	$0.243275\pi$
$240$			169610.i			2.94461i
$241$		−	43509.5i		−	0.749117i	−0.927203	−	0.374558i	$-0.877794\pi$
$241$		−	43509.5i		−	0.749117i	0.927203	−	0.374558i	$-0.122206\pi$
$242$	−77360.7			−1.32096
$243$	−3788.00			−0.0641500
$244$		−	135660.i		−	2.27863i
$245$		−	111778.i		−	1.86218i
$246$	−43243.9			−0.714587
$247$		−	681.864i		−	0.0111765i
$248$	63064.3			1.02537
$249$			34109.3i			0.550141i
$250$			424520.i			6.79232i
$251$			9819.06i			0.155856i	0.996959	+	0.0779278i	$0.0248304\pi$
$251$			9819.06i			0.155856i	−0.996959	+	0.0779278i	$0.975170\pi$
$252$			11880.8i			0.187087i
$253$	−51761.0	+	65780.3i	−0.808651	+	1.02767i
$254$	63451.3			0.983497
$255$	−69799.1			−1.07342
$256$	−42533.6			−0.649011
$257$	25987.7			0.393461			0.196731	−	0.980458i	$-0.436968\pi$
$257$	25987.7			0.393461			0.196731	+	0.980458i	$0.436968\pi$
$258$		−	96742.8i		−	1.45338i
$259$	15483.3			0.230816
$260$		−	6741.37i		−	0.0997245i
$261$	−29634.5			−0.435027
$262$	−212225.			−3.09167
$263$			61615.4i			0.890795i	0.895333	+	0.445398i	$0.146938\pi$
$263$			61615.4i			0.890795i	−0.895333	+	0.445398i	$0.853062\pi$
$264$		−	143048.i		−	2.05245i
$265$	135476.			1.92917
$266$	16265.3			0.229878
$267$		−	41947.1i		−	0.588410i
$268$		−	234993.i		−	3.27178i
$269$	−37018.9			−0.511587			−0.255793	−	0.966731i	$-0.582337\pi$
$269$	−37018.9			−0.511587			−0.255793	+	0.966731i	$0.582337\pi$
$270$			51271.5i			0.703313i
$271$	52584.0			0.716003			0.358002	−	0.933721i	$-0.383458\pi$
$271$	52584.0			0.716003			0.358002	+	0.933721i	$0.383458\pi$
$272$			181813.i			2.45747i
$273$		−	202.401i		−	0.00271574i
$274$			45958.5i			0.612159i
$275$		−	282697.i		−	3.73814i
$276$	−85067.2	−	66937.4i	−1.11672	−	0.878720i
$277$	−79595.0			−1.03735			−0.518676	−	0.854971i	$-0.673575\pi$
$277$	−79595.0			−1.03735			−0.518676	+	0.854971i	$0.673575\pi$
$278$	112930.			1.46123
$279$	9786.66			0.125726
$280$	95472.3			1.21776
$281$			49952.4i			0.632621i	0.948656	+	0.316310i	$0.102444\pi$
$281$			49952.4i			0.632621i	−0.948656	+	0.316310i	$0.897556\pi$
$282$	16267.1			0.204555
$283$		−	37069.8i		−	0.462858i	−0.972852	−	0.231429i	$-0.925660\pi$
$283$		−	37069.8i		−	0.462858i	0.972852	−	0.231429i	$-0.0743400\pi$
$284$	134770.			1.67093
$285$	49913.0			0.614503
$286$			4104.72i			0.0501824i
$287$			12496.2i			0.151710i
$288$	58390.8			0.703979
$289$	8699.71			0.104162
$290$			401110.i			4.76944i
$291$			49743.3i			0.587419i
$292$	−192163.			−2.25374
$293$		−	40120.2i		−	0.467334i	−0.972317	−	0.233667i	$-0.924927\pi$
$293$		−	40120.2i		−	0.467334i	0.972317	−	0.233667i	$-0.0750726\pi$
$294$	−88014.9			−1.01827
$295$			156979.i			1.80384i
$296$		−	241084.i		−	2.75159i
$297$		−	22199.0i		−	0.251663i
$298$			162699.i			1.83212i
$299$	1449.21	+	1140.35i	0.0162102	+	0.0127554i
$300$	365585.			4.06205
$301$	−27955.8			−0.308559
$302$	−57104.4			−0.626117
$303$	37125.8			0.404381
$304$		−	130014.i		−	1.40683i
$305$	−169175.			−1.81860
$306$			54960.6i			0.586960i
$307$	57405.8			0.609087			0.304543	−	0.952499i	$-0.401496\pi$
$307$	57405.8			0.609087			0.304543	+	0.952499i	$0.401496\pi$
$308$	−69625.5			−0.733950
$309$		−	40382.2i		−	0.422934i
$310$		−	132465.i		−	1.37841i
$311$	84357.4			0.872173			0.436086	−	0.899905i	$-0.356364\pi$
$311$	84357.4			0.872173			0.436086	+	0.899905i	$0.356364\pi$
$312$	−3151.49			−0.0323748
$313$		−	48772.2i		−	0.497833i	−0.968525	−	0.248916i	$-0.919926\pi$
$313$		−	48772.2i		−	0.497833i	0.968525	−	0.248916i	$-0.0800744\pi$
$314$			29087.7i			0.295019i
$315$	14815.9			0.149316
$316$		−	38158.2i		−	0.382132i
$317$	175587.			1.74733			0.873664	−	0.486531i	$-0.161738\pi$
$317$	175587.			1.74733			0.873664	+	0.486531i	$0.161738\pi$
$318$		−	106675.i		−	1.05489i
$319$		−	173668.i		−	1.70663i
$320$		−	268073.i		−	2.61790i
$321$			86685.5i			0.841272i
$322$	−27202.0	+	34569.5i	−0.262354	+	0.333412i
$323$	53504.3			0.512842
$324$	28707.7			0.273470
$325$	−6228.11			−0.0589643
$326$	−684.341			−0.00643928
$327$			10773.1i			0.100750i
$328$	194572.			1.80856
$329$		−	4700.69i		−	0.0434280i
$330$	−300469.			−2.75912
$331$	−86667.7			−0.791045			−0.395523	−	0.918456i	$-0.629436\pi$
$331$	−86667.7			−0.791045			−0.395523	+	0.918456i	$0.629436\pi$
$332$		−	258501.i		−	2.34523i
$333$		−	37412.7i		−	0.337389i
$334$	28453.5			0.255060
$335$	−293047.			−2.61125
$336$		−	38592.7i		−	0.341843i
$337$			146348.i			1.28862i	0.764762	+	0.644312i	$0.222856\pi$
$337$			146348.i			1.28862i	−0.764762	+	0.644312i	$0.777144\pi$
$338$	−212453.			−1.85965
$339$			56081.3i			0.487999i
$340$	528980.			4.57595
$341$			57353.2i			0.493229i
$342$		−	39302.0i		−	0.336018i
$343$			52262.5i			0.444224i
$344$			435286.i			3.67839i
$345$	−83474.3	+	106083.i	−0.701317	+	0.891267i
$346$	71292.0			0.595510
$347$	172085.			1.42917			0.714586	−	0.699548i	$-0.246615\pi$
$347$	172085.			1.42917			0.714586	+	0.699548i	$0.246615\pi$
$348$	224588.			1.85451
$349$	−68489.2			−0.562304			−0.281152	−	0.959663i	$-0.590717\pi$
$349$	−68489.2			−0.562304			−0.281152	+	0.959663i	$0.590717\pi$
$350$		−	148566.i		−	1.21278i
$351$	−489.066			−0.00396966
$352$			342190.i			2.76174i
$353$	−156170.			−1.25328			−0.626641	−	0.779308i	$-0.715571\pi$
$353$	−156170.			−1.25328			−0.626641	+	0.779308i	$0.715571\pi$
$354$	123607.			0.986365
$355$		−	168065.i		−	1.33359i
$356$			317901.i			2.50837i
$357$	15882.0			0.124614
$358$	−202790.			−1.58227
$359$		−	60879.0i		−	0.472366i	−0.971709	−	0.236183i	$-0.924103\pi$
$359$		−	60879.0i		−	0.472366i	0.971709	−	0.236183i	$-0.0758965\pi$
$360$		−	230692.i		−	1.78003i
$361$	92060.3			0.706412
$362$		−	450471.i		−	3.43756i
$363$	54016.6			0.409934
$364$			1533.92i			0.0115771i
$365$			239636.i			1.79874i
$366$			133210.i			0.994434i
$367$		−	189642.i		−	1.40800i	−0.710201	−	0.703999i	$-0.751396\pi$
$367$		−	189642.i		−	1.40800i	0.710201	−	0.703999i	$-0.248604\pi$
$368$	276326.	+	217435.i	2.04045	+	1.60558i
$369$	30194.8			0.221758
$370$	−506390.			−3.69898
$371$	−30825.9			−0.223959
$372$	−74169.3			−0.535967
$373$		−	27864.3i		−	0.200277i	−0.994974	−	0.100138i	$-0.968071\pi$
$373$		−	27864.3i		−	0.200277i	0.994974	−	0.100138i	$-0.0319285\pi$
$374$	−322088.			−2.30267
$375$		−	296418.i		−	2.10786i
$376$	−73192.2			−0.517713
$377$	−3826.09			−0.0269198
$378$		−	11666.2i		−	0.0816482i
$379$			43587.9i			0.303450i	0.988423	+	0.151725i	$0.0484829\pi$
$379$			43587.9i			0.303450i	−0.988423	+	0.151725i	$0.951517\pi$
$380$	−378271.			−2.61961
$381$	−44304.5			−0.305209
$382$			267477.i			1.83299i
$383$			240133.i			1.63702i	0.574490	+	0.818512i	$0.305200\pi$
$383$			240133.i			1.63702i	−0.574490	+	0.818512i	$0.694800\pi$
$384$	−31286.4			−0.212174
$385$			86826.4i			0.585774i
$386$	61938.0			0.415703
$387$			67550.0i			0.451028i
$388$		−	376985.i		−	2.50415i
$389$			187297.i			1.23775i	0.785491	+	0.618873i	$0.212410\pi$
$389$			187297.i			1.23775i	−0.785491	+	0.618873i	$0.787590\pi$
$390$			6619.63i			0.0435216i
$391$	−89480.4	+	113716.i	−0.585295	+	0.743820i
$392$	396015.			2.57715
$393$	148185.			0.959441
$394$	297699.			1.91772
$395$	−47585.1			−0.304984
$396$			168237.i			1.07283i
$397$	145407.			0.922580			0.461290	−	0.887249i	$-0.347387\pi$
$397$	145407.			0.922580			0.461290	+	0.887249i	$0.347387\pi$
$398$			168863.i			1.06603i
$399$	−11357.1			−0.0713382
$400$	−1.18754e6			−7.42212
$401$			159568.i			0.992330i	0.868228	+	0.496165i	$0.165259\pi$
$401$			159568.i			0.992330i	−0.868228	+	0.496165i	$0.834741\pi$
$402$			230749.i			1.42787i
$403$	1263.55			0.00778005
$404$	−281362.			−1.72386
$405$		−	35800.0i		−	0.218259i
$406$		−	91267.9i		−	0.553689i
$407$	219251.			1.32359
$408$		−	247290.i		−	1.48555i
$409$	−150045.			−0.896963			−0.448482	−	0.893792i	$-0.648035\pi$
$409$	−150045.			−0.896963			−0.448482	+	0.893792i	$0.648035\pi$
$410$		−	408695.i		−	2.43126i
$411$		−	32090.2i		−	0.189972i
$412$			306041.i			1.80296i
$413$		−	35718.8i		−	0.209410i
$414$	83531.0	+	65728.6i	0.487357	+	0.383490i
$415$	−322364.			−1.87176
$416$	7538.80			0.0435628
$417$	−78852.4			−0.453464
$418$	230324.			1.31821
$419$			242660.i			1.38220i	0.722759	+	0.691100i	$0.242873\pi$
$419$			242660.i			1.38220i	−0.722759	+	0.691100i	$0.757127\pi$
$420$	−112284.			−0.636531
$421$			132304.i			0.746465i	0.927738	+	0.373232i	$0.121751\pi$
$421$			132304.i			0.746465i	−0.927738	+	0.373232i	$0.878249\pi$
$422$	308061.			1.72986
$423$	−11358.4			−0.0634798
$424$			479975.i			2.66985i
$425$		−	488706.i		−	2.70564i
$426$	−132336.			−0.729223
$427$	38493.8			0.211123
$428$		−	656956.i		−	3.58631i
$429$		−	2866.09i		−	0.0155731i
$430$	914307.			4.94487
$431$		−	283741.i		−	1.52745i	−0.645540	−	0.763726i	$-0.723368\pi$
$431$		−	283741.i		−	1.52745i	0.645540	−	0.763726i	$-0.276632\pi$
$432$	−93252.1			−0.499679
$433$		−	74552.8i		−	0.397638i	−0.980036	−	0.198819i	$-0.936289\pi$
$433$		−	74552.8i		−	0.397638i	0.980036	−	0.198819i	$-0.0637106\pi$
$434$			30140.9i			0.160021i
$435$		−	280072.i		−	1.48010i
$436$		−	81644.8i		−	0.429492i
$437$	63987.0	−	81317.7i	0.335065	−	0.425816i
$438$	188692.			0.983572
$439$	−252584.			−1.31062			−0.655310	−	0.755360i	$-0.727462\pi$
$439$	−252584.			−1.31062			−0.655310	+	0.755360i	$0.727462\pi$
$440$	1.35193e6			6.98312
$441$	61455.8			0.315999
$442$			7095.93i			0.0363216i
$443$	292478.			1.49034			0.745172	−	0.666872i	$-0.232367\pi$
$443$	292478.			1.49034			0.745172	+	0.666872i	$0.232367\pi$
$444$			283536.i			1.43828i
$445$	396439.			2.00196
$446$	229775.			1.15514
$447$		−	113604.i		−	0.568561i
$448$			60996.8i			0.303914i
$449$	−144789.			−0.718195			−0.359097	−	0.933300i	$-0.616915\pi$
$449$	−144789.			−0.718195			−0.359097	+	0.933300i	$0.616915\pi$
$450$	−358983.			−1.77275
$451$			176952.i			0.869966i
$452$		−	425019.i		−	2.08032i
$453$	39872.8			0.194303
$454$		−	133006.i		−	0.645297i
$455$	1912.87			0.00923983
$456$			176836.i			0.850435i
$457$		−	298954.i		−	1.43144i	−0.698388	−	0.715719i	$-0.746099\pi$
$457$		−	298954.i		−	1.43144i	0.698388	−	0.715719i	$-0.253901\pi$
$458$			163968.i			0.781677i
$459$		−	38375.9i		−	0.182152i
$460$	632619.	−	803962.i	2.98969	−	3.79944i
$461$	−156956.			−0.738543			−0.369271	−	0.929322i	$-0.620393\pi$
$461$	−156956.			−0.738543			−0.369271	+	0.929322i	$0.620393\pi$
$462$	68368.1			0.320309
$463$	197172.			0.919776			0.459888	−	0.887977i	$-0.347890\pi$
$463$	197172.			0.919776			0.459888	+	0.887977i	$0.347890\pi$
$464$	−729535.			−3.38852
$465$			92492.8i			0.427762i
$466$	−284768.			−1.31135
$467$			83794.7i			0.384223i	0.981373	+	0.192111i	$0.0615335\pi$
$467$			83794.7i			0.384223i	−0.981373	+	0.192111i	$0.938467\pi$
$468$	3706.44			0.0169225
$469$	66679.5			0.303142
$470$			153738.i			0.695964i
$471$		−	20310.3i		−	0.0915535i
$472$	−556160.			−2.49641
$473$	−395866.			−1.76940
$474$			37469.1i			0.166769i
$475$			349471.i			1.54890i
$476$	−120363.			−0.531227
$477$			74485.2i			0.327366i
$478$	613719.			2.68605
$479$			28862.5i			0.125795i	0.998020	+	0.0628975i	$0.0200341\pi$
$479$			28862.5i			0.125795i	−0.998020	+	0.0628975i	$0.979966\pi$
$480$			551846.i			2.39517i
$481$		−	4830.33i		−	0.0208779i
$482$		−	323786.i		−	1.39368i
$483$	18993.6	−	24137.9i	0.0814166	−	0.103468i
$484$	−409371.			−1.74754
$485$	−470119.			−1.99859
$486$	−28189.3			−0.119347
$487$	35373.9			0.149150			0.0745752	−	0.997215i	$-0.476240\pi$
$487$	35373.9			0.149150			0.0745752	+	0.997215i	$0.476240\pi$
$488$		−	599368.i		−	2.51683i
$489$	477.837			0.00199831
$490$		−	831820.i		−	3.46447i
$491$	−224491.			−0.931183			−0.465592	−	0.885000i	$-0.654158\pi$
$491$	−224491.			−0.931183			−0.465592	+	0.885000i	$0.654158\pi$
$492$	−228835.			−0.945348
$493$		−	300224.i		−	1.23524i
$494$		−	5074.26i		−	0.0207931i
$495$	209800.			0.856240
$496$	240926.			0.979311
$497$			38241.3i			0.154817i
$498$			253833.i			1.02350i
$499$	−59361.4			−0.238398			−0.119199	−	0.992870i	$-0.538033\pi$
$499$	−59361.4			−0.238398			−0.119199	+	0.992870i	$0.538033\pi$
$500$			2.24644e6i			8.98576i
$501$	−19867.4			−0.0791528
$502$			73071.0i			0.289960i
$503$			267762.i			1.05831i	0.848526	+	0.529154i	$0.177491\pi$
$503$			267762.i			1.05831i	−0.848526	+	0.529154i	$0.822509\pi$
$504$			52491.1i			0.206645i
$505$			350873.i			1.37584i
$506$	−385192.	+	489520.i	−1.50444	+	1.91192i
$507$	148344.			0.577105
$508$	335766.			1.30110
$509$	462239.			1.78415			0.892075	−	0.451887i	$-0.149249\pi$
$509$	462239.			1.78415			0.892075	+	0.451887i	$0.149249\pi$
$510$	−519427.			−1.99703
$511$		−	54526.5i		−	0.208817i
$512$	−412861.			−1.57494
$513$			27442.4i			0.104277i
$514$	193394.			0.732010
$515$	381648.			1.43896
$516$		−	511935.i		−	1.92272i
$517$		−	66563.9i		−	0.249034i
$518$	115223.			0.429418
$519$	−49779.2			−0.184805
$520$		−	29784.4i		−	0.110150i
$521$		−	467093.i		−	1.72079i	−0.509629	−	0.860394i	$-0.670217\pi$
$521$		−	467093.i		−	1.72079i	0.509629	−	0.860394i	$-0.329783\pi$
$522$	−220532.			−0.809340
$523$			133920.i			0.489600i	0.969574	+	0.244800i	$0.0787223\pi$
$523$			133920.i			0.489600i	−0.969574	+	0.244800i	$0.921278\pi$
$524$	−1.12303e6			−4.09007
$525$			103735.i			0.376364i
$526$			458527.i			1.65727i
$527$			99147.9i			0.356995i
$528$		−	546489.i		−	1.96026i
$529$	65817.8	+	271991.i	0.235197	+	0.971948i
$530$	1.00818e6			3.58909
$531$	−86308.0			−0.306099
$532$	86071.2			0.304113
$533$	3898.43			0.0137226
$534$		−	312160.i		−	1.09470i
$535$	−819256.			−2.86228
$536$		−	1.03823e6i		−	3.61381i
$537$	141597.			0.491027
$538$	−275486.			−0.951775
$539$			360152.i			1.23968i
$540$			271314.i			0.930433i
$541$	273776.			0.935406			0.467703	−	0.883886i	$-0.345082\pi$
$541$	273776.			0.935406			0.467703	+	0.883886i	$0.345082\pi$
$542$	391317.			1.33208
$543$			314539.i			1.06678i
$544$			591552.i			1.99892i
$545$	−101815.			−0.342783
$546$		−	1506.22i		−	0.00505246i
$547$	217018.			0.725305			0.362652	−	0.931925i	$-0.381871\pi$
$547$	217018.			0.725305			0.362652	+	0.931925i	$0.381871\pi$
$548$			243199.i			0.809843i
$549$		−	93013.3i		−	0.308603i
$550$		−	2.10376e6i		−	6.95458i
$551$			214689.i			0.707142i
$552$	−375840.	−	295740.i	−1.23346	−	0.970581i
$553$	10827.4			0.0354059
$554$	−592326.			−1.92993
$555$	353584.			1.14791
$556$	597592.			1.93310
$557$			65560.8i			0.211317i	0.994402	+	0.105658i	$0.0336950\pi$
$557$			65560.8i			0.211317i	−0.994402	+	0.105658i	$0.966305\pi$
$558$	72829.9			0.233906
$559$			8721.34i			0.0279100i
$560$	364736.			1.16306
$561$	224896.			0.714588
$562$			371733.i			1.17695i
$563$			56808.8i			0.179225i	0.995977	+	0.0896126i	$0.0285629\pi$
$563$			56808.8i			0.179225i	−0.995977	+	0.0896126i	$0.971437\pi$
$564$	86080.7			0.270612
$565$	−530019.			−1.66033
$566$		−	275864.i		−	0.861118i
$567$			8145.86i			0.0253379i
$568$	595436.			1.84560
$569$		−	227173.i		−	0.701668i	−0.936438	−	0.350834i	$-0.885898\pi$
$569$		−	227173.i		−	0.701668i	0.936438	−	0.350834i	$-0.114102\pi$
$570$	371440.			1.14324
$571$		−	314063.i		−	0.963261i	−0.876374	−	0.481631i	$-0.840045\pi$
$571$		−	314063.i		−	0.963261i	0.876374	−	0.481631i	$-0.159955\pi$
$572$			21721.0i			0.0663877i
$573$		−	186764.i		−	0.568832i
$574$			92993.6i			0.282247i
$575$	−742751.	−	584454.i	−2.24651	−	1.76772i
$576$	147388.			0.444238
$577$	−75780.9			−0.227619			−0.113809	−	0.993503i	$-0.536305\pi$
$577$	−75780.9			−0.227619			−0.113809	+	0.993503i	$0.536305\pi$
$578$	64741.0			0.193787
$579$	−43247.8			−0.129005
$580$			2.12256e6i			6.30963i
$581$	73350.0			0.217294
$582$			370177.i			1.09286i
$583$	−436509.			−1.28427
$584$	−849005.			−2.48934
$585$		−	4622.11i		−	0.0135061i
$586$		−	298564.i		−	0.869446i
$587$	467136.			1.35571			0.677856	−	0.735195i	$-0.262909\pi$
$587$	467136.			1.35571			0.677856	+	0.735195i	$0.262909\pi$
$588$	−465750.			−1.34709
$589$		−	70900.2i		−	0.204370i
$590$			1.16820e6i			3.35594i
$591$	−207866.			−0.595126
$592$		−	921018.i		−	2.62800i
$593$	523323.			1.48820			0.744098	−	0.668070i	$-0.232879\pi$
$593$	523323.			1.48820			0.744098	+	0.668070i	$0.232879\pi$
$594$		−	165199.i		−	0.468203i
$595$			150099.i			0.423978i
$596$			860958.i			2.42376i
$597$		−	117907.i		−	0.330820i
$598$	10784.6	+	8486.17i	0.0301580	+	0.0237306i
$599$	−597662.			−1.66572			−0.832860	−	0.553484i	$-0.813298\pi$
$599$	−597662.			−1.66572			−0.832860	+	0.553484i	$0.813298\pi$
$600$	1.61521e6			4.48669
$601$	−699830.			−1.93751			−0.968754	−	0.248025i	$-0.920218\pi$
$601$	−699830.			−1.93751			−0.968754	+	0.248025i	$0.920218\pi$
$602$	−208040.			−0.574055
$603$		−	161119.i		−	0.443110i
$604$	−302180.			−0.828309
$605$			510506.i			1.39473i
$606$	276281.			0.752326
$607$	−19307.5			−0.0524021			−0.0262010	−	0.999657i	$-0.508341\pi$
$607$	−19307.5			−0.0524021			−0.0262010	+	0.999657i	$0.508341\pi$
$608$		−	423016.i		−	1.14433i
$609$			63727.2i			0.171827i
$610$	−1.25896e6			−3.38339
$611$	−1466.47			−0.00392818
$612$			290836.i			0.776506i
$613$		−	664451.i		−	1.76824i	−0.467256	−	0.884122i	$-0.654757\pi$
$613$		−	664451.i		−	1.76824i	0.467256	−	0.884122i	$-0.345243\pi$
$614$	427199.			1.13317
$615$			285368.i			0.754493i
$616$	−307616.			−0.810676
$617$		−	480320.i		−	1.26171i	−0.775900	−	0.630856i	$-0.782704\pi$
$617$		−	480320.i		−	1.26171i	0.775900	−	0.630856i	$-0.217296\pi$
$618$		−	300514.i		−	0.786843i
$619$			677097.i			1.76713i	0.468304	+	0.883567i	$0.344865\pi$
$619$			677097.i			1.76713i	−0.468304	+	0.883567i	$0.655135\pi$
$620$		−	700967.i		−	1.82354i
$621$	−58325.0	−	45894.5i	−0.151242	−	0.119008i
$622$	627767.			1.62262
$623$	−90204.9			−0.232410
$624$	−12039.7			−0.0309206
$625$	1.68478e6			4.31303
$626$		−	362950.i		−	0.926186i
$627$	−160822.			−0.409082
$628$			153924.i			0.390290i
$629$	379025.			0.958002
$630$	110256.			0.277794
$631$			483637.i			1.21468i	0.794443	+	0.607338i	$0.207763\pi$
$631$			483637.i			1.21468i	−0.794443	+	0.607338i	$0.792237\pi$
$632$		−	168589.i		−	0.422080i
$633$	−215102.			−0.536830
$634$	1.30668e6			3.25079
$635$		−	418717.i		−	1.03842i
$636$		−	564494.i		−	1.39555i
$637$	7934.52			0.0195543
$638$		−	1.29239e6i		−	3.17507i
$639$	92403.1			0.226300
$640$		−	295684.i		−	0.721886i
$641$			607233.i			1.47788i	0.673771	+	0.738940i	$0.264673\pi$
$641$			607233.i			1.47788i	−0.673771	+	0.738940i	$0.735327\pi$
$642$			645091.i			1.56513i
$643$		−	20493.9i		−	0.0495681i	−0.999693	−	0.0247840i	$-0.992110\pi$
$643$		−	20493.9i		−	0.0495681i	0.999693	−	0.0247840i	$-0.00788982\pi$
$644$	−143945.	+	182932.i	−0.347076	+	0.441081i
$645$	−638409.			−1.53454
$646$	398166.			0.954111
$647$	−254027.			−0.606836			−0.303418	−	0.952858i	$-0.598128\pi$
$647$	−254027.			−0.606836			−0.303418	+	0.952858i	$0.598128\pi$
$648$	126835.			0.302058
$649$		−	505795.i		−	1.20084i
$650$	−46348.0			−0.109699
$651$		−	21045.7i		−	0.0496593i
$652$	−3621.34			−0.00851871
$653$	−69362.6			−0.162667			−0.0813334	−	0.996687i	$-0.525918\pi$
$653$	−69362.6			−0.162667			−0.0813334	+	0.996687i	$0.525918\pi$
$654$			80170.3i			0.187438i
$655$			1.40048e6i			3.26433i
$656$	743330.			1.72732
$657$	−131753.			−0.305232
$658$		−	34981.4i		−	0.0807951i
$659$		−	438121.i		−	1.00884i	−0.863458	−	0.504421i	$-0.831706\pi$
$659$		−	438121.i		−	1.00884i	0.863458	−	0.504421i	$-0.168294\pi$
$660$	−1.58999e6			−3.65013
$661$		−	182990.i		−	0.418817i	−0.977828	−	0.209409i	$-0.932846\pi$
$661$		−	182990.i		−	0.418817i	0.977828	−	0.209409i	$-0.0671539\pi$
$662$	−644959.			−1.47169
$663$		−	4954.68i		−	0.0112717i
$664$		−	1.14210e6i		−	2.59040i
$665$		−	107335.i		−	0.242716i
$666$		−	278416.i		−	0.627690i
$667$	−456291.	−	359045.i	−1.02563	−	0.807044i
$668$	150568.			0.337426
$669$	−160439.			−0.358474
$670$	−2.18079e6			−4.85806
$671$	545090.			1.21066
$672$		−	125566.i		−	0.278057i
$673$	205678.			0.454108			0.227054	−	0.973882i	$-0.427091\pi$
$673$	205678.			0.454108			0.227054	+	0.973882i	$0.427091\pi$
$674$			1.08908e6i			2.39741i
$675$	250657.			0.550139
$676$	−1.12424e6			−2.46018
$677$		−	252173.i		−	0.550200i	−0.961416	−	0.275100i	$-0.911289\pi$
$677$		−	252173.i		−	0.550200i	0.961416	−	0.275100i	$-0.0887110\pi$
$678$			417343.i			0.907891i
$679$	106970.			0.232018
$680$	2.33712e6			5.05432
$681$			92870.5i			0.200255i
$682$			426808.i			0.917622i
$683$	645549.			1.38385			0.691923	−	0.721971i	$-0.256764\pi$
$683$	645549.			1.38385			0.691923	+	0.721971i	$0.256764\pi$
$684$		−	207975.i		−	0.444528i
$685$	303281.			0.646345
$686$			388924.i			0.826451i
$687$		−	114489.i		−	0.242578i
$688$			1.66293e6i			3.51316i
$689$			9616.74i			0.0202577i
$690$	−621195.	+	789443.i	−1.30476	+	1.65815i
$691$	−403276.			−0.844591			−0.422295	−	0.906458i	$-0.638775\pi$
$691$	−403276.			−0.844591			−0.422295	+	0.906458i	$0.638775\pi$
$692$	377257.			0.787817
$693$	−47737.6			−0.0994017
$694$	1.28061e6			2.65888
$695$		−	745227.i		−	1.54283i
$696$	992265.			2.04837
$697$			305901.i			0.629674i
$698$	−509680.			−1.04613
$699$	198837.			0.406953
$700$		−	786169.i		−	1.60443i
$701$			594481.i			1.20977i	0.796314	+	0.604883i	$0.206780\pi$
$701$			594481.i			1.20977i	−0.796314	+	0.604883i	$0.793220\pi$
$702$	−3639.50			−0.00738530
$703$	−271039.			−0.548429
$704$			863742.i			1.74276i
$705$		−	107347.i		−	0.215979i
$706$	−1.16218e6			−2.33165
$707$		−	79837.0i		−	0.159722i
$708$	654095.			1.30489
$709$		−	931797.i		−	1.85365i	−0.375489	−	0.926827i	$-0.622525\pi$
$709$		−	931797.i		−	1.85365i	0.375489	−	0.926827i	$-0.377475\pi$
$710$		−	1.25070e6i		−	2.48105i
$711$		−	26162.5i		−	0.0517536i
$712$			1.40454e6i			2.77060i
$713$	150688.	+	118573.i	0.296415	+	0.233242i
$714$	118190.			0.231837
$715$	27087.2			0.0529848
$716$	−1.07311e6			−2.09323
$717$	−428526.			−0.833563
$718$		−	453046.i		−	0.878808i
$719$	−125813.			−0.243371			−0.121686	−	0.992569i	$-0.538830\pi$
$719$	−125813.			−0.243371			−0.121686	+	0.992569i	$0.538830\pi$
$720$		−	881317.i		−	1.70007i
$721$	−86839.6			−0.167050
$722$	685090.			1.31423
$723$			226082.i			0.432503i
$724$		−	2.38377e6i		−	4.54765i
$725$	1.96096e6			3.73071
$726$	401978.			0.762657
$727$		−	390925.i		−	0.739648i	−0.929102	−	0.369824i	$-0.879418\pi$
$727$		−	390925.i		−	0.739648i	0.929102	−	0.369824i	$-0.120582\pi$
$728$			6777.10i			0.0127874i
$729$	19683.0			0.0370370
$730$			1.78331e6i			3.34643i
$731$	−684344.			−1.28068
$732$			704911.i			1.31557i
$733$		−	483475.i		−	0.899842i	−0.893068	−	0.449921i	$-0.851452\pi$
$733$		−	483475.i		−	0.899842i	0.893068	−	0.449921i	$-0.148548\pi$
$734$		−	1.41127e6i		−	2.61949i
$735$			580813.i			1.07513i
$736$	899062.	+	707451.i	1.65972	+	1.30599i
$737$	944212.			1.73834
$738$	224702.			0.412567
$739$	721370.			1.32090			0.660449	−	0.750871i	$-0.270366\pi$
$739$	721370.			1.32090			0.660449	+	0.750871i	$0.270366\pi$
$740$	−2.67967e6			−4.89349
$741$			3543.07i			0.00645273i
$742$	−229399.			−0.416661
$743$		−	859654.i		−	1.55721i	−0.627517	−	0.778603i	$-0.715929\pi$
$743$		−	859654.i		−	1.55721i	0.627517	−	0.778603i	$-0.284071\pi$
$744$	−327692.			−0.591997
$745$	1.07366e6			1.93443
$746$		−	207359.i		−	0.372602i
$747$		−	177237.i		−	0.317624i
$748$	−1.70440e6			−3.04627
$749$	186412.			0.332285
$750$		−	2.20587e6i		−	3.92155i
$751$			528747.i			0.937492i	0.883333	+	0.468746i	$0.155294\pi$
$751$			528747.i			0.937492i	−0.883333	+	0.468746i	$0.844706\pi$
$752$	−279618.			−0.494458
$753$		−	51021.4i		−	0.0899833i
$754$	−28472.8			−0.0500826
$755$			376834.i			0.661083i
$756$		−	61734.3i		−	0.108015i
$757$		−	273108.i		−	0.476588i	−0.971193	−	0.238294i	$-0.923412\pi$
$757$		−	273108.i		−	0.476588i	0.971193	−	0.238294i	$-0.0765881\pi$
$758$			324370.i			0.564550i
$759$	268958.	−	341804.i	0.466875	−	0.593327i
$760$	−1.67126e6			−2.89346
$761$	−265044.			−0.457666			−0.228833	−	0.973466i	$-0.573491\pi$
$761$	−265044.			−0.457666			−0.228833	+	0.973466i	$0.573491\pi$
$762$	−329703.			−0.567822
$763$	23166.8			0.0397940
$764$			1.41541e6i			2.42491i
$765$	362687.			0.619739
$766$			1.78701e6i			3.04558i
$767$	−11143.2			−0.0189417
$768$	221011.			0.374707
$769$		−	771637.i		−	1.30485i	−0.757853	−	0.652425i	$-0.773752\pi$
$769$		−	771637.i		−	1.30485i	0.757853	−	0.652425i	$-0.226248\pi$
$770$			646140.i			1.08980i
$771$	−135036.			−0.227165
$772$	327758.			0.549945
$773$			1.09414e6i			1.83111i	0.402192	+	0.915555i	$0.368248\pi$
$773$			1.09414e6i			1.83111i	−0.402192	+	0.915555i	$0.631752\pi$
$774$			502690.i			0.839109i
$775$	−647598.			−1.07821
$776$		−	1.66558e6i		−	2.76593i
$777$	−80453.8			−0.133261
$778$			1.39382e6i			2.30275i
$779$		−	218748.i		−	0.360471i
$780$			35029.2i			0.0575759i
$781$			541514.i			0.887784i
$782$	−665891.	+	846246.i	−1.08890	+	1.38383i
$783$	153985.			0.251163
$784$	1.51291e6			2.46139
$785$	191951.			0.311495
$786$	1.10275e6			1.78498
$787$		−	820354.i		−	1.32450i	−0.749283	−	0.662250i	$-0.769602\pi$
$787$		−	820354.i		−	1.32450i	0.749283	−	0.662250i	$-0.230398\pi$
$788$	1.57534e6			2.53701
$789$		−	320163.i		−	0.514301i
$790$	−354117.			−0.567404
$791$	120600.			0.192749
$792$			743298.i			1.18498i
$793$		−	12008.9i		−	0.0190966i
$794$	1.08208e6			1.71640
$795$	−703953.			−1.11380
$796$			893574.i			1.41028i
$797$		−	1.05531e6i		−	1.66135i	−0.556755	−	0.830676i	$-0.687954\pi$
$797$		−	1.05531e6i		−	1.66135i	0.556755	−	0.830676i	$-0.312046\pi$
$798$	−84516.8			−0.132720
$799$		−	115071.i		−	0.180248i
$800$	−3.86381e6			−6.03720
$801$			217964.i			0.339719i
$802$			1.18746e6i			1.84617i
$803$		−	772119.i		−	1.19744i
$804$			1.22106e6i			1.88896i
$805$	228126.	+	179507.i	0.352032	+	0.277006i
$806$	9403.02			0.0144743
$807$	192356.			0.295365
$808$	−1.24310e6			−1.90408
$809$	−134570.			−0.205613			−0.102806	−	0.994701i	$-0.532782\pi$
$809$	−134570.			−0.205613			−0.102806	+	0.994701i	$0.532782\pi$
$810$		−	266415.i		−	0.406058i
$811$	−966167.			−1.46896			−0.734481	−	0.678629i	$-0.762574\pi$
$811$	−966167.			−1.46896			−0.734481	+	0.678629i	$0.762574\pi$
$812$		−	482964.i		−	0.732491i
$813$	−273234.			−0.413385
$814$	1.63161e6			2.46245
$815$			4515.99i			0.00679889i
$816$		−	944730.i		−	1.41882i
$817$	489371.			0.733152
$818$	−1.11660e6			−1.66874
$819$			1051.71i			0.00156793i
$820$		−	2.16270e6i		−	3.21638i
$821$	−1.06373e6			−1.57814			−0.789068	−	0.614305i	$-0.789436\pi$
$821$	−1.06373e6			−1.57814			−0.789068	+	0.614305i	$0.789436\pi$
$822$		−	238807.i		−	0.353430i
$823$	−133222.			−0.196687			−0.0983434	−	0.995153i	$-0.531354\pi$
$823$	−133222.			−0.196687			−0.0983434	+	0.995153i	$0.531354\pi$
$824$			1.35214e6i			1.99144i
$825$			1.46894e6i			2.15822i
$826$		−	265810.i		−	0.389594i
$827$			543898.i			0.795256i	0.917547	+	0.397628i	$0.130166\pi$
$827$			543898.i			0.795256i	−0.917547	+	0.397628i	$0.869834\pi$
$828$	442022.	+	347817.i	0.644738	+	0.507329i
$829$	−1.17996e6			−1.71695			−0.858473	−	0.512859i	$-0.828586\pi$
$829$	−1.17996e6			−1.71695			−0.858473	+	0.512859i	$0.828586\pi$
$830$	−2.39895e6			−3.48229
$831$	413588.			0.598915
$832$	19029.1			0.0274898
$833$			622604.i			0.897267i
$834$	−586800.			−0.843641
$835$		−	187765.i		−	0.269304i
$836$	1.21881e6			1.74390
$837$	−50853.0			−0.0725881
$838$			1.80582e6i			2.57150i
$839$			429976.i			0.610829i	0.952219	+	0.305415i	$0.0987951\pi$
$839$			429976.i			0.610829i	−0.952219	+	0.305415i	$0.901205\pi$
$840$	−496089.			−0.703074
$841$	497384.			0.703234
$842$			984575.i			1.38875i
$843$		−	259560.i		−	0.365244i
$844$	1.63017e6			2.28849
$845$			1.40199e6i			1.96350i
$846$	−84526.1			−0.118100
$847$		−	116160.i		−	0.161915i
$848$			1.83366e6i			2.54993i
$849$			192620.i			0.267231i
$850$		−	3.63682e6i		−	5.03367i
$851$	453284.	−	576055.i	0.625909	−	0.795435i
$852$	−700287.			−0.964710
$853$	−767502.			−1.05483			−0.527414	−	0.849609i	$-0.676838\pi$
$853$	−767502.			−1.05483			−0.527414	+	0.849609i	$0.676838\pi$
$854$	286461.			0.392781
$855$	−259356.			−0.354783
$856$		−	2.90253e6i		−	3.96122i
$857$	−1.18822e6			−1.61784			−0.808920	−	0.587918i	$-0.799948\pi$
$857$	−1.18822e6			−1.61784			−0.808920	+	0.587918i	$0.799948\pi$
$858$		−	21328.7i		−	0.0289728i
$859$	703322.			0.953165			0.476582	−	0.879130i	$-0.341875\pi$
$859$	703322.			0.953165			0.476582	+	0.879130i	$0.341875\pi$
$860$	4.83825e6			6.54172
$861$		−	64932.2i		−	0.0875898i
$862$		−	2.11153e6i		−	2.84173i
$863$	−708330.			−0.951073			−0.475536	−	0.879696i	$-0.657746\pi$
$863$	−708330.			−0.951073			−0.475536	+	0.879696i	$0.657746\pi$
$864$	−303408.			−0.406442
$865$		−	470459.i		−	0.628766i
$866$		−	554803.i		−	0.739781i
$867$	−45205.0			−0.0601379
$868$			159497.i			0.211696i
$869$	153321.			0.203031
$870$		−	2.08423e6i		−	2.75364i
$871$		−	20802.0i		−	0.0274200i
$872$		−	360719.i		−	0.474391i
$873$		−	258474.i		−	0.339147i
$874$	476175.	−	605146.i	0.623367	−	0.792205i
$875$	−637431.			−0.832562
$876$	998507.			1.30120
$877$	718291.			0.933901			0.466951	−	0.884283i	$-0.345353\pi$
$877$	718291.			0.933901			0.466951	+	0.884283i	$0.345353\pi$
$878$	−1.87967e6			−2.43833
$879$			208470.i			0.269815i
$880$	5.16482e6			6.66945
$881$		−	425210.i		−	0.547837i	−0.961753	−	0.273919i	$-0.911680\pi$
$881$		−	425210.i		−	0.547837i	0.961753	−	0.273919i	$-0.0883199\pi$
$882$	457339.			0.587896
$883$	326938.			0.419318			0.209659	−	0.977775i	$-0.432765\pi$
$883$	326938.			0.419318			0.209659	+	0.977775i	$0.432765\pi$
$884$			37549.6i			0.0480509i
$885$		−	815689.i		−	1.04145i
$886$	2.17655e6			2.77269
$887$	−612556.			−0.778572			−0.389286	−	0.921117i	$-0.627278\pi$
$887$	−612556.			−0.778572			−0.389286	+	0.921117i	$0.627278\pi$
$888$			1.25271e6i			1.58863i
$889$			95274.2i			0.120551i
$890$	2.95020e6			3.72452
$891$			115349.i			0.145298i
$892$	1.21591e6			1.52816
$893$			82286.5i			0.103187i
$894$		−	845410.i		−	1.05777i
$895$			1.33822e6i			1.67063i
$896$			67279.5i			0.0838044i
$897$	−7530.30	−	5925.42i	−0.00935895	−	0.00736434i
$898$	−1.07748e6			−1.33616
$899$	−397836.			−0.492249
$900$	−1.89963e6			−2.34523
$901$	−754603.			−0.929542
$902$			1.31683e6i			1.61852i
$903$	145262.			0.178147
$904$		−	1.87780e6i		−	2.29780i
$905$	−2.97268e6			−3.62953
$906$	296723.			0.361489
$907$			913097.i			1.10995i	0.831868	+	0.554974i	$0.187272\pi$
$907$			913097.i			1.10995i	−0.831868	+	0.554974i	$0.812728\pi$
$908$		−	703830.i		−	0.853682i
$909$	−192912.			−0.233470
$910$	14235.1			0.0171901
$911$			936548.i			1.12848i	0.825611	+	0.564239i	$0.190830\pi$
$911$			936548.i			1.12848i	−0.825611	+	0.564239i	$0.809170\pi$
$912$			675572.i			0.812235i
$913$	1.03867e6			1.24605
$914$		−	2.22474e6i		−	2.66310i
$915$	879060.			1.04997
$916$			867671.i			1.03410i
$917$		−	318663.i		−	0.378959i
$918$		−	285584.i		−	0.338882i
$919$			794529.i			0.940760i	0.882464	+	0.470380i	$0.155883\pi$
$919$			794529.i			0.940760i	−0.882464	+	0.470380i	$0.844117\pi$
$920$	2.79501e6	−	3.55203e6i	3.30223	−	4.19663i
$921$	−298289.			−0.351656
$922$	−1.16803e6			−1.37401
$923$	11930.1			0.0140036
$924$	361784.			0.423746
$925$			2.47565e6i			2.89338i
$926$	1.46730e6			1.71119
$927$			209832.i			0.244181i
$928$	−2.37363e6			−2.75625
$929$	100991.			0.117017			0.0585086	−	0.998287i	$-0.481366\pi$
$929$	100991.			0.117017			0.0585086	+	0.998287i	$0.481366\pi$
$930$			688308.i			0.795824i
$931$		−	445221.i		−	0.513661i
$932$	−1.50691e6			−1.73483
$933$	−438334.			−0.503549
$934$			623579.i			0.714822i
$935$			2.12547e6i			2.43126i
$936$	16375.6			0.0186916
$937$		−	808400.i		−	0.920761i	−0.887722	−	0.460381i	$-0.847713\pi$
$937$		−	808400.i		−	0.920761i	0.887722	−	0.460381i	$-0.152287\pi$
$938$	496212.			0.563977
$939$			253428.i			0.287424i
$940$			813540.i			0.920711i
$941$			25875.4i			0.0292219i	0.999893	+	0.0146110i	$0.00465098\pi$
$941$			25875.4i			0.0292219i	−0.999893	+	0.0146110i	$0.995349\pi$
$942$		−	151144.i		−	0.170329i
$943$	464919.	+	365834.i	0.522822	+	0.411396i
$944$	−2.12471e6			−2.38428
$945$	−76985.8			−0.0862079
$946$	−2.94594e6			−3.29186
$947$	1.27426e6			1.42089			0.710444	−	0.703754i	$-0.248494\pi$
$947$	1.27426e6			1.42089			0.710444	+	0.703754i	$0.248494\pi$
$948$			198276.i			0.220624i
$949$	−17010.6			−0.0188880
$950$			2.60068e6i			2.88163i
$951$	−912378.			−1.00882
$952$	−531783.			−0.586760
$953$		−	1.57859e6i		−	1.73813i	−0.494696	−	0.869066i	$-0.664721\pi$
$953$		−	1.57859e6i		−	1.73813i	0.494696	−	0.869066i	$-0.335279\pi$
$954$			554300.i			0.609043i
$955$	1.76509e6			1.93535
$956$	3.24763e6			3.55345
$957$			902406.i			0.985322i
$958$			214788.i			0.234034i
$959$	−69008.1			−0.0750348
$960$			1.39295e6i			1.51144i
$961$	−792137.			−0.857736
$962$		−	35946.1i		−	0.0388420i
$963$		−	450431.i		−	0.485708i
$964$		−	1.71339e6i		−	1.84375i
$965$		−	408731.i		−	0.438918i
$966$	141346.	−	179629.i	0.151470	−	0.192496i
$967$	68383.2			0.0731301			0.0365651	−	0.999331i	$-0.488358\pi$
$967$	68383.2			0.0731301			0.0365651	+	0.999331i	$0.488358\pi$
$968$	−1.80866e6			−1.93022
$969$	−278017.			−0.296090
$970$	−3.49851e6			−3.71826
$971$		−	902024.i		−	0.956709i	−0.878167	−	0.478354i	$-0.841233\pi$
$971$		−	902024.i		−	0.956709i	0.878167	−	0.478354i	$-0.158767\pi$
$972$	−149170.			−0.157888
$973$			169568.i			0.179109i
$974$	263243.			0.277485
$975$	32362.2			0.0340431
$976$		−	2.28978e6i		−	2.40378i
$977$		−	917308.i		−	0.961006i	−0.876993	−	0.480503i	$-0.840454\pi$
$977$		−	917308.i		−	0.961006i	0.876993	−	0.480503i	$-0.159546\pi$
$978$	3555.94			0.00371772
$979$	−1.27734e6			−1.33273
$980$		−	4.40176e6i		−	4.58325i
$981$		−	55978.4i		−	0.0581678i
$982$	−1.67060e6			−1.73241
$983$			1.81657e6i			1.87994i	0.341257	+	0.939970i	$0.389147\pi$
$983$			1.81657e6i			1.87994i	−0.341257	+	0.939970i	$0.610853\pi$
$984$	−1.01103e6			−1.04417
$985$		−	1.96453e6i		−	2.02481i
$986$		−	2.23419e6i		−	2.29809i
$987$			24425.5i			0.0250732i
$988$		−	26851.6i		−	0.0275078i
$989$	−818422.	+	1.04009e6i	−0.836729	+	1.06335i
$990$	1.56128e6			1.59298
$991$	−242.686			−0.000247114			−0.000123557	−	1.00000i	$-0.500039\pi$
$991$	−242.686			−0.000247114			−0.000123557		1.00000i	$0.500039\pi$
$992$	783884.			0.796578
$993$	450338.			0.456710
$994$			284582.i			0.288028i
$995$	1.11433e6			1.12556
$996$			1.34321e6i			1.35402i
$997$	1.34734e6			1.35546			0.677731	−	0.735310i	$-0.262963\pi$
$997$	1.34734e6			1.35546			0.677731	+	0.735310i	$0.262963\pi$
$998$	−441753.			−0.443525
$999$			194402.i			0.194791i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.5.d.a.22.15	✓	16
3.2	odd	2		207.5.d.c.91.2		16
4.3	odd	2		1104.5.c.c.1057.9		16
23.22	odd	2	inner	69.5.d.a.22.16	yes	16
69.68	even	2		207.5.d.c.91.1		16
92.91	even	2		1104.5.c.c.1057.16		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.5.d.a.22.15	✓	16	1.1	even	1	trivial
69.5.d.a.22.16	yes	16	23.22	odd	2	inner
207.5.d.c.91.1		16	69.68	even	2
207.5.d.c.91.2		16	3.2	odd	2
1104.5.c.c.1057.9		16	4.3	odd	2
1104.5.c.c.1057.16		16	92.91	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	69.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.44175 q^{2} -5.19615 q^{3} +39.3796 q^{4} -49.1084i q^{5} -38.6685 q^{6} +11.1740i q^{7} +173.985 q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q+7.44175 q^{2} -5.19615 q^{3} +39.3796 q^{4} -49.1084i q^{5} -38.6685 q^{6} +11.1740i q^{7} +173.985 q^{8} +27.0000 q^{9} -365.452i q^{10} +158.229i q^{11} -204.623 q^{12} +3.48595 q^{13} +83.1543i q^{14} +255.174i q^{15} +664.681 q^{16} +273.535i q^{17} +200.927 q^{18} -195.603i q^{19} -1933.87i q^{20} -58.0619i q^{21} +1177.50i q^{22} +(415.728 + 327.126i) q^{23} -904.054 q^{24} -1786.63 q^{25} +25.9416 q^{26} -140.296 q^{27} +440.029i q^{28} -1097.57 q^{29} +1898.94i q^{30} +362.469 q^{31} +2162.62 q^{32} -822.183i q^{33} +2035.58i q^{34} +548.738 q^{35} +1063.25 q^{36} -1385.65i q^{37} -1455.63i q^{38} -18.1135 q^{39} -8544.13i q^{40} +1118.33 q^{41} -432.082i q^{42} +2501.85i q^{43} +6231.01i q^{44} -1325.93i q^{45} +(3093.74 + 2434.39i) q^{46} -420.680 q^{47} -3453.78 q^{48} +2276.14 q^{49} -13295.7 q^{50} -1421.33i q^{51} +137.276 q^{52} +2758.71i q^{53} -1044.05 q^{54} +7770.38 q^{55} +1944.12i q^{56} +1016.39i q^{57} -8167.86 q^{58} -3196.59 q^{59} +10048.7i q^{60} -3444.94i q^{61} +2697.40 q^{62} +301.699i q^{63} +5458.80 q^{64} -171.189i q^{65} -6118.48i q^{66} -5967.36i q^{67} +10771.7i q^{68} +(-2160.18 - 1699.80i) q^{69} +4083.57 q^{70} +3422.34 q^{71} +4697.60 q^{72} -4879.75 q^{73} -10311.7i q^{74} +9283.60 q^{75} -7702.79i q^{76} -1768.06 q^{77} -134.796 q^{78} -968.983i q^{79} -32641.4i q^{80} +729.000 q^{81} +8322.30 q^{82} -6564.33i q^{83} -2286.46i q^{84} +13432.8 q^{85} +18618.2i q^{86} +5703.15 q^{87} +27529.6i q^{88} +8072.73i q^{89} -9867.20i q^{90} +38.9521i q^{91} +(16371.2 + 12882.1i) q^{92} -1883.44 q^{93} -3130.60 q^{94} -9605.76 q^{95} -11237.3 q^{96} -9573.09i q^{97} +16938.5 q^{98} +4272.19i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{2} + 144 q^{4} - 36 q^{6} + 372 q^{8} + 432 q^{9}+O(q^{10}) \) 16 * q + 12 * q^2 + 144 * q^4 - 36 * q^6 + 372 * q^8 + 432 * q^9	\( 16 q + 12 q^{2} + 144 q^{4} - 36 q^{6} + 372 q^{8} + 432 q^{9} + 104 q^{13} + 680 q^{16} + 324 q^{18} - 732 q^{23} - 1764 q^{24} - 2984 q^{25} + 1800 q^{26} - 3528 q^{29} - 400 q^{31} + 5244 q^{32} + 912 q^{35} + 3888 q^{36} + 2016 q^{39} + 1008 q^{41} - 1168 q^{46} - 8664 q^{47} - 2016 q^{48} + 7240 q^{49} - 18852 q^{50} - 20952 q^{52} - 972 q^{54} + 6816 q^{55} - 13352 q^{58} + 20112 q^{59} + 4248 q^{62} - 896 q^{64} - 10044 q^{69} - 10680 q^{70} + 40368 q^{71} + 10044 q^{72} - 9568 q^{73} + 7560 q^{75} + 2952 q^{77} - 6912 q^{78} + 11664 q^{81} + 71800 q^{82} + 42744 q^{85} + 8352 q^{87} - 9876 q^{92} - 10008 q^{93} + 73720 q^{94} + 33312 q^{95} - 24948 q^{96} - 59052 q^{98}+O(q^{100}) \) 16 * q + 12 * q^2 + 144 * q^4 - 36 * q^6 + 372 * q^8 + 432 * q^9 + 104 * q^13 + 680 * q^16 + 324 * q^18 - 732 * q^23 - 1764 * q^24 - 2984 * q^25 + 1800 * q^26 - 3528 * q^29 - 400 * q^31 + 5244 * q^32 + 912 * q^35 + 3888 * q^36 + 2016 * q^39 + 1008 * q^41 - 1168 * q^46 - 8664 * q^47 - 2016 * q^48 + 7240 * q^49 - 18852 * q^50 - 20952 * q^52 - 972 * q^54 + 6816 * q^55 - 13352 * q^58 + 20112 * q^59 + 4248 * q^62 - 896 * q^64 - 10044 * q^69 - 10680 * q^70 + 40368 * q^71 + 10044 * q^72 - 9568 * q^73 + 7560 * q^75 + 2952 * q^77 - 6912 * q^78 + 11664 * q^81 + 71800 * q^82 + 42744 * q^85 + 8352 * q^87 - 9876 * q^92 - 10008 * q^93 + 73720 * q^94 + 33312 * q^95 - 24948 * q^96 - 59052 * q^98

Introduction

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