LMFDB - Embedded newform 23.25.b.c.22.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [23,25,Mod(22,23)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(23, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 25, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("23.22"); S:= CuspForms(chi, 25); N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 25 $
Character orbit:	$[\chi]$	$=$	23.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$83.9424450193$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.3
Character	$\chi$	$=$	23.22
Dual form			23.25.b.c.22.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-7352.04 q^{2} -139223. q^{3} +3.72753e7 q^{4} -1.97315e8i q^{5} +1.02357e9 q^{6} -1.56753e10i q^{7} -1.50703e11 q^{8} -2.63047e11 q^{9} +1.45067e12i q^{10} +1.45224e12i q^{11} -5.18958e12 q^{12} -1.36856e12 q^{13} +1.15245e14i q^{14} +2.74707e13i q^{15} +4.82599e14 q^{16} -4.49232e14i q^{17} +1.93393e15 q^{18} -1.46317e15i q^{19} -7.35497e15i q^{20} +2.18236e15i q^{21} -1.06769e16i q^{22} +(1.29280e16 + 1.76951e16i) q^{23} +2.09813e16 q^{24} +2.06715e16 q^{25} +1.00617e16 q^{26} +7.59428e16 q^{27} -5.84301e17i q^{28} -5.63231e16 q^{29} -2.01966e17i q^{30} +1.19892e18 q^{31} -1.01971e18 q^{32} -2.02185e17i q^{33} +3.30278e18i q^{34} -3.09296e18 q^{35} -9.80514e18 q^{36} +7.78529e18i q^{37} +1.07573e19i q^{38} +1.90535e17 q^{39} +2.97359e19i q^{40} -1.13622e19 q^{41} -1.60448e19i q^{42} +3.84229e19i q^{43} +5.41327e19i q^{44} +5.19030e19i q^{45} +(-9.50470e19 - 1.30095e20i) q^{46} +5.65599e19 q^{47} -6.71888e19 q^{48} -5.41326e19 q^{49} -1.51978e20 q^{50} +6.25434e19i q^{51} -5.10135e19 q^{52} +1.91474e20i q^{53} -5.58334e20 q^{54} +2.86548e20 q^{55} +2.36231e21i q^{56} +2.03706e20i q^{57} +4.14090e20 q^{58} -3.57457e20 q^{59} +1.02398e21i q^{60} +4.02074e21i q^{61} -8.81453e21 q^{62} +4.12332e21i q^{63} -5.99709e20 q^{64} +2.70037e20i q^{65} +1.48647e21i q^{66} -1.40778e22i q^{67} -1.67453e22i q^{68} +(-1.79987e21 - 2.46357e21i) q^{69} +2.27396e22 q^{70} +8.98848e21 q^{71} +3.96419e22 q^{72} +6.61720e21 q^{73} -5.72378e22i q^{74} -2.87795e21 q^{75} -5.45400e22i q^{76} +2.27642e22 q^{77} -1.40082e21 q^{78} -2.56926e22i q^{79} -9.52239e22i q^{80} +6.37191e22 q^{81} +8.35351e22 q^{82} +1.76633e23i q^{83} +8.13480e22i q^{84} -8.86402e22 q^{85} -2.82487e23i q^{86} +7.84146e21 q^{87} -2.18857e23i q^{88} +7.50483e22i q^{89} -3.81593e23i q^{90} +2.14525e22i q^{91} +(4.81895e23 + 6.59592e23i) q^{92} -1.66918e23 q^{93} -4.15831e23 q^{94} -2.88704e23 q^{95} +1.41967e23 q^{96} -1.29911e23i q^{97} +3.97986e23 q^{98} -3.82007e23i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4232 q^{2} - 434562 q^{3} + 317760360 q^{4} - 8460029520 q^{6} - 198307023760 q^{8} + 4220041988298 q^{9} - 67439597688792 q^{12} + 5771152551358 q^{13} + 18\!\cdots\!92 q^{16} + 18\!\cdots\!68 q^{18}+ \cdots - 20\!\cdots\!92 q^{98}+O(q^{100}) $ 44 * q - 4232 * q^2 - 434562 * q^3 + 317760360 * q^4 - 8460029520 * q^6 - 198307023760 * q^8 + 4220041988298 * q^9 - 67439597688792 * q^12 + 5771152551358 * q^13 + 1801550792530192 * q^16 + 1842830558165568 * q^18 - 117204213263210644 * q^23 - 186493005437450784 * q^24 - 689533084157602612 * q^25 - 548650296251047576 * q^26 + 824037505793640630 * q^27 - 1583078054945709602 * q^29 - 77249813729122274 * q^31 - 2838716832733708512 * q^32 + 3418569899175948192 * q^35 + 71181245116739335248 * q^36 + 47106572467817156310 * q^39 - 9150679195666184738 * q^41 - 153049487518536134480 * q^46 + 22735751396127769438 * q^47 - 1317482931964495330032 * q^48 - 1842110772006482246836 * q^49 + 284936617115007894520 * q^50 + 207129175371045152568 * q^52 - 381774025126375304544 * q^54 - 5430040696094013021792 * q^55 - 2168901871328366220760 * q^58 + 16048018791948427185304 * q^59 + 14054450375064171442400 * q^62 + 20310379126091866412192 * q^64 + 24476879859710435470878 * q^69 - 118028399895318845638416 * q^70 - 68204988047209908888578 * q^71 - 106777935582637627887744 * q^72 - 7319375899831086678722 * q^73 + 12913295383070041819998 * q^75 + 111116199573084838704480 * q^77 - 422583593213386075404144 * q^78 - 331350288185757013618680 * q^81 + 1000764218021118649437320 * q^82 - 113385735487939722862944 * q^85 - 248828610584854760185290 * q^87 - 4238455111972177116774696 * q^92 + 4492289396741153284965270 * q^93 - 1602622056539190351258592 * q^94 - 1668692394291208047471936 * q^95 - 12092663288748189077973504 * q^96 - 2067304853876486831513192 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/23\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

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 $

$2$	−7352.04			−1.79493			−0.897466	−	0.441083i	$-0.854594\pi$
$2$	−7352.04			−1.79493			−0.897466	+	0.441083i	$0.854594\pi$
$3$	−139223.			−0.261972			−0.130986	−	0.991384i	$-0.541814\pi$
$3$	−139223.			−0.261972			−0.130986	+	0.991384i	$0.541814\pi$
$4$	3.72753e7			2.22178
$5$		−	1.97315e8i		−	0.808201i	−0.914715	−	0.404101i	$-0.867585\pi$
$5$		−	1.97315e8i		−	0.808201i	0.914715	−	0.404101i	$-0.132415\pi$
$6$	1.02357e9			0.470223
$7$		−	1.56753e10i		−	1.13250i	−0.824233	−	0.566250i	$-0.808394\pi$
$7$		−	1.56753e10i		−	1.13250i	0.824233	−	0.566250i	$-0.191606\pi$
$8$	−1.50703e11			−2.19302
$9$	−2.63047e11			−0.931370
$10$			1.45067e12i			1.45067i
$11$			1.45224e12i			0.462728i	0.972867	+	0.231364i	$0.0743189\pi$
$11$			1.45224e12i			0.462728i	−0.972867	+	0.231364i	$0.925681\pi$
$12$	−5.18958e12			−0.582046
$13$	−1.36856e12			−0.0587413			−0.0293706	−	0.999569i	$-0.509350\pi$
$13$	−1.36856e12			−0.0587413			−0.0293706	+	0.999569i	$0.509350\pi$
$14$			1.15245e14i			2.03276i
$15$			2.74707e13i			0.211726i
$16$	4.82599e14			1.71454
$17$		−	4.49232e14i		−	0.771053i	−0.922697	−	0.385526i	$-0.874020\pi$
$17$		−	4.49232e14i		−	0.771053i	0.922697	−	0.385526i	$-0.125980\pi$
$18$	1.93393e15			1.67175
$19$		−	1.46317e15i		−	0.661075i	−0.943793	−	0.330537i	$-0.892770\pi$
$19$		−	1.46317e15i		−	0.661075i	0.943793	−	0.330537i	$-0.107230\pi$
$20$		−	7.35497e15i		−	1.79565i
$21$			2.18236e15i			0.296684i
$22$		−	1.06769e16i		−	0.830566i
$23$	1.29280e16	+	1.76951e16i	0.589925	+	0.807458i
$23$	1.29280e16	+	1.76951e16i	0.589925	+	0.807458i
$24$	2.09813e16			0.574510
$25$	2.06715e16			0.346811
$26$	1.00617e16			0.105437
$27$	7.59428e16			0.505966
$28$		−	5.84301e17i		−	2.51617i
$29$	−5.63231e16			−0.159188			−0.0795940	−	0.996827i	$-0.525362\pi$
$29$	−5.63231e16			−0.159188			−0.0795940	+	0.996827i	$0.525362\pi$
$30$		−	2.01966e17i		−	0.380035i
$31$	1.19892e18			1.52213			0.761064	−	0.648677i	$-0.224677\pi$
$31$	1.19892e18			1.52213			0.761064	+	0.648677i	$0.224677\pi$
$32$	−1.01971e18			−0.884458
$33$		−	2.02185e17i		−	0.121222i
$34$			3.30278e18i			1.38399i
$35$	−3.09296e18			−0.915288
$36$	−9.80514e18			−2.06930
$37$			7.78529e18i			1.18264i	0.806435	+	0.591322i	$0.201394\pi$
$37$			7.78529e18i			1.18264i	−0.806435	+	0.591322i	$0.798606\pi$
$38$			1.07573e19i			1.18658i
$39$	1.90535e17			0.0153886
$40$			2.97359e19i			1.77240i
$41$	−1.13622e19			−0.503564			−0.251782	−	0.967784i	$-0.581017\pi$
$41$	−1.13622e19			−0.503564			−0.251782	+	0.967784i	$0.581017\pi$
$42$		−	1.60448e19i		−	0.532528i
$43$			3.84229e19i			0.961544i	0.876846	+	0.480772i	$0.159643\pi$
$43$			3.84229e19i			0.961544i	−0.876846	+	0.480772i	$0.840357\pi$
$44$			5.41327e19i			1.02808i
$45$			5.19030e19i			0.752735i
$46$	−9.50470e19	−	1.30095e20i	−1.05887	−	1.44933i
$47$	5.65599e19			0.486782			0.243391	−	0.969928i	$-0.421740\pi$
$47$	5.65599e19			0.486782			0.243391	+	0.969928i	$0.421740\pi$
$48$	−6.71888e19			−0.449161
$49$	−5.41326e19			−0.282557
$50$	−1.51978e20			−0.622502
$51$			6.25434e19i			0.201995i
$52$	−5.10135e19			−0.130510
$53$			1.91474e20i			0.389762i	0.980827	+	0.194881i	$0.0624321\pi$
$53$			1.91474e20i			0.389762i	−0.980827	+	0.194881i	$0.937568\pi$
$54$	−5.58334e20			−0.908174
$55$	2.86548e20			0.373978
$56$			2.36231e21i			2.48359i
$57$			2.03706e20i			0.173183i
$58$	4.14090e20			0.285732
$59$	−3.57457e20			−0.200909			−0.100455	−	0.994942i	$-0.532030\pi$
$59$	−3.57457e20			−0.200909			−0.100455	+	0.994942i	$0.532030\pi$
$60$			1.02398e21i			0.470410i
$61$			4.02074e21i			1.51477i	0.652966	+	0.757387i	$0.273525\pi$
$61$			4.02074e21i			1.51477i	−0.652966	+	0.757387i	$0.726475\pi$
$62$	−8.81453e21			−2.73212
$63$			4.12332e21i			1.05478i
$64$	−5.99709e20			−0.126993
$65$			2.70037e20i			0.0474748i
$66$			1.48647e21i			0.217585i
$67$		−	1.40778e22i		−	1.72043i	−0.509928	−	0.860217i	$-0.670328\pi$
$67$		−	1.40778e22i		−	1.72043i	0.509928	−	0.860217i	$-0.329672\pi$
$68$		−	1.67453e22i		−	1.71311i
$69$	−1.79987e21	−	2.46357e21i	−0.154544	−	0.211532i
$70$	2.27396e22			1.64288
$71$	8.98848e21			0.547754			0.273877	−	0.961765i	$-0.411694\pi$
$71$	8.98848e21			0.547754			0.273877	+	0.961765i	$0.411694\pi$
$72$	3.96419e22			2.04251
$73$	6.61720e21			0.288935			0.144467	−	0.989510i	$-0.453853\pi$
$73$	6.61720e21			0.288935			0.144467	+	0.989510i	$0.453853\pi$
$74$		−	5.72378e22i		−	2.12277i
$75$	−2.87795e21			−0.0908549
$76$		−	5.45400e22i		−	1.46876i
$77$	2.27642e22			0.524040
$78$	−1.40082e21			−0.0276215
$79$		−	2.56926e22i		−	0.434794i	−0.976083	−	0.217397i	$-0.930243\pi$
$79$		−	2.56926e22i		−	0.434794i	0.976083	−	0.217397i	$-0.0697566\pi$
$80$		−	9.52239e22i		−	1.38569i
$81$	6.37191e22			0.798821
$82$	8.35351e22			0.903863
$83$			1.76633e23i			1.65247i	0.563325	+	0.826236i	$0.309522\pi$
$83$			1.76633e23i			1.65247i	−0.563325	+	0.826236i	$0.690478\pi$
$84$			8.13480e22i			0.659167i
$85$	−8.86402e22			−0.623166
$86$		−	2.82487e23i		−	1.72591i
$87$	7.84146e21			0.0417029
$88$		−	2.18857e23i		−	1.01477i
$89$			7.50483e22i			0.303851i	0.988392	+	0.151926i	$0.0485474\pi$
$89$			7.50483e22i			0.303851i	−0.988392	+	0.151926i	$0.951453\pi$
$90$		−	3.81593e23i		−	1.35111i
$91$			2.14525e22i			0.0665245i
$92$	4.81895e23	+	6.59592e23i	1.31068	+	1.79400i
$93$	−1.66918e23			−0.398755
$94$	−4.15831e23			−0.873740
$95$	−2.88704e23			−0.534281
$96$	1.41967e23			0.231704
$97$		−	1.29911e23i		−	0.187234i	−0.995608	−	0.0936172i	$-0.970157\pi$
$97$		−	1.29911e23i		−	0.187234i	0.995608	−	0.0936172i	$-0.0298430\pi$
$98$	3.97986e23			0.507171
$99$		−	3.82007e23i		−	0.430971i
$100$	7.70538e23			0.770538
$101$	5.61088e22			0.0497937			0.0248969	−	0.999690i	$-0.492074\pi$
$101$	5.61088e22			0.0497937			0.0248969	+	0.999690i	$0.492074\pi$
$102$		−	4.59822e23i		−	0.362566i
$103$		−	1.58786e24i		−	1.11369i	−0.830616	−	0.556845i	$-0.812012\pi$
$103$		−	1.58786e24i		−	1.11369i	0.830616	−	0.556845i	$-0.187988\pi$
$104$	2.06246e23			0.128821
$105$	4.30611e23			0.239780
$106$		−	1.40773e24i		−	0.699597i
$107$			2.84640e24i			1.26384i	0.775035	+	0.631918i	$0.217732\pi$
$107$			2.84640e24i			1.26384i	−0.775035	+	0.631918i	$0.782268\pi$
$108$	2.83079e24			1.12415
$109$			1.27003e23i			0.0451541i	0.999745	+	0.0225770i	$0.00718711\pi$
$109$			1.27003e23i			0.0451541i	−0.999745	+	0.0225770i	$0.992813\pi$
$110$	−2.10672e24			−0.671264
$111$		−	1.08389e24i		−	0.309820i
$112$		−	7.56486e24i		−	1.94171i
$113$			6.47061e24i			1.49281i	0.665493	+	0.746404i	$0.268221\pi$
$113$			6.47061e24i			1.49281i	−0.665493	+	0.746404i	$0.731779\pi$
$114$		−	1.49766e24i		−	0.310852i
$115$	3.49151e24	−	2.55088e24i	0.652589	−	0.476778i
$116$	−2.09946e24			−0.353681
$117$	3.59995e23			0.0547099
$118$	2.62804e24			0.360618
$119$	−7.04184e24			−0.873217
$120$		−	4.13992e24i		−	0.464320i
$121$	7.74073e24			0.785883
$122$		−	2.95607e25i		−	2.71892i
$123$	1.58187e24			0.131920
$124$	4.46902e25			3.38184
$125$		−	1.58397e25i		−	1.08849i
$126$		−	3.03149e25i		−	1.89325i
$127$	1.25578e25			0.713295			0.356648	−	0.934239i	$-0.383920\pi$
$127$	1.25578e25			0.713295			0.356648	+	0.934239i	$0.383920\pi$
$128$	2.15170e25			1.11240
$129$		−	5.34935e24i		−	0.251898i
$130$		−	1.98532e24i		−	0.0852140i
$131$	−3.27228e25			−1.28113			−0.640567	−	0.767902i	$-0.721301\pi$
$131$	−3.27228e25			−1.28113			−0.640567	+	0.767902i	$0.721301\pi$
$132$		−	7.53651e24i		−	0.269329i
$133$	−2.29355e25			−0.748667
$134$			1.03501e26i			3.08806i
$135$		−	1.49846e25i		−	0.408922i
$136$			6.77007e25i			1.69093i
$137$			2.75417e25i			0.630006i	0.949091	+	0.315003i	$0.102005\pi$
$137$			2.75417e25i			0.630006i	−0.949091	+	0.315003i	$0.897995\pi$
$138$	1.32327e25	+	1.81123e25i	0.277396	+	0.379685i
$139$	−3.83947e25			−0.738063			−0.369032	−	0.929417i	$-0.620311\pi$
$139$	−3.83947e25			−0.738063			−0.369032	+	0.929417i	$0.620311\pi$
$140$	−1.15291e26			−2.03357
$141$	−7.87443e24			−0.127523
$142$	−6.60837e25			−0.983182
$143$		−	1.98748e24i		−	0.0271813i
$144$	−1.26946e26			−1.59687
$145$			1.11134e25i			0.128656i
$146$	−4.86499e25			−0.518618
$147$	7.53650e24			0.0740222
$148$			2.90199e26i			2.62758i
$149$			6.16463e25i			0.514840i	0.966300	+	0.257420i	$0.0828723\pi$
$149$			6.16463e25i			0.514840i	−0.966300	+	0.257420i	$0.917128\pi$
$150$	2.11588e25			0.163078
$151$	2.42167e26			1.72342			0.861712	−	0.507397i	$-0.169392\pi$
$151$	2.42167e26			1.72342			0.861712	+	0.507397i	$0.169392\pi$
$152$			2.20504e26i			1.44975i
$153$			1.18169e26i			0.718136i
$154$	−1.67364e26			−0.940616
$155$		−	2.36565e26i		−	1.23019i
$156$	7.10225e24			0.0341901
$157$		−	1.62002e26i		−	0.722313i	−0.932505	−	0.361156i	$-0.882382\pi$
$157$		−	1.62002e26i		−	0.722313i	0.932505	−	0.361156i	$-0.117618\pi$
$158$			1.88893e26i			0.780426i
$159$		−	2.66576e25i		−	0.102107i
$160$			2.01204e26i			0.714820i
$161$	2.77376e26	−	2.02649e26i	0.914447	−	0.668090i
$162$	−4.68466e26			−1.43383
$163$	1.00358e26			0.285300			0.142650	−	0.989773i	$-0.454438\pi$
$163$	1.00358e26			0.285300			0.142650	+	0.989773i	$0.454438\pi$
$164$	−4.23528e26			−1.11881
$165$	−3.98941e25			−0.0979718
$166$		−	1.29861e27i		−	2.96607i
$167$	7.22944e26			1.53641			0.768205	−	0.640204i	$-0.221150\pi$
$167$	7.22944e26			1.53641			0.768205	+	0.640204i	$0.221150\pi$
$168$		−	3.28887e26i		−	0.650633i
$169$	−5.40928e26			−0.996549
$170$	6.51686e26			1.11854
$171$			3.84881e26i			0.615706i
$172$			1.43223e27i			2.13634i
$173$	6.08198e25			0.0846237			0.0423118	−	0.999104i	$-0.486528\pi$
$173$	6.08198e25			0.0846237			0.0423118	+	0.999104i	$0.486528\pi$
$174$	−5.76508e25			−0.0748539
$175$		−	3.24032e26i		−	0.392763i
$176$			7.00849e26i			0.793364i
$177$	4.97662e25			0.0526326
$178$		−	5.51759e26i		−	0.545392i
$179$	−3.19029e26			−0.294845			−0.147423	−	0.989074i	$-0.547098\pi$
$179$	−3.19029e26			−0.294845			−0.147423	+	0.989074i	$0.547098\pi$
$180$			1.93470e27i			1.67241i
$181$		−	9.70535e26i		−	0.784997i	−0.919753	−	0.392499i	$-0.871611\pi$
$181$		−	9.70535e26i		−	0.784997i	0.919753	−	0.392499i	$-0.128389\pi$
$182$		−	1.57720e26i		−	0.119407i
$183$		−	5.59779e26i		−	0.396829i
$184$	−1.94828e27	−	2.66671e27i	−1.29371	−	1.77077i
$185$	1.53615e27			0.955815
$186$	1.22718e27			0.715739
$187$	6.52393e26			0.356788
$188$	2.10829e27			1.08152
$189$		−	1.19042e27i		−	0.573006i
$190$	2.12257e27			0.958999
$191$		−	3.19292e27i		−	1.35453i	−0.735741	−	0.677263i	$-0.763166\pi$
$191$		−	3.19292e27i		−	1.35453i	0.735741	−	0.677263i	$-0.236834\pi$
$192$	8.34933e25			0.0332688
$193$	2.07936e26			0.0778471			0.0389235	−	0.999242i	$-0.487607\pi$
$193$	2.07936e26			0.0778471			0.0389235	+	0.999242i	$0.487607\pi$
$194$			9.55113e26i			0.336073i
$195$		−	3.75953e25i		−	0.0124371i
$196$	−2.01781e27			−0.627781
$197$	−2.67306e27			−0.782373			−0.391187	−	0.920311i	$-0.627935\pi$
$197$	−2.67306e27			−0.782373			−0.391187	+	0.920311i	$0.627935\pi$
$198$			2.80853e27i			0.773565i
$199$		−	4.23746e27i		−	1.09867i	−0.835601	−	0.549337i	$-0.814880\pi$
$199$		−	4.23746e27i		−	1.09867i	0.835601	−	0.549337i	$-0.185120\pi$
$200$	−3.11526e27			−0.760562
$201$			1.95996e27i			0.450706i
$202$	−4.12515e26			−0.0893764
$203$			8.82879e26i			0.180281i
$204$			2.33133e27i			0.448788i
$205$			2.24192e27i			0.406981i
$206$			1.16740e28i			1.99900i
$207$	−3.40066e27	−	4.65465e27i	−0.549438	−	0.752043i
$208$	−6.60465e26			−0.100714
$209$	2.12487e27			0.305898
$210$	−3.16587e27			−0.430389
$211$	1.51989e28			1.95175			0.975874	−	0.218336i	$-0.0700627\pi$
$211$	1.51989e28			1.95175			0.975874	+	0.218336i	$0.0700627\pi$
$212$			7.13726e27i			0.865966i
$213$	−1.25140e27			−0.143497
$214$		−	2.09269e28i		−	2.26850i
$215$	7.58141e27			0.777121
$216$	−1.14448e28			−1.10959
$217$		−	1.87934e28i		−	1.72381i
$218$		−	9.33734e26i		−	0.0810485i
$219$	−9.21265e26			−0.0756929
$220$	1.06812e28			0.830897
$221$			6.14801e26i			0.0452926i
$222$			7.96881e27i			0.556106i
$223$	2.52153e28			1.66727			0.833633	−	0.552318i	$-0.186257\pi$
$223$	2.52153e28			1.66727			0.833633	+	0.552318i	$0.186257\pi$
$224$			1.59842e28i			1.00165i
$225$	−5.43757e27			−0.323009
$226$		−	4.75722e28i		−	2.67949i
$227$			8.83304e27i			0.471846i	0.971772	+	0.235923i	$0.0758113\pi$
$227$			8.83304e27i			0.471846i	−0.971772	+	0.235923i	$0.924189\pi$
$228$			7.59322e27i			0.384776i
$229$		−	3.58905e28i		−	1.72565i	−0.505501	−	0.862826i	$-0.668692\pi$
$229$		−	3.58905e28i		−	1.72565i	0.505501	−	0.862826i	$-0.331308\pi$
$230$	−2.56698e28	+	1.87542e28i	−1.17135	+	0.855784i
$231$	−3.16930e27			−0.137284
$232$	8.48806e27			0.349102
$233$	1.42644e28			0.557165			0.278582	−	0.960412i	$-0.410135\pi$
$233$	1.42644e28			0.557165			0.278582	+	0.960412i	$0.410135\pi$
$234$	−2.64670e27			−0.0982006
$235$		−	1.11601e28i		−	0.393418i
$236$	−1.33243e28			−0.446376
$237$			3.57700e27i			0.113904i
$238$	5.17719e28			1.56737
$239$	2.78475e28			0.801699			0.400850	−	0.916144i	$-0.368715\pi$
$239$	2.78475e28			0.801699			0.400850	+	0.916144i	$0.368715\pi$
$240$			1.32573e28i			0.363012i
$241$		−	2.81924e28i		−	0.734392i	−0.930144	−	0.367196i	$-0.880318\pi$
$241$		−	2.81924e28i		−	0.734392i	0.930144	−	0.367196i	$-0.119682\pi$
$242$	−5.69102e28			−1.41061
$243$	−3.03196e28			−0.715235
$244$			1.49874e29i			3.36550i
$245$			1.06812e28i			0.228363i
$246$	−1.16300e28			−0.236787
$247$			2.00243e27i			0.0388324i
$248$	−1.80681e29			−3.33805
$249$		−	2.45913e28i		−	0.432902i
$250$			1.16454e29i			1.95377i
$251$			7.92520e28i			1.26744i	0.773564	+	0.633718i	$0.218472\pi$
$251$			7.92520e28i			1.26744i	−0.773564	+	0.633718i	$0.781528\pi$
$252$			1.53698e29i			2.34349i
$253$	−2.56976e28	+	1.87745e28i	−0.373634	+	0.272975i
$254$	−9.23256e28			−1.28032
$255$	1.23407e28			0.163252
$256$	−1.48132e29			−1.86969
$257$	−6.32074e28			−0.761326			−0.380663	−	0.924714i	$-0.624304\pi$
$257$	−6.32074e28			−0.761326			−0.380663	+	0.924714i	$0.624304\pi$
$258$			3.93287e28i			0.452140i
$259$	1.22036e29			1.33935
$260$			1.00657e28i			0.105479i
$261$	1.48156e28			0.148263
$262$	2.40579e29			2.29955
$263$		−	9.29056e28i		−	0.848346i	−0.905581	−	0.424173i	$-0.860565\pi$
$263$		−	9.29056e28i		−	0.848346i	0.905581	−	0.424173i	$-0.139435\pi$
$264$			3.04699e28i			0.265842i
$265$	3.77807e28			0.315006
$266$	1.68623e29			1.34381
$267$		−	1.04484e28i		−	0.0796006i
$268$		−	5.24756e29i		−	3.82243i
$269$	1.29223e29			0.900147			0.450074	−	0.892991i	$-0.351398\pi$
$269$	1.29223e29			0.900147			0.450074	+	0.892991i	$0.351398\pi$
$270$			1.10168e29i			0.733988i
$271$	−2.60376e29			−1.65947			−0.829736	−	0.558156i	$-0.811509\pi$
$271$	−2.60376e29			−1.65947			−0.829736	+	0.558156i	$0.811509\pi$
$272$		−	2.16799e29i		−	1.32200i
$273$		−	2.98668e27i		−	0.0174276i
$274$		−	2.02488e29i		−	1.13082i
$275$			3.00200e28i			0.160479i
$276$	−6.70907e28	−	9.18304e28i	−0.343363	−	0.469978i
$277$	−2.04072e29			−1.00006			−0.500030	−	0.866008i	$-0.666678\pi$
$277$	−2.04072e29			−1.00006			−0.500030	+	0.866008i	$0.666678\pi$
$278$	2.82279e29			1.32477
$279$	−3.15373e29			−1.41766
$280$	4.66118e29			2.00724
$281$			3.24893e28i			0.134049i	0.997751	+	0.0670247i	$0.0213506\pi$
$281$			3.24893e28i			0.134049i	−0.997751	+	0.0670247i	$0.978649\pi$
$282$	5.78932e28			0.228896
$283$			6.28219e27i			0.0238053i	0.999929	+	0.0119027i	$0.00378882\pi$
$283$			6.28219e27i			0.0238053i	−0.999929	+	0.0119027i	$0.996211\pi$
$284$	3.35048e29			1.21699
$285$	4.01943e28			0.139967
$286$			1.46120e28i			0.0487885i
$287$			1.78105e29i			0.570286i
$288$	2.68231e29			0.823758
$289$	1.37639e29			0.405478
$290$		−	8.17060e28i		−	0.230929i
$291$			1.80866e28i			0.0490503i
$292$	2.46658e29			0.641950
$293$			1.26162e29i			0.315150i	0.987507	+	0.157575i	$0.0503676\pi$
$293$			1.26162e29i			0.315150i	−0.987507	+	0.157575i	$0.949632\pi$
$294$	−5.54087e28			−0.132865
$295$			7.05315e28i			0.162375i
$296$		−	1.17327e30i		−	2.59356i
$297$			1.10287e29i			0.234125i
$298$		−	4.53226e29i		−	0.924102i
$299$	−1.76927e28	−	2.42169e28i	−0.0346529	−	0.0474311i
$300$	−1.07277e29			−0.201860
$301$	6.02290e29			1.08895
$302$	−1.78043e30			−3.09343
$303$	−7.81163e27			−0.0130446
$304$		−	7.06122e29i		−	1.13344i
$305$	7.93352e29			1.22424
$306$		−	8.68784e29i		−	1.28900i
$307$	−6.07154e29			−0.866240			−0.433120	−	0.901336i	$-0.642587\pi$
$307$	−6.07154e29			−0.866240			−0.433120	+	0.901336i	$0.642587\pi$
$308$	8.48544e29			1.16430
$309$			2.21066e29i			0.291756i
$310$			1.73924e30i			2.20810i
$311$	−8.94154e28			−0.109216			−0.0546082	−	0.998508i	$-0.517391\pi$
$311$	−8.94154e28			−0.109216			−0.0546082	+	0.998508i	$0.517391\pi$
$312$	−2.87142e28			−0.0337475
$313$			1.34911e30i			1.52586i	0.646483	+	0.762929i	$0.276239\pi$
$313$			1.34911e30i			1.52586i	−0.646483	+	0.762929i	$0.723761\pi$
$314$			1.19105e30i			1.29650i
$315$	8.13593e29			0.852472
$316$		−	9.57701e29i		−	0.966018i
$317$	−1.75827e30			−1.70756			−0.853779	−	0.520635i	$-0.825695\pi$
$317$	−1.75827e30			−1.70756			−0.853779	+	0.520635i	$0.825695\pi$
$318$			1.95988e29i			0.183275i
$319$		−	8.17946e28i		−	0.0736608i
$320$			1.18332e29i			0.102636i
$321$		−	3.96284e29i		−	0.331090i
$322$	−2.03928e30	+	1.48989e30i	−1.64137	+	1.19918i
$323$	−6.57302e29			−0.509723
$324$	2.37515e30			1.77481
$325$	−2.82902e28			−0.0203721
$326$	−7.37837e29			−0.512094
$327$		−	1.76818e28i		−	0.0118291i
$328$	1.71231e30			1.10432
$329$		−	8.86591e29i		−	0.551280i
$330$	2.93303e29			0.175853
$331$	5.48345e29			0.317043			0.158521	−	0.987356i	$-0.449327\pi$
$331$	5.48345e29			0.317043			0.158521	+	0.987356i	$0.449327\pi$
$332$			6.58404e30i			3.67143i
$333$		−	2.04789e30i		−	1.10148i
$334$	−5.31512e30			−2.75775
$335$	−2.77776e30			−1.39046
$336$			1.05320e30i			0.508675i
$337$			2.73336e30i			1.27391i	0.770902	+	0.636954i	$0.219806\pi$
$337$			2.73336e30i			1.27391i	−0.770902	+	0.636954i	$0.780194\pi$
$338$	3.97692e30			1.78874
$339$		−	9.00857e29i		−	0.391074i
$340$	−3.30409e30			−1.38454
$341$			1.74112e30i			0.704331i
$342$		−	2.82966e30i		−	1.10515i
$343$		−	2.15454e30i		−	0.812504i
$344$		−	5.79045e30i		−	2.10868i
$345$	−4.86099e29	+	3.55141e29i	−0.170960	+	0.124903i
$346$	−4.47150e29			−0.151894
$347$	8.01257e29			0.262917			0.131458	−	0.991322i	$-0.458034\pi$
$347$	8.01257e29			0.262917			0.131458	+	0.991322i	$0.458034\pi$
$348$	2.92293e29			0.0926547
$349$	4.08973e30			1.25253			0.626266	−	0.779609i	$-0.284582\pi$
$349$	4.08973e30			1.25253			0.626266	+	0.779609i	$0.284582\pi$
$350$			2.38230e30i			0.704984i
$351$	−1.03932e29			−0.0297211
$352$		−	1.48086e30i		−	0.409264i
$353$	4.31101e30			1.15155			0.575775	−	0.817609i	$-0.304701\pi$
$353$	4.31101e30			1.15155			0.575775	+	0.817609i	$0.304701\pi$
$354$	−3.65883e29			−0.0944720
$355$		−	1.77356e30i		−	0.442696i
$356$			2.79745e30i			0.675091i
$357$	9.80385e29			0.228759
$358$	2.34552e30			0.529228
$359$		−	8.89180e30i		−	1.94024i	−0.242619	−	0.970122i	$-0.578007\pi$
$359$		−	8.89180e30i		−	1.94024i	0.242619	−	0.970122i	$-0.421993\pi$
$360$		−	7.82193e30i		−	1.65076i
$361$	2.75791e30			0.562980
$362$			7.13541e30i			1.40902i
$363$	−1.07769e30			−0.205880
$364$			7.99650e29i			0.147803i
$365$		−	1.30567e30i		−	0.233517i
$366$			4.11552e30i			0.712282i
$367$		−	2.88453e30i		−	0.483150i	−0.970382	−	0.241575i	$-0.922336\pi$
$367$		−	2.88453e30i		−	0.483150i	0.970382	−	0.241575i	$-0.0776639\pi$
$368$	6.23902e30	+	8.53966e30i	1.01145	+	1.38442i
$369$	2.98878e30			0.469005
$370$	−1.12939e31			−1.71562
$371$	3.00141e30			0.441406
$372$	−6.22191e30			−0.885948
$373$			4.45050e30i			0.613624i	0.951770	+	0.306812i	$0.0992624\pi$
$373$			4.45050e30i			0.613624i	−0.951770	+	0.306812i	$0.900738\pi$
$374$	−4.79642e30			−0.640410
$375$			2.20525e30i			0.285155i
$376$	−8.52374e30			−1.06752
$377$	7.70815e28			0.00935091
$378$			8.75204e30i			1.02851i
$379$			5.39612e30i			0.614343i	0.951654	+	0.307172i	$0.0993825\pi$
$379$			5.39612e30i			0.614343i	−0.951654	+	0.307172i	$0.900617\pi$
$380$	−1.07615e31			−1.18706
$381$	−1.74833e30			−0.186864
$382$			2.34745e31i			2.43128i
$383$		−	1.10935e31i		−	1.11348i	−0.830685	−	0.556742i	$-0.812051\pi$
$383$		−	1.10935e31i		−	1.11348i	0.830685	−	0.556742i	$-0.187949\pi$
$384$	−2.99566e30			−0.291419
$385$		−	4.49172e30i		−	0.423530i
$386$	−1.52876e30			−0.139730
$387$		−	1.01070e31i		−	0.895553i
$388$		−	4.84248e30i		−	0.415994i
$389$		−	1.73769e31i		−	1.44736i	−0.690134	−	0.723682i	$-0.742448\pi$
$389$		−	1.73769e31i		−	1.44736i	0.690134	−	0.723682i	$-0.257552\pi$
$390$			2.76402e29i			0.0223237i
$391$	7.94923e30	−	5.80767e30i	0.622593	−	0.454863i
$392$	8.15795e30			0.619653
$393$	4.55576e30			0.335622
$394$	1.96525e31			1.40431
$395$	−5.06954e30			−0.351401
$396$		−	1.42394e31i		−	0.957525i
$397$	7.75510e30			0.505942			0.252971	−	0.967474i	$-0.418592\pi$
$397$	7.75510e30			0.505942			0.252971	+	0.967474i	$0.418592\pi$
$398$			3.11540e31i			1.97205i
$399$	3.19315e30			0.196130
$400$	9.97606e30			0.594619
$401$		−	2.00092e30i		−	0.115743i	−0.998324	−	0.0578717i	$-0.981569\pi$
$401$		−	2.00092e30i		−	0.115743i	0.998324	−	0.0578717i	$-0.0184314\pi$
$402$		−	1.44097e31i		−	0.808987i
$403$	−1.64080e30			−0.0894117
$404$	2.09148e30			0.110631
$405$		−	1.25727e31i		−	0.645608i
$406$		−	6.49097e30i		−	0.323591i
$407$	−1.13061e31			−0.547243
$408$		−	9.42548e30i		−	0.442977i
$409$	2.78372e31			1.27042			0.635209	−	0.772340i	$-0.280914\pi$
$409$	2.78372e31			1.27042			0.635209	+	0.772340i	$0.280914\pi$
$410$		−	1.64827e31i		−	0.730503i
$411$		−	3.83444e30i		−	0.165044i
$412$		−	5.91878e31i		−	2.47438i
$413$			5.60323e30i			0.227530i
$414$	2.50018e31	+	3.42212e31i	0.986205	+	1.34987i
$415$	3.48522e31			1.33553
$416$	1.39553e30			0.0519542
$417$	5.34542e30			0.193352
$418$	−1.56221e31			−0.549066
$419$		−	1.17113e31i		−	0.399977i	−0.979798	−	0.199989i	$-0.935910\pi$
$419$		−	1.17113e31i		−	0.399977i	0.979798	−	0.199989i	$-0.0640905\pi$
$420$	1.60512e31			0.532740
$421$		−	3.98197e31i		−	1.28444i	−0.766522	−	0.642218i	$-0.778014\pi$
$421$		−	3.98197e31i		−	1.28444i	0.766522	−	0.642218i	$-0.221986\pi$
$422$	−1.11743e32			−3.50325
$423$	−1.48779e31			−0.453374
$424$		−	2.88557e31i		−	0.854755i
$425$		−	9.28632e30i		−	0.267409i
$426$	9.20036e30			0.257567
$427$	6.30262e31			1.71548
$428$			1.06101e32i			2.80797i
$429$			2.76702e29i			0.00712074i
$430$	−5.57389e31			−1.39488
$431$			4.07217e31i			0.991057i	0.868592	+	0.495528i	$0.165026\pi$
$431$			4.07217e31i			0.991057i	−0.868592	+	0.495528i	$0.834974\pi$
$432$	3.66499e31			0.867496
$433$		−	4.55378e31i		−	1.04838i	−0.851602	−	0.524189i	$-0.824369\pi$
$433$		−	4.55378e31i		−	1.04838i	0.851602	−	0.524189i	$-0.175631\pi$
$434$			1.38170e32i			3.09412i
$435$		−	1.54724e30i		−	0.0337043i
$436$			4.73409e30i			0.100323i
$437$	2.58909e31	−	1.89158e31i	0.533790	−	0.389984i
$438$	6.77318e30			0.135864
$439$	2.21602e31			0.432514			0.216257	−	0.976337i	$-0.430615\pi$
$439$	2.21602e31			0.432514			0.216257	+	0.976337i	$0.430615\pi$
$440$	−4.31837e31			−0.820139
$441$	1.42394e31			0.263165
$442$		−	4.52004e30i		−	0.0812972i
$443$	−6.22469e31			−1.08961			−0.544807	−	0.838561i	$-0.683397\pi$
$443$	−6.22469e31			−1.08961			−0.544807	+	0.838561i	$0.683397\pi$
$444$		−	4.04024e31i		−	0.688353i
$445$	1.48081e31			0.245573
$446$	−1.85384e32			−2.99263
$447$		−	8.58258e30i		−	0.134874i
$448$			9.40060e30i			0.143820i
$449$	−7.49916e31			−1.11701			−0.558504	−	0.829502i	$-0.688625\pi$
$449$	−7.49916e31			−1.11701			−0.558504	+	0.829502i	$0.688625\pi$
$450$	3.99773e31			0.579780
$451$		−	1.65006e31i		−	0.233013i
$452$			2.41194e32i			3.31669i
$453$	−3.37152e31			−0.451490
$454$		−	6.49409e31i		−	0.846931i
$455$	4.23290e30			0.0537652
$456$		−	3.06991e31i		−	0.379794i
$457$		−	5.04143e31i		−	0.607518i	−0.952749	−	0.303759i	$-0.901758\pi$
$457$		−	5.04143e31i		−	0.607518i	0.952749	−	0.303759i	$-0.0982417\pi$
$458$			2.63868e32i			3.09743i
$459$		−	3.41159e31i		−	0.390126i
$460$	1.30147e32	−	9.50849e31i	1.44991	−	1.05930i
$461$	1.21143e32			1.31488			0.657442	−	0.753505i	$-0.271639\pi$
$461$	1.21143e32			1.31488			0.657442	+	0.753505i	$0.271639\pi$
$462$	2.33009e31			0.246416
$463$	1.14897e32			1.18395			0.591977	−	0.805955i	$-0.298348\pi$
$463$	1.14897e32			1.18395			0.591977	+	0.805955i	$0.298348\pi$
$464$	−2.71815e31			−0.272934
$465$			3.29353e31i			0.322275i
$466$	−1.04873e32			−1.00007
$467$			8.69199e30i			0.0807824i	0.999184	+	0.0403912i	$0.0128604\pi$
$467$			8.69199e30i			0.0807824i	−0.999184	+	0.0403912i	$0.987140\pi$
$468$	1.34189e31			0.121553
$469$	−2.20674e32			−1.94839
$470$			8.20495e31i			0.706158i
$471$			2.25544e31i			0.189226i
$472$	5.38698e31			0.440597
$473$	−5.57993e31			−0.444933
$474$		−	2.62983e31i		−	0.204450i
$475$		−	3.02459e31i		−	0.229268i
$476$	−2.62487e32			−1.94010
$477$		−	5.03666e31i		−	0.363013i
$478$	−2.04736e32			−1.43900
$479$			2.16497e32i			1.48397i	0.670416	+	0.741985i	$0.266116\pi$
$479$			2.16497e32i			1.48397i	−0.670416	+	0.741985i	$0.733884\pi$
$480$		−	2.80122e31i		−	0.187263i
$481$		−	1.06546e31i		−	0.0694700i
$482$			2.07272e32i			1.31818i
$483$	−3.86171e31	+	2.82134e31i	−0.239560	+	0.175021i
$484$	2.88538e32			1.74606
$485$	−2.56334e31			−0.151323
$486$	2.22911e32			1.28380
$487$	−1.52864e32			−0.858933			−0.429466	−	0.903083i	$-0.641298\pi$
$487$	−1.52864e32			−0.858933			−0.429466	+	0.903083i	$0.641298\pi$
$488$		−	6.05938e32i		−	3.32193i
$489$	−1.39722e31			−0.0747407
$490$		−	7.85284e31i		−	0.409896i
$491$	1.70534e32			0.868626			0.434313	−	0.900762i	$-0.356991\pi$
$491$	1.70534e32			0.868626			0.434313	+	0.900762i	$0.356991\pi$
$492$	5.89648e31			0.293097
$493$			2.53022e31i			0.122742i
$494$		−	1.47220e31i		−	0.0697015i
$495$	−7.53755e31			−0.348312
$496$	5.78599e32			2.60974
$497$		−	1.40897e32i		−	0.620332i
$498$			1.80796e32i			0.777030i
$499$	2.64439e32			1.10948			0.554739	−	0.832025i	$-0.312818\pi$
$499$	2.64439e32			1.10948			0.554739	+	0.832025i	$0.312818\pi$
$500$		−	5.90429e32i		−	2.41840i
$501$	−1.00650e32			−0.402497
$502$		−	5.82664e32i		−	2.27496i
$503$			3.50896e32i			1.33771i	0.743393	+	0.668855i	$0.233215\pi$
$503$			3.50896e32i			1.33771i	−0.743393	+	0.668855i	$0.766785\pi$
$504$		−	6.21397e32i		−	2.31314i
$505$		−	1.10711e31i		−	0.0402434i
$506$	1.88930e32	−	1.38031e32i	0.670647	−	0.489971i
$507$	7.53095e31			0.261068
$508$	4.68096e32			1.58479
$509$	2.03318e32			0.672300			0.336150	−	0.941809i	$-0.390875\pi$
$509$	2.03318e32			0.672300			0.336150	+	0.941809i	$0.390875\pi$
$510$	−9.07297e31			−0.293027
$511$		−	1.03726e32i		−	0.327219i
$512$	7.28081e32			2.24357
$513$		−	1.11117e32i		−	0.334481i
$514$	4.64704e32			1.36653
$515$	−3.13307e32			−0.900086
$516$		−	1.99399e32i		−	0.559662i
$517$			8.21385e31i			0.225248i
$518$	−8.97218e32			−2.40403
$519$	−8.46751e30			−0.0221691
$520$		−	4.06954e31i		−	0.104113i
$521$			4.34185e32i			1.08548i	0.839900	+	0.542740i	$0.182613\pi$
$521$			4.34185e32i			1.08548i	−0.839900	+	0.542740i	$0.817387\pi$
$522$	−1.08925e32			−0.266122
$523$			3.81243e32i			0.910295i	0.890416	+	0.455147i	$0.150413\pi$
$523$			3.81243e32i			0.910295i	−0.890416	+	0.455147i	$0.849587\pi$
$524$	−1.21975e33			−2.84640
$525$			4.51126e31i			0.102893i
$526$			6.83046e32i			1.52272i
$527$		−	5.38595e32i		−	1.17364i
$528$		−	9.75742e31i		−	0.207839i
$529$	−1.45986e32	+	4.57525e32i	−0.303978	+	0.952679i
$530$	−2.77765e32			−0.565415
$531$	9.40278e31			0.187121
$532$	−8.54929e32			−1.66338
$533$	1.55498e31			0.0295800
$534$			7.68174e31i			0.142878i
$535$	5.61637e32			1.02143
$536$			2.12157e33i			3.77294i
$537$	4.44162e31			0.0772414
$538$	−9.50055e32			−1.61570
$539$		−	7.86136e31i		−	0.130747i
$540$		−	5.58557e32i		−	0.908536i
$541$	−7.14744e32			−1.13706			−0.568530	−	0.822663i	$-0.692488\pi$
$541$	−7.14744e32			−1.13706			−0.568530	+	0.822663i	$0.692488\pi$
$542$	1.91430e33			2.97864
$543$			1.35121e32i			0.205648i
$544$			4.58087e32i			0.681963i
$545$	2.50596e31			0.0364936
$546$			2.19582e31i			0.0312813i
$547$	−1.05184e33			−1.46589			−0.732947	−	0.680285i	$-0.761856\pi$
$547$	−1.05184e33			−1.46589			−0.732947	+	0.680285i	$0.761856\pi$
$548$			1.02663e33i			1.39974i
$549$		−	1.05764e33i		−	1.41082i
$550$		−	2.20708e32i		−	0.288049i
$551$			8.24101e31i			0.105235i
$552$	2.71246e32	+	3.71267e32i	0.338918	+	0.463893i
$553$	−4.02739e32			−0.492404
$554$	1.50035e33			1.79504
$555$	−2.13868e32			−0.250397
$556$	−1.43117e33			−1.63982
$557$			1.17367e33i			1.31608i	0.752983	+	0.658040i	$0.228614\pi$
$557$			1.17367e33i			1.31608i	−0.752983	+	0.658040i	$0.771386\pi$
$558$	2.31863e33			2.54461
$559$		−	5.25841e31i		−	0.0564823i
$560$	−1.49266e33			−1.56929
$561$	−9.08280e31			−0.0934686
$562$		−	2.38863e32i		−	0.240610i
$563$		−	4.51304e32i		−	0.445008i	−0.974932	−	0.222504i	$-0.928577\pi$
$563$		−	4.51304e32i		−	0.445008i	0.974932	−	0.222504i	$-0.0714231\pi$
$564$	−2.93522e32			−0.283329
$565$	1.27675e33			1.20649
$566$		−	4.61870e31i		−	0.0427289i
$567$		−	9.98814e32i		−	0.904666i
$568$	−1.35459e33			−1.20123
$569$		−	1.44946e33i		−	1.25852i	−0.777196	−	0.629258i	$-0.783359\pi$
$569$		−	1.44946e33i		−	1.25852i	0.777196	−	0.629258i	$-0.216641\pi$
$570$	−2.95510e32			−0.251231
$571$		−	7.43370e32i		−	0.618830i	−0.950927	−	0.309415i	$-0.899867\pi$
$571$		−	7.43370e32i		−	0.618830i	0.950927	−	0.309415i	$-0.100133\pi$
$572$		−	7.40838e31i		−	0.0603908i
$573$			4.44527e32i			0.354848i
$574$		−	1.30943e33i		−	1.02363i
$575$	2.67241e32	+	3.65786e32i	0.204592	+	0.280035i
$576$	1.57751e32			0.118278
$577$	2.57420e33			1.89031			0.945153	−	0.326629i	$-0.105913\pi$
$577$	2.57420e33			1.89031			0.945153	+	0.326629i	$0.105913\pi$
$578$	−1.01193e33			−0.727806
$579$	−2.89495e31			−0.0203938
$580$			4.14255e32i			0.285846i
$581$	2.76876e33			1.87142
$582$		−	1.32974e32i		−	0.0880419i
$583$	−2.78066e32			−0.180354
$584$	−9.97231e32			−0.633639
$585$		−	7.10323e31i		−	0.0442166i
$586$		−	9.27548e32i		−	0.565673i
$587$	−2.78661e33			−1.66502			−0.832511	−	0.554009i	$-0.813097\pi$
$587$	−2.78661e33			−1.66502			−0.832511	+	0.554009i	$0.813097\pi$
$588$	2.80926e32			0.164461
$589$		−	1.75422e33i		−	1.00624i
$590$		−	5.18551e32i		−	0.291452i
$591$	3.72151e32			0.204960
$592$			3.75717e33i			2.02769i
$593$	−3.04043e33			−1.60797			−0.803986	−	0.594648i	$-0.797291\pi$
$593$	−3.04043e33			−1.60797			−0.803986	+	0.594648i	$0.797291\pi$
$594$		−	8.10835e32i		−	0.420238i
$595$			1.38946e33i			0.705735i
$596$			2.29789e33i			1.14386i
$597$			5.89952e32i			0.287822i
$598$	1.30078e32	+	1.78043e32i	0.0621997	+	0.0851357i
$599$	1.49194e33			0.699244			0.349622	−	0.936891i	$-0.386310\pi$
$599$	1.49194e33			0.699244			0.349622	+	0.936891i	$0.386310\pi$
$600$	4.33716e32			0.199246
$601$	1.01941e33			0.459045			0.229522	−	0.973303i	$-0.426284\pi$
$601$	1.01941e33			0.459045			0.229522	+	0.973303i	$0.426284\pi$
$602$	−4.42806e33			−1.95459
$603$			3.70312e33i			1.60236i
$604$	9.02687e33			3.82907
$605$		−	1.52736e33i		−	0.635151i
$606$	5.74315e31			0.0234142
$607$	−3.36394e33			−1.34457			−0.672286	−	0.740291i	$-0.734688\pi$
$607$	−3.36394e33			−1.34457			−0.672286	+	0.740291i	$0.734688\pi$
$608$			1.49201e33i			0.584693i
$609$		−	1.22917e32i		−	0.0472285i
$610$	−5.83276e33			−2.19743
$611$	−7.74056e31			−0.0285942
$612$			4.40479e33i			1.59554i
$613$			1.70666e33i			0.606206i	0.952958	+	0.303103i	$0.0980226\pi$
$613$			1.70666e33i			0.606206i	−0.952958	+	0.303103i	$0.901977\pi$
$614$	4.46382e33			1.55484
$615$		−	3.12127e32i		−	0.106618i
$616$	−3.43064e33			−1.14923
$617$			2.86514e33i			0.941290i	0.882323	+	0.470645i	$0.155979\pi$
$617$			2.86514e33i			0.941290i	−0.882323	+	0.470645i	$0.844021\pi$
$618$		−	1.62529e33i		−	0.523682i
$619$		−	5.95866e33i		−	1.88305i	−0.336948	−	0.941523i	$-0.609395\pi$
$619$		−	5.95866e33i		−	1.88305i	0.336948	−	0.941523i	$-0.390605\pi$
$620$		−	8.81805e33i		−	2.73320i
$621$	9.81786e32	+	1.34382e33i	0.298482	+	0.408546i
$622$	6.57386e32			0.196036
$623$	1.17640e33			0.344112
$624$	9.19518e31			0.0263843
$625$	−1.89328e33			−0.532912
$626$		−	9.91868e33i		−	2.73881i
$627$	−2.95830e32			−0.0801368
$628$		−	6.03869e33i		−	1.60482i
$629$	3.49740e33			0.911881
$630$	−5.98157e33			−1.53013
$631$			5.33244e33i			1.33836i	0.743099	+	0.669182i	$0.233355\pi$
$631$			5.33244e33i			1.33836i	−0.743099	+	0.669182i	$0.766645\pi$
$632$			3.87196e33i			0.953511i
$633$	−2.11604e33			−0.511304
$634$	1.29269e34			3.06495
$635$		−	2.47784e33i		−	0.576486i
$636$		−	9.93670e32i		−	0.226859i
$637$	7.40837e31			0.0165978
$638$			6.01358e32i			0.132216i
$639$	−2.36439e33			−0.510162
$640$		−	4.24562e33i		−	0.899045i
$641$			4.70437e33i			0.977698i	0.872368	+	0.488849i	$0.162583\pi$
$641$			4.70437e33i			0.977698i	−0.872368	+	0.488849i	$0.837417\pi$
$642$			2.91350e33i			0.594285i
$643$			9.63282e33i			1.92851i	0.264978	+	0.964254i	$0.414635\pi$
$643$			9.63282e33i			1.92851i	−0.264978	+	0.964254i	$0.585365\pi$
$644$	1.03393e34	−	7.55382e33i	2.03170	−	1.48435i
$645$	−1.05551e33			−0.203584
$646$	4.83251e33			0.914919
$647$	4.02928e33			0.748817			0.374408	−	0.927264i	$-0.377846\pi$
$647$	4.02928e33			0.748817			0.374408	+	0.927264i	$0.377846\pi$
$648$	−9.60266e33			−1.75183
$649$		−	5.19113e32i		−	0.0929663i
$650$	2.07991e32			0.0365666
$651$			2.61648e33i			0.451591i
$652$	3.74088e33			0.633874
$653$	−8.10068e33			−1.34761			−0.673804	−	0.738910i	$-0.735341\pi$
$653$	−8.10068e33			−1.34761			−0.673804	+	0.738910i	$0.735341\pi$
$654$			1.29997e32i			0.0212325i
$655$			6.45669e33i			1.03541i
$656$	−5.48336e33			−0.863378
$657$	−1.74063e33			−0.269105
$658$			6.51826e33i			0.989511i
$659$		−	9.21730e33i		−	1.37398i	−0.726669	−	0.686988i	$-0.758933\pi$
$659$		−	9.21730e33i		−	1.37398i	0.726669	−	0.686988i	$-0.241067\pi$
$660$	−1.48706e33			−0.217672
$661$		−	7.08811e33i		−	1.01886i	−0.860513	−	0.509428i	$-0.829857\pi$
$661$		−	7.08811e33i		−	1.01886i	0.860513	−	0.509428i	$-0.170143\pi$
$662$	−4.03146e33			−0.569070
$663$		−	8.55944e31i		−	0.0118654i
$664$		−	2.66191e34i		−	3.62390i
$665$			4.52552e33i			0.605074i
$666$			1.50562e34i			1.97708i
$667$	−7.28143e32	−	9.96645e32i	−0.0939089	−	0.128538i
$668$	2.69480e34			3.41357
$669$	−3.51054e33			−0.436778
$670$	2.04222e34			2.49578
$671$	−5.83908e33			−0.700929
$672$		−	2.22537e33i		−	0.262404i
$673$	−5.53511e33			−0.641129			−0.320565	−	0.947227i	$-0.603873\pi$
$673$	−5.53511e33			−0.641129			−0.320565	+	0.947227i	$0.603873\pi$
$674$		−	2.00958e34i		−	2.28658i
$675$	1.56985e33			0.175474
$676$	−2.01633e34			−2.21412
$677$		−	4.11028e33i		−	0.443411i	−0.975114	−	0.221706i	$-0.928838\pi$
$677$		−	4.11028e33i		−	0.443411i	0.975114	−	0.221706i	$-0.0711624\pi$
$678$			6.62314e33i			0.701952i
$679$	−2.03639e33			−0.212043
$680$	1.33583e34			1.36661
$681$		−	1.22976e33i		−	0.123611i
$682$		−	1.28008e34i		−	1.26423i
$683$	−1.39662e34			−1.35529			−0.677644	−	0.735391i	$-0.736999\pi$
$683$	−1.39662e34			−1.35529			−0.677644	+	0.735391i	$0.736999\pi$
$684$			1.43466e34i			1.36796i
$685$	5.43439e33			0.509171
$686$			1.58403e34i			1.45839i
$687$			4.99677e33i			0.452073i
$688$			1.85429e34i			1.64860i
$689$		−	2.62044e32i		−	0.0228951i
$690$	3.57382e33	−	2.61101e33i	0.306862	−	0.224192i
$691$	6.62281e33			0.558864			0.279432	−	0.960166i	$-0.409854\pi$
$691$	6.62281e33			0.558864			0.279432	+	0.960166i	$0.409854\pi$
$692$	2.26708e33			0.188015
$693$	−5.98805e33			−0.488075
$694$	−5.89087e33			−0.471918
$695$			7.57584e33i			0.596504i
$696$	−1.18173e33			−0.0914551
$697$			5.10425e33i			0.388274i
$698$	−3.00678e34			−2.24821
$699$	−1.98594e33			−0.145962
$700$		−	1.20784e34i		−	0.872635i
$701$			1.39821e34i			0.993015i	0.868032	+	0.496507i	$0.165384\pi$
$701$			1.39821e34i			0.993015i	−0.868032	+	0.496507i	$0.834616\pi$
$702$	7.64114e32			0.0533473
$703$	1.13912e34			0.781816
$704$		−	8.70922e32i		−	0.0587634i
$705$			1.55374e33i			0.103065i
$706$	−3.16947e34			−2.06695
$707$		−	8.79521e32i		−	0.0563914i
$708$	1.85505e33			0.116938
$709$		−	6.33399e33i		−	0.392575i	−0.980546	−	0.196288i	$-0.937111\pi$
$709$		−	6.33399e33i		−	0.392575i	0.980546	−	0.196288i	$-0.0628886\pi$
$710$			1.30393e34i			0.794609i
$711$			6.75836e33i			0.404954i
$712$		−	1.13100e34i		−	0.666351i
$713$	1.54996e34	+	2.12151e34i	0.897940	+	1.22905i
$714$	−7.20783e33			−0.410607
$715$	−3.92158e32			−0.0219679
$716$	−1.18919e34			−0.655082
$717$	−3.87701e33			−0.210023
$718$			6.53729e34i			3.48261i
$719$	1.00625e34			0.527183			0.263591	−	0.964634i	$-0.415093\pi$
$719$	1.00625e34			0.527183			0.263591	+	0.964634i	$0.415093\pi$
$720$			2.50483e34i			1.29059i
$721$	−2.48901e34			−1.26125
$722$	−2.02762e34			−1.01051
$723$			3.92503e33i			0.192390i
$724$		−	3.61770e34i		−	1.74409i
$725$	−1.16428e33			−0.0552081
$726$	7.92320e33			0.369540
$727$			9.85886e33i			0.452287i	0.974094	+	0.226143i	$0.0726118\pi$
$727$			9.85886e33i			0.452287i	−0.974094	+	0.226143i	$0.927388\pi$
$728$		−	3.23296e33i		−	0.145889i
$729$	−1.37750e34			−0.611450
$730$			9.59935e33i			0.419148i
$731$	1.72608e34			0.741401
$732$		−	2.08660e34i		−	0.881668i
$733$			1.77623e34i			0.738330i	0.929364	+	0.369165i	$0.120356\pi$
$733$			1.77623e34i			0.738330i	−0.929364	+	0.369165i	$0.879644\pi$
$734$			2.12072e34i			0.867221i
$735$		−	1.48706e33i		−	0.0598248i
$736$	−1.31828e34	−	1.80439e34i	−0.521763	−	0.714163i
$737$	2.04444e34			0.796093
$738$	−2.19736e34			−0.841831
$739$	−2.24280e34			−0.845390			−0.422695	−	0.906272i	$-0.638916\pi$
$739$	−2.24280e34			−0.845390			−0.422695	+	0.906272i	$0.638916\pi$
$740$	5.72606e34			2.12361
$741$		−	2.78784e32i		−	0.0101730i
$742$	−2.20665e34			−0.792293
$743$		−	2.48588e34i		−	0.878242i	−0.898428	−	0.439121i	$-0.855290\pi$
$743$		−	2.48588e34i		−	0.878242i	0.898428	−	0.439121i	$-0.144710\pi$
$744$	2.51550e34			0.874477
$745$	1.21637e34			0.416094
$746$		−	3.27203e34i		−	1.10141i
$747$		−	4.64626e34i		−	1.53906i
$748$	2.43182e34			0.792705
$749$	4.46181e34			1.43130
$750$		−	1.62131e34i		−	0.511835i
$751$		−	1.94374e34i		−	0.603891i	−0.953325	−	0.301946i	$-0.902364\pi$
$751$		−	1.94374e34i		−	0.603891i	0.953325	−	0.301946i	$-0.0976361\pi$
$752$	2.72957e34			0.834604
$753$		−	1.10337e34i		−	0.332033i
$754$	−5.66706e32			−0.0167843
$755$		−	4.77832e34i		−	1.39287i
$756$		−	4.43734e34i		−	1.27310i
$757$		−	6.30901e34i		−	1.78160i	−0.454395	−	0.890800i	$-0.650145\pi$
$757$		−	6.30901e34i		−	1.78160i	0.454395	−	0.890800i	$-0.349855\pi$
$758$		−	3.96725e34i		−	1.10270i
$759$	3.57769e33	−	2.61384e33i	0.0978818	−	0.0715119i
$760$	4.35086e34			1.17169
$761$	9.12067e33			0.241775			0.120887	−	0.992666i	$-0.461426\pi$
$761$	9.12067e33			0.241775			0.120887	+	0.992666i	$0.461426\pi$
$762$	1.28538e34			0.335408
$763$	1.99081e33			0.0511370
$764$		−	1.19017e35i		−	3.00946i
$765$	2.33165e34			0.580398
$766$			8.15600e34i			1.99863i
$767$	4.89201e32			0.0118017
$768$	2.06234e34			0.489808
$769$		−	1.55307e34i		−	0.363140i	−0.983378	−	0.181570i	$-0.941882\pi$
$769$		−	1.55307e34i		−	0.363140i	0.983378	−	0.181570i	$-0.0581180\pi$
$770$			3.30233e34i			0.760207i
$771$	8.79992e33			0.199446
$772$	7.75089e33			0.172959
$773$		−	6.62488e34i		−	1.45554i	−0.685821	−	0.727770i	$-0.740557\pi$
$773$		−	6.62488e34i		−	1.45554i	0.685821	−	0.727770i	$-0.259443\pi$
$774$			7.43072e34i			1.60746i
$775$	2.47836e34			0.527890
$776$			1.95780e34i			0.410608i
$777$	−1.69903e34			−0.350872
$778$			1.27756e35i			2.59792i
$779$			1.66247e34i			0.332893i
$780$		−	1.40138e33i		−	0.0276325i
$781$			1.30534e34i			0.253461i
$782$	−5.84431e34	+	4.26982e34i	−1.11751	+	0.816448i
$783$	−4.27733e33			−0.0805437
$784$	−2.61243e34			−0.484454
$785$	−3.19654e34			−0.583774
$786$	−3.34941e34			−0.602419
$787$			6.19053e34i			1.09656i	0.836296	+	0.548278i	$0.184717\pi$
$787$			6.19053e34i			1.09656i	−0.836296	+	0.548278i	$0.815283\pi$
$788$	−9.96392e34			−1.73826
$789$			1.29346e34i			0.222243i
$790$	3.72715e34			0.630741
$791$	1.01428e35			1.69060
$792$			5.75695e34i			0.945128i
$793$		−	5.50262e33i		−	0.0889798i
$794$	−5.70158e34			−0.908132
$795$	−5.25993e33			−0.0825229
$796$		−	1.57953e35i		−	2.44101i
$797$		−	4.99097e34i		−	0.759776i	−0.925033	−	0.379888i	$-0.875963\pi$
$797$		−	4.99097e34i		−	0.759776i	0.925033	−	0.379888i	$-0.124037\pi$
$798$	−2.34762e34			−0.352041
$799$		−	2.54085e34i		−	0.375334i
$800$	−2.10790e34			−0.306739
$801$		−	1.97412e34i		−	0.282998i
$802$			1.47108e34i			0.207752i
$803$			9.60976e33i			0.133698i
$804$			7.30580e34i			1.00137i
$805$	−3.99857e34	−	5.47304e34i	−0.539951	−	0.739057i
$806$	1.20632e34			0.160488
$807$	−1.79908e34			−0.235814
$808$	−8.45577e33			−0.109199
$809$	2.16610e34			0.275611			0.137806	−	0.990459i	$-0.455995\pi$
$809$	2.16610e34			0.275611			0.137806	+	0.990459i	$0.455995\pi$
$810$			9.24352e34i			1.15882i
$811$	−1.88524e34			−0.232871			−0.116436	−	0.993198i	$-0.537147\pi$
$811$	−1.88524e34			−0.232871			−0.116436	+	0.993198i	$0.537147\pi$
$812$			3.29096e34i			0.400544i
$813$	3.62503e34			0.434736
$814$	8.31230e34			0.982264
$815$		−	1.98021e34i		−	0.230580i
$816$			3.01834e34i			0.346327i
$817$	5.62192e34			0.635652
$818$	−2.04661e35			−2.28031
$819$		−	5.64301e33i		−	0.0619590i
$820$			8.35684e34i			0.904223i
$821$	1.63041e35			1.73852			0.869258	−	0.494359i	$-0.164597\pi$
$821$	1.63041e35			1.73852			0.869258	+	0.494359i	$0.164597\pi$
$822$			2.81910e34i			0.296243i
$823$	2.00527e34			0.207670			0.103835	−	0.994595i	$-0.466889\pi$
$823$	2.00527e34			0.207670			0.103835	+	0.994595i	$0.466889\pi$
$824$			2.39295e35i			2.44234i
$825$		−	4.17947e33i		−	0.0420411i
$826$		−	4.11952e34i		−	0.408400i
$827$		−	3.35667e34i		−	0.327977i	−0.986462	−	0.163988i	$-0.947564\pi$
$827$		−	3.35667e34i		−	0.327977i	0.986462	−	0.163988i	$-0.0524359\pi$
$828$	−1.26761e35	−	1.73503e35i	−1.22073	−	1.67088i
$829$	−1.38413e35			−1.31378			−0.656888	−	0.753988i	$-0.728128\pi$
$829$	−1.38413e35			−1.31378			−0.656888	+	0.753988i	$0.728128\pi$
$830$	−2.56235e35			−2.39718
$831$	2.84115e34			0.261988
$832$	8.20738e32			0.00745976
$833$			2.43181e34i			0.217866i
$834$	−3.92998e34			−0.347054
$835$		−	1.42648e35i		−	1.24173i
$836$	7.92052e34			0.679639
$837$	9.10495e34			0.770144
$838$			8.61016e34i			0.717932i
$839$			7.18650e34i			0.590709i	0.955388	+	0.295355i	$0.0954378\pi$
$839$			7.18650e34i			0.590709i	−0.955388	+	0.295355i	$0.904562\pi$
$840$	−6.48944e34			−0.525842
$841$	−1.22013e35			−0.974659
$842$			2.92756e35i			2.30548i
$843$		−	4.52326e33i		−	0.0351172i
$844$	5.66545e35			4.33636
$845$			1.06733e35i			0.805413i
$846$	1.09383e35			0.813776
$847$		−	1.21338e35i		−	0.890012i
$848$			9.24051e34i			0.668261i
$849$		−	8.74625e32i		−	0.00623634i
$850$			6.82734e34i			0.479982i
$851$	−1.37762e35	+	1.00648e35i	−0.954936	+	0.697671i
$852$	−4.66464e34			−0.318818
$853$	5.15530e34			0.347429			0.173714	−	0.984796i	$-0.444423\pi$
$853$	5.15530e34			0.347429			0.173714	+	0.984796i	$0.444423\pi$
$854$	−4.63371e35			−3.07918
$855$	7.59427e34			0.497614
$856$		−	4.28961e35i		−	2.77161i
$857$	−1.36367e34			−0.0868841			−0.0434421	−	0.999056i	$-0.513832\pi$
$857$	−1.36367e34			−0.0868841			−0.0434421	+	0.999056i	$0.513832\pi$
$858$		−	2.03433e33i		−	0.0127812i
$859$	8.13039e34			0.503725			0.251863	−	0.967763i	$-0.418957\pi$
$859$	8.13039e34			0.503725			0.251863	+	0.967763i	$0.418957\pi$
$860$	2.82600e35			1.72659
$861$		−	2.47963e34i		−	0.149399i
$862$		−	2.99388e35i		−	1.77888i
$863$	1.08801e35			0.637537			0.318768	−	0.947833i	$-0.396731\pi$
$863$	1.08801e35			0.637537			0.318768	+	0.947833i	$0.396731\pi$
$864$	−7.74396e34			−0.447505
$865$		−	1.20006e34i		−	0.0683930i
$866$			3.34796e35i			1.88177i
$867$	−1.91625e34			−0.106224
$868$		−	7.00531e35i		−	3.82993i
$869$	3.73119e34			0.201192
$870$			1.13753e34i			0.0604970i
$871$			1.92663e34i			0.101060i
$872$		−	1.91398e34i		−	0.0990237i
$873$			3.41727e34i			0.174385i
$874$	−1.90351e35	+	1.39070e35i	−0.958118	+	0.699995i
$875$	−2.48291e35			−1.23272
$876$	−3.43405e34			−0.168173
$877$	−3.20564e35			−1.54853			−0.774265	−	0.632862i	$-0.781880\pi$
$877$	−3.20564e35			−1.54853			−0.774265	+	0.632862i	$0.781880\pi$
$878$	−1.62923e35			−0.776333
$879$		−	1.75646e34i		−	0.0825606i
$880$	1.38288e35			0.641198
$881$			3.49445e35i			1.59833i	0.601109	+	0.799167i	$0.294726\pi$
$881$			3.49445e35i			1.59833i	−0.601109	+	0.799167i	$0.705274\pi$
$882$	−1.04689e35			−0.472364
$883$	3.58606e35			1.59620			0.798102	−	0.602522i	$-0.205837\pi$
$883$	3.58606e35			1.59620			0.798102	+	0.602522i	$0.205837\pi$
$884$			2.29169e34i			0.100630i
$885$		−	9.81960e33i		−	0.0425378i
$886$	4.57642e35			1.95578
$887$	−1.98079e35			−0.835133			−0.417567	−	0.908646i	$-0.637117\pi$
$887$	−1.98079e35			−0.835133			−0.417567	+	0.908646i	$0.637117\pi$
$888$			1.63346e35i			0.679441i
$889$		−	1.96847e35i		−	0.807807i
$890$	−1.08870e35			−0.440787
$891$			9.25355e34i			0.369637i
$892$	9.39907e35			3.70430
$893$		−	8.27565e34i		−	0.321799i
$894$			6.30995e34i			0.242089i
$895$			6.29492e34i			0.238294i
$896$		−	3.37284e35i		−	1.25980i
$897$	2.46323e33	+	3.37154e33i	0.00907811	+	0.0124256i
$898$	5.51341e35			2.00495
$899$	−6.75270e34			−0.242304
$900$	−2.02687e35			−0.717656
$901$	8.60163e34			0.300527
$902$			1.21313e35i			0.418243i
$903$	−8.38525e34			−0.285275
$904$		−	9.75140e35i		−	3.27375i
$905$	−1.91501e35			−0.634436
$906$	2.47876e35			0.810394
$907$		−	9.92296e34i		−	0.320150i	−0.987105	−	0.160075i	$-0.948826\pi$
$907$		−	9.92296e34i		−	0.320150i	0.987105	−	0.160075i	$-0.0511736\pi$
$908$			3.29254e35i			1.04834i
$909$	−1.47592e34			−0.0463764
$910$	−3.11205e34			−0.0965049
$911$			3.72506e35i			1.14002i	0.821637	+	0.570011i	$0.193061\pi$
$911$			3.72506e35i			1.14002i	−0.821637	+	0.570011i	$0.806939\pi$
$912$			9.83084e34i			0.296929i
$913$	−2.56513e35			−0.764645
$914$			3.70648e35i			1.09045i
$915$	−1.10453e35			−0.320718
$916$		−	1.33783e36i		−	3.83402i
$917$			5.12938e35i			1.45089i
$918$			2.50822e35i			0.700250i
$919$			3.40459e35i			0.938164i	0.883155	+	0.469082i	$0.155415\pi$
$919$			3.40459e35i			0.938164i	−0.883155	+	0.469082i	$0.844585\pi$
$920$	−5.26182e35	+	3.84425e35i	−1.43114	+	1.04558i
$921$	8.45297e34			0.226931
$922$	−8.90649e35			−2.36013
$923$	−1.23013e34			−0.0321758
$924$	−1.18137e35			−0.305015
$925$			1.60934e35i			0.410154i
$926$	−8.44724e35			−2.12512
$927$			4.17680e35i			1.03726i
$928$	5.74332e34			0.140795
$929$	−2.05508e35			−0.497324			−0.248662	−	0.968590i	$-0.579991\pi$
$929$	−2.05508e35			−0.497324			−0.248662	+	0.968590i	$0.579991\pi$
$930$		−	2.42142e35i		−	0.578461i
$931$			7.92051e34i			0.186791i
$932$	5.31712e35			1.23790
$933$	1.24487e34			0.0286117
$934$		−	6.39039e34i		−	0.144999i
$935$		−	1.28727e35i		−	0.288356i
$936$	−5.42523e34			−0.119980
$937$			6.63894e35i			1.44952i	0.689003	+	0.724759i	$0.258049\pi$
$937$			6.63894e35i			1.44952i	−0.689003	+	0.724759i	$0.741951\pi$
$938$	1.62240e36			3.49723
$939$		−	1.87826e35i		−	0.399733i
$940$		−	4.15996e35i		−	0.874088i
$941$		−	2.52330e35i		−	0.523471i	−0.965140	−	0.261736i	$-0.915705\pi$
$941$		−	2.52330e35i		−	0.523471i	0.965140	−	0.261736i	$-0.0842948\pi$
$942$		−	1.65821e35i		−	0.339648i
$943$	−1.46890e35	−	2.01055e35i	−0.297065	−	0.406607i
$944$	−1.72508e35			−0.344466
$945$	−2.34888e35			−0.463105
$946$	4.10239e35			0.798625
$947$	−3.21098e35			−0.617217			−0.308608	−	0.951189i	$-0.599863\pi$
$947$	−3.21098e35			−0.617217			−0.308608	+	0.951189i	$0.599863\pi$
$948$			1.33334e35i			0.253070i
$949$	−9.05603e33			−0.0169724
$950$			2.22369e35i			0.411520i
$951$	2.44792e35			0.447333
$952$	1.06123e36			1.91498
$953$		−	6.95989e35i		−	1.24019i	−0.784527	−	0.620094i	$-0.787094\pi$
$953$		−	6.95989e35i		−	1.24019i	0.784527	−	0.620094i	$-0.212906\pi$
$954$			3.70297e35i			0.651584i
$955$	−6.30010e35			−1.09473
$956$	1.03802e36			1.78120
$957$			1.13877e34i			0.0192971i
$958$		−	1.59169e36i		−	2.66363i
$959$	4.31724e35			0.713482
$960$		−	1.64745e34i		−	0.0268879i
$961$	8.17004e35			1.31687
$962$			7.83333e34i			0.124694i
$963$		−	7.48736e35i		−	1.17710i
$964$		−	1.05088e36i		−	1.63166i
$965$		−	4.10289e34i		−	0.0629161i
$966$	2.83915e35	−	2.07426e35i	0.429994	−	0.314151i
$967$	4.66305e35			0.697514			0.348757	−	0.937213i	$-0.386604\pi$
$967$	4.66305e35			0.697514			0.348757	+	0.937213i	$0.386604\pi$
$968$	−1.16655e36			−1.72345
$969$	9.15115e34			0.133533
$970$	1.88458e35			0.271615
$971$		−	7.72733e35i		−	1.10002i	−0.835159	−	0.550008i	$-0.814625\pi$
$971$		−	7.72733e35i		−	1.10002i	0.835159	−	0.550008i	$-0.185375\pi$
$972$	−1.13017e36			−1.58910
$973$			6.01847e35i			0.835857i
$974$	1.12387e36			1.54173
$975$	3.93865e33			0.00533693
$976$			1.94040e36i			2.59713i
$977$		−	9.50855e35i		−	1.25713i	−0.777758	−	0.628564i	$-0.783643\pi$
$977$		−	9.50855e35i		−	1.25713i	0.777758	−	0.628564i	$-0.216357\pi$
$978$	1.02724e35			0.134154
$979$	−1.08988e35			−0.140601
$980$			3.98144e35i			0.507373i
$981$		−	3.34078e34i		−	0.0420552i
$982$	−1.25377e36			−1.55912
$983$			1.08487e36i			1.33271i	0.745635	+	0.666355i	$0.232146\pi$
$983$			1.08487e36i			1.33271i	−0.745635	+	0.666355i	$0.767854\pi$
$984$	−2.38393e35			−0.289302
$985$			5.27434e35i			0.632315i
$986$		−	1.86023e35i		−	0.220314i
$987$			1.23434e35i			0.144420i
$988$			7.46412e34i			0.0862771i
$989$	−6.79899e35	+	4.96731e35i	−0.776406	+	0.567238i
$990$	5.54164e35			0.625196
$991$	−1.31036e36			−1.46052			−0.730260	−	0.683170i	$-0.760601\pi$
$991$	−1.31036e36			−1.46052			−0.730260	+	0.683170i	$0.760601\pi$
$992$	−1.22255e36			−1.34626
$993$	−7.63422e34			−0.0830565
$994$			1.03588e36i			1.11345i
$995$	−8.36114e35			−0.887950
$996$		−	9.16649e35i		−	0.961814i
$997$	1.26389e36			1.31029			0.655144	−	0.755504i	$-0.272608\pi$
$997$	1.26389e36			1.31029			0.655144	+	0.755504i	$0.272608\pi$
$998$	−1.94417e36			−1.99144
$999$			5.91236e35i			0.598378i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.25.b.c.22.3	✓	44
23.22	odd	2	inner	23.25.b.c.22.4	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.25.b.c.22.3	✓	44	1.1	even	1	trivial
23.25.b.c.22.4	yes	44	23.22	odd	2	inner

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:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 25 \)
Character orbit:	\([\chi]\)	\(=\)	23.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7352.04 q^{2} -139223. q^{3} +3.72753e7 q^{4} -1.97315e8i q^{5} +1.02357e9 q^{6} -1.56753e10i q^{7} -1.50703e11 q^{8} -2.63047e11 q^{9} +1.45067e12i q^{10} +1.45224e12i q^{11} -5.18958e12 q^{12} -1.36856e12 q^{13} +1.15245e14i q^{14} +2.74707e13i q^{15} +4.82599e14 q^{16} -4.49232e14i q^{17} +1.93393e15 q^{18} -1.46317e15i q^{19} -7.35497e15i q^{20} +2.18236e15i q^{21} -1.06769e16i q^{22} +(1.29280e16 + 1.76951e16i) q^{23} +2.09813e16 q^{24} +2.06715e16 q^{25} +1.00617e16 q^{26} +7.59428e16 q^{27} -5.84301e17i q^{28} -5.63231e16 q^{29} -2.01966e17i q^{30} +1.19892e18 q^{31} -1.01971e18 q^{32} -2.02185e17i q^{33} +3.30278e18i q^{34} -3.09296e18 q^{35} -9.80514e18 q^{36} +7.78529e18i q^{37} +1.07573e19i q^{38} +1.90535e17 q^{39} +2.97359e19i q^{40} -1.13622e19 q^{41} -1.60448e19i q^{42} +3.84229e19i q^{43} +5.41327e19i q^{44} +5.19030e19i q^{45} +(-9.50470e19 - 1.30095e20i) q^{46} +5.65599e19 q^{47} -6.71888e19 q^{48} -5.41326e19 q^{49} -1.51978e20 q^{50} +6.25434e19i q^{51} -5.10135e19 q^{52} +1.91474e20i q^{53} -5.58334e20 q^{54} +2.86548e20 q^{55} +2.36231e21i q^{56} +2.03706e20i q^{57} +4.14090e20 q^{58} -3.57457e20 q^{59} +1.02398e21i q^{60} +4.02074e21i q^{61} -8.81453e21 q^{62} +4.12332e21i q^{63} -5.99709e20 q^{64} +2.70037e20i q^{65} +1.48647e21i q^{66} -1.40778e22i q^{67} -1.67453e22i q^{68} +(-1.79987e21 - 2.46357e21i) q^{69} +2.27396e22 q^{70} +8.98848e21 q^{71} +3.96419e22 q^{72} +6.61720e21 q^{73} -5.72378e22i q^{74} -2.87795e21 q^{75} -5.45400e22i q^{76} +2.27642e22 q^{77} -1.40082e21 q^{78} -2.56926e22i q^{79} -9.52239e22i q^{80} +6.37191e22 q^{81} +8.35351e22 q^{82} +1.76633e23i q^{83} +8.13480e22i q^{84} -8.86402e22 q^{85} -2.82487e23i q^{86} +7.84146e21 q^{87} -2.18857e23i q^{88} +7.50483e22i q^{89} -3.81593e23i q^{90} +2.14525e22i q^{91} +(4.81895e23 + 6.59592e23i) q^{92} -1.66918e23 q^{93} -4.15831e23 q^{94} -2.88704e23 q^{95} +1.41967e23 q^{96} -1.29911e23i q^{97} +3.97986e23 q^{98} -3.82007e23i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4232 q^{2} - 434562 q^{3} + 317760360 q^{4} - 8460029520 q^{6} - 198307023760 q^{8} + 4220041988298 q^{9} - 67439597688792 q^{12} + 5771152551358 q^{13} + 18\!\cdots\!92 q^{16} + 18\!\cdots\!68 q^{18}+ \cdots - 20\!\cdots\!92 q^{98}+O(q^{100}) \) 44 * q - 4232 * q^2 - 434562 * q^3 + 317760360 * q^4 - 8460029520 * q^6 - 198307023760 * q^8 + 4220041988298 * q^9 - 67439597688792 * q^12 + 5771152551358 * q^13 + 1801550792530192 * q^16 + 1842830558165568 * q^18 - 117204213263210644 * q^23 - 186493005437450784 * q^24 - 689533084157602612 * q^25 - 548650296251047576 * q^26 + 824037505793640630 * q^27 - 1583078054945709602 * q^29 - 77249813729122274 * q^31 - 2838716832733708512 * q^32 + 3418569899175948192 * q^35 + 71181245116739335248 * q^36 + 47106572467817156310 * q^39 - 9150679195666184738 * q^41 - 153049487518536134480 * q^46 + 22735751396127769438 * q^47 - 1317482931964495330032 * q^48 - 1842110772006482246836 * q^49 + 284936617115007894520 * q^50 + 207129175371045152568 * q^52 - 381774025126375304544 * q^54 - 5430040696094013021792 * q^55 - 2168901871328366220760 * q^58 + 16048018791948427185304 * q^59 + 14054450375064171442400 * q^62 + 20310379126091866412192 * q^64 + 24476879859710435470878 * q^69 - 118028399895318845638416 * q^70 - 68204988047209908888578 * q^71 - 106777935582637627887744 * q^72 - 7319375899831086678722 * q^73 + 12913295383070041819998 * q^75 + 111116199573084838704480 * q^77 - 422583593213386075404144 * q^78 - 331350288185757013618680 * q^81 + 1000764218021118649437320 * q^82 - 113385735487939722862944 * q^85 - 248828610584854760185290 * q^87 - 4238455111972177116774696 * q^92 + 4492289396741153284965270 * q^93 - 1602622056539190351258592 * q^94 - 1668692394291208047471936 * q^95 - 12092663288748189077973504 * q^96 - 2067304853876486831513192 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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