LMFDB - Embedded newform 690.3.c.a.91.25

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,3,Mod(91,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("690.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	690.c (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:	$18.8011382409$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.25
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949 q^{6} -7.54509i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949 q^{6} -7.54509i q^{7} +2.82843 q^{8} +3.00000 q^{9} -3.16228i q^{10} -8.65106i q^{11} +3.46410 q^{12} -19.9643 q^{13} -10.6704i q^{14} -3.87298i q^{15} +4.00000 q^{16} +9.27071i q^{17} +4.24264 q^{18} -35.9882i q^{19} -4.47214i q^{20} -13.0685i q^{21} -12.2344i q^{22} +(22.9761 - 1.04751i) q^{23} +4.89898 q^{24} -5.00000 q^{25} -28.2338 q^{26} +5.19615 q^{27} -15.0902i q^{28} -27.4794 q^{29} -5.47723i q^{30} +22.3027 q^{31} +5.65685 q^{32} -14.9841i q^{33} +13.1108i q^{34} -16.8713 q^{35} +6.00000 q^{36} +42.9329i q^{37} -50.8949i q^{38} -34.5792 q^{39} -6.32456i q^{40} -39.2309 q^{41} -18.4816i q^{42} +20.8570i q^{43} -17.3021i q^{44} -6.70820i q^{45} +(32.4932 - 1.48140i) q^{46} +66.5989 q^{47} +6.92820 q^{48} -7.92839 q^{49} -7.07107 q^{50} +16.0573i q^{51} -39.9286 q^{52} -64.5799i q^{53} +7.34847 q^{54} -19.3444 q^{55} -21.3407i q^{56} -62.3333i q^{57} -38.8617 q^{58} +90.0416 q^{59} -7.74597i q^{60} +10.7927i q^{61} +31.5407 q^{62} -22.6353i q^{63} +8.00000 q^{64} +44.6416i q^{65} -21.1907i q^{66} -58.7797i q^{67} +18.5414i q^{68} +(39.7958 - 1.81434i) q^{69} -23.8597 q^{70} -29.3496 q^{71} +8.48528 q^{72} +95.8335 q^{73} +60.7163i q^{74} -8.66025 q^{75} -71.9763i q^{76} -65.2730 q^{77} -48.9024 q^{78} +72.0229i q^{79} -8.94427i q^{80} +9.00000 q^{81} -55.4809 q^{82} -106.235i q^{83} -26.1370i q^{84} +20.7299 q^{85} +29.4963i q^{86} -47.5956 q^{87} -24.4689i q^{88} +80.0068i q^{89} -9.48683i q^{90} +150.633i q^{91} +(45.9523 - 2.09502i) q^{92} +38.6294 q^{93} +94.1850 q^{94} -80.4720 q^{95} +9.79796 q^{96} +103.236i q^{97} -11.2124 q^{98} -25.9532i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) $ 32 * q + 64 * q^4 + 96 * q^9	$ 32 q + 64 q^{4} + 96 q^{9} - 48 q^{13} + 128 q^{16} - 80 q^{23} - 160 q^{25} + 120 q^{29} + 248 q^{31} - 120 q^{35} + 192 q^{36} - 48 q^{39} + 72 q^{41} + 160 q^{46} + 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} + 120 q^{59} + 160 q^{62} + 256 q^{64} + 192 q^{69} + 104 q^{71} + 16 q^{73} + 240 q^{77} + 192 q^{78} + 288 q^{81} + 64 q^{82} - 120 q^{85} + 144 q^{87} - 160 q^{92} - 192 q^{93} + 96 q^{94} - 160 q^{95} + 64 q^{98}+O(q^{100}) $ 32 * q + 64 * q^4 + 96 * q^9 - 48 * q^13 + 128 * q^16 - 80 * q^23 - 160 * q^25 + 120 * q^29 + 248 * q^31 - 120 * q^35 + 192 * q^36 - 48 * q^39 + 72 * q^41 + 160 * q^46 + 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 + 120 * q^59 + 160 * q^62 + 256 * q^64 + 192 * q^69 + 104 * q^71 + 16 * q^73 + 240 * q^77 + 192 * q^78 + 288 * q^81 + 64 * q^82 - 120 * q^85 + 144 * q^87 - 160 * q^92 - 192 * q^93 + 96 * q^94 - 160 * q^95 + 64 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$511$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	2.00000			0.500000
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$	2.44949			0.408248
$7$		−	7.54509i		−	1.07787i	−0.842347	−	0.538935i	$-0.818827\pi$
$7$		−	7.54509i		−	1.07787i	0.842347	−	0.538935i	$-0.181173\pi$
$8$	2.82843			0.353553
$9$	3.00000			0.333333
$10$		−	3.16228i		−	0.316228i
$11$		−	8.65106i		−	0.786460i	−0.919440	−	0.393230i	$-0.871358\pi$
$11$		−	8.65106i		−	0.786460i	0.919440	−	0.393230i	$-0.128642\pi$
$12$	3.46410			0.288675
$13$	−19.9643			−1.53572			−0.767858	−	0.640620i	$-0.778678\pi$
$13$	−19.9643			−1.53572			−0.767858	+	0.640620i	$0.778678\pi$
$14$		−	10.6704i		−	0.762169i
$15$		−	3.87298i		−	0.258199i
$16$	4.00000			0.250000
$17$			9.27071i			0.545336i	0.962108	+	0.272668i	$0.0879061\pi$
$17$			9.27071i			0.545336i	−0.962108	+	0.272668i	$0.912094\pi$
$18$	4.24264			0.235702
$19$		−	35.9882i		−	1.89411i	−0.321066	−	0.947057i	$-0.604041\pi$
$19$		−	35.9882i		−	1.89411i	0.321066	−	0.947057i	$-0.395959\pi$
$20$		−	4.47214i		−	0.223607i
$21$		−	13.0685i		−	0.622309i
$22$		−	12.2344i		−	0.556111i
$23$	22.9761	−	1.04751i	0.998962	−	0.0455439i
$23$	22.9761	−	1.04751i	0.998962	−	0.0455439i
$24$	4.89898			0.204124
$25$	−5.00000			−0.200000
$26$	−28.2338			−1.08592
$27$	5.19615			0.192450
$28$		−	15.0902i		−	0.538935i
$29$	−27.4794			−0.947564			−0.473782	−	0.880642i	$-0.657112\pi$
$29$	−27.4794			−0.947564			−0.473782	+	0.880642i	$0.657112\pi$
$30$		−	5.47723i		−	0.182574i
$31$	22.3027			0.719441			0.359720	−	0.933060i	$-0.382872\pi$
$31$	22.3027			0.719441			0.359720	+	0.933060i	$0.382872\pi$
$32$	5.65685			0.176777
$33$		−	14.9841i		−	0.454063i
$34$			13.1108i			0.385611i
$35$	−16.8713			−0.482038
$36$	6.00000			0.166667
$37$			42.9329i			1.16035i	0.814493	+	0.580174i	$0.197015\pi$
$37$			42.9329i			1.16035i	−0.814493	+	0.580174i	$0.802985\pi$
$38$		−	50.8949i		−	1.33934i
$39$	−34.5792			−0.886647
$40$		−	6.32456i		−	0.158114i
$41$	−39.2309			−0.956851			−0.478426	−	0.878128i	$-0.658792\pi$
$41$	−39.2309			−0.956851			−0.478426	+	0.878128i	$0.658792\pi$
$42$		−	18.4816i		−	0.440039i
$43$			20.8570i			0.485047i	0.970146	+	0.242523i	$0.0779751\pi$
$43$			20.8570i			0.485047i	−0.970146	+	0.242523i	$0.922025\pi$
$44$		−	17.3021i		−	0.393230i
$45$		−	6.70820i		−	0.149071i
$46$	32.4932	−	1.48140i	0.706373	−	0.0322044i
$47$	66.5989			1.41700			0.708499	−	0.705712i	$-0.249373\pi$
$47$	66.5989			1.41700			0.708499	+	0.705712i	$0.249373\pi$
$48$	6.92820			0.144338
$49$	−7.92839			−0.161804
$50$	−7.07107			−0.141421
$51$			16.0573i			0.314850i
$52$	−39.9286			−0.767858
$53$		−	64.5799i		−	1.21849i	−0.792982	−	0.609245i	$-0.791473\pi$
$53$		−	64.5799i		−	1.21849i	0.792982	−	0.609245i	$-0.208527\pi$
$54$	7.34847			0.136083
$55$	−19.3444			−0.351716
$56$		−	21.3407i		−	0.381085i
$57$		−	62.3333i		−	1.09357i
$58$	−38.8617			−0.670029
$59$	90.0416			1.52613			0.763064	−	0.646323i	$-0.223694\pi$
$59$	90.0416			1.52613			0.763064	+	0.646323i	$0.223694\pi$
$60$		−	7.74597i		−	0.129099i
$61$			10.7927i			0.176930i	0.996079	+	0.0884650i	$0.0281961\pi$
$61$			10.7927i			0.176930i	−0.996079	+	0.0884650i	$0.971804\pi$
$62$	31.5407			0.508722
$63$		−	22.6353i		−	0.359290i
$64$	8.00000			0.125000
$65$			44.6416i			0.686793i
$66$		−	21.1907i		−	0.321071i
$67$		−	58.7797i		−	0.877309i	−0.898656	−	0.438655i	$-0.855455\pi$
$67$		−	58.7797i		−	0.877309i	0.898656	−	0.438655i	$-0.144545\pi$
$68$			18.5414i			0.272668i
$69$	39.7958	−	1.81434i	0.576751	−	0.0262948i
$70$	−23.8597			−0.340852
$71$	−29.3496			−0.413375			−0.206688	−	0.978407i	$-0.566268\pi$
$71$	−29.3496			−0.413375			−0.206688	+	0.978407i	$0.566268\pi$
$72$	8.48528			0.117851
$73$	95.8335			1.31279			0.656394	−	0.754418i	$-0.272081\pi$
$73$	95.8335			1.31279			0.656394	+	0.754418i	$0.272081\pi$
$74$			60.7163i			0.820490i
$75$	−8.66025			−0.115470
$76$		−	71.9763i		−	0.947057i
$77$	−65.2730			−0.847701
$78$	−48.9024			−0.626954
$79$			72.0229i			0.911682i	0.890061	+	0.455841i	$0.150661\pi$
$79$			72.0229i			0.911682i	−0.890061	+	0.455841i	$0.849339\pi$
$80$		−	8.94427i		−	0.111803i
$81$	9.00000			0.111111
$82$	−55.4809			−0.676596
$83$		−	106.235i		−	1.27994i	−0.768401	−	0.639969i	$-0.778947\pi$
$83$		−	106.235i		−	1.27994i	0.768401	−	0.639969i	$-0.221053\pi$
$84$		−	26.1370i		−	0.311154i
$85$	20.7299			0.243882
$86$			29.4963i			0.342980i
$87$	−47.5956			−0.547076
$88$		−	24.4689i		−	0.278056i
$89$			80.0068i			0.898952i	0.893292	+	0.449476i	$0.148389\pi$
$89$			80.0068i			0.898952i	−0.893292	+	0.449476i	$0.851611\pi$
$90$		−	9.48683i		−	0.105409i
$91$			150.633i			1.65530i
$92$	45.9523	−	2.09502i	0.499481	−	0.0227720i
$93$	38.6294			0.415369
$94$	94.1850			1.00197
$95$	−80.4720			−0.847073
$96$	9.79796			0.102062
$97$			103.236i			1.06428i	0.846655	+	0.532142i	$0.178613\pi$
$97$			103.236i			1.06428i	−0.846655	+	0.532142i	$0.821387\pi$
$98$	−11.2124			−0.114413
$99$		−	25.9532i		−	0.262153i
$100$	−10.0000			−0.100000
$101$	85.8682			0.850180			0.425090	−	0.905151i	$-0.360242\pi$
$101$	85.8682			0.850180			0.425090	+	0.905151i	$0.360242\pi$
$102$			22.7085i			0.222633i
$103$			178.631i			1.73429i	0.498060	+	0.867143i	$0.334046\pi$
$103$			178.631i			1.73429i	−0.498060	+	0.867143i	$0.665954\pi$
$104$	−56.4676			−0.542958
$105$	−29.2220			−0.278305
$106$		−	91.3298i		−	0.861602i
$107$		−	21.2445i		−	0.198547i	−0.995060	−	0.0992734i	$-0.968348\pi$
$107$		−	21.2445i		−	0.198547i	0.995060	−	0.0992734i	$-0.0316519\pi$
$108$	10.3923			0.0962250
$109$		−	187.493i		−	1.72012i	−0.510195	−	0.860059i	$-0.670427\pi$
$109$		−	187.493i		−	1.72012i	0.510195	−	0.860059i	$-0.329573\pi$
$110$	−27.3570			−0.248700
$111$			74.3619i			0.669927i
$112$		−	30.1804i		−	0.269468i
$113$		−	5.20875i		−	0.0460952i	−0.999734	−	0.0230476i	$-0.992663\pi$
$113$		−	5.20875i		−	0.0460952i	0.999734	−	0.0230476i	$-0.00733692\pi$
$114$		−	88.1526i		−	0.773269i
$115$	−2.34231	−	51.3762i	−0.0203679	−	0.446750i
$116$	−54.9587			−0.473782
$117$	−59.8930			−0.511906
$118$	127.338			1.07914
$119$	69.9484			0.587801
$120$		−	10.9545i		−	0.0912871i
$121$	46.1592			0.381481
$122$			15.2632i			0.125108i
$123$	−67.9499			−0.552438
$124$	44.6053			0.359720
$125$			11.1803i			0.0894427i
$126$		−	32.0111i		−	0.254056i
$127$	213.909			1.68432			0.842161	−	0.539227i	$-0.181283\pi$
$127$	213.909			1.68432			0.842161	+	0.539227i	$0.181283\pi$
$128$	11.3137			0.0883883
$129$			36.1254i			0.280042i
$130$			63.1327i			0.485636i
$131$	−209.839			−1.60183			−0.800913	−	0.598780i	$-0.795652\pi$
$131$	−209.839			−1.60183			−0.800913	+	0.598780i	$0.795652\pi$
$132$		−	29.9681i		−	0.227031i
$133$	−271.534			−2.04161
$134$		−	83.1271i		−	0.620351i
$135$		−	11.6190i		−	0.0860663i
$136$			26.2215i			0.192805i
$137$		−	37.7195i		−	0.275325i	−0.990479	−	0.137662i	$-0.956041\pi$
$137$		−	37.7195i		−	0.275325i	0.990479	−	0.137662i	$-0.0439589\pi$
$138$	56.2798	−	2.56587i	0.407825	−	0.0185932i
$139$	−8.08981			−0.0582001			−0.0291000	−	0.999577i	$-0.509264\pi$
$139$	−8.08981			−0.0582001			−0.0291000	+	0.999577i	$0.509264\pi$
$140$	−33.7427			−0.241019
$141$	115.353			0.818104
$142$	−41.5067			−0.292300
$143$			172.712i			1.20778i
$144$	12.0000			0.0833333
$145$			61.4457i			0.423764i
$146$	135.529			0.928281
$147$	−13.7324			−0.0934175
$148$			85.8658i			0.580174i
$149$			112.471i			0.754837i	0.926043	+	0.377419i	$0.123188\pi$
$149$			112.471i			0.754837i	−0.926043	+	0.377419i	$0.876812\pi$
$150$	−12.2474			−0.0816497
$151$	−214.567			−1.42098			−0.710488	−	0.703709i	$-0.751526\pi$
$151$	−214.567			−1.42098			−0.710488	+	0.703709i	$0.751526\pi$
$152$		−	101.790i		−	0.669670i
$153$			27.8121i			0.181779i
$154$	−92.3100			−0.599415
$155$		−	49.8703i		−	0.321744i
$156$	−69.1584			−0.443323
$157$			57.5819i			0.366764i	0.983042	+	0.183382i	$0.0587045\pi$
$157$			57.5819i			0.366764i	−0.983042	+	0.183382i	$0.941295\pi$
$158$			101.856i			0.644657i
$159$		−	111.856i		−	0.703495i
$160$		−	12.6491i		−	0.0790569i
$161$	−7.90356	−	173.357i	−0.0490905	−	1.07675i
$162$	12.7279			0.0785674
$163$	−208.382			−1.27842			−0.639208	−	0.769034i	$-0.720738\pi$
$163$	−208.382			−1.27842			−0.639208	+	0.769034i	$0.720738\pi$
$164$	−78.4618			−0.478426
$165$	−33.5054			−0.203063
$166$		−	150.239i		−	0.905053i
$167$	253.727			1.51932			0.759661	−	0.650319i	$-0.225365\pi$
$167$	253.727			1.51932			0.759661	+	0.650319i	$0.225365\pi$
$168$		−	36.9632i		−	0.220019i
$169$	229.574			1.35843
$170$	29.3166			0.172450
$171$		−	107.964i		−	0.631371i
$172$			41.7140i			0.242523i
$173$	−90.9046			−0.525460			−0.262730	−	0.964869i	$-0.584623\pi$
$173$	−90.9046			−0.525460			−0.262730	+	0.964869i	$0.584623\pi$
$174$	−67.3104			−0.386841
$175$			37.7255i			0.215574i
$176$		−	34.6042i		−	0.196615i
$177$	155.957			0.881111
$178$			113.147i			0.635655i
$179$	27.7233			0.154879			0.0774393	−	0.996997i	$-0.475326\pi$
$179$	27.7233			0.154879			0.0774393	+	0.996997i	$0.475326\pi$
$180$		−	13.4164i		−	0.0745356i
$181$			321.337i			1.77534i	0.460479	+	0.887671i	$0.347678\pi$
$181$			321.337i			1.77534i	−0.460479	+	0.887671i	$0.652322\pi$
$182$			213.027i			1.17048i
$183$			18.6936i			0.102151i
$184$	64.9863	−	2.96281i	0.353187	−	0.0161022i
$185$	96.0008			0.518923
$186$	54.6302			0.293710
$187$	80.2015			0.428885
$188$	133.198			0.708499
$189$		−	39.2054i		−	0.207436i
$190$	−113.805			−0.598971
$191$		−	32.2385i		−	0.168788i	−0.996432	−	0.0843940i	$-0.973105\pi$
$191$		−	32.2385i		−	0.168788i	0.996432	−	0.0843940i	$-0.0268954\pi$
$192$	13.8564			0.0721688
$193$	−30.8524			−0.159857			−0.0799285	−	0.996801i	$-0.525469\pi$
$193$	−30.8524			−0.159857			−0.0799285	+	0.996801i	$0.525469\pi$
$194$			145.997i			0.752562i
$195$			77.3215i			0.396520i
$196$	−15.8568			−0.0809019
$197$	284.323			1.44326			0.721632	−	0.692277i	$-0.243392\pi$
$197$	284.323			1.44326			0.721632	+	0.692277i	$0.243392\pi$
$198$		−	36.7033i		−	0.185370i
$199$			296.034i			1.48761i	0.668397	+	0.743805i	$0.266981\pi$
$199$			296.034i			1.48761i	−0.668397	+	0.743805i	$0.733019\pi$
$200$	−14.1421			−0.0707107
$201$		−	101.809i		−	0.506515i
$202$	121.436			0.601168
$203$			207.334i			1.02135i
$204$			32.1147i			0.157425i
$205$			87.7230i			0.427917i
$206$			252.623i			1.22633i
$207$	68.9284	−	3.14253i	0.332987	−	0.0151813i
$208$	−79.8573			−0.383929
$209$	−311.336			−1.48964
$210$	−41.3262			−0.196791
$211$	−232.465			−1.10173			−0.550866	−	0.834594i	$-0.685703\pi$
$211$	−232.465			−1.10173			−0.550866	+	0.834594i	$0.685703\pi$
$212$		−	129.160i		−	0.609245i
$213$	−50.8351			−0.238662
$214$		−	30.0443i		−	0.140394i
$215$	46.6377			0.216920
$216$	14.6969			0.0680414
$217$		−	168.276i		−	0.775464i
$218$		−	265.155i		−	1.21631i
$219$	165.988			0.757938
$220$	−38.6887			−0.175858
$221$		−	185.084i		−	0.837482i
$222$			105.164i			0.473710i
$223$	−196.516			−0.881237			−0.440618	−	0.897694i	$-0.645241\pi$
$223$	−196.516			−0.881237			−0.440618	+	0.897694i	$0.645241\pi$
$224$		−	42.6815i		−	0.190542i
$225$	−15.0000			−0.0666667
$226$		−	7.36629i		−	0.0325942i
$227$		−	302.870i		−	1.33423i	−0.744955	−	0.667115i	$-0.767529\pi$
$227$		−	302.870i		−	1.33423i	0.744955	−	0.667115i	$-0.232471\pi$
$228$		−	124.667i		−	0.546783i
$229$			416.009i			1.81663i	0.418285	+	0.908316i	$0.362631\pi$
$229$			416.009i			1.81663i	−0.418285	+	0.908316i	$0.637369\pi$
$230$	−3.31252	−	72.6569i	−0.0144023	−	0.315900i
$231$	−113.056			−0.489421
$232$	−77.7234			−0.335014
$233$	−98.6862			−0.423546			−0.211773	−	0.977319i	$-0.567924\pi$
$233$	−98.6862			−0.423546			−0.211773	+	0.977319i	$0.567924\pi$
$234$	−84.7014			−0.361972
$235$		−	148.920i		−	0.633701i
$236$	180.083			0.763064
$237$			124.747i			0.526360i
$238$	98.9219			0.415638
$239$	−24.2213			−0.101344			−0.0506722	−	0.998715i	$-0.516136\pi$
$239$	−24.2213			−0.101344			−0.0506722	+	0.998715i	$0.516136\pi$
$240$		−	15.4919i		−	0.0645497i
$241$			44.0207i			0.182658i	0.995821	+	0.0913292i	$0.0291115\pi$
$241$			44.0207i			0.182658i	−0.995821	+	0.0913292i	$0.970888\pi$
$242$	65.2790			0.269748
$243$	15.5885			0.0641500
$244$			21.5855i			0.0884650i
$245$			17.7284i			0.0723609i
$246$	−96.0957			−0.390633
$247$			718.479i			2.90882i
$248$	63.0815			0.254361
$249$		−	184.004i		−	0.738973i
$250$			15.8114i			0.0632456i
$251$			18.0472i			0.0719013i	0.999354	+	0.0359507i	$0.0114459\pi$
$251$			18.0472i			0.0719013i	−0.999354	+	0.0359507i	$0.988554\pi$
$252$		−	45.2705i		−	0.179645i
$253$	−9.06208	−	198.768i	−0.0358185	−	0.785644i
$254$	302.513			1.19100
$255$	35.9053			0.140805
$256$	16.0000			0.0625000
$257$	−348.964			−1.35784			−0.678919	−	0.734213i	$-0.737551\pi$
$257$	−348.964			−1.35784			−0.678919	+	0.734213i	$0.737551\pi$
$258$			51.0890i			0.198020i
$259$	323.932			1.25070
$260$			89.2832i			0.343397i
$261$	−82.4381			−0.315855
$262$	−296.758			−1.13266
$263$		−	138.601i		−	0.526999i	−0.964660	−	0.263500i	$-0.915123\pi$
$263$		−	138.601i		−	0.526999i	0.964660	−	0.263500i	$-0.0848768\pi$
$264$		−	42.3814i		−	0.160535i
$265$	−144.405			−0.544925
$266$	−384.007			−1.44364
$267$			138.576i			0.519010i
$268$		−	117.559i		−	0.438655i
$269$	175.666			0.653034			0.326517	−	0.945191i	$-0.394125\pi$
$269$	175.666			0.653034			0.326517	+	0.945191i	$0.394125\pi$
$270$		−	16.4317i		−	0.0608581i
$271$	245.596			0.906259			0.453130	−	0.891445i	$-0.350307\pi$
$271$	245.596			0.906259			0.453130	+	0.891445i	$0.350307\pi$
$272$			37.0829i			0.136334i
$273$			260.903i			0.955690i
$274$		−	53.3434i		−	0.194684i
$275$			43.2553i			0.157292i
$276$	79.5917	−	3.62868i	0.288376	−	0.0131474i
$277$	319.260			1.15256			0.576281	−	0.817251i	$-0.304503\pi$
$277$	319.260			1.15256			0.576281	+	0.817251i	$0.304503\pi$
$278$	−11.4407			−0.0411537
$279$	66.9080			0.239814
$280$	−47.7193			−0.170426
$281$		−	84.2955i		−	0.299984i	−0.988687	−	0.149992i	$-0.952075\pi$
$281$		−	84.2955i		−	0.299984i	0.988687	−	0.149992i	$-0.0479248\pi$
$282$	163.133			0.578487
$283$			140.725i			0.497260i	0.968598	+	0.248630i	$0.0799804\pi$
$283$			140.725i			0.497260i	−0.968598	+	0.248630i	$0.920020\pi$
$284$	−58.6993			−0.206688
$285$	−139.382			−0.489058
$286$			244.252i			0.854029i
$287$			296.001i			1.03136i
$288$	16.9706			0.0589256
$289$	203.054			0.702608
$290$			86.8974i			0.299646i
$291$			178.809i			0.614464i
$292$	191.667			0.656394
$293$		−	271.891i		−	0.927956i	−0.885847	−	0.463978i	$-0.846422\pi$
$293$		−	271.891i		−	0.927956i	0.885847	−	0.463978i	$-0.153578\pi$
$294$	−19.4205			−0.0660561
$295$		−	201.339i		−	0.682505i
$296$			121.433i			0.410245i
$297$		−	44.9522i		−	0.151354i
$298$			159.058i			0.533751i
$299$	−458.703	+	20.9128i	−1.53412	+	0.0699426i
$300$	−17.3205			−0.0577350
$301$	157.368			0.522817
$302$	−303.444			−1.00478
$303$	148.728			0.490852
$304$		−	143.953i		−	0.473528i
$305$	24.1333			0.0791255
$306$			39.3323i			0.128537i
$307$	249.583			0.812975			0.406487	−	0.913656i	$-0.366753\pi$
$307$	249.583			0.812975			0.406487	+	0.913656i	$0.366753\pi$
$308$	−130.546			−0.423851
$309$			309.399i			1.00129i
$310$		−	70.5272i		−	0.227507i
$311$	−266.239			−0.856074			−0.428037	−	0.903761i	$-0.640795\pi$
$311$	−266.239			−0.856074			−0.428037	+	0.903761i	$0.640795\pi$
$312$	−97.8048			−0.313477
$313$			252.371i			0.806297i	0.915135	+	0.403148i	$0.132084\pi$
$313$			252.371i			0.806297i	−0.915135	+	0.403148i	$0.867916\pi$
$314$			81.4331i			0.259341i
$315$	−50.6140			−0.160679
$316$			144.046i			0.455841i
$317$	9.32174			0.0294061			0.0147031	−	0.999892i	$-0.495320\pi$
$317$	9.32174			0.0294061			0.0147031	+	0.999892i	$0.495320\pi$
$318$		−	158.188i		−	0.497446i
$319$			237.726i			0.745221i
$320$		−	17.8885i		−	0.0559017i
$321$		−	36.7966i		−	0.114631i
$322$	−11.1773	−	245.164i	−0.0347122	−	0.761378i
$323$	333.636			1.03293
$324$	18.0000			0.0555556
$325$	99.8216			0.307143
$326$	−294.696			−0.903976
$327$		−	324.747i		−	0.993110i
$328$	−110.962			−0.338298
$329$		−	502.495i		−	1.52734i
$330$	−47.3838			−0.143587
$331$	−545.022			−1.64659			−0.823297	−	0.567611i	$-0.807868\pi$
$331$	−545.022			−1.64659			−0.823297	+	0.567611i	$0.807868\pi$
$332$		−	212.470i		−	0.639969i
$333$			128.799i			0.386783i
$334$	358.824			1.07432
$335$	−131.435			−0.392345
$336$		−	52.2739i		−	0.155577i
$337$		−	9.80857i		−	0.0291055i	−0.999894	−	0.0145528i	$-0.995368\pi$
$337$		−	9.80857i		−	0.0291055i	0.999894	−	0.0145528i	$-0.00463245\pi$
$338$	324.667			0.960552
$339$		−	9.02182i		−	0.0266131i
$340$	41.4599			0.121941
$341$		−	192.942i		−	0.565811i
$342$		−	152.685i		−	0.446447i
$343$		−	309.889i		−	0.903467i
$344$			58.9925i			0.171490i
$345$	−4.05699	−	88.9862i	−0.0117594	−	0.257931i
$346$	−128.559			−0.371557
$347$	507.867			1.46359			0.731797	−	0.681522i	$-0.238682\pi$
$347$	507.867			1.46359			0.731797	+	0.681522i	$0.238682\pi$
$348$	−95.1913			−0.273538
$349$	182.157			0.521941			0.260971	−	0.965347i	$-0.415957\pi$
$349$	182.157			0.521941			0.260971	+	0.965347i	$0.415957\pi$
$350$			53.3518i			0.152434i
$351$	−103.738			−0.295549
$352$		−	48.9378i		−	0.139028i
$353$	148.258			0.419994			0.209997	−	0.977702i	$-0.432655\pi$
$353$	148.258			0.419994			0.209997	+	0.977702i	$0.432655\pi$
$354$	220.556			0.623039
$355$			65.6278i			0.184867i
$356$			160.014i			0.449476i
$357$	121.154			0.339367
$358$	39.2066			0.109516
$359$		−	473.762i		−	1.31967i	−0.751410	−	0.659836i	$-0.770626\pi$
$359$		−	473.762i		−	1.31967i	0.751410	−	0.659836i	$-0.229374\pi$
$360$		−	18.9737i		−	0.0527046i
$361$	−934.147			−2.58767
$362$			454.439i			1.25536i
$363$	79.9501			0.220248
$364$			301.265i			0.827652i
$365$		−	214.290i		−	0.587096i
$366$			26.4367i			0.0722314i
$367$		−	25.7019i		−	0.0700324i	−0.999387	−	0.0350162i	$-0.988852\pi$
$367$		−	25.7019i		−	0.0700324i	0.999387	−	0.0350162i	$-0.0111483\pi$
$368$	91.9045	−	4.19004i	0.249741	−	0.0113860i
$369$	−117.693			−0.318950
$370$	135.766			0.366934
$371$	−487.261			−1.31337
$372$	77.2587			0.207685
$373$			103.774i			0.278213i	0.990277	+	0.139107i	$0.0444231\pi$
$373$			103.774i			0.278213i	−0.990277	+	0.139107i	$0.955577\pi$
$374$	113.422			0.303267
$375$			19.3649i			0.0516398i
$376$	188.370			0.500984
$377$	548.607			1.45519
$378$		−	55.4449i		−	0.146680i
$379$		−	60.4401i		−	0.159473i	−0.996816	−	0.0797363i	$-0.974592\pi$
$379$		−	60.4401i		−	0.159473i	0.996816	−	0.0797363i	$-0.0254078\pi$
$380$	−160.944			−0.423537
$381$	370.501			0.972443
$382$		−	45.5921i		−	0.119351i
$383$		−	228.703i		−	0.597135i	−0.954389	−	0.298567i	$-0.903491\pi$
$383$		−	228.703i		−	0.597135i	0.954389	−	0.298567i	$-0.0965087\pi$
$384$	19.5959			0.0510310
$385$			145.955i			0.379104i
$386$	−43.6319			−0.113036
$387$			62.5710i			0.161682i
$388$			206.471i			0.532142i
$389$		−	88.9467i		−	0.228655i	−0.993443	−	0.114327i	$-0.963529\pi$
$389$		−	88.9467i		−	0.228655i	0.993443	−	0.114327i	$-0.0364713\pi$
$390$			109.349i			0.280382i
$391$	9.71117	+	213.005i	0.0248368	+	0.544770i
$392$	−22.4249			−0.0572063
$393$	−363.452			−0.924815
$394$	402.093			1.02054
$395$	161.048			0.407717
$396$		−	51.9063i		−	0.131077i
$397$	480.726			1.21090			0.605449	−	0.795884i	$-0.292994\pi$
$397$	480.726			1.21090			0.605449	+	0.795884i	$0.292994\pi$
$398$			418.656i			1.05190i
$399$	−470.310			−1.17872
$400$	−20.0000			−0.0500000
$401$		−	619.704i		−	1.54540i	−0.634773	−	0.772699i	$-0.718906\pi$
$401$		−	619.704i		−	1.54540i	0.634773	−	0.772699i	$-0.281094\pi$
$402$		−	143.980i		−	0.358160i
$403$	−445.258			−1.10486
$404$	171.736			0.425090
$405$		−	20.1246i		−	0.0496904i
$406$			293.215i			0.722204i
$407$	371.415			0.912567
$408$			45.4170i			0.111316i
$409$	−79.9943			−0.195585			−0.0977925	−	0.995207i	$-0.531178\pi$
$409$	−79.9943			−0.195585			−0.0977925	+	0.995207i	$0.531178\pi$
$410$			124.059i			0.302583i
$411$		−	65.3320i		−	0.158959i
$412$			357.263i			0.867143i
$413$		−	679.372i		−	1.64497i
$414$	97.4795	−	4.44421i	0.235458	−	0.0107348i
$415$	−237.548			−0.572406
$416$	−112.935			−0.271479
$417$	−14.0120			−0.0336018
$418$	−440.295			−1.05334
$419$		−	255.496i		−	0.609777i	−0.952388	−	0.304888i	$-0.901381\pi$
$419$		−	255.496i		−	0.609777i	0.952388	−	0.304888i	$-0.0986192\pi$
$420$	−58.4440			−0.139152
$421$			306.663i			0.728416i	0.931318	+	0.364208i	$0.118660\pi$
$421$			306.663i			0.728416i	−0.931318	+	0.364208i	$0.881340\pi$
$422$	−328.756			−0.779042
$423$	199.797			0.472332
$424$		−	182.660i		−	0.430801i
$425$		−	46.3536i		−	0.109067i
$426$	−71.8917			−0.168760
$427$	81.4321			0.190708
$428$		−	42.4890i		−	0.0992734i
$429$			299.147i			0.697312i
$430$	65.9557			0.153385
$431$			580.271i			1.34634i	0.739489	+	0.673168i	$0.235067\pi$
$431$			580.271i			1.34634i	−0.739489	+	0.673168i	$0.764933\pi$
$432$	20.7846			0.0481125
$433$		−	695.400i		−	1.60600i	−0.595976	−	0.803002i	$-0.703235\pi$
$433$		−	695.400i		−	1.60600i	0.595976	−	0.803002i	$-0.296765\pi$
$434$		−	237.978i		−	0.548336i
$435$			106.427i			0.244660i
$436$		−	374.986i		−	0.860059i
$437$	−37.6980	−	826.869i	−0.0862654	−	1.89215i
$438$	234.743			0.535943
$439$	−315.512			−0.718705			−0.359353	−	0.933202i	$-0.617002\pi$
$439$	−315.512			−0.718705			−0.359353	+	0.933202i	$0.617002\pi$
$440$	−54.7141			−0.124350
$441$	−23.7852			−0.0539346
$442$		−	261.748i		−	0.592189i
$443$	536.137			1.21024			0.605121	−	0.796134i	$-0.293125\pi$
$443$	536.137			1.21024			0.605121	+	0.796134i	$0.293125\pi$
$444$			148.724i			0.334964i
$445$	178.901			0.402024
$446$	−277.915			−0.623129
$447$			194.805i			0.435805i
$448$		−	60.3607i		−	0.134734i
$449$	−676.452			−1.50658			−0.753288	−	0.657691i	$-0.771533\pi$
$449$	−676.452			−1.50658			−0.753288	+	0.657691i	$0.771533\pi$
$450$	−21.2132			−0.0471405
$451$			339.389i			0.752525i
$452$		−	10.4175i		−	0.0230476i
$453$	−371.642			−0.820401
$454$		−	428.323i		−	0.943443i
$455$	336.825			0.740274
$456$		−	176.305i		−	0.386634i
$457$		−	171.828i		−	0.375992i	−0.982170	−	0.187996i	$-0.939801\pi$
$457$		−	171.828i		−	0.375992i	0.982170	−	0.187996i	$-0.0601992\pi$
$458$			588.325i			1.28455i
$459$			48.1720i			0.104950i
$460$	−4.68461	−	102.752i	−0.0101839	−	0.223375i
$461$	−744.326			−1.61459			−0.807295	−	0.590148i	$-0.799070\pi$
$461$	−744.326			−1.61459			−0.807295	+	0.590148i	$0.799070\pi$
$462$	−159.886			−0.346073
$463$	−115.174			−0.248757			−0.124378	−	0.992235i	$-0.539694\pi$
$463$	−115.174			−0.248757			−0.124378	+	0.992235i	$0.539694\pi$
$464$	−109.917			−0.236891
$465$		−	86.3779i		−	0.185759i
$466$	−139.563			−0.299492
$467$			563.253i			1.20611i	0.797700	+	0.603055i	$0.206050\pi$
$467$			563.253i			1.20611i	−0.797700	+	0.603055i	$0.793950\pi$
$468$	−119.786			−0.255953
$469$	−443.498			−0.945626
$470$		−	210.604i		−	0.448094i
$471$			99.7348i			0.211751i
$472$	254.676			0.539568
$473$	180.435			0.381470
$474$			176.419i			0.372193i
$475$			179.941i			0.378823i
$476$	139.897			0.293901
$477$		−	193.740i		−	0.406163i
$478$	−34.2541			−0.0716613
$479$		−	422.812i		−	0.882698i	−0.897336	−	0.441349i	$-0.854500\pi$
$479$		−	422.812i		−	0.882698i	0.897336	−	0.441349i	$-0.145500\pi$
$480$		−	21.9089i		−	0.0456435i
$481$		−	857.126i		−	1.78197i
$482$			62.2546i			0.129159i
$483$	−13.6894	−	300.263i	−0.0283424	−	0.621663i
$484$	92.3184			0.190740
$485$	230.842			0.475962
$486$	22.0454			0.0453609
$487$	813.888			1.67123			0.835614	−	0.549317i	$-0.185112\pi$
$487$	813.888			1.67123			0.835614	+	0.549317i	$0.185112\pi$
$488$			30.5264i			0.0625542i
$489$	−360.928			−0.738094
$490$			25.0718i			0.0511669i
$491$	162.937			0.331848			0.165924	−	0.986139i	$-0.446939\pi$
$491$	162.937			0.331848			0.165924	+	0.986139i	$0.446939\pi$
$492$	−135.900			−0.276219
$493$		−	254.753i		−	0.516741i
$494$			1016.08i			2.05685i
$495$	−58.0331			−0.117239
$496$	89.2107			0.179860
$497$			221.446i			0.445565i
$498$		−	260.221i		−	0.522533i
$499$	410.699			0.823044			0.411522	−	0.911400i	$-0.364997\pi$
$499$	410.699			0.823044			0.411522	+	0.911400i	$0.364997\pi$
$500$			22.3607i			0.0447214i
$501$	439.468			0.877181
$502$			25.5226i			0.0508419i
$503$			817.224i			1.62470i	0.583170	+	0.812350i	$0.301812\pi$
$503$			817.224i			1.62470i	−0.583170	+	0.812350i	$0.698188\pi$
$504$		−	64.0222i		−	0.127028i
$505$		−	192.007i		−	0.380212i
$506$	−12.8157	−	281.100i	−0.0253275	−	0.555534i
$507$	397.634			0.784288
$508$	427.818			0.842161
$509$	411.147			0.807755			0.403878	−	0.914813i	$-0.367662\pi$
$509$	411.147			0.807755			0.403878	+	0.914813i	$0.367662\pi$
$510$	50.7778			0.0995643
$511$		−	723.072i		−	1.41501i
$512$	22.6274			0.0441942
$513$		−	187.000i		−	0.364522i
$514$	−493.510			−0.960136
$515$	399.432			0.775596
$516$			72.2508i			0.140021i
$517$		−	576.151i		−	1.11441i
$518$	458.110			0.884382
$519$	−157.451			−0.303375
$520$			126.265i			0.242818i
$521$		−	848.413i		−	1.62843i	−0.580562	−	0.814216i	$-0.697167\pi$
$521$		−	848.413i		−	1.62843i	0.580562	−	0.814216i	$-0.302833\pi$
$522$	−116.585			−0.223343
$523$		−	527.585i		−	1.00877i	−0.863480	−	0.504383i	$-0.831720\pi$
$523$		−	527.585i		−	1.00877i	0.863480	−	0.504383i	$-0.168280\pi$
$524$	−419.679			−0.800913
$525$			65.3424i			0.124462i
$526$		−	196.011i		−	0.372645i
$527$			206.762i			0.392337i
$528$		−	59.9363i		−	0.113516i
$529$	526.805	−	48.1355i	0.995851	−	0.0909934i
$530$	−204.220			−0.385320
$531$	270.125			0.508710
$532$	−543.068			−1.02080
$533$	783.218			1.46945
$534$			195.976i			0.366996i
$535$	−47.5042			−0.0887929
$536$		−	166.254i		−	0.310176i
$537$	48.0181			0.0894192
$538$	248.429			0.461765
$539$			68.5889i			0.127252i
$540$		−	23.2379i		−	0.0430331i
$541$	−693.810			−1.28246			−0.641230	−	0.767349i	$-0.721575\pi$
$541$	−693.810			−1.28246			−0.641230	+	0.767349i	$0.721575\pi$
$542$	347.326			0.640822
$543$			556.572i			1.02499i
$544$			52.4431i			0.0964027i
$545$	−419.247			−0.769260
$546$			368.973i			0.675775i
$547$	−235.004			−0.429624			−0.214812	−	0.976655i	$-0.568914\pi$
$547$	−235.004			−0.429624			−0.214812	+	0.976655i	$0.568914\pi$
$548$		−	75.4389i		−	0.137662i
$549$			32.3782i			0.0589767i
$550$			61.1722i			0.111222i
$551$			988.931i			1.79479i
$552$	112.560	−	5.13173i	0.203912	−	0.00929662i
$553$	543.419			0.982675
$554$	451.502			0.814985
$555$	166.278			0.299601
$556$	−16.1796			−0.0291000
$557$		−	894.637i		−	1.60617i	−0.595864	−	0.803085i	$-0.703190\pi$
$557$		−	894.637i		−	1.60617i	0.595864	−	0.803085i	$-0.296810\pi$
$558$	94.6222			0.169574
$559$		−	416.396i		−	0.744895i
$560$	−67.4853			−0.120510
$561$	138.913			0.247617
$562$		−	119.212i		−	0.212121i
$563$			718.198i			1.27566i	0.770176	+	0.637831i	$0.220168\pi$
$563$			718.198i			1.27566i	−0.770176	+	0.637831i	$0.779832\pi$
$564$	230.705			0.409052
$565$	−11.6471			−0.0206144
$566$			199.015i			0.351616i
$567$		−	67.9058i		−	0.119763i
$568$	−83.0133			−0.146150
$569$		−	644.903i		−	1.13340i	−0.823925	−	0.566699i	$-0.808220\pi$
$569$		−	644.903i		−	1.13340i	0.823925	−	0.566699i	$-0.191780\pi$
$570$	−197.115			−0.345816
$571$		−	908.958i		−	1.59187i	−0.605381	−	0.795935i	$-0.706979\pi$
$571$		−	908.958i		−	1.59187i	0.605381	−	0.795935i	$-0.293021\pi$
$572$			345.425i			0.603890i
$573$		−	55.8387i		−	0.0974498i
$574$			418.608i			0.729283i
$575$	−114.881	+	5.23755i	−0.199792	+	0.00910879i
$576$	24.0000			0.0416667
$577$	−156.784			−0.271723			−0.135861	−	0.990728i	$-0.543380\pi$
$577$	−156.784			−0.271723			−0.135861	+	0.990728i	$0.543380\pi$
$578$	287.162			0.496819
$579$	−53.4379			−0.0922935
$580$			122.891i			0.211882i
$581$	−801.552			−1.37961
$582$			252.874i			0.434492i
$583$	−558.685			−0.958293
$584$	271.058			0.464140
$585$			133.925i			0.228931i
$586$		−	384.512i		−	0.656164i
$587$	−68.2770			−0.116315			−0.0581576	−	0.998307i	$-0.518523\pi$
$587$	−68.2770			−0.116315			−0.0581576	+	0.998307i	$0.518523\pi$
$588$	−27.4647			−0.0467087
$589$		−	802.632i		−	1.36270i
$590$		−	284.736i		−	0.482604i
$591$	492.462			0.833269
$592$			171.732i			0.290087i
$593$	−167.137			−0.281849			−0.140925	−	0.990020i	$-0.545008\pi$
$593$	−167.137			−0.281849			−0.140925	+	0.990020i	$0.545008\pi$
$594$		−	63.5720i		−	0.107024i
$595$		−	156.409i		−	0.262873i
$596$			224.941i			0.377419i
$597$			512.746i			0.858872i
$598$	−648.704	+	29.5752i	−1.08479	+	0.0494569i
$599$	−603.116			−1.00687			−0.503436	−	0.864033i	$-0.667931\pi$
$599$	−603.116			−1.00687			−0.503436	+	0.864033i	$0.667931\pi$
$600$	−24.4949			−0.0408248
$601$	270.772			0.450536			0.225268	−	0.974297i	$-0.427674\pi$
$601$	270.772			0.450536			0.225268	+	0.974297i	$0.427674\pi$
$602$	222.552			0.369688
$603$		−	176.339i		−	0.292436i
$604$	−429.135			−0.710488
$605$		−	103.215i		−	0.170603i
$606$	210.333			0.347084
$607$	279.365			0.460239			0.230119	−	0.973162i	$-0.426088\pi$
$607$	279.365			0.460239			0.230119	+	0.973162i	$0.426088\pi$
$608$		−	203.580i		−	0.334835i
$609$			359.113i			0.589677i
$610$	34.1296			0.0559502
$611$	−1329.60			−2.17611
$612$			55.6243i			0.0908894i
$613$		−	361.882i		−	0.590345i	−0.955444	−	0.295173i	$-0.904623\pi$
$613$		−	361.882i		−	0.590345i	0.955444	−	0.295173i	$-0.0953771\pi$
$614$	352.964			0.574860
$615$			151.941i			0.247058i
$616$	−184.620			−0.299708
$617$		−	132.515i		−	0.214773i	−0.994217	−	0.107387i	$-0.965752\pi$
$617$		−	132.515i		−	0.214773i	0.994217	−	0.107387i	$-0.0342483\pi$
$618$			437.556i			0.708019i
$619$		−	76.7590i		−	0.124005i	−0.998076	−	0.0620024i	$-0.980251\pi$
$619$		−	76.7590i		−	0.124005i	0.998076	−	0.0620024i	$-0.0197487\pi$
$620$		−	99.7406i		−	0.160872i
$621$	119.387	−	5.44303i	0.192250	−	0.00876494i
$622$	−376.519			−0.605336
$623$	603.658			0.968954
$624$	−138.317			−0.221662
$625$	25.0000			0.0400000
$626$			356.906i			0.570138i
$627$	−539.249			−0.860046
$628$			115.164i			0.183382i
$629$	−398.018			−0.632780
$630$	−71.5790			−0.113617
$631$		−	832.399i		−	1.31917i	−0.751628	−	0.659587i	$-0.770731\pi$
$631$		−	832.399i		−	1.31917i	0.751628	−	0.659587i	$-0.229269\pi$
$632$			203.712i			0.322328i
$633$	−402.642			−0.636085
$634$	13.1829			0.0207933
$635$		−	478.315i		−	0.753251i
$636$		−	223.711i		−	0.351748i
$637$	158.285			0.248485
$638$			336.195i			0.526951i
$639$	−88.0489			−0.137792
$640$		−	25.2982i		−	0.0395285i
$641$			139.943i			0.218319i	0.994024	+	0.109160i	$0.0348160\pi$
$641$			139.943i			0.218319i	−0.994024	+	0.109160i	$0.965184\pi$
$642$		−	52.0382i		−	0.0810564i
$643$			973.488i			1.51398i	0.653428	+	0.756989i	$0.273330\pi$
$643$			973.488i			1.51398i	−0.653428	+	0.756989i	$0.726670\pi$
$644$	−15.8071	−	346.714i	−0.0245452	−	0.538376i
$645$	80.7789			0.125239
$646$	471.832			0.730391
$647$	−465.146			−0.718928			−0.359464	−	0.933159i	$-0.617040\pi$
$647$	−465.146			−0.718928			−0.359464	+	0.933159i	$0.617040\pi$
$648$	25.4558			0.0392837
$649$		−	778.955i		−	1.20024i
$650$	141.169			0.217183
$651$		−	291.462i		−	0.447714i
$652$	−416.763			−0.639208
$653$	333.454			0.510649			0.255325	−	0.966855i	$-0.417818\pi$
$653$	333.454			0.510649			0.255325	+	0.966855i	$0.417818\pi$
$654$		−	459.262i		−	0.702235i
$655$			469.215i			0.716359i
$656$	−156.924			−0.239213
$657$	287.500			0.437596
$658$		−	710.635i		−	1.07999i
$659$			608.381i			0.923188i	0.887091	+	0.461594i	$0.152722\pi$
$659$			608.381i			0.923188i	−0.887091	+	0.461594i	$0.847278\pi$
$660$	−67.0108			−0.101532
$661$			62.9560i			0.0952435i	0.998865	+	0.0476218i	$0.0151642\pi$
$661$			62.9560i			0.0952435i	−0.998865	+	0.0476218i	$0.984836\pi$
$662$	−770.778			−1.16432
$663$		−	320.574i		−	0.483520i
$664$		−	300.478i		−	0.452527i
$665$			607.168i			0.913035i
$666$			182.149i			0.273497i
$667$	−631.369	+	28.7849i	−0.946581	+	0.0431558i
$668$	507.453			0.759661
$669$	−340.375			−0.508782
$670$	−185.878			−0.277430
$671$	93.3685			0.139148
$672$		−	73.9265i		−	0.110010i
$673$	−910.479			−1.35287			−0.676433	−	0.736504i	$-0.736475\pi$
$673$	−910.479			−1.35287			−0.676433	+	0.736504i	$0.736475\pi$
$674$		−	13.8714i		−	0.0205807i
$675$	−25.9808			−0.0384900
$676$	459.148			0.679213
$677$		−	470.376i		−	0.694794i	−0.937718	−	0.347397i	$-0.887066\pi$
$677$		−	470.376i		−	0.694794i	0.937718	−	0.347397i	$-0.112934\pi$
$678$		−	12.7588i		−	0.0188183i
$679$	778.921			1.14716
$680$	58.6331			0.0862252
$681$		−	524.587i		−	0.770318i
$682$		−	272.861i		−	0.400089i
$683$	764.068			1.11869			0.559347	−	0.828933i	$-0.311052\pi$
$683$	764.068			1.11869			0.559347	+	0.828933i	$0.311052\pi$
$684$		−	215.929i		−	0.315686i
$685$	−84.3433			−0.123129
$686$		−	438.249i		−	0.638847i
$687$			720.548i			1.04883i
$688$			83.4281i			0.121262i
$689$			1289.29i			1.87125i
$690$	−5.73745	−	125.845i	−0.00831515	−	0.182385i
$691$	106.591			0.154257			0.0771284	−	0.997021i	$-0.475425\pi$
$691$	106.591			0.154257			0.0771284	+	0.997021i	$0.475425\pi$
$692$	−181.809			−0.262730
$693$	−195.819			−0.282567
$694$	718.233			1.03492
$695$			18.0894i			0.0260279i
$696$	−134.621			−0.193421
$697$		−	363.698i		−	0.521806i
$698$	257.610			0.369068
$699$	−170.929			−0.244534
$700$			75.4509i			0.107787i
$701$			792.708i			1.13082i	0.824808	+	0.565412i	$0.191283\pi$
$701$			792.708i			1.13082i	−0.824808	+	0.565412i	$0.808717\pi$
$702$	−146.707			−0.208985
$703$	1545.08			2.19783
$704$		−	69.2085i		−	0.0983075i
$705$		−	257.936i		−	0.365867i
$706$	209.668			0.296981
$707$		−	647.883i		−	0.916383i
$708$	311.913			0.440555
$709$			89.5195i			0.126262i	0.998005	+	0.0631309i	$0.0201085\pi$
$709$			89.5195i			0.126262i	−0.998005	+	0.0631309i	$0.979891\pi$
$710$			92.8117i			0.130721i
$711$			216.069i			0.303894i
$712$			226.293i			0.317828i
$713$	512.429	−	23.3623i	0.718694	−	0.0327662i
$714$	171.338			0.239969
$715$	386.197			0.540135
$716$	55.4465			0.0774393
$717$	−41.9525			−0.0585112
$718$		−	670.001i		−	0.933149i
$719$	−277.614			−0.386112			−0.193056	−	0.981188i	$-0.561840\pi$
$719$	−277.614			−0.386112			−0.193056	+	0.981188i	$0.561840\pi$
$720$		−	26.8328i		−	0.0372678i
$721$	1347.79			1.86933
$722$	−1321.08			−1.82976
$723$			76.2460i			0.105458i
$724$			642.674i			0.887671i
$725$	137.397			0.189513
$726$	113.066			0.155739
$727$		−	1240.96i		−	1.70696i	−0.521123	−	0.853481i	$-0.674487\pi$
$727$		−	1240.96i		−	1.70696i	0.521123	−	0.853481i	$-0.325513\pi$
$728$			426.053i			0.585238i
$729$	27.0000			0.0370370
$730$		−	303.052i		−	0.415140i
$731$	−193.359			−0.264514
$732$			37.3871i			0.0510753i
$733$			996.086i			1.35892i	0.733714	+	0.679459i	$0.237785\pi$
$733$			996.086i			1.35892i	−0.733714	+	0.679459i	$0.762215\pi$
$734$		−	36.3480i		−	0.0495204i
$735$			30.7065i			0.0417776i
$736$	129.973	−	5.92562i	0.176593	−	0.00805111i
$737$	−508.507			−0.689969
$738$	−166.443			−0.225532
$739$	−413.675			−0.559776			−0.279888	−	0.960033i	$-0.590297\pi$
$739$	−413.675			−0.559776			−0.279888	+	0.960033i	$0.590297\pi$
$740$	192.002			0.259462
$741$			1244.44i			1.67941i
$742$	−689.092			−0.928695
$743$			704.731i			0.948494i	0.880392	+	0.474247i	$0.157280\pi$
$743$			704.731i			0.948494i	−0.880392	+	0.474247i	$0.842720\pi$
$744$	109.260			0.146855
$745$	251.492			0.337573
$746$			146.758i			0.196726i
$747$		−	318.705i		−	0.426646i
$748$	160.403			0.214442
$749$	−160.292			−0.214008
$750$			27.3861i			0.0365148i
$751$			996.202i			1.32650i	0.748398	+	0.663250i	$0.230824\pi$
$751$			996.202i			1.32650i	−0.748398	+	0.663250i	$0.769176\pi$
$752$	266.396			0.354249
$753$			31.2587i			0.0415123i
$754$	775.847			1.02897
$755$			479.787i			0.635480i
$756$		−	78.4109i		−	0.103718i
$757$			532.958i			0.704040i	0.935992	+	0.352020i	$0.114505\pi$
$757$			532.958i			0.704040i	−0.935992	+	0.352020i	$0.885495\pi$
$758$		−	85.4752i		−	0.112764i
$759$	−15.6960	−	344.276i	−0.0206798	−	0.453592i
$760$	−227.609			−0.299486
$761$	1294.59			1.70117			0.850586	−	0.525835i	$-0.176247\pi$
$761$	1294.59			1.70117			0.850586	+	0.525835i	$0.176247\pi$
$762$	523.967			0.687621
$763$	−1414.65			−1.85406
$764$		−	64.4770i		−	0.0843940i
$765$	62.1898			0.0812939
$766$		−	323.434i		−	0.422238i
$767$	−1797.62			−2.34370
$768$	27.7128			0.0360844
$769$			691.918i			0.899764i	0.893088	+	0.449882i	$0.148534\pi$
$769$			691.918i			0.899764i	−0.893088	+	0.449882i	$0.851466\pi$
$770$			206.411i			0.268067i
$771$	−604.424			−0.783948
$772$	−61.7048			−0.0799285
$773$			721.253i			0.933057i	0.884506	+	0.466529i	$0.154496\pi$
$773$			721.253i			0.933057i	−0.884506	+	0.466529i	$0.845504\pi$
$774$			88.4888i			0.114327i
$775$	−111.513			−0.143888
$776$			291.994i			0.376281i
$777$	561.067			0.722095
$778$		−	125.790i		−	0.161683i
$779$			1411.85i			1.81238i
$780$			154.643i			0.198260i
$781$			253.905i			0.325103i
$782$	13.7337	+	301.235i	0.0175622	+	0.385211i
$783$	−142.787			−0.182359
$784$	−31.7135			−0.0404510
$785$	128.757			0.164022
$786$	−513.999			−0.653943
$787$			1085.29i			1.37903i	0.724274	+	0.689513i	$0.242175\pi$
$787$			1085.29i			1.37903i	−0.724274	+	0.689513i	$0.757825\pi$
$788$	568.646			0.721632
$789$		−	240.064i		−	0.304263i
$790$	227.756			0.288299
$791$	−39.3005			−0.0496846
$792$		−	73.4067i		−	0.0926852i
$793$		−	215.470i		−	0.271714i
$794$	679.849			0.856234
$795$	−250.117			−0.314613
$796$			592.069i			0.743805i
$797$			518.555i			0.650634i	0.945605	+	0.325317i	$0.105471\pi$
$797$			518.555i			0.650634i	−0.945605	+	0.325317i	$0.894529\pi$
$798$	−665.119			−0.833483
$799$			617.419i			0.772740i
$800$	−28.2843			−0.0353553
$801$			240.020i			0.299651i
$802$		−	876.394i		−	1.09276i
$803$		−	829.061i		−	1.03245i
$804$		−	203.619i		−	0.253257i
$805$	−387.638	+	17.6729i	−0.481538	+	0.0219539i
$806$	−629.689			−0.781252
$807$	304.263			0.377029
$808$	242.872			0.300584
$809$	−890.746			−1.10105			−0.550523	−	0.834820i	$-0.685572\pi$
$809$	−890.746			−1.10105			−0.550523	+	0.834820i	$0.685572\pi$
$810$		−	28.4605i		−	0.0351364i
$811$	584.404			0.720597			0.360298	−	0.932837i	$-0.382675\pi$
$811$	584.404			0.720597			0.360298	+	0.932837i	$0.382675\pi$
$812$			414.668i			0.510675i
$813$	425.385			0.523229
$814$	525.260			0.645282
$815$			465.956i			0.571725i
$816$			64.2294i			0.0787125i
$817$	750.605			0.918734
$818$	−113.129			−0.138299
$819$			451.898i			0.551768i
$820$			175.446i			0.213958i
$821$	599.765			0.730529			0.365265	−	0.930904i	$-0.380978\pi$
$821$	599.765			0.730529			0.365265	+	0.930904i	$0.380978\pi$
$822$		−	92.3935i		−	0.112401i
$823$	878.550			1.06750			0.533749	−	0.845643i	$-0.320783\pi$
$823$	878.550			1.06750			0.533749	+	0.845643i	$0.320783\pi$
$824$			505.246i			0.613163i
$825$			74.9204i			0.0908126i
$826$		−	960.777i		−	1.16317i
$827$			1493.08i			1.80541i	0.430258	+	0.902706i	$0.358423\pi$
$827$			1493.08i			1.80541i	−0.430258	+	0.902706i	$0.641577\pi$
$828$	137.857	−	6.28506i	0.166494	−	0.00759066i
$829$	−614.423			−0.741161			−0.370581	−	0.928800i	$-0.620841\pi$
$829$	−614.423			−0.741161			−0.370581	+	0.928800i	$0.620841\pi$
$830$	−335.944			−0.404752
$831$	552.974			0.665432
$832$	−159.715			−0.191965
$833$		−	73.5018i		−	0.0882375i
$834$	−19.8159			−0.0237601
$835$		−	567.350i		−	0.679461i
$836$	−622.671			−0.744822
$837$	115.888			0.138456
$838$		−	361.327i		−	0.431177i
$839$		−	20.0377i		−	0.0238829i	−0.999929	−	0.0119414i	$-0.996199\pi$
$839$		−	20.0377i		−	0.0238829i	0.999929	−	0.0119414i	$-0.00380117\pi$
$840$	−82.6523			−0.0983956
$841$	−85.8849			−0.102122
$842$			433.687i			0.515068i
$843$		−	146.004i		−	0.173196i
$844$	−464.931			−0.550866
$845$		−	513.343i		−	0.607507i
$846$	282.555			0.333989
$847$		−	348.275i		−	0.411187i
$848$		−	258.320i		−	0.304622i
$849$			243.742i			0.287093i
$850$		−	65.5539i		−	0.0771222i
$851$	44.9727	+	986.432i	0.0528468	+	1.15914i
$852$	−101.670			−0.119331
$853$	−342.579			−0.401617			−0.200808	−	0.979631i	$-0.564357\pi$
$853$	−342.579			−0.401617			−0.200808	+	0.979631i	$0.564357\pi$
$854$	115.162			0.134851
$855$	−241.416			−0.282358
$856$		−	60.0886i		−	0.0701969i
$857$	206.677			0.241163			0.120582	−	0.992703i	$-0.461524\pi$
$857$	206.677			0.241163			0.120582	+	0.992703i	$0.461524\pi$
$858$			423.057i			0.493074i
$859$	−131.130			−0.152655			−0.0763274	−	0.997083i	$-0.524319\pi$
$859$	−131.130			−0.152655			−0.0763274	+	0.997083i	$0.524319\pi$
$860$	93.2754			0.108460
$861$			512.688i			0.595457i
$862$			820.627i			0.952004i
$863$	506.478			0.586881			0.293440	−	0.955977i	$-0.405200\pi$
$863$	506.478			0.586881			0.293440	+	0.955977i	$0.405200\pi$
$864$	29.3939			0.0340207
$865$			203.269i			0.234993i
$866$		−	983.444i		−	1.13562i
$867$	351.700			0.405651
$868$		−	336.551i		−	0.387732i
$869$	623.074			0.717001
$870$			150.511i			0.173001i
$871$			1173.50i			1.34730i
$872$		−	530.310i		−	0.608153i
$873$			309.707i			0.354761i
$874$	−53.3130	−	1169.37i	−0.0609989	−	1.33795i
$875$	84.3567			0.0964076
$876$	331.977			0.378969
$877$	605.650			0.690593			0.345297	−	0.938494i	$-0.387778\pi$
$877$	605.650			0.690593			0.345297	+	0.938494i	$0.387778\pi$
$878$	−446.201			−0.508201
$879$		−	470.929i		−	0.535756i
$880$	−77.3774			−0.0879289
$881$			870.965i			0.988609i	0.869289	+	0.494305i	$0.164577\pi$
$881$			870.965i			0.988609i	−0.869289	+	0.494305i	$0.835423\pi$
$882$	−33.6373			−0.0381375
$883$	−790.016			−0.894696			−0.447348	−	0.894360i	$-0.647631\pi$
$883$	−790.016			−0.894696			−0.447348	+	0.894360i	$0.647631\pi$
$884$		−	370.167i		−	0.418741i
$885$		−	348.730i		−	0.394045i
$886$	758.212			0.855770
$887$	−329.981			−0.372020			−0.186010	−	0.982548i	$-0.559556\pi$
$887$	−329.981			−0.372020			−0.186010	+	0.982548i	$0.559556\pi$
$888$			210.327i			0.236855i
$889$		−	1613.96i		−	1.81548i
$890$	253.004			0.284274
$891$		−	77.8595i		−	0.0873844i
$892$	−393.032			−0.440618
$893$		−	2396.77i		−	2.68395i
$894$			275.496i			0.308161i
$895$		−	61.9911i		−	0.0692638i
$896$		−	85.3630i		−	0.0952712i
$897$	−794.497	+	36.2221i	−0.885727	+	0.0403814i
$898$	−956.648			−1.06531
$899$	−612.863			−0.681716
$900$	−30.0000			−0.0333333
$901$	598.702			0.664486
$902$			479.968i			0.532116i
$903$	272.569			0.301849
$904$		−	14.7326i		−	0.0162971i
$905$	718.531			0.793957
$906$	−525.581			−0.580111
$907$		−	990.130i		−	1.09165i	−0.837898	−	0.545827i	$-0.816216\pi$
$907$		−	990.130i		−	1.09165i	0.837898	−	0.545827i	$-0.183784\pi$
$908$		−	605.740i		−	0.667115i
$909$	257.604			0.283393
$910$	476.342			0.523453
$911$			399.780i			0.438836i	0.975631	+	0.219418i	$0.0704158\pi$
$911$			399.780i			0.438836i	−0.975631	+	0.219418i	$0.929584\pi$
$912$		−	249.333i		−	0.273392i
$913$	−919.044			−1.00662
$914$		−	243.002i		−	0.265867i
$915$	41.8001			0.0456831
$916$			832.017i			0.908316i
$917$			1583.26i			1.72656i
$918$			68.1256i			0.0742108i
$919$			1471.91i			1.60164i	0.598903	+	0.800822i	$0.295604\pi$
$919$			1471.91i			1.60164i	−0.598903	+	0.800822i	$0.704396\pi$
$920$	−6.62504	−	145.314i	−0.00720113	−	0.157950i
$921$	432.291			0.469371
$922$	−1052.64			−1.14169
$923$	585.946			0.634827
$924$	−226.112			−0.244710
$925$		−	214.664i		−	0.232070i
$926$	−162.881			−0.175897
$927$			535.894i			0.578095i
$928$	−155.447			−0.167507
$929$	407.172			0.438291			0.219145	−	0.975692i	$-0.429673\pi$
$929$	407.172			0.438291			0.219145	+	0.975692i	$0.429673\pi$
$930$		−	122.157i		−	0.131351i
$931$			285.328i			0.306475i
$932$	−197.372			−0.211773
$933$	−461.140			−0.494255
$934$			796.560i			0.852848i
$935$		−	179.336i		−	0.191803i
$936$	−169.403			−0.180986
$937$			981.778i			1.04779i	0.851783	+	0.523895i	$0.175521\pi$
$937$			981.778i			1.04779i	−0.851783	+	0.523895i	$0.824479\pi$
$938$	−627.201			−0.668658
$939$			437.119i			0.465516i
$940$		−	297.839i		−	0.316850i
$941$			1305.07i			1.38690i	0.720506	+	0.693449i	$0.243910\pi$
$941$			1305.07i			1.38690i	−0.720506	+	0.693449i	$0.756090\pi$
$942$			141.046i			0.149731i
$943$	−901.374	+	41.0948i	−0.955858	+	0.0435788i
$944$	360.166			0.381532
$945$	−87.6660			−0.0927683
$946$	255.174			0.269740
$947$	−363.792			−0.384152			−0.192076	−	0.981380i	$-0.561522\pi$
$947$	−363.792			−0.384152			−0.192076	+	0.981380i	$0.561522\pi$
$948$			249.495i			0.263180i
$949$	−1913.25			−2.01607
$950$			254.475i			0.267868i
$951$	16.1457			0.0169776
$952$	197.844			0.207819
$953$			995.226i			1.04431i	0.852851	+	0.522154i	$0.174872\pi$
$953$			995.226i			1.04431i	−0.852851	+	0.522154i	$0.825128\pi$
$954$		−	273.989i		−	0.287201i
$955$	−72.0875			−0.0754843
$956$	−48.4426			−0.0506722
$957$			411.753i			0.430254i
$958$		−	597.947i		−	0.624162i
$959$	−284.597			−0.296764
$960$		−	30.9839i		−	0.0322749i
$961$	−463.591			−0.482405
$962$		−	1212.16i		−	1.26004i
$963$		−	63.7336i		−	0.0661823i
$964$			88.0413i			0.0913292i
$965$			68.9881i			0.0714902i
$966$	−19.3597	−	424.636i	−0.0200411	−	0.439582i
$967$	1577.54			1.63138			0.815688	−	0.578493i	$-0.196359\pi$
$967$	1577.54			1.63138			0.815688	+	0.578493i	$0.196359\pi$
$968$	130.558			0.134874
$969$	577.874			0.596362
$970$	326.459			0.336556
$971$		−	1397.09i		−	1.43882i	−0.694586	−	0.719410i	$-0.744412\pi$
$971$		−	1397.09i		−	1.43882i	0.694586	−	0.719410i	$-0.255588\pi$
$972$	31.1769			0.0320750
$973$			61.0384i			0.0627321i
$974$	1151.01			1.18174
$975$	172.896			0.177329
$976$			43.1709i			0.0442325i
$977$			1329.91i			1.36122i	0.732648	+	0.680608i	$0.238284\pi$
$977$			1329.91i			1.36122i	−0.732648	+	0.680608i	$0.761716\pi$
$978$	−510.429			−0.521911
$979$	692.143			0.706990
$980$			35.4568i			0.0361804i
$981$		−	562.478i		−	0.573373i
$982$	230.428			0.234652
$983$			1362.46i			1.38603i	0.720925	+	0.693013i	$0.243717\pi$
$983$			1362.46i			1.38603i	−0.720925	+	0.693013i	$0.756283\pi$
$984$	−192.191			−0.195316
$985$		−	635.766i		−	0.645447i
$986$		−	360.276i		−	0.365391i
$987$		−	870.346i		−	0.881810i
$988$			1436.96i			1.45441i
$989$	21.8479	+	479.214i	0.0220909	+	0.484543i
$990$	−82.0711			−0.0829001
$991$	−1614.70			−1.62937			−0.814684	−	0.579905i	$-0.803090\pi$
$991$	−1614.70			−1.62937			−0.814684	+	0.579905i	$0.803090\pi$
$992$	126.163			0.127180
$993$	−944.006			−0.950661
$994$			313.172i			0.315062i
$995$	661.953			0.665279
$996$		−	368.008i		−	0.369486i
$997$	−931.722			−0.934525			−0.467263	−	0.884119i	$-0.654760\pi$
$997$	−931.722			−0.934525			−0.467263	+	0.884119i	$0.654760\pi$
$998$	580.816			0.581980
$999$			223.086i			0.223309i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.c.a.91.25	✓	32
3.2	odd	2		2070.3.c.b.91.11		32
23.22	odd	2	inner	690.3.c.a.91.32	yes	32
69.68	even	2		2070.3.c.b.91.6		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.25	✓	32	1.1	even	1	trivial
690.3.c.a.91.32	yes	32	23.22	odd	2	inner
2070.3.c.b.91.6		32	69.68	even	2
2070.3.c.b.91.11		32	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	690.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949 q^{6} -7.54509i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949 q^{6} -7.54509i q^{7} +2.82843 q^{8} +3.00000 q^{9} -3.16228i q^{10} -8.65106i q^{11} +3.46410 q^{12} -19.9643 q^{13} -10.6704i q^{14} -3.87298i q^{15} +4.00000 q^{16} +9.27071i q^{17} +4.24264 q^{18} -35.9882i q^{19} -4.47214i q^{20} -13.0685i q^{21} -12.2344i q^{22} +(22.9761 - 1.04751i) q^{23} +4.89898 q^{24} -5.00000 q^{25} -28.2338 q^{26} +5.19615 q^{27} -15.0902i q^{28} -27.4794 q^{29} -5.47723i q^{30} +22.3027 q^{31} +5.65685 q^{32} -14.9841i q^{33} +13.1108i q^{34} -16.8713 q^{35} +6.00000 q^{36} +42.9329i q^{37} -50.8949i q^{38} -34.5792 q^{39} -6.32456i q^{40} -39.2309 q^{41} -18.4816i q^{42} +20.8570i q^{43} -17.3021i q^{44} -6.70820i q^{45} +(32.4932 - 1.48140i) q^{46} +66.5989 q^{47} +6.92820 q^{48} -7.92839 q^{49} -7.07107 q^{50} +16.0573i q^{51} -39.9286 q^{52} -64.5799i q^{53} +7.34847 q^{54} -19.3444 q^{55} -21.3407i q^{56} -62.3333i q^{57} -38.8617 q^{58} +90.0416 q^{59} -7.74597i q^{60} +10.7927i q^{61} +31.5407 q^{62} -22.6353i q^{63} +8.00000 q^{64} +44.6416i q^{65} -21.1907i q^{66} -58.7797i q^{67} +18.5414i q^{68} +(39.7958 - 1.81434i) q^{69} -23.8597 q^{70} -29.3496 q^{71} +8.48528 q^{72} +95.8335 q^{73} +60.7163i q^{74} -8.66025 q^{75} -71.9763i q^{76} -65.2730 q^{77} -48.9024 q^{78} +72.0229i q^{79} -8.94427i q^{80} +9.00000 q^{81} -55.4809 q^{82} -106.235i q^{83} -26.1370i q^{84} +20.7299 q^{85} +29.4963i q^{86} -47.5956 q^{87} -24.4689i q^{88} +80.0068i q^{89} -9.48683i q^{90} +150.633i q^{91} +(45.9523 - 2.09502i) q^{92} +38.6294 q^{93} +94.1850 q^{94} -80.4720 q^{95} +9.79796 q^{96} +103.236i q^{97} -11.2124 q^{98} -25.9532i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) 32 * q + 64 * q^4 + 96 * q^9	\( 32 q + 64 q^{4} + 96 q^{9} - 48 q^{13} + 128 q^{16} - 80 q^{23} - 160 q^{25} + 120 q^{29} + 248 q^{31} - 120 q^{35} + 192 q^{36} - 48 q^{39} + 72 q^{41} + 160 q^{46} + 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} + 120 q^{59} + 160 q^{62} + 256 q^{64} + 192 q^{69} + 104 q^{71} + 16 q^{73} + 240 q^{77} + 192 q^{78} + 288 q^{81} + 64 q^{82} - 120 q^{85} + 144 q^{87} - 160 q^{92} - 192 q^{93} + 96 q^{94} - 160 q^{95} + 64 q^{98}+O(q^{100}) \) 32 * q + 64 * q^4 + 96 * q^9 - 48 * q^13 + 128 * q^16 - 80 * q^23 - 160 * q^25 + 120 * q^29 + 248 * q^31 - 120 * q^35 + 192 * q^36 - 48 * q^39 + 72 * q^41 + 160 * q^46 + 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 + 120 * q^59 + 160 * q^62 + 256 * q^64 + 192 * q^69 + 104 * q^71 + 16 * q^73 + 240 * q^77 + 192 * q^78 + 288 * q^81 + 64 * q^82 - 120 * q^85 + 144 * q^87 - 160 * q^92 - 192 * q^93 + 96 * q^94 - 160 * q^95 + 64 * q^98

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