LMFDB - Embedded newform 690.3.c.a.91.24

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			91.24
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} +11.8194i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} +11.8194i q^{7} +2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} -20.7652i q^{11} -3.46410 q^{12} +7.75559 q^{13} +16.7152i q^{14} -3.87298i q^{15} +4.00000 q^{16} +22.9081i q^{17} +4.24264 q^{18} +21.0098i q^{19} +4.47214i q^{20} -20.4719i q^{21} -29.3664i q^{22} +(-21.9968 + 6.71856i) q^{23} -4.89898 q^{24} -5.00000 q^{25} +10.9681 q^{26} -5.19615 q^{27} +23.6389i q^{28} -38.2483 q^{29} -5.47723i q^{30} +33.3493 q^{31} +5.65685 q^{32} +35.9663i q^{33} +32.3970i q^{34} -26.4291 q^{35} +6.00000 q^{36} +49.4326i q^{37} +29.7124i q^{38} -13.4331 q^{39} +6.32456i q^{40} -0.679308 q^{41} -28.9516i q^{42} -7.99696i q^{43} -41.5303i q^{44} +6.70820i q^{45} +(-31.1082 + 9.50148i) q^{46} -85.9436 q^{47} -6.92820 q^{48} -90.6993 q^{49} -7.07107 q^{50} -39.6780i q^{51} +15.5112 q^{52} +66.6981i q^{53} -7.34847 q^{54} +46.4323 q^{55} +33.4304i q^{56} -36.3901i q^{57} -54.0913 q^{58} +110.632 q^{59} -7.74597i q^{60} +16.6780i q^{61} +47.1630 q^{62} +35.4583i q^{63} +8.00000 q^{64} +17.3420i q^{65} +50.8640i q^{66} +117.857i q^{67} +45.8163i q^{68} +(38.0996 - 11.6369i) q^{69} -37.3764 q^{70} +28.9967 q^{71} +8.48528 q^{72} +31.0752 q^{73} +69.9083i q^{74} +8.66025 q^{75} +42.0196i q^{76} +245.433 q^{77} -18.9972 q^{78} -105.069i q^{79} +8.94427i q^{80} +9.00000 q^{81} -0.960687 q^{82} -43.1979i q^{83} -40.9438i q^{84} -51.2241 q^{85} -11.3094i q^{86} +66.2480 q^{87} -58.7327i q^{88} +49.1782i q^{89} +9.48683i q^{90} +91.6668i q^{91} +(-43.9937 + 13.4371i) q^{92} -57.7627 q^{93} -121.543 q^{94} -46.9794 q^{95} -9.79796 q^{96} +69.6738i q^{97} -128.268 q^{98} -62.2955i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			11.8194i			1.68849i	0.535955	+	0.844246i	$0.319951\pi$
$7$			11.8194i			1.68849i	−0.535955	+	0.844246i	$0.680049\pi$
$8$	2.82843			0.353553
$9$	3.00000			0.333333
$10$			3.16228i			0.316228i
$11$		−	20.7652i		−	1.88774i	−0.330315	−	0.943871i	$-0.607155\pi$
$11$		−	20.7652i		−	1.88774i	0.330315	−	0.943871i	$-0.392845\pi$
$12$	−3.46410			−0.288675
$13$	7.75559			0.596584			0.298292	−	0.954475i	$-0.403583\pi$
$13$	7.75559			0.596584			0.298292	+	0.954475i	$0.403583\pi$
$14$			16.7152i			1.19394i
$15$		−	3.87298i		−	0.258199i
$16$	4.00000			0.250000
$17$			22.9081i			1.34754i	0.738943	+	0.673768i	$0.235325\pi$
$17$			22.9081i			1.34754i	−0.738943	+	0.673768i	$0.764675\pi$
$18$	4.24264			0.235702
$19$			21.0098i			1.10578i	0.833255	+	0.552890i	$0.186475\pi$
$19$			21.0098i			1.10578i	−0.833255	+	0.552890i	$0.813525\pi$
$20$			4.47214i			0.223607i
$21$		−	20.4719i		−	0.974852i
$22$		−	29.3664i		−	1.33483i
$23$	−21.9968	+	6.71856i	−0.956384	+	0.292111i
$23$	−21.9968	+	6.71856i	−0.956384	+	0.292111i
$24$	−4.89898			−0.204124
$25$	−5.00000			−0.200000
$26$	10.9681			0.421848
$27$	−5.19615			−0.192450
$28$			23.6389i			0.844246i
$29$	−38.2483			−1.31891			−0.659454	−	0.751745i	$-0.729212\pi$
$29$	−38.2483			−1.31891			−0.659454	+	0.751745i	$0.729212\pi$
$30$		−	5.47723i		−	0.182574i
$31$	33.3493			1.07578			0.537892	−	0.843014i	$-0.319221\pi$
$31$	33.3493			1.07578			0.537892	+	0.843014i	$0.319221\pi$
$32$	5.65685			0.176777
$33$			35.9663i			1.08989i
$34$			32.3970i			0.952852i
$35$	−26.4291			−0.755117
$36$	6.00000			0.166667
$37$			49.4326i			1.33602i	0.744154	+	0.668009i	$0.232853\pi$
$37$			49.4326i			1.33602i	−0.744154	+	0.668009i	$0.767147\pi$
$38$			29.7124i			0.781904i
$39$	−13.4331			−0.344438
$40$			6.32456i			0.158114i
$41$	−0.679308			−0.0165685			−0.00828424	−	0.999966i	$-0.502637\pi$
$41$	−0.679308			−0.0165685			−0.00828424	+	0.999966i	$0.502637\pi$
$42$		−	28.9516i		−	0.689324i
$43$		−	7.99696i		−	0.185976i	−0.995667	−	0.0929879i	$-0.970358\pi$
$43$		−	7.99696i		−	0.185976i	0.995667	−	0.0929879i	$-0.0296418\pi$
$44$		−	41.5303i		−	0.943871i
$45$			6.70820i			0.149071i
$46$	−31.1082	+	9.50148i	−0.676266	+	0.206554i
$47$	−85.9436			−1.82859			−0.914294	−	0.405052i	$-0.867253\pi$
$47$	−85.9436			−1.82859			−0.914294	+	0.405052i	$0.867253\pi$
$48$	−6.92820			−0.144338
$49$	−90.6993			−1.85101
$50$	−7.07107			−0.141421
$51$		−	39.6780i		−	0.778001i
$52$	15.5112			0.298292
$53$			66.6981i			1.25845i	0.777221	+	0.629227i	$0.216629\pi$
$53$			66.6981i			1.25845i	−0.777221	+	0.629227i	$0.783371\pi$
$54$	−7.34847			−0.136083
$55$	46.4323			0.844224
$56$			33.4304i			0.596972i
$57$		−	36.3901i		−	0.638422i
$58$	−54.0913			−0.932608
$59$	110.632			1.87512			0.937560	−	0.347823i	$-0.113079\pi$
$59$	110.632			1.87512			0.937560	+	0.347823i	$0.113079\pi$
$60$		−	7.74597i		−	0.129099i
$61$			16.6780i			0.273409i	0.990612	+	0.136705i	$0.0436511\pi$
$61$			16.6780i			0.273409i	−0.990612	+	0.136705i	$0.956349\pi$
$62$	47.1630			0.760694
$63$			35.4583i			0.562831i
$64$	8.00000			0.125000
$65$			17.3420i			0.266800i
$66$			50.8640i			0.770667i
$67$			117.857i			1.75906i	0.475840	+	0.879532i	$0.342144\pi$
$67$			117.857i			1.75906i	−0.475840	+	0.879532i	$0.657856\pi$
$68$			45.8163i			0.673768i
$69$	38.0996	−	11.6369i	0.552169	−	0.168651i
$70$	−37.3764			−0.533948
$71$	28.9967			0.408405			0.204202	−	0.978929i	$-0.434540\pi$
$71$	28.9967			0.408405			0.204202	+	0.978929i	$0.434540\pi$
$72$	8.48528			0.117851
$73$	31.0752			0.425688			0.212844	−	0.977086i	$-0.431727\pi$
$73$	31.0752			0.425688			0.212844	+	0.977086i	$0.431727\pi$
$74$			69.9083i			0.944707i
$75$	8.66025			0.115470
$76$			42.0196i			0.552890i
$77$	245.433			3.18744
$78$	−18.9972			−0.243554
$79$		−	105.069i		−	1.32998i	−0.746850	−	0.664992i	$-0.768435\pi$
$79$		−	105.069i		−	1.32998i	0.746850	−	0.664992i	$-0.231565\pi$
$80$			8.94427i			0.111803i
$81$	9.00000			0.111111
$82$	−0.960687			−0.0117157
$83$		−	43.1979i		−	0.520456i	−0.965547	−	0.260228i	$-0.916202\pi$
$83$		−	43.1979i		−	0.520456i	0.965547	−	0.260228i	$-0.0837977\pi$
$84$		−	40.9438i		−	0.487426i
$85$	−51.2241			−0.602637
$86$		−	11.3094i		−	0.131505i
$87$	66.2480			0.761471
$88$		−	58.7327i		−	0.667417i
$89$			49.1782i			0.552564i	0.961077	+	0.276282i	$0.0891023\pi$
$89$			49.1782i			0.552564i	−0.961077	+	0.276282i	$0.910898\pi$
$90$			9.48683i			0.105409i
$91$			91.6668i			1.00733i
$92$	−43.9937	+	13.4371i	−0.478192	+	0.146056i
$93$	−57.7627			−0.621104
$94$	−121.543			−1.29301
$95$	−46.9794			−0.494520
$96$	−9.79796			−0.102062
$97$			69.6738i			0.718287i	0.933282	+	0.359143i	$0.116931\pi$
$97$			69.6738i			0.718287i	−0.933282	+	0.359143i	$0.883069\pi$
$98$	−128.268			−1.30886
$99$		−	62.2955i		−	0.629247i
$100$	−10.0000			−0.100000
$101$	−17.2732			−0.171022			−0.0855111	−	0.996337i	$-0.527252\pi$
$101$	−17.2732			−0.171022			−0.0855111	+	0.996337i	$0.527252\pi$
$102$		−	56.1132i		−	0.550130i
$103$		−	51.7425i		−	0.502355i	−0.967941	−	0.251177i	$-0.919182\pi$
$103$		−	51.7425i		−	0.502355i	0.967941	−	0.251177i	$-0.0808177\pi$
$104$	21.9361			0.210924
$105$	45.7765			0.435967
$106$			94.3254i			0.889862i
$107$		−	54.2161i		−	0.506693i	−0.967376	−	0.253346i	$-0.918469\pi$
$107$		−	54.2161i		−	0.506693i	0.967376	−	0.253346i	$-0.0815313\pi$
$108$	−10.3923			−0.0962250
$109$		−	0.0647920i		−	0.000594422i	−1.00000		0.000297211i	$-0.999905\pi$
$109$		−	0.0647920i		−	0.000594422i	1.00000		0.000297211i	$-9.46053e-5\pi$
$110$	65.6652			0.596956
$111$		−	85.6198i		−	0.771350i
$112$			47.2778i			0.422123i
$113$		−	157.283i		−	1.39188i	−0.718098	−	0.695942i	$-0.754987\pi$
$113$		−	157.283i		−	1.39188i	0.718098	−	0.695942i	$-0.245013\pi$
$114$		−	51.4633i		−	0.451432i
$115$	−15.0232	−	49.1864i	−0.130636	−	0.427708i
$116$	−76.4966			−0.659454
$117$	23.2668			0.198861
$118$	156.457			1.32591
$119$	−270.761			−2.27531
$120$		−	10.9545i		−	0.0912871i
$121$	−310.192			−2.56357
$122$			23.5862i			0.193329i
$123$	1.17660			0.00956582
$124$	66.6986			0.537892
$125$		−	11.1803i		−	0.0894427i
$126$			50.1457i			0.397981i
$127$	133.370			1.05016			0.525080	−	0.851053i	$-0.324035\pi$
$127$	133.370			1.05016			0.525080	+	0.851053i	$0.324035\pi$
$128$	11.3137			0.0883883
$129$			13.8511i			0.107373i
$130$			24.5253i			0.188656i
$131$	62.9089			0.480221			0.240110	−	0.970746i	$-0.422816\pi$
$131$	62.9089			0.480221			0.240110	+	0.970746i	$0.422816\pi$
$132$			71.9326i			0.544944i
$133$	−248.324			−1.86710
$134$			166.675i			1.24385i
$135$		−	11.6190i		−	0.0860663i
$136$			64.7940i			0.476426i
$137$		−	152.155i		−	1.11062i	−0.831643	−	0.555311i	$-0.812599\pi$
$137$		−	152.155i		−	1.11062i	0.831643	−	0.555311i	$-0.187401\pi$
$138$	53.8810	−	16.4570i	0.390442	−	0.119254i
$139$	63.6449			0.457877			0.228938	−	0.973441i	$-0.426475\pi$
$139$	63.6449			0.457877			0.228938	+	0.973441i	$0.426475\pi$
$140$	−52.8582			−0.377558
$141$	148.859			1.05574
$142$	41.0076			0.288786
$143$		−	161.046i		−	1.12620i
$144$	12.0000			0.0833333
$145$		−	85.5258i		−	0.589833i
$146$	43.9470			0.301007
$147$	157.096			1.06868
$148$			98.8653i			0.668009i
$149$			133.418i			0.895421i	0.894178	+	0.447711i	$0.147761\pi$
$149$			133.418i			0.895421i	−0.894178	+	0.447711i	$0.852239\pi$
$150$	12.2474			0.0816497
$151$	133.515			0.884204			0.442102	−	0.896965i	$-0.354233\pi$
$151$	133.515			0.884204			0.442102	+	0.896965i	$0.354233\pi$
$152$			59.4247i			0.390952i
$153$			68.7244i			0.449179i
$154$	347.094			2.25386
$155$			74.5713i			0.481105i
$156$	−26.8662			−0.172219
$157$		−	117.137i		−	0.746094i	−0.927812	−	0.373047i	$-0.878313\pi$
$157$		−	117.137i		−	0.746094i	0.927812	−	0.373047i	$-0.121687\pi$
$158$		−	148.590i		−	0.940441i
$159$		−	115.524i		−	0.726569i
$160$			12.6491i			0.0790569i
$161$	−79.4097	−	259.990i	−0.493228	−	1.61485i
$162$	12.7279			0.0785674
$163$	34.3108			0.210496			0.105248	−	0.994446i	$-0.466436\pi$
$163$	34.3108			0.210496			0.105248	+	0.994446i	$0.466436\pi$
$164$	−1.35862			−0.00828424
$165$	−80.4231			−0.487413
$166$		−	61.0910i		−	0.368018i
$167$	−33.8574			−0.202739			−0.101370	−	0.994849i	$-0.532322\pi$
$167$	−33.8574			−0.202739			−0.101370	+	0.994849i	$0.532322\pi$
$168$		−	57.9032i		−	0.344662i
$169$	−108.851			−0.644088
$170$	−72.4419			−0.426129
$171$			63.0294i			0.368593i
$172$		−	15.9939i		−	0.0929879i
$173$	207.140			1.19734			0.598670	−	0.800996i	$-0.295696\pi$
$173$	207.140			1.19734			0.598670	+	0.800996i	$0.295696\pi$
$174$	93.6888			0.538442
$175$		−	59.0972i		−	0.337698i
$176$		−	83.0606i		−	0.471935i
$177$	−191.620			−1.08260
$178$			69.5484i			0.390722i
$179$	−74.8759			−0.418301			−0.209151	−	0.977883i	$-0.567070\pi$
$179$	−74.8759			−0.418301			−0.209151	+	0.977883i	$0.567070\pi$
$180$			13.4164i			0.0745356i
$181$		−	28.3027i		−	0.156369i	−0.996939	−	0.0781844i	$-0.975088\pi$
$181$		−	28.3027i		−	0.156369i	0.996939	−	0.0781844i	$-0.0249123\pi$
$182$			129.636i			0.712288i
$183$		−	28.8871i		−	0.157853i
$184$	−62.2165	+	19.0030i	−0.338133	+	0.103277i
$185$	−110.535			−0.597485
$186$	−81.6888			−0.439187
$187$	475.691			2.54380
$188$	−171.887			−0.914294
$189$		−	61.4157i		−	0.324951i
$190$	−66.4388			−0.349678
$191$		−	33.2592i		−	0.174132i	−0.996203	−	0.0870659i	$-0.972251\pi$
$191$		−	33.2592i		−	0.174132i	0.996203	−	0.0870659i	$-0.0277491\pi$
$192$	−13.8564			−0.0721688
$193$	93.6041			0.484995			0.242498	−	0.970152i	$-0.422033\pi$
$193$	93.6041			0.484995			0.242498	+	0.970152i	$0.422033\pi$
$194$			98.5336i			0.507905i
$195$		−	30.0373i		−	0.154037i
$196$	−181.399			−0.925503
$197$	372.604			1.89139			0.945696	−	0.325053i	$-0.105382\pi$
$197$	372.604			1.89139			0.945696	+	0.325053i	$0.105382\pi$
$198$		−	88.0991i		−	0.444945i
$199$		−	59.8348i		−	0.300678i	−0.988635	−	0.150339i	$-0.951964\pi$
$199$		−	59.8348i		−	0.300678i	0.988635	−	0.150339i	$-0.0480364\pi$
$200$	−14.1421			−0.0707107
$201$		−	204.135i		−	1.01560i
$202$	−24.4281			−0.120931
$203$		−	452.074i		−	2.22696i
$204$		−	79.3561i		−	0.389000i
$205$		−	1.51898i		−	0.00740965i
$206$		−	73.1750i		−	0.355218i
$207$	−65.9905	+	20.1557i	−0.318795	+	0.0973704i
$208$	31.0224			0.149146
$209$	436.272			2.08743
$210$	64.7378			0.308275
$211$	−152.649			−0.723455			−0.361728	−	0.932284i	$-0.617813\pi$
$211$	−152.649			−0.723455			−0.361728	+	0.932284i	$0.617813\pi$
$212$			133.396i			0.629227i
$213$	−50.2238			−0.235793
$214$		−	76.6732i		−	0.358286i
$215$	17.8817			0.0831709
$216$	−14.6969			−0.0680414
$217$			394.170i			1.81645i
$218$		−	0.0916298i		−	0.000420320i
$219$	−53.8239			−0.245771
$220$	92.8646			0.422112
$221$			177.666i			0.803919i
$222$		−	121.085i		−	0.545427i
$223$	205.883			0.923243			0.461622	−	0.887077i	$-0.347268\pi$
$223$	205.883			0.923243			0.461622	+	0.887077i	$0.347268\pi$
$224$			66.8609i			0.298486i
$225$	−15.0000			−0.0666667
$226$		−	222.432i		−	0.984210i
$227$		−	308.272i		−	1.35802i	−0.734127	−	0.679012i	$-0.762408\pi$
$227$		−	308.272i		−	1.35802i	0.734127	−	0.679012i	$-0.237592\pi$
$228$		−	72.7801i		−	0.319211i
$229$		−	26.6431i		−	0.116346i	−0.998307	−	0.0581728i	$-0.981473\pi$
$229$		−	26.6431i		−	0.116346i	0.998307	−	0.0581728i	$-0.0185274\pi$
$230$	−21.2460	−	69.5601i	−0.0923737	−	0.302435i
$231$	−425.102			−1.84027
$232$	−108.183			−0.466304
$233$	−362.883			−1.55744			−0.778720	−	0.627372i	$-0.784131\pi$
$233$	−362.883			−1.55744			−0.778720	+	0.627372i	$0.784131\pi$
$234$	32.9042			0.140616
$235$		−	192.176i		−	0.817769i
$236$	221.264			0.937560
$237$			181.984i			0.767867i
$238$	−382.914			−1.60888
$239$	375.990			1.57318			0.786591	−	0.617474i	$-0.211844\pi$
$239$	375.990			1.57318			0.786591	+	0.617474i	$0.211844\pi$
$240$		−	15.4919i		−	0.0645497i
$241$			214.703i			0.890882i	0.895311	+	0.445441i	$0.146953\pi$
$241$			214.703i			0.890882i	−0.895311	+	0.445441i	$0.853047\pi$
$242$	−438.677			−1.81272
$243$	−15.5885			−0.0641500
$244$			33.3559i			0.136705i
$245$		−	202.810i		−	0.827795i
$246$	1.66396			0.00676406
$247$			162.943i			0.659690i
$248$	94.3261			0.380347
$249$			74.8209i			0.300485i
$250$		−	15.8114i		−	0.0632456i
$251$		−	164.214i		−	0.654237i	−0.944983	−	0.327119i	$-0.893922\pi$
$251$		−	164.214i		−	0.654237i	0.944983	−	0.327119i	$-0.106078\pi$
$252$			70.9167i			0.281415i
$253$	139.512	+	456.768i	0.551431	+	1.80541i
$254$	188.614			0.742576
$255$	88.7228			0.347933
$256$	16.0000			0.0625000
$257$	92.0757			0.358271			0.179136	−	0.983824i	$-0.442670\pi$
$257$	92.0757			0.358271			0.179136	+	0.983824i	$0.442670\pi$
$258$			19.5885i			0.0759243i
$259$	−584.266			−2.25586
$260$			34.6841i			0.133400i
$261$	−114.745			−0.439636
$262$	88.9667			0.339567
$263$		−	340.726i		−	1.29554i	−0.761837	−	0.647769i	$-0.775702\pi$
$263$		−	340.726i		−	1.29554i	0.761837	−	0.647769i	$-0.224298\pi$
$264$			101.728i			0.385334i
$265$	−149.141			−0.562798
$266$	−351.184			−1.32024
$267$		−	85.1791i		−	0.319023i
$268$			235.715i			0.879532i
$269$	253.330			0.941748			0.470874	−	0.882200i	$-0.343939\pi$
$269$	253.330			0.941748			0.470874	+	0.882200i	$0.343939\pi$
$270$		−	16.4317i		−	0.0608581i
$271$	−21.6513			−0.0798939			−0.0399470	−	0.999202i	$-0.512719\pi$
$271$	−21.6513			−0.0798939			−0.0399470	+	0.999202i	$0.512719\pi$
$272$			91.6325i			0.336884i
$273$		−	158.772i		−	0.581581i
$274$		−	215.180i		−	0.785328i
$275$			103.826i			0.377548i
$276$	76.1993	−	23.2738i	0.276084	−	0.0843253i
$277$	459.389			1.65844			0.829222	−	0.558919i	$-0.188784\pi$
$277$	459.389			1.65844			0.829222	+	0.558919i	$0.188784\pi$
$278$	90.0074			0.323768
$279$	100.048			0.358595
$280$	−74.7527			−0.266974
$281$			318.671i			1.13406i	0.823697	+	0.567030i	$0.191908\pi$
$281$			318.671i			1.13406i	−0.823697	+	0.567030i	$0.808092\pi$
$282$	210.518			0.746518
$283$			5.36172i			0.0189460i	0.999955	+	0.00947301i	$0.00301540\pi$
$283$			5.36172i			0.0189460i	−0.999955	+	0.00947301i	$0.996985\pi$
$284$	57.9935			0.204202
$285$	81.3706			0.285511
$286$		−	227.753i		−	0.796341i
$287$		−	8.02904i		−	0.0279758i
$288$	16.9706			0.0589256
$289$	−235.782			−0.815856
$290$		−	120.952i		−	0.417075i
$291$		−	120.679i		−	0.414703i
$292$	62.1505			0.212844
$293$			519.450i			1.77287i	0.462854	+	0.886434i	$0.346825\pi$
$293$			519.450i			1.77287i	−0.462854	+	0.886434i	$0.653175\pi$
$294$	222.167			0.755670
$295$			247.381i			0.838579i
$296$			139.817i			0.472353i
$297$			107.899i			0.363296i
$298$			188.681i			0.633159i
$299$	−170.598	+	52.1064i	−0.570563	+	0.174269i
$300$	17.3205			0.0577350
$301$	94.5196			0.314019
$302$	188.818			0.625227
$303$	29.9181			0.0987397
$304$			84.0392i			0.276445i
$305$	−37.2930			−0.122272
$306$			97.1910i			0.317617i
$307$	−465.601			−1.51662			−0.758308	−	0.651897i	$-0.773974\pi$
$307$	−465.601			−1.51662			−0.758308	+	0.651897i	$0.773974\pi$
$308$	490.865			1.59372
$309$			89.6207i			0.290035i
$310$			105.460i			0.340193i
$311$	299.817			0.964043			0.482021	−	0.876159i	$-0.339903\pi$
$311$	299.817			0.964043			0.482021	+	0.876159i	$0.339903\pi$
$312$	−37.9945			−0.121777
$313$			473.837i			1.51386i	0.653498	+	0.756928i	$0.273301\pi$
$313$			473.837i			1.51386i	−0.653498	+	0.756928i	$0.726699\pi$
$314$		−	165.656i		−	0.527568i
$315$	−79.2873			−0.251706
$316$		−	210.138i		−	0.664992i
$317$	217.165			0.685065			0.342532	−	0.939506i	$-0.388715\pi$
$317$	217.165			0.685065			0.342532	+	0.939506i	$0.388715\pi$
$318$		−	163.376i		−	0.513762i
$319$			794.232i			2.48976i
$320$			17.8885i			0.0559017i
$321$			93.9051i			0.292539i
$322$	−112.302	−	367.682i	−0.348765	−	1.14187i
$323$	−481.295			−1.49008
$324$	18.0000			0.0555556
$325$	−38.7779			−0.119317
$326$	48.5229			0.148843
$327$			0.112223i			0.000343190i
$328$	−1.92137			−0.00585784
$329$		−	1015.81i		−	3.08756i
$330$	−113.735			−0.344653
$331$	−275.838			−0.833348			−0.416674	−	0.909056i	$-0.636804\pi$
$331$	−275.838			−0.833348			−0.416674	+	0.909056i	$0.636804\pi$
$332$		−	86.3957i		−	0.260228i
$333$			148.298i			0.445339i
$334$	−47.8816			−0.143358
$335$	−263.537			−0.786677
$336$		−	81.8875i		−	0.243713i
$337$		−	201.315i		−	0.597374i	−0.954351	−	0.298687i	$-0.903451\pi$
$337$		−	201.315i		−	0.597374i	0.954351	−	0.298687i	$-0.0965487\pi$
$338$	−153.938			−0.455439
$339$			272.422i			0.803604i
$340$	−102.448			−0.301318
$341$		−	692.503i		−	2.03080i
$342$			89.1371i			0.260635i
$343$		−	492.863i		−	1.43692i
$344$		−	22.6188i		−	0.0657524i
$345$	26.0209	+	85.1934i	0.0754228	+	0.246937i
$346$	292.940			0.846647
$347$	152.784			0.440300			0.220150	−	0.975466i	$-0.429345\pi$
$347$	152.784			0.440300			0.220150	+	0.975466i	$0.429345\pi$
$348$	132.496			0.380736
$349$	599.141			1.71674			0.858368	−	0.513035i	$-0.171479\pi$
$349$	599.141			1.71674			0.858368	+	0.513035i	$0.171479\pi$
$350$		−	83.5761i		−	0.238789i
$351$	−40.2992			−0.114813
$352$		−	117.465i		−	0.333709i
$353$	−574.915			−1.62865			−0.814327	−	0.580407i	$-0.802894\pi$
$353$	−574.915			−1.62865			−0.814327	+	0.580407i	$0.802894\pi$
$354$	−270.992			−0.765515
$355$			64.8387i			0.182644i
$356$			98.3563i			0.276282i
$357$	468.973			1.31365
$358$	−105.890			−0.295783
$359$		−	165.976i		−	0.462330i	−0.972915	−	0.231165i	$-0.925746\pi$
$359$		−	165.976i		−	0.462330i	0.972915	−	0.231165i	$-0.0742537\pi$
$360$			18.9737i			0.0527046i
$361$	−80.4119			−0.222748
$362$		−	40.0261i		−	0.110569i
$363$	537.268			1.48008
$364$			183.334i			0.503664i
$365$			69.4863i			0.190373i
$366$		−	40.8525i		−	0.111619i
$367$		−	433.965i		−	1.18247i	−0.806501	−	0.591233i	$-0.798642\pi$
$367$		−	433.965i		−	1.18247i	0.806501	−	0.591233i	$-0.201358\pi$
$368$	−87.9874	+	26.8742i	−0.239096	+	0.0730278i
$369$	−2.03792			−0.00552283
$370$	−156.320			−0.422486
$371$	−788.335			−2.12489
$372$	−115.525			−0.310552
$373$			342.059i			0.917048i	0.888682	+	0.458524i	$0.151622\pi$
$373$			342.059i			0.917048i	−0.888682	+	0.458524i	$0.848378\pi$
$374$	672.728			1.79874
$375$			19.3649i			0.0516398i
$376$	−243.085			−0.646503
$377$	−296.638			−0.786839
$378$		−	86.8548i		−	0.229775i
$379$			76.7922i			0.202618i	0.994855	+	0.101309i	$0.0323031\pi$
$379$			76.7922i			0.202618i	−0.994855	+	0.101309i	$0.967697\pi$
$380$	−93.9587			−0.247260
$381$	−231.004			−0.606310
$382$		−	47.0356i		−	0.123130i
$383$		−	78.7390i		−	0.205585i	−0.994703	−	0.102792i	$-0.967222\pi$
$383$		−	78.7390i		−	0.205585i	0.994703	−	0.102792i	$-0.0327777\pi$
$384$	−19.5959			−0.0510310
$385$			548.804i			1.42547i
$386$	132.376			0.342944
$387$		−	23.9909i		−	0.0619919i
$388$			139.348i			0.359143i
$389$			65.8586i			0.169302i	0.996411	+	0.0846511i	$0.0269776\pi$
$389$			65.8586i			0.169302i	−0.996411	+	0.0846511i	$0.973022\pi$
$390$		−	42.4791i		−	0.108921i
$391$	−153.910	−	503.906i	−0.393631	−	1.28876i
$392$	−256.536			−0.654430
$393$	−108.961			−0.277256
$394$	526.942			1.33742
$395$	234.941			0.594787
$396$		−	124.591i		−	0.314624i
$397$	−261.799			−0.659443			−0.329721	−	0.944078i	$-0.606955\pi$
$397$	−261.799			−0.659443			−0.329721	+	0.944078i	$0.606955\pi$
$398$		−	84.6192i		−	0.212611i
$399$	430.110			1.07797
$400$	−20.0000			−0.0500000
$401$			136.192i			0.339630i	0.985476	+	0.169815i	$0.0543171\pi$
$401$			136.192i			0.339630i	−0.985476	+	0.169815i	$0.945683\pi$
$402$		−	288.690i		−	0.718135i
$403$	258.644			0.641795
$404$	−34.5465			−0.0855111
$405$			20.1246i			0.0496904i
$406$		−	639.329i		−	1.57470i
$407$	1026.48			2.52205
$408$		−	112.226i		−	0.275065i
$409$	−162.202			−0.396583			−0.198291	−	0.980143i	$-0.563539\pi$
$409$	−162.202			−0.396583			−0.198291	+	0.980143i	$0.563539\pi$
$410$		−	2.14816i		−	0.00523942i
$411$			263.540i			0.641217i
$412$		−	103.485i		−	0.251177i
$413$			1307.61i			3.16613i
$414$	−93.3247	+	28.5044i	−0.225422	+	0.0688513i
$415$	96.5933			0.232755
$416$	43.8722			0.105462
$417$	−110.236			−0.264355
$418$	616.982			1.47603
$419$		−	746.726i		−	1.78216i	−0.453844	−	0.891081i	$-0.649948\pi$
$419$		−	746.726i		−	1.78216i	0.453844	−	0.891081i	$-0.350052\pi$
$420$	91.5530			0.217983
$421$			675.466i			1.60443i	0.597033	+	0.802216i	$0.296346\pi$
$421$			675.466i			1.60443i	−0.597033	+	0.802216i	$0.703654\pi$
$422$	−215.878			−0.511560
$423$	−257.831			−0.609529
$424$			188.651i			0.444931i
$425$		−	114.541i		−	0.269507i
$426$	−71.0272			−0.166730
$427$	−197.124			−0.461649
$428$		−	108.432i		−	0.253346i
$429$			278.940i			0.650210i
$430$	25.2886			0.0588107
$431$			270.979i			0.628721i	0.949304	+	0.314360i	$0.101790\pi$
$431$			270.979i			0.628721i	−0.949304	+	0.314360i	$0.898210\pi$
$432$	−20.7846			−0.0481125
$433$		−	549.884i		−	1.26994i	−0.772537	−	0.634970i	$-0.781012\pi$
$433$		−	549.884i		−	1.26994i	0.772537	−	0.634970i	$-0.218988\pi$
$434$			557.441i			1.28443i
$435$			148.135i			0.340540i
$436$		−	0.129584i		−	0.000297211i
$437$	−141.156	−	462.149i	−0.323011	−	1.05755i
$438$	−76.1184			−0.173786
$439$	342.257			0.779628			0.389814	−	0.920894i	$-0.372539\pi$
$439$	342.257			0.779628			0.389814	+	0.920894i	$0.372539\pi$
$440$	131.330			0.298478
$441$	−272.098			−0.617002
$442$			251.258i			0.568456i
$443$	476.091			1.07470			0.537349	−	0.843360i	$-0.319426\pi$
$443$	476.091			1.07470			0.537349	+	0.843360i	$0.319426\pi$
$444$		−	171.240i		−	0.385675i
$445$	−109.966			−0.247114
$446$	291.163			0.652832
$447$		−	231.086i		−	0.516972i
$448$			94.5556i			0.211062i
$449$	554.189			1.23427			0.617137	−	0.786855i	$-0.288292\pi$
$449$	554.189			1.23427			0.617137	+	0.786855i	$0.288292\pi$
$450$	−21.2132			−0.0471405
$451$			14.1059i			0.0312770i
$452$		−	314.566i		−	0.695942i
$453$	−231.254			−0.510496
$454$		−	435.962i		−	0.960269i
$455$	−204.973			−0.450490
$456$		−	102.927i		−	0.225716i
$457$		−	142.050i		−	0.310832i	−0.987849	−	0.155416i	$-0.950328\pi$
$457$		−	142.050i		−	0.310832i	0.987849	−	0.155416i	$-0.0496718\pi$
$458$		−	37.6791i		−	0.0822688i
$459$		−	119.034i		−	0.259334i
$460$	−30.0463	−	98.3729i	−0.0653181	−	0.213854i
$461$	−618.196			−1.34099			−0.670495	−	0.741914i	$-0.733918\pi$
$461$	−618.196			−1.34099			−0.670495	+	0.741914i	$0.733918\pi$
$462$	−601.185			−1.30127
$463$	115.319			0.249069			0.124534	−	0.992215i	$-0.460256\pi$
$463$	115.319			0.249069			0.124534	+	0.992215i	$0.460256\pi$
$464$	−152.993			−0.329727
$465$		−	129.161i		−	0.277766i
$466$	−513.195			−1.10128
$467$			778.924i			1.66793i	0.551817	+	0.833966i	$0.313935\pi$
$467$			778.924i			1.66793i	−0.551817	+	0.833966i	$0.686065\pi$
$468$	46.5335			0.0994306
$469$	−1393.01			−2.97017
$470$		−	271.777i		−	0.578250i
$471$			202.887i			0.430757i
$472$	312.915			0.662955
$473$	−166.058			−0.351074
$474$			257.365i			0.542964i
$475$		−	105.049i		−	0.221156i
$476$	−541.523			−1.13765
$477$			200.094i			0.419485i
$478$	531.731			1.11241
$479$			18.0817i			0.0377488i	0.999822	+	0.0188744i	$0.00600826\pi$
$479$			18.0817i			0.0377488i	−0.999822	+	0.0188744i	$0.993992\pi$
$480$		−	21.9089i		−	0.0456435i
$481$			383.379i			0.797046i
$482$			303.635i			0.629949i
$483$	137.542	+	450.317i	0.284765	+	0.932333i
$484$	−620.383			−1.28178
$485$	−155.795			−0.321228
$486$	−22.0454			−0.0453609
$487$	522.588			1.07308			0.536538	−	0.843876i	$-0.319732\pi$
$487$	522.588			1.07308			0.536538	+	0.843876i	$0.319732\pi$
$488$			47.1724i			0.0966647i
$489$	−59.4281			−0.121530
$490$		−	286.816i		−	0.585340i
$491$	524.138			1.06749			0.533746	−	0.845645i	$-0.320784\pi$
$491$	524.138			1.06749			0.533746	+	0.845645i	$0.320784\pi$
$492$	2.35319			0.00478291
$493$		−	876.197i		−	1.77728i
$494$			230.437i			0.466471i
$495$	139.297			0.281408
$496$	133.397			0.268946
$497$			342.725i			0.689588i
$498$			105.813i			0.212475i
$499$	−384.131			−0.769801			−0.384900	−	0.922958i	$-0.625764\pi$
$499$	−384.131			−0.769801			−0.384900	+	0.922958i	$0.625764\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	58.6428			0.117052
$502$		−	232.233i		−	0.462616i
$503$		−	815.894i		−	1.62206i	−0.585007	−	0.811028i	$-0.698908\pi$
$503$		−	815.894i		−	1.62206i	0.585007	−	0.811028i	$-0.301092\pi$
$504$			100.291i			0.198991i
$505$		−	38.6241i		−	0.0764835i
$506$	197.300	+	645.967i	0.389920	+	1.27661i
$507$	188.535			0.371864
$508$	266.741			0.525080
$509$	−138.733			−0.272560			−0.136280	−	0.990670i	$-0.543515\pi$
$509$	−138.733			−0.272560			−0.136280	+	0.990670i	$0.543515\pi$
$510$	125.473			0.246025
$511$			367.292i			0.718771i
$512$	22.6274			0.0441942
$513$		−	109.170i		−	0.212807i
$514$	130.215			0.253336
$515$	115.700			0.224660
$516$			27.7023i			0.0536866i
$517$			1784.63i			3.45190i
$518$	−826.278			−1.59513
$519$	−358.777			−0.691285
$520$			49.0507i			0.0943282i
$521$		−	79.0746i		−	0.151775i	−0.997116	−	0.0758873i	$-0.975821\pi$
$521$		−	79.0746i		−	0.151775i	0.997116	−	0.0758873i	$-0.0241789\pi$
$522$	−162.274			−0.310869
$523$			188.376i			0.360184i	0.983650	+	0.180092i	$0.0576396\pi$
$523$			188.376i			0.360184i	−0.983650	+	0.180092i	$0.942360\pi$
$524$	125.818			0.240110
$525$			102.359i			0.194970i
$526$		−	481.860i		−	0.916083i
$527$			763.970i			1.44966i
$528$			143.865i			0.272472i
$529$	438.722	−	295.574i	0.829342	−	0.558741i
$530$	−210.918			−0.397958
$531$	331.896			0.625040
$532$	−496.649			−0.933550
$533$	−5.26843			−0.00988449
$534$		−	120.461i		−	0.225583i
$535$	121.231			0.226600
$536$			333.351i			0.621923i
$537$	129.689			0.241506
$538$	358.263			0.665917
$539$			1883.39i			3.49422i
$540$		−	23.2379i		−	0.0430331i
$541$	−362.416			−0.669900			−0.334950	−	0.942236i	$-0.608719\pi$
$541$	−362.416			−0.669900			−0.334950	+	0.942236i	$0.608719\pi$
$542$	−30.6195			−0.0564935
$543$			49.0218i			0.0902795i
$544$			129.588i			0.238213i
$545$	0.144879			0.000265834
$546$		−	224.537i		−	0.411240i
$547$	−593.659			−1.08530			−0.542650	−	0.839959i	$-0.682579\pi$
$547$	−593.659			−1.08530			−0.542650	+	0.839959i	$0.682579\pi$
$548$		−	304.310i		−	0.555311i
$549$			50.0339i			0.0911364i
$550$			146.832i			0.266967i
$551$		−	803.589i		−	1.45842i
$552$	107.762	−	32.9141i	0.195221	−	0.0596270i
$553$	1241.85			2.24567
$554$	649.674			1.17270
$555$	191.452			0.344958
$556$	127.290			0.228938
$557$			183.937i			0.330229i	0.986274	+	0.165114i	$0.0527993\pi$
$557$			183.937i			0.330229i	−0.986274	+	0.165114i	$0.947201\pi$
$558$	141.489			0.253565
$559$		−	62.0211i		−	0.110950i
$560$	−105.716			−0.188779
$561$	−823.921			−1.46866
$562$			450.668i			0.801901i
$563$			447.708i			0.795218i	0.917555	+	0.397609i	$0.130160\pi$
$563$			447.708i			0.795218i	−0.917555	+	0.397609i	$0.869840\pi$
$564$	297.717			0.527868
$565$	351.695			0.622469
$566$			7.58262i			0.0133969i
$567$			106.375i			0.187610i
$568$	82.0151			0.144393
$569$		−	334.865i		−	0.588515i	−0.955726	−	0.294258i	$-0.904928\pi$
$569$		−	334.865i		−	0.588515i	0.955726	−	0.294258i	$-0.0950724\pi$
$570$	115.075			0.201887
$571$		−	644.554i		−	1.12882i	−0.825496	−	0.564408i	$-0.809104\pi$
$571$		−	644.554i		−	1.12882i	0.825496	−	0.564408i	$-0.190896\pi$
$572$		−	322.092i		−	0.563098i
$573$			57.6066i			0.100535i
$574$		−	11.3548i		−	0.0197819i
$575$	109.984	−	33.5928i	0.191277	−	0.0584223i
$576$	24.0000			0.0416667
$577$	82.7980			0.143497			0.0717487	−	0.997423i	$-0.477142\pi$
$577$	82.7980			0.143497			0.0717487	+	0.997423i	$0.477142\pi$
$578$	−333.447			−0.576897
$579$	−162.127			−0.280012
$580$		−	171.052i		−	0.294917i
$581$	510.575			0.878786
$582$		−	170.665i		−	0.293239i
$583$	1385.00			2.37564
$584$	87.8940			0.150503
$585$			52.0261i			0.0889335i
$586$			734.614i			1.25361i
$587$	48.0652			0.0818827			0.0409414	−	0.999162i	$-0.486964\pi$
$587$	48.0652			0.0818827			0.0409414	+	0.999162i	$0.486964\pi$
$588$	314.192			0.534340
$589$			700.662i			1.18958i
$590$			349.849i			0.592965i
$591$	−645.369			−1.09200
$592$			197.731i			0.334004i
$593$	−115.147			−0.194176			−0.0970882	−	0.995276i	$-0.530953\pi$
$593$	−115.147			−0.194176			−0.0970882	+	0.995276i	$0.530953\pi$
$594$			152.592i			0.256889i
$595$		−	605.441i		−	1.01755i
$596$			266.836i			0.447711i
$597$			103.637i			0.173596i
$598$	−241.263	+	73.6896i	−0.403449	+	0.123227i
$599$	−813.749			−1.35851			−0.679256	−	0.733901i	$-0.737697\pi$
$599$	−813.749			−1.35851			−0.679256	+	0.733901i	$0.737697\pi$
$600$	24.4949			0.0408248
$601$	−550.200			−0.915474			−0.457737	−	0.889088i	$-0.651340\pi$
$601$	−550.200			−0.915474			−0.457737	+	0.889088i	$0.651340\pi$
$602$	133.671			0.222045
$603$			353.572i			0.586355i
$604$	267.030			0.442102
$605$		−	693.609i		−	1.14646i
$606$	42.3106			0.0698195
$607$	287.538			0.473703			0.236851	−	0.971546i	$-0.423885\pi$
$607$	287.538			0.473703			0.236851	+	0.971546i	$0.423885\pi$
$608$			118.849i			0.195476i
$609$			783.015i			1.28574i
$610$	−52.7403			−0.0864596
$611$	−666.543			−1.09091
$612$			137.449i			0.224589i
$613$		−	573.173i		−	0.935029i	−0.883985	−	0.467514i	$-0.845150\pi$
$613$		−	573.173i		−	0.935029i	0.883985	−	0.467514i	$-0.154850\pi$
$614$	−658.459			−1.07241
$615$			2.63095i			0.00427796i
$616$	694.188			1.12693
$617$			1156.17i			1.87385i	0.349525	+	0.936927i	$0.386343\pi$
$617$			1156.17i			1.87385i	−0.349525	+	0.936927i	$0.613657\pi$
$618$			126.743i			0.205085i
$619$		−	14.9600i		−	0.0241680i	−0.999927	−	0.0120840i	$-0.996153\pi$
$619$		−	14.9600i		−	0.0241680i	0.999927	−	0.0120840i	$-0.00384655\pi$
$620$			149.143i			0.240553i
$621$	114.299	−	34.9107i	0.184056	−	0.0562168i
$622$	424.006			0.681681
$623$	−581.259			−0.933000
$624$	−53.7323			−0.0861095
$625$	25.0000			0.0400000
$626$			670.107i			1.07046i
$627$	−755.645			−1.20518
$628$		−	234.273i		−	0.373047i
$629$	−1132.41			−1.80033
$630$	−112.129			−0.177983
$631$			361.056i			0.572196i	0.958200	+	0.286098i	$0.0923583\pi$
$631$			361.056i			0.572196i	−0.958200	+	0.286098i	$0.907642\pi$
$632$		−	297.179i		−	0.470220i
$633$	264.396			0.417687
$634$	307.118			0.484414
$635$			298.225i			0.469646i
$636$		−	231.049i		−	0.363285i
$637$	−703.427			−1.10428
$638$			1123.21i			1.76052i
$639$	86.9902			0.136135
$640$			25.2982i			0.0395285i
$641$			1132.34i			1.76652i	0.468881	+	0.883261i	$0.344657\pi$
$641$			1132.34i			1.76652i	−0.468881	+	0.883261i	$0.655343\pi$
$642$			132.802i			0.206856i
$643$			753.140i			1.17129i	0.810567	+	0.585645i	$0.199159\pi$
$643$			753.140i			1.17129i	−0.810567	+	0.585645i	$0.800841\pi$
$644$	−158.819	−	519.981i	−0.246614	−	0.807424i
$645$	−30.9721			−0.0480187
$646$	−680.654			−1.05364
$647$	−32.8271			−0.0507375			−0.0253687	−	0.999678i	$-0.508076\pi$
$647$	−32.8271			−0.0507375			−0.0253687	+	0.999678i	$0.508076\pi$
$648$	25.4558			0.0392837
$649$		−	2297.29i		−	3.53974i
$650$	−54.8403			−0.0843697
$651$		−	682.723i		−	1.04873i
$652$	68.6217			0.105248
$653$	−1288.05			−1.97251			−0.986257	−	0.165219i	$-0.947167\pi$
$653$	−1288.05			−1.97251			−0.986257	+	0.165219i	$0.947167\pi$
$654$			0.158707i			0.000242672i
$655$			140.669i			0.214761i
$656$	−2.71723			−0.00414212
$657$	93.2257			0.141896
$658$		−	1436.57i		−	2.18323i
$659$			101.045i			0.153330i	0.997057	+	0.0766652i	$0.0244273\pi$
$659$			101.045i			0.153330i	−0.997057	+	0.0766652i	$0.975573\pi$
$660$	−160.846			−0.243706
$661$		−	597.593i		−	0.904074i	−0.891999	−	0.452037i	$-0.850698\pi$
$661$		−	597.593i		−	0.904074i	0.891999	−	0.452037i	$-0.149302\pi$
$662$	−390.094			−0.589266
$663$		−	307.727i		−	0.464143i
$664$		−	122.182i		−	0.184009i
$665$		−	555.270i		−	0.834992i
$666$			209.725i			0.314902i
$667$	841.342	−	256.974i	1.26138	−	0.385268i
$668$	−67.7149			−0.101370
$669$	−356.600			−0.533035
$670$	−372.697			−0.556265
$671$	346.320			0.516126
$672$		−	115.806i		−	0.172331i
$673$	178.729			0.265570			0.132785	−	0.991145i	$-0.457608\pi$
$673$	178.729			0.265570			0.132785	+	0.991145i	$0.457608\pi$
$674$		−	284.703i		−	0.422407i
$675$	25.9808			0.0384900
$676$	−217.702			−0.322044
$677$		−	1197.98i		−	1.76955i	−0.466023	−	0.884773i	$-0.654314\pi$
$677$		−	1197.98i		−	1.76955i	0.466023	−	0.884773i	$-0.345686\pi$
$678$			385.263i			0.568234i
$679$	−823.506			−1.21282
$680$	−144.884			−0.213064
$681$			533.942i			0.784056i
$682$		−	979.348i		−	1.43599i
$683$	264.345			0.387035			0.193518	−	0.981097i	$-0.438010\pi$
$683$	264.345			0.387035			0.193518	+	0.981097i	$0.438010\pi$
$684$			126.059i			0.184297i
$685$	340.229			0.496685
$686$		−	697.014i		−	1.01605i
$687$			46.1473i			0.0671722i
$688$		−	31.9878i		−	0.0464939i
$689$			517.283i			0.750774i
$690$	36.7991	+	120.482i	0.0533320	+	0.174611i
$691$	236.685			0.342525			0.171262	−	0.985225i	$-0.445215\pi$
$691$	236.685			0.342525			0.171262	+	0.985225i	$0.445215\pi$
$692$	414.280			0.598670
$693$	736.298			1.06248
$694$	216.069			0.311339
$695$			142.314i			0.204769i
$696$	187.378			0.269221
$697$		−	15.5617i		−	0.0223266i
$698$	847.313			1.21392
$699$	628.533			0.899188
$700$		−	118.194i		−	0.168849i
$701$		−	322.720i		−	0.460371i	−0.973147	−	0.230186i	$-0.926067\pi$
$701$		−	322.720i		−	0.460371i	0.973147	−	0.230186i	$-0.0739334\pi$
$702$	−56.9917			−0.0811848
$703$	−1038.57			−1.47734
$704$		−	166.121i		−	0.235968i
$705$			332.858i			0.472139i
$706$	−813.052			−1.15163
$707$		−	204.160i		−	0.288770i
$708$	−383.241			−0.541301
$709$		−	100.401i		−	0.141609i	−0.997490	−	0.0708046i	$-0.977443\pi$
$709$		−	100.401i		−	0.141609i	0.997490	−	0.0708046i	$-0.0225567\pi$
$710$			91.6957i			0.129149i
$711$		−	315.206i		−	0.443328i
$712$			139.097i			0.195361i
$713$	−733.579	+	224.059i	−1.02886	+	0.314249i
$714$	663.227			0.928890
$715$	360.110			0.503650
$716$	−149.752			−0.209151
$717$	−651.235			−0.908277
$718$		−	234.726i		−	0.326917i
$719$	1250.73			1.73953			0.869767	−	0.493462i	$-0.164269\pi$
$719$	1250.73			1.73953			0.869767	+	0.493462i	$0.164269\pi$
$720$			26.8328i			0.0372678i
$721$	611.568			0.848222
$722$	−113.720			−0.157506
$723$		−	371.876i		−	0.514351i
$724$		−	56.6055i		−	0.0781844i
$725$	191.242			0.263781
$726$	759.811			1.04657
$727$		−	528.487i		−	0.726942i	−0.931605	−	0.363471i	$-0.881592\pi$
$727$		−	528.487i		−	0.726942i	0.931605	−	0.363471i	$-0.118408\pi$
$728$			259.273i			0.356144i
$729$	27.0000			0.0370370
$730$			98.2685i			0.134614i
$731$	183.195			0.250609
$732$		−	57.7741i		−	0.0789264i
$733$		−	37.5960i		−	0.0512905i	−0.999671	−	0.0256453i	$-0.991836\pi$
$733$		−	37.5960i		−	0.0512905i	0.999671	−	0.0256453i	$-0.00816403\pi$
$734$		−	613.719i		−	0.836129i
$735$			351.277i			0.477928i
$736$	−124.433	+	38.0059i	−0.169066	+	0.0516385i
$737$	2447.32			3.32066
$738$	−2.88206			−0.00390523
$739$	−1383.68			−1.87237			−0.936187	−	0.351502i	$-0.885671\pi$
$739$	−1383.68			−1.87237			−0.936187	+	0.351502i	$0.885671\pi$
$740$	−221.069			−0.298743
$741$		−	282.226i		−	0.380872i
$742$	−1114.87			−1.50253
$743$			143.332i			0.192909i	0.995337	+	0.0964547i	$0.0307503\pi$
$743$			143.332i			0.192909i	−0.995337	+	0.0964547i	$0.969250\pi$
$744$	−163.378			−0.219593
$745$	−298.331			−0.400445
$746$			483.745i			0.648451i
$747$		−	129.594i		−	0.173485i
$748$	951.382			1.27190
$749$	640.805			0.855547
$750$			27.3861i			0.0365148i
$751$		−	1057.29i		−	1.40784i	−0.710277	−	0.703922i	$-0.751431\pi$
$751$		−	1057.29i		−	1.40784i	0.710277	−	0.703922i	$-0.248569\pi$
$752$	−343.774			−0.457147
$753$			284.426i			0.377724i
$754$	−419.510			−0.556379
$755$			298.548i			0.395428i
$756$		−	122.831i		−	0.162475i
$757$			85.5898i			0.113065i	0.998401	+	0.0565323i	$0.0180044\pi$
$757$			85.5898i			0.113065i	−0.998401	+	0.0565323i	$0.981996\pi$
$758$			108.601i			0.143272i
$759$	−241.642	−	791.145i	−0.318369	−	1.04235i
$760$	−132.878			−0.174839
$761$	−751.679			−0.987752			−0.493876	−	0.869532i	$-0.664420\pi$
$761$	−751.679			−0.987752			−0.493876	+	0.869532i	$0.664420\pi$
$762$	−326.689			−0.428726
$763$	0.765806			0.00100368
$764$		−	66.5184i		−	0.0870659i
$765$	−153.672			−0.200879
$766$		−	111.354i		−	0.145370i
$767$	858.017			1.11867
$768$	−27.7128			−0.0360844
$769$			80.5080i			0.104692i	0.998629	+	0.0523459i	$0.0166698\pi$
$769$			80.5080i			0.104692i	−0.998629	+	0.0523459i	$0.983330\pi$
$770$			776.126i			1.00796i
$771$	−159.480			−0.206848
$772$	187.208			0.242498
$773$			1038.03i			1.34286i	0.741070	+	0.671428i	$0.234319\pi$
$773$			1038.03i			1.34286i	−0.741070	+	0.671428i	$0.765681\pi$
$774$		−	33.9282i		−	0.0438349i
$775$	−166.747			−0.215157
$776$			197.067i			0.253953i
$777$	1011.98			1.30242
$778$			93.1381i			0.119715i
$779$		−	14.2721i		−	0.0183211i
$780$		−	60.0745i		−	0.0770186i
$781$		−	602.122i		−	0.770962i
$782$	−217.661	−	712.631i	−0.278339	−	0.911293i
$783$	198.744			0.253824
$784$	−362.797			−0.462752
$785$	261.926			0.333663
$786$	−154.095			−0.196049
$787$		−	1078.55i		−	1.37045i	−0.728330	−	0.685227i	$-0.759703\pi$
$787$		−	1078.55i		−	1.37045i	0.728330	−	0.685227i	$-0.240297\pi$
$788$	745.208			0.945696
$789$			590.155i			0.747979i
$790$	332.257			0.420578
$791$	1859.00			2.35019
$792$		−	176.198i		−	0.222472i
$793$			129.347i			0.163111i
$794$	−370.239			−0.466297
$795$	258.321			0.324932
$796$		−	119.670i		−	0.150339i
$797$		−	407.609i		−	0.511429i	−0.966752	−	0.255714i	$-0.917689\pi$
$797$		−	407.609i		−	0.511429i	0.966752	−	0.255714i	$-0.0823106\pi$
$798$	608.268			0.762240
$799$		−	1968.81i		−	2.46409i
$800$	−28.2843			−0.0353553
$801$			147.535i			0.184188i
$802$			192.604i			0.240155i
$803$		−	645.282i		−	0.803589i
$804$		−	408.270i		−	0.507798i
$805$	581.356	−	177.565i	0.722182	−	0.220578i
$806$	365.777			0.453818
$807$	−438.781			−0.543719
$808$	−48.8561			−0.0604655
$809$	788.093			0.974158			0.487079	−	0.873358i	$-0.338062\pi$
$809$	788.093			0.974158			0.487079	+	0.873358i	$0.338062\pi$
$810$			28.4605i			0.0351364i
$811$	779.565			0.961239			0.480619	−	0.876929i	$-0.340412\pi$
$811$	779.565			0.961239			0.480619	+	0.876929i	$0.340412\pi$
$812$		−	904.148i		−	1.11348i
$813$	37.5011			0.0461268
$814$	1451.66			1.78336
$815$			76.7214i			0.0941366i
$816$		−	158.712i		−	0.194500i
$817$	168.015			0.205648
$818$	−229.389			−0.280426
$819$			275.000i			0.335776i
$820$		−	3.03796i		−	0.00370483i
$821$	1297.12			1.57992			0.789962	−	0.613156i	$-0.210100\pi$
$821$	1297.12			1.57992			0.789962	+	0.613156i	$0.210100\pi$
$822$			372.702i			0.453409i
$823$	−562.355			−0.683299			−0.341649	−	0.939828i	$-0.610986\pi$
$823$	−562.355			−0.683299			−0.341649	+	0.939828i	$0.610986\pi$
$824$		−	146.350i		−	0.177609i
$825$		−	179.832i		−	0.217978i
$826$			1849.24i			2.23879i
$827$			1203.99i			1.45585i	0.685656	+	0.727926i	$0.259515\pi$
$827$			1203.99i			1.45585i	−0.685656	+	0.727926i	$0.740485\pi$
$828$	−131.981	+	40.3114i	−0.159397	+	0.0486852i
$829$	1175.99			1.41856			0.709282	−	0.704925i	$-0.249019\pi$
$829$	1175.99			1.41856			0.709282	+	0.704925i	$0.249019\pi$
$830$	136.604			0.164583
$831$	−795.685			−0.957503
$832$	62.0447			0.0745730
$833$		−	2077.75i		−	2.49430i
$834$	−155.897			−0.186927
$835$		−	75.7075i		−	0.0906677i
$836$	872.544			1.04371
$837$	−173.288			−0.207035
$838$		−	1056.03i		−	1.26018i
$839$			620.713i			0.739825i	0.929067	+	0.369912i	$0.120612\pi$
$839$			620.713i			0.739825i	−0.929067	+	0.369912i	$0.879388\pi$
$840$	129.476			0.154138
$841$	621.933			0.739516
$842$			955.254i			1.13451i
$843$		−	551.954i		−	0.654749i
$844$	−305.298			−0.361728
$845$		−	243.398i		−	0.288045i
$846$	−364.628			−0.431002
$847$		−	3666.29i		−	4.32856i
$848$			266.792i			0.314614i
$849$		−	9.28678i		−	0.0109385i
$850$		−	161.985i		−	0.190570i
$851$	−332.116	−	1087.36i	−0.390266	−	1.27775i
$852$	−100.448			−0.117896
$853$	−576.042			−0.675314			−0.337657	−	0.941269i	$-0.609634\pi$
$853$	−576.042			−0.675314			−0.337657	+	0.941269i	$0.609634\pi$
$854$	−278.776			−0.326435
$855$	−140.938			−0.164840
$856$		−	153.346i		−	0.179143i
$857$	1079.53			1.25966			0.629832	−	0.776731i	$-0.283124\pi$
$857$	1079.53			1.25966			0.629832	+	0.776731i	$0.283124\pi$
$858$			394.481i			0.459768i
$859$	−314.625			−0.366269			−0.183134	−	0.983088i	$-0.558624\pi$
$859$	−314.625			−0.366269			−0.183134	+	0.983088i	$0.558624\pi$
$860$	35.7635			0.0415854
$861$			13.9067i			0.0161518i
$862$			383.222i			0.444573i
$863$	−1073.21			−1.24358			−0.621791	−	0.783183i	$-0.713595\pi$
$863$	−1073.21			−1.24358			−0.621791	+	0.783183i	$0.713595\pi$
$864$	−29.3939			−0.0340207
$865$			463.179i			0.535467i
$866$		−	777.654i		−	0.897983i
$867$	408.387			0.471035
$868$			788.341i			0.908227i
$869$	−2181.77			−2.51067
$870$			209.495i			0.240798i
$871$			914.053i			1.04943i
$872$		−	0.183260i		−	0.000210160i
$873$			209.021i			0.239429i
$874$	−199.624	−	653.578i	−0.228403	−	0.747801i
$875$	132.145			0.151023
$876$	−107.648			−0.122886
$877$	−26.5292			−0.0302500			−0.0151250	−	0.999886i	$-0.504815\pi$
$877$	−26.5292			−0.0302500			−0.0151250	+	0.999886i	$0.504815\pi$
$878$	484.024			0.551280
$879$		−	899.715i		−	1.02357i
$880$	185.729			0.211056
$881$			528.674i			0.600084i	0.953926	+	0.300042i	$0.0970007\pi$
$881$			528.674i			0.600084i	−0.953926	+	0.300042i	$0.902999\pi$
$882$	−384.805			−0.436287
$883$	1219.35			1.38092			0.690461	−	0.723370i	$-0.257408\pi$
$883$	1219.35			1.38092			0.690461	+	0.723370i	$0.257408\pi$
$884$			355.332i			0.401959i
$885$		−	428.476i		−	0.484154i
$886$	673.294			0.759926
$887$	−1586.65			−1.78878			−0.894390	−	0.447289i	$-0.852390\pi$
$887$	−1586.65			−1.78878			−0.894390	+	0.447289i	$0.852390\pi$
$888$		−	242.169i		−	0.272713i
$889$			1576.36i			1.77319i
$890$	−155.515			−0.174736
$891$		−	186.886i		−	0.209749i
$892$	411.767			0.461622
$893$		−	1805.66i		−	2.02201i
$894$		−	326.805i		−	0.365554i
$895$		−	167.428i		−	0.187070i
$896$			133.722i			0.149243i
$897$	295.485	−	90.2509i	0.329415	−	0.100614i
$898$	783.742			0.872764
$899$	−1275.55			−1.41886
$900$	−30.0000			−0.0333333
$901$	−1527.93			−1.69581
$902$			19.9488i			0.0221162i
$903$	−163.713			−0.181299
$904$		−	444.863i		−	0.492105i
$905$	63.2868			0.0699302
$906$	−327.043			−0.360975
$907$		−	107.955i		−	0.119024i	−0.998228	−	0.0595122i	$-0.981045\pi$
$907$		−	107.955i		−	0.119024i	0.998228	−	0.0595122i	$-0.0189545\pi$
$908$		−	616.543i		−	0.679012i
$909$	−51.8197			−0.0570074
$910$	−289.876			−0.318545
$911$			949.537i			1.04230i	0.853464	+	0.521151i	$0.174497\pi$
$911$			949.537i			1.04230i	−0.853464	+	0.521151i	$0.825503\pi$
$912$		−	145.560i		−	0.159605i
$913$	−897.010			−0.982486
$914$		−	200.889i		−	0.219791i
$915$	64.5935			0.0705939
$916$		−	53.2863i		−	0.0581728i
$917$			743.549i			0.810849i
$918$		−	168.340i		−	0.183377i
$919$			88.6314i			0.0964433i	0.998837	+	0.0482217i	$0.0153554\pi$
$919$			88.6314i			0.0964433i	−0.998837	+	0.0482217i	$0.984645\pi$
$920$	−42.4919	−	139.120i	−0.0461869	−	0.151218i
$921$	806.445			0.875619
$922$	−874.261			−0.948223
$923$	224.887			0.243648
$924$	−850.204			−0.920134
$925$		−	247.163i		−	0.267203i
$926$	163.085			0.176118
$927$		−	155.228i		−	0.167452i
$928$	−216.365			−0.233152
$929$	−614.940			−0.661938			−0.330969	−	0.943642i	$-0.607375\pi$
$929$	−614.940			−0.661938			−0.330969	+	0.943642i	$0.607375\pi$
$930$		−	182.662i		−	0.196410i
$931$		−	1905.58i		−	2.04680i
$932$	−725.767			−0.778720
$933$	−519.299			−0.556590
$934$			1101.56i			1.17941i
$935$			1063.68i			1.13762i
$936$	65.8084			0.0703081
$937$			79.7852i			0.0851496i	0.999093	+	0.0425748i	$0.0135561\pi$
$937$			79.7852i			0.0851496i	−0.999093	+	0.0425748i	$0.986444\pi$
$938$	−1970.01			−2.10022
$939$		−	820.710i		−	0.874025i
$940$		−	384.351i		−	0.408884i
$941$		−	382.492i		−	0.406474i	−0.979130	−	0.203237i	$-0.934854\pi$
$941$		−	382.492i		−	0.406474i	0.979130	−	0.203237i	$-0.0651462\pi$
$942$			286.925i			0.304591i
$943$	14.9426	−	4.56397i	0.0158458	−	0.00483984i
$944$	442.528			0.468780
$945$	137.330			0.145322
$946$	−234.842			−0.248247
$947$	−1415.40			−1.49462			−0.747309	−	0.664476i	$-0.768655\pi$
$947$	−1415.40			−1.49462			−0.747309	+	0.664476i	$0.768655\pi$
$948$			363.969i			0.383933i
$949$	241.007			0.253959
$950$		−	148.562i		−	0.156381i
$951$	−376.142			−0.395522
$952$	−765.829			−0.804442
$953$		−	1298.22i		−	1.36224i	−0.732170	−	0.681122i	$-0.761492\pi$
$953$		−	1298.22i		−	1.36224i	0.732170	−	0.681122i	$-0.238508\pi$
$954$			282.976i			0.296621i
$955$	74.3698			0.0778741
$956$	751.981			0.786591
$957$		−	1375.65i		−	1.43746i
$958$			25.5713i			0.0266924i
$959$	1798.39			1.87528
$960$		−	30.9839i		−	0.0322749i
$961$	151.176			0.157311
$962$			542.180i			0.563597i
$963$		−	162.648i		−	0.168898i
$964$			429.405i			0.445441i
$965$			209.305i			0.216897i
$966$	194.513	+	636.844i	0.201359	+	0.659259i
$967$	−1707.55			−1.76582			−0.882912	−	0.469538i	$-0.844420\pi$
$967$	−1707.55			−1.76582			−0.882912	+	0.469538i	$0.844420\pi$
$968$	−877.354			−0.906358
$969$	833.628			0.860297
$970$	−220.328			−0.227142
$971$		−	195.262i		−	0.201094i	−0.994932	−	0.100547i	$-0.967941\pi$
$971$		−	195.262i		−	0.201094i	0.994932	−	0.100547i	$-0.0320593\pi$
$972$	−31.1769			−0.0320750
$973$			752.247i			0.773121i
$974$	739.051			0.758779
$975$	67.1654			0.0688876
$976$			66.7118i			0.0683523i
$977$		−	216.403i		−	0.221497i	−0.993848	−	0.110748i	$-0.964675\pi$
$977$		−	216.403i		−	0.221497i	0.993848	−	0.110748i	$-0.0353248\pi$
$978$	−84.0440			−0.0859346
$979$	1021.19			1.04310
$980$		−	405.620i		−	0.413898i
$981$		−	0.194376i		−	0.000198141i
$982$	741.243			0.754830
$983$			1395.17i			1.41929i	0.704557	+	0.709647i	$0.251146\pi$
$983$			1395.17i			1.41929i	−0.704557	+	0.709647i	$0.748854\pi$
$984$	3.32792			0.00338203
$985$			833.168i			0.845856i
$986$		−	1239.13i		−	1.25672i
$987$			1759.43i			1.78260i
$988$			325.887i			0.329845i
$989$	53.7280	+	175.908i	0.0543256	+	0.177864i
$990$	196.996			0.198985
$991$	1736.34			1.75211			0.876054	−	0.482213i	$-0.160167\pi$
$991$	1736.34			1.75211			0.876054	+	0.482213i	$0.160167\pi$
$992$	188.652			0.190174
$993$	477.766			0.481134
$994$			484.687i			0.487612i
$995$	133.795			0.134467
$996$			149.642i			0.150243i
$997$	705.759			0.707883			0.353941	−	0.935268i	$-0.384841\pi$
$997$	705.759			0.707883			0.353941	+	0.935268i	$0.384841\pi$
$998$	−543.243			−0.544331
$999$		−	256.860i		−	0.257117i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.c.a.91.24	yes	32
3.2	odd	2		2070.3.c.b.91.8		32
23.22	odd	2	inner	690.3.c.a.91.17	✓	32
69.68	even	2		2070.3.c.b.91.9		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.17	✓	32	23.22	odd	2	inner
690.3.c.a.91.24	yes	32	1.1	even	1	trivial
2070.3.c.b.91.8		32	3.2	odd	2
2070.3.c.b.91.9		32	69.68	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} +11.8194i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} +11.8194i q^{7} +2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} -20.7652i q^{11} -3.46410 q^{12} +7.75559 q^{13} +16.7152i q^{14} -3.87298i q^{15} +4.00000 q^{16} +22.9081i q^{17} +4.24264 q^{18} +21.0098i q^{19} +4.47214i q^{20} -20.4719i q^{21} -29.3664i q^{22} +(-21.9968 + 6.71856i) q^{23} -4.89898 q^{24} -5.00000 q^{25} +10.9681 q^{26} -5.19615 q^{27} +23.6389i q^{28} -38.2483 q^{29} -5.47723i q^{30} +33.3493 q^{31} +5.65685 q^{32} +35.9663i q^{33} +32.3970i q^{34} -26.4291 q^{35} +6.00000 q^{36} +49.4326i q^{37} +29.7124i q^{38} -13.4331 q^{39} +6.32456i q^{40} -0.679308 q^{41} -28.9516i q^{42} -7.99696i q^{43} -41.5303i q^{44} +6.70820i q^{45} +(-31.1082 + 9.50148i) q^{46} -85.9436 q^{47} -6.92820 q^{48} -90.6993 q^{49} -7.07107 q^{50} -39.6780i q^{51} +15.5112 q^{52} +66.6981i q^{53} -7.34847 q^{54} +46.4323 q^{55} +33.4304i q^{56} -36.3901i q^{57} -54.0913 q^{58} +110.632 q^{59} -7.74597i q^{60} +16.6780i q^{61} +47.1630 q^{62} +35.4583i q^{63} +8.00000 q^{64} +17.3420i q^{65} +50.8640i q^{66} +117.857i q^{67} +45.8163i q^{68} +(38.0996 - 11.6369i) q^{69} -37.3764 q^{70} +28.9967 q^{71} +8.48528 q^{72} +31.0752 q^{73} +69.9083i q^{74} +8.66025 q^{75} +42.0196i q^{76} +245.433 q^{77} -18.9972 q^{78} -105.069i q^{79} +8.94427i q^{80} +9.00000 q^{81} -0.960687 q^{82} -43.1979i q^{83} -40.9438i q^{84} -51.2241 q^{85} -11.3094i q^{86} +66.2480 q^{87} -58.7327i q^{88} +49.1782i q^{89} +9.48683i q^{90} +91.6668i q^{91} +(-43.9937 + 13.4371i) q^{92} -57.7627 q^{93} -121.543 q^{94} -46.9794 q^{95} -9.79796 q^{96} +69.6738i q^{97} -128.268 q^{98} -62.2955i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) 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

Introduction

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