LMFDB - Embedded newform 690.3.c.a.91.13

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			91.13
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} -12.3097i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} -12.3097i q^{7} -2.82843 q^{8} +3.00000 q^{9} -3.16228i q^{10} +16.3902i q^{11} +3.46410 q^{12} -0.736671 q^{13} +17.4085i q^{14} +3.87298i q^{15} +4.00000 q^{16} +27.6301i q^{17} -4.24264 q^{18} +16.9936i q^{19} +4.47214i q^{20} -21.3210i q^{21} -23.1793i q^{22} +(-11.5510 - 19.8890i) q^{23} -4.89898 q^{24} -5.00000 q^{25} +1.04181 q^{26} +5.19615 q^{27} -24.6194i q^{28} +45.7427 q^{29} -5.47723i q^{30} +55.8779 q^{31} -5.65685 q^{32} +28.3887i q^{33} -39.0749i q^{34} +27.5253 q^{35} +6.00000 q^{36} -4.22262i q^{37} -24.0326i q^{38} -1.27595 q^{39} -6.32456i q^{40} +23.5007 q^{41} +30.1525i q^{42} -8.31673i q^{43} +32.7805i q^{44} +6.70820i q^{45} +(16.3356 + 28.1273i) q^{46} +19.7428 q^{47} +6.92820 q^{48} -102.528 q^{49} +7.07107 q^{50} +47.8568i q^{51} -1.47334 q^{52} -28.0875i q^{53} -7.34847 q^{54} -36.6497 q^{55} +34.8171i q^{56} +29.4338i q^{57} -64.6899 q^{58} +61.9670 q^{59} +7.74597i q^{60} +56.2784i q^{61} -79.0233 q^{62} -36.9291i q^{63} +8.00000 q^{64} -1.64725i q^{65} -40.1477i q^{66} +63.8492i q^{67} +55.2603i q^{68} +(-20.0070 - 34.4488i) q^{69} -38.9266 q^{70} +22.3876 q^{71} -8.48528 q^{72} +93.7225 q^{73} +5.97168i q^{74} -8.66025 q^{75} +33.9872i q^{76} +201.759 q^{77} +1.80447 q^{78} -29.5036i q^{79} +8.94427i q^{80} +9.00000 q^{81} -33.2350 q^{82} -119.769i q^{83} -42.6420i q^{84} -61.7829 q^{85} +11.7616i q^{86} +79.2287 q^{87} -46.3586i q^{88} +151.467i q^{89} -9.48683i q^{90} +9.06819i q^{91} +(-23.1021 - 39.7781i) q^{92} +96.7834 q^{93} -27.9206 q^{94} -37.9989 q^{95} -9.79796 q^{96} +89.5915i q^{97} +144.997 q^{98} +49.1707i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	12.3097i		−	1.75853i	−0.476336	−	0.879263i	$-0.658035\pi$
$7$		−	12.3097i		−	1.75853i	0.476336	−	0.879263i	$-0.341965\pi$
$8$	−2.82843			−0.353553
$9$	3.00000			0.333333
$10$		−	3.16228i		−	0.316228i
$11$			16.3902i			1.49002i	0.667053	+	0.745011i	$0.267556\pi$
$11$			16.3902i			1.49002i	−0.667053	+	0.745011i	$0.732444\pi$
$12$	3.46410			0.288675
$13$	−0.736671			−0.0566670			−0.0283335	−	0.999599i	$-0.509020\pi$
$13$	−0.736671			−0.0566670			−0.0283335	+	0.999599i	$0.509020\pi$
$14$			17.4085i			1.24347i
$15$			3.87298i			0.258199i
$16$	4.00000			0.250000
$17$			27.6301i			1.62530i	0.582750	+	0.812651i	$0.301977\pi$
$17$			27.6301i			1.62530i	−0.582750	+	0.812651i	$0.698023\pi$
$18$	−4.24264			−0.235702
$19$			16.9936i			0.894401i	0.894434	+	0.447200i	$0.147579\pi$
$19$			16.9936i			0.894401i	−0.894434	+	0.447200i	$0.852421\pi$
$20$			4.47214i			0.223607i
$21$		−	21.3210i		−	1.01529i
$22$		−	23.1793i		−	1.05360i
$23$	−11.5510	−	19.8890i	−0.502219	−	0.864741i
$23$	−11.5510	−	19.8890i	−0.502219	−	0.864741i
$24$	−4.89898			−0.204124
$25$	−5.00000			−0.200000
$26$	1.04181			0.0400696
$27$	5.19615			0.192450
$28$		−	24.6194i		−	0.879263i
$29$	45.7427			1.57733			0.788667	−	0.614821i	$-0.210772\pi$
$29$	45.7427			1.57733			0.788667	+	0.614821i	$0.210772\pi$
$30$		−	5.47723i		−	0.182574i
$31$	55.8779			1.80251			0.901256	−	0.433286i	$-0.142646\pi$
$31$	55.8779			1.80251			0.901256	+	0.433286i	$0.142646\pi$
$32$	−5.65685			−0.176777
$33$			28.3887i			0.860264i
$34$		−	39.0749i		−	1.14926i
$35$	27.5253			0.786437
$36$	6.00000			0.166667
$37$		−	4.22262i		−	0.114125i	−0.998371	−	0.0570624i	$-0.981827\pi$
$37$		−	4.22262i		−	0.114125i	0.998371	−	0.0570624i	$-0.0181734\pi$
$38$		−	24.0326i		−	0.632437i
$39$	−1.27595			−0.0327167
$40$		−	6.32456i		−	0.158114i
$41$	23.5007			0.573188			0.286594	−	0.958052i	$-0.407477\pi$
$41$	23.5007			0.573188			0.286594	+	0.958052i	$0.407477\pi$
$42$			30.1525i			0.717916i
$43$		−	8.31673i		−	0.193412i	−0.995313	−	0.0967061i	$-0.969169\pi$
$43$		−	8.31673i		−	0.193412i	0.995313	−	0.0967061i	$-0.0308307\pi$
$44$			32.7805i			0.745011i
$45$			6.70820i			0.149071i
$46$	16.3356	+	28.1273i	0.355122	+	0.611464i
$47$	19.7428			0.420061			0.210030	−	0.977695i	$-0.432644\pi$
$47$	19.7428			0.420061			0.210030	+	0.977695i	$0.432644\pi$
$48$	6.92820			0.144338
$49$	−102.528			−2.09242
$50$	7.07107			0.141421
$51$			47.8568i			0.938369i
$52$	−1.47334			−0.0283335
$53$		−	28.0875i		−	0.529953i	−0.964255	−	0.264976i	$-0.914636\pi$
$53$		−	28.0875i		−	0.529953i	0.964255	−	0.264976i	$-0.0853642\pi$
$54$	−7.34847			−0.136083
$55$	−36.6497			−0.666358
$56$			34.8171i			0.621733i
$57$			29.4338i			0.516383i
$58$	−64.6899			−1.11534
$59$	61.9670			1.05029			0.525144	−	0.851014i	$-0.324012\pi$
$59$	61.9670			1.05029			0.525144	+	0.851014i	$0.324012\pi$
$60$			7.74597i			0.129099i
$61$			56.2784i			0.922596i	0.887245	+	0.461298i	$0.152616\pi$
$61$			56.2784i			0.922596i	−0.887245	+	0.461298i	$0.847384\pi$
$62$	−79.0233			−1.27457
$63$		−	36.9291i		−	0.586176i
$64$	8.00000			0.125000
$65$		−	1.64725i		−	0.0253422i
$66$		−	40.1477i		−	0.608299i
$67$			63.8492i			0.952973i	0.879182	+	0.476486i	$0.158090\pi$
$67$			63.8492i			0.952973i	−0.879182	+	0.476486i	$0.841910\pi$
$68$			55.2603i			0.812651i
$69$	−20.0070	−	34.4488i	−0.289956	−	0.499258i
$70$	−38.9266			−0.556095
$71$	22.3876			0.315318			0.157659	−	0.987494i	$-0.449605\pi$
$71$	22.3876			0.315318			0.157659	+	0.987494i	$0.449605\pi$
$72$	−8.48528			−0.117851
$73$	93.7225			1.28387			0.641935	−	0.766759i	$-0.278132\pi$
$73$	93.7225			1.28387			0.641935	+	0.766759i	$0.278132\pi$
$74$			5.97168i			0.0806984i
$75$	−8.66025			−0.115470
$76$			33.9872i			0.447200i
$77$	201.759			2.62024
$78$	1.80447			0.0231342
$79$		−	29.5036i		−	0.373463i	−0.982411	−	0.186732i	$-0.940211\pi$
$79$		−	29.5036i		−	0.373463i	0.982411	−	0.186732i	$-0.0597894\pi$
$80$			8.94427i			0.111803i
$81$	9.00000			0.111111
$82$	−33.2350			−0.405305
$83$		−	119.769i		−	1.44300i	−0.692412	−	0.721502i	$-0.743452\pi$
$83$		−	119.769i		−	1.44300i	0.692412	−	0.721502i	$-0.256548\pi$
$84$		−	42.6420i		−	0.507643i
$85$	−61.7829			−0.726858
$86$			11.7616i			0.136763i
$87$	79.2287			0.910674
$88$		−	46.3586i		−	0.526802i
$89$			151.467i			1.70188i	0.525261	+	0.850941i	$0.323968\pi$
$89$			151.467i			1.70188i	−0.525261	+	0.850941i	$0.676032\pi$
$90$		−	9.48683i		−	0.105409i
$91$			9.06819i			0.0996504i
$92$	−23.1021	−	39.7781i	−0.251109	−	0.432370i
$93$	96.7834			1.04068
$94$	−27.9206			−0.297028
$95$	−37.9989			−0.399988
$96$	−9.79796			−0.102062
$97$			89.5915i			0.923624i	0.886978	+	0.461812i	$0.152800\pi$
$97$			89.5915i			0.923624i	−0.886978	+	0.461812i	$0.847200\pi$
$98$	144.997			1.47956
$99$			49.1707i			0.496674i
$100$	−10.0000			−0.100000
$101$	−139.120			−1.37743			−0.688715	−	0.725032i	$-0.741825\pi$
$101$	−139.120			−1.37743			−0.688715	+	0.725032i	$0.741825\pi$
$102$		−	67.6798i		−	0.663527i
$103$		−	100.537i		−	0.976083i	−0.872820	−	0.488042i	$-0.837711\pi$
$103$		−	100.537i		−	0.976083i	0.872820	−	0.488042i	$-0.162289\pi$
$104$	2.08362			0.0200348
$105$	47.6752			0.454050
$106$			39.7217i			0.374733i
$107$			114.130i			1.06664i	0.845914	+	0.533319i	$0.179055\pi$
$107$			114.130i			1.06664i	−0.845914	+	0.533319i	$0.820945\pi$
$108$	10.3923			0.0962250
$109$		−	42.6647i		−	0.391420i	−0.980662	−	0.195710i	$-0.937299\pi$
$109$		−	42.6647i		−	0.391420i	0.980662	−	0.195710i	$-0.0627011\pi$
$110$	51.8305			0.471186
$111$		−	7.31379i		−	0.0658900i
$112$		−	49.2387i		−	0.439632i
$113$			113.101i			1.00090i	0.865767	+	0.500448i	$0.166831\pi$
$113$			113.101i			1.00090i	−0.865767	+	0.500448i	$0.833169\pi$
$114$		−	41.6257i		−	0.365138i
$115$	44.4732	−	25.8289i	0.386724	−	0.224599i
$116$	91.4854			0.788667
$117$	−2.21001			−0.0188890
$118$	−87.6345			−0.742665
$119$	340.119			2.85814
$120$		−	10.9545i		−	0.0912871i
$121$	−147.640			−1.22016
$122$		−	79.5896i		−	0.652374i
$123$	40.7044			0.330930
$124$	111.756			0.901256
$125$		−	11.1803i		−	0.0894427i
$126$			52.2256i			0.414489i
$127$	56.8211			0.447410			0.223705	−	0.974657i	$-0.428185\pi$
$127$	56.8211			0.447410			0.223705	+	0.974657i	$0.428185\pi$
$128$	−11.3137			−0.0883883
$129$		−	14.4050i		−	0.111667i
$130$			2.32956i			0.0179197i
$131$	−41.8455			−0.319431			−0.159716	−	0.987163i	$-0.551058\pi$
$131$	−41.8455			−0.319431			−0.159716	+	0.987163i	$0.551058\pi$
$132$			56.7774i			0.430132i
$133$	209.186			1.57283
$134$		−	90.2964i		−	0.673853i
$135$			11.6190i			0.0860663i
$136$		−	78.1499i		−	0.574631i
$137$			42.0689i			0.307072i	0.988143	+	0.153536i	$0.0490661\pi$
$137$			42.0689i			0.307072i	−0.988143	+	0.153536i	$0.950934\pi$
$138$	28.2941	+	48.7180i	0.205030	+	0.353029i
$139$	232.680			1.67395			0.836977	−	0.547237i	$-0.184321\pi$
$139$	232.680			1.67395			0.836977	+	0.547237i	$0.184321\pi$
$140$	55.0506			0.393219
$141$	34.1956			0.242522
$142$	−31.6608			−0.222964
$143$		−	12.0742i		−	0.0844350i
$144$	12.0000			0.0833333
$145$			102.284i			0.705405i
$146$	−132.544			−0.907833
$147$	−177.584			−1.20806
$148$		−	8.44523i		−	0.0570624i
$149$		−	69.9994i		−	0.469795i	−0.972020	−	0.234897i	$-0.924525\pi$
$149$		−	69.9994i		−	0.469795i	0.972020	−	0.234897i	$-0.0754754\pi$
$150$	12.2474			0.0816497
$151$	−243.908			−1.61528			−0.807641	−	0.589675i	$-0.799256\pi$
$151$	−243.908			−1.61528			−0.807641	+	0.589675i	$0.799256\pi$
$152$		−	48.0652i		−	0.316218i
$153$			82.8904i			0.541768i
$154$	−285.330			−1.85279
$155$			124.947i			0.806108i
$156$	−2.55190			−0.0163583
$157$		−	54.9962i		−	0.350294i	−0.984542	−	0.175147i	$-0.943960\pi$
$157$		−	54.9962i		−	0.350294i	0.984542	−	0.175147i	$-0.0560401\pi$
$158$			41.7244i			0.264078i
$159$		−	48.6490i		−	0.305968i
$160$		−	12.6491i		−	0.0790569i
$161$	−244.828	+	142.190i	−1.52067	+	0.883165i
$162$	−12.7279			−0.0785674
$163$	−61.4656			−0.377090			−0.188545	−	0.982065i	$-0.560377\pi$
$163$	−61.4656			−0.377090			−0.188545	+	0.982065i	$0.560377\pi$
$164$	47.0014			0.286594
$165$	−63.4791			−0.384722
$166$			169.380i			1.02036i
$167$	203.105			1.21620			0.608099	−	0.793862i	$-0.291933\pi$
$167$	203.105			1.21620			0.608099	+	0.793862i	$0.291933\pi$
$168$			60.3049i			0.358958i
$169$	−168.457			−0.996789
$170$	87.3742			0.513966
$171$			50.9808i			0.298134i
$172$		−	16.6335i		−	0.0967061i
$173$	157.267			0.909059			0.454530	−	0.890732i	$-0.349807\pi$
$173$	157.267			0.909059			0.454530	+	0.890732i	$0.349807\pi$
$174$	−112.046			−0.643944
$175$			61.5484i			0.351705i
$176$			65.5609i			0.372505i
$177$	107.330			0.606384
$178$		−	214.207i		−	1.20341i
$179$	−142.176			−0.794280			−0.397140	−	0.917758i	$-0.629997\pi$
$179$	−142.176			−0.794280			−0.397140	+	0.917758i	$0.629997\pi$
$180$			13.4164i			0.0745356i
$181$		−	104.944i		−	0.579803i	−0.957057	−	0.289901i	$-0.906377\pi$
$181$		−	104.944i		−	0.579803i	0.957057	−	0.289901i	$-0.0936225\pi$
$182$		−	12.8244i		−	0.0704635i
$183$			97.4770i			0.532661i
$184$	32.6713	+	56.2547i	0.177561	+	0.305732i
$185$	9.44206			0.0510381
$186$	−136.872			−0.735873
$187$	−452.865			−2.42174
$188$	39.4857			0.210030
$189$		−	63.9630i		−	0.338429i
$190$	53.7385			0.282834
$191$			225.646i			1.18139i	0.806894	+	0.590697i	$0.201147\pi$
$191$			225.646i			1.18139i	−0.806894	+	0.590697i	$0.798853\pi$
$192$	13.8564			0.0721688
$193$	156.066			0.808634			0.404317	−	0.914619i	$-0.367509\pi$
$193$	156.066			0.808634			0.404317	+	0.914619i	$0.367509\pi$
$194$		−	126.701i		−	0.653100i
$195$		−	2.85311i		−	0.0146314i
$196$	−205.057			−1.04621
$197$	−318.292			−1.61569			−0.807847	−	0.589393i	$-0.799367\pi$
$197$	−318.292			−1.61569			−0.807847	+	0.589393i	$0.799367\pi$
$198$		−	69.5379i		−	0.351201i
$199$		−	50.9274i		−	0.255917i	−0.991780	−	0.127958i	$-0.959158\pi$
$199$		−	50.9274i		−	0.255917i	0.991780	−	0.127958i	$-0.0408424\pi$
$200$	14.1421			0.0707107
$201$			110.590i			0.550199i
$202$	196.746			0.973990
$203$		−	563.078i		−	2.77378i
$204$			95.7136i			0.469185i
$205$			52.5492i			0.256338i
$206$			142.180i			0.690195i
$207$	−34.6531	−	59.6671i	−0.167406	−	0.288247i
$208$	−2.94668			−0.0141667
$209$	−278.529			−1.33268
$210$	−67.4229			−0.321062
$211$	61.4496			0.291230			0.145615	−	0.989341i	$-0.453484\pi$
$211$	61.4496			0.291230			0.145615	+	0.989341i	$0.453484\pi$
$212$		−	56.1750i		−	0.264976i
$213$	38.7764			0.182049
$214$		−	161.405i		−	0.754227i
$215$	18.5968			0.0864966
$216$	−14.6969			−0.0680414
$217$		−	687.839i		−	3.16977i
$218$			60.3371i			0.276775i
$219$	162.332			0.741243
$220$	−73.2993			−0.333179
$221$		−	20.3543i		−	0.0921010i
$222$			10.3433i			0.0465912i
$223$	−245.051			−1.09888			−0.549441	−	0.835533i	$-0.685159\pi$
$223$	−245.051			−1.09888			−0.549441	+	0.835533i	$0.685159\pi$
$224$			69.6341i			0.310867i
$225$	−15.0000			−0.0666667
$226$		−	159.949i		−	0.707740i
$227$		−	319.342i		−	1.40679i	−0.710797	−	0.703397i	$-0.751666\pi$
$227$		−	319.342i		−	1.40679i	0.710797	−	0.703397i	$-0.248334\pi$
$228$			58.8676i			0.258191i
$229$		−	393.219i		−	1.71711i	−0.512717	−	0.858557i	$-0.671361\pi$
$229$		−	393.219i		−	1.71711i	0.512717	−	0.858557i	$-0.328639\pi$
$230$	−62.8946	+	36.5276i	−0.273455	+	0.158816i
$231$	349.456			1.51280
$232$	−129.380			−0.557672
$233$	−226.065			−0.970235			−0.485117	−	0.874449i	$-0.661223\pi$
$233$	−226.065			−0.970235			−0.485117	+	0.874449i	$0.661223\pi$
$234$	3.12543			0.0133565
$235$			44.1463i			0.187857i
$236$	123.934			0.525144
$237$		−	51.1017i		−	0.215619i
$238$	−481.000			−2.02101
$239$	−119.938			−0.501831			−0.250915	−	0.968009i	$-0.580732\pi$
$239$	−119.938			−0.501831			−0.250915	+	0.968009i	$0.580732\pi$
$240$			15.4919i			0.0645497i
$241$			239.516i			0.993844i	0.867795	+	0.496922i	$0.165536\pi$
$241$			239.516i			0.993844i	−0.867795	+	0.496922i	$0.834464\pi$
$242$	208.794			0.862785
$243$	15.5885			0.0641500
$244$			112.557i			0.461298i
$245$		−	229.260i		−	0.935757i
$246$	−57.5648			−0.234003
$247$		−	12.5187i		−	0.0506830i
$248$	−158.047			−0.637285
$249$		−	207.447i		−	0.833119i
$250$			15.8114i			0.0632456i
$251$			186.568i			0.743301i	0.928373	+	0.371650i	$0.121208\pi$
$251$			186.568i			0.743301i	−0.928373	+	0.371650i	$0.878792\pi$
$252$		−	73.8581i		−	0.293088i
$253$	325.986	−	189.324i	1.28848	−	0.748317i
$254$	−80.3571			−0.316367
$255$	−107.011			−0.419651
$256$	16.0000			0.0625000
$257$	−252.617			−0.982944			−0.491472	−	0.870893i	$-0.663541\pi$
$257$	−252.617			−0.982944			−0.491472	+	0.870893i	$0.663541\pi$
$258$			20.3717i			0.0789602i
$259$	−51.9791			−0.200691
$260$		−	3.29449i		−	0.0126711i
$261$	137.228			0.525778
$262$	59.1785			0.225872
$263$			257.480i			0.979010i	0.872001	+	0.489505i	$0.162822\pi$
$263$			257.480i			0.979010i	−0.872001	+	0.489505i	$0.837178\pi$
$264$		−	80.2954i		−	0.304149i
$265$	62.8055			0.237002
$266$	−295.834			−1.11216
$267$			262.349i			0.982582i
$268$			127.698i			0.476486i
$269$	334.855			1.24481			0.622406	−	0.782694i	$-0.286155\pi$
$269$	334.855			1.24481			0.622406	+	0.782694i	$0.286155\pi$
$270$		−	16.4317i		−	0.0608581i
$271$	−324.994			−1.19924			−0.599620	−	0.800285i	$-0.704682\pi$
$271$	−324.994			−1.19924			−0.599620	+	0.800285i	$0.704682\pi$
$272$			110.521i			0.406326i
$273$			15.7066i			0.0575332i
$274$		−	59.4944i		−	0.217133i
$275$		−	81.9512i		−	0.298004i
$276$	−40.0140	−	68.8976i	−0.144978	−	0.249629i
$277$	139.848			0.504865			0.252433	−	0.967614i	$-0.418769\pi$
$277$	139.848			0.504865			0.252433	+	0.967614i	$0.418769\pi$
$278$	−329.059			−1.18366
$279$	167.634			0.600838
$280$	−77.8533			−0.278047
$281$			180.786i			0.643367i	0.946847	+	0.321683i	$0.104249\pi$
$281$			180.786i			0.643367i	−0.946847	+	0.321683i	$0.895751\pi$
$282$	−48.3599			−0.171489
$283$		−	313.878i		−	1.10911i	−0.832148	−	0.554554i	$-0.812889\pi$
$283$		−	313.878i		−	1.10911i	0.832148	−	0.554554i	$-0.187111\pi$
$284$	44.7752			0.157659
$285$	−65.8160			−0.230933
$286$			17.0755i			0.0597046i
$287$		−	289.286i		−	1.00797i
$288$	−16.9706			−0.0589256
$289$	−474.425			−1.64161
$290$		−	144.651i		−	0.498797i
$291$			155.177i			0.533254i
$292$	187.445			0.641935
$293$		−	358.632i		−	1.22400i	−0.790858	−	0.612000i	$-0.790365\pi$
$293$		−	358.632i		−	1.22400i	0.790858	−	0.612000i	$-0.209635\pi$
$294$	251.142			0.854225
$295$			138.562i			0.469703i
$296$			11.9434i			0.0403492i
$297$			85.1661i			0.286755i
$298$			98.9941i			0.332195i
$299$	8.50931	+	14.6517i	0.0284592	+	0.0490022i
$300$	−17.3205			−0.0577350
$301$	−102.376			−0.340121
$302$	344.937			1.14218
$303$	−240.964			−0.795259
$304$			67.9745i			0.223600i
$305$	−125.842			−0.412598
$306$		−	117.225i		−	0.383088i
$307$	238.617			0.777255			0.388627	−	0.921395i	$-0.372949\pi$
$307$	238.617			0.777255			0.388627	+	0.921395i	$0.372949\pi$
$308$	403.517			1.31012
$309$		−	174.134i		−	0.563542i
$310$		−	176.701i		−	0.570005i
$311$	−48.3746			−0.155545			−0.0777727	−	0.996971i	$-0.524781\pi$
$311$	−48.3746			−0.155545			−0.0777727	+	0.996971i	$0.524781\pi$
$312$	3.60893			0.0115671
$313$		−	156.768i		−	0.500857i	−0.968135	−	0.250429i	$-0.919428\pi$
$313$		−	156.768i		−	0.500857i	0.968135	−	0.250429i	$-0.0805716\pi$
$314$			77.7764i			0.247696i
$315$	82.5759			0.262146
$316$		−	59.0072i		−	0.186732i
$317$	−469.006			−1.47951			−0.739757	−	0.672874i	$-0.765059\pi$
$317$	−469.006			−1.47951			−0.739757	+	0.672874i	$0.765059\pi$
$318$			68.8000i			0.216352i
$319$			749.733i			2.35026i
$320$			17.8885i			0.0559017i
$321$			197.679i			0.615824i
$322$	346.239	−	201.086i	1.07528	−	0.624492i
$323$	−469.536			−1.45367
$324$	18.0000			0.0555556
$325$	3.68335			0.0113334
$326$	86.9255			0.266643
$327$		−	73.8975i		−	0.225986i
$328$	−66.4701			−0.202653
$329$		−	243.028i		−	0.738688i
$330$	89.7730			0.272039
$331$	59.9419			0.181093			0.0905467	−	0.995892i	$-0.471139\pi$
$331$	59.9419			0.181093			0.0905467	+	0.995892i	$0.471139\pi$
$332$		−	239.539i		−	0.721502i
$333$		−	12.6678i		−	0.0380416i
$334$	−287.234			−0.859981
$335$	−142.771			−0.426182
$336$		−	85.2840i		−	0.253821i
$337$			88.6746i			0.263129i	0.991308	+	0.131565i	$0.0420001\pi$
$337$			88.6746i			0.263129i	−0.991308	+	0.131565i	$0.958000\pi$
$338$	238.235			0.704836
$339$			195.897i			0.577867i
$340$	−123.566			−0.363429
$341$			915.852i			2.68578i
$342$		−	72.0978i		−	0.210812i
$343$			658.918i			1.92104i
$344$			23.5233i			0.0683816i
$345$	77.0299	−	44.7370i	0.223275	−	0.129672i
$346$	−222.410			−0.642802
$347$	560.347			1.61483			0.807417	−	0.589981i	$-0.200865\pi$
$347$	560.347			1.61483			0.807417	+	0.589981i	$0.200865\pi$
$348$	158.457			0.455337
$349$	−204.499			−0.585958			−0.292979	−	0.956119i	$-0.594647\pi$
$349$	−204.499			−0.585958			−0.292979	+	0.956119i	$0.594647\pi$
$350$		−	87.0426i		−	0.248693i
$351$	−3.82785			−0.0109056
$352$		−	92.7172i		−	0.263401i
$353$	37.1800			0.105326			0.0526629	−	0.998612i	$-0.483229\pi$
$353$	37.1800			0.105326			0.0526629	+	0.998612i	$0.483229\pi$
$354$	−151.787			−0.428778
$355$			50.0602i			0.141015i
$356$			302.935i			0.850941i
$357$	589.103			1.65015
$358$	201.068			0.561641
$359$			165.775i			0.461770i	0.972981	+	0.230885i	$0.0741620\pi$
$359$			165.775i			0.461770i	−0.972981	+	0.230885i	$0.925838\pi$
$360$		−	18.9737i		−	0.0527046i
$361$	72.2170			0.200047
$362$			148.414i			0.409982i
$363$	−255.719			−0.704461
$364$			18.1364i			0.0498252i
$365$			209.570i			0.574164i
$366$		−	137.853i		−	0.376648i
$367$			607.172i			1.65442i	0.561893	+	0.827210i	$0.310073\pi$
$367$			607.172i			1.65442i	−0.561893	+	0.827210i	$0.689927\pi$
$368$	−46.2041	−	79.5561i	−0.125555	−	0.216185i
$369$	70.5021			0.191063
$370$	−13.3531			−0.0360894
$371$	−345.748			−0.931936
$372$	193.567			0.520341
$373$			105.223i			0.282099i	0.990003	+	0.141050i	$0.0450477\pi$
$373$			105.223i			0.282099i	−0.990003	+	0.141050i	$0.954952\pi$
$374$	640.447			1.71243
$375$		−	19.3649i		−	0.0516398i
$376$	−55.8412			−0.148514
$377$	−33.6973			−0.0893827
$378$			90.4574i			0.239305i
$379$		−	490.558i		−	1.29435i	−0.762342	−	0.647175i	$-0.775950\pi$
$379$		−	490.558i		−	1.29435i	0.762342	−	0.647175i	$-0.224050\pi$
$380$	−75.9978			−0.199994
$381$	98.4170			0.258312
$382$		−	319.112i		−	0.835372i
$383$			504.773i			1.31795i	0.752167	+	0.658973i	$0.229009\pi$
$383$			504.773i			1.31795i	−0.752167	+	0.658973i	$0.770991\pi$
$384$	−19.5959			−0.0510310
$385$			451.146i			1.17181i
$386$	−220.711			−0.571790
$387$		−	24.9502i		−	0.0644708i
$388$			179.183i			0.461812i
$389$		−	763.205i		−	1.96197i	−0.194090	−	0.980984i	$-0.562175\pi$
$389$		−	763.205i		−	1.96197i	0.194090	−	0.980984i	$-0.437825\pi$
$390$			4.03491i			0.0103459i
$391$	549.537	−	319.157i	1.40547	−	0.816258i
$392$	289.994			0.739781
$393$	−72.4786			−0.184424
$394$	450.132			1.14247
$395$	65.9720			0.167018
$396$			98.3414i			0.248337i
$397$	−753.911			−1.89902			−0.949510	−	0.313737i	$-0.898419\pi$
$397$	−753.911			−1.89902			−0.949510	+	0.313737i	$0.898419\pi$
$398$			72.0222i			0.180960i
$399$	362.321			0.908073
$400$	−20.0000			−0.0500000
$401$		−	701.572i		−	1.74956i	−0.484523	−	0.874778i	$-0.661007\pi$
$401$		−	701.572i		−	1.74956i	0.484523	−	0.874778i	$-0.338993\pi$
$402$		−	156.398i		−	0.389049i
$403$	−41.1636			−0.102143
$404$	−278.241			−0.688715
$405$			20.1246i			0.0496904i
$406$			796.313i			1.96136i
$407$	69.2097			0.170048
$408$		−	135.360i		−	0.331764i
$409$	214.043			0.523333			0.261666	−	0.965158i	$-0.415728\pi$
$409$	214.043			0.523333			0.261666	+	0.965158i	$0.415728\pi$
$410$		−	74.3158i		−	0.181258i
$411$			72.8655i			0.177288i
$412$		−	201.073i		−	0.488042i
$413$		−	762.794i		−	1.84696i
$414$	49.0069	+	84.3820i	0.118374	+	0.203821i
$415$	267.813			0.645331
$416$	4.16724			0.0100174
$417$	403.013			0.966458
$418$	393.900			0.942344
$419$			210.362i			0.502057i	0.967980	+	0.251029i	$0.0807688\pi$
$419$			210.362i			0.502057i	−0.967980	+	0.251029i	$0.919231\pi$
$420$	95.3504			0.227025
$421$		−	176.903i		−	0.420196i	−0.977680	−	0.210098i	$-0.932622\pi$
$421$		−	176.903i		−	0.420196i	0.977680	−	0.210098i	$-0.0673783\pi$
$422$	−86.9029			−0.205931
$423$	59.2285			0.140020
$424$			79.4434i			0.187367i
$425$		−	138.151i		−	0.325061i
$426$	−54.8382			−0.128728
$427$	692.769			1.62241
$428$			228.261i			0.533319i
$429$		−	20.9131i		−	0.0487486i
$430$	−26.2998			−0.0611623
$431$		−	3.50520i		−	0.00813272i	−0.999992	−	0.00406636i	$-0.998706\pi$
$431$		−	3.50520i		−	0.00813272i	0.999992	−	0.00406636i	$-0.00129437\pi$
$432$	20.7846			0.0481125
$433$		−	456.481i		−	1.05423i	−0.849794	−	0.527114i	$-0.823274\pi$
$433$		−	456.481i		−	1.05423i	0.849794	−	0.527114i	$-0.176726\pi$
$434$			972.752i			2.24136i
$435$			177.161i			0.407266i
$436$		−	85.3295i		−	0.195710i
$437$	337.987	−	196.294i	0.773425	−	0.449185i
$438$	−229.572			−0.524138
$439$	412.180			0.938907			0.469453	−	0.882957i	$-0.344451\pi$
$439$	412.180			0.938907			0.469453	+	0.882957i	$0.344451\pi$
$440$	103.661			0.235593
$441$	−307.585			−0.697472
$442$			28.7854i			0.0651252i
$443$	695.794			1.57064			0.785321	−	0.619089i	$-0.212498\pi$
$443$	695.794			1.57064			0.785321	+	0.619089i	$0.212498\pi$
$444$		−	14.6276i		−	0.0329450i
$445$	−338.692			−0.761105
$446$	346.554			0.777027
$447$		−	121.243i		−	0.271236i
$448$		−	98.4775i		−	0.219816i
$449$	−220.418			−0.490908			−0.245454	−	0.969408i	$-0.578937\pi$
$449$	−220.418			−0.490908			−0.245454	+	0.969408i	$0.578937\pi$
$450$	21.2132			0.0471405
$451$			385.182i			0.854062i
$452$			226.202i			0.500448i
$453$	−422.460			−0.932583
$454$			451.618i			0.994754i
$455$	−20.2771			−0.0445650
$456$		−	83.2514i		−	0.182569i
$457$		−	192.452i		−	0.421120i	−0.977581	−	0.210560i	$-0.932471\pi$
$457$		−	192.452i		−	0.421120i	0.977581	−	0.210560i	$-0.0675287\pi$
$458$			556.096i			1.21418i
$459$			143.570i			0.312790i
$460$	88.9465	−	51.6578i	0.193362	−	0.112300i
$461$	−426.613			−0.925408			−0.462704	−	0.886513i	$-0.653121\pi$
$461$	−426.613			−0.925408			−0.462704	+	0.886513i	$0.653121\pi$
$462$	−494.206			−1.06971
$463$	157.129			0.339371			0.169686	−	0.985498i	$-0.445725\pi$
$463$	157.129			0.339371			0.169686	+	0.985498i	$0.445725\pi$
$464$	182.971			0.394333
$465$			216.414i			0.465407i
$466$	319.704			0.686060
$467$		−	418.986i		−	0.897187i	−0.893736	−	0.448594i	$-0.851925\pi$
$467$		−	418.986i		−	0.897187i	0.893736	−	0.448594i	$-0.148075\pi$
$468$	−4.42002			−0.00944450
$469$	785.963			1.67583
$470$		−	62.4324i		−	0.132835i
$471$		−	95.2563i		−	0.202243i
$472$	−175.269			−0.371333
$473$	136.313			0.288188
$474$			72.2687i			0.152466i
$475$		−	84.9681i		−	0.178880i
$476$	680.237			1.42907
$477$		−	84.2625i		−	0.176651i
$478$	169.617			0.354848
$479$		−	393.407i		−	0.821308i	−0.911791	−	0.410654i	$-0.865300\pi$
$479$		−	393.407i		−	0.821308i	0.911791	−	0.410654i	$-0.134700\pi$
$480$		−	21.9089i		−	0.0456435i
$481$			3.11068i			0.00646711i
$482$		−	338.727i		−	0.702754i
$483$	−424.054	+	246.280i	−0.877959	+	0.509896i
$484$	−295.279			−0.610081
$485$	−200.333			−0.413057
$486$	−22.0454			−0.0453609
$487$	−130.137			−0.267221			−0.133611	−	0.991034i	$-0.542657\pi$
$487$	−130.137			−0.267221			−0.133611	+	0.991034i	$0.542657\pi$
$488$		−	159.179i		−	0.326187i
$489$	−106.462			−0.217713
$490$			324.223i			0.661680i
$491$	779.481			1.58754			0.793769	−	0.608219i	$-0.208116\pi$
$491$	779.481			1.58754			0.793769	+	0.608219i	$0.208116\pi$
$492$	81.4089			0.165465
$493$			1263.88i			2.56365i
$494$			17.7041i			0.0358383i
$495$	−109.949			−0.222119
$496$	223.512			0.450628
$497$		−	275.584i		−	0.554495i
$498$			293.374i			0.589104i
$499$	181.903			0.364535			0.182267	−	0.983249i	$-0.441656\pi$
$499$	181.903			0.364535			0.182267	+	0.983249i	$0.441656\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	351.788			0.702172
$502$		−	263.848i		−	0.525593i
$503$		−	306.185i		−	0.608719i	−0.952557	−	0.304359i	$-0.901558\pi$
$503$		−	306.185i		−	0.608719i	0.952557	−	0.304359i	$-0.0984423\pi$
$504$			104.451i			0.207244i
$505$		−	311.083i		−	0.616005i
$506$	−461.014	+	267.745i	−0.911094	+	0.529140i
$507$	−291.777			−0.575496
$508$	113.642			0.223705
$509$	−736.428			−1.44681			−0.723406	−	0.690422i	$-0.757425\pi$
$509$	−736.428			−1.44681			−0.723406	+	0.690422i	$0.757425\pi$
$510$	151.337			0.296738
$511$		−	1153.70i		−	2.25772i
$512$	−22.6274			−0.0441942
$513$			88.3014i			0.172128i
$514$	357.254			0.695046
$515$	224.807			0.436518
$516$		−	28.8100i		−	0.0558333i
$517$			323.590i			0.625899i
$518$	73.5095			0.141910
$519$	272.395			0.524846
$520$			4.65911i			0.00895984i
$521$			638.384i			1.22531i	0.790352	+	0.612653i	$0.209898\pi$
$521$			638.384i			1.22531i	−0.790352	+	0.612653i	$0.790102\pi$
$522$	−194.070			−0.371781
$523$		−	545.787i		−	1.04357i	−0.853077	−	0.521785i	$-0.825266\pi$
$523$		−	545.787i		−	1.04357i	0.853077	−	0.521785i	$-0.174734\pi$
$524$	−83.6910			−0.159716
$525$			106.605i			0.203057i
$526$		−	364.131i		−	0.692265i
$527$			1543.91i			2.92963i
$528$			113.555i			0.215066i
$529$	−262.147	+	459.478i	−0.495552	+	0.868578i
$530$	−88.8205			−0.167586
$531$	185.901			0.350096
$532$	418.372			0.786414
$533$	−17.3123			−0.0324808
$534$		−	371.018i		−	0.694790i
$535$	−255.203			−0.477015
$536$		−	180.593i		−	0.336927i
$537$	−246.256			−0.458578
$538$	−473.556			−0.880215
$539$		−	1680.46i		−	3.11774i
$540$			23.2379i			0.0430331i
$541$	500.231			0.924641			0.462320	−	0.886713i	$-0.347017\pi$
$541$	500.231			0.924641			0.462320	+	0.886713i	$0.347017\pi$
$542$	459.611			0.847991
$543$		−	181.769i		−	0.334749i
$544$		−	156.300i		−	0.287316i
$545$	95.4013			0.175048
$546$		−	22.2124i		−	0.0406821i
$547$	698.533			1.27702			0.638512	−	0.769612i	$-0.279550\pi$
$547$	698.533			1.27702			0.638512	+	0.769612i	$0.279550\pi$
$548$			84.1378i			0.153536i
$549$			168.835i			0.307532i
$550$			115.896i			0.210721i
$551$			777.334i			1.41077i
$552$	56.5883	+	97.4360i	0.102515	+	0.176514i
$553$	−363.180			−0.656745
$554$	−197.775			−0.356994
$555$	16.3541			0.0294669
$556$	465.359			0.836977
$557$		−	299.862i		−	0.538352i	−0.963091	−	0.269176i	$-0.913249\pi$
$557$		−	299.862i		−	0.538352i	0.963091	−	0.269176i	$-0.0867514\pi$
$558$	−237.070			−0.424856
$559$			6.12669i			0.0109601i
$560$	110.101			0.196609
$561$	−784.384			−1.39819
$562$		−	255.670i		−	0.454929i
$563$			721.083i			1.28079i	0.768047	+	0.640394i	$0.221229\pi$
$563$			721.083i			1.28079i	−0.768047	+	0.640394i	$0.778771\pi$
$564$	68.3912			0.121261
$565$	−252.902			−0.447614
$566$			443.890i			0.784258i
$567$		−	110.787i		−	0.195392i
$568$	−63.3217			−0.111482
$569$		−	295.012i		−	0.518474i	−0.965814	−	0.259237i	$-0.916529\pi$
$569$		−	295.012i		−	0.518474i	0.965814	−	0.259237i	$-0.0834711\pi$
$570$	93.0779			0.163295
$571$		−	240.613i		−	0.421389i	−0.977552	−	0.210694i	$-0.932428\pi$
$571$		−	240.613i		−	0.421389i	0.977552	−	0.210694i	$-0.0675725\pi$
$572$		−	24.1484i		−	0.0422175i
$573$			390.831i			0.682078i
$574$			409.113i			0.712740i
$575$	57.7552	+	99.4452i	0.100444	+	0.172948i
$576$	24.0000			0.0416667
$577$	−781.267			−1.35402			−0.677008	−	0.735976i	$-0.736724\pi$
$577$	−781.267			−1.35402			−0.677008	+	0.735976i	$0.736724\pi$
$578$	670.939			1.16079
$579$	270.315			0.466865
$580$			204.568i			0.352703i
$581$	−1474.32			−2.53756
$582$		−	219.453i		−	0.377068i
$583$	460.361			0.789641
$584$	−265.087			−0.453917
$585$		−	4.94174i		−	0.00844741i
$586$			507.182i			0.865499i
$587$	−825.864			−1.40692			−0.703462	−	0.710733i	$-0.748363\pi$
$587$	−825.864			−1.40692			−0.703462	+	0.710733i	$0.748363\pi$
$588$	−355.169			−0.604029
$589$			949.568i			1.61217i
$590$		−	195.957i		−	0.332130i
$591$	−551.297			−0.932821
$592$		−	16.8905i		−	0.0285312i
$593$	−257.273			−0.433849			−0.216925	−	0.976188i	$-0.569603\pi$
$593$	−257.273			−0.433849			−0.216925	+	0.976188i	$0.569603\pi$
$594$		−	120.443i		−	0.202766i
$595$			760.528i			1.27820i
$596$		−	139.999i		−	0.234897i
$597$		−	88.2088i		−	0.147754i
$598$	−12.0340	−	20.7206i	−0.0201237	−	0.0346498i
$599$	−1081.22			−1.80505			−0.902523	−	0.430641i	$-0.858288\pi$
$599$	−1081.22			−1.80505			−0.902523	+	0.430641i	$0.858288\pi$
$600$	24.4949			0.0408248
$601$	481.001			0.800334			0.400167	−	0.916442i	$-0.368952\pi$
$601$	481.001			0.800334			0.400167	+	0.916442i	$0.368952\pi$
$602$	144.782			0.240502
$603$			191.547i			0.317658i
$604$	−487.815			−0.807641
$605$		−	330.132i		−	0.545673i
$606$	340.774			0.562333
$607$	878.744			1.44768			0.723842	−	0.689966i	$-0.242375\pi$
$607$	878.744			1.44768			0.723842	+	0.689966i	$0.242375\pi$
$608$		−	96.1304i		−	0.158109i
$609$		−	975.280i		−	1.60144i
$610$	177.968			0.291751
$611$	−14.5440			−0.0238036
$612$			165.781i			0.270884i
$613$			249.513i			0.407036i	0.979071	+	0.203518i	$0.0652375\pi$
$613$			249.513i			0.407036i	−0.979071	+	0.203518i	$0.934762\pi$
$614$	−337.456			−0.549602
$615$			91.0179i			0.147997i
$616$	−570.660			−0.926395
$617$		−	399.806i		−	0.647984i	−0.946060	−	0.323992i	$-0.894975\pi$
$617$		−	399.806i		−	0.647984i	0.946060	−	0.323992i	$-0.105025\pi$
$618$			246.263i			0.398484i
$619$			38.5190i			0.0622278i	0.999516	+	0.0311139i	$0.00990547\pi$
$619$			38.5190i			0.0622278i	−0.999516	+	0.0311139i	$0.990095\pi$
$620$			249.894i			0.403054i
$621$	−60.0209	−	103.346i	−0.0966521	−	0.166419i
$622$	68.4120			0.109987
$623$	1864.52			2.99280
$624$	−5.10380			−0.00817917
$625$	25.0000			0.0400000
$626$			221.704i			0.354160i
$627$	−482.427			−0.769421
$628$		−	109.992i		−	0.175147i
$629$	116.672			0.185487
$630$	−116.780			−0.185365
$631$		−	950.121i		−	1.50574i	−0.658170	−	0.752870i	$-0.728669\pi$
$631$		−	950.121i		−	1.50574i	0.658170	−	0.752870i	$-0.271331\pi$
$632$			83.4487i			0.132039i
$633$	106.434			0.168142
$634$	663.274			1.04617
$635$			127.056i			0.200088i
$636$		−	97.2979i		−	0.152984i
$637$	75.5297			0.118571
$638$		−	1060.28i		−	1.66189i
$639$	67.1628			0.105106
$640$		−	25.2982i		−	0.0395285i
$641$			423.980i			0.661435i	0.943730	+	0.330717i	$0.107291\pi$
$641$			423.980i			0.661435i	−0.943730	+	0.330717i	$0.892709\pi$
$642$		−	279.561i		−	0.435453i
$643$		−	311.699i		−	0.484758i	−0.970182	−	0.242379i	$-0.922072\pi$
$643$		−	311.699i		−	0.484758i	0.970182	−	0.242379i	$-0.0779278\pi$
$644$	−489.656	+	284.379i	−0.760335	+	0.441583i
$645$	32.2106			0.0499388
$646$	664.024			1.02790
$647$	−1127.57			−1.74277			−0.871387	−	0.490597i	$-0.836779\pi$
$647$	−1127.57			−1.74277			−0.871387	+	0.490597i	$0.836779\pi$
$648$	−25.4558			−0.0392837
$649$			1015.65i			1.56495i
$650$	−5.20905			−0.00801392
$651$		−	1191.37i		−	1.83007i
$652$	−122.931			−0.188545
$653$	−483.611			−0.740599			−0.370299	−	0.928912i	$-0.620745\pi$
$653$	−483.611			−0.740599			−0.370299	+	0.928912i	$0.620745\pi$
$654$			104.507i			0.159796i
$655$		−	93.5694i		−	0.142854i
$656$	94.0029			0.143297
$657$	281.168			0.427957
$658$			343.694i			0.522331i
$659$		−	375.574i		−	0.569915i	−0.958540	−	0.284958i	$-0.908021\pi$
$659$		−	375.574i		−	0.569915i	0.958540	−	0.284958i	$-0.0919795\pi$
$660$	−126.958			−0.192361
$661$		−	249.264i		−	0.377101i	−0.982064	−	0.188550i	$-0.939621\pi$
$661$		−	249.264i		−	0.377101i	0.982064	−	0.188550i	$-0.0603789\pi$
$662$	−84.7707			−0.128052
$663$		−	35.2547i		−	0.0531745i
$664$			338.759i			0.510179i
$665$			467.754i			0.703390i
$666$			17.9150i			0.0268995i
$667$	−528.375	−	909.778i	−0.792167	−	1.36398i
$668$	406.210			0.608099
$669$	−424.440			−0.634440
$670$	201.909			0.301356
$671$	−922.415			−1.37469
$672$			120.610i			0.179479i
$673$	−393.290			−0.584384			−0.292192	−	0.956360i	$-0.594385\pi$
$673$	−393.290			−0.584384			−0.292192	+	0.956360i	$0.594385\pi$
$674$		−	125.405i		−	0.186061i
$675$	−25.9808			−0.0384900
$676$	−336.915			−0.498394
$677$		−	78.7429i		−	0.116311i	−0.998308	−	0.0581557i	$-0.981478\pi$
$677$		−	78.7429i		−	0.116311i	0.998308	−	0.0581557i	$-0.0185220\pi$
$678$		−	277.040i		−	0.408614i
$679$	1102.84			1.62422
$680$	174.748			0.256983
$681$		−	553.117i		−	0.812213i
$682$		−	1295.21i		−	1.89913i
$683$	773.666			1.13275			0.566374	−	0.824149i	$-0.308346\pi$
$683$	773.666			1.13275			0.566374	+	0.824149i	$0.308346\pi$
$684$			101.962i			0.149067i
$685$	−94.0689			−0.137327
$686$		−	931.851i		−	1.35838i
$687$		−	681.076i		−	0.991377i
$688$		−	33.2669i		−	0.0483531i
$689$			20.6912i			0.0300308i
$690$	−108.937	+	63.2676i	−0.157879	+	0.0916922i
$691$	780.457			1.12946			0.564730	−	0.825276i	$-0.308980\pi$
$691$	780.457			1.12946			0.564730	+	0.825276i	$0.308980\pi$
$692$	314.535			0.454530
$693$	605.276			0.873414
$694$	−792.451			−1.14186
$695$			520.288i			0.748615i
$696$	−224.092			−0.321972
$697$			649.328i			0.931604i
$698$	289.206			0.414335
$699$	−391.556			−0.560165
$700$			123.097i			0.175853i
$701$		−	690.190i		−	0.984579i	−0.870431	−	0.492290i	$-0.836160\pi$
$701$		−	690.190i		−	0.984579i	0.870431	−	0.492290i	$-0.163840\pi$
$702$	5.41340			0.00771140
$703$	71.7575			0.102073
$704$			131.122i			0.186253i
$705$			76.4637i			0.108459i
$706$	−52.5805			−0.0744766
$707$			1712.53i			2.42225i
$708$	214.660			0.303192
$709$		−	691.256i		−	0.974973i	−0.873131	−	0.487486i	$-0.837914\pi$
$709$		−	691.256i		−	0.974973i	0.873131	−	0.487486i	$-0.162086\pi$
$710$		−	70.7958i		−	0.0997123i
$711$		−	88.5107i		−	0.124488i
$712$		−	428.415i		−	0.601706i
$713$	−645.447	−	1111.36i	−0.905256	−	1.55871i
$714$	−833.117			−1.16683
$715$	26.9987			0.0377605
$716$	−284.352			−0.397140
$717$	−207.738			−0.289732
$718$		−	234.442i		−	0.326520i
$719$	591.124			0.822148			0.411074	−	0.911602i	$-0.365154\pi$
$719$	591.124			0.822148			0.411074	+	0.911602i	$0.365154\pi$
$720$			26.8328i			0.0372678i
$721$	−1237.57			−1.71647
$722$	−102.130			−0.141455
$723$			414.855i			0.573796i
$724$		−	209.889i		−	0.289901i
$725$	−228.713			−0.315467
$726$	361.642			0.498129
$727$		−	380.059i		−	0.522777i	−0.965234	−	0.261389i	$-0.915820\pi$
$727$		−	380.059i		−	0.522777i	0.965234	−	0.261389i	$-0.0841804\pi$
$728$		−	25.6487i		−	0.0352317i
$729$	27.0000			0.0370370
$730$		−	296.377i		−	0.405995i
$731$	229.792			0.314354
$732$			194.954i			0.266331i
$733$			438.730i			0.598540i	0.954169	+	0.299270i	$0.0967431\pi$
$733$			438.730i			0.598540i	−0.954169	+	0.299270i	$0.903257\pi$
$734$		−	858.671i		−	1.16985i
$735$		−	397.091i		−	0.540260i
$736$	65.3425	+	112.509i	0.0887806	+	0.152866i
$737$	−1046.50			−1.41995
$738$	−99.7051			−0.135102
$739$	−515.703			−0.697839			−0.348920	−	0.937153i	$-0.613451\pi$
$739$	−515.703			−0.697839			−0.348920	+	0.937153i	$0.613451\pi$
$740$	18.8841			0.0255191
$741$		−	21.6830i		−	0.0292618i
$742$	488.962			0.658978
$743$		−	413.330i		−	0.556298i	−0.960538	−	0.278149i	$-0.910279\pi$
$743$		−	413.330i		−	0.556298i	0.960538	−	0.278149i	$-0.0897209\pi$
$744$	−273.745			−0.367936
$745$	156.523			0.210099
$746$		−	148.808i		−	0.199474i
$747$		−	359.308i		−	0.481002i
$748$	−905.729			−1.21087
$749$	1404.91			1.87571
$750$			27.3861i			0.0365148i
$751$			863.045i			1.14919i	0.818436	+	0.574597i	$0.194841\pi$
$751$			863.045i			1.14919i	−0.818436	+	0.574597i	$0.805159\pi$
$752$	78.9714			0.105015
$753$			323.146i			0.429145i
$754$	47.6552			0.0632031
$755$		−	545.394i		−	0.722376i
$756$		−	127.926i		−	0.169214i
$757$			1048.70i			1.38534i	0.721254	+	0.692671i	$0.243566\pi$
$757$			1048.70i			1.38534i	−0.721254	+	0.692671i	$0.756434\pi$
$758$			693.754i			0.915243i
$759$	564.624	−	327.919i	0.743905	−	0.432041i
$760$	107.477			0.141417
$761$	−113.408			−0.149025			−0.0745126	−	0.997220i	$-0.523740\pi$
$761$	−113.408			−0.149025			−0.0745126	+	0.997220i	$0.523740\pi$
$762$	−139.183			−0.182654
$763$	−525.190			−0.688322
$764$			451.292i			0.590697i
$765$	−185.349			−0.242286
$766$		−	713.857i		−	0.931929i
$767$	−45.6492			−0.0595166
$768$	27.7128			0.0360844
$769$		−	1143.62i		−	1.48716i	−0.668648	−	0.743579i	$-0.733127\pi$
$769$		−	1143.62i		−	1.48716i	0.668648	−	0.743579i	$-0.266873\pi$
$770$		−	638.017i		−	0.828593i
$771$	−437.545			−0.567503
$772$	312.133			0.404317
$773$		−	149.788i		−	0.193775i	−0.995295	−	0.0968877i	$-0.969111\pi$
$773$		−	149.788i		−	0.193775i	0.995295	−	0.0968877i	$-0.0308888\pi$
$774$			35.2849i			0.0455877i
$775$	−279.389			−0.360503
$776$		−	253.403i		−	0.326550i
$777$	−90.0304			−0.115869
$778$			1079.34i			1.38732i
$779$			399.362i			0.512660i
$780$		−	5.70623i		−	0.00731568i
$781$			366.938i			0.469831i
$782$	−777.163	+	451.356i	−0.993814	+	0.577181i
$783$	237.686			0.303558
$784$	−410.114			−0.523104
$785$	122.975			0.156656
$786$	102.500			0.130407
$787$			923.698i			1.17370i	0.809697	+	0.586848i	$0.199631\pi$
$787$			923.698i			1.17370i	−0.809697	+	0.586848i	$0.800369\pi$
$788$	−636.583			−0.807847
$789$			445.968i			0.565232i
$790$	−93.2985			−0.118099
$791$	1392.24			1.76010
$792$		−	139.076i		−	0.175601i
$793$		−	41.4586i		−	0.0522807i
$794$	1066.19			1.34281
$795$	108.782			0.136833
$796$		−	101.855i		−	0.127958i
$797$			892.269i			1.11953i	0.828650	+	0.559767i	$0.189109\pi$
$797$			892.269i			1.11953i	−0.828650	+	0.559767i	$0.810891\pi$
$798$	−512.399			−0.642104
$799$			545.498i			0.682726i
$800$	28.2843			0.0353553
$801$			454.402i			0.567294i
$802$			992.173i			1.23712i
$803$			1536.13i			1.91299i
$804$			221.180i			0.275099i
$805$	−317.946	−	547.452i	−0.394964	−	0.680064i
$806$	58.2141			0.0722260
$807$	579.985			0.718693
$808$	393.492			0.486995
$809$	−1134.01			−1.40174			−0.700872	−	0.713288i	$-0.747205\pi$
$809$	−1134.01			−1.40174			−0.700872	+	0.713288i	$0.747205\pi$
$810$		−	28.4605i		−	0.0351364i
$811$	48.6571			0.0599965			0.0299982	−	0.999550i	$-0.490450\pi$
$811$	48.6571			0.0599965			0.0299982	+	0.999550i	$0.490450\pi$
$812$		−	1126.16i		−	1.38689i
$813$	−562.906			−0.692382
$814$	−97.8772			−0.120242
$815$		−	137.441i		−	0.168640i
$816$			191.427i			0.234592i
$817$	141.331			0.172988
$818$	−302.703			−0.370052
$819$			27.2046i			0.0332168i
$820$			105.098i			0.128169i
$821$	224.131			0.272998			0.136499	−	0.990640i	$-0.456415\pi$
$821$	224.131			0.272998			0.136499	+	0.990640i	$0.456415\pi$
$822$		−	103.047i		−	0.125362i
$823$	−34.5010			−0.0419210			−0.0209605	−	0.999780i	$-0.506672\pi$
$823$	−34.5010			−0.0419210			−0.0209605	+	0.999780i	$0.506672\pi$
$824$			284.360i			0.345098i
$825$		−	141.944i		−	0.172053i
$826$			1078.75i			1.30600i
$827$			544.216i			0.658061i	0.944319	+	0.329030i	$0.106722\pi$
$827$			544.216i			0.658061i	−0.944319	+	0.329030i	$0.893278\pi$
$828$	−69.3062	−	119.334i	−0.0837031	−	0.144123i
$829$	1503.82			1.81402			0.907010	−	0.421108i	$-0.138359\pi$
$829$	1503.82			1.81402			0.907010	+	0.421108i	$0.138359\pi$
$830$	−378.744			−0.456318
$831$	242.223			0.291484
$832$	−5.89337			−0.00708337
$833$		−	2832.88i		−	3.40081i
$834$	−569.947			−0.683389
$835$			454.156i			0.543900i
$836$	−557.059			−0.666338
$837$	290.350			0.346894
$838$		−	297.497i		−	0.355008i
$839$			1027.98i			1.22524i	0.790378	+	0.612620i	$0.209884\pi$
$839$			1027.98i			1.22524i	−0.790378	+	0.612620i	$0.790116\pi$
$840$	−134.846			−0.160531
$841$	1251.39			1.48798
$842$			250.178i			0.297124i
$843$			313.131i			0.371448i
$844$	122.899			0.145615
$845$		−	376.682i		−	0.445778i
$846$	−83.7618			−0.0990092
$847$			1817.40i			2.14569i
$848$		−	112.350i		−	0.132488i
$849$		−	543.652i		−	0.640344i
$850$			195.375i			0.229853i
$851$	−83.9838	+	48.7756i	−0.0986883	+	0.0573156i
$852$	77.5529			0.0910245
$853$	−1412.71			−1.65616			−0.828082	−	0.560607i	$-0.810568\pi$
$853$	−1412.71			−1.65616			−0.828082	+	0.560607i	$0.810568\pi$
$854$	−979.723			−1.14722
$855$	−113.997			−0.133329
$856$		−	322.809i		−	0.377114i
$857$	979.071			1.14244			0.571220	−	0.820797i	$-0.306470\pi$
$857$	979.071			1.14244			0.571220	+	0.820797i	$0.306470\pi$
$858$			29.5756i			0.0344704i
$859$	−962.695			−1.12072			−0.560358	−	0.828251i	$-0.689336\pi$
$859$	−962.695			−1.12072			−0.560358	+	0.828251i	$0.689336\pi$
$860$	37.1935			0.0432483
$861$		−	501.059i		−	0.581950i
$862$			4.95710i			0.00575070i
$863$	−1389.35			−1.60991			−0.804953	−	0.593338i	$-0.797810\pi$
$863$	−1389.35			−1.60991			−0.804953	+	0.593338i	$0.797810\pi$
$864$	−29.3939			−0.0340207
$865$			351.660i			0.406544i
$866$			645.562i			0.745452i
$867$	−821.728			−0.947784
$868$		−	1375.68i		−	1.58488i
$869$	483.571			0.556468
$870$		−	250.543i		−	0.287980i
$871$		−	47.0358i		−	0.0540021i
$872$			120.674i			0.138388i
$873$			268.774i			0.307875i
$874$	−477.985	+	277.601i	−0.546894	+	0.317622i
$875$	−137.626			−0.157287
$876$	324.664			0.370621
$877$	1394.09			1.58961			0.794805	−	0.606865i	$-0.207573\pi$
$877$	1394.09			1.58961			0.794805	+	0.606865i	$0.207573\pi$
$878$	−582.911			−0.663907
$879$		−	621.169i		−	0.706677i
$880$	−146.599			−0.166589
$881$		−	963.638i		−	1.09380i	−0.837198	−	0.546900i	$-0.815808\pi$
$881$		−	963.638i		−	1.09380i	0.837198	−	0.546900i	$-0.184192\pi$
$882$	434.991			0.493187
$883$	−1057.87			−1.19804			−0.599018	−	0.800735i	$-0.704442\pi$
$883$	−1057.87			−1.19804			−0.599018	+	0.800735i	$0.704442\pi$
$884$		−	40.7086i		−	0.0460505i
$885$			239.997i			0.271183i
$886$	−984.002			−1.11061
$887$	570.846			0.643569			0.321785	−	0.946813i	$-0.395717\pi$
$887$	570.846			0.643569			0.321785	+	0.946813i	$0.395717\pi$
$888$			20.6865i			0.0232956i
$889$		−	699.450i		−	0.786782i
$890$	478.982			0.538182
$891$			147.512i			0.165558i
$892$	−490.101			−0.549441
$893$			335.502i			0.375702i
$894$			171.463i			0.191793i
$895$		−	317.916i		−	0.355213i
$896$			139.268i			0.155433i
$897$	14.7386	+	25.3774i	0.0164309	+	0.0282915i
$898$	311.718			0.347125
$899$	2556.01			2.84316
$900$	−30.0000			−0.0333333
$901$	776.062			0.861334
$902$		−	544.730i		−	0.603913i
$903$	−177.321			−0.196369
$904$		−	319.898i		−	0.353870i
$905$	234.663			0.259296
$906$	597.449			0.659436
$907$			361.229i			0.398268i	0.979972	+	0.199134i	$0.0638129\pi$
$907$			361.229i			0.398268i	−0.979972	+	0.199134i	$0.936187\pi$
$908$		−	638.685i		−	0.703397i
$909$	−417.361			−0.459143
$910$	28.6761			0.0315122
$911$			709.529i			0.778847i	0.921059	+	0.389423i	$0.127326\pi$
$911$			709.529i			0.778847i	−0.921059	+	0.389423i	$0.872674\pi$
$912$			117.735i			0.129096i
$913$	1963.05			2.15011
$914$			272.168i			0.297777i
$915$	−217.965			−0.238213
$916$		−	786.439i		−	0.858557i
$917$			515.105i			0.561729i
$918$		−	203.039i		−	0.221176i
$919$			35.4970i			0.0386257i	0.999813	+	0.0193128i	$0.00614785\pi$
$919$			35.4970i			0.0386257i	−0.999813	+	0.0193128i	$0.993852\pi$
$920$	−125.789	+	73.0552i	−0.136727	+	0.0794078i
$921$	413.297			0.448748
$922$	603.322			0.654363
$923$	−16.4923			−0.0178681
$924$	698.912			0.756399
$925$			21.1131i			0.0228250i
$926$	−222.214			−0.239972
$927$		−	301.610i		−	0.325361i
$928$	−258.760			−0.278836
$929$	1212.59			1.30526			0.652630	−	0.757677i	$-0.273666\pi$
$929$	1212.59			1.30526			0.652630	+	0.757677i	$0.273666\pi$
$930$		−	306.056i		−	0.329092i
$931$		−	1742.33i		−	1.87146i
$932$	−452.129			−0.485117
$933$	−83.7873			−0.0898041
$934$			592.536i			0.634407i
$935$		−	1012.64i		−	1.08303i
$936$	6.25086			0.00667827
$937$		−	1636.00i		−	1.74600i	−0.487719	−	0.873001i	$-0.662171\pi$
$937$		−	1636.00i		−	1.74600i	0.487719	−	0.873001i	$-0.337829\pi$
$938$	−1111.52			−1.18499
$939$		−	271.531i		−	0.289170i
$940$			88.2927i			0.0939284i
$941$			1374.81i			1.46101i	0.682908	+	0.730504i	$0.260715\pi$
$941$			1374.81i			1.46101i	−0.682908	+	0.730504i	$0.739285\pi$
$942$			134.713i			0.143007i
$943$	−271.458	−	467.406i	−0.287866	−	0.495659i
$944$	247.868			0.262572
$945$	143.026			0.151350
$946$	−192.776			−0.203780
$947$	−493.454			−0.521071			−0.260535	−	0.965464i	$-0.583899\pi$
$947$	−493.454			−0.521071			−0.260535	+	0.965464i	$0.583899\pi$
$948$		−	102.203i		−	0.107809i
$949$	−69.0426			−0.0727531
$950$			120.163i			0.126487i
$951$	−812.342			−0.854197
$952$	−962.000			−1.01050
$953$		−	1098.73i		−	1.15291i	−0.817127	−	0.576457i	$-0.804435\pi$
$953$		−	1098.73i		−	1.15291i	0.817127	−	0.576457i	$-0.195565\pi$
$954$			119.165i			0.124911i
$955$	−504.560			−0.528335
$956$	−239.875			−0.250915
$957$			1298.58i			1.35692i
$958$			556.361i			0.580753i
$959$	517.855			0.539995
$960$			30.9839i			0.0322749i
$961$	2161.34			2.24905
$962$		−	4.39916i		−	0.00457293i
$963$			342.391i			0.355546i
$964$			479.033i			0.496922i
$965$			348.975i			0.361632i
$966$	599.703	−	348.292i	0.620811	−	0.360551i
$967$	−445.512			−0.460716			−0.230358	−	0.973106i	$-0.573990\pi$
$967$	−445.512			−0.460716			−0.230358	+	0.973106i	$0.573990\pi$
$968$	417.588			0.431393
$969$	−813.260			−0.839278
$970$	283.313			0.292075
$971$		−	1456.91i		−	1.50042i	−0.661199	−	0.750211i	$-0.729952\pi$
$971$		−	1456.91i		−	1.50042i	0.661199	−	0.750211i	$-0.270048\pi$
$972$	31.1769			0.0320750
$973$		−	2864.21i		−	2.94369i
$974$	184.041			0.188954
$975$	6.37976			0.00654334
$976$			225.113i			0.230649i
$977$			787.556i			0.806096i	0.915179	+	0.403048i	$0.132049\pi$
$977$			787.556i			0.806096i	−0.915179	+	0.403048i	$0.867951\pi$
$978$	150.559			0.153946
$979$	−2482.59			−2.53584
$980$		−	458.521i		−	0.467879i
$981$		−	127.994i		−	0.130473i
$982$	−1102.35			−1.12256
$983$		−	591.646i		−	0.601878i	−0.953643	−	0.300939i	$-0.902700\pi$
$983$		−	591.646i		−	0.601878i	0.953643	−	0.300939i	$-0.0973000\pi$
$984$	−115.130			−0.117002
$985$		−	711.722i		−	0.722560i
$986$		−	1787.39i		−	1.81277i
$987$		−	420.937i		−	0.426482i
$988$		−	25.0374i		−	0.0253415i
$989$	−165.412	+	96.0668i	−0.167251	+	0.0971353i
$990$	155.491			0.157062
$991$	931.463			0.939922			0.469961	−	0.882687i	$-0.344268\pi$
$991$	931.463			0.939922			0.469961	+	0.882687i	$0.344268\pi$
$992$	−316.093			−0.318642
$993$	103.822			0.104554
$994$			389.735i			0.392087i
$995$	113.877			0.114449
$996$		−	414.893i		−	0.416560i
$997$	−1004.27			−1.00729			−0.503645	−	0.863911i	$-0.668008\pi$
$997$	−1004.27			−1.00729			−0.503645	+	0.863911i	$0.668008\pi$
$998$	−257.249			−0.257765
$999$		−	21.9414i		−	0.0219633i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.c.a.91.13	yes	32
3.2	odd	2		2070.3.c.b.91.17		32
23.22	odd	2	inner	690.3.c.a.91.12	✓	32
69.68	even	2		2070.3.c.b.91.32		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.12	✓	32	23.22	odd	2	inner
690.3.c.a.91.13	yes	32	1.1	even	1	trivial
2070.3.c.b.91.17		32	3.2	odd	2
2070.3.c.b.91.32		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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} -12.3097i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} -12.3097i q^{7} -2.82843 q^{8} +3.00000 q^{9} -3.16228i q^{10} +16.3902i q^{11} +3.46410 q^{12} -0.736671 q^{13} +17.4085i q^{14} +3.87298i q^{15} +4.00000 q^{16} +27.6301i q^{17} -4.24264 q^{18} +16.9936i q^{19} +4.47214i q^{20} -21.3210i q^{21} -23.1793i q^{22} +(-11.5510 - 19.8890i) q^{23} -4.89898 q^{24} -5.00000 q^{25} +1.04181 q^{26} +5.19615 q^{27} -24.6194i q^{28} +45.7427 q^{29} -5.47723i q^{30} +55.8779 q^{31} -5.65685 q^{32} +28.3887i q^{33} -39.0749i q^{34} +27.5253 q^{35} +6.00000 q^{36} -4.22262i q^{37} -24.0326i q^{38} -1.27595 q^{39} -6.32456i q^{40} +23.5007 q^{41} +30.1525i q^{42} -8.31673i q^{43} +32.7805i q^{44} +6.70820i q^{45} +(16.3356 + 28.1273i) q^{46} +19.7428 q^{47} +6.92820 q^{48} -102.528 q^{49} +7.07107 q^{50} +47.8568i q^{51} -1.47334 q^{52} -28.0875i q^{53} -7.34847 q^{54} -36.6497 q^{55} +34.8171i q^{56} +29.4338i q^{57} -64.6899 q^{58} +61.9670 q^{59} +7.74597i q^{60} +56.2784i q^{61} -79.0233 q^{62} -36.9291i q^{63} +8.00000 q^{64} -1.64725i q^{65} -40.1477i q^{66} +63.8492i q^{67} +55.2603i q^{68} +(-20.0070 - 34.4488i) q^{69} -38.9266 q^{70} +22.3876 q^{71} -8.48528 q^{72} +93.7225 q^{73} +5.97168i q^{74} -8.66025 q^{75} +33.9872i q^{76} +201.759 q^{77} +1.80447 q^{78} -29.5036i q^{79} +8.94427i q^{80} +9.00000 q^{81} -33.2350 q^{82} -119.769i q^{83} -42.6420i q^{84} -61.7829 q^{85} +11.7616i q^{86} +79.2287 q^{87} -46.3586i q^{88} +151.467i q^{89} -9.48683i q^{90} +9.06819i q^{91} +(-23.1021 - 39.7781i) q^{92} +96.7834 q^{93} -27.9206 q^{94} -37.9989 q^{95} -9.79796 q^{96} +89.5915i q^{97} +144.997 q^{98} +49.1707i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) 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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