LMFDB - Embedded newform 90.12.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [90,12,Mod(1,90)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("90.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	90.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-64,0,2048,-6250,0,14092] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$69.1508862504$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1969}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 492 $ x^2 - x - 492
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$22.6867$ of defining polynomial
Character	$\chi$	$=$	90.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-32.0000 q^{2} +1024.00 q^{4} -3125.00 q^{5} +85586.9 q^{7} -32768.0 q^{8} +100000. q^{10} -767235. q^{11} +220960. q^{13} -2.73878e6 q^{14} +1.04858e6 q^{16} +930719. q^{17} -1.77341e7 q^{19} -3.20000e6 q^{20} +2.45515e7 q^{22} +3.99596e7 q^{23} +9.76562e6 q^{25} -7.07072e6 q^{26} +8.76410e7 q^{28} -7.68554e7 q^{29} -2.96314e7 q^{31} -3.35544e7 q^{32} -2.97830e7 q^{34} -2.67459e8 q^{35} +5.40911e7 q^{37} +5.67491e8 q^{38} +1.02400e8 q^{40} -1.26006e8 q^{41} -2.88676e8 q^{43} -7.85648e8 q^{44} -1.27871e9 q^{46} +1.57008e9 q^{47} +5.34780e9 q^{49} -3.12500e8 q^{50} +2.26263e8 q^{52} +4.09006e9 q^{53} +2.39761e9 q^{55} -2.80451e9 q^{56} +2.45937e9 q^{58} -3.77882e9 q^{59} -9.64103e9 q^{61} +9.48205e8 q^{62} +1.07374e9 q^{64} -6.90500e8 q^{65} -1.63819e10 q^{67} +9.53056e8 q^{68} +8.55869e9 q^{70} -1.03471e10 q^{71} +4.27149e9 q^{73} -1.73091e9 q^{74} -1.81597e10 q^{76} -6.56653e10 q^{77} -1.96636e10 q^{79} -3.27680e9 q^{80} +4.03220e9 q^{82} +1.35791e10 q^{83} -2.90850e9 q^{85} +9.23762e9 q^{86} +2.51407e10 q^{88} -2.25058e10 q^{89} +1.89113e10 q^{91} +4.09186e10 q^{92} -5.02424e10 q^{94} +5.54191e10 q^{95} -1.08976e11 q^{97} -1.71130e11 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 64 q^{2} + 2048 q^{4} - 6250 q^{5} + 14092 q^{7} - 65536 q^{8} + 200000 q^{10} - 421584 q^{11} + 1730524 q^{13} - 450944 q^{14} + 2097152 q^{16} + 6323628 q^{17} - 28897400 q^{19} - 6400000 q^{20}+ \cdots - 271423966272 q^{98}+O(q^{100}) $ 2 * q - 64 * q^2 + 2048 * q^4 - 6250 * q^5 + 14092 * q^7 - 65536 * q^8 + 200000 * q^10 - 421584 * q^11 + 1730524 * q^13 - 450944 * q^14 + 2097152 * q^16 + 6323628 * q^17 - 28897400 * q^19 - 6400000 * q^20 + 13490688 * q^22 + 45236076 * q^23 + 19531250 * q^25 - 55376768 * q^26 + 14430208 * q^28 - 58226220 * q^29 + 41413384 * q^31 - 67108864 * q^32 - 202356096 * q^34 - 44037500 * q^35 + 377255452 * q^37 + 924716800 * q^38 + 204800000 * q^40 + 785271036 * q^41 - 1452987236 * q^43 - 431702016 * q^44 - 1447554432 * q^46 + 1288127748 * q^47 + 8481998946 * q^49 - 625000000 * q^50 + 1772056576 * q^52 + 30490836 * q^53 + 1317450000 * q^55 - 461766656 * q^56 + 1863239040 * q^58 - 8677102440 * q^59 + 1115498764 * q^61 - 1325228288 * q^62 + 2147483648 * q^64 - 5407887500 * q^65 - 12673769708 * q^67 + 6475395072 * q^68 + 1409200000 * q^70 - 13799832984 * q^71 - 17842079516 * q^73 - 12072174464 * q^74 - 29590937600 * q^76 - 90377541264 * q^77 - 12636930320 * q^79 - 6553600000 * q^80 - 25128673152 * q^82 - 41986488924 * q^83 - 19761337500 * q^85 + 46495591552 * q^86 + 13814464512 * q^88 - 13208740020 * q^89 - 89014903096 * q^91 + 46321741824 * q^92 - 41220087936 * q^94 + 90304375000 * q^95 - 61787462828 * q^97 - 271423966272 * q^98

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 $

$2$	−32.0000	−0.707107
$2$	−32.0000	−0.707107
$3$	0	0
$3$	0	0
$4$	1024.00	0.500000
$5$	−3125.00	−0.447214
$5$	−3125.00	−0.447214
$6$	0	0
$7$	85586.9	1.92472	0.962362	−	0.271772i	$-0.0876097\pi$
$7$	85586.9	1.92472	0.962362	+	0.271772i	$0.0876097\pi$
$8$	−32768.0	−0.353553
$9$	0	0
$10$	100000.	0.316228
$11$	−767235.	−1.43638	−0.718188	−	0.695849i	$-0.755028\pi$
$11$	−767235.	−1.43638	−0.718188	+	0.695849i	$0.755028\pi$
$12$	0	0
$13$	220960.	0.165054	0.0825268	−	0.996589i	$-0.473701\pi$
$13$	220960.	0.165054	0.0825268	+	0.996589i	$0.473701\pi$
$14$	−2.73878e6	−1.36098
$15$	0	0
$16$	1.04858e6	0.250000
$17$	930719.	0.158983	0.0794913	−	0.996836i	$-0.474670\pi$
$17$	930719.	0.158983	0.0794913	+	0.996836i	$0.474670\pi$
$18$	0	0
$19$	−1.77341e7	−1.64310	−0.821551	−	0.570135i	$-0.806891\pi$
$19$	−1.77341e7	−1.64310	−0.821551	+	0.570135i	$0.806891\pi$
$20$	−3.20000e6	−0.223607
$21$	0	0
$22$	2.45515e7	1.01567
$23$	3.99596e7	1.29455	0.647274	−	0.762257i	$-0.275909\pi$
$23$	3.99596e7	1.29455	0.647274	+	0.762257i	$0.275909\pi$
$24$	0	0
$25$	9.76562e6	0.200000
$26$	−7.07072e6	−0.116711
$27$	0	0
$28$	8.76410e7	0.962362
$29$	−7.68554e7	−0.695801	−0.347901	−	0.937531i	$-0.613105\pi$
$29$	−7.68554e7	−0.695801	−0.347901	+	0.937531i	$0.613105\pi$
$30$	0	0
$31$	−2.96314e7	−0.185893	−0.0929465	−	0.995671i	$-0.529629\pi$
$31$	−2.96314e7	−0.185893	−0.0929465	+	0.995671i	$0.529629\pi$
$32$	−3.35544e7	−0.176777
$33$	0	0
$34$	−2.97830e7	−0.112418
$35$	−2.67459e8	−0.860762
$36$	0	0
$37$	5.40911e7	0.128238	0.0641189	−	0.997942i	$-0.479576\pi$
$37$	5.40911e7	0.128238	0.0641189	+	0.997942i	$0.479576\pi$
$38$	5.67491e8	1.16185
$39$	0	0
$40$	1.02400e8	0.158114
$41$	−1.26006e8	−0.169856	−0.0849280	−	0.996387i	$-0.527066\pi$
$41$	−1.26006e8	−0.169856	−0.0849280	+	0.996387i	$0.527066\pi$
$42$	0	0
$43$	−2.88676e8	−0.299457	−0.149728	−	0.988727i	$-0.547840\pi$
$43$	−2.88676e8	−0.299457	−0.149728	+	0.988727i	$0.547840\pi$
$44$	−7.85648e8	−0.718188
$45$	0	0
$46$	−1.27871e9	−0.915384
$47$	1.57008e9	0.998579	0.499290	−	0.866435i	$-0.333594\pi$
$47$	1.57008e9	0.998579	0.499290	+	0.866435i	$0.333594\pi$
$48$	0	0
$49$	5.34780e9	2.70456
$50$	−3.12500e8	−0.141421
$51$	0	0
$52$	2.26263e8	0.0825268
$53$	4.09006e9	1.34342	0.671711	−	0.740813i	$-0.265560\pi$
$53$	4.09006e9	1.34342	0.671711	+	0.740813i	$0.265560\pi$
$54$	0	0
$55$	2.39761e9	0.642367
$56$	−2.80451e9	−0.680492
$57$	0	0
$58$	2.45937e9	0.492006
$59$	−3.77882e9	−0.688129	−0.344065	−	0.938946i	$-0.611804\pi$
$59$	−3.77882e9	−0.688129	−0.344065	+	0.938946i	$0.611804\pi$
$60$	0	0
$61$	−9.64103e9	−1.46154	−0.730768	−	0.682626i	$-0.760838\pi$
$61$	−9.64103e9	−1.46154	−0.730768	+	0.682626i	$0.760838\pi$
$62$	9.48205e8	0.131446
$63$	0	0
$64$	1.07374e9	0.125000
$65$	−6.90500e8	−0.0738143
$66$	0	0
$67$	−1.63819e10	−1.48236	−0.741178	−	0.671308i	$-0.765733\pi$
$67$	−1.63819e10	−1.48236	−0.741178	+	0.671308i	$0.765733\pi$
$68$	9.53056e8	0.0794913
$69$	0	0
$70$	8.55869e9	0.608651
$71$	−1.03471e10	−0.680609	−0.340304	−	0.940315i	$-0.610530\pi$
$71$	−1.03471e10	−0.680609	−0.340304	+	0.940315i	$0.610530\pi$
$72$	0	0
$73$	4.27149e9	0.241159	0.120580	−	0.992704i	$-0.461525\pi$
$73$	4.27149e9	0.241159	0.120580	+	0.992704i	$0.461525\pi$
$74$	−1.73091e9	−0.0906778
$75$	0	0
$76$	−1.81597e10	−0.821551
$77$	−6.56653e10	−2.76463
$78$	0	0
$79$	−1.96636e10	−0.718977	−0.359489	−	0.933149i	$-0.617049\pi$
$79$	−1.96636e10	−0.718977	−0.359489	+	0.933149i	$0.617049\pi$
$80$	−3.27680e9	−0.111803
$81$	0	0
$82$	4.03220e9	0.120106
$83$	1.35791e10	0.378391	0.189196	−	0.981939i	$-0.439412\pi$
$83$	1.35791e10	0.378391	0.189196	+	0.981939i	$0.439412\pi$
$84$	0	0
$85$	−2.90850e9	−0.0710991
$86$	9.23762e9	0.211748
$87$	0	0
$88$	2.51407e10	0.507836
$89$	−2.25058e10	−0.427218	−0.213609	−	0.976919i	$-0.568522\pi$
$89$	−2.25058e10	−0.427218	−0.213609	+	0.976919i	$0.568522\pi$
$90$	0	0
$91$	1.89113e10	0.317683
$92$	4.09186e10	0.647274
$93$	0	0
$94$	−5.02424e10	−0.706102
$95$	5.54191e10	0.734818
$96$	0	0
$97$	−1.08976e11	−1.28851	−0.644255	−	0.764811i	$-0.722832\pi$
$97$	−1.08976e11	−1.28851	−0.644255	+	0.764811i	$0.722832\pi$
$98$	−1.71130e11	−1.91241
$99$	0	0
$100$	1.00000e10	0.100000
$101$	−1.63516e11	−1.54807	−0.774037	−	0.633140i	$-0.781766\pi$
$101$	−1.63516e11	−1.54807	−0.774037	+	0.633140i	$0.781766\pi$
$102$	0	0
$103$	−7.69876e10	−0.654359	−0.327180	−	0.944962i	$-0.606098\pi$
$103$	−7.69876e10	−0.654359	−0.327180	+	0.944962i	$0.606098\pi$
$104$	−7.24042e9	−0.0583553
$105$	0	0
$106$	−1.30882e11	−0.949943
$107$	−1.19891e11	−0.826376	−0.413188	−	0.910646i	$-0.635585\pi$
$107$	−1.19891e11	−0.826376	−0.413188	+	0.910646i	$0.635585\pi$
$108$	0	0
$109$	−7.33426e10	−0.456573	−0.228287	−	0.973594i	$-0.573312\pi$
$109$	−7.33426e10	−0.456573	−0.228287	+	0.973594i	$0.573312\pi$
$110$	−7.67235e10	−0.454222
$111$	0	0
$112$	8.97444e10	0.481181
$113$	−2.18835e11	−1.11734	−0.558669	−	0.829391i	$-0.688688\pi$
$113$	−2.18835e11	−1.11734	−0.558669	+	0.829391i	$0.688688\pi$
$114$	0	0
$115$	−1.24874e11	−0.578939
$116$	−7.86999e10	−0.347901
$117$	0	0
$118$	1.20922e11	0.486581
$119$	7.96574e10	0.305997
$120$	0	0
$121$	3.03337e11	1.06318
$122$	3.08513e11	1.03346
$123$	0	0
$124$	−3.03426e10	−0.0929465
$125$	−3.05176e10	−0.0894427
$126$	0	0
$127$	3.93053e11	1.05568	0.527838	−	0.849345i	$-0.323003\pi$
$127$	3.93053e11	1.05568	0.527838	+	0.849345i	$0.323003\pi$
$128$	−3.43597e10	−0.0883883
$129$	0	0
$130$	2.20960e10	0.0521946
$131$	2.63175e11	0.596008	0.298004	−	0.954565i	$-0.403679\pi$
$131$	2.63175e11	0.596008	0.298004	+	0.954565i	$0.403679\pi$
$132$	0	0
$133$	−1.51781e12	−3.16252
$134$	5.24220e11	1.04818
$135$	0	0
$136$	−3.04978e10	−0.0562088
$137$	−5.20521e11	−0.921457	−0.460728	−	0.887541i	$-0.652412\pi$
$137$	−5.20521e11	−0.921457	−0.460728	+	0.887541i	$0.652412\pi$
$138$	0	0
$139$	6.82675e11	1.11592	0.557960	−	0.829868i	$-0.311584\pi$
$139$	6.82675e11	1.11592	0.557960	+	0.829868i	$0.311584\pi$
$140$	−2.73878e11	−0.430381
$141$	0	0
$142$	3.31107e11	0.481263
$143$	−1.69528e11	−0.237079
$144$	0	0
$145$	2.40173e11	0.311172
$146$	−1.36688e11	−0.170525
$147$	0	0
$148$	5.53893e10	0.0641189
$149$	−1.01566e12	−1.13299	−0.566493	−	0.824066i	$-0.691700\pi$
$149$	−1.01566e12	−1.13299	−0.566493	+	0.824066i	$0.691700\pi$
$150$	0	0
$151$	5.34040e11	0.553605	0.276803	−	0.960927i	$-0.410725\pi$
$151$	5.34040e11	0.553605	0.276803	+	0.960927i	$0.410725\pi$
$152$	5.81111e11	0.580924
$153$	0	0
$154$	2.10129e12	1.95489
$155$	9.25981e10	0.0831338
$156$	0	0
$157$	−7.29675e11	−0.610494	−0.305247	−	0.952273i	$-0.598739\pi$
$157$	−7.29675e11	−0.610494	−0.305247	+	0.952273i	$0.598739\pi$
$158$	6.29237e11	0.508394
$159$	0	0
$160$	1.04858e11	0.0790569
$161$	3.42002e12	2.49165
$162$	0	0
$163$	−2.12171e11	−0.144429	−0.0722143	−	0.997389i	$-0.523007\pi$
$163$	−2.12171e11	−0.144429	−0.0722143	+	0.997389i	$0.523007\pi$
$164$	−1.29030e11	−0.0849280
$165$	0	0
$166$	−4.34531e11	−0.267563
$167$	2.36613e11	0.140961	0.0704804	−	0.997513i	$-0.477547\pi$
$167$	2.36613e11	0.140961	0.0704804	+	0.997513i	$0.477547\pi$
$168$	0	0
$169$	−1.74334e12	−0.972757
$170$	9.30719e10	0.0502747
$171$	0	0
$172$	−2.95604e11	−0.149728
$173$	7.36983e11	0.361579	0.180790	−	0.983522i	$-0.442135\pi$
$173$	7.36983e11	0.361579	0.180790	+	0.983522i	$0.442135\pi$
$174$	0	0
$175$	8.35810e11	0.384945
$176$	−8.04504e11	−0.359094
$177$	0	0
$178$	7.20185e11	0.302089
$179$	−1.35344e11	−0.0550487	−0.0275244	−	0.999621i	$-0.508762\pi$
$179$	−1.35344e11	−0.0550487	−0.0275244	+	0.999621i	$0.508762\pi$
$180$	0	0
$181$	−1.35881e12	−0.519907	−0.259953	−	0.965621i	$-0.583707\pi$
$181$	−1.35881e12	−0.519907	−0.259953	+	0.965621i	$0.583707\pi$
$182$	−6.05161e11	−0.224636
$183$	0	0
$184$	−1.30940e12	−0.457692
$185$	−1.69035e11	−0.0573497
$186$	0	0
$187$	−7.14080e11	−0.228359
$188$	1.60776e12	0.499290
$189$	0	0
$190$	−1.77341e12	−0.519594
$191$	−1.10648e12	−0.314964	−0.157482	−	0.987522i	$-0.550338\pi$
$191$	−1.10648e12	−0.314964	−0.157482	+	0.987522i	$0.550338\pi$
$192$	0	0
$193$	6.17025e12	1.65858	0.829292	−	0.558816i	$-0.188744\pi$
$193$	6.17025e12	1.65858	0.829292	+	0.558816i	$0.188744\pi$
$194$	3.48724e12	0.911114
$195$	0	0
$196$	5.47615e12	1.35228
$197$	−2.00333e12	−0.481047	−0.240523	−	0.970643i	$-0.577319\pi$
$197$	−2.00333e12	−0.481047	−0.240523	+	0.970643i	$0.577319\pi$
$198$	0	0
$199$	−7.73897e12	−1.75789	−0.878944	−	0.476925i	$-0.841751\pi$
$199$	−7.73897e12	−1.75789	−0.878944	+	0.476925i	$0.841751\pi$
$200$	−3.20000e11	−0.0707107
$201$	0	0
$202$	5.23250e12	1.09465
$203$	−6.57782e12	−1.33922
$204$	0	0
$205$	3.93770e11	0.0759619
$206$	2.46360e12	0.462702
$207$	0	0
$208$	2.31693e11	0.0412634
$209$	1.36062e13	2.36011
$210$	0	0
$211$	−9.01879e12	−1.48455	−0.742275	−	0.670096i	$-0.766253\pi$
$211$	−9.01879e12	−1.48455	−0.742275	+	0.670096i	$0.766253\pi$
$212$	4.18822e12	0.671711
$213$	0	0
$214$	3.83653e12	0.584336
$215$	9.02112e11	0.133921
$216$	0	0
$217$	−2.53606e12	−0.357792
$218$	2.34696e12	0.322846
$219$	0	0
$220$	2.45515e12	0.321184
$221$	2.05652e11	0.0262407
$222$	0	0
$223$	2.06620e12	0.250897	0.125448	−	0.992100i	$-0.459963\pi$
$223$	2.06620e12	0.250897	0.125448	+	0.992100i	$0.459963\pi$
$224$	−2.87182e12	−0.340246
$225$	0	0
$226$	7.00271e12	0.790077
$227$	1.56806e13	1.72672	0.863359	−	0.504591i	$-0.168357\pi$
$227$	1.56806e13	1.72672	0.863359	+	0.504591i	$0.168357\pi$
$228$	0	0
$229$	6.02309e12	0.632010	0.316005	−	0.948758i	$-0.397658\pi$
$229$	6.02309e12	0.632010	0.316005	+	0.948758i	$0.397658\pi$
$230$	3.99596e12	0.409372
$231$	0	0
$232$	2.51840e12	0.246003
$233$	−1.43129e13	−1.36544	−0.682718	−	0.730682i	$-0.739202\pi$
$233$	−1.43129e13	−1.36544	−0.682718	+	0.730682i	$0.739202\pi$
$234$	0	0
$235$	−4.90649e12	−0.446578
$236$	−3.86951e12	−0.344065
$237$	0	0
$238$	−2.54904e12	−0.216373
$239$	−2.22886e13	−1.84881	−0.924407	−	0.381407i	$-0.875440\pi$
$239$	−2.22886e13	−1.84881	−0.924407	+	0.381407i	$0.875440\pi$
$240$	0	0
$241$	2.06638e13	1.63726	0.818629	−	0.574323i	$-0.194735\pi$
$241$	2.06638e13	1.63726	0.818629	+	0.574323i	$0.194735\pi$
$242$	−9.70679e12	−0.751781
$243$	0	0
$244$	−9.87241e12	−0.730768
$245$	−1.67119e13	−1.20952
$246$	0	0
$247$	−3.91853e12	−0.271200
$248$	9.70962e11	0.0657231
$249$	0	0
$250$	9.76562e11	0.0632456
$251$	1.23384e13	0.781721	0.390861	−	0.920450i	$-0.372177\pi$
$251$	1.23384e13	0.781721	0.390861	+	0.920450i	$0.372177\pi$
$252$	0	0
$253$	−3.06584e13	−1.85946
$254$	−1.25777e13	−0.746475
$255$	0	0
$256$	1.09951e12	0.0625000
$257$	−1.19160e13	−0.662975	−0.331487	−	0.943460i	$-0.607550\pi$
$257$	−1.19160e13	−0.662975	−0.331487	+	0.943460i	$0.607550\pi$
$258$	0	0
$259$	4.62949e12	0.246822
$260$	−7.07072e11	−0.0369071
$261$	0	0
$262$	−8.42159e12	−0.421441
$263$	−1.54645e12	−0.0757845	−0.0378923	−	0.999282i	$-0.512064\pi$
$263$	−1.54645e12	−0.0757845	−0.0378923	+	0.999282i	$0.512064\pi$
$264$	0	0
$265$	−1.27814e13	−0.600797
$266$	4.85699e13	2.23624
$267$	0	0
$268$	−1.67751e13	−0.741178
$269$	−1.43195e13	−0.619855	−0.309927	−	0.950760i	$-0.600305\pi$
$269$	−1.43195e13	−0.619855	−0.309927	+	0.950760i	$0.600305\pi$
$270$	0	0
$271$	2.26776e13	0.942468	0.471234	−	0.882008i	$-0.343809\pi$
$271$	2.26776e13	0.942468	0.471234	+	0.882008i	$0.343809\pi$
$272$	9.75929e11	0.0397456
$273$	0	0
$274$	1.66567e13	0.651568
$275$	−7.49253e12	−0.287275
$276$	0	0
$277$	9.25283e12	0.340907	0.170454	−	0.985366i	$-0.445477\pi$
$277$	9.25283e12	0.340907	0.170454	+	0.985366i	$0.445477\pi$
$278$	−2.18456e13	−0.789074
$279$	0	0
$280$	8.76410e12	0.304325
$281$	2.26511e12	0.0771265	0.0385633	−	0.999256i	$-0.487722\pi$
$281$	2.26511e12	0.0771265	0.0385633	+	0.999256i	$0.487722\pi$
$282$	0	0
$283$	−3.60409e13	−1.18024	−0.590121	−	0.807315i	$-0.700920\pi$
$283$	−3.60409e13	−1.18024	−0.590121	+	0.807315i	$0.700920\pi$
$284$	−1.05954e13	−0.340304
$285$	0	0
$286$	5.42490e12	0.167640
$287$	−1.07845e13	−0.326926
$288$	0	0
$289$	−3.34057e13	−0.974725
$290$	−7.68554e12	−0.220032
$291$	0	0
$292$	4.37401e12	0.120580
$293$	−1.63348e13	−0.441918	−0.220959	−	0.975283i	$-0.570919\pi$
$293$	−1.63348e13	−0.441918	−0.220959	+	0.975283i	$0.570919\pi$
$294$	0	0
$295$	1.18088e13	0.307741
$296$	−1.77246e12	−0.0453389
$297$	0	0
$298$	3.25012e13	0.801142
$299$	8.82948e12	0.213670
$300$	0	0
$301$	−2.47069e13	−0.576371
$302$	−1.70893e13	−0.391458
$303$	0	0
$304$	−1.85956e13	−0.410776
$305$	3.01282e13	0.653618
$306$	0	0
$307$	−5.61322e13	−1.17476	−0.587382	−	0.809310i	$-0.699842\pi$
$307$	−5.61322e13	−1.17476	−0.587382	+	0.809310i	$0.699842\pi$
$308$	−6.72412e13	−1.38231
$309$	0	0
$310$	−2.96314e12	−0.0587845
$311$	−6.92340e13	−1.34939	−0.674694	−	0.738097i	$-0.735725\pi$
$311$	−6.92340e13	−1.34939	−0.674694	+	0.738097i	$0.735725\pi$
$312$	0	0
$313$	−3.23822e13	−0.609273	−0.304637	−	0.952469i	$-0.598535\pi$
$313$	−3.23822e13	−0.609273	−0.304637	+	0.952469i	$0.598535\pi$
$314$	2.33496e13	0.431684
$315$	0	0
$316$	−2.01356e13	−0.359489
$317$	1.84023e13	0.322885	0.161442	−	0.986882i	$-0.448385\pi$
$317$	1.84023e13	0.322885	0.161442	+	0.986882i	$0.448385\pi$
$318$	0	0
$319$	5.89661e13	0.999433
$320$	−3.35544e12	−0.0559017
$321$	0	0
$322$	−1.09441e14	−1.76186
$323$	−1.65055e13	−0.261224
$324$	0	0
$325$	2.15781e12	0.0330107
$326$	6.78946e12	0.102126
$327$	0	0
$328$	4.12897e12	0.0600532
$329$	1.34378e14	1.92199
$330$	0	0
$331$	6.74629e13	0.933277	0.466639	−	0.884448i	$-0.345465\pi$
$331$	6.74629e13	0.933277	0.466639	+	0.884448i	$0.345465\pi$
$332$	1.39050e13	0.189196
$333$	0	0
$334$	−7.57163e12	−0.0996744
$335$	5.11934e13	0.662930
$336$	0	0
$337$	1.07897e14	1.35222	0.676109	−	0.736802i	$-0.263665\pi$
$337$	1.07897e14	1.35222	0.676109	+	0.736802i	$0.263665\pi$
$338$	5.57868e13	0.687843
$339$	0	0
$340$	−2.97830e12	−0.0355496
$341$	2.27342e13	0.267012
$342$	0	0
$343$	2.88468e14	3.28081
$344$	9.45933e12	0.105874
$345$	0	0
$346$	−2.35834e13	−0.255675
$347$	1.29756e13	0.138458	0.0692288	−	0.997601i	$-0.477946\pi$
$347$	1.29756e13	0.138458	0.0692288	+	0.997601i	$0.477946\pi$
$348$	0	0
$349$	−7.76358e13	−0.802643	−0.401321	−	0.915937i	$-0.631449\pi$
$349$	−7.76358e13	−0.802643	−0.401321	+	0.915937i	$0.631449\pi$
$350$	−2.67459e13	−0.272197
$351$	0	0
$352$	2.57441e13	0.253918
$353$	4.21031e13	0.408839	0.204420	−	0.978883i	$-0.434469\pi$
$353$	4.21031e13	0.408839	0.204420	+	0.978883i	$0.434469\pi$
$354$	0	0
$355$	3.23347e13	0.304377
$356$	−2.30459e13	−0.213609
$357$	0	0
$358$	4.33101e12	0.0389253
$359$	−1.84963e14	−1.63707	−0.818533	−	0.574460i	$-0.805212\pi$
$359$	−1.84963e14	−1.63707	−0.818533	+	0.574460i	$0.805212\pi$
$360$	0	0
$361$	1.98008e14	1.69978
$362$	4.34818e13	0.367630
$363$	0	0
$364$	1.93652e13	0.158841
$365$	−1.33484e13	−0.107850
$366$	0	0
$367$	−1.68959e14	−1.32470	−0.662351	−	0.749194i	$-0.730441\pi$
$367$	−1.68959e14	−1.32470	−0.662351	+	0.749194i	$0.730441\pi$
$368$	4.19007e13	0.323637
$369$	0	0
$370$	5.40911e12	0.0405524
$371$	3.50056e14	2.58572
$372$	0	0
$373$	−1.87187e14	−1.34238	−0.671191	−	0.741284i	$-0.734217\pi$
$373$	−1.87187e14	−1.34238	−0.671191	+	0.741284i	$0.734217\pi$
$374$	2.28505e13	0.161474
$375$	0	0
$376$	−5.14483e13	−0.353051
$377$	−1.69820e13	−0.114845
$378$	0	0
$379$	−7.47613e13	−0.491090	−0.245545	−	0.969385i	$-0.578967\pi$
$379$	−7.47613e13	−0.491090	−0.245545	+	0.969385i	$0.578967\pi$
$380$	5.67491e13	0.367409
$381$	0	0
$382$	3.54074e13	0.222713
$383$	1.55521e14	0.964264	0.482132	−	0.876099i	$-0.339863\pi$
$383$	1.55521e14	0.964264	0.482132	+	0.876099i	$0.339863\pi$
$384$	0	0
$385$	2.05204e14	1.23638
$386$	−1.97448e14	−1.17280
$387$	0	0
$388$	−1.11592e14	−0.644255
$389$	8.76448e13	0.498889	0.249444	−	0.968389i	$-0.419752\pi$
$389$	8.76448e13	0.498889	0.249444	+	0.968389i	$0.419752\pi$
$390$	0	0
$391$	3.71912e13	0.205811
$392$	−1.75237e14	−0.956206
$393$	0	0
$394$	6.41064e13	0.340151
$395$	6.14489e13	0.321536
$396$	0	0
$397$	5.19767e13	0.264521	0.132261	−	0.991215i	$-0.457776\pi$
$397$	5.19767e13	0.264521	0.132261	+	0.991215i	$0.457776\pi$
$398$	2.47647e14	1.24301
$399$	0	0
$400$	1.02400e13	0.0500000
$401$	3.40753e14	1.64114	0.820570	−	0.571546i	$-0.193656\pi$
$401$	3.40753e14	1.64114	0.820570	+	0.571546i	$0.193656\pi$
$402$	0	0
$403$	−6.54735e12	−0.0306823
$404$	−1.67440e14	−0.774037
$405$	0	0
$406$	2.10490e14	0.946975
$407$	−4.15005e13	−0.184198
$408$	0	0
$409$	3.34635e14	1.44575	0.722875	−	0.690979i	$-0.242820\pi$
$409$	3.34635e14	1.44575	0.722875	+	0.690979i	$0.242820\pi$
$410$	−1.26006e13	−0.0537132
$411$	0	0
$412$	−7.88353e13	−0.327180
$413$	−3.23417e14	−1.32446
$414$	0	0
$415$	−4.24347e13	−0.169222
$416$	−7.41419e12	−0.0291776
$417$	0	0
$418$	−4.35399e14	−1.66885
$419$	−1.37830e14	−0.521396	−0.260698	−	0.965420i	$-0.583953\pi$
$419$	−1.37830e14	−0.521396	−0.260698	+	0.965420i	$0.583953\pi$
$420$	0	0
$421$	4.35453e14	1.60468	0.802342	−	0.596864i	$-0.203587\pi$
$421$	4.35453e14	1.60468	0.802342	+	0.596864i	$0.203587\pi$
$422$	2.88601e14	1.04974
$423$	0	0
$424$	−1.34023e14	−0.474972
$425$	9.08905e12	0.0317965
$426$	0	0
$427$	−8.25146e14	−2.81305
$428$	−1.22769e14	−0.413188
$429$	0	0
$430$	−2.88676e13	−0.0946965
$431$	−1.51185e14	−0.489647	−0.244824	−	0.969568i	$-0.578730\pi$
$431$	−1.51185e14	−0.489647	−0.244824	+	0.969568i	$0.578730\pi$
$432$	0	0
$433$	−3.75044e13	−0.118413	−0.0592065	−	0.998246i	$-0.518857\pi$
$433$	−3.75044e13	−0.118413	−0.0592065	+	0.998246i	$0.518857\pi$
$434$	8.11539e13	0.252997
$435$	0	0
$436$	−7.51028e13	−0.228287
$437$	−7.08648e14	−2.12707
$438$	0	0
$439$	−1.61934e14	−0.474004	−0.237002	−	0.971509i	$-0.576165\pi$
$439$	−1.61934e14	−0.474004	−0.237002	+	0.971509i	$0.576165\pi$
$440$	−7.85648e13	−0.227111
$441$	0	0
$442$	−6.58085e12	−0.0185549
$443$	1.29342e14	0.360179	0.180090	−	0.983650i	$-0.442361\pi$
$443$	1.29342e14	0.360179	0.180090	+	0.983650i	$0.442361\pi$
$444$	0	0
$445$	7.03306e13	0.191058
$446$	−6.61183e13	−0.177411
$447$	0	0
$448$	9.18983e13	0.240590
$449$	−1.30255e13	−0.0336852	−0.0168426	−	0.999858i	$-0.505361\pi$
$449$	−1.30255e13	−0.0336852	−0.0168426	+	0.999858i	$0.505361\pi$
$450$	0	0
$451$	9.66764e13	0.243977
$452$	−2.24087e14	−0.558669
$453$	0	0
$454$	−5.01780e14	−1.22097
$455$	−5.90978e13	−0.142072
$456$	0	0
$457$	3.45724e14	0.811318	0.405659	−	0.914025i	$-0.367042\pi$
$457$	3.45724e14	0.811318	0.405659	+	0.914025i	$0.367042\pi$
$458$	−1.92739e14	−0.446899
$459$	0	0
$460$	−1.27871e14	−0.289470
$461$	−2.96945e14	−0.664234	−0.332117	−	0.943238i	$-0.607763\pi$
$461$	−2.96945e14	−0.664234	−0.332117	+	0.943238i	$0.607763\pi$
$462$	0	0
$463$	4.55258e14	0.994402	0.497201	−	0.867635i	$-0.334361\pi$
$463$	4.55258e14	0.994402	0.497201	+	0.867635i	$0.334361\pi$
$464$	−8.05887e13	−0.173950
$465$	0	0
$466$	4.58014e14	0.965509
$467$	3.66499e14	0.763537	0.381768	−	0.924258i	$-0.375315\pi$
$467$	3.66499e14	0.763537	0.381768	+	0.924258i	$0.375315\pi$
$468$	0	0
$469$	−1.40208e15	−2.85313
$470$	1.57008e14	0.315778
$471$	0	0
$472$	1.23824e14	0.243290
$473$	2.21482e14	0.430132
$474$	0	0
$475$	−1.73185e14	−0.328620
$476$	8.15691e13	0.152999
$477$	0	0
$478$	7.13234e14	1.30731
$479$	−5.78377e14	−1.04801	−0.524005	−	0.851715i	$-0.675563\pi$
$479$	−5.78377e14	−1.04801	−0.524005	+	0.851715i	$0.675563\pi$
$480$	0	0
$481$	1.19520e13	0.0211661
$482$	−6.61242e14	−1.15772
$483$	0	0
$484$	3.10617e14	0.531589
$485$	3.40551e14	0.576239
$486$	0	0
$487$	−4.75203e14	−0.786086	−0.393043	−	0.919520i	$-0.628578\pi$
$487$	−4.75203e14	−0.786086	−0.393043	+	0.919520i	$0.628578\pi$
$488$	3.15917e14	0.516731
$489$	0	0
$490$	5.34780e14	0.855257
$491$	−7.59641e14	−1.20132	−0.600662	−	0.799503i	$-0.705096\pi$
$491$	−7.59641e14	−1.20132	−0.600662	+	0.799503i	$0.705096\pi$
$492$	0	0
$493$	−7.15307e13	−0.110620
$494$	1.25393e14	0.191767
$495$	0	0
$496$	−3.10708e13	−0.0464732
$497$	−8.85576e14	−1.30998
$498$	0	0
$499$	−4.91039e14	−0.710499	−0.355249	−	0.934772i	$-0.615604\pi$
$499$	−4.91039e14	−0.710499	−0.355249	+	0.934772i	$0.615604\pi$
$500$	−3.12500e13	−0.0447214
$501$	0	0
$502$	−3.94828e14	−0.552760
$503$	4.25544e14	0.589278	0.294639	−	0.955609i	$-0.404801\pi$
$503$	4.25544e14	0.589278	0.294639	+	0.955609i	$0.404801\pi$
$504$	0	0
$505$	5.10987e14	0.692320
$506$	9.81069e14	1.31484
$507$	0	0
$508$	4.02486e14	0.527838
$509$	1.41466e15	1.83529	0.917644	−	0.397404i	$-0.130089\pi$
$509$	1.41466e15	1.83529	0.917644	+	0.397404i	$0.130089\pi$
$510$	0	0
$511$	3.65584e14	0.464165
$512$	−3.51844e13	−0.0441942
$513$	0	0
$514$	3.81311e14	0.468794
$515$	2.40586e14	0.292638
$516$	0	0
$517$	−1.20462e15	−1.43434
$518$	−1.48144e14	−0.174530
$519$	0	0
$520$	2.26263e13	0.0260973
$521$	−6.27267e14	−0.715888	−0.357944	−	0.933743i	$-0.616522\pi$
$521$	−6.27267e14	−0.715888	−0.357944	+	0.933743i	$0.616522\pi$
$522$	0	0
$523$	2.23774e14	0.250063	0.125032	−	0.992153i	$-0.460097\pi$
$523$	2.23774e14	0.250063	0.125032	+	0.992153i	$0.460097\pi$
$524$	2.69491e14	0.298004
$525$	0	0
$526$	4.94865e13	0.0535877
$527$	−2.75785e13	−0.0295537
$528$	0	0
$529$	6.43961e14	0.675855
$530$	4.09006e14	0.424827
$531$	0	0
$532$	−1.55424e15	−1.58126
$533$	−2.78424e13	−0.0280354
$534$	0	0
$535$	3.74661e14	0.369566
$536$	5.36802e14	0.524092
$537$	0	0
$538$	4.58224e14	0.438303
$539$	−4.10302e15	−3.88477
$540$	0	0
$541$	5.47756e14	0.508162	0.254081	−	0.967183i	$-0.418227\pi$
$541$	5.47756e14	0.508162	0.254081	+	0.967183i	$0.418227\pi$
$542$	−7.25684e14	−0.666425
$543$	0	0
$544$	−3.12297e13	−0.0281044
$545$	2.29196e14	0.204186
$546$	0	0
$547$	4.33518e14	0.378509	0.189255	−	0.981928i	$-0.439393\pi$
$547$	4.33518e14	0.378509	0.189255	+	0.981928i	$0.439393\pi$
$548$	−5.33013e14	−0.460728
$549$	0	0
$550$	2.39761e14	0.203134
$551$	1.36296e15	1.14327
$552$	0	0
$553$	−1.68295e15	−1.38383
$554$	−2.96091e14	−0.241058
$555$	0	0
$556$	6.99060e14	0.557960
$557$	7.36061e14	0.581715	0.290857	−	0.956766i	$-0.406059\pi$
$557$	7.36061e14	0.581715	0.290857	+	0.956766i	$0.406059\pi$
$558$	0	0
$559$	−6.37858e13	−0.0494264
$560$	−2.80451e14	−0.215191
$561$	0	0
$562$	−7.24834e13	−0.0545367
$563$	9.03199e14	0.672957	0.336478	−	0.941691i	$-0.390764\pi$
$563$	9.03199e14	0.672957	0.336478	+	0.941691i	$0.390764\pi$
$564$	0	0
$565$	6.83858e14	0.499689
$566$	1.15331e15	0.834557
$567$	0	0
$568$	3.39054e14	0.240632
$569$	2.25786e15	1.58701	0.793503	−	0.608566i	$-0.208255\pi$
$569$	2.25786e15	1.58701	0.793503	+	0.608566i	$0.208255\pi$
$570$	0	0
$571$	9.15100e14	0.630913	0.315457	−	0.948940i	$-0.397842\pi$
$571$	9.15100e14	0.630913	0.315457	+	0.948940i	$0.397842\pi$
$572$	−1.73597e14	−0.118540
$573$	0	0
$574$	3.45104e14	0.231172
$575$	3.90231e14	0.258910
$576$	0	0
$577$	2.84484e14	0.185179	0.0925893	−	0.995704i	$-0.470486\pi$
$577$	2.84484e14	0.185179	0.0925893	+	0.995704i	$0.470486\pi$
$578$	1.06898e15	0.689234
$579$	0	0
$580$	2.45937e14	0.155586
$581$	1.16219e15	0.728299
$582$	0	0
$583$	−3.13804e15	−1.92966
$584$	−1.39968e14	−0.0852626
$585$	0	0
$586$	5.22714e14	0.312484
$587$	1.99879e15	1.18374	0.591871	−	0.806032i	$-0.298389\pi$
$587$	1.99879e15	1.18374	0.591871	+	0.806032i	$0.298389\pi$
$588$	0	0
$589$	5.25486e14	0.305441
$590$	−3.77882e14	−0.217606
$591$	0	0
$592$	5.67186e13	0.0320595
$593$	−8.52303e14	−0.477302	−0.238651	−	0.971105i	$-0.576705\pi$
$593$	−8.52303e14	−0.477302	−0.238651	+	0.971105i	$0.576705\pi$
$594$	0	0
$595$	−2.48929e14	−0.136846
$596$	−1.04004e15	−0.566493
$597$	0	0
$598$	−2.82543e14	−0.151087
$599$	−1.08984e14	−0.0577449	−0.0288725	−	0.999583i	$-0.509192\pi$
$599$	−1.08984e14	−0.0577449	−0.0288725	+	0.999583i	$0.509192\pi$
$600$	0	0
$601$	3.21315e14	0.167156	0.0835778	−	0.996501i	$-0.473365\pi$
$601$	3.21315e14	0.167156	0.0835778	+	0.996501i	$0.473365\pi$
$602$	7.90620e14	0.407556
$603$	0	0
$604$	5.46856e14	0.276803
$605$	−9.47929e14	−0.475468
$606$	0	0
$607$	−3.08680e15	−1.52044	−0.760222	−	0.649664i	$-0.774910\pi$
$607$	−3.08680e15	−1.52044	−0.760222	+	0.649664i	$0.774910\pi$
$608$	5.95058e14	0.290462
$609$	0	0
$610$	−9.64103e14	−0.462178
$611$	3.46924e14	0.164819
$612$	0	0
$613$	8.70867e14	0.406368	0.203184	−	0.979141i	$-0.434871\pi$
$613$	8.70867e14	0.406368	0.203184	+	0.979141i	$0.434871\pi$
$614$	1.79623e15	0.830684
$615$	0	0
$616$	2.15172e15	0.977444
$617$	−3.06108e15	−1.37818	−0.689091	−	0.724675i	$-0.741990\pi$
$617$	−3.06108e15	−1.37818	−0.689091	+	0.724675i	$0.741990\pi$
$618$	0	0
$619$	−1.21317e15	−0.536566	−0.268283	−	0.963340i	$-0.586456\pi$
$619$	−1.21317e15	−0.536566	−0.268283	+	0.963340i	$0.586456\pi$
$620$	9.48205e13	0.0415669
$621$	0	0
$622$	2.21549e15	0.954162
$623$	−1.92620e15	−0.822276
$624$	0	0
$625$	9.53674e13	0.0400000
$626$	1.03623e15	0.430821
$627$	0	0
$628$	−7.47187e14	−0.305247
$629$	5.03436e13	0.0203876
$630$	0	0
$631$	2.95354e15	1.17539	0.587693	−	0.809084i	$-0.300036\pi$
$631$	2.95354e15	1.17539	0.587693	+	0.809084i	$0.300036\pi$
$632$	6.44338e14	0.254197
$633$	0	0
$634$	−5.88875e14	−0.228314
$635$	−1.22829e15	−0.472112
$636$	0	0
$637$	1.18165e15	0.446398
$638$	−1.88691e15	−0.706706
$639$	0	0
$640$	1.07374e14	0.0395285
$641$	1.24519e15	0.454483	0.227241	−	0.973838i	$-0.427029\pi$
$641$	1.24519e15	0.454483	0.227241	+	0.973838i	$0.427029\pi$
$642$	0	0
$643$	2.82493e14	0.101356	0.0506778	−	0.998715i	$-0.483862\pi$
$643$	2.82493e14	0.101356	0.0506778	+	0.998715i	$0.483862\pi$
$644$	3.50210e15	1.24582
$645$	0	0
$646$	5.28175e14	0.184714
$647$	3.45042e15	1.19646	0.598230	−	0.801324i	$-0.295871\pi$
$647$	3.45042e15	1.19646	0.598230	+	0.801324i	$0.295871\pi$
$648$	0	0
$649$	2.89924e15	0.988413
$650$	−6.90500e13	−0.0233421
$651$	0	0
$652$	−2.17263e14	−0.0722143
$653$	−3.55303e15	−1.17105	−0.585527	−	0.810653i	$-0.699112\pi$
$653$	−3.55303e15	−1.17105	−0.585527	+	0.810653i	$0.699112\pi$
$654$	0	0
$655$	−8.22421e14	−0.266543
$656$	−1.32127e14	−0.0424640
$657$	0	0
$658$	−4.30010e15	−1.35905
$659$	−1.76602e14	−0.0553511	−0.0276756	−	0.999617i	$-0.508811\pi$
$659$	−1.76602e14	−0.0553511	−0.0276756	+	0.999617i	$0.508811\pi$
$660$	0	0
$661$	3.46137e15	1.06694	0.533470	−	0.845819i	$-0.320888\pi$
$661$	3.46137e15	1.06694	0.533470	+	0.845819i	$0.320888\pi$
$662$	−2.15881e15	−0.659927
$663$	0	0
$664$	−4.44960e14	−0.133782
$665$	4.74315e15	1.41432
$666$	0	0
$667$	−3.07111e15	−0.900748
$668$	2.42292e14	0.0704804
$669$	0	0
$670$	−1.63819e15	−0.468762
$671$	7.39693e15	2.09932
$672$	0	0
$673$	6.92295e15	1.93290	0.966448	−	0.256864i	$-0.0826893\pi$
$673$	6.92295e15	1.93290	0.966448	+	0.256864i	$0.0826893\pi$
$674$	−3.45272e15	−0.956162
$675$	0	0
$676$	−1.78518e15	−0.486379
$677$	3.89173e15	1.05173	0.525865	−	0.850568i	$-0.323742\pi$
$677$	3.89173e15	1.05173	0.525865	+	0.850568i	$0.323742\pi$
$678$	0	0
$679$	−9.32695e15	−2.48002
$680$	9.53056e13	0.0251373
$681$	0	0
$682$	−7.27496e14	−0.188806
$683$	−4.79007e15	−1.23318	−0.616592	−	0.787283i	$-0.711487\pi$
$683$	−4.79007e15	−1.23318	−0.616592	+	0.787283i	$0.711487\pi$
$684$	0	0
$685$	1.62663e15	0.412088
$686$	−9.23099e15	−2.31988
$687$	0	0
$688$	−3.02698e14	−0.0748641
$689$	9.03740e14	0.221737
$690$	0	0
$691$	1.60990e15	0.388749	0.194374	−	0.980927i	$-0.437732\pi$
$691$	1.60990e15	0.388749	0.194374	+	0.980927i	$0.437732\pi$
$692$	7.54670e14	0.180790
$693$	0	0
$694$	−4.15220e14	−0.0979043
$695$	−2.13336e15	−0.499054
$696$	0	0
$697$	−1.17276e14	−0.0270041
$698$	2.48435e15	0.567554
$699$	0	0
$700$	8.55869e14	0.192472
$701$	2.46356e15	0.549686	0.274843	−	0.961489i	$-0.411374\pi$
$701$	2.46356e15	0.549686	0.274843	+	0.961489i	$0.411374\pi$
$702$	0	0
$703$	−9.59257e14	−0.210708
$704$	−8.23812e14	−0.179547
$705$	0	0
$706$	−1.34730e15	−0.289093
$707$	−1.39948e16	−2.97962
$708$	0	0
$709$	6.21323e15	1.30246	0.651228	−	0.758882i	$-0.274254\pi$
$709$	6.21323e15	1.30246	0.651228	+	0.758882i	$0.274254\pi$
$710$	−1.03471e15	−0.215227
$711$	0	0
$712$	7.37470e14	0.151044
$713$	−1.18406e15	−0.240647
$714$	0	0
$715$	5.29776e14	0.106025
$716$	−1.38592e14	−0.0275244
$717$	0	0
$718$	5.91883e15	1.15758
$719$	−7.12825e15	−1.38348	−0.691742	−	0.722145i	$-0.743156\pi$
$719$	−7.12825e15	−1.38348	−0.691742	+	0.722145i	$0.743156\pi$
$720$	0	0
$721$	−6.58914e15	−1.25946
$722$	−6.33627e15	−1.20193
$723$	0	0
$724$	−1.39142e15	−0.259953
$725$	−7.50541e14	−0.139160
$726$	0	0
$727$	5.59273e15	1.02137	0.510686	−	0.859767i	$-0.329391\pi$
$727$	5.59273e15	1.02137	0.510686	+	0.859767i	$0.329391\pi$
$728$	−6.19685e14	−0.112318
$729$	0	0
$730$	4.27149e14	0.0762612
$731$	−2.68676e14	−0.0476084
$732$	0	0
$733$	−2.75558e15	−0.480995	−0.240498	−	0.970650i	$-0.577311\pi$
$733$	−2.75558e15	−0.480995	−0.240498	+	0.970650i	$0.577311\pi$
$734$	5.40669e15	0.936705
$735$	0	0
$736$	−1.34082e15	−0.228846
$737$	1.25688e16	2.12922
$738$	0	0
$739$	1.99004e15	0.332138	0.166069	−	0.986114i	$-0.446893\pi$
$739$	1.99004e15	0.332138	0.166069	+	0.986114i	$0.446893\pi$
$740$	−1.73091e14	−0.0286749
$741$	0	0
$742$	−1.12018e16	−1.82838
$743$	7.01045e15	1.13581	0.567907	−	0.823093i	$-0.307753\pi$
$743$	7.01045e15	1.13581	0.567907	+	0.823093i	$0.307753\pi$
$744$	0	0
$745$	3.17394e15	0.506687
$746$	5.98997e15	0.949208
$747$	0	0
$748$	−7.31218e14	−0.114179
$749$	−1.02611e16	−1.59054
$750$	0	0
$751$	−9.37589e15	−1.43216	−0.716082	−	0.698016i	$-0.754066\pi$
$751$	−9.37589e15	−1.43216	−0.716082	+	0.698016i	$0.754066\pi$
$752$	1.64634e15	0.249645
$753$	0	0
$754$	5.43423e14	0.0812074
$755$	−1.66887e15	−0.247580
$756$	0	0
$757$	1.88289e15	0.275294	0.137647	−	0.990481i	$-0.456046\pi$
$757$	1.88289e15	0.275294	0.137647	+	0.990481i	$0.456046\pi$
$758$	2.39236e15	0.347253
$759$	0	0
$760$	−1.81597e15	−0.259797
$761$	−6.71695e15	−0.954018	−0.477009	−	0.878898i	$-0.658279\pi$
$761$	−6.71695e15	−0.954018	−0.477009	+	0.878898i	$0.658279\pi$
$762$	0	0
$763$	−6.27717e15	−0.878777
$764$	−1.13304e15	−0.157482
$765$	0	0
$766$	−4.97667e15	−0.681837
$767$	−8.34968e14	−0.113578
$768$	0	0
$769$	−5.72271e15	−0.767374	−0.383687	−	0.923463i	$-0.625346\pi$
$769$	−5.72271e15	−0.767374	−0.383687	+	0.923463i	$0.625346\pi$
$770$	−6.56653e15	−0.874252
$771$	0	0
$772$	6.31833e15	0.829292
$773$	3.24936e15	0.423458	0.211729	−	0.977328i	$-0.432091\pi$
$773$	3.24936e15	0.423458	0.211729	+	0.977328i	$0.432091\pi$
$774$	0	0
$775$	−2.89369e14	−0.0371786
$776$	3.57094e15	0.455557
$777$	0	0
$778$	−2.80463e15	−0.352767
$779$	2.23461e15	0.279091
$780$	0	0
$781$	7.93865e15	0.977611
$782$	−1.19012e15	−0.145530
$783$	0	0
$784$	5.60757e15	0.676140
$785$	2.28023e15	0.273021
$786$	0	0
$787$	1.29754e16	1.53201	0.766003	−	0.642838i	$-0.222243\pi$
$787$	1.29754e16	1.53201	0.766003	+	0.642838i	$0.222243\pi$
$788$	−2.05140e15	−0.240523
$789$	0	0
$790$	−1.96636e15	−0.227361
$791$	−1.87294e16	−2.15057
$792$	0	0
$793$	−2.13028e15	−0.241232
$794$	−1.66325e15	−0.187045
$795$	0	0
$796$	−7.92470e15	−0.878944
$797$	1.21467e16	1.33795	0.668974	−	0.743286i	$-0.266734\pi$
$797$	1.21467e16	1.33795	0.668974	+	0.743286i	$0.266734\pi$
$798$	0	0
$799$	1.46130e15	0.158757
$800$	−3.27680e14	−0.0353553
$801$	0	0
$802$	−1.09041e16	−1.16046
$803$	−3.27724e15	−0.346395
$804$	0	0
$805$	−1.06876e16	−1.11430
$806$	2.09515e14	0.0216957
$807$	0	0
$808$	5.35808e15	0.547327
$809$	−3.06008e15	−0.310468	−0.155234	−	0.987878i	$-0.549613\pi$
$809$	−3.06008e15	−0.310468	−0.155234	+	0.987878i	$0.549613\pi$
$810$	0	0
$811$	1.63765e15	0.163910	0.0819552	−	0.996636i	$-0.473884\pi$
$811$	1.63765e15	0.163910	0.0819552	+	0.996636i	$0.473884\pi$
$812$	−6.73568e15	−0.669612
$813$	0	0
$814$	1.32802e15	0.130248
$815$	6.63033e14	0.0645904
$816$	0	0
$817$	5.11941e15	0.492038
$818$	−1.07083e16	−1.02230
$819$	0	0
$820$	4.03220e14	0.0379810
$821$	−1.07044e15	−0.100156	−0.0500778	−	0.998745i	$-0.515947\pi$
$821$	−1.07044e15	−0.100156	−0.0500778	+	0.998745i	$0.515947\pi$
$822$	0	0
$823$	−3.24575e15	−0.299651	−0.149825	−	0.988712i	$-0.547871\pi$
$823$	−3.24575e15	−0.299651	−0.149825	+	0.988712i	$0.547871\pi$
$824$	2.52273e15	0.231351
$825$	0	0
$826$	1.03494e16	0.936533
$827$	2.05440e16	1.84674	0.923369	−	0.383913i	$-0.125424\pi$
$827$	2.05440e16	1.84674	0.923369	+	0.383913i	$0.125424\pi$
$828$	0	0
$829$	5.85872e15	0.519700	0.259850	−	0.965649i	$-0.416327\pi$
$829$	5.85872e15	0.519700	0.259850	+	0.965649i	$0.416327\pi$
$830$	1.35791e15	0.119658
$831$	0	0
$832$	2.37254e14	0.0206317
$833$	4.97730e15	0.429978
$834$	0	0
$835$	−7.39417e14	−0.0630396
$836$	1.39328e16	1.18006
$837$	0	0
$838$	4.41057e15	0.368683
$839$	2.25383e16	1.87168	0.935838	−	0.352430i	$-0.114645\pi$
$839$	2.25383e16	1.87168	0.935838	+	0.352430i	$0.114645\pi$
$840$	0	0
$841$	−6.29376e15	−0.515861
$842$	−1.39345e16	−1.13468
$843$	0	0
$844$	−9.23524e15	−0.742275
$845$	5.44793e15	0.435030
$846$	0	0
$847$	2.59617e16	2.04632
$848$	4.28874e15	0.335856
$849$	0	0
$850$	−2.90850e14	−0.0224835
$851$	2.16146e15	0.166010
$852$	0	0
$853$	1.84471e16	1.39865	0.699326	−	0.714803i	$-0.253484\pi$
$853$	1.84471e16	1.39865	0.699326	+	0.714803i	$0.253484\pi$
$854$	2.64047e16	1.98913
$855$	0	0
$856$	3.92860e15	0.292168
$857$	1.53957e15	0.113764	0.0568820	−	0.998381i	$-0.481884\pi$
$857$	1.53957e15	0.113764	0.0568820	+	0.998381i	$0.481884\pi$
$858$	0	0
$859$	−2.14973e16	−1.56827	−0.784136	−	0.620589i	$-0.786893\pi$
$859$	−2.14973e16	−1.56827	−0.784136	+	0.620589i	$0.786893\pi$
$860$	9.23762e14	0.0669605
$861$	0	0
$862$	4.83792e15	0.346233
$863$	1.16923e16	0.831462	0.415731	−	0.909488i	$-0.363526\pi$
$863$	1.16923e16	0.831462	0.415731	+	0.909488i	$0.363526\pi$
$864$	0	0
$865$	−2.30307e15	−0.161703
$866$	1.20014e15	0.0837306
$867$	0	0
$868$	−2.59693e15	−0.178896
$869$	1.50866e16	1.03272
$870$	0	0
$871$	−3.61974e15	−0.244668
$872$	2.40329e15	0.161423
$873$	0	0
$874$	2.26767e16	1.50407
$875$	−2.61191e15	−0.172152
$876$	0	0
$877$	1.13573e16	0.739225	0.369613	−	0.929186i	$-0.379490\pi$
$877$	1.13573e16	0.739225	0.369613	+	0.929186i	$0.379490\pi$
$878$	5.18188e15	0.335172
$879$	0	0
$880$	2.51407e15	0.160592
$881$	−1.12847e16	−0.716346	−0.358173	−	0.933655i	$-0.616600\pi$
$881$	−1.12847e16	−0.716346	−0.358173	+	0.933655i	$0.616600\pi$
$882$	0	0
$883$	−5.36286e15	−0.336211	−0.168106	−	0.985769i	$-0.553765\pi$
$883$	−5.36286e15	−0.336211	−0.168106	+	0.985769i	$0.553765\pi$
$884$	2.10587e14	0.0131203
$885$	0	0
$886$	−4.13894e15	−0.254685
$887$	−8.12685e15	−0.496984	−0.248492	−	0.968634i	$-0.579935\pi$
$887$	−8.12685e15	−0.496984	−0.248492	+	0.968634i	$0.579935\pi$
$888$	0	0
$889$	3.36402e16	2.03188
$890$	−2.25058e15	−0.135098
$891$	0	0
$892$	2.11579e15	0.125448
$893$	−2.78439e16	−1.64077
$894$	0	0
$895$	4.22950e14	0.0246185
$896$	−2.94075e15	−0.170123
$897$	0	0
$898$	4.16815e14	0.0238190
$899$	2.27733e15	0.129345
$900$	0	0
$901$	3.80670e15	0.213581
$902$	−3.09364e15	−0.172518
$903$	0	0
$904$	7.17077e15	0.395039
$905$	4.24627e15	0.232509
$906$	0	0
$907$	−1.18191e16	−0.639361	−0.319680	−	0.947525i	$-0.603576\pi$
$907$	−1.18191e16	−0.639361	−0.319680	+	0.947525i	$0.603576\pi$
$908$	1.60570e16	0.863359
$909$	0	0
$910$	1.89113e15	0.100460
$911$	−1.50443e16	−0.794366	−0.397183	−	0.917739i	$-0.630012\pi$
$911$	−1.50443e16	−0.794366	−0.397183	+	0.917739i	$0.630012\pi$
$912$	0	0
$913$	−1.04183e16	−0.543513
$914$	−1.10632e16	−0.573688
$915$	0	0
$916$	6.16764e15	0.316005
$917$	2.25243e16	1.14715
$918$	0	0
$919$	−3.98208e15	−0.200389	−0.100195	−	0.994968i	$-0.531947\pi$
$919$	−3.98208e15	−0.200389	−0.100195	+	0.994968i	$0.531947\pi$
$920$	4.09186e15	0.204686
$921$	0	0
$922$	9.50225e15	0.469685
$923$	−2.28629e15	−0.112337
$924$	0	0
$925$	5.28233e14	0.0256476
$926$	−1.45682e16	−0.703148
$927$	0	0
$928$	2.57884e15	0.123001
$929$	−4.06546e16	−1.92763	−0.963814	−	0.266575i	$-0.914108\pi$
$929$	−4.06546e16	−1.92763	−0.963814	+	0.266575i	$0.914108\pi$
$930$	0	0
$931$	−9.48384e16	−4.44387
$932$	−1.46564e16	−0.682718
$933$	0	0
$934$	−1.17280e16	−0.539902
$935$	2.23150e15	0.102125
$936$	0	0
$937$	−1.18458e15	−0.0535792	−0.0267896	−	0.999641i	$-0.508528\pi$
$937$	−1.18458e15	−0.0535792	−0.0267896	+	0.999641i	$0.508528\pi$
$938$	4.48664e16	2.01746
$939$	0	0
$940$	−5.02424e15	−0.223289
$941$	1.02223e16	0.451652	0.225826	−	0.974168i	$-0.427492\pi$
$941$	1.02223e16	0.451652	0.225826	+	0.974168i	$0.427492\pi$
$942$	0	0
$943$	−5.03516e15	−0.219887
$944$	−3.96238e15	−0.172032
$945$	0	0
$946$	−7.08743e15	−0.304150
$947$	1.68932e16	0.720755	0.360378	−	0.932807i	$-0.382648\pi$
$947$	1.68932e16	0.720755	0.360378	+	0.932807i	$0.382648\pi$
$948$	0	0
$949$	9.43829e14	0.0398042
$950$	5.54191e15	0.232370
$951$	0	0
$952$	−2.61021e15	−0.108186
$953$	3.76078e16	1.54977	0.774884	−	0.632103i	$-0.217808\pi$
$953$	3.76078e16	1.54977	0.774884	+	0.632103i	$0.217808\pi$
$954$	0	0
$955$	3.45775e15	0.140856
$956$	−2.28235e16	−0.924407
$957$	0	0
$958$	1.85081e16	0.741055
$959$	−4.45498e16	−1.77355
$960$	0	0
$961$	−2.45305e16	−0.965444
$962$	−3.82463e14	−0.0149667
$963$	0	0
$964$	2.11598e16	0.818629
$965$	−1.92820e16	−0.741741
$966$	0	0
$967$	−3.52262e16	−1.33974	−0.669869	−	0.742479i	$-0.733650\pi$
$967$	−3.52262e16	−1.33974	−0.669869	+	0.742479i	$0.733650\pi$
$968$	−9.93976e15	−0.375890
$969$	0	0
$970$	−1.08976e16	−0.407462
$971$	−3.56726e16	−1.32626	−0.663131	−	0.748504i	$-0.730773\pi$
$971$	−3.56726e16	−1.32626	−0.663131	+	0.748504i	$0.730773\pi$
$972$	0	0
$973$	5.84281e16	2.14784
$974$	1.52065e16	0.555847
$975$	0	0
$976$	−1.01094e16	−0.365384
$977$	4.79343e16	1.72277	0.861383	−	0.507957i	$-0.169599\pi$
$977$	4.79343e16	1.72277	0.861383	+	0.507957i	$0.169599\pi$
$978$	0	0
$979$	1.72672e16	0.613646
$980$	−1.71130e16	−0.604758
$981$	0	0
$982$	2.43085e16	0.849464
$983$	−2.05131e16	−0.712833	−0.356416	−	0.934327i	$-0.616002\pi$
$983$	−2.05131e16	−0.712833	−0.356416	+	0.934327i	$0.616002\pi$
$984$	0	0
$985$	6.26039e15	0.215131
$986$	2.28898e15	0.0782203
$987$	0	0
$988$	−4.01257e15	−0.135600
$989$	−1.15354e16	−0.387661
$990$	0	0
$991$	−2.11314e16	−0.702301	−0.351150	−	0.936319i	$-0.614209\pi$
$991$	−2.11314e16	−0.702301	−0.351150	+	0.936319i	$0.614209\pi$
$992$	9.94265e14	0.0328615
$993$	0	0
$994$	2.83384e16	0.926298
$995$	2.41843e16	0.786151
$996$	0	0
$997$	4.57566e16	1.47106	0.735531	−	0.677491i	$-0.236933\pi$
$997$	4.57566e16	1.47106	0.735531	+	0.677491i	$0.236933\pi$
$998$	1.57133e16	0.502398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.12.a.l.1.2		2
3.2	odd	2		10.12.a.d.1.1	✓	2
12.11	even	2		80.12.a.g.1.2		2
15.2	even	4		50.12.b.f.49.4		4
15.8	even	4		50.12.b.f.49.1		4
15.14	odd	2		50.12.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.12.a.d.1.1	✓	2	3.2	odd	2
50.12.a.f.1.2		2	15.14	odd	2
50.12.b.f.49.1		4	15.8	even	4
50.12.b.f.49.4		4	15.2	even	4
80.12.a.g.1.2		2	12.11	even	2
90.12.a.l.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

\(f(q)\)	\(=\)	\(q-32.0000 q^{2} +1024.00 q^{4} -3125.00 q^{5} +85586.9 q^{7} -32768.0 q^{8} +100000. q^{10} -767235. q^{11} +220960. q^{13} -2.73878e6 q^{14} +1.04858e6 q^{16} +930719. q^{17} -1.77341e7 q^{19} -3.20000e6 q^{20} +2.45515e7 q^{22} +3.99596e7 q^{23} +9.76562e6 q^{25} -7.07072e6 q^{26} +8.76410e7 q^{28} -7.68554e7 q^{29} -2.96314e7 q^{31} -3.35544e7 q^{32} -2.97830e7 q^{34} -2.67459e8 q^{35} +5.40911e7 q^{37} +5.67491e8 q^{38} +1.02400e8 q^{40} -1.26006e8 q^{41} -2.88676e8 q^{43} -7.85648e8 q^{44} -1.27871e9 q^{46} +1.57008e9 q^{47} +5.34780e9 q^{49} -3.12500e8 q^{50} +2.26263e8 q^{52} +4.09006e9 q^{53} +2.39761e9 q^{55} -2.80451e9 q^{56} +2.45937e9 q^{58} -3.77882e9 q^{59} -9.64103e9 q^{61} +9.48205e8 q^{62} +1.07374e9 q^{64} -6.90500e8 q^{65} -1.63819e10 q^{67} +9.53056e8 q^{68} +8.55869e9 q^{70} -1.03471e10 q^{71} +4.27149e9 q^{73} -1.73091e9 q^{74} -1.81597e10 q^{76} -6.56653e10 q^{77} -1.96636e10 q^{79} -3.27680e9 q^{80} +4.03220e9 q^{82} +1.35791e10 q^{83} -2.90850e9 q^{85} +9.23762e9 q^{86} +2.51407e10 q^{88} -2.25058e10 q^{89} +1.89113e10 q^{91} +4.09186e10 q^{92} -5.02424e10 q^{94} +5.54191e10 q^{95} -1.08976e11 q^{97} -1.71130e11 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 64 q^{2} + 2048 q^{4} - 6250 q^{5} + 14092 q^{7} - 65536 q^{8} + 200000 q^{10} - 421584 q^{11} + 1730524 q^{13} - 450944 q^{14} + 2097152 q^{16} + 6323628 q^{17} - 28897400 q^{19} - 6400000 q^{20}+ \cdots - 271423966272 q^{98}+O(q^{100}) \) 2 * q - 64 * q^2 + 2048 * q^4 - 6250 * q^5 + 14092 * q^7 - 65536 * q^8 + 200000 * q^10 - 421584 * q^11 + 1730524 * q^13 - 450944 * q^14 + 2097152 * q^16 + 6323628 * q^17 - 28897400 * q^19 - 6400000 * q^20 + 13490688 * q^22 + 45236076 * q^23 + 19531250 * q^25 - 55376768 * q^26 + 14430208 * q^28 - 58226220 * q^29 + 41413384 * q^31 - 67108864 * q^32 - 202356096 * q^34 - 44037500 * q^35 + 377255452 * q^37 + 924716800 * q^38 + 204800000 * q^40 + 785271036 * q^41 - 1452987236 * q^43 - 431702016 * q^44 - 1447554432 * q^46 + 1288127748 * q^47 + 8481998946 * q^49 - 625000000 * q^50 + 1772056576 * q^52 + 30490836 * q^53 + 1317450000 * q^55 - 461766656 * q^56 + 1863239040 * q^58 - 8677102440 * q^59 + 1115498764 * q^61 - 1325228288 * q^62 + 2147483648 * q^64 - 5407887500 * q^65 - 12673769708 * q^67 + 6475395072 * q^68 + 1409200000 * q^70 - 13799832984 * q^71 - 17842079516 * q^73 - 12072174464 * q^74 - 29590937600 * q^76 - 90377541264 * q^77 - 12636930320 * q^79 - 6553600000 * q^80 - 25128673152 * q^82 - 41986488924 * q^83 - 19761337500 * q^85 + 46495591552 * q^86 + 13814464512 * q^88 - 13208740020 * q^89 - 89014903096 * q^91 + 46321741824 * q^92 - 41220087936 * q^94 + 90304375000 * q^95 - 61787462828 * q^97 - 271423966272 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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