LMFDB - Embedded newform 431.2.a.f.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [431,2,Mod(1,431)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(431, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("431.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 431 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	431.a (trivial)

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:	yes
Analytic conductor:	$3.44155232712$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.43862 q^{2} +2.31815 q^{3} +0.0696413 q^{4} +1.01873 q^{5} +3.33494 q^{6} +0.326151 q^{7} -2.77706 q^{8} +2.37380 q^{9} +1.46556 q^{10} -1.08039 q^{11} +0.161439 q^{12} +3.08765 q^{13} +0.469209 q^{14} +2.36155 q^{15} -4.13443 q^{16} +7.14719 q^{17} +3.41500 q^{18} +0.624685 q^{19} +0.0709454 q^{20} +0.756065 q^{21} -1.55427 q^{22} -8.86619 q^{23} -6.43763 q^{24} -3.96220 q^{25} +4.44197 q^{26} -1.45163 q^{27} +0.0227136 q^{28} -1.70194 q^{29} +3.39739 q^{30} -9.40429 q^{31} -0.393774 q^{32} -2.50449 q^{33} +10.2821 q^{34} +0.332258 q^{35} +0.165314 q^{36} +10.1698 q^{37} +0.898687 q^{38} +7.15762 q^{39} -2.82906 q^{40} -8.92892 q^{41} +1.08769 q^{42} +8.42056 q^{43} -0.0752396 q^{44} +2.41825 q^{45} -12.7551 q^{46} -10.4407 q^{47} -9.58422 q^{48} -6.89363 q^{49} -5.70012 q^{50} +16.5682 q^{51} +0.215028 q^{52} +7.05404 q^{53} -2.08835 q^{54} -1.10062 q^{55} -0.905741 q^{56} +1.44811 q^{57} -2.44845 q^{58} +4.59980 q^{59} +0.164462 q^{60} -0.252961 q^{61} -13.5293 q^{62} +0.774216 q^{63} +7.70237 q^{64} +3.14547 q^{65} -3.60303 q^{66} -4.69041 q^{67} +0.497740 q^{68} -20.5531 q^{69} +0.477995 q^{70} +9.47783 q^{71} -6.59218 q^{72} +11.5900 q^{73} +14.6305 q^{74} -9.18495 q^{75} +0.0435039 q^{76} -0.352369 q^{77} +10.2971 q^{78} -4.05934 q^{79} -4.21185 q^{80} -10.4865 q^{81} -12.8454 q^{82} +10.0605 q^{83} +0.0526534 q^{84} +7.28103 q^{85} +12.1140 q^{86} -3.94534 q^{87} +3.00030 q^{88} +6.53272 q^{89} +3.47895 q^{90} +1.00704 q^{91} -0.617453 q^{92} -21.8005 q^{93} -15.0202 q^{94} +0.636382 q^{95} -0.912825 q^{96} -11.1189 q^{97} -9.91734 q^{98} -2.56462 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{2} + q^{3} + 33 q^{4} + 13 q^{5} + 17 q^{6} + 8 q^{7} - 3 q^{8} + 31 q^{9} - 6 q^{10} + 15 q^{11} - 12 q^{12} + 11 q^{13} + 16 q^{14} - 5 q^{15} + 43 q^{16} + 6 q^{17} - 8 q^{18} + 18 q^{19}+ \cdots - 9 q^{99}+O(q^{100}) $ 24 * q + q^2 + q^3 + 33 * q^4 + 13 * q^5 + 17 * q^6 + 8 * q^7 - 3 * q^8 + 31 * q^9 - 6 * q^10 + 15 * q^11 - 12 * q^12 + 11 * q^13 + 16 * q^14 - 5 * q^15 + 43 * q^16 + 6 * q^17 - 8 * q^18 + 18 * q^19 - 10 * q^20 - 7 * q^21 - 5 * q^22 - 8 * q^23 + 15 * q^24 + 51 * q^25 + 5 * q^26 + 13 * q^27 + 2 * q^28 + 17 * q^29 - 54 * q^30 + 37 * q^31 - 20 * q^32 + 16 * q^33 + 50 * q^34 + 3 * q^35 + 54 * q^36 + 7 * q^37 - 25 * q^38 - 38 * q^39 + 2 * q^40 + 24 * q^41 + 18 * q^42 + 10 * q^43 + 78 * q^45 + 9 * q^46 - 6 * q^47 - 25 * q^48 + 84 * q^49 - 65 * q^50 - 21 * q^51 - 31 * q^52 - 22 * q^53 + 23 * q^54 + 12 * q^55 - 45 * q^56 + 9 * q^57 - 12 * q^58 + 21 * q^59 - 82 * q^60 + 46 * q^61 - 68 * q^62 - 47 * q^63 + 53 * q^64 - 42 * q^65 - 56 * q^66 - 25 * q^67 + 3 * q^68 - 33 * q^69 - 18 * q^70 - 6 * q^71 - 115 * q^72 + 86 * q^73 - 60 * q^74 - 34 * q^75 + 48 * q^76 - 31 * q^77 - 84 * q^78 - 6 * q^79 - 24 * q^80 + 44 * q^81 - 3 * q^82 - 49 * q^83 - 126 * q^84 + 75 * q^85 - 56 * q^86 - 31 * q^87 - 91 * q^88 + 61 * q^89 - 161 * q^90 + 32 * q^91 - 130 * q^92 - 6 * q^93 + 8 * q^94 - 23 * q^95 - 5 * q^96 + 88 * q^97 - 120 * q^98 - 9 * q^99

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 $

$2$	1.43862	1.01726	0.508631	−	0.860985i	$-0.330152\pi$
$2$	1.43862	1.01726	0.508631	+	0.860985i	$0.330152\pi$
$3$	2.31815	1.33838	0.669191	−	0.743091i	$-0.266641\pi$
$3$	2.31815	1.33838	0.669191	+	0.743091i	$0.266641\pi$
$4$	0.0696413	0.0348207
$5$	1.01873	0.455588	0.227794	−	0.973709i	$-0.426849\pi$
$5$	1.01873	0.455588	0.227794	+	0.973709i	$0.426849\pi$
$6$	3.33494	1.36148
$7$	0.326151	0.123273	0.0616367	−	0.998099i	$-0.480368\pi$
$7$	0.326151	0.123273	0.0616367	+	0.998099i	$0.480368\pi$
$8$	−2.77706	−0.981840
$9$	2.37380	0.791266
$10$	1.46556	0.463452
$11$	−1.08039	−0.325749	−0.162875	−	0.986647i	$-0.552077\pi$
$11$	−1.08039	−0.325749	−0.162875	+	0.986647i	$0.552077\pi$
$12$	0.161439	0.0466034
$13$	3.08765	0.856360	0.428180	−	0.903693i	$-0.359155\pi$
$13$	3.08765	0.856360	0.428180	+	0.903693i	$0.359155\pi$
$14$	0.469209	0.125401
$15$	2.36155	0.609750
$16$	−4.13443	−1.03361
$17$	7.14719	1.73345	0.866724	−	0.498787i	$-0.166221\pi$
$17$	7.14719	1.73345	0.866724	+	0.498787i	$0.166221\pi$
$18$	3.41500	0.804924
$19$	0.624685	0.143312	0.0716562	−	0.997429i	$-0.477172\pi$
$19$	0.624685	0.143312	0.0716562	+	0.997429i	$0.477172\pi$
$20$	0.0709454	0.0158639
$21$	0.756065	0.164987
$22$	−1.55427	−0.331372
$23$	−8.86619	−1.84873	−0.924364	−	0.381511i	$-0.875404\pi$
$23$	−8.86619	−1.84873	−0.924364	+	0.381511i	$0.875404\pi$
$24$	−6.43763	−1.31408
$25$	−3.96220	−0.792440
$26$	4.44197	0.871142
$27$	−1.45163	−0.279366
$28$	0.0227136	0.00429246
$29$	−1.70194	−0.316042	−0.158021	−	0.987436i	$-0.550511\pi$
$29$	−1.70194	−0.316042	−0.158021	+	0.987436i	$0.550511\pi$
$30$	3.39739	0.620276
$31$	−9.40429	−1.68906	−0.844531	−	0.535507i	$-0.820120\pi$
$31$	−9.40429	−1.68906	−0.844531	+	0.535507i	$0.820120\pi$
$32$	−0.393774	−0.0696100
$33$	−2.50449	−0.435977
$34$	10.2821	1.76337
$35$	0.332258	0.0561619
$36$	0.165314	0.0275524
$37$	10.1698	1.67190	0.835950	−	0.548806i	$-0.184917\pi$
$37$	10.1698	1.67190	0.835950	+	0.548806i	$0.184917\pi$
$38$	0.898687	0.145786
$39$	7.15762	1.14614
$40$	−2.82906	−0.447314
$41$	−8.92892	−1.39446	−0.697232	−	0.716846i	$-0.745585\pi$
$41$	−8.92892	−1.39446	−0.697232	+	0.716846i	$0.745585\pi$
$42$	1.08769	0.167835
$43$	8.42056	1.28412	0.642062	−	0.766653i	$-0.278079\pi$
$43$	8.42056	1.28412	0.642062	+	0.766653i	$0.278079\pi$
$44$	−0.0752396	−0.0113428
$45$	2.41825	0.360491
$46$	−12.7551	−1.88064
$47$	−10.4407	−1.52293	−0.761463	−	0.648208i	$-0.775519\pi$
$47$	−10.4407	−1.52293	−0.761463	+	0.648208i	$0.775519\pi$
$48$	−9.58422	−1.38336
$49$	−6.89363	−0.984804
$50$	−5.70012	−0.806118
$51$	16.5682	2.32002
$52$	0.215028	0.0298190
$53$	7.05404	0.968947	0.484473	−	0.874806i	$-0.339011\pi$
$53$	7.05404	0.968947	0.484473	+	0.874806i	$0.339011\pi$
$54$	−2.08835	−0.284188
$55$	−1.10062	−0.148407
$56$	−0.905741	−0.121035
$57$	1.44811	0.191807
$58$	−2.44845	−0.321497
$59$	4.59980	0.598843	0.299421	−	0.954121i	$-0.403206\pi$
$59$	4.59980	0.598843	0.299421	+	0.954121i	$0.403206\pi$
$60$	0.164462	0.0212319
$61$	−0.252961	−0.0323883	−0.0161942	−	0.999869i	$-0.505155\pi$
$61$	−0.252961	−0.0323883	−0.0161942	+	0.999869i	$0.505155\pi$
$62$	−13.5293	−1.71822
$63$	0.774216	0.0975420
$64$	7.70237	0.962797
$65$	3.14547	0.390147
$66$	−3.60303	−0.443502
$67$	−4.69041	−0.573024	−0.286512	−	0.958077i	$-0.592496\pi$
$67$	−4.69041	−0.573024	−0.286512	+	0.958077i	$0.592496\pi$
$68$	0.497740	0.0603598
$69$	−20.5531	−2.47430
$70$	0.477995	0.0571313
$71$	9.47783	1.12481	0.562406	−	0.826861i	$-0.309876\pi$
$71$	9.47783	1.12481	0.562406	+	0.826861i	$0.309876\pi$
$72$	−6.59218	−0.776896
$73$	11.5900	1.35651	0.678253	−	0.734828i	$-0.262737\pi$
$73$	11.5900	1.35651	0.678253	+	0.734828i	$0.262737\pi$
$74$	14.6305	1.70076
$75$	−9.18495	−1.06059
$76$	0.0435039	0.00499024
$77$	−0.352369	−0.0401562
$78$	10.2971	1.16592
$79$	−4.05934	−0.456712	−0.228356	−	0.973578i	$-0.573335\pi$
$79$	−4.05934	−0.456712	−0.228356	+	0.973578i	$0.573335\pi$
$80$	−4.21185	−0.470899
$81$	−10.4865	−1.16516
$82$	−12.8454	−1.41853
$83$	10.0605	1.10428	0.552141	−	0.833751i	$-0.313811\pi$
$83$	10.0605	1.10428	0.552141	+	0.833751i	$0.313811\pi$
$84$	0.0526534	0.00574495
$85$	7.28103	0.789738
$86$	12.1140	1.30629
$87$	−3.94534	−0.422985
$88$	3.00030	0.319833
$89$	6.53272	0.692467	0.346234	−	0.938148i	$-0.387460\pi$
$89$	6.53272	0.692467	0.346234	+	0.938148i	$0.387460\pi$
$90$	3.47895	0.366714
$91$	1.00704	0.105566
$92$	−0.617453	−0.0643740
$93$	−21.8005	−2.26061
$94$	−15.0202	−1.54921
$95$	0.636382	0.0652914
$96$	−0.912825	−0.0931648
$97$	−11.1189	−1.12895	−0.564474	−	0.825451i	$-0.690921\pi$
$97$	−11.1189	−1.12895	−0.564474	+	0.825451i	$0.690921\pi$
$98$	−9.91734	−1.00180
$99$	−2.56462	−0.257754
$100$	−0.275933	−0.0275933
$101$	−2.46775	−0.245551	−0.122775	−	0.992434i	$-0.539179\pi$
$101$	−2.46775	−0.245551	−0.122775	+	0.992434i	$0.539179\pi$
$102$	23.8355	2.36006
$103$	5.46494	0.538477	0.269238	−	0.963074i	$-0.413228\pi$
$103$	5.46494	0.538477	0.269238	+	0.963074i	$0.413228\pi$
$104$	−8.57459	−0.840808
$105$	0.770222	0.0751660
$106$	10.1481	0.985672
$107$	−2.33849	−0.226071	−0.113035	−	0.993591i	$-0.536057\pi$
$107$	−2.33849	−0.226071	−0.113035	+	0.993591i	$0.536057\pi$
$108$	−0.101093	−0.00972772
$109$	5.54254	0.530879	0.265440	−	0.964127i	$-0.414483\pi$
$109$	5.54254	0.530879	0.265440	+	0.964127i	$0.414483\pi$
$110$	−1.58338	−0.150969
$111$	23.5750	2.23764
$112$	−1.34845	−0.127416
$113$	19.0918	1.79600	0.898002	−	0.439992i	$-0.145019\pi$
$113$	19.0918	1.79600	0.898002	+	0.439992i	$0.145019\pi$
$114$	2.08329	0.195118
$115$	−9.03221	−0.842258
$116$	−0.118525	−0.0110048
$117$	7.32945	0.677608
$118$	6.61739	0.609180
$119$	2.33106	0.213688
$120$	−6.55818	−0.598677
$121$	−9.83276	−0.893888
$122$	−0.363916	−0.0329474
$123$	−20.6985	−1.86632
$124$	−0.654928	−0.0588142
$125$	−9.13002	−0.816614
$126$	1.11381	0.0992257
$127$	4.70117	0.417161	0.208581	−	0.978005i	$-0.433116\pi$
$127$	4.70117	0.417161	0.208581	+	0.978005i	$0.433116\pi$
$128$	11.8684	1.04903
$129$	19.5201	1.71865
$130$	4.52515	0.396882
$131$	−6.15951	−0.538159	−0.269079	−	0.963118i	$-0.586719\pi$
$131$	−6.15951	−0.538159	−0.269079	+	0.963118i	$0.586719\pi$
$132$	−0.174416	−0.0151810
$133$	0.203741	0.0176666
$134$	−6.74774	−0.582916
$135$	−1.47881	−0.127276
$136$	−19.8482	−1.70197
$137$	2.30414	0.196856	0.0984280	−	0.995144i	$-0.468619\pi$
$137$	2.30414	0.196856	0.0984280	+	0.995144i	$0.468619\pi$
$138$	−29.5682	−2.51701
$139$	16.8924	1.43280	0.716399	−	0.697690i	$-0.245789\pi$
$139$	16.8924	1.43280	0.716399	+	0.697690i	$0.245789\pi$
$140$	0.0231389	0.00195559
$141$	−24.2030	−2.03826
$142$	13.6350	1.14423
$143$	−3.33586	−0.278958
$144$	−9.81430	−0.817859
$145$	−1.73381	−0.143985
$146$	16.6737	1.37992
$147$	−15.9804	−1.31804
$148$	0.708236	0.0582167
$149$	1.76922	0.144940	0.0724699	−	0.997371i	$-0.476912\pi$
$149$	1.76922	0.144940	0.0724699	+	0.997371i	$0.476912\pi$
$150$	−13.2137	−1.07889
$151$	−7.74117	−0.629968	−0.314984	−	0.949097i	$-0.601999\pi$
$151$	−7.74117	−0.629968	−0.314984	+	0.949097i	$0.601999\pi$
$152$	−1.73479	−0.140710
$153$	16.9660	1.37162
$154$	−0.506927	−0.0408493
$155$	−9.58039	−0.769516
$156$	0.498466	0.0399092
$157$	7.32976	0.584979	0.292489	−	0.956269i	$-0.405516\pi$
$157$	7.32976	0.584979	0.292489	+	0.956269i	$0.405516\pi$
$158$	−5.83987	−0.464595
$159$	16.3523	1.29682
$160$	−0.401147	−0.0317135
$161$	−2.89171	−0.227899
$162$	−15.0861	−1.18528
$163$	21.4421	1.67947	0.839736	−	0.542995i	$-0.182710\pi$
$163$	21.4421	1.67947	0.839736	+	0.542995i	$0.182710\pi$
$164$	−0.621822	−0.0485562
$165$	−2.55139	−0.198626
$166$	14.4733	1.12334
$167$	−16.7125	−1.29325	−0.646627	−	0.762806i	$-0.723821\pi$
$167$	−16.7125	−1.29325	−0.646627	+	0.762806i	$0.723821\pi$
$168$	−2.09964	−0.161991
$169$	−3.46642	−0.266648
$170$	10.4747	0.803370
$171$	1.48287	0.113398
$172$	0.586419	0.0447141
$173$	−3.83728	−0.291743	−0.145871	−	0.989304i	$-0.546599\pi$
$173$	−3.83728	−0.291743	−0.145871	+	0.989304i	$0.546599\pi$
$174$	−5.67586	−0.430286
$175$	−1.29227	−0.0976868
$176$	4.46679	0.336697
$177$	10.6630	0.801481
$178$	9.39814	0.704420
$179$	−21.3060	−1.59249	−0.796244	−	0.604976i	$-0.793183\pi$
$179$	−21.3060	−1.59249	−0.796244	+	0.604976i	$0.793183\pi$
$180$	0.168410	0.0125525
$181$	−2.59746	−0.193068	−0.0965339	−	0.995330i	$-0.530776\pi$
$181$	−2.59746	−0.193068	−0.0965339	+	0.995330i	$0.530776\pi$
$182$	1.44875	0.107389
$183$	−0.586400	−0.0433479
$184$	24.6220	1.81515
$185$	10.3602	0.761697
$186$	−31.3628	−2.29963
$187$	−7.72174	−0.564669
$188$	−0.727102	−0.0530293
$189$	−0.473450	−0.0344384
$190$	0.915515	0.0664184
$191$	4.20693	0.304403	0.152201	−	0.988349i	$-0.451364\pi$
$191$	4.20693	0.304403	0.152201	+	0.988349i	$0.451364\pi$
$192$	17.8552	1.28859
$193$	20.3372	1.46390	0.731952	−	0.681357i	$-0.238610\pi$
$193$	20.3372	1.46390	0.731952	+	0.681357i	$0.238610\pi$
$194$	−15.9959	−1.14844
$195$	7.29165	0.522166
$196$	−0.480081	−0.0342915
$197$	−16.2124	−1.15509	−0.577544	−	0.816360i	$-0.695989\pi$
$197$	−16.2124	−1.15509	−0.577544	+	0.816360i	$0.695989\pi$
$198$	−3.68953	−0.262203
$199$	2.02312	0.143415	0.0717077	−	0.997426i	$-0.477155\pi$
$199$	2.02312	0.143415	0.0717077	+	0.997426i	$0.477155\pi$
$200$	11.0033	0.778049
$201$	−10.8730	−0.766926
$202$	−3.55017	−0.249789
$203$	−0.555088	−0.0389596
$204$	1.15383	0.0807845
$205$	−9.09612	−0.635301
$206$	7.86200	0.547772
$207$	−21.0465	−1.46284
$208$	−12.7657	−0.885141
$209$	−0.674901	−0.0466839
$210$	1.10806	0.0764635
$211$	21.5719	1.48507	0.742536	−	0.669807i	$-0.233623\pi$
$211$	21.5719	1.48507	0.742536	+	0.669807i	$0.233623\pi$
$212$	0.491253	0.0337394
$213$	21.9710	1.50543
$214$	−3.36422	−0.229973
$215$	8.57824	0.585031
$216$	4.03126	0.274293
$217$	−3.06722	−0.208216
$218$	7.97364	0.540043
$219$	26.8673	1.81552
$220$	−0.0766485	−0.00516764
$221$	22.0680	1.48446
$222$	33.9156	2.27626
$223$	18.6279	1.24742	0.623708	−	0.781657i	$-0.285625\pi$
$223$	18.6279	1.24742	0.623708	+	0.781657i	$0.285625\pi$
$224$	−0.128430	−0.00858107
$225$	−9.40546	−0.627030
$226$	27.4659	1.82701
$227$	28.5686	1.89617	0.948083	−	0.318022i	$-0.103019\pi$
$227$	28.5686	1.89617	0.948083	+	0.318022i	$0.103019\pi$
$228$	0.100848	0.00667884
$229$	18.8683	1.24686	0.623428	−	0.781881i	$-0.285740\pi$
$229$	18.8683	1.24686	0.623428	+	0.781881i	$0.285740\pi$
$230$	−12.9940	−0.856797
$231$	−0.816843	−0.0537443
$232$	4.72639	0.310302
$233$	−2.97606	−0.194968	−0.0974839	−	0.995237i	$-0.531079\pi$
$233$	−2.97606	−0.194968	−0.0974839	+	0.995237i	$0.531079\pi$
$234$	10.5443	0.689305
$235$	−10.6362	−0.693827
$236$	0.320336	0.0208521
$237$	−9.41015	−0.611255
$238$	3.35352	0.217377
$239$	−28.4068	−1.83748	−0.918741	−	0.394860i	$-0.870793\pi$
$239$	−28.4068	−1.83748	−0.918741	+	0.394860i	$0.870793\pi$
$240$	−9.76368	−0.630243
$241$	−10.7554	−0.692816	−0.346408	−	0.938084i	$-0.612599\pi$
$241$	−10.7554	−0.692816	−0.346408	+	0.938084i	$0.612599\pi$
$242$	−14.1457	−0.909317
$243$	−19.9543	−1.28007
$244$	−0.0176165	−0.00112778
$245$	−7.02271	−0.448665
$246$	−29.7774	−1.89854
$247$	1.92881	0.122727
$248$	26.1163	1.65839
$249$	23.3217	1.47795
$250$	−13.1347	−0.830710
$251$	−9.63194	−0.607963	−0.303981	−	0.952678i	$-0.598316\pi$
$251$	−9.63194	−0.607963	−0.303981	+	0.952678i	$0.598316\pi$
$252$	0.0539174	0.00339648
$253$	9.57892	0.602221
$254$	6.76321	0.424362
$255$	16.8785	1.05697
$256$	1.66939	0.104337
$257$	−1.50503	−0.0938813	−0.0469406	−	0.998898i	$-0.514947\pi$
$257$	−1.50503	−0.0938813	−0.0469406	+	0.998898i	$0.514947\pi$
$258$	28.0821	1.74831
$259$	3.31688	0.206101
$260$	0.219055	0.0135852
$261$	−4.04005	−0.250073
$262$	−8.86122	−0.547448
$263$	−20.5402	−1.26656	−0.633280	−	0.773923i	$-0.718292\pi$
$263$	−20.5402	−1.26656	−0.633280	+	0.773923i	$0.718292\pi$
$264$	6.95514	0.428059
$265$	7.18613	0.441440
$266$	0.293107	0.0179716
$267$	15.1438	0.926786
$268$	−0.326646	−0.0199531
$269$	−16.1641	−0.985540	−0.492770	−	0.870160i	$-0.664016\pi$
$269$	−16.1641	−0.985540	−0.492770	+	0.870160i	$0.664016\pi$
$270$	−2.12746	−0.129473
$271$	−5.93854	−0.360741	−0.180370	−	0.983599i	$-0.557730\pi$
$271$	−5.93854	−0.360741	−0.180370	+	0.983599i	$0.557730\pi$
$272$	−29.5496	−1.79171
$273$	2.33446	0.141288
$274$	3.31479	0.200254
$275$	4.28071	0.258136
$276$	−1.43135	−0.0861569
$277$	5.34856	0.321364	0.160682	−	0.987006i	$-0.448631\pi$
$277$	5.34856	0.321364	0.160682	+	0.987006i	$0.448631\pi$
$278$	24.3019	1.45753
$279$	−22.3239	−1.33650
$280$	−0.922701	−0.0551419
$281$	4.30380	0.256743	0.128372	−	0.991726i	$-0.459025\pi$
$281$	4.30380	0.256743	0.128372	+	0.991726i	$0.459025\pi$
$282$	−34.8190	−2.07344
$283$	15.9059	0.945509	0.472754	−	0.881194i	$-0.343260\pi$
$283$	15.9059	0.945509	0.472754	+	0.881194i	$0.343260\pi$
$284$	0.660049	0.0391667
$285$	1.47523	0.0873848
$286$	−4.79905	−0.283774
$287$	−2.91218	−0.171900
$288$	−0.934739	−0.0550800
$289$	34.0824	2.00484
$290$	−2.49430	−0.146470
$291$	−25.7751	−1.51096
$292$	0.807143	0.0472345
$293$	−6.39472	−0.373583	−0.186792	−	0.982400i	$-0.559809\pi$
$293$	−6.39472	−0.373583	−0.186792	+	0.982400i	$0.559809\pi$
$294$	−22.9898	−1.34079
$295$	4.68593	0.272826
$296$	−28.2421	−1.64154
$297$	1.56832	0.0910033
$298$	2.54524	0.147442
$299$	−27.3757	−1.58318
$300$	−0.639652	−0.0369303
$301$	2.74637	0.158298
$302$	−11.1366	−0.640842
$303$	−5.72061	−0.328640
$304$	−2.58272	−0.148129
$305$	−0.257697	−0.0147557
$306$	24.4077	1.39529
$307$	14.5848	0.832397	0.416198	−	0.909274i	$-0.363362\pi$
$307$	14.5848	0.832397	0.416198	+	0.909274i	$0.363362\pi$
$308$	−0.0245395	−0.00139827
$309$	12.6685	0.720688
$310$	−13.7826	−0.782799
$311$	−12.4793	−0.707634	−0.353817	−	0.935315i	$-0.615116\pi$
$311$	−12.4793	−0.707634	−0.353817	+	0.935315i	$0.615116\pi$
$312$	−19.8772	−1.12532
$313$	−14.0853	−0.796148	−0.398074	−	0.917353i	$-0.630321\pi$
$313$	−14.0853	−0.796148	−0.398074	+	0.917353i	$0.630321\pi$
$314$	10.5448	0.595076
$315$	0.788713	0.0444390
$316$	−0.282698	−0.0159030
$317$	25.0164	1.40506	0.702530	−	0.711654i	$-0.252054\pi$
$317$	25.0164	1.40506	0.702530	+	0.711654i	$0.252054\pi$
$318$	23.5248	1.31921
$319$	1.83875	0.102950
$320$	7.84660	0.438638
$321$	−5.42097	−0.302569
$322$	−4.16009	−0.231833
$323$	4.46474	0.248425
$324$	−0.730292	−0.0405718
$325$	−12.2339	−0.678614
$326$	30.8471	1.70846
$327$	12.8484	0.710519
$328$	24.7962	1.36914
$329$	−3.40523	−0.187736
$330$	−3.67050	−0.202054
$331$	−33.3448	−1.83280	−0.916398	−	0.400269i	$-0.868917\pi$
$331$	−33.3448	−1.83280	−0.916398	+	0.400269i	$0.868917\pi$
$332$	0.700626	0.0384518
$333$	24.1410	1.32292
$334$	−24.0430	−1.31558
$335$	−4.77824	−0.261063
$336$	−3.12590	−0.170532
$337$	6.55685	0.357174	0.178587	−	0.983924i	$-0.442847\pi$
$337$	6.55685	0.357174	0.178587	+	0.983924i	$0.442847\pi$
$338$	−4.98688	−0.271250
$339$	44.2575	2.40374
$340$	0.507060	0.0274992
$341$	10.1603	0.550210
$342$	2.13330	0.115356
$343$	−4.53142	−0.244674
$344$	−23.3844	−1.26080
$345$	−20.9380	−1.12726
$346$	−5.52040	−0.296779
$347$	28.5119	1.53060	0.765298	−	0.643676i	$-0.222591\pi$
$347$	28.5119	1.53060	0.765298	+	0.643676i	$0.222591\pi$
$348$	−0.274759	−0.0147286
$349$	19.9621	1.06855	0.534274	−	0.845312i	$-0.320585\pi$
$349$	19.9621	1.06855	0.534274	+	0.845312i	$0.320585\pi$
$350$	−1.85910	−0.0993730
$351$	−4.48212	−0.239238
$352$	0.425428	0.0226754
$353$	3.95728	0.210625	0.105312	−	0.994439i	$-0.466416\pi$
$353$	3.95728	0.210625	0.105312	+	0.994439i	$0.466416\pi$
$354$	15.3401	0.815315
$355$	9.65531	0.512451
$356$	0.454948	0.0241122
$357$	5.40374	0.285996
$358$	−30.6514	−1.61998
$359$	−1.04479	−0.0551417	−0.0275708	−	0.999620i	$-0.508777\pi$
$359$	−1.04479	−0.0551417	−0.0275708	+	0.999620i	$0.508777\pi$
$360$	−6.71562	−0.353944
$361$	−18.6098	−0.979462
$362$	−3.73677	−0.196400
$363$	−22.7938	−1.19636
$364$	0.0701316	0.00367589
$365$	11.8070	0.618008
$366$	−0.843609	−0.0440962
$367$	6.58474	0.343720	0.171860	−	0.985121i	$-0.445022\pi$
$367$	6.58474	0.343720	0.171860	+	0.985121i	$0.445022\pi$
$368$	36.6567	1.91086
$369$	−21.1955	−1.10339
$370$	14.9044	0.774845
$371$	2.30068	0.119445
$372$	−1.51822	−0.0787159
$373$	11.8140	0.611704	0.305852	−	0.952079i	$-0.401059\pi$
$373$	11.8140	0.611704	0.305852	+	0.952079i	$0.401059\pi$
$374$	−11.1087	−0.574416
$375$	−21.1647	−1.09294
$376$	28.9944	1.49527
$377$	−5.25499	−0.270646
$378$	−0.681117	−0.0350329
$379$	−36.7729	−1.88890	−0.944450	−	0.328656i	$-0.893404\pi$
$379$	−36.7729	−1.88890	−0.944450	+	0.328656i	$0.893404\pi$
$380$	0.0443185	0.00227349
$381$	10.8980	0.558321
$382$	6.05220	0.309657
$383$	−19.6707	−1.00513	−0.502564	−	0.864540i	$-0.667610\pi$
$383$	−19.6707	−1.00513	−0.502564	+	0.864540i	$0.667610\pi$
$384$	27.5126	1.40400
$385$	−0.358967	−0.0182947
$386$	29.2576	1.48917
$387$	19.9887	1.01608
$388$	−0.774332	−0.0393107
$389$	−15.8147	−0.801837	−0.400919	−	0.916114i	$-0.631309\pi$
$389$	−15.8147	−0.801837	−0.400919	+	0.916114i	$0.631309\pi$
$390$	10.4899	0.531179
$391$	−63.3684	−3.20468
$392$	19.1440	0.966919
$393$	−14.2786	−0.720262
$394$	−23.3236	−1.17503
$395$	−4.13536	−0.208072
$396$	−0.178604	−0.00897517
$397$	1.94092	0.0974121	0.0487060	−	0.998813i	$-0.484490\pi$
$397$	1.94092	0.0974121	0.0487060	+	0.998813i	$0.484490\pi$
$398$	2.91052	0.145891
$399$	0.472302	0.0236447
$400$	16.3814	0.819072
$401$	−19.1816	−0.957883	−0.478942	−	0.877847i	$-0.658979\pi$
$401$	−19.1816	−0.957883	−0.478942	+	0.877847i	$0.658979\pi$
$402$	−15.6422	−0.780164
$403$	−29.0372	−1.44644
$404$	−0.171858	−0.00855024
$405$	−10.6828	−0.530835
$406$	−0.798564	−0.0396321
$407$	−10.9873	−0.544620
$408$	−46.0110	−2.27788
$409$	−8.60497	−0.425488	−0.212744	−	0.977108i	$-0.568240\pi$
$409$	−8.60497	−0.425488	−0.212744	+	0.977108i	$0.568240\pi$
$410$	−13.0859	−0.646267
$411$	5.34133	0.263468
$412$	0.380586	0.0187501
$413$	1.50023	0.0738214
$414$	−30.2781	−1.48809
$415$	10.2489	0.503097
$416$	−1.21584	−0.0596112
$417$	39.1591	1.91763
$418$	−0.970930	−0.0474897
$419$	−34.1094	−1.66635	−0.833176	−	0.553008i	$-0.813480\pi$
$419$	−34.1094	−1.66635	−0.833176	+	0.553008i	$0.813480\pi$
$420$	0.0536393	0.00261733
$421$	9.64398	0.470019	0.235009	−	0.971993i	$-0.424488\pi$
$421$	9.64398	0.470019	0.235009	+	0.971993i	$0.424488\pi$
$422$	31.0339	1.51071
$423$	−24.7840	−1.20504
$424$	−19.5895	−0.951350
$425$	−28.3186	−1.37365
$426$	31.6080	1.53141
$427$	−0.0825033	−0.00399262
$428$	−0.162856	−0.00787194
$429$	−7.73300	−0.373353
$430$	12.3409	0.595130
$431$	1.00000	0.0481683
$431$	1.00000	0.0481683
$432$	6.00166	0.288755
$433$	9.86132	0.473905	0.236952	−	0.971521i	$-0.423851\pi$
$433$	9.86132	0.473905	0.236952	+	0.971521i	$0.423851\pi$
$434$	−4.41258	−0.211810
$435$	−4.01922	−0.192707
$436$	0.385990	0.0184856
$437$	−5.53857	−0.264946
$438$	38.6520	1.84686
$439$	−16.9448	−0.808730	−0.404365	−	0.914598i	$-0.632508\pi$
$439$	−16.9448	−0.808730	−0.404365	+	0.914598i	$0.632508\pi$
$440$	3.05648	0.145712
$441$	−16.3641	−0.779241
$442$	31.7476	1.51008
$443$	18.7700	0.891788	0.445894	−	0.895086i	$-0.352886\pi$
$443$	18.7700	0.891788	0.445894	+	0.895086i	$0.352886\pi$
$444$	1.64179	0.0779161
$445$	6.65505	0.315480
$446$	26.7986	1.26895
$447$	4.10130	0.193985
$448$	2.51214	0.118687
$449$	17.1910	0.811292	0.405646	−	0.914030i	$-0.367047\pi$
$449$	17.1910	0.811292	0.405646	+	0.914030i	$0.367047\pi$
$450$	−13.5309	−0.637854
$451$	9.64670	0.454245
$452$	1.32958	0.0625381
$453$	−17.9452	−0.843137
$454$	41.0995	1.92890
$455$	1.02590	0.0480948
$456$	−4.02149	−0.188323
$457$	−2.35330	−0.110083	−0.0550415	−	0.998484i	$-0.517529\pi$
$457$	−2.35330	−0.110083	−0.0550415	+	0.998484i	$0.517529\pi$
$458$	27.1445	1.26838
$459$	−10.3751	−0.484267
$460$	−0.629015	−0.0293280
$461$	−28.9484	−1.34826	−0.674131	−	0.738611i	$-0.735482\pi$
$461$	−28.9484	−1.34826	−0.674131	+	0.738611i	$0.735482\pi$
$462$	−1.17513	−0.0546720
$463$	−17.2701	−0.802607	−0.401304	−	0.915945i	$-0.631443\pi$
$463$	−17.2701	−0.802607	−0.401304	+	0.915945i	$0.631443\pi$
$464$	7.03655	0.326663
$465$	−22.2087	−1.02991
$466$	−4.28143	−0.198333
$467$	−16.4101	−0.759368	−0.379684	−	0.925116i	$-0.623967\pi$
$467$	−16.4101	−0.759368	−0.379684	+	0.925116i	$0.623967\pi$
$468$	0.510433	0.0235948
$469$	−1.52978	−0.0706387
$470$	−15.3014	−0.705803
$471$	16.9915	0.782925
$472$	−12.7739	−0.587968
$473$	−9.09747	−0.418302
$474$	−13.5377	−0.621806
$475$	−2.47512	−0.113566
$476$	0.162338	0.00744076
$477$	16.7449	0.766694
$478$	−40.8667	−1.86920
$479$	−28.4166	−1.29839	−0.649193	−	0.760624i	$-0.724893\pi$
$479$	−28.4166	−1.29839	−0.649193	+	0.760624i	$0.724893\pi$
$480$	−0.929918	−0.0424447
$481$	31.4007	1.43175
$482$	−15.4730	−0.704775
$483$	−6.70341	−0.305016
$484$	−0.684767	−0.0311258
$485$	−11.3271	−0.514335
$486$	−28.7067	−1.30216
$487$	3.06565	0.138918	0.0694589	−	0.997585i	$-0.477873\pi$
$487$	3.06565	0.138918	0.0694589	+	0.997585i	$0.477873\pi$
$488$	0.702488	0.0318001
$489$	49.7058	2.24777
$490$	−10.1030	−0.456409
$491$	−17.5739	−0.793098	−0.396549	−	0.918014i	$-0.629792\pi$
$491$	−17.5739	−0.793098	−0.396549	+	0.918014i	$0.629792\pi$
$492$	−1.44147	−0.0649867
$493$	−12.1641	−0.547842
$494$	2.77483	0.124845
$495$	−2.61264	−0.117430
$496$	38.8814	1.74583
$497$	3.09120	0.138659
$498$	33.5511	1.50346
$499$	37.4409	1.67609	0.838043	−	0.545605i	$-0.183700\pi$
$499$	37.4409	1.67609	0.838043	+	0.545605i	$0.183700\pi$
$500$	−0.635827	−0.0284350
$501$	−38.7420	−1.73087
$502$	−13.8567	−0.618457
$503$	−0.167059	−0.00744878	−0.00372439	−	0.999993i	$-0.501186\pi$
$503$	−0.167059	−0.00744878	−0.00372439	+	0.999993i	$0.501186\pi$
$504$	−2.15005	−0.0957706
$505$	−2.51396	−0.111870
$506$	13.7805	0.612617
$507$	−8.03566	−0.356876
$508$	0.327396	0.0145258
$509$	−6.72523	−0.298091	−0.149045	−	0.988830i	$-0.547620\pi$
$509$	−6.72523	−0.298091	−0.149045	+	0.988830i	$0.547620\pi$
$510$	24.2818	1.07522
$511$	3.78009	0.167221
$512$	−21.3351	−0.942888
$513$	−0.906810	−0.0400367
$514$	−2.16518	−0.0955018
$515$	5.56728	0.245323
$516$	1.35941	0.0598445
$517$	11.2800	0.496092
$518$	4.77174	0.209658
$519$	−8.89537	−0.390463
$520$	−8.73516	−0.383062
$521$	−6.51176	−0.285285	−0.142643	−	0.989774i	$-0.545560\pi$
$521$	−6.51176	−0.285285	−0.142643	+	0.989774i	$0.545560\pi$
$522$	−5.81212	−0.254390
$523$	11.2692	0.492766	0.246383	−	0.969173i	$-0.420758\pi$
$523$	11.2692	0.492766	0.246383	+	0.969173i	$0.420758\pi$
$524$	−0.428957	−0.0187391
$525$	−2.99568	−0.130742
$526$	−29.5496	−1.28842
$527$	−67.2143	−2.92790
$528$	10.3547	0.450629
$529$	55.6093	2.41780
$530$	10.3381	0.449060
$531$	10.9190	0.473844
$532$	0.0141888	0.000615163 0
$533$	−27.5694	−1.19416
$534$	21.7862	0.942783
$535$	−2.38228	−0.102995
$536$	13.0256	0.562618
$537$	−49.3905	−2.13136
$538$	−23.2540	−1.00255
$539$	7.44779	0.320799
$540$	−0.102986	−0.00443183
$541$	8.12108	0.349152	0.174576	−	0.984644i	$-0.444144\pi$
$541$	8.12108	0.349152	0.174576	+	0.984644i	$0.444144\pi$
$542$	−8.54334	−0.366968
$543$	−6.02129	−0.258398
$544$	−2.81438	−0.120665
$545$	5.64633	0.241862
$546$	3.35842	0.143727
$547$	−23.9265	−1.02302	−0.511512	−	0.859276i	$-0.670914\pi$
$547$	−23.9265	−1.02302	−0.511512	+	0.859276i	$0.670914\pi$
$548$	0.160463	0.00685466
$549$	−0.600477	−0.0256278
$550$	6.15833	0.262592
$551$	−1.06317	−0.0452927
$552$	57.0773	2.42937
$553$	−1.32396	−0.0563004
$554$	7.69457	0.326911
$555$	24.0164	1.01944
$556$	1.17641	0.0498910
$557$	−14.1560	−0.599807	−0.299903	−	0.953970i	$-0.596954\pi$
$557$	−14.1560	−0.599807	−0.299903	+	0.953970i	$0.596954\pi$
$558$	−32.1157	−1.35957
$559$	25.9997	1.09967
$560$	−1.37370	−0.0580494
$561$	−17.9001	−0.755743
$562$	6.19155	0.261175
$563$	−27.2173	−1.14707	−0.573536	−	0.819180i	$-0.694429\pi$
$563$	−27.2173	−1.14707	−0.573536	+	0.819180i	$0.694429\pi$
$564$	−1.68553	−0.0709735
$565$	19.4493	0.818237
$566$	22.8827	0.961830
$567$	−3.42017	−0.143634
$568$	−26.3205	−1.10438
$569$	19.1424	0.802489	0.401245	−	0.915971i	$-0.368578\pi$
$569$	19.1424	0.802489	0.401245	+	0.915971i	$0.368578\pi$
$570$	2.12230	0.0888932
$571$	−13.4624	−0.563385	−0.281693	−	0.959505i	$-0.590896\pi$
$571$	−13.4624	−0.563385	−0.281693	+	0.959505i	$0.590896\pi$
$572$	−0.232314	−0.00971352
$573$	9.75228	0.407407
$574$	−4.18953	−0.174868
$575$	35.1296	1.46501
$576$	18.2839	0.761828
$577$	44.3284	1.84541	0.922707	−	0.385502i	$-0.125972\pi$
$577$	44.3284	1.84541	0.922707	+	0.385502i	$0.125972\pi$
$578$	49.0317	2.03945
$579$	47.1446	1.95926
$580$	−0.120745	−0.00501365
$581$	3.28124	0.136129
$582$	−37.0807	−1.53705
$583$	−7.62109	−0.315633
$584$	−32.1861	−1.33187
$585$	7.46670	0.308710
$586$	−9.19960	−0.380032
$587$	7.24535	0.299048	0.149524	−	0.988758i	$-0.452226\pi$
$587$	7.24535	0.299048	0.149524	+	0.988758i	$0.452226\pi$
$588$	−1.11290	−0.0458952
$589$	−5.87472	−0.242064
$590$	6.74130	0.277535
$591$	−37.5828	−1.54595
$592$	−42.0462	−1.72809
$593$	7.45718	0.306230	0.153115	−	0.988208i	$-0.451070\pi$
$593$	7.45718	0.306230	0.153115	+	0.988208i	$0.451070\pi$
$594$	2.25623	0.0925741
$595$	2.37471	0.0973537
$596$	0.123211	0.00504690
$597$	4.68989	0.191945
$598$	−39.3833	−1.61050
$599$	11.6897	0.477627	0.238814	−	0.971065i	$-0.423242\pi$
$599$	11.6897	0.477627	0.238814	+	0.971065i	$0.423242\pi$
$600$	25.5072	1.04133
$601$	15.2410	0.621692	0.310846	−	0.950460i	$-0.399388\pi$
$601$	15.2410	0.621692	0.310846	+	0.950460i	$0.399388\pi$
$602$	3.95100	0.161031
$603$	−11.1341	−0.453415
$604$	−0.539106	−0.0219359
$605$	−10.0169	−0.407244
$606$	−8.22981	−0.334313
$607$	−15.4309	−0.626323	−0.313161	−	0.949700i	$-0.601388\pi$
$607$	−15.4309	−0.626323	−0.313161	+	0.949700i	$0.601388\pi$
$608$	−0.245984	−0.00997599
$609$	−1.28678	−0.0521428
$610$	−0.370730	−0.0150104
$611$	−32.2371	−1.30417
$612$	1.18153	0.0477607
$613$	−33.7895	−1.36474	−0.682372	−	0.731005i	$-0.739052\pi$
$613$	−33.7895	−1.36474	−0.682372	+	0.731005i	$0.739052\pi$
$614$	20.9820	0.846765
$615$	−21.0861	−0.850275
$616$	0.978551	0.0394269
$617$	8.87856	0.357438	0.178719	−	0.983900i	$-0.442805\pi$
$617$	8.87856	0.357438	0.178719	+	0.983900i	$0.442805\pi$
$618$	18.2253	0.733128
$619$	25.9511	1.04306	0.521532	−	0.853232i	$-0.325361\pi$
$619$	25.9511	1.04306	0.521532	+	0.853232i	$0.325361\pi$
$620$	−0.667191	−0.0267951
$621$	12.8704	0.516472
$622$	−17.9530	−0.719848
$623$	2.13065	0.0853628
$624$	−29.5927	−1.18466
$625$	10.5100	0.420401
$626$	−20.2635	−0.809891
$627$	−1.56452	−0.0624809
$628$	0.510455	0.0203694
$629$	72.6853	2.89815
$630$	1.13466	0.0452060
$631$	25.1431	1.00093	0.500466	−	0.865756i	$-0.333162\pi$
$631$	25.1431	1.00093	0.500466	+	0.865756i	$0.333162\pi$
$632$	11.2731	0.448418
$633$	50.0068	1.98759
$634$	35.9892	1.42931
$635$	4.78920	0.190053
$636$	1.13880	0.0451562
$637$	−21.2851	−0.843346
$638$	2.64527	0.104727
$639$	22.4985	0.890025
$640$	12.0906	0.477923
$641$	25.9517	1.02503	0.512514	−	0.858679i	$-0.328714\pi$
$641$	25.9517	1.02503	0.512514	+	0.858679i	$0.328714\pi$
$642$	−7.79874	−0.307792
$643$	−20.1346	−0.794031	−0.397015	−	0.917812i	$-0.629954\pi$
$643$	−20.1346	−0.794031	−0.397015	+	0.917812i	$0.629954\pi$
$644$	−0.201383	−0.00793560
$645$	19.8856	0.782995
$646$	6.42309	0.252713
$647$	−29.7747	−1.17056	−0.585282	−	0.810830i	$-0.699016\pi$
$647$	−29.7747	−1.17056	−0.585282	+	0.810830i	$0.699016\pi$
$648$	29.1216	1.14400
$649$	−4.96957	−0.195073
$650$	−17.6000	−0.690327
$651$	−7.11026	−0.278673
$652$	1.49325	0.0584803
$653$	−26.8913	−1.05234	−0.526169	−	0.850380i	$-0.676372\pi$
$653$	−26.8913	−1.05234	−0.526169	+	0.850380i	$0.676372\pi$
$654$	18.4841	0.722784
$655$	−6.27485	−0.245179
$656$	36.9160	1.44133
$657$	27.5123	1.07336
$658$	−4.89885	−0.190977
$659$	25.7643	1.00363	0.501816	−	0.864974i	$-0.332665\pi$
$659$	25.7643	1.00363	0.501816	+	0.864974i	$0.332665\pi$
$660$	−0.177682	−0.00691628
$661$	30.4913	1.18597	0.592987	−	0.805212i	$-0.297949\pi$
$661$	30.4913	1.18597	0.592987	+	0.805212i	$0.297949\pi$
$662$	−47.9706	−1.86443
$663$	51.1569	1.98677
$664$	−27.9386	−1.08423
$665$	0.207556	0.00804869
$666$	34.7298	1.34575
$667$	15.0897	0.584275
$668$	−1.16388	−0.0450320
$669$	43.1822	1.66952
$670$	−6.87409	−0.265569
$671$	0.273296	0.0105505
$672$	−0.297719	−0.0114847
$673$	−8.45445	−0.325895	−0.162948	−	0.986635i	$-0.552100\pi$
$673$	−8.45445	−0.325895	−0.162948	+	0.986635i	$0.552100\pi$
$674$	9.43285	0.363340
$675$	5.75164	0.221381
$676$	−0.241406	−0.00928485
$677$	−33.1288	−1.27324	−0.636622	−	0.771176i	$-0.719669\pi$
$677$	−33.1288	−1.27324	−0.636622	+	0.771176i	$0.719669\pi$
$678$	63.6700	2.44523
$679$	−3.62642	−0.139169
$680$	−20.2199	−0.775396
$681$	66.2262	2.53779
$682$	14.6168	0.559707
$683$	32.2944	1.23571	0.617856	−	0.786291i	$-0.288001\pi$
$683$	32.2944	1.23571	0.617856	+	0.786291i	$0.288001\pi$
$684$	0.103269	0.00394860
$685$	2.34729	0.0896852
$686$	−6.51901	−0.248897
$687$	43.7396	1.66877
$688$	−34.8143	−1.32728
$689$	21.7804	0.829767
$690$	−30.1219	−1.14672
$691$	−13.8209	−0.525771	−0.262885	−	0.964827i	$-0.584674\pi$
$691$	−13.8209	−0.525771	−0.262885	+	0.964827i	$0.584674\pi$
$692$	−0.267233	−0.0101587
$693$	−0.836453	−0.0317742
$694$	41.0179	1.55702
$695$	17.2088	0.652766
$696$	10.9564	0.415303
$697$	−63.8167	−2.41723
$698$	28.7180	1.08699
$699$	−6.89893	−0.260941
$700$	−0.0899957	−0.00340152
$701$	−35.8917	−1.35561	−0.677805	−	0.735241i	$-0.737069\pi$
$701$	−35.8917	−1.35561	−0.677805	+	0.735241i	$0.737069\pi$
$702$	−6.44809	−0.243368
$703$	6.35290	0.239604
$704$	−8.32155	−0.313630
$705$	−24.6562	−0.928605
$706$	5.69305	0.214261
$707$	−0.804860	−0.0302699
$708$	0.742586	0.0279081
$709$	−35.2159	−1.32256	−0.661281	−	0.750138i	$-0.729987\pi$
$709$	−35.2159	−1.32256	−0.661281	+	0.750138i	$0.729987\pi$
$710$	13.8904	0.521296
$711$	−9.63606	−0.361380
$712$	−18.1418	−0.679892
$713$	83.3803	3.12262
$714$	7.77396	0.290933
$715$	−3.39832	−0.127090
$716$	−1.48378	−0.0554515
$717$	−65.8511	−2.45925
$718$	−1.50305	−0.0560935
$719$	4.34475	0.162032	0.0810159	−	0.996713i	$-0.474184\pi$
$719$	4.34475	0.162032	0.0810159	+	0.996713i	$0.474184\pi$
$720$	−9.99808	−0.372606
$721$	1.78240	0.0663799
$722$	−26.7725	−0.996368
$723$	−24.9326	−0.927252
$724$	−0.180891	−0.00672275
$725$	6.74341	0.250444
$726$	−32.7917	−1.21701
$727$	−29.0267	−1.07654	−0.538270	−	0.842773i	$-0.680922\pi$
$727$	−29.0267	−1.07654	−0.538270	+	0.842773i	$0.680922\pi$
$728$	−2.79661	−0.103649
$729$	−14.7975	−0.548056
$730$	16.9859	0.628676
$731$	60.1834	2.22596
$732$	−0.0408377	−0.00150940
$733$	−14.1267	−0.521781	−0.260890	−	0.965368i	$-0.584016\pi$
$733$	−14.1267	−0.521781	−0.260890	+	0.965368i	$0.584016\pi$
$734$	9.47296	0.349653
$735$	−16.2797	−0.600484
$736$	3.49127	0.128690
$737$	5.06746	0.186662
$738$	−30.4923	−1.12244
$739$	1.46866	0.0540257	0.0270128	−	0.999635i	$-0.491400\pi$
$739$	1.46866	0.0540257	0.0270128	+	0.999635i	$0.491400\pi$
$740$	0.721498	0.0265228
$741$	4.47125	0.164256
$742$	3.30982	0.121507
$743$	26.5034	0.972314	0.486157	−	0.873871i	$-0.338398\pi$
$743$	26.5034	0.972314	0.486157	+	0.873871i	$0.338398\pi$
$744$	60.5414	2.21956
$745$	1.80235	0.0660328
$746$	16.9959	0.622263
$747$	23.8816	0.873780
$748$	−0.537752	−0.0196622
$749$	−0.762702	−0.0278685
$750$	−30.4481	−1.11181
$751$	−23.9023	−0.872208	−0.436104	−	0.899896i	$-0.643642\pi$
$751$	−23.9023	−0.872208	−0.436104	+	0.899896i	$0.643642\pi$
$752$	43.1662	1.57411
$753$	−22.3282	−0.813686
$754$	−7.55995	−0.275317
$755$	−7.88613	−0.287006
$756$	−0.0329717	−0.00119917
$757$	30.0435	1.09195	0.545975	−	0.837801i	$-0.316159\pi$
$757$	30.0435	1.09195	0.545975	+	0.837801i	$0.316159\pi$
$758$	−52.9025	−1.92150
$759$	22.2053	0.806002
$760$	−1.76727	−0.0641057
$761$	−18.6862	−0.677373	−0.338686	−	0.940899i	$-0.609983\pi$
$761$	−18.6862	−0.677373	−0.338686	+	0.940899i	$0.609983\pi$
$762$	15.6781	0.567958
$763$	1.80771	0.0654433
$764$	0.292976	0.0105995
$765$	17.2837	0.624893
$766$	−28.2988	−1.02248
$767$	14.2026	0.512825
$768$	3.86989	0.139643
$769$	40.0310	1.44355	0.721777	−	0.692126i	$-0.243326\pi$
$769$	40.0310	1.44355	0.721777	+	0.692126i	$0.243326\pi$
$770$	−0.516419	−0.0186105
$771$	−3.48888	−0.125649
$772$	1.41631	0.0509741
$773$	−0.862526	−0.0310229	−0.0155115	−	0.999880i	$-0.504938\pi$
$773$	−0.862526	−0.0310229	−0.0155115	+	0.999880i	$0.504938\pi$
$774$	28.7563	1.03362
$775$	37.2617	1.33848
$776$	30.8777	1.10845
$777$	7.68900	0.275841
$778$	−22.7514	−0.815678
$779$	−5.57776	−0.199844
$780$	0.507800	0.0181822
$781$	−10.2397	−0.366406
$782$	−91.1633	−3.25999
$783$	2.47058	0.0882914
$784$	28.5012	1.01790
$785$	7.46702	0.266509
$786$	−20.5416	−0.732695
$787$	36.6867	1.30774	0.653869	−	0.756608i	$-0.273145\pi$
$787$	36.6867	1.30774	0.653869	+	0.756608i	$0.273145\pi$
$788$	−1.12906	−0.0402209
$789$	−47.6151	−1.69514
$790$	−5.94923	−0.211664
$791$	6.22680	0.221400
$792$	7.12211	0.253073
$793$	−0.781054	−0.0277360
$794$	2.79226	0.0990935
$795$	16.6585	0.590816
$796$	0.140893	0.00499382
$797$	9.44101	0.334418	0.167209	−	0.985921i	$-0.446525\pi$
$797$	9.44101	0.334418	0.167209	+	0.985921i	$0.446525\pi$
$798$	0.679465	0.0240528
$799$	−74.6214	−2.63992
$800$	1.56021	0.0551618
$801$	15.5074	0.547926
$802$	−27.5951	−0.974418
$803$	−12.5217	−0.441881
$804$	−0.757214	−0.0267049
$805$	−2.94586	−0.103828
$806$	−41.7736	−1.47141
$807$	−37.4706	−1.31903
$808$	6.85310	0.241091
$809$	−23.9366	−0.841565	−0.420783	−	0.907161i	$-0.638245\pi$
$809$	−23.9366	−0.841565	−0.420783	+	0.907161i	$0.638245\pi$
$810$	−15.3686	−0.539998
$811$	−1.77926	−0.0624783	−0.0312392	−	0.999512i	$-0.509945\pi$
$811$	−1.77926	−0.0624783	−0.0312392	+	0.999512i	$0.509945\pi$
$812$	−0.0386571	−0.00135660
$813$	−13.7664	−0.482809
$814$	−15.8066	−0.554021
$815$	21.8436	0.765147
$816$	−68.5002	−2.39799
$817$	5.26020	0.184031
$818$	−12.3793	−0.432833
$819$	2.39051	0.0835311
$820$	−0.633466	−0.0221216
$821$	9.62802	0.336020	0.168010	−	0.985785i	$-0.446266\pi$
$821$	9.62802	0.336020	0.168010	+	0.985785i	$0.446266\pi$
$822$	7.68417	0.268016
$823$	−12.9874	−0.452711	−0.226356	−	0.974045i	$-0.572681\pi$
$823$	−12.9874	−0.452711	−0.226356	+	0.974045i	$0.572681\pi$
$824$	−15.1765	−0.528698
$825$	9.92331	0.345485
$826$	2.15827	0.0750957
$827$	25.3943	0.883045	0.441522	−	0.897250i	$-0.354439\pi$
$827$	25.3943	0.883045	0.441522	+	0.897250i	$0.354439\pi$
$828$	−1.46571	−0.0509369
$829$	35.8474	1.24503	0.622515	−	0.782608i	$-0.286111\pi$
$829$	35.8474	1.24503	0.622515	+	0.782608i	$0.286111\pi$
$830$	14.7443	0.511782
$831$	12.3987	0.430107
$832$	23.7822	0.824500
$833$	−49.2701	−1.70711
$834$	56.3353	1.95073
$835$	−17.0255	−0.589191
$836$	−0.0470010	−0.00162556
$837$	13.6516	0.471867
$838$	−49.0706	−1.69512
$839$	57.1095	1.97164	0.985819	−	0.167810i	$-0.0536695\pi$
$839$	57.1095	1.97164	0.985819	+	0.167810i	$0.0536695\pi$
$840$	−2.13896	−0.0738010
$841$	−26.1034	−0.900118
$842$	13.8741	0.478132
$843$	9.97683	0.343620
$844$	1.50230	0.0517112
$845$	−3.53133	−0.121481
$846$	−35.6549	−1.22584
$847$	−3.20696	−0.110193
$848$	−29.1644	−1.00151
$849$	36.8722	1.26545
$850$	−40.7398	−1.39736
$851$	−90.1671	−3.09089
$852$	1.53009	0.0524200
$853$	0.277200	0.00949113	0.00474557	−	0.999989i	$-0.498489\pi$
$853$	0.277200	0.00949113	0.00474557	+	0.999989i	$0.498489\pi$
$854$	−0.118691	−0.00406153
$855$	1.51064	0.0516628
$856$	6.49414	0.221965
$857$	−48.8638	−1.66916	−0.834578	−	0.550890i	$-0.814288\pi$
$857$	−48.8638	−1.66916	−0.834578	+	0.550890i	$0.814288\pi$
$858$	−11.1249	−0.379797
$859$	7.43773	0.253772	0.126886	−	0.991917i	$-0.459502\pi$
$859$	7.43773	0.253772	0.126886	+	0.991917i	$0.459502\pi$
$860$	0.597400	0.0203712
$861$	−6.75085	−0.230068
$862$	1.43862	0.0489998
$863$	−31.7148	−1.07958	−0.539792	−	0.841799i	$-0.681497\pi$
$863$	−31.7148	−1.07958	−0.539792	+	0.841799i	$0.681497\pi$
$864$	0.571614	0.0194467
$865$	−3.90913	−0.132914
$866$	14.1867	0.482085
$867$	79.0078	2.68325
$868$	−0.213605	−0.00725023
$869$	4.38566	0.148773
$870$	−5.78214	−0.196033
$871$	−14.4823	−0.490715
$872$	−15.3920	−0.521238
$873$	−26.3939	−0.893298
$874$	−7.96793	−0.269519
$875$	−2.97776	−0.100667
$876$	1.87108	0.0632178
$877$	−25.9726	−0.877033	−0.438517	−	0.898723i	$-0.644496\pi$
$877$	−25.9726	−0.877033	−0.438517	+	0.898723i	$0.644496\pi$
$878$	−24.3772	−0.822690
$879$	−14.8239	−0.499997
$880$	4.55043	0.153395
$881$	13.1590	0.443338	0.221669	−	0.975122i	$-0.428850\pi$
$881$	13.1590	0.443338	0.221669	+	0.975122i	$0.428850\pi$
$882$	−23.5418	−0.792692
$883$	14.2968	0.481125	0.240563	−	0.970634i	$-0.422668\pi$
$883$	14.2968	0.481125	0.240563	+	0.970634i	$0.422668\pi$
$884$	1.53685	0.0516898
$885$	10.8627	0.365145
$886$	27.0029	0.907181
$887$	−51.0373	−1.71367	−0.856833	−	0.515595i	$-0.827571\pi$
$887$	−51.0373	−1.71367	−0.856833	+	0.515595i	$0.827571\pi$
$888$	−65.4692	−2.19700
$889$	1.53329	0.0514249
$890$	9.57412	0.320925
$891$	11.3295	0.379551
$892$	1.29727	0.0434359
$893$	−6.52212	−0.218254
$894$	5.90023	0.197333
$895$	−21.7050	−0.725518
$896$	3.87088	0.129317
$897$	−63.4608	−2.11890
$898$	24.7314	0.825296
$899$	16.0055	0.533814
$900$	−0.655009	−0.0218336
$901$	50.4166	1.67962
$902$	13.8780	0.462086
$903$	6.36649	0.211864
$904$	−53.0191	−1.76339
$905$	−2.64610	−0.0879593
$906$	−25.8164	−0.857691
$907$	0.440127	0.0146142	0.00730708	−	0.999973i	$-0.497674\pi$
$907$	0.440127	0.0146142	0.00730708	+	0.999973i	$0.497674\pi$
$908$	1.98956	0.0660258
$909$	−5.85794	−0.194296
$910$	1.47588	0.0489250
$911$	23.7372	0.786448	0.393224	−	0.919443i	$-0.371360\pi$
$911$	23.7372	0.786448	0.393224	+	0.919443i	$0.371360\pi$
$912$	−5.98711	−0.198253
$913$	−10.8692	−0.359719
$914$	−3.38552	−0.111983
$915$	−0.597380	−0.0197488
$916$	1.31402	0.0434163
$917$	−2.00893	−0.0663407
$918$	−14.9258	−0.492626
$919$	−56.0191	−1.84790	−0.923950	−	0.382513i	$-0.875059\pi$
$919$	−56.0191	−1.84790	−0.923950	+	0.382513i	$0.875059\pi$
$920$	25.0830	0.826962
$921$	33.8096	1.11406
$922$	−41.6459	−1.37154
$923$	29.2642	0.963244
$924$	−0.0568860	−0.00187141
$925$	−40.2946	−1.32488
$926$	−24.8451	−0.816462
$927$	12.9727	0.426078
$928$	0.670178	0.0219997
$929$	20.9799	0.688328	0.344164	−	0.938910i	$-0.388162\pi$
$929$	20.9799	0.688328	0.344164	+	0.938910i	$0.388162\pi$
$930$	−31.9500	−1.04768
$931$	−4.30634	−0.141135
$932$	−0.207256	−0.00678891
$933$	−28.9287	−0.947084
$934$	−23.6079	−0.772475
$935$	−7.86633	−0.257256
$936$	−20.3543	−0.665303
$937$	−39.1285	−1.27827	−0.639137	−	0.769093i	$-0.720708\pi$
$937$	−39.1285	−1.27827	−0.639137	+	0.769093i	$0.720708\pi$
$938$	−2.20078	−0.0718580
$939$	−32.6518	−1.06555
$940$	−0.740717	−0.0241595
$941$	9.05841	0.295296	0.147648	−	0.989040i	$-0.452830\pi$
$941$	9.05841	0.295296	0.147648	+	0.989040i	$0.452830\pi$
$942$	24.4443	0.796439
$943$	79.1655	2.57798
$944$	−19.0176	−0.618969
$945$	−0.482316	−0.0156897
$946$	−13.0878	−0.425523
$947$	4.02287	0.130726	0.0653629	−	0.997862i	$-0.479180\pi$
$947$	4.02287	0.130726	0.0653629	+	0.997862i	$0.479180\pi$
$948$	−0.655335	−0.0212843
$949$	35.7859	1.16166
$950$	−3.56078	−0.115527
$951$	57.9916	1.88051
$952$	−6.47350	−0.209807
$953$	−19.6804	−0.637511	−0.318756	−	0.947837i	$-0.603265\pi$
$953$	−19.6804	−0.637511	−0.318756	+	0.947837i	$0.603265\pi$
$954$	24.0896	0.779928
$955$	4.28571	0.138682
$956$	−1.97829	−0.0639824
$957$	4.26249	0.137787
$958$	−40.8808	−1.32080
$959$	0.751497	0.0242671
$960$	18.1896	0.587066
$961$	57.4408	1.85293
$962$	45.1738	1.45646
$963$	−5.55111	−0.178882
$964$	−0.749020	−0.0241243
$965$	20.7180	0.666936
$966$	−9.64370	−0.310281
$967$	−30.2562	−0.972973	−0.486486	−	0.873688i	$-0.661722\pi$
$967$	−30.2562	−0.972973	−0.486486	+	0.873688i	$0.661722\pi$
$968$	27.3062	0.877654
$969$	10.3499	0.332487
$970$	−16.2954	−0.523213
$971$	6.41013	0.205711	0.102855	−	0.994696i	$-0.467202\pi$
$971$	6.41013	0.205711	0.102855	+	0.994696i	$0.467202\pi$
$972$	−1.38964	−0.0445728
$973$	5.50949	0.176626
$974$	4.41032	0.141316
$975$	−28.3599	−0.908244
$976$	1.04585	0.0334768
$977$	55.7983	1.78515	0.892573	−	0.450903i	$-0.148898\pi$
$977$	55.7983	1.78515	0.892573	+	0.450903i	$0.148898\pi$
$978$	71.5080	2.28657
$979$	−7.05787	−0.225571
$980$	−0.489071	−0.0156228
$981$	13.1569	0.420067
$982$	−25.2822	−0.806788
$983$	−8.26764	−0.263697	−0.131848	−	0.991270i	$-0.542091\pi$
$983$	−8.26764	−0.263697	−0.131848	+	0.991270i	$0.542091\pi$
$984$	57.4811	1.83243
$985$	−16.5160	−0.526244
$986$	−17.4995	−0.557299
$987$	−7.89382	−0.251263
$988$	0.134325	0.00427344
$989$	−74.6583	−2.37400
$990$	−3.75861	−0.119457
$991$	−9.02386	−0.286652	−0.143326	−	0.989676i	$-0.545780\pi$
$991$	−9.02386	−0.286652	−0.143326	+	0.989676i	$0.545780\pi$
$992$	3.70317	0.117576
$993$	−77.2981	−2.45298
$994$	4.44708	0.141053
$995$	2.06101	0.0653383
$996$	1.62415	0.0514632
$997$	45.0871	1.42792	0.713962	−	0.700185i	$-0.246899\pi$
$997$	45.0871	1.42792	0.713962	+	0.700185i	$0.246899\pi$
$998$	53.8634	1.70502
$999$	−14.7627	−0.467072

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	431.2.a.f.1.17	✓	24
3.2	odd	2		3879.2.a.r.1.8		24
4.3	odd	2		6896.2.a.w.1.5		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
431.2.a.f.1.17	✓	24	1.1	even	1	trivial
3879.2.a.r.1.8		24	3.2	odd	2
6896.2.a.w.1.5		24	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 431 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	431.a (trivial)

\(f(q)\)	\(=\)	\(q+1.43862 q^{2} +2.31815 q^{3} +0.0696413 q^{4} +1.01873 q^{5} +3.33494 q^{6} +0.326151 q^{7} -2.77706 q^{8} +2.37380 q^{9} +1.46556 q^{10} -1.08039 q^{11} +0.161439 q^{12} +3.08765 q^{13} +0.469209 q^{14} +2.36155 q^{15} -4.13443 q^{16} +7.14719 q^{17} +3.41500 q^{18} +0.624685 q^{19} +0.0709454 q^{20} +0.756065 q^{21} -1.55427 q^{22} -8.86619 q^{23} -6.43763 q^{24} -3.96220 q^{25} +4.44197 q^{26} -1.45163 q^{27} +0.0227136 q^{28} -1.70194 q^{29} +3.39739 q^{30} -9.40429 q^{31} -0.393774 q^{32} -2.50449 q^{33} +10.2821 q^{34} +0.332258 q^{35} +0.165314 q^{36} +10.1698 q^{37} +0.898687 q^{38} +7.15762 q^{39} -2.82906 q^{40} -8.92892 q^{41} +1.08769 q^{42} +8.42056 q^{43} -0.0752396 q^{44} +2.41825 q^{45} -12.7551 q^{46} -10.4407 q^{47} -9.58422 q^{48} -6.89363 q^{49} -5.70012 q^{50} +16.5682 q^{51} +0.215028 q^{52} +7.05404 q^{53} -2.08835 q^{54} -1.10062 q^{55} -0.905741 q^{56} +1.44811 q^{57} -2.44845 q^{58} +4.59980 q^{59} +0.164462 q^{60} -0.252961 q^{61} -13.5293 q^{62} +0.774216 q^{63} +7.70237 q^{64} +3.14547 q^{65} -3.60303 q^{66} -4.69041 q^{67} +0.497740 q^{68} -20.5531 q^{69} +0.477995 q^{70} +9.47783 q^{71} -6.59218 q^{72} +11.5900 q^{73} +14.6305 q^{74} -9.18495 q^{75} +0.0435039 q^{76} -0.352369 q^{77} +10.2971 q^{78} -4.05934 q^{79} -4.21185 q^{80} -10.4865 q^{81} -12.8454 q^{82} +10.0605 q^{83} +0.0526534 q^{84} +7.28103 q^{85} +12.1140 q^{86} -3.94534 q^{87} +3.00030 q^{88} +6.53272 q^{89} +3.47895 q^{90} +1.00704 q^{91} -0.617453 q^{92} -21.8005 q^{93} -15.0202 q^{94} +0.636382 q^{95} -0.912825 q^{96} -11.1189 q^{97} -9.91734 q^{98} -2.56462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + q^{3} + 33 q^{4} + 13 q^{5} + 17 q^{6} + 8 q^{7} - 3 q^{8} + 31 q^{9} - 6 q^{10} + 15 q^{11} - 12 q^{12} + 11 q^{13} + 16 q^{14} - 5 q^{15} + 43 q^{16} + 6 q^{17} - 8 q^{18} + 18 q^{19}+ \cdots - 9 q^{99}+O(q^{100}) \) 24 * q + q^2 + q^3 + 33 * q^4 + 13 * q^5 + 17 * q^6 + 8 * q^7 - 3 * q^8 + 31 * q^9 - 6 * q^10 + 15 * q^11 - 12 * q^12 + 11 * q^13 + 16 * q^14 - 5 * q^15 + 43 * q^16 + 6 * q^17 - 8 * q^18 + 18 * q^19 - 10 * q^20 - 7 * q^21 - 5 * q^22 - 8 * q^23 + 15 * q^24 + 51 * q^25 + 5 * q^26 + 13 * q^27 + 2 * q^28 + 17 * q^29 - 54 * q^30 + 37 * q^31 - 20 * q^32 + 16 * q^33 + 50 * q^34 + 3 * q^35 + 54 * q^36 + 7 * q^37 - 25 * q^38 - 38 * q^39 + 2 * q^40 + 24 * q^41 + 18 * q^42 + 10 * q^43 + 78 * q^45 + 9 * q^46 - 6 * q^47 - 25 * q^48 + 84 * q^49 - 65 * q^50 - 21 * q^51 - 31 * q^52 - 22 * q^53 + 23 * q^54 + 12 * q^55 - 45 * q^56 + 9 * q^57 - 12 * q^58 + 21 * q^59 - 82 * q^60 + 46 * q^61 - 68 * q^62 - 47 * q^63 + 53 * q^64 - 42 * q^65 - 56 * q^66 - 25 * q^67 + 3 * q^68 - 33 * q^69 - 18 * q^70 - 6 * q^71 - 115 * q^72 + 86 * q^73 - 60 * q^74 - 34 * q^75 + 48 * q^76 - 31 * q^77 - 84 * q^78 - 6 * q^79 - 24 * q^80 + 44 * q^81 - 3 * q^82 - 49 * q^83 - 126 * q^84 + 75 * q^85 - 56 * q^86 - 31 * q^87 - 91 * q^88 + 61 * q^89 - 161 * q^90 + 32 * q^91 - 130 * q^92 - 6 * q^93 + 8 * q^94 - 23 * q^95 - 5 * q^96 + 88 * q^97 - 120 * q^98 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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