LMFDB - Embedded newform 431.2.a.f.1.8

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			1.8
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)

$f(q)$	$=$	\(q-1.40794 q^{2} -2.74104 q^{3} -0.0177091 q^{4} +4.09865 q^{5} +3.85922 q^{6} -4.31281 q^{7} +2.84081 q^{8} +4.51331 q^{9} -5.77065 q^{10} -0.743484 q^{11} +0.0485414 q^{12} -6.34810 q^{13} +6.07217 q^{14} -11.2346 q^{15} -3.96427 q^{16} +2.20015 q^{17} -6.35446 q^{18} -2.13671 q^{19} -0.0725834 q^{20} +11.8216 q^{21} +1.04678 q^{22} +6.65760 q^{23} -7.78678 q^{24} +11.7990 q^{25} +8.93773 q^{26} -4.14803 q^{27} +0.0763760 q^{28} +4.63554 q^{29} +15.8176 q^{30} +7.85410 q^{31} -0.100175 q^{32} +2.03792 q^{33} -3.09768 q^{34} -17.6767 q^{35} -0.0799265 q^{36} -5.84050 q^{37} +3.00836 q^{38} +17.4004 q^{39} +11.6435 q^{40} +0.253530 q^{41} -16.6441 q^{42} -0.0120649 q^{43} +0.0131664 q^{44} +18.4985 q^{45} -9.37349 q^{46} +2.90991 q^{47} +10.8662 q^{48} +11.6003 q^{49} -16.6122 q^{50} -6.03070 q^{51} +0.112419 q^{52} +4.36617 q^{53} +5.84017 q^{54} -3.04728 q^{55} -12.2519 q^{56} +5.85682 q^{57} -6.52656 q^{58} +14.4731 q^{59} +0.198954 q^{60} +10.1341 q^{61} -11.0581 q^{62} -19.4650 q^{63} +8.06958 q^{64} -26.0186 q^{65} -2.86927 q^{66} -3.92556 q^{67} -0.0389627 q^{68} -18.2488 q^{69} +24.8877 q^{70} -0.836961 q^{71} +12.8214 q^{72} +5.05132 q^{73} +8.22306 q^{74} -32.3414 q^{75} +0.0378393 q^{76} +3.20651 q^{77} -24.4987 q^{78} -4.55354 q^{79} -16.2482 q^{80} -2.16999 q^{81} -0.356954 q^{82} +2.32619 q^{83} -0.209350 q^{84} +9.01765 q^{85} +0.0169866 q^{86} -12.7062 q^{87} -2.11210 q^{88} +2.29653 q^{89} -26.0447 q^{90} +27.3781 q^{91} -0.117900 q^{92} -21.5284 q^{93} -4.09697 q^{94} -8.75765 q^{95} +0.274583 q^{96} -0.401922 q^{97} -16.3326 q^{98} -3.35557 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.40794	−0.995563	−0.497781	−	0.867303i	$-0.665852\pi$
$2$	−1.40794	−0.995563	−0.497781	+	0.867303i	$0.665852\pi$
$3$	−2.74104	−1.58254	−0.791270	−	0.611466i	$-0.790580\pi$
$3$	−2.74104	−1.58254	−0.791270	+	0.611466i	$0.790580\pi$
$4$	−0.0177091	−0.00885455
$5$	4.09865	1.83297	0.916487	−	0.400065i	$-0.131013\pi$
$5$	4.09865	1.83297	0.916487	+	0.400065i	$0.131013\pi$
$6$	3.85922	1.57552
$7$	−4.31281	−1.63009	−0.815045	−	0.579398i	$-0.803288\pi$
$7$	−4.31281	−1.63009	−0.815045	+	0.579398i	$0.803288\pi$
$8$	2.84081	1.00438
$9$	4.51331	1.50444
$10$	−5.77065	−1.82484
$11$	−0.743484	−0.224169	−0.112084	−	0.993699i	$-0.535753\pi$
$11$	−0.743484	−0.224169	−0.112084	+	0.993699i	$0.535753\pi$
$12$	0.0485414	0.0140127
$13$	−6.34810	−1.76065	−0.880323	−	0.474376i	$-0.842674\pi$
$13$	−6.34810	−1.76065	−0.880323	+	0.474376i	$0.842674\pi$
$14$	6.07217	1.62286
$15$	−11.2346	−2.90075
$16$	−3.96427	−0.991067
$17$	2.20015	0.533615	0.266807	−	0.963750i	$-0.414031\pi$
$17$	2.20015	0.533615	0.266807	+	0.963750i	$0.414031\pi$
$18$	−6.35446	−1.49776
$19$	−2.13671	−0.490196	−0.245098	−	0.969498i	$-0.578820\pi$
$19$	−2.13671	−0.490196	−0.245098	+	0.969498i	$0.578820\pi$
$20$	−0.0725834	−0.0162301
$21$	11.8216	2.57968
$22$	1.04678	0.223174
$23$	6.65760	1.38821	0.694103	−	0.719876i	$-0.255801\pi$
$23$	6.65760	1.38821	0.694103	+	0.719876i	$0.255801\pi$
$24$	−7.78678	−1.58947
$25$	11.7990	2.35979
$26$	8.93773	1.75283
$27$	−4.14803	−0.798289
$28$	0.0763760	0.0144337
$29$	4.63554	0.860799	0.430399	−	0.902639i	$-0.358373\pi$
$29$	4.63554	0.860799	0.430399	+	0.902639i	$0.358373\pi$
$30$	15.8176	2.88788
$31$	7.85410	1.41064	0.705319	−	0.708890i	$-0.250804\pi$
$31$	7.85410	1.41064	0.705319	+	0.708890i	$0.250804\pi$
$32$	−0.100175	−0.0177086
$33$	2.03792	0.354757
$34$	−3.09768	−0.531247
$35$	−17.6767	−2.98791
$36$	−0.0799265	−0.0133211
$37$	−5.84050	−0.960172	−0.480086	−	0.877222i	$-0.659394\pi$
$37$	−5.84050	−0.960172	−0.480086	+	0.877222i	$0.659394\pi$
$38$	3.00836	0.488021
$39$	17.4004	2.78629
$40$	11.6435	1.84100
$41$	0.253530	0.0395947	0.0197973	−	0.999804i	$-0.493698\pi$
$41$	0.253530	0.0395947	0.0197973	+	0.999804i	$0.493698\pi$
$42$	−16.6441	−2.56824
$43$	−0.0120649	−0.00183988	−0.000919940	−	1.00000i	$-0.500293\pi$
$43$	−0.0120649	−0.00183988	−0.000919940		1.00000i	$0.500293\pi$
$44$	0.0131664	0.00198491
$45$	18.4985	2.75759
$46$	−9.37349	−1.38205
$47$	2.90991	0.424454	0.212227	−	0.977220i	$-0.431928\pi$
$47$	2.90991	0.424454	0.212227	+	0.977220i	$0.431928\pi$
$48$	10.8662	1.56840
$49$	11.6003	1.65719
$50$	−16.6122	−2.34932
$51$	−6.03070	−0.844467
$52$	0.112419	0.0155897
$53$	4.36617	0.599740	0.299870	−	0.953980i	$-0.403057\pi$
$53$	4.36617	0.599740	0.299870	+	0.953980i	$0.403057\pi$
$54$	5.84017	0.794747
$55$	−3.04728	−0.410896
$56$	−12.2519	−1.63723
$57$	5.85682	0.775755
$58$	−6.52656	−0.856979
$59$	14.4731	1.88424	0.942122	−	0.335271i	$-0.108828\pi$
$59$	14.4731	1.88424	0.942122	+	0.335271i	$0.108828\pi$
$60$	0.198954	0.0256849
$61$	10.1341	1.29754	0.648770	−	0.760984i	$-0.275284\pi$
$61$	10.1341	1.29754	0.648770	+	0.760984i	$0.275284\pi$
$62$	−11.0581	−1.40438
$63$	−19.4650	−2.45236
$64$	8.06958	1.00870
$65$	−26.0186	−3.22722
$66$	−2.86927	−0.353182
$67$	−3.92556	−0.479583	−0.239792	−	0.970824i	$-0.577079\pi$
$67$	−3.92556	−0.479583	−0.239792	+	0.970824i	$0.577079\pi$
$68$	−0.0389627	−0.00472492
$69$	−18.2488	−2.19689
$70$	24.8877	2.97465
$71$	−0.836961	−0.0993290	−0.0496645	−	0.998766i	$-0.515815\pi$
$71$	−0.836961	−0.0993290	−0.0496645	+	0.998766i	$0.515815\pi$
$72$	12.8214	1.51102
$73$	5.05132	0.591213	0.295606	−	0.955310i	$-0.404478\pi$
$73$	5.05132	0.591213	0.295606	+	0.955310i	$0.404478\pi$
$74$	8.22306	0.955911
$75$	−32.3414	−3.73446
$76$	0.0378393	0.00434046
$77$	3.20651	0.365415
$78$	−24.4987	−2.77393
$79$	−4.55354	−0.512313	−0.256156	−	0.966635i	$-0.582456\pi$
$79$	−4.55354	−0.512313	−0.256156	+	0.966635i	$0.582456\pi$
$80$	−16.2482	−1.81660
$81$	−2.16999	−0.241110
$82$	−0.356954	−0.0394190
$83$	2.32619	0.255333	0.127666	−	0.991817i	$-0.459251\pi$
$83$	2.32619	0.255333	0.127666	+	0.991817i	$0.459251\pi$
$84$	−0.209350	−0.0228419
$85$	9.01765	0.978102
$86$	0.0169866	0.00183172
$87$	−12.7062	−1.36225
$88$	−2.11210	−0.225150
$89$	2.29653	0.243432	0.121716	−	0.992565i	$-0.461160\pi$
$89$	2.29653	0.243432	0.121716	+	0.992565i	$0.461160\pi$
$90$	−26.0447	−2.74535
$91$	27.3781	2.87001
$92$	−0.117900	−0.0122919
$93$	−21.5284	−2.23239
$94$	−4.09697	−0.422571
$95$	−8.75765	−0.898516
$96$	0.274583	0.0280245
$97$	−0.401922	−0.0408090	−0.0204045	−	0.999792i	$-0.506495\pi$
$97$	−0.401922	−0.0408090	−0.0204045	+	0.999792i	$0.506495\pi$
$98$	−16.3326	−1.64984
$99$	−3.35557	−0.337248
$100$	−0.208949	−0.0208949
$101$	9.79978	0.975115	0.487557	−	0.873091i	$-0.337888\pi$
$101$	9.79978	0.975115	0.487557	+	0.873091i	$0.337888\pi$
$102$	8.49086	0.840720
$103$	3.53984	0.348791	0.174396	−	0.984676i	$-0.444203\pi$
$103$	3.53984	0.348791	0.174396	+	0.984676i	$0.444203\pi$
$104$	−18.0337	−1.76835
$105$	48.4526	4.72849
$106$	−6.14730	−0.597079
$107$	5.40089	0.522124	0.261062	−	0.965322i	$-0.415927\pi$
$107$	5.40089	0.522124	0.261062	+	0.965322i	$0.415927\pi$
$108$	0.0734579	0.00706849
$109$	−14.2013	−1.36023	−0.680117	−	0.733104i	$-0.738071\pi$
$109$	−14.2013	−1.36023	−0.680117	+	0.733104i	$0.738071\pi$
$110$	4.29039	0.409073
$111$	16.0090	1.51951
$112$	17.0971	1.61553
$113$	11.0146	1.03617	0.518085	−	0.855329i	$-0.326645\pi$
$113$	11.0146	1.03617	0.518085	+	0.855329i	$0.326645\pi$
$114$	−8.24605	−0.772313
$115$	27.2872	2.54454
$116$	−0.0820913	−0.00762198
$117$	−28.6509	−2.64878
$118$	−20.3773	−1.87588
$119$	−9.48883	−0.869840
$120$	−31.9153	−2.91345
$121$	−10.4472	−0.949748
$122$	−14.2682	−1.29178
$123$	−0.694935	−0.0626602
$124$	−0.139089	−0.0124906
$125$	27.8665	2.49246
$126$	27.4056	2.44148
$127$	−17.2307	−1.52898	−0.764489	−	0.644636i	$-0.777009\pi$
$127$	−17.2307	−1.52898	−0.764489	+	0.644636i	$0.777009\pi$
$128$	−11.1611	−0.986513
$129$	0.0330704	0.00291168
$130$	36.6326	3.21290
$131$	−6.03573	−0.527345	−0.263672	−	0.964612i	$-0.584934\pi$
$131$	−6.03573	−0.527345	−0.263672	+	0.964612i	$0.584934\pi$
$132$	−0.0360897	−0.00314121
$133$	9.21525	0.799063
$134$	5.52694	0.477455
$135$	−17.0013	−1.46324
$136$	6.25021	0.535951
$137$	8.15179	0.696454	0.348227	−	0.937410i	$-0.386784\pi$
$137$	8.15179	0.696454	0.348227	+	0.937410i	$0.386784\pi$
$138$	25.6931	2.18714
$139$	1.69634	0.143882	0.0719410	−	0.997409i	$-0.477081\pi$
$139$	1.69634	0.143882	0.0719410	+	0.997409i	$0.477081\pi$
$140$	0.313039	0.0264566
$141$	−7.97618	−0.671716
$142$	1.17839	0.0988882
$143$	4.71971	0.394682
$144$	−17.8920	−1.49100
$145$	18.9995	1.57782
$146$	−7.11195	−0.588589
$147$	−31.7970	−2.62257
$148$	0.103430	0.00850188
$149$	12.4258	1.01796	0.508980	−	0.860779i	$-0.330023\pi$
$149$	12.4258	1.01796	0.508980	+	0.860779i	$0.330023\pi$
$150$	45.5347	3.71789
$151$	23.4238	1.90620	0.953100	−	0.302655i	$-0.0978729\pi$
$151$	23.4238	1.90620	0.953100	+	0.302655i	$0.0978729\pi$
$152$	−6.07000	−0.492342
$153$	9.92995	0.802789
$154$	−4.51457	−0.363794
$155$	32.1912	2.58566
$156$	−0.308145	−0.0246714
$157$	−7.31771	−0.584017	−0.292008	−	0.956416i	$-0.594324\pi$
$157$	−7.31771	−0.584017	−0.292008	+	0.956416i	$0.594324\pi$
$158$	6.41110	0.510040
$159$	−11.9679	−0.949113
$160$	−0.410582	−0.0324593
$161$	−28.7130	−2.26290
$162$	3.05521	0.240040
$163$	−3.55408	−0.278377	−0.139189	−	0.990266i	$-0.544449\pi$
$163$	−3.55408	−0.278377	−0.139189	+	0.990266i	$0.544449\pi$
$164$	−0.00448978	−0.000350593 0
$165$	8.35273	0.650259
$166$	−3.27513	−0.254200
$167$	−5.37100	−0.415620	−0.207810	−	0.978169i	$-0.566634\pi$
$167$	−5.37100	−0.415620	−0.207810	+	0.978169i	$0.566634\pi$
$168$	33.5829	2.59098
$169$	27.2983	2.09987
$170$	−12.6963	−0.973762
$171$	−9.64365	−0.737468
$172$	0.000213658 0	1.62913e−5 0
$173$	−14.8892	−1.13201	−0.566004	−	0.824402i	$-0.691511\pi$
$173$	−14.8892	−1.13201	−0.566004	+	0.824402i	$0.691511\pi$
$174$	17.8896	1.35620
$175$	−50.8866	−3.84667
$176$	2.94737	0.222167
$177$	−39.6715	−2.98189
$178$	−3.23338	−0.242352
$179$	2.25449	0.168508	0.0842541	−	0.996444i	$-0.473149\pi$
$179$	2.25449	0.168508	0.0842541	+	0.996444i	$0.473149\pi$
$180$	−0.327591	−0.0244172
$181$	−20.9380	−1.55631	−0.778156	−	0.628071i	$-0.783845\pi$
$181$	−20.9380	−1.55631	−0.778156	+	0.628071i	$0.783845\pi$
$182$	−38.5467	−2.85727
$183$	−27.7780	−2.05341
$184$	18.9130	1.39428
$185$	−23.9382	−1.75997
$186$	30.3107	2.22249
$187$	−1.63578	−0.119620
$188$	−0.0515319	−0.00375835
$189$	17.8897	1.30128
$190$	12.3302	0.894529
$191$	−7.10699	−0.514244	−0.257122	−	0.966379i	$-0.582774\pi$
$191$	−7.10699	−0.514244	−0.257122	+	0.966379i	$0.582774\pi$
$192$	−22.1190	−1.59630
$193$	13.5928	0.978431	0.489216	−	0.872163i	$-0.337283\pi$
$193$	13.5928	0.978431	0.489216	+	0.872163i	$0.337283\pi$
$194$	0.565882	0.0406280
$195$	71.3182	5.10720
$196$	−0.205431	−0.0146737
$197$	10.3656	0.738517	0.369258	−	0.929327i	$-0.379612\pi$
$197$	10.3656	0.738517	0.369258	+	0.929327i	$0.379612\pi$
$198$	4.72444	0.335751
$199$	−12.0810	−0.856401	−0.428201	−	0.903684i	$-0.640852\pi$
$199$	−12.0810	−0.856401	−0.428201	+	0.903684i	$0.640852\pi$
$200$	33.5186	2.37012
$201$	10.7601	0.758960
$202$	−13.7975	−0.970788
$203$	−19.9922	−1.40318
$204$	0.106798	0.00747737
$205$	1.03913	0.0725759
$206$	−4.98388	−0.347243
$207$	30.0478	2.08847
$208$	25.1656	1.74492
$209$	1.58861	0.109887
$210$	−68.2183	−4.70751
$211$	−0.353078	−0.0243069	−0.0121534	−	0.999926i	$-0.503869\pi$
$211$	−0.353078	−0.0243069	−0.0121534	+	0.999926i	$0.503869\pi$
$212$	−0.0773210	−0.00531043
$213$	2.29414	0.157192
$214$	−7.60412	−0.519807
$215$	−0.0494498	−0.00337245
$216$	−11.7838	−0.801784
$217$	−33.8733	−2.29947
$218$	19.9945	1.35420
$219$	−13.8459	−0.935618
$220$	0.0539646	0.00363830
$221$	−13.9668	−0.939506
$222$	−22.5397	−1.51277
$223$	18.1208	1.21346	0.606728	−	0.794910i	$-0.292482\pi$
$223$	18.1208	1.21346	0.606728	+	0.794910i	$0.292482\pi$
$224$	0.432035	0.0288666
$225$	53.2523	3.55015
$226$	−15.5079	−1.03157
$227$	7.44033	0.493832	0.246916	−	0.969037i	$-0.420583\pi$
$227$	7.44033	0.493832	0.246916	+	0.969037i	$0.420583\pi$
$228$	−0.103719	−0.00686896
$229$	17.6697	1.16765	0.583823	−	0.811881i	$-0.301556\pi$
$229$	17.6697	1.16765	0.583823	+	0.811881i	$0.301556\pi$
$230$	−38.4187	−2.53325
$231$	−8.78917	−0.578285
$232$	13.1687	0.864568
$233$	8.53445	0.559110	0.279555	−	0.960130i	$-0.409813\pi$
$233$	8.53445	0.559110	0.279555	+	0.960130i	$0.409813\pi$
$234$	40.3387	2.63702
$235$	11.9267	0.778013
$236$	−0.256306	−0.0166841
$237$	12.4814	0.810756
$238$	13.3597	0.865980
$239$	−0.869775	−0.0562611	−0.0281305	−	0.999604i	$-0.508955\pi$
$239$	−0.869775	−0.0562611	−0.0281305	+	0.999604i	$0.508955\pi$
$240$	44.5369	2.87484
$241$	−18.2278	−1.17416	−0.587078	−	0.809531i	$-0.699722\pi$
$241$	−18.2278	−1.17416	−0.587078	+	0.809531i	$0.699722\pi$
$242$	14.7091	0.945534
$243$	18.3921	1.17986
$244$	−0.179466	−0.0114891
$245$	47.5457	3.03759
$246$	0.978425	0.0623821
$247$	13.5641	0.863061
$248$	22.3120	1.41681
$249$	−6.37618	−0.404074
$250$	−39.2344	−2.48140
$251$	−7.84856	−0.495397	−0.247698	−	0.968837i	$-0.579674\pi$
$251$	−7.84856	−0.495397	−0.247698	+	0.968837i	$0.579674\pi$
$252$	0.344708	0.0217146
$253$	−4.94982	−0.311193
$254$	24.2598	1.52219
$255$	−24.7178	−1.54789
$256$	−0.424985	−0.0265616
$257$	−13.8444	−0.863590	−0.431795	−	0.901972i	$-0.642120\pi$
$257$	−13.8444	−0.863590	−0.431795	+	0.901972i	$0.642120\pi$
$258$	−0.0465611	−0.00289876
$259$	25.1890	1.56517
$260$	0.460767	0.0285755
$261$	20.9216	1.29502
$262$	8.49794	0.525005
$263$	25.3007	1.56011	0.780053	−	0.625714i	$-0.215192\pi$
$263$	25.3007	1.56011	0.780053	+	0.625714i	$0.215192\pi$
$264$	5.78935	0.356310
$265$	17.8954	1.09931
$266$	−12.9745	−0.795518
$267$	−6.29489	−0.385241
$268$	0.0695180	0.00424649
$269$	−7.08548	−0.432009	−0.216005	−	0.976392i	$-0.569303\pi$
$269$	−7.08548	−0.432009	−0.216005	+	0.976392i	$0.569303\pi$
$270$	23.9368	1.45675
$271$	7.54164	0.458122	0.229061	−	0.973412i	$-0.426435\pi$
$271$	7.54164	0.458122	0.229061	+	0.973412i	$0.426435\pi$
$272$	−8.72199	−0.528848
$273$	−75.0446	−4.54191
$274$	−11.4772	−0.693364
$275$	−8.77234	−0.528992
$276$	0.323169	0.0194525
$277$	−32.8036	−1.97098	−0.985490	−	0.169734i	$-0.945709\pi$
$277$	−32.8036	−1.97098	−0.985490	+	0.169734i	$0.945709\pi$
$278$	−2.38835	−0.143244
$279$	35.4480	2.12221
$280$	−50.2162	−3.00099
$281$	−16.1057	−0.960788	−0.480394	−	0.877053i	$-0.659506\pi$
$281$	−16.1057	−0.960788	−0.480394	+	0.877053i	$0.659506\pi$
$282$	11.2300	0.668735
$283$	11.2287	0.667475	0.333737	−	0.942666i	$-0.391690\pi$
$283$	11.2287	0.667475	0.333737	+	0.942666i	$0.391690\pi$
$284$	0.0148218	0.000879513 0
$285$	24.0051	1.42194
$286$	−6.64506	−0.392931
$287$	−1.09342	−0.0645428
$288$	−0.452120	−0.0266414
$289$	−12.1593	−0.715255
$290$	−26.7501	−1.57082
$291$	1.10169	0.0645820
$292$	−0.0894543	−0.00523492
$293$	−9.96356	−0.582077	−0.291039	−	0.956711i	$-0.594001\pi$
$293$	−9.96356	−0.582077	−0.291039	+	0.956711i	$0.594001\pi$
$294$	44.7682	2.61094
$295$	59.3204	3.45377
$296$	−16.5917	−0.964375
$297$	3.08400	0.178952
$298$	−17.4947	−1.01344
$299$	−42.2631	−2.44414
$300$	0.572737	0.0330670
$301$	0.0520336	0.00299917
$302$	−32.9792	−1.89774
$303$	−26.8616	−1.54316
$304$	8.47051	0.485817
$305$	41.5362	2.37836
$306$	−13.9808	−0.799227
$307$	0.141606	0.00808188	0.00404094	−	0.999992i	$-0.498714\pi$
$307$	0.141606	0.00808188	0.00404094	+	0.999992i	$0.498714\pi$
$308$	−0.0567843	−0.00323559
$309$	−9.70285	−0.551976
$310$	−45.3233	−2.57419
$311$	23.3642	1.32486	0.662430	−	0.749124i	$-0.269525\pi$
$311$	23.3642	1.32486	0.662430	+	0.749124i	$0.269525\pi$
$312$	49.4312	2.79849
$313$	21.9828	1.24254	0.621269	−	0.783597i	$-0.286617\pi$
$313$	21.9828	1.24254	0.621269	+	0.783597i	$0.286617\pi$
$314$	10.3029	0.581426
$315$	−79.7804	−4.49512
$316$	0.0806390	0.00453630
$317$	−18.1088	−1.01709	−0.508547	−	0.861034i	$-0.669817\pi$
$317$	−18.1088	−1.01709	−0.508547	+	0.861034i	$0.669817\pi$
$318$	16.8500	0.944902
$319$	−3.44645	−0.192964
$320$	33.0744	1.84891
$321$	−14.8041	−0.826282
$322$	40.4261	2.25286
$323$	−4.70109	−0.261576
$324$	0.0384286	0.00213492
$325$	−74.9009	−4.15475
$326$	5.00393	0.277142
$327$	38.9262	2.15262
$328$	0.720229	0.0397680
$329$	−12.5499	−0.691898
$330$	−11.7601	−0.647374
$331$	−3.77675	−0.207589	−0.103795	−	0.994599i	$-0.533098\pi$
$331$	−3.77675	−0.207589	−0.103795	+	0.994599i	$0.533098\pi$
$332$	−0.0411947	−0.00226085
$333$	−26.3599	−1.44452
$334$	7.56204	0.413776
$335$	−16.0895	−0.879063
$336$	−46.8640	−2.55664
$337$	−25.9975	−1.41617	−0.708087	−	0.706125i	$-0.750442\pi$
$337$	−25.9975	−1.41617	−0.708087	+	0.706125i	$0.750442\pi$
$338$	−38.4344	−2.09055
$339$	−30.1916	−1.63978
$340$	−0.159694	−0.00866065
$341$	−5.83940	−0.316221
$342$	13.5777	0.734196
$343$	−19.8404	−1.07128
$344$	−0.0342741	−0.00184793
$345$	−74.7953	−4.02684
$346$	20.9631	1.12699
$347$	22.0927	1.18600	0.592999	−	0.805203i	$-0.297944\pi$
$347$	22.0927	1.18600	0.592999	+	0.805203i	$0.297944\pi$
$348$	0.225016	0.0120621
$349$	−24.8105	−1.32808	−0.664038	−	0.747699i	$-0.731159\pi$
$349$	−24.8105	−1.32808	−0.664038	+	0.747699i	$0.731159\pi$
$350$	71.6453	3.82960
$351$	26.3321	1.40550
$352$	0.0744784	0.00396971
$353$	32.9268	1.75251	0.876257	−	0.481844i	$-0.160033\pi$
$353$	32.9268	1.75251	0.876257	+	0.481844i	$0.160033\pi$
$354$	55.8550	2.96866
$355$	−3.43041	−0.182067
$356$	−0.0406695	−0.00215548
$357$	26.0093	1.37656
$358$	−3.17418	−0.167761
$359$	−4.80706	−0.253707	−0.126853	−	0.991921i	$-0.540488\pi$
$359$	−4.80706	−0.253707	−0.126853	+	0.991921i	$0.540488\pi$
$360$	52.5506	2.76966
$361$	−14.4345	−0.759708
$362$	29.4795	1.54941
$363$	28.6363	1.50302
$364$	−0.484842	−0.0254126
$365$	20.7036	1.08368
$366$	39.1097	2.04430
$367$	−5.71120	−0.298122	−0.149061	−	0.988828i	$-0.547625\pi$
$367$	−5.71120	−0.298122	−0.149061	+	0.988828i	$0.547625\pi$
$368$	−26.3925	−1.37580
$369$	1.14426	0.0595676
$370$	33.7035	1.75216
$371$	−18.8305	−0.977630
$372$	0.381249	0.0197668
$373$	10.5496	0.546239	0.273119	−	0.961980i	$-0.411945\pi$
$373$	10.5496	0.546239	0.273119	+	0.961980i	$0.411945\pi$
$374$	2.30307	0.119089
$375$	−76.3833	−3.94442
$376$	8.26650	0.426312
$377$	−29.4269	−1.51556
$378$	−25.1876	−1.29551
$379$	−25.2773	−1.29841	−0.649204	−	0.760614i	$-0.724898\pi$
$379$	−25.2773	−1.29841	−0.649204	+	0.760614i	$0.724898\pi$
$380$	0.155090	0.00795595
$381$	47.2301	2.41967
$382$	10.0062	0.511962
$383$	36.5453	1.86738	0.933690	−	0.358083i	$-0.116569\pi$
$383$	36.5453	1.86738	0.933690	+	0.358083i	$0.116569\pi$
$384$	30.5931	1.56120
$385$	13.1424	0.669797
$386$	−19.1378	−0.974090
$387$	−0.0544526	−0.00276798
$388$	0.00711768	0.000361346 0
$389$	18.5474	0.940392	0.470196	−	0.882562i	$-0.344183\pi$
$389$	18.5474	0.940392	0.470196	+	0.882562i	$0.344183\pi$
$390$	−100.412	−5.08454
$391$	14.6477	0.740767
$392$	32.9544	1.66445
$393$	16.5442	0.834544
$394$	−14.5941	−0.735240
$395$	−18.6634	−0.939056
$396$	0.0594241	0.00298618
$397$	10.1070	0.507254	0.253627	−	0.967302i	$-0.418376\pi$
$397$	10.1070	0.507254	0.253627	+	0.967302i	$0.418376\pi$
$398$	17.0093	0.852601
$399$	−25.2594	−1.26455
$400$	−46.7742	−2.33871
$401$	13.1768	0.658017	0.329009	−	0.944327i	$-0.393286\pi$
$401$	13.1768	0.658017	0.329009	+	0.944327i	$0.393286\pi$
$402$	−15.1496	−0.755592
$403$	−49.8586	−2.48363
$404$	−0.173545	−0.00863420
$405$	−8.89403	−0.441948
$406$	28.1478	1.39695
$407$	4.34232	0.215241
$408$	−17.1321	−0.848164
$409$	23.7895	1.17631	0.588157	−	0.808747i	$-0.299854\pi$
$409$	23.7895	1.17631	0.588157	+	0.808747i	$0.299854\pi$
$410$	−1.46303	−0.0722539
$411$	−22.3444	−1.10217
$412$	−0.0626874	−0.00308839
$413$	−62.4199	−3.07148
$414$	−42.3054	−2.07920
$415$	9.53425	0.468018
$416$	0.635919	0.0311785
$417$	−4.64975	−0.227699
$418$	−2.23667	−0.109399
$419$	27.2731	1.33238	0.666188	−	0.745784i	$-0.267925\pi$
$419$	27.2731	1.33238	0.666188	+	0.745784i	$0.267925\pi$
$420$	−0.858052	−0.0418686
$421$	−11.0899	−0.540491	−0.270245	−	0.962791i	$-0.587105\pi$
$421$	−11.0899	−0.540491	−0.270245	+	0.962791i	$0.587105\pi$
$422$	0.497111	0.0241990
$423$	13.1333	0.638563
$424$	12.4035	0.602366
$425$	25.9595	1.25922
$426$	−3.23001	−0.156495
$427$	−43.7065	−2.11511
$428$	−0.0956449	−0.00462317
$429$	−12.9369	−0.624600
$430$	0.0696223	0.00335749
$431$	1.00000	0.0481683
$431$	1.00000	0.0481683
$432$	16.4439	0.791158
$433$	13.2780	0.638102	0.319051	−	0.947738i	$-0.396636\pi$
$433$	13.2780	0.638102	0.319051	+	0.947738i	$0.396636\pi$
$434$	47.6915	2.28926
$435$	−52.0784	−2.49697
$436$	0.251491	0.0120442
$437$	−14.2254	−0.680493
$438$	19.4941	0.931466
$439$	32.3962	1.54619	0.773093	−	0.634293i	$-0.218709\pi$
$439$	32.3962	1.54619	0.773093	+	0.634293i	$0.218709\pi$
$440$	−8.65676	−0.412695
$441$	52.3559	2.49314
$442$	19.6643	0.935338
$443$	−10.3828	−0.493302	−0.246651	−	0.969104i	$-0.579330\pi$
$443$	−10.3828	−0.493302	−0.246651	+	0.969104i	$0.579330\pi$
$444$	−0.283506	−0.0134546
$445$	9.41269	0.446204
$446$	−25.5129	−1.20807
$447$	−34.0596	−1.61096
$448$	−34.8026	−1.64427
$449$	−2.65069	−0.125094	−0.0625468	−	0.998042i	$-0.519922\pi$
$449$	−2.65069	−0.125094	−0.0625468	+	0.998042i	$0.519922\pi$
$450$	−74.9759	−3.53440
$451$	−0.188495	−0.00887589
$452$	−0.195059	−0.00917482
$453$	−64.2055	−3.01664
$454$	−10.4755	−0.491641
$455$	112.213	5.26065
$456$	16.6381	0.779151
$457$	24.0000	1.12267	0.561336	−	0.827588i	$-0.310288\pi$
$457$	24.0000	1.12267	0.561336	+	0.827588i	$0.310288\pi$
$458$	−24.8778	−1.16247
$459$	−9.12630	−0.425979
$460$	−0.483231	−0.0225308
$461$	21.5533	1.00384	0.501919	−	0.864915i	$-0.332628\pi$
$461$	21.5533	1.00384	0.501919	+	0.864915i	$0.332628\pi$
$462$	12.3746	0.575719
$463$	−4.69695	−0.218286	−0.109143	−	0.994026i	$-0.534811\pi$
$463$	−4.69695	−0.218286	−0.109143	+	0.994026i	$0.534811\pi$
$464$	−18.3765	−0.853109
$465$	−88.2375	−4.09192
$466$	−12.0160	−0.556629
$467$	1.73937	0.0804885	0.0402443	−	0.999190i	$-0.487186\pi$
$467$	1.73937	0.0804885	0.0402443	+	0.999190i	$0.487186\pi$
$468$	0.507381	0.0234537
$469$	16.9302	0.781763
$470$	−16.7921	−0.774560
$471$	20.0581	0.924231
$472$	41.1155	1.89249
$473$	0.00897006	0.000412444 0
$474$	−17.5731	−0.807158
$475$	−25.2110	−1.15676
$476$	0.168039	0.00770204
$477$	19.7059	0.902270
$478$	1.22459	0.0560114
$479$	−16.7558	−0.765593	−0.382797	−	0.923833i	$-0.625039\pi$
$479$	−16.7558	−0.765593	−0.382797	+	0.923833i	$0.625039\pi$
$480$	1.12542	0.0513682
$481$	37.0760	1.69052
$482$	25.6636	1.16895
$483$	78.7034	3.58113
$484$	0.185011	0.00840959
$485$	−1.64734	−0.0748019
$486$	−25.8950	−1.17462
$487$	5.58275	0.252978	0.126489	−	0.991968i	$-0.459629\pi$
$487$	5.58275	0.252978	0.126489	+	0.991968i	$0.459629\pi$
$488$	28.7891	1.30322
$489$	9.74188	0.440543
$490$	−66.9415	−3.02411
$491$	−22.2316	−1.00330	−0.501649	−	0.865072i	$-0.667273\pi$
$491$	−22.2316	−1.00330	−0.501649	+	0.865072i	$0.667273\pi$
$492$	0.0123067	0.000554827 0
$493$	10.1989	0.459335
$494$	−19.0974	−0.859232
$495$	−13.7533	−0.618166
$496$	−31.1358	−1.39804
$497$	3.60965	0.161915
$498$	8.97727	0.402281
$499$	−17.4266	−0.780123	−0.390062	−	0.920789i	$-0.627546\pi$
$499$	−17.4266	−0.780123	−0.390062	+	0.920789i	$0.627546\pi$
$500$	−0.493491	−0.0220696
$501$	14.7221	0.657736
$502$	11.0503	0.493199
$503$	30.1617	1.34484	0.672422	−	0.740168i	$-0.265254\pi$
$503$	30.1617	1.34484	0.672422	+	0.740168i	$0.265254\pi$
$504$	−55.2965	−2.46310
$505$	40.1659	1.78736
$506$	6.96904	0.309812
$507$	−74.8258	−3.32313
$508$	0.305140	0.0135384
$509$	30.4592	1.35008	0.675041	−	0.737780i	$-0.264126\pi$
$509$	30.4592	1.35008	0.675041	+	0.737780i	$0.264126\pi$
$510$	34.8011	1.54102
$511$	−21.7854	−0.963729
$512$	22.9206	1.01296
$513$	8.86316	0.391318
$514$	19.4921	0.859758
$515$	14.5086	0.639325
$516$	−0.000585646 0	−2.57816e−5 0
$517$	−2.16347	−0.0951494
$518$	−35.4645	−1.55822
$519$	40.8120	1.79145
$520$	−73.9140	−3.24134
$521$	−28.4023	−1.24433	−0.622164	−	0.782887i	$-0.713746\pi$
$521$	−28.4023	−1.24433	−0.622164	+	0.782887i	$0.713746\pi$
$522$	−29.4564	−1.28927
$523$	−2.25695	−0.0986897	−0.0493449	−	0.998782i	$-0.515713\pi$
$523$	−2.25695	−0.0986897	−0.0493449	+	0.998782i	$0.515713\pi$
$524$	0.106887	0.00466940
$525$	139.482	6.08751
$526$	−35.6218	−1.55318
$527$	17.2802	0.752738
$528$	−8.07887	−0.351588
$529$	21.3236	0.927114
$530$	−25.1957	−1.09443
$531$	65.3217	2.83472
$532$	−0.163194	−0.00707534
$533$	−1.60943	−0.0697121
$534$	8.86282	0.383532
$535$	22.1364	0.957039
$536$	−11.1518	−0.481683
$537$	−6.17964	−0.266671
$538$	9.97592	0.430092
$539$	−8.62467	−0.371491
$540$	0.301078	0.0129564
$541$	1.78328	0.0766692	0.0383346	−	0.999265i	$-0.487795\pi$
$541$	1.78328	0.0766692	0.0383346	+	0.999265i	$0.487795\pi$
$542$	−10.6182	−0.456089
$543$	57.3920	2.46293
$544$	−0.220400	−0.00944956
$545$	−58.2060	−2.49327
$546$	105.658	4.52175
$547$	24.8280	1.06157	0.530785	−	0.847506i	$-0.321897\pi$
$547$	24.8280	1.06157	0.530785	+	0.847506i	$0.321897\pi$
$548$	−0.144361	−0.00616679
$549$	45.7383	1.95207
$550$	12.3509	0.526645
$551$	−9.90483	−0.421960
$552$	−51.8412	−2.20651
$553$	19.6385	0.835116
$554$	46.1855	1.96223
$555$	65.6155	2.78522
$556$	−0.0300407	−0.00127401
$557$	−4.79171	−0.203031	−0.101516	−	0.994834i	$-0.532369\pi$
$557$	−4.79171	−0.203031	−0.101516	+	0.994834i	$0.532369\pi$
$558$	−49.9086	−2.11280
$559$	0.0765891	0.00323937
$560$	70.0752	2.96122
$561$	4.48373	0.189303
$562$	22.6759	0.956525
$563$	9.97703	0.420482	0.210241	−	0.977650i	$-0.432575\pi$
$563$	9.97703	0.420482	0.210241	+	0.977650i	$0.432575\pi$
$564$	0.141251	0.00594774
$565$	45.1452	1.89927
$566$	−15.8093	−0.664513
$567$	9.35876	0.393031
$568$	−2.37765	−0.0997639
$569$	−37.1026	−1.55542	−0.777711	−	0.628622i	$-0.783619\pi$
$569$	−37.1026	−1.55542	−0.777711	+	0.628622i	$0.783619\pi$
$570$	−33.7977	−1.41563
$571$	23.4808	0.982640	0.491320	−	0.870979i	$-0.336514\pi$
$571$	23.4808	0.982640	0.491320	+	0.870979i	$0.336514\pi$
$572$	−0.0835818	−0.00349473
$573$	19.4806	0.813812
$574$	1.53947	0.0642564
$575$	78.5527	3.27587
$576$	36.4205	1.51752
$577$	5.30944	0.221035	0.110517	−	0.993874i	$-0.464749\pi$
$577$	5.30944	0.221035	0.110517	+	0.993874i	$0.464749\pi$
$578$	17.1196	0.712082
$579$	−37.2584	−1.54841
$580$	−0.336464	−0.0139709
$581$	−10.0324	−0.416215
$582$	−1.55111	−0.0642954
$583$	−3.24618	−0.134443
$584$	14.3498	0.593801
$585$	−117.430	−4.85514
$586$	14.0281	0.579495
$587$	−33.5260	−1.38377	−0.691884	−	0.722009i	$-0.743219\pi$
$587$	−33.5260	−1.38377	−0.691884	+	0.722009i	$0.743219\pi$
$588$	0.563096	0.0232217
$589$	−16.7820	−0.691489
$590$	−83.5195	−3.43844
$591$	−28.4125	−1.16873
$592$	23.1533	0.951594
$593$	21.4972	0.882783	0.441392	−	0.897315i	$-0.354485\pi$
$593$	21.4972	0.882783	0.441392	+	0.897315i	$0.354485\pi$
$594$	−4.34208	−0.178158
$595$	−38.8914	−1.59439
$596$	−0.220049	−0.00901357
$597$	33.1146	1.35529
$598$	59.5038	2.43329
$599$	−10.1885	−0.416290	−0.208145	−	0.978098i	$-0.566743\pi$
$599$	−10.1885	−0.416290	−0.208145	+	0.978098i	$0.566743\pi$
$600$	−91.8758	−3.75081
$601$	22.9830	0.937496	0.468748	−	0.883332i	$-0.344705\pi$
$601$	22.9830	0.937496	0.468748	+	0.883332i	$0.344705\pi$
$602$	−0.0732601	−0.00298586
$603$	−17.7172	−0.721502
$604$	−0.414814	−0.0168785
$605$	−42.8196	−1.74086
$606$	37.8195	1.53631
$607$	−37.3345	−1.51536	−0.757680	−	0.652627i	$-0.773667\pi$
$607$	−37.3345	−1.51536	−0.757680	+	0.652627i	$0.773667\pi$
$608$	0.214045	0.00868067
$609$	54.7995	2.22059
$610$	−58.4804	−2.36780
$611$	−18.4724	−0.747313
$612$	−0.175850	−0.00710833
$613$	−34.0902	−1.37689	−0.688446	−	0.725288i	$-0.741707\pi$
$613$	−34.0902	−1.37689	−0.688446	+	0.725288i	$0.741707\pi$
$614$	−0.199372	−0.00804602
$615$	−2.84830	−0.114854
$616$	9.10908	0.367015
$617$	−2.96312	−0.119291	−0.0596454	−	0.998220i	$-0.518997\pi$
$617$	−2.96312	−0.119291	−0.0596454	+	0.998220i	$0.518997\pi$
$618$	13.6610	0.549527
$619$	−17.2928	−0.695056	−0.347528	−	0.937670i	$-0.612979\pi$
$619$	−17.2928	−0.695056	−0.347528	+	0.937670i	$0.612979\pi$
$620$	−0.570078	−0.0228949
$621$	−27.6159	−1.10819
$622$	−32.8953	−1.31898
$623$	−9.90451	−0.396816
$624$	−68.9798	−2.76140
$625$	55.2205	2.20882
$626$	−30.9504	−1.23703
$627$	−4.35446	−0.173900
$628$	0.129590	0.00517121
$629$	−12.8500	−0.512362
$630$	112.326	4.47517
$631$	−6.26408	−0.249369	−0.124685	−	0.992196i	$-0.539792\pi$
$631$	−6.26408	−0.249369	−0.124685	+	0.992196i	$0.539792\pi$
$632$	−12.9357	−0.514556
$633$	0.967800	0.0384666
$634$	25.4961	1.01258
$635$	−70.6227	−2.80258
$636$	0.211940	0.00840397
$637$	−73.6401	−2.91773
$638$	4.85240	0.192108
$639$	−3.77746	−0.149434
$640$	−45.7455	−1.80825
$641$	29.3399	1.15886	0.579429	−	0.815023i	$-0.303276\pi$
$641$	29.3399	1.15886	0.579429	+	0.815023i	$0.303276\pi$
$642$	20.8432	0.822616
$643$	10.9848	0.433199	0.216599	−	0.976261i	$-0.430503\pi$
$643$	10.9848	0.433199	0.216599	+	0.976261i	$0.430503\pi$
$644$	0.508481	0.0200369
$645$	0.135544	0.00533704
$646$	6.61885	0.260415
$647$	−8.65498	−0.340262	−0.170131	−	0.985421i	$-0.554419\pi$
$647$	−8.65498	−0.340262	−0.170131	+	0.985421i	$0.554419\pi$
$648$	−6.16453	−0.242166
$649$	−10.7606	−0.422389
$650$	105.456	4.13632
$651$	92.8480	3.63900
$652$	0.0629395	0.00246490
$653$	−6.64752	−0.260137	−0.130069	−	0.991505i	$-0.541520\pi$
$653$	−6.64752	−0.260137	−0.130069	+	0.991505i	$0.541520\pi$
$654$	−54.8057	−2.14307
$655$	−24.7384	−0.966608
$656$	−1.00506	−0.0392410
$657$	22.7982	0.889441
$658$	17.6695	0.688828
$659$	−17.0334	−0.663526	−0.331763	−	0.943363i	$-0.607643\pi$
$659$	−17.0334	−0.663526	−0.331763	+	0.943363i	$0.607643\pi$
$660$	−0.147919	−0.00575775
$661$	28.0538	1.09117	0.545583	−	0.838057i	$-0.316308\pi$
$661$	28.0538	1.09117	0.545583	+	0.838057i	$0.316308\pi$
$662$	5.31743	0.206668
$663$	38.2835	1.48681
$664$	6.60827	0.256450
$665$	37.7701	1.46466
$666$	37.1132	1.43811
$667$	30.8616	1.19497
$668$	0.0951155	0.00368013
$669$	−49.6697	−1.92034
$670$	22.6530	0.875162
$671$	−7.53455	−0.290868
$672$	−1.18423	−0.0456825
$673$	−5.81377	−0.224104	−0.112052	−	0.993702i	$-0.535742\pi$
$673$	−5.81377	−0.224104	−0.112052	+	0.993702i	$0.535742\pi$
$674$	36.6029	1.40989
$675$	−48.9424	−1.88380
$676$	−0.483429	−0.0185934
$677$	25.1421	0.966289	0.483144	−	0.875541i	$-0.339495\pi$
$677$	25.1421	0.966289	0.483144	+	0.875541i	$0.339495\pi$
$678$	42.5079	1.63251
$679$	1.73342	0.0665224
$680$	25.6174	0.982384
$681$	−20.3942	−0.781509
$682$	8.22152	0.314818
$683$	−33.4105	−1.27842	−0.639208	−	0.769034i	$-0.720738\pi$
$683$	−33.4105	−1.27842	−0.639208	+	0.769034i	$0.720738\pi$
$684$	0.170780	0.00652995
$685$	33.4114	1.27658
$686$	27.9340	1.06653
$687$	−48.4333	−1.84785
$688$	0.0478285	0.00182344
$689$	−27.7169	−1.05593
$690$	105.307	4.00898
$691$	−31.5283	−1.19939	−0.599697	−	0.800227i	$-0.704712\pi$
$691$	−31.5283	−1.19939	−0.599697	+	0.800227i	$0.704712\pi$
$692$	0.263675	0.0100234
$693$	14.4719	0.549744
$694$	−31.1051	−1.18073
$695$	6.95272	0.263732
$696$	−36.0959	−1.36821
$697$	0.557803	0.0211283
$698$	34.9317	1.32218
$699$	−23.3933	−0.884815
$700$	0.901156	0.0340605
$701$	43.1201	1.62862	0.814312	−	0.580428i	$-0.197115\pi$
$701$	43.1201	1.62862	0.814312	+	0.580428i	$0.197115\pi$
$702$	−37.0740	−1.39927
$703$	12.4795	0.470672
$704$	−5.99960	−0.226119
$705$	−32.6916	−1.23124
$706$	−46.3588	−1.74474
$707$	−42.2646	−1.58952
$708$	0.702546	0.0264033
$709$	28.8045	1.08178	0.540888	−	0.841095i	$-0.318088\pi$
$709$	28.8045	1.08178	0.540888	+	0.841095i	$0.318088\pi$
$710$	4.82981	0.181260
$711$	−20.5515	−0.770741
$712$	6.52402	0.244498
$713$	52.2895	1.95826
$714$	−36.6195	−1.37045
$715$	19.3445	0.723442
$716$	−0.0399249	−0.00149206
$717$	2.38409	0.0890354
$718$	6.76804	0.252581
$719$	−2.34460	−0.0874390	−0.0437195	−	0.999044i	$-0.513921\pi$
$719$	−2.34460	−0.0874390	−0.0437195	+	0.999044i	$0.513921\pi$
$720$	−73.3329	−2.73296
$721$	−15.2667	−0.568560
$722$	20.3228	0.756337
$723$	49.9631	1.85815
$724$	0.370794	0.0137804
$725$	54.6946	2.03131
$726$	−40.3181	−1.49635
$727$	−40.8010	−1.51323	−0.756613	−	0.653862i	$-0.773147\pi$
$727$	−40.8010	−1.51323	−0.756613	+	0.653862i	$0.773147\pi$
$728$	77.7761	2.88257
$729$	−43.9036	−1.62606
$730$	−29.1494	−1.07887
$731$	−0.0265446	−0.000981787 0
$732$	0.491924	0.0181820
$733$	−36.1463	−1.33509	−0.667547	−	0.744568i	$-0.732655\pi$
$733$	−36.1463	−1.33509	−0.667547	+	0.744568i	$0.732655\pi$
$734$	8.04102	0.296799
$735$	−130.325	−4.80710
$736$	−0.666924	−0.0245831
$737$	2.91859	0.107508
$738$	−1.61104	−0.0593033
$739$	7.44981	0.274046	0.137023	−	0.990568i	$-0.456247\pi$
$739$	7.44981	0.274046	0.137023	+	0.990568i	$0.456247\pi$
$740$	0.423923	0.0155837
$741$	−37.1797	−1.36583
$742$	26.5122	0.973292
$743$	11.0337	0.404786	0.202393	−	0.979304i	$-0.435128\pi$
$743$	11.0337	0.404786	0.202393	+	0.979304i	$0.435128\pi$
$744$	−61.1581	−2.24217
$745$	50.9289	1.86589
$746$	−14.8532	−0.543815
$747$	10.4988	0.384131
$748$	0.0289681	0.00105918
$749$	−23.2930	−0.851108
$750$	107.543	3.92692
$751$	16.2487	0.592922	0.296461	−	0.955045i	$-0.404193\pi$
$751$	16.2487	0.592922	0.296461	+	0.955045i	$0.404193\pi$
$752$	−11.5357	−0.420662
$753$	21.5132	0.783986
$754$	41.4312	1.50884
$755$	96.0059	3.49401
$756$	−0.316810	−0.0115223
$757$	7.45713	0.271034	0.135517	−	0.990775i	$-0.456730\pi$
$757$	7.45713	0.271034	0.135517	+	0.990775i	$0.456730\pi$
$758$	35.5889	1.29265
$759$	13.5677	0.492475
$760$	−24.8788	−0.902450
$761$	−48.9997	−1.77624	−0.888120	−	0.459612i	$-0.847988\pi$
$761$	−48.9997	−1.77624	−0.888120	+	0.459612i	$0.847988\pi$
$762$	−66.4971	−2.40893
$763$	61.2473	2.21730
$764$	0.125858	0.00455340
$765$	40.6994	1.47149
$766$	−51.4536	−1.85909
$767$	−91.8769	−3.31748
$768$	1.16490	0.0420347
$769$	45.9000	1.65519	0.827597	−	0.561322i	$-0.189707\pi$
$769$	45.9000	1.65519	0.827597	+	0.561322i	$0.189707\pi$
$770$	−18.5036	−0.666825
$771$	37.9481	1.36667
$772$	−0.240716	−0.00866357
$773$	−26.8764	−0.966676	−0.483338	−	0.875434i	$-0.660576\pi$
$773$	−26.8764	−0.966676	−0.483338	+	0.875434i	$0.660576\pi$
$774$	0.0766659	0.00275570
$775$	92.6702	3.32881
$776$	−1.14179	−0.0409877
$777$	−69.0439	−2.47694
$778$	−26.1136	−0.936220
$779$	−0.541720	−0.0194091
$780$	−1.26298	−0.0452219
$781$	0.622267	0.0222665
$782$	−20.6231	−0.737480
$783$	−19.2284	−0.687166
$784$	−45.9868	−1.64239
$785$	−29.9928	−1.07049
$786$	−23.2932	−0.830841
$787$	48.2008	1.71817	0.859087	−	0.511830i	$-0.171032\pi$
$787$	48.2008	1.71817	0.859087	+	0.511830i	$0.171032\pi$
$788$	−0.183565	−0.00653923
$789$	−69.3502	−2.46893
$790$	26.2769	0.934889
$791$	−47.5041	−1.68905
$792$	−9.53254	−0.338724
$793$	−64.3323	−2.28451
$794$	−14.2300	−0.505003
$795$	−49.0521	−1.73970
$796$	0.213944	0.00758305
$797$	−22.2963	−0.789775	−0.394888	−	0.918729i	$-0.629216\pi$
$797$	−22.2963	−0.789775	−0.394888	+	0.918729i	$0.629216\pi$
$798$	35.5636	1.25894
$799$	6.40224	0.226495
$800$	−1.18196	−0.0417885
$801$	10.3650	0.366228
$802$	−18.5521	−0.655097
$803$	−3.75558	−0.132532
$804$	−0.190552	−0.00672024
$805$	−117.684	−4.14783
$806$	70.1978	2.47261
$807$	19.4216	0.683672
$808$	27.8393	0.979384
$809$	21.9715	0.772477	0.386239	−	0.922399i	$-0.373774\pi$
$809$	21.9715	0.772477	0.386239	+	0.922399i	$0.373774\pi$
$810$	12.5223	0.439987
$811$	39.4534	1.38540	0.692698	−	0.721228i	$-0.256422\pi$
$811$	39.4534	1.38540	0.692698	+	0.721228i	$0.256422\pi$
$812$	0.354044	0.0124245
$813$	−20.6719	−0.724996
$814$	−6.11372	−0.214286
$815$	−14.5669	−0.510258
$816$	23.9073	0.836924
$817$	0.0257792	0.000901901 0
$818$	−33.4941	−1.17109
$819$	123.566	4.31774
$820$	−0.0184020	−0.000642627 0
$821$	−15.8584	−0.553463	−0.276732	−	0.960947i	$-0.589251\pi$
$821$	−15.8584	−0.553463	−0.276732	+	0.960947i	$0.589251\pi$
$822$	31.4595	1.09728
$823$	−32.6788	−1.13911	−0.569556	−	0.821953i	$-0.692885\pi$
$823$	−32.6788	−1.13911	−0.569556	+	0.821953i	$0.692885\pi$
$824$	10.0560	0.350318
$825$	24.0453	0.837151
$826$	87.8835	3.05786
$827$	19.7152	0.685566	0.342783	−	0.939415i	$-0.388630\pi$
$827$	19.7152	0.685566	0.342783	+	0.939415i	$0.388630\pi$
$828$	−0.532119	−0.0184924
$829$	−8.57402	−0.297788	−0.148894	−	0.988853i	$-0.547571\pi$
$829$	−8.57402	−0.297788	−0.148894	+	0.988853i	$0.547571\pi$
$830$	−13.4236	−0.465941
$831$	89.9161	3.11916
$832$	−51.2264	−1.77596
$833$	25.5225	0.884302
$834$	6.54656	0.226689
$835$	−22.0139	−0.761821
$836$	−0.0281329	−0.000972997 0
$837$	−32.5791	−1.12610
$838$	−38.3988	−1.32646
$839$	−40.1585	−1.38643	−0.693213	−	0.720733i	$-0.743805\pi$
$839$	−40.1585	−1.38643	−0.693213	+	0.720733i	$0.743805\pi$
$840$	137.645	4.74919
$841$	−7.51173	−0.259025
$842$	15.6140	0.538092
$843$	44.1465	1.52049
$844$	0.00625268	0.000215226 0
$845$	111.886	3.84901
$846$	−18.4909	−0.635730
$847$	45.0569	1.54817
$848$	−17.3087	−0.594383
$849$	−30.7782	−1.05631
$850$	−36.5493	−1.25363
$851$	−38.8837	−1.33292
$852$	−0.0406272	−0.00139187
$853$	46.8346	1.60359	0.801793	−	0.597602i	$-0.203880\pi$
$853$	46.8346	1.60359	0.801793	+	0.597602i	$0.203880\pi$
$854$	61.5361	2.10572
$855$	−39.5260	−1.35176
$856$	15.3429	0.524410
$857$	34.1142	1.16532	0.582659	−	0.812716i	$-0.302012\pi$
$857$	34.1142	1.16532	0.582659	+	0.812716i	$0.302012\pi$
$858$	18.2144	0.621829
$859$	−14.3320	−0.489000	−0.244500	−	0.969649i	$-0.578624\pi$
$859$	−14.3320	−0.489000	−0.244500	+	0.969649i	$0.578624\pi$
$860$	0.000875711 0	2.98615e−5 0
$861$	2.99712	0.102142
$862$	−1.40794	−0.0479546
$863$	33.3755	1.13612	0.568058	−	0.822989i	$-0.307695\pi$
$863$	33.3755	1.13612	0.568058	+	0.822989i	$0.307695\pi$
$864$	0.415528	0.0141366
$865$	−61.0258	−2.07494
$866$	−18.6947	−0.635271
$867$	33.3292	1.13192
$868$	0.599865	0.0203607
$869$	3.38548	0.114845
$870$	73.3231	2.48589
$871$	24.9198	0.844376
$872$	−40.3431	−1.36619
$873$	−1.81400	−0.0613945
$874$	20.0285	0.677473
$875$	−120.183	−4.06293
$876$	0.245198	0.00828447
$877$	−22.2208	−0.750344	−0.375172	−	0.926955i	$-0.622416\pi$
$877$	−22.2208	−0.750344	−0.375172	+	0.926955i	$0.622416\pi$
$878$	−45.6119	−1.53933
$879$	27.3105	0.921161
$880$	12.0803	0.407225
$881$	−8.14558	−0.274431	−0.137216	−	0.990541i	$-0.543815\pi$
$881$	−8.14558	−0.274431	−0.137216	+	0.990541i	$0.543815\pi$
$882$	−73.7138	−2.48207
$883$	−30.0987	−1.01290	−0.506451	−	0.862269i	$-0.669043\pi$
$883$	−30.0987	−1.01290	−0.506451	+	0.862269i	$0.669043\pi$
$884$	0.247339	0.00831890
$885$	−162.600	−5.46573
$886$	14.6184	0.491113
$887$	−9.26328	−0.311031	−0.155515	−	0.987833i	$-0.549704\pi$
$887$	−9.26328	−0.311031	−0.155515	+	0.987833i	$0.549704\pi$
$888$	45.4786	1.52616
$889$	74.3128	2.49237
$890$	−13.2525	−0.444225
$891$	1.61335	0.0540494
$892$	−0.320902	−0.0107446
$893$	−6.21765	−0.208066
$894$	47.9538	1.60381
$895$	9.24036	0.308871
$896$	48.1358	1.60810
$897$	115.845	3.86795
$898$	3.73201	0.124539
$899$	36.4080	1.21428
$900$	−0.943050	−0.0314350
$901$	9.60624	0.320030
$902$	0.265390	0.00883651
$903$	−0.142626	−0.00474630
$904$	31.2905	1.04071
$905$	−85.8178	−2.85268
$906$	90.3975	3.00325
$907$	−8.63333	−0.286665	−0.143332	−	0.989675i	$-0.545782\pi$
$907$	−8.63333	−0.286665	−0.143332	+	0.989675i	$0.545782\pi$
$908$	−0.131761	−0.00437266
$909$	44.2294	1.46700
$910$	−157.990	−5.23731
$911$	−27.6270	−0.915325	−0.457662	−	0.889126i	$-0.651313\pi$
$911$	−27.6270	−0.915325	−0.457662	+	0.889126i	$0.651313\pi$
$912$	−23.2180	−0.768825
$913$	−1.72949	−0.0572376
$914$	−33.7905	−1.11769
$915$	−113.852	−3.76385
$916$	−0.312914	−0.0103390
$917$	26.0310	0.859619
$918$	12.8493	0.424089
$919$	−11.9774	−0.395099	−0.197549	−	0.980293i	$-0.563298\pi$
$919$	−11.9774	−0.395099	−0.197549	+	0.980293i	$0.563298\pi$
$920$	77.5177	2.55568
$921$	−0.388148	−0.0127899
$922$	−30.3457	−0.999383
$923$	5.31311	0.174883
$924$	0.155648	0.00512045
$925$	−68.9117	−2.26580
$926$	6.61302	0.217317
$927$	15.9764	0.524733
$928$	−0.464365	−0.0152435
$929$	51.8723	1.70188	0.850938	−	0.525266i	$-0.176034\pi$
$929$	51.8723	1.70188	0.850938	+	0.525266i	$0.176034\pi$
$930$	124.233	4.07376
$931$	−24.7866	−0.812348
$932$	−0.151137	−0.00495067
$933$	−64.0421	−2.09664
$934$	−2.44893	−0.0801314
$935$	−6.70448	−0.219260
$936$	−81.3918	−2.66037
$937$	17.6441	0.576407	0.288203	−	0.957569i	$-0.406942\pi$
$937$	17.6441	0.576407	0.288203	+	0.957569i	$0.406942\pi$
$938$	−23.8367	−0.778294
$939$	−60.2556	−1.96637
$940$	−0.211211	−0.00688895
$941$	14.8231	0.483218	0.241609	−	0.970374i	$-0.422325\pi$
$941$	14.8231	0.483218	0.241609	+	0.970374i	$0.422325\pi$
$942$	−28.2406	−0.920130
$943$	1.68790	0.0549655
$944$	−57.3754	−1.86741
$945$	73.3236	2.38522
$946$	−0.0126293	−0.000410614 0
$947$	54.5890	1.77390	0.886952	−	0.461862i	$-0.152818\pi$
$947$	54.5890	1.77390	0.886952	+	0.461862i	$0.152818\pi$
$948$	−0.221035	−0.00717888
$949$	−32.0663	−1.04092
$950$	35.4955	1.15163
$951$	49.6370	1.60959
$952$	−26.9560	−0.873648
$953$	26.2485	0.850273	0.425136	−	0.905129i	$-0.360226\pi$
$953$	26.2485	0.850273	0.425136	+	0.905129i	$0.360226\pi$
$954$	−27.7447	−0.898267
$955$	−29.1291	−0.942595
$956$	0.0154029	0.000498166 0
$957$	9.44687	0.305374
$958$	23.5912	0.762196
$959$	−35.1571	−1.13528
$960$	−90.6583	−2.92598
$961$	30.6869	0.989901
$962$	−52.2008	−1.68302
$963$	24.3759	0.785501
$964$	0.322798	0.0103966
$965$	55.7122	1.79344
$966$	−110.810	−3.56524
$967$	−33.7099	−1.08404	−0.542019	−	0.840366i	$-0.682340\pi$
$967$	−33.7099	−1.08404	−0.542019	+	0.840366i	$0.682340\pi$
$968$	−29.6786	−0.953906
$969$	12.8859	0.413954
$970$	2.31935	0.0744700
$971$	−13.6275	−0.437326	−0.218663	−	0.975800i	$-0.570170\pi$
$971$	−13.6275	−0.437326	−0.218663	+	0.975800i	$0.570170\pi$
$972$	−0.325708	−0.0104471
$973$	−7.31601	−0.234541
$974$	−7.86016	−0.251856
$975$	205.306	6.57507
$976$	−40.1743	−1.28595
$977$	−2.75551	−0.0881565	−0.0440782	−	0.999028i	$-0.514035\pi$
$977$	−2.75551	−0.0881565	−0.0440782	+	0.999028i	$0.514035\pi$
$978$	−13.7160	−0.438588
$979$	−1.70744	−0.0545699
$980$	−0.841992	−0.0268965
$981$	−64.0946	−2.04638
$982$	31.3007	0.998845
$983$	−16.1627	−0.515510	−0.257755	−	0.966210i	$-0.582983\pi$
$983$	−16.1627	−0.515510	−0.257755	+	0.966210i	$0.582983\pi$
$984$	−1.97418	−0.0629345
$985$	42.4849	1.35368
$986$	−14.3594	−0.457297
$987$	34.3998	1.09496
$988$	−0.240207	−0.00764202
$989$	−0.0803232	−0.00255413
$990$	19.3638	0.615423
$991$	24.6281	0.782336	0.391168	−	0.920319i	$-0.372071\pi$
$991$	24.6281	0.782336	0.391168	+	0.920319i	$0.372071\pi$
$992$	−0.786783	−0.0249804
$993$	10.3522	0.328518
$994$	−5.08217	−0.161197
$995$	−49.5159	−1.56976
$996$	0.112916	0.00357789
$997$	14.9726	0.474186	0.237093	−	0.971487i	$-0.423805\pi$
$997$	14.9726	0.474186	0.237093	+	0.971487i	$0.423805\pi$
$998$	24.5356	0.776662
$999$	24.2266	0.766495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	431.2.a.f.1.8	✓	24
3.2	odd	2		3879.2.a.r.1.17		24
4.3	odd	2		6896.2.a.w.1.22		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
431.2.a.f.1.8	✓	24	1.1	even	1	trivial
3879.2.a.r.1.17		24	3.2	odd	2
6896.2.a.w.1.22		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)

\(f(q)\)	\(=\)	\(q-1.40794 q^{2} -2.74104 q^{3} -0.0177091 q^{4} +4.09865 q^{5} +3.85922 q^{6} -4.31281 q^{7} +2.84081 q^{8} +4.51331 q^{9} -5.77065 q^{10} -0.743484 q^{11} +0.0485414 q^{12} -6.34810 q^{13} +6.07217 q^{14} -11.2346 q^{15} -3.96427 q^{16} +2.20015 q^{17} -6.35446 q^{18} -2.13671 q^{19} -0.0725834 q^{20} +11.8216 q^{21} +1.04678 q^{22} +6.65760 q^{23} -7.78678 q^{24} +11.7990 q^{25} +8.93773 q^{26} -4.14803 q^{27} +0.0763760 q^{28} +4.63554 q^{29} +15.8176 q^{30} +7.85410 q^{31} -0.100175 q^{32} +2.03792 q^{33} -3.09768 q^{34} -17.6767 q^{35} -0.0799265 q^{36} -5.84050 q^{37} +3.00836 q^{38} +17.4004 q^{39} +11.6435 q^{40} +0.253530 q^{41} -16.6441 q^{42} -0.0120649 q^{43} +0.0131664 q^{44} +18.4985 q^{45} -9.37349 q^{46} +2.90991 q^{47} +10.8662 q^{48} +11.6003 q^{49} -16.6122 q^{50} -6.03070 q^{51} +0.112419 q^{52} +4.36617 q^{53} +5.84017 q^{54} -3.04728 q^{55} -12.2519 q^{56} +5.85682 q^{57} -6.52656 q^{58} +14.4731 q^{59} +0.198954 q^{60} +10.1341 q^{61} -11.0581 q^{62} -19.4650 q^{63} +8.06958 q^{64} -26.0186 q^{65} -2.86927 q^{66} -3.92556 q^{67} -0.0389627 q^{68} -18.2488 q^{69} +24.8877 q^{70} -0.836961 q^{71} +12.8214 q^{72} +5.05132 q^{73} +8.22306 q^{74} -32.3414 q^{75} +0.0378393 q^{76} +3.20651 q^{77} -24.4987 q^{78} -4.55354 q^{79} -16.2482 q^{80} -2.16999 q^{81} -0.356954 q^{82} +2.32619 q^{83} -0.209350 q^{84} +9.01765 q^{85} +0.0169866 q^{86} -12.7062 q^{87} -2.11210 q^{88} +2.29653 q^{89} -26.0447 q^{90} +27.3781 q^{91} -0.117900 q^{92} -21.5284 q^{93} -4.09697 q^{94} -8.75765 q^{95} +0.274583 q^{96} -0.401922 q^{97} -16.3326 q^{98} -3.35557 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + q^{3} + 33 q^{4} + 13 q^{5} + 17 q^{6} + 8 q^{7} - 3 q^{8} + 31 q^{9} - 6 q^{10} + 15 q^{11} - 12 q^{12} + 11 q^{13} + 16 q^{14} - 5 q^{15} + 43 q^{16} + 6 q^{17} - 8 q^{18} + 18 q^{19}+ \cdots - 9 q^{99}+O(q^{100}) \) 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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