LMFDB - Embedded newform 7942.2.a.bv.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.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:	$63.4171892853$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + x^{10} + 48 x^{9} - 47 x^{8} - 138 x^{7} + 150 x^{6} + 172 x^{5} - 139 x^{4} + \cdots + 1 $ x^12 - 6x^11 + x^10 + 48x^9 - 47x^8 - 138x^7 + 150x^6 + 172x^5 - 139x^4 - 86x^3 + 14x^2 + 10x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.220619$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.22062 q^{3} +1.00000 q^{4} -2.91567 q^{5} -1.22062 q^{6} -2.48215 q^{7} +1.00000 q^{8} -1.51009 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.22062 q^{3} +1.00000 q^{4} -2.91567 q^{5} -1.22062 q^{6} -2.48215 q^{7} +1.00000 q^{8} -1.51009 q^{9} -2.91567 q^{10} +1.00000 q^{11} -1.22062 q^{12} -3.41090 q^{13} -2.48215 q^{14} +3.55892 q^{15} +1.00000 q^{16} +5.24378 q^{17} -1.51009 q^{18} -2.91567 q^{20} +3.02976 q^{21} +1.00000 q^{22} +2.47751 q^{23} -1.22062 q^{24} +3.50113 q^{25} -3.41090 q^{26} +5.50510 q^{27} -2.48215 q^{28} +5.00578 q^{29} +3.55892 q^{30} -7.12227 q^{31} +1.00000 q^{32} -1.22062 q^{33} +5.24378 q^{34} +7.23713 q^{35} -1.51009 q^{36} +9.20624 q^{37} +4.16341 q^{39} -2.91567 q^{40} +9.14491 q^{41} +3.02976 q^{42} -1.40023 q^{43} +1.00000 q^{44} +4.40292 q^{45} +2.47751 q^{46} +8.43053 q^{47} -1.22062 q^{48} -0.838927 q^{49} +3.50113 q^{50} -6.40066 q^{51} -3.41090 q^{52} +9.77997 q^{53} +5.50510 q^{54} -2.91567 q^{55} -2.48215 q^{56} +5.00578 q^{58} -9.98989 q^{59} +3.55892 q^{60} -12.9497 q^{61} -7.12227 q^{62} +3.74827 q^{63} +1.00000 q^{64} +9.94506 q^{65} -1.22062 q^{66} +2.54199 q^{67} +5.24378 q^{68} -3.02410 q^{69} +7.23713 q^{70} -1.91862 q^{71} -1.51009 q^{72} -12.7727 q^{73} +9.20624 q^{74} -4.27355 q^{75} -2.48215 q^{77} +4.16341 q^{78} -12.5722 q^{79} -2.91567 q^{80} -2.18937 q^{81} +9.14491 q^{82} +12.7850 q^{83} +3.02976 q^{84} -15.2891 q^{85} -1.40023 q^{86} -6.11016 q^{87} +1.00000 q^{88} +6.64714 q^{89} +4.40292 q^{90} +8.46637 q^{91} +2.47751 q^{92} +8.69358 q^{93} +8.43053 q^{94} -1.22062 q^{96} -12.6360 q^{97} -0.838927 q^{98} -1.51009 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} - 6 q^{3} + 12 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 12 q^{8} - 2 q^{9}+O(q^{10}) $ 12 * q + 12 * q^2 - 6 * q^3 + 12 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^7 + 12 * q^8 - 2 * q^9	$ 12 q + 12 q^{2} - 6 q^{3} + 12 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 12 q^{8} - 2 q^{9} - 6 q^{10} + 12 q^{11} - 6 q^{12} - 26 q^{13} - 6 q^{14} + 16 q^{15} + 12 q^{16} - 2 q^{18} - 6 q^{20} - 10 q^{21} + 12 q^{22} + 4 q^{23} - 6 q^{24} - 26 q^{26} - 6 q^{27} - 6 q^{28} - 24 q^{29} + 16 q^{30} - 16 q^{31} + 12 q^{32} - 6 q^{33} + 6 q^{35} - 2 q^{36} - 8 q^{37} - 6 q^{40} - 16 q^{41} - 10 q^{42} - 32 q^{43} + 12 q^{44} - 2 q^{45} + 4 q^{46} - 2 q^{47} - 6 q^{48} - 4 q^{49} + 2 q^{51} - 26 q^{52} - 28 q^{53} - 6 q^{54} - 6 q^{55} - 6 q^{56} - 24 q^{58} - 6 q^{59} + 16 q^{60} - 16 q^{62} + 4 q^{63} + 12 q^{64} - 14 q^{65} - 6 q^{66} - 12 q^{67} - 16 q^{69} + 6 q^{70} - 4 q^{71} - 2 q^{72} - 22 q^{73} - 8 q^{74} - 20 q^{75} - 6 q^{77} - 6 q^{80} - 32 q^{81} - 16 q^{82} + 40 q^{83} - 10 q^{84} - 16 q^{85} - 32 q^{86} + 12 q^{88} - 16 q^{89} - 2 q^{90} + 60 q^{91} + 4 q^{92} - 2 q^{94} - 6 q^{96} - 20 q^{97} - 4 q^{98} - 2 q^{99}+O(q^{100}) $ 12 * q + 12 * q^2 - 6 * q^3 + 12 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^7 + 12 * q^8 - 2 * q^9 - 6 * q^10 + 12 * q^11 - 6 * q^12 - 26 * q^13 - 6 * q^14 + 16 * q^15 + 12 * q^16 - 2 * q^18 - 6 * q^20 - 10 * q^21 + 12 * q^22 + 4 * q^23 - 6 * q^24 - 26 * q^26 - 6 * q^27 - 6 * q^28 - 24 * q^29 + 16 * q^30 - 16 * q^31 + 12 * q^32 - 6 * q^33 + 6 * q^35 - 2 * q^36 - 8 * q^37 - 6 * q^40 - 16 * q^41 - 10 * q^42 - 32 * q^43 + 12 * q^44 - 2 * q^45 + 4 * q^46 - 2 * q^47 - 6 * q^48 - 4 * q^49 + 2 * q^51 - 26 * q^52 - 28 * q^53 - 6 * q^54 - 6 * q^55 - 6 * q^56 - 24 * q^58 - 6 * q^59 + 16 * q^60 - 16 * q^62 + 4 * q^63 + 12 * q^64 - 14 * q^65 - 6 * q^66 - 12 * q^67 - 16 * q^69 + 6 * q^70 - 4 * q^71 - 2 * q^72 - 22 * q^73 - 8 * q^74 - 20 * q^75 - 6 * q^77 - 6 * q^80 - 32 * q^81 - 16 * q^82 + 40 * q^83 - 10 * q^84 - 16 * q^85 - 32 * q^86 + 12 * q^88 - 16 * q^89 - 2 * q^90 + 60 * q^91 + 4 * q^92 - 2 * q^94 - 6 * q^96 - 20 * q^97 - 4 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.22062	−0.704725	−0.352362	−	0.935864i	$-0.614622\pi$
$3$	−1.22062	−0.704725	−0.352362	+	0.935864i	$0.614622\pi$
$4$	1.00000	0.500000
$5$	−2.91567	−1.30393	−0.651964	−	0.758250i	$-0.726055\pi$
$5$	−2.91567	−1.30393	−0.651964	+	0.758250i	$0.726055\pi$
$6$	−1.22062	−0.498316
$7$	−2.48215	−0.938165	−0.469082	−	0.883154i	$-0.655415\pi$
$7$	−2.48215	−0.938165	−0.469082	+	0.883154i	$0.655415\pi$
$8$	1.00000	0.353553
$9$	−1.51009	−0.503363
$10$	−2.91567	−0.922016
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.22062	−0.352362
$13$	−3.41090	−0.946013	−0.473007	−	0.881059i	$-0.656831\pi$
$13$	−3.41090	−0.946013	−0.473007	+	0.881059i	$0.656831\pi$
$14$	−2.48215	−0.663383
$15$	3.55892	0.918910
$16$	1.00000	0.250000
$17$	5.24378	1.27180	0.635901	−	0.771770i	$-0.280628\pi$
$17$	5.24378	1.27180	0.635901	+	0.771770i	$0.280628\pi$
$18$	−1.51009	−0.355931
$19$	0	0
$19$	0	0
$20$	−2.91567	−0.651964
$21$	3.02976	0.661148
$22$	1.00000	0.213201
$23$	2.47751	0.516597	0.258298	−	0.966065i	$-0.416838\pi$
$23$	2.47751	0.516597	0.258298	+	0.966065i	$0.416838\pi$
$24$	−1.22062	−0.249158
$25$	3.50113	0.700227
$26$	−3.41090	−0.668932
$27$	5.50510	1.05946
$28$	−2.48215	−0.469082
$29$	5.00578	0.929551	0.464775	−	0.885429i	$-0.346135\pi$
$29$	5.00578	0.929551	0.464775	+	0.885429i	$0.346135\pi$
$30$	3.55892	0.649768
$31$	−7.12227	−1.27920	−0.639598	−	0.768709i	$-0.720899\pi$
$31$	−7.12227	−1.27920	−0.639598	+	0.768709i	$0.720899\pi$
$32$	1.00000	0.176777
$33$	−1.22062	−0.212483
$34$	5.24378	0.899300
$35$	7.23713	1.22330
$36$	−1.51009	−0.251681
$37$	9.20624	1.51350	0.756748	−	0.653706i	$-0.226787\pi$
$37$	9.20624	1.51350	0.756748	+	0.653706i	$0.226787\pi$
$38$	0	0
$39$	4.16341	0.666679
$40$	−2.91567	−0.461008
$41$	9.14491	1.42820	0.714098	−	0.700046i	$-0.246837\pi$
$41$	9.14491	1.42820	0.714098	+	0.700046i	$0.246837\pi$
$42$	3.02976	0.467502
$43$	−1.40023	−0.213533	−0.106766	−	0.994284i	$-0.534050\pi$
$43$	−1.40023	−0.213533	−0.106766	+	0.994284i	$0.534050\pi$
$44$	1.00000	0.150756
$45$	4.40292	0.656348
$46$	2.47751	0.365289
$47$	8.43053	1.22972	0.614859	−	0.788637i	$-0.289213\pi$
$47$	8.43053	1.22972	0.614859	+	0.788637i	$0.289213\pi$
$48$	−1.22062	−0.176181
$49$	−0.838927	−0.119847
$50$	3.50113	0.495135
$51$	−6.40066	−0.896271
$52$	−3.41090	−0.473007
$53$	9.77997	1.34338	0.671691	−	0.740832i	$-0.265568\pi$
$53$	9.77997	1.34338	0.671691	+	0.740832i	$0.265568\pi$
$54$	5.50510	0.749149
$55$	−2.91567	−0.393149
$56$	−2.48215	−0.331691
$57$	0	0
$58$	5.00578	0.657292
$59$	−9.98989	−1.30057	−0.650287	−	0.759689i	$-0.725351\pi$
$59$	−9.98989	−1.30057	−0.650287	+	0.759689i	$0.725351\pi$
$60$	3.55892	0.459455
$61$	−12.9497	−1.65804	−0.829018	−	0.559222i	$-0.811100\pi$
$61$	−12.9497	−1.65804	−0.829018	+	0.559222i	$0.811100\pi$
$62$	−7.12227	−0.904529
$63$	3.74827	0.472237
$64$	1.00000	0.125000
$65$	9.94506	1.23353
$66$	−1.22062	−0.150248
$67$	2.54199	0.310554	0.155277	−	0.987871i	$-0.450373\pi$
$67$	2.54199	0.310554	0.155277	+	0.987871i	$0.450373\pi$
$68$	5.24378	0.635901
$69$	−3.02410	−0.364059
$70$	7.23713	0.865003
$71$	−1.91862	−0.227698	−0.113849	−	0.993498i	$-0.536318\pi$
$71$	−1.91862	−0.227698	−0.113849	+	0.993498i	$0.536318\pi$
$72$	−1.51009	−0.177966
$73$	−12.7727	−1.49494	−0.747468	−	0.664298i	$-0.768731\pi$
$73$	−12.7727	−1.49494	−0.747468	+	0.664298i	$0.768731\pi$
$74$	9.20624	1.07020
$75$	−4.27355	−0.493467
$76$	0	0
$77$	−2.48215	−0.282867
$78$	4.16341	0.471413
$79$	−12.5722	−1.41448	−0.707242	−	0.706972i	$-0.750061\pi$
$79$	−12.5722	−1.41448	−0.707242	+	0.706972i	$0.750061\pi$
$80$	−2.91567	−0.325982
$81$	−2.18937	−0.243263
$82$	9.14491	1.00989
$83$	12.7850	1.40334	0.701669	−	0.712503i	$-0.252438\pi$
$83$	12.7850	1.40334	0.701669	+	0.712503i	$0.252438\pi$
$84$	3.02976	0.330574
$85$	−15.2891	−1.65834
$86$	−1.40023	−0.150991
$87$	−6.11016	−0.655078
$88$	1.00000	0.106600
$89$	6.64714	0.704595	0.352297	−	0.935888i	$-0.385401\pi$
$89$	6.64714	0.704595	0.352297	+	0.935888i	$0.385401\pi$
$90$	4.40292	0.464108
$91$	8.46637	0.887516
$92$	2.47751	0.258298
$93$	8.69358	0.901482
$94$	8.43053	0.869542
$95$	0	0
$96$	−1.22062	−0.124579
$97$	−12.6360	−1.28300	−0.641498	−	0.767125i	$-0.721687\pi$
$97$	−12.6360	−1.28300	−0.641498	+	0.767125i	$0.721687\pi$
$98$	−0.838927	−0.0847445
$99$	−1.51009	−0.151770
$100$	3.50113	0.350113
$101$	3.20491	0.318901	0.159450	−	0.987206i	$-0.449028\pi$
$101$	3.20491	0.318901	0.159450	+	0.987206i	$0.449028\pi$
$102$	−6.40066	−0.633759
$103$	0.811802	0.0799892	0.0399946	−	0.999200i	$-0.487266\pi$
$103$	0.811802	0.0799892	0.0399946	+	0.999200i	$0.487266\pi$
$104$	−3.41090	−0.334466
$105$	−8.83379	−0.862089
$106$	9.77997	0.949914
$107$	−6.41779	−0.620432	−0.310216	−	0.950666i	$-0.600401\pi$
$107$	−6.41779	−0.620432	−0.310216	+	0.950666i	$0.600401\pi$
$108$	5.50510	0.529729
$109$	−15.7612	−1.50965	−0.754823	−	0.655929i	$-0.772277\pi$
$109$	−15.7612	−1.50965	−0.754823	+	0.655929i	$0.772277\pi$
$110$	−2.91567	−0.277998
$111$	−11.2373	−1.06660
$112$	−2.48215	−0.234541
$113$	9.08566	0.854707	0.427354	−	0.904085i	$-0.359446\pi$
$113$	9.08566	0.854707	0.427354	+	0.904085i	$0.359446\pi$
$114$	0	0
$115$	−7.22360	−0.673605
$116$	5.00578	0.464775
$117$	5.15076	0.476188
$118$	−9.98989	−0.919644
$119$	−13.0158	−1.19316
$120$	3.55892	0.324884
$121$	1.00000	0.0909091
$122$	−12.9497	−1.17241
$123$	−11.1625	−1.00648
$124$	−7.12227	−0.639598
$125$	4.37020	0.390883
$126$	3.74827	0.333922
$127$	−9.80066	−0.869668	−0.434834	−	0.900511i	$-0.643193\pi$
$127$	−9.80066	−0.869668	−0.434834	+	0.900511i	$0.643193\pi$
$128$	1.00000	0.0883883
$129$	1.70915	0.150482
$130$	9.94506	0.872239
$131$	−12.5430	−1.09589	−0.547945	−	0.836514i	$-0.684590\pi$
$131$	−12.5430	−1.09589	−0.547945	+	0.836514i	$0.684590\pi$
$132$	−1.22062	−0.106241
$133$	0	0
$134$	2.54199	0.219595
$135$	−16.0511	−1.38146
$136$	5.24378	0.449650
$137$	−17.3914	−1.48585	−0.742924	−	0.669376i	$-0.766562\pi$
$137$	−17.3914	−1.48585	−0.742924	+	0.669376i	$0.766562\pi$
$138$	−3.02410	−0.257428
$139$	−0.427620	−0.0362702	−0.0181351	−	0.999836i	$-0.505773\pi$
$139$	−0.427620	−0.0362702	−0.0181351	+	0.999836i	$0.505773\pi$
$140$	7.23713	0.611649
$141$	−10.2905	−0.866613
$142$	−1.91862	−0.161007
$143$	−3.41090	−0.285234
$144$	−1.51009	−0.125841
$145$	−14.5952	−1.21207
$146$	−12.7727	−1.05708
$147$	1.02401	0.0844590
$148$	9.20624	0.756748
$149$	7.00469	0.573847	0.286923	−	0.957954i	$-0.407367\pi$
$149$	7.00469	0.573847	0.286923	+	0.957954i	$0.407367\pi$
$150$	−4.27355	−0.348934
$151$	14.3716	1.16955	0.584773	−	0.811197i	$-0.301183\pi$
$151$	14.3716	1.16955	0.584773	+	0.811197i	$0.301183\pi$
$152$	0	0
$153$	−7.91857	−0.640178
$154$	−2.48215	−0.200017
$155$	20.7662	1.66798
$156$	4.16341	0.333340
$157$	−4.21439	−0.336345	−0.168173	−	0.985758i	$-0.553787\pi$
$157$	−4.21439	−0.336345	−0.168173	+	0.985758i	$0.553787\pi$
$158$	−12.5722	−1.00019
$159$	−11.9376	−0.946715
$160$	−2.91567	−0.230504
$161$	−6.14955	−0.484653
$162$	−2.18937	−0.172013
$163$	1.09203	0.0855344	0.0427672	−	0.999085i	$-0.486383\pi$
$163$	1.09203	0.0855344	0.0427672	+	0.999085i	$0.486383\pi$
$164$	9.14491	0.714098
$165$	3.55892	0.277062
$166$	12.7850	0.992310
$167$	9.44952	0.731226	0.365613	−	0.930767i	$-0.380859\pi$
$167$	9.44952	0.731226	0.365613	+	0.930767i	$0.380859\pi$
$168$	3.02976	0.233751
$169$	−1.36576	−0.105059
$170$	−15.2891	−1.17262
$171$	0	0
$172$	−1.40023	−0.106766
$173$	−4.08734	−0.310755	−0.155377	−	0.987855i	$-0.549659\pi$
$173$	−4.08734	−0.310755	−0.155377	+	0.987855i	$0.549659\pi$
$174$	−6.11016	−0.463210
$175$	−8.69034	−0.656928
$176$	1.00000	0.0753778
$177$	12.1939	0.916546
$178$	6.64714	0.498224
$179$	−15.1369	−1.13138	−0.565692	−	0.824617i	$-0.691391\pi$
$179$	−15.1369	−1.13138	−0.565692	+	0.824617i	$0.691391\pi$
$180$	4.40292	0.328174
$181$	6.92966	0.515078	0.257539	−	0.966268i	$-0.417088\pi$
$181$	6.92966	0.515078	0.257539	+	0.966268i	$0.417088\pi$
$182$	8.46637	0.627569
$183$	15.8066	1.16846
$184$	2.47751	0.182645
$185$	−26.8424	−1.97349
$186$	8.69358	0.637444
$187$	5.24378	0.383463
$188$	8.43053	0.614859
$189$	−13.6645	−0.993946
$190$	0	0
$191$	22.6058	1.63570	0.817848	−	0.575434i	$-0.195167\pi$
$191$	22.6058	1.63570	0.817848	+	0.575434i	$0.195167\pi$
$192$	−1.22062	−0.0880906
$193$	2.76023	0.198686	0.0993428	−	0.995053i	$-0.468326\pi$
$193$	2.76023	0.198686	0.0993428	+	0.995053i	$0.468326\pi$
$194$	−12.6360	−0.907215
$195$	−12.1391	−0.869301
$196$	−0.838927	−0.0599234
$197$	8.88610	0.633109	0.316554	−	0.948574i	$-0.397474\pi$
$197$	8.88610	0.633109	0.316554	+	0.948574i	$0.397474\pi$
$198$	−1.51009	−0.107317
$199$	4.00998	0.284260	0.142130	−	0.989848i	$-0.454605\pi$
$199$	4.00998	0.284260	0.142130	+	0.989848i	$0.454605\pi$
$200$	3.50113	0.247568
$201$	−3.10281	−0.218855
$202$	3.20491	0.225497
$203$	−12.4251	−0.872072
$204$	−6.40066	−0.448136
$205$	−26.6635	−1.86226
$206$	0.811802	0.0565609
$207$	−3.74126	−0.260035
$208$	−3.41090	−0.236503
$209$	0	0
$210$	−8.83379	−0.609589
$211$	−14.2257	−0.979337	−0.489669	−	0.871909i	$-0.662882\pi$
$211$	−14.2257	−0.979337	−0.489669	+	0.871909i	$0.662882\pi$
$212$	9.77997	0.671691
$213$	2.34190	0.160465
$214$	−6.41779	−0.438711
$215$	4.08260	0.278431
$216$	5.50510	0.374575
$217$	17.6785	1.20010
$218$	−15.7612	−1.06748
$219$	15.5907	1.05352
$220$	−2.91567	−0.196574
$221$	−17.8860	−1.20314
$222$	−11.2373	−0.754199
$223$	0.671657	0.0449775	0.0224887	−	0.999747i	$-0.492841\pi$
$223$	0.671657	0.0449775	0.0224887	+	0.999747i	$0.492841\pi$
$224$	−2.48215	−0.165846
$225$	−5.28702	−0.352468
$226$	9.08566	0.604369
$227$	−21.4896	−1.42631	−0.713157	−	0.701004i	$-0.752736\pi$
$227$	−21.4896	−1.42631	−0.713157	+	0.701004i	$0.752736\pi$
$228$	0	0
$229$	11.9348	0.788671	0.394335	−	0.918967i	$-0.370975\pi$
$229$	11.9348	0.788671	0.394335	+	0.918967i	$0.370975\pi$
$230$	−7.22360	−0.476310
$231$	3.02976	0.199344
$232$	5.00578	0.328646
$233$	15.0254	0.984347	0.492173	−	0.870497i	$-0.336203\pi$
$233$	15.0254	0.984347	0.492173	+	0.870497i	$0.336203\pi$
$234$	5.15076	0.336716
$235$	−24.5806	−1.60346
$236$	−9.98989	−0.650287
$237$	15.3459	0.996822
$238$	−13.0158	−0.843692
$239$	−3.48819	−0.225632	−0.112816	−	0.993616i	$-0.535987\pi$
$239$	−3.48819	−0.225632	−0.112816	+	0.993616i	$0.535987\pi$
$240$	3.55892	0.229728
$241$	15.5710	1.00302	0.501509	−	0.865152i	$-0.332778\pi$
$241$	15.5710	1.00302	0.501509	+	0.865152i	$0.332778\pi$
$242$	1.00000	0.0642824
$243$	−13.8429	−0.888024
$244$	−12.9497	−0.829018
$245$	2.44604	0.156271
$246$	−11.1625	−0.711692
$247$	0	0
$248$	−7.12227	−0.452264
$249$	−15.6056	−0.988968
$250$	4.37020	0.276396
$251$	−1.89329	−0.119504	−0.0597518	−	0.998213i	$-0.519031\pi$
$251$	−1.89329	−0.119504	−0.0597518	+	0.998213i	$0.519031\pi$
$252$	3.74827	0.236119
$253$	2.47751	0.155760
$254$	−9.80066	−0.614948
$255$	18.6622	1.16867
$256$	1.00000	0.0625000
$257$	−20.1079	−1.25429	−0.627147	−	0.778901i	$-0.715778\pi$
$257$	−20.1079	−1.25429	−0.627147	+	0.778901i	$0.715778\pi$
$258$	1.70915	0.106407
$259$	−22.8513	−1.41991
$260$	9.94506	0.616766
$261$	−7.55917	−0.467901
$262$	−12.5430	−0.774912
$263$	15.4349	0.951758	0.475879	−	0.879511i	$-0.342130\pi$
$263$	15.4349	0.951758	0.475879	+	0.879511i	$0.342130\pi$
$264$	−1.22062	−0.0751239
$265$	−28.5152	−1.75167
$266$	0	0
$267$	−8.11362	−0.496546
$268$	2.54199	0.155277
$269$	−28.7064	−1.75026	−0.875130	−	0.483889i	$-0.839224\pi$
$269$	−28.7064	−1.75026	−0.875130	+	0.483889i	$0.839224\pi$
$270$	−16.0511	−0.976836
$271$	−4.61695	−0.280460	−0.140230	−	0.990119i	$-0.544784\pi$
$271$	−4.61695	−0.280460	−0.140230	+	0.990119i	$0.544784\pi$
$272$	5.24378	0.317951
$273$	−10.3342	−0.625455
$274$	−17.3914	−1.05065
$275$	3.50113	0.211126
$276$	−3.02410	−0.182029
$277$	24.4081	1.46654	0.733271	−	0.679937i	$-0.237993\pi$
$277$	24.4081	1.46654	0.733271	+	0.679937i	$0.237993\pi$
$278$	−0.427620	−0.0256469
$279$	10.7553	0.643900
$280$	7.23713	0.432501
$281$	−13.3974	−0.799222	−0.399611	−	0.916685i	$-0.630855\pi$
$281$	−13.3974	−0.799222	−0.399611	+	0.916685i	$0.630855\pi$
$282$	−10.2905	−0.612788
$283$	−23.5302	−1.39873	−0.699364	−	0.714766i	$-0.746533\pi$
$283$	−23.5302	−1.39873	−0.699364	+	0.714766i	$0.746533\pi$
$284$	−1.91862	−0.113849
$285$	0	0
$286$	−3.41090	−0.201691
$287$	−22.6991	−1.33988
$288$	−1.51009	−0.0889828
$289$	10.4972	0.617482
$290$	−14.5952	−0.857060
$291$	15.4238	0.904159
$292$	−12.7727	−0.747468
$293$	−26.4133	−1.54308	−0.771540	−	0.636181i	$-0.780513\pi$
$293$	−26.4133	−1.54308	−0.771540	+	0.636181i	$0.780513\pi$
$294$	1.02401	0.0597215
$295$	29.1272	1.69585
$296$	9.20624	0.535102
$297$	5.50510	0.319438
$298$	7.00469	0.405771
$299$	−8.45054	−0.488707
$300$	−4.27355	−0.246734
$301$	3.47558	0.200329
$302$	14.3716	0.826994
$303$	−3.91198	−0.224737
$304$	0	0
$305$	37.7570	2.16196
$306$	−7.91857	−0.452674
$307$	14.6785	0.837746	0.418873	−	0.908045i	$-0.362425\pi$
$307$	14.6785	0.837746	0.418873	+	0.908045i	$0.362425\pi$
$308$	−2.48215	−0.141434
$309$	−0.990901	−0.0563704
$310$	20.7662	1.17944
$311$	25.2257	1.43042	0.715209	−	0.698910i	$-0.246331\pi$
$311$	25.2257	1.43042	0.715209	+	0.698910i	$0.246331\pi$
$312$	4.16341	0.235707
$313$	−16.8134	−0.950350	−0.475175	−	0.879891i	$-0.657615\pi$
$313$	−16.8134	−0.950350	−0.475175	+	0.879891i	$0.657615\pi$
$314$	−4.21439	−0.237832
$315$	−10.9287	−0.615763
$316$	−12.5722	−0.707242
$317$	7.83724	0.440183	0.220092	−	0.975479i	$-0.429364\pi$
$317$	7.83724	0.440183	0.220092	+	0.975479i	$0.429364\pi$
$318$	−11.9376	−0.669428
$319$	5.00578	0.280270
$320$	−2.91567	−0.162991
$321$	7.83368	0.437234
$322$	−6.14955	−0.342701
$323$	0	0
$324$	−2.18937	−0.121632
$325$	−11.9420	−0.662424
$326$	1.09203	0.0604820
$327$	19.2384	1.06389
$328$	9.14491	0.504943
$329$	−20.9258	−1.15368
$330$	3.55892	0.195912
$331$	−8.53126	−0.468920	−0.234460	−	0.972126i	$-0.575332\pi$
$331$	−8.53126	−0.468920	−0.234460	+	0.972126i	$0.575332\pi$
$332$	12.7850	0.701669
$333$	−13.9022	−0.761838
$334$	9.44952	0.517055
$335$	−7.41161	−0.404940
$336$	3.02976	0.165287
$337$	−9.79270	−0.533443	−0.266721	−	0.963774i	$-0.585940\pi$
$337$	−9.79270	−0.533443	−0.266721	+	0.963774i	$0.585940\pi$
$338$	−1.36576	−0.0742877
$339$	−11.0901	−0.602334
$340$	−15.2891	−0.829169
$341$	−7.12227	−0.385692
$342$	0	0
$343$	19.4574	1.05060
$344$	−1.40023	−0.0754953
$345$	8.81727	0.474706
$346$	−4.08734	−0.219737
$347$	−19.4188	−1.04246	−0.521229	−	0.853417i	$-0.674526\pi$
$347$	−19.4188	−1.04246	−0.521229	+	0.853417i	$0.674526\pi$
$348$	−6.11016	−0.327539
$349$	−22.2851	−1.19289	−0.596447	−	0.802652i	$-0.703422\pi$
$349$	−22.2851	−1.19289	−0.596447	+	0.802652i	$0.703422\pi$
$350$	−8.69034	−0.464518
$351$	−18.7773	−1.00226
$352$	1.00000	0.0533002
$353$	−23.5821	−1.25515	−0.627575	−	0.778556i	$-0.715952\pi$
$353$	−23.5821	−1.25515	−0.627575	+	0.778556i	$0.715952\pi$
$354$	12.1939	0.648096
$355$	5.59406	0.296902
$356$	6.64714	0.352297
$357$	15.8874	0.840850
$358$	−15.1369	−0.800009
$359$	−7.27265	−0.383836	−0.191918	−	0.981411i	$-0.561471\pi$
$359$	−7.27265	−0.383836	−0.191918	+	0.981411i	$0.561471\pi$
$360$	4.40292	0.232054
$361$	0	0
$362$	6.92966	0.364215
$363$	−1.22062	−0.0640659
$364$	8.46637	0.443758
$365$	37.2411	1.94929
$366$	15.8066	0.826226
$367$	25.2796	1.31958	0.659792	−	0.751448i	$-0.270644\pi$
$367$	25.2796	1.31958	0.659792	+	0.751448i	$0.270644\pi$
$368$	2.47751	0.129149
$369$	−13.8096	−0.718900
$370$	−26.8424	−1.39547
$371$	−24.2754	−1.26031
$372$	8.69358	0.450741
$373$	−0.761437	−0.0394257	−0.0197129	−	0.999806i	$-0.506275\pi$
$373$	−0.761437	−0.0394257	−0.0197129	+	0.999806i	$0.506275\pi$
$374$	5.24378	0.271149
$375$	−5.33435	−0.275465
$376$	8.43053	0.434771
$377$	−17.0742	−0.879367
$378$	−13.6645	−0.702826
$379$	−4.89041	−0.251204	−0.125602	−	0.992081i	$-0.540086\pi$
$379$	−4.89041	−0.251204	−0.125602	+	0.992081i	$0.540086\pi$
$380$	0	0
$381$	11.9629	0.612877
$382$	22.6058	1.15661
$383$	−28.4474	−1.45360	−0.726798	−	0.686851i	$-0.758992\pi$
$383$	−28.4474	−1.45360	−0.726798	+	0.686851i	$0.758992\pi$
$384$	−1.22062	−0.0622895
$385$	7.23713	0.368838
$386$	2.76023	0.140492
$387$	2.11447	0.107484
$388$	−12.6360	−0.641498
$389$	−13.9047	−0.704998	−0.352499	−	0.935812i	$-0.614668\pi$
$389$	−13.9047	−0.704998	−0.352499	+	0.935812i	$0.614668\pi$
$390$	−12.1391	−0.614689
$391$	12.9915	0.657009
$392$	−0.838927	−0.0423722
$393$	15.3103	0.772302
$394$	8.88610	0.447675
$395$	36.6564	1.84438
$396$	−1.51009	−0.0758848
$397$	24.1973	1.21443	0.607215	−	0.794538i	$-0.292287\pi$
$397$	24.1973	1.21443	0.607215	+	0.794538i	$0.292287\pi$
$398$	4.00998	0.201002
$399$	0	0
$400$	3.50113	0.175057
$401$	−0.173946	−0.00868643	−0.00434322	−	0.999991i	$-0.501382\pi$
$401$	−0.173946	−0.00868643	−0.00434322	+	0.999991i	$0.501382\pi$
$402$	−3.10281	−0.154754
$403$	24.2933	1.21014
$404$	3.20491	0.159450
$405$	6.38348	0.317198
$406$	−12.4251	−0.616648
$407$	9.20624	0.456336
$408$	−6.40066	−0.316880
$409$	9.77181	0.483185	0.241593	−	0.970378i	$-0.422330\pi$
$409$	9.77181	0.483185	0.241593	+	0.970378i	$0.422330\pi$
$410$	−26.6635	−1.31682
$411$	21.2283	1.04711
$412$	0.811802	0.0399946
$413$	24.7964	1.22015
$414$	−3.74126	−0.183873
$415$	−37.2769	−1.82985
$416$	−3.41090	−0.167233
$417$	0.521961	0.0255605
$418$	0	0
$419$	15.8445	0.774053	0.387027	−	0.922069i	$-0.373502\pi$
$419$	15.8445	0.774053	0.387027	+	0.922069i	$0.373502\pi$
$420$	−8.83379	−0.431045
$421$	−18.4656	−0.899960	−0.449980	−	0.893039i	$-0.648569\pi$
$421$	−18.4656	−0.899960	−0.449980	+	0.893039i	$0.648569\pi$
$422$	−14.2257	−0.692496
$423$	−12.7308	−0.618994
$424$	9.77997	0.474957
$425$	18.3592	0.890550
$426$	2.34190	0.113466
$427$	32.1431	1.55551
$428$	−6.41779	−0.310216
$429$	4.16341	0.201011
$430$	4.08260	0.196881
$431$	−6.20546	−0.298906	−0.149453	−	0.988769i	$-0.547751\pi$
$431$	−6.20546	−0.298906	−0.149453	+	0.988769i	$0.547751\pi$
$432$	5.50510	0.264864
$433$	−7.97143	−0.383083	−0.191541	−	0.981485i	$-0.561349\pi$
$433$	−7.97143	−0.383083	−0.191541	+	0.981485i	$0.561349\pi$
$434$	17.6785	0.848597
$435$	17.8152	0.854174
$436$	−15.7612	−0.754823
$437$	0	0
$438$	15.5907	0.744950
$439$	−34.9963	−1.67028	−0.835142	−	0.550034i	$-0.814615\pi$
$439$	−34.9963	−1.67028	−0.835142	+	0.550034i	$0.814615\pi$
$440$	−2.91567	−0.138999
$441$	1.26685	0.0603264
$442$	−17.8860	−0.850750
$443$	35.4410	1.68385	0.841927	−	0.539591i	$-0.181421\pi$
$443$	35.4410	1.68385	0.841927	+	0.539591i	$0.181421\pi$
$444$	−11.2373	−0.533299
$445$	−19.3809	−0.918741
$446$	0.671657	0.0318039
$447$	−8.55007	−0.404404
$448$	−2.48215	−0.117271
$449$	−16.4353	−0.775631	−0.387815	−	0.921737i	$-0.626770\pi$
$449$	−16.4353	−0.775631	−0.387815	+	0.921737i	$0.626770\pi$
$450$	−5.28702	−0.249233
$451$	9.14491	0.430617
$452$	9.08566	0.427354
$453$	−17.5423	−0.824209
$454$	−21.4896	−1.00856
$455$	−24.6851	−1.15726
$456$	0	0
$457$	17.2189	0.805468	0.402734	−	0.915317i	$-0.368060\pi$
$457$	17.2189	0.805468	0.402734	+	0.915317i	$0.368060\pi$
$458$	11.9348	0.557674
$459$	28.8675	1.34742
$460$	−7.22360	−0.336802
$461$	37.6193	1.75211	0.876054	−	0.482214i	$-0.160167\pi$
$461$	37.6193	1.75211	0.876054	+	0.482214i	$0.160167\pi$
$462$	3.02976	0.140957
$463$	−36.3871	−1.69105	−0.845525	−	0.533936i	$-0.820712\pi$
$463$	−36.3871	−1.69105	−0.845525	+	0.533936i	$0.820712\pi$
$464$	5.00578	0.232388
$465$	−25.3476	−1.17547
$466$	15.0254	0.696038
$467$	4.64737	0.215055	0.107527	−	0.994202i	$-0.465707\pi$
$467$	4.64737	0.215055	0.107527	+	0.994202i	$0.465707\pi$
$468$	5.15076	0.238094
$469$	−6.30961	−0.291351
$470$	−24.5806	−1.13382
$471$	5.14417	0.237031
$472$	−9.98989	−0.459822
$473$	−1.40023	−0.0643826
$474$	15.3459	0.704859
$475$	0	0
$476$	−13.0158	−0.596580
$477$	−14.7686	−0.676208
$478$	−3.48819	−0.159546
$479$	−35.8149	−1.63643	−0.818213	−	0.574915i	$-0.805035\pi$
$479$	−35.8149	−1.63643	−0.818213	+	0.574915i	$0.805035\pi$
$480$	3.55892	0.162442
$481$	−31.4016	−1.43179
$482$	15.5710	0.709241
$483$	7.50627	0.341547
$484$	1.00000	0.0454545
$485$	36.8425	1.67293
$486$	−13.8429	−0.627927
$487$	7.78969	0.352984	0.176492	−	0.984302i	$-0.443525\pi$
$487$	7.78969	0.352984	0.176492	+	0.984302i	$0.443525\pi$
$488$	−12.9497	−0.586204
$489$	−1.33295	−0.0602782
$490$	2.44604	0.110501
$491$	23.4941	1.06027	0.530136	−	0.847912i	$-0.322141\pi$
$491$	23.4941	1.06027	0.530136	+	0.847912i	$0.322141\pi$
$492$	−11.1625	−0.503242
$493$	26.2492	1.18221
$494$	0	0
$495$	4.40292	0.197896
$496$	−7.12227	−0.319799
$497$	4.76230	0.213618
$498$	−15.6056	−0.699306
$499$	43.2397	1.93567	0.967837	−	0.251578i	$-0.0809496\pi$
$499$	43.2397	1.93567	0.967837	+	0.251578i	$0.0809496\pi$
$500$	4.37020	0.195441
$501$	−11.5343	−0.515313
$502$	−1.89329	−0.0845018
$503$	13.5552	0.604396	0.302198	−	0.953245i	$-0.402280\pi$
$503$	13.5552	0.604396	0.302198	+	0.953245i	$0.402280\pi$
$504$	3.74827	0.166961
$505$	−9.34447	−0.415823
$506$	2.47751	0.110139
$507$	1.66708	0.0740375
$508$	−9.80066	−0.434834
$509$	35.3212	1.56558	0.782792	−	0.622283i	$-0.213795\pi$
$509$	35.3212	1.56558	0.782792	+	0.622283i	$0.213795\pi$
$510$	18.6622	0.826376
$511$	31.7039	1.40250
$512$	1.00000	0.0441942
$513$	0	0
$514$	−20.1079	−0.886920
$515$	−2.36695	−0.104300
$516$	1.70915	0.0752410
$517$	8.43053	0.370774
$518$	−22.8513	−1.00403
$519$	4.98909	0.218997
$520$	9.94506	0.436120
$521$	21.7442	0.952631	0.476315	−	0.879275i	$-0.341972\pi$
$521$	21.7442	0.952631	0.476315	+	0.879275i	$0.341972\pi$
$522$	−7.55917	−0.330856
$523$	−23.5994	−1.03193	−0.515964	−	0.856610i	$-0.672566\pi$
$523$	−23.5994	−1.03193	−0.515964	+	0.856610i	$0.672566\pi$
$524$	−12.5430	−0.547945
$525$	10.6076	0.462954
$526$	15.4349	0.672994
$527$	−37.3476	−1.62689
$528$	−1.22062	−0.0531206
$529$	−16.8619	−0.733128
$530$	−28.5152	−1.23862
$531$	15.0856	0.654660
$532$	0	0
$533$	−31.1924	−1.35109
$534$	−8.11362	−0.351111
$535$	18.7122	0.808998
$536$	2.54199	0.109797
$537$	18.4764	0.797314
$538$	−28.7064	−1.23762
$539$	−0.838927	−0.0361352
$540$	−16.0511	−0.690728
$541$	15.0947	0.648973	0.324486	−	0.945890i	$-0.394809\pi$
$541$	15.0947	0.648973	0.324486	+	0.945890i	$0.394809\pi$
$542$	−4.61695	−0.198315
$543$	−8.45848	−0.362988
$544$	5.24378	0.224825
$545$	45.9544	1.96847
$546$	−10.3342	−0.442264
$547$	31.3699	1.34128	0.670641	−	0.741782i	$-0.266019\pi$
$547$	31.3699	1.34128	0.670641	+	0.741782i	$0.266019\pi$
$548$	−17.3914	−0.742924
$549$	19.5552	0.834594
$550$	3.50113	0.149289
$551$	0	0
$552$	−3.02410	−0.128714
$553$	31.2061	1.32702
$554$	24.4081	1.03700
$555$	32.7643	1.39077
$556$	−0.427620	−0.0181351
$557$	−20.3693	−0.863076	−0.431538	−	0.902095i	$-0.642029\pi$
$557$	−20.3693	−0.863076	−0.431538	+	0.902095i	$0.642029\pi$
$558$	10.7553	0.455306
$559$	4.77604	0.202005
$560$	7.23713	0.305825
$561$	−6.40066	−0.270236
$562$	−13.3974	−0.565135
$563$	−21.5912	−0.909959	−0.454980	−	0.890502i	$-0.650353\pi$
$563$	−21.5912	−0.909959	−0.454980	+	0.890502i	$0.650353\pi$
$564$	−10.2905	−0.433307
$565$	−26.4908	−1.11448
$566$	−23.5302	−0.989050
$567$	5.43435	0.228221
$568$	−1.91862	−0.0805034
$569$	−4.61198	−0.193344	−0.0966721	−	0.995316i	$-0.530820\pi$
$569$	−4.61198	−0.193344	−0.0966721	+	0.995316i	$0.530820\pi$
$570$	0	0
$571$	19.0552	0.797436	0.398718	−	0.917074i	$-0.369455\pi$
$571$	19.0552	0.797436	0.398718	+	0.917074i	$0.369455\pi$
$572$	−3.41090	−0.142617
$573$	−27.5931	−1.15272
$574$	−22.6991	−0.947440
$575$	8.67410	0.361735
$576$	−1.51009	−0.0629203
$577$	−13.9173	−0.579385	−0.289693	−	0.957120i	$-0.593553\pi$
$577$	−13.9173	−0.579385	−0.289693	+	0.957120i	$0.593553\pi$
$578$	10.4972	0.436626
$579$	−3.36919	−0.140019
$580$	−14.5952	−0.606033
$581$	−31.7344	−1.31656
$582$	15.4238	0.639337
$583$	9.77997	0.405045
$584$	−12.7727	−0.528540
$585$	−15.0179	−0.620914
$586$	−26.4133	−1.09112
$587$	1.49825	0.0618393	0.0309196	−	0.999522i	$-0.490156\pi$
$587$	1.49825	0.0618393	0.0309196	+	0.999522i	$0.490156\pi$
$588$	1.02401	0.0422295
$589$	0	0
$590$	29.1272	1.19915
$591$	−10.8466	−0.446167
$592$	9.20624	0.378374
$593$	−31.2678	−1.28401	−0.642007	−	0.766699i	$-0.721898\pi$
$593$	−31.2678	−1.28401	−0.642007	+	0.766699i	$0.721898\pi$
$594$	5.50510	0.225877
$595$	37.9499	1.55579
$596$	7.00469	0.286923
$597$	−4.89465	−0.200325
$598$	−8.45054	−0.345568
$599$	−38.0107	−1.55307	−0.776537	−	0.630071i	$-0.783026\pi$
$599$	−38.0107	−1.55307	−0.776537	+	0.630071i	$0.783026\pi$
$600$	−4.27355	−0.174467
$601$	−39.6414	−1.61701	−0.808503	−	0.588492i	$-0.799722\pi$
$601$	−39.6414	−1.61701	−0.808503	+	0.588492i	$0.799722\pi$
$602$	3.47558	0.141654
$603$	−3.83863	−0.156321
$604$	14.3716	0.584773
$605$	−2.91567	−0.118539
$606$	−3.91198	−0.158913
$607$	−14.4630	−0.587036	−0.293518	−	0.955954i	$-0.594826\pi$
$607$	−14.4630	−0.587036	−0.293518	+	0.955954i	$0.594826\pi$
$608$	0	0
$609$	15.1663	0.614571
$610$	37.7570	1.52874
$611$	−28.7557	−1.16333
$612$	−7.91857	−0.320089
$613$	1.50789	0.0609030	0.0304515	−	0.999536i	$-0.490305\pi$
$613$	1.50789	0.0609030	0.0304515	+	0.999536i	$0.490305\pi$
$614$	14.6785	0.592376
$615$	32.5460	1.31238
$616$	−2.48215	−0.100009
$617$	−9.45001	−0.380443	−0.190222	−	0.981741i	$-0.560921\pi$
$617$	−9.45001	−0.380443	−0.190222	+	0.981741i	$0.560921\pi$
$618$	−0.990901	−0.0398599
$619$	−37.6772	−1.51438	−0.757188	−	0.653197i	$-0.773427\pi$
$619$	−37.6772	−1.51438	−0.757188	+	0.653197i	$0.773427\pi$
$620$	20.7662	0.833990
$621$	13.6389	0.547312
$622$	25.2257	1.01146
$623$	−16.4992	−0.661026
$624$	4.16341	0.166670
$625$	−30.2477	−1.20991
$626$	−16.8134	−0.671999
$627$	0	0
$628$	−4.21439	−0.168173
$629$	48.2755	1.92487
$630$	−10.9287	−0.435410
$631$	−12.1029	−0.481808	−0.240904	−	0.970549i	$-0.577444\pi$
$631$	−12.1029	−0.481808	−0.240904	+	0.970549i	$0.577444\pi$
$632$	−12.5722	−0.500095
$633$	17.3642	0.690163
$634$	7.83724	0.311257
$635$	28.5755	1.13398
$636$	−11.9376	−0.473357
$637$	2.86150	0.113377
$638$	5.00578	0.198181
$639$	2.89728	0.114615
$640$	−2.91567	−0.115252
$641$	44.2907	1.74938	0.874689	−	0.484685i	$-0.161066\pi$
$641$	44.2907	1.74938	0.874689	+	0.484685i	$0.161066\pi$
$642$	7.83368	0.309171
$643$	−38.9617	−1.53650	−0.768250	−	0.640150i	$-0.778872\pi$
$643$	−38.9617	−1.53650	−0.768250	+	0.640150i	$0.778872\pi$
$644$	−6.14955	−0.242326
$645$	−4.98331	−0.196218
$646$	0	0
$647$	−28.7612	−1.13072	−0.565360	−	0.824844i	$-0.691263\pi$
$647$	−28.7612	−1.13072	−0.565360	+	0.824844i	$0.691263\pi$
$648$	−2.18937	−0.0860066
$649$	−9.98989	−0.392138
$650$	−11.9420	−0.468404
$651$	−21.5788	−0.845739
$652$	1.09203	0.0427672
$653$	−15.9352	−0.623591	−0.311795	−	0.950149i	$-0.600930\pi$
$653$	−15.9352	−0.623591	−0.311795	+	0.950149i	$0.600930\pi$
$654$	19.2384	0.752280
$655$	36.5714	1.42896
$656$	9.14491	0.357049
$657$	19.2880	0.752495
$658$	−20.9258	−0.815774
$659$	−5.76246	−0.224473	−0.112237	−	0.993681i	$-0.535801\pi$
$659$	−5.76246	−0.224473	−0.112237	+	0.993681i	$0.535801\pi$
$660$	3.55892	0.138531
$661$	41.1392	1.60013	0.800065	−	0.599913i	$-0.204798\pi$
$661$	41.1392	1.60013	0.800065	+	0.599913i	$0.204798\pi$
$662$	−8.53126	−0.331577
$663$	21.8320	0.847885
$664$	12.7850	0.496155
$665$	0	0
$666$	−13.9022	−0.538701
$667$	12.4019	0.480203
$668$	9.44952	0.365613
$669$	−0.819837	−0.0316967
$670$	−7.41161	−0.286336
$671$	−12.9497	−0.499917
$672$	3.02976	0.116876
$673$	−32.1249	−1.23832	−0.619162	−	0.785263i	$-0.712528\pi$
$673$	−32.1249	−1.23832	−0.619162	+	0.785263i	$0.712528\pi$
$674$	−9.79270	−0.377201
$675$	19.2741	0.741860
$676$	−1.36576	−0.0525293
$677$	−33.0149	−1.26887	−0.634434	−	0.772977i	$-0.718767\pi$
$677$	−33.0149	−1.26887	−0.634434	+	0.772977i	$0.718767\pi$
$678$	−11.0901	−0.425914
$679$	31.3645	1.20366
$680$	−15.2891	−0.586311
$681$	26.2306	1.00516
$682$	−7.12227	−0.272726
$683$	16.8628	0.645238	0.322619	−	0.946529i	$-0.395437\pi$
$683$	16.8628	0.645238	0.322619	+	0.946529i	$0.395437\pi$
$684$	0	0
$685$	50.7076	1.93744
$686$	19.4574	0.742887
$687$	−14.5678	−0.555796
$688$	−1.40023	−0.0533832
$689$	−33.3585	−1.27086
$690$	8.81727	0.335668
$691$	−30.7755	−1.17075	−0.585377	−	0.810761i	$-0.699053\pi$
$691$	−30.7755	−1.17075	−0.585377	+	0.810761i	$0.699053\pi$
$692$	−4.08734	−0.155377
$693$	3.74827	0.142385
$694$	−19.4188	−0.737129
$695$	1.24680	0.0472937
$696$	−6.11016	−0.231605
$697$	47.9539	1.81638
$698$	−22.2851	−0.843504
$699$	−18.3403	−0.693694
$700$	−8.69034	−0.328464
$701$	18.9234	0.714727	0.357364	−	0.933965i	$-0.383676\pi$
$701$	18.9234	0.714727	0.357364	+	0.933965i	$0.383676\pi$
$702$	−18.7773	−0.708705
$703$	0	0
$704$	1.00000	0.0376889
$705$	30.0036	1.13000
$706$	−23.5821	−0.887525
$707$	−7.95507	−0.299181
$708$	12.1939	0.458273
$709$	25.3404	0.951678	0.475839	−	0.879532i	$-0.342144\pi$
$709$	25.3404	0.951678	0.475839	+	0.879532i	$0.342144\pi$
$710$	5.59406	0.209941
$711$	18.9851	0.711998
$712$	6.64714	0.249112
$713$	−17.6455	−0.660829
$714$	15.8874	0.594571
$715$	9.94506	0.371924
$716$	−15.1369	−0.565692
$717$	4.25775	0.159009
$718$	−7.27265	−0.271413
$719$	17.5841	0.655777	0.327889	−	0.944716i	$-0.393663\pi$
$719$	17.5841	0.655777	0.327889	+	0.944716i	$0.393663\pi$
$720$	4.40292	0.164087
$721$	−2.01501	−0.0750431
$722$	0	0
$723$	−19.0063	−0.706852
$724$	6.92966	0.257539
$725$	17.5259	0.650896
$726$	−1.22062	−0.0453014
$727$	−46.6791	−1.73123	−0.865617	−	0.500707i	$-0.833073\pi$
$727$	−46.6791	−1.73123	−0.865617	+	0.500707i	$0.833073\pi$
$728$	8.46637	0.313784
$729$	23.4650	0.869076
$730$	37.2411	1.37836
$731$	−7.34249	−0.271572
$732$	15.8066	0.584230
$733$	21.6591	0.799996	0.399998	−	0.916516i	$-0.369011\pi$
$733$	21.6591	0.799996	0.399998	+	0.916516i	$0.369011\pi$
$734$	25.2796	0.933087
$735$	−2.98568	−0.110128
$736$	2.47751	0.0913223
$737$	2.54199	0.0936355
$738$	−13.8096	−0.508339
$739$	−11.5029	−0.423143	−0.211571	−	0.977363i	$-0.567858\pi$
$739$	−11.5029	−0.423143	−0.211571	+	0.977363i	$0.567858\pi$
$740$	−26.8424	−0.986745
$741$	0	0
$742$	−24.2754	−0.891176
$743$	2.45842	0.0901908	0.0450954	−	0.998983i	$-0.485641\pi$
$743$	2.45842	0.0901908	0.0450954	+	0.998983i	$0.485641\pi$
$744$	8.69358	0.318722
$745$	−20.4234	−0.748255
$746$	−0.761437	−0.0278782
$747$	−19.3065	−0.706388
$748$	5.24378	0.191731
$749$	15.9299	0.582067
$750$	−5.33435	−0.194783
$751$	−21.7733	−0.794520	−0.397260	−	0.917706i	$-0.630039\pi$
$751$	−21.7733	−0.794520	−0.397260	+	0.917706i	$0.630039\pi$
$752$	8.43053	0.307430
$753$	2.31099	0.0842172
$754$	−17.0742	−0.621807
$755$	−41.9029	−1.52500
$756$	−13.6645	−0.496973
$757$	25.0091	0.908972	0.454486	−	0.890754i	$-0.349823\pi$
$757$	25.0091	0.908972	0.454486	+	0.890754i	$0.349823\pi$
$758$	−4.89041	−0.177628
$759$	−3.02410	−0.109768
$760$	0	0
$761$	37.5749	1.36209	0.681045	−	0.732242i	$-0.261526\pi$
$761$	37.5749	1.36209	0.681045	+	0.732242i	$0.261526\pi$
$762$	11.9629	0.433369
$763$	39.1216	1.41630
$764$	22.6058	0.817848
$765$	23.0879	0.834746
$766$	−28.4474	−1.02785
$767$	34.0745	1.23036
$768$	−1.22062	−0.0440453
$769$	−47.9520	−1.72919	−0.864596	−	0.502467i	$-0.832426\pi$
$769$	−47.9520	−1.72919	−0.864596	+	0.502467i	$0.832426\pi$
$770$	7.23713	0.260808
$771$	24.5441	0.883933
$772$	2.76023	0.0993428
$773$	−13.3899	−0.481601	−0.240800	−	0.970575i	$-0.577410\pi$
$773$	−13.3899	−0.481601	−0.240800	+	0.970575i	$0.577410\pi$
$774$	2.11447	0.0760030
$775$	−24.9360	−0.895728
$776$	−12.6360	−0.453607
$777$	27.8927	1.00065
$778$	−13.9047	−0.498509
$779$	0	0
$780$	−12.1391	−0.434651
$781$	−1.91862	−0.0686535
$782$	12.9915	0.464576
$783$	27.5573	0.984819
$784$	−0.838927	−0.0299617
$785$	12.2878	0.438570
$786$	15.3103	0.546100
$787$	−11.2681	−0.401666	−0.200833	−	0.979626i	$-0.564365\pi$
$787$	−11.2681	−0.401666	−0.200833	+	0.979626i	$0.564365\pi$
$788$	8.88610	0.316554
$789$	−18.8402	−0.670727
$790$	36.6564	1.30418
$791$	−22.5520	−0.801856
$792$	−1.51009	−0.0536586
$793$	44.1701	1.56852
$794$	24.1973	0.858732
$795$	34.8062	1.23445
$796$	4.00998	0.142130
$797$	−15.0764	−0.534035	−0.267017	−	0.963692i	$-0.586038\pi$
$797$	−15.0764	−0.534035	−0.267017	+	0.963692i	$0.586038\pi$
$798$	0	0
$799$	44.2078	1.56396
$800$	3.50113	0.123784
$801$	−10.0378	−0.354667
$802$	−0.173946	−0.00614224
$803$	−12.7727	−0.450740
$804$	−3.10281	−0.109428
$805$	17.9301	0.631952
$806$	24.2933	0.855696
$807$	35.0396	1.23345
$808$	3.20491	0.112748
$809$	9.61166	0.337928	0.168964	−	0.985622i	$-0.445958\pi$
$809$	9.61166	0.337928	0.168964	+	0.985622i	$0.445958\pi$
$810$	6.38348	0.224293
$811$	−48.3615	−1.69820	−0.849101	−	0.528230i	$-0.822856\pi$
$811$	−48.3615	−1.69820	−0.849101	+	0.528230i	$0.822856\pi$
$812$	−12.4251	−0.436036
$813$	5.63554	0.197647
$814$	9.20624	0.322679
$815$	−3.18400	−0.111531
$816$	−6.40066	−0.224068
$817$	0	0
$818$	9.77181	0.341663
$819$	−12.7850	−0.446743
$820$	−26.6635	−0.931131
$821$	31.7268	1.10727	0.553637	−	0.832758i	$-0.313239\pi$
$821$	31.7268	1.10727	0.553637	+	0.832758i	$0.313239\pi$
$822$	21.2283	0.740421
$823$	−35.1658	−1.22580	−0.612901	−	0.790159i	$-0.709998\pi$
$823$	−35.1658	−1.22580	−0.612901	+	0.790159i	$0.709998\pi$
$824$	0.811802	0.0282805
$825$	−4.27355	−0.148786
$826$	24.7964	0.862778
$827$	−36.2558	−1.26074	−0.630369	−	0.776296i	$-0.717096\pi$
$827$	−36.2558	−1.26074	−0.630369	+	0.776296i	$0.717096\pi$
$828$	−3.74126	−0.130018
$829$	55.6274	1.93202	0.966010	−	0.258505i	$-0.0832298\pi$
$829$	55.6274	1.93202	0.966010	+	0.258505i	$0.0832298\pi$
$830$	−37.2769	−1.29390
$831$	−29.7930	−1.03351
$832$	−3.41090	−0.118252
$833$	−4.39915	−0.152421
$834$	0.521961	0.0180740
$835$	−27.5517	−0.953465
$836$	0	0
$837$	−39.2088	−1.35525
$838$	15.8445	0.547338
$839$	−43.9897	−1.51869	−0.759346	−	0.650687i	$-0.774481\pi$
$839$	−43.9897	−1.51869	−0.759346	+	0.650687i	$0.774481\pi$
$840$	−8.83379	−0.304795
$841$	−3.94214	−0.135936
$842$	−18.4656	−0.636368
$843$	16.3531	0.563232
$844$	−14.2257	−0.489669
$845$	3.98211	0.136989
$846$	−12.7308	−0.437695
$847$	−2.48215	−0.0852877
$848$	9.77997	0.335845
$849$	28.7215	0.985718
$850$	18.3592	0.629714
$851$	22.8086	0.781867
$852$	2.34190	0.0802323
$853$	−5.93360	−0.203163	−0.101581	−	0.994827i	$-0.532390\pi$
$853$	−5.93360	−0.203163	−0.101581	+	0.994827i	$0.532390\pi$
$854$	32.1431	1.09991
$855$	0	0
$856$	−6.41779	−0.219356
$857$	−2.97073	−0.101478	−0.0507392	−	0.998712i	$-0.516158\pi$
$857$	−2.97073	−0.101478	−0.0507392	+	0.998712i	$0.516158\pi$
$858$	4.16341	0.142136
$859$	−57.1910	−1.95133	−0.975666	−	0.219261i	$-0.929636\pi$
$859$	−57.1910	−1.95133	−0.975666	+	0.219261i	$0.929636\pi$
$860$	4.08260	0.139216
$861$	27.7069	0.944249
$862$	−6.20546	−0.211359
$863$	30.8484	1.05009	0.525046	−	0.851074i	$-0.324048\pi$
$863$	30.8484	1.05009	0.525046	+	0.851074i	$0.324048\pi$
$864$	5.50510	0.187287
$865$	11.9173	0.405202
$866$	−7.97143	−0.270880
$867$	−12.8131	−0.435155
$868$	17.6785	0.600049
$869$	−12.5722	−0.426483
$870$	17.8152	0.603992
$871$	−8.67048	−0.293788
$872$	−15.7612	−0.533740
$873$	19.0815	0.645812
$874$	0	0
$875$	−10.8475	−0.366712
$876$	15.5907	0.526760
$877$	−14.1416	−0.477529	−0.238765	−	0.971077i	$-0.576742\pi$
$877$	−14.1416	−0.477529	−0.238765	+	0.971077i	$0.576742\pi$
$878$	−34.9963	−1.18107
$879$	32.2406	1.08745
$880$	−2.91567	−0.0982872
$881$	−15.1138	−0.509198	−0.254599	−	0.967047i	$-0.581943\pi$
$881$	−15.1138	−0.509198	−0.254599	+	0.967047i	$0.581943\pi$
$882$	1.26685	0.0426572
$883$	−20.6923	−0.696350	−0.348175	−	0.937430i	$-0.613198\pi$
$883$	−20.6923	−0.696350	−0.348175	+	0.937430i	$0.613198\pi$
$884$	−17.8860	−0.601571
$885$	−35.5533	−1.19511
$886$	35.4410	1.19066
$887$	26.9618	0.905290	0.452645	−	0.891691i	$-0.350481\pi$
$887$	26.9618	0.905290	0.452645	+	0.891691i	$0.350481\pi$
$888$	−11.2373	−0.377100
$889$	24.3267	0.815892
$890$	−19.3809	−0.649648
$891$	−2.18937	−0.0733467
$892$	0.671657	0.0224887
$893$	0	0
$894$	−8.55007	−0.285957
$895$	44.1341	1.47524
$896$	−2.48215	−0.0829228
$897$	10.3149	0.344404
$898$	−16.4353	−0.548454
$899$	−35.6525	−1.18908
$900$	−5.28702	−0.176234
$901$	51.2840	1.70852
$902$	9.14491	0.304492
$903$	−4.24236	−0.141177
$904$	9.08566	0.302185
$905$	−20.2046	−0.671624
$906$	−17.5423	−0.582804
$907$	−29.7989	−0.989457	−0.494728	−	0.869048i	$-0.664732\pi$
$907$	−29.7989	−0.989457	−0.494728	+	0.869048i	$0.664732\pi$
$908$	−21.4896	−0.713157
$909$	−4.83970	−0.160523
$910$	−24.6851	−0.818304
$911$	30.9855	1.02660	0.513298	−	0.858210i	$-0.328424\pi$
$911$	30.9855	1.02660	0.513298	+	0.858210i	$0.328424\pi$
$912$	0	0
$913$	12.7850	0.423122
$914$	17.2189	0.569552
$915$	−46.0869	−1.52359
$916$	11.9348	0.394335
$917$	31.1337	1.02813
$918$	28.8675	0.952770
$919$	26.2076	0.864510	0.432255	−	0.901751i	$-0.357718\pi$
$919$	26.2076	0.864510	0.432255	+	0.901751i	$0.357718\pi$
$920$	−7.22360	−0.238155
$921$	−17.9169	−0.590381
$922$	37.6193	1.23893
$923$	6.54421	0.215405
$924$	3.02976	0.0996718
$925$	32.2323	1.05979
$926$	−36.3871	−1.19575
$927$	−1.22589	−0.0402636
$928$	5.00578	0.164323
$929$	54.2279	1.77916	0.889579	−	0.456781i	$-0.150998\pi$
$929$	54.2279	1.77916	0.889579	+	0.456781i	$0.150998\pi$
$930$	−25.3476	−0.831181
$931$	0	0
$932$	15.0254	0.492173
$933$	−30.7910	−1.00805
$934$	4.64737	0.152067
$935$	−15.2891	−0.500008
$936$	5.15076	0.168358
$937$	−21.2213	−0.693269	−0.346634	−	0.938000i	$-0.612676\pi$
$937$	−21.2213	−0.693269	−0.346634	+	0.938000i	$0.612676\pi$
$938$	−6.30961	−0.206016
$939$	20.5228	0.669735
$940$	−24.5806	−0.801732
$941$	−18.9436	−0.617544	−0.308772	−	0.951136i	$-0.599918\pi$
$941$	−18.9436	−0.617544	−0.308772	+	0.951136i	$0.599918\pi$
$942$	5.14417	0.167606
$943$	22.6566	0.737801
$944$	−9.98989	−0.325143
$945$	39.8412	1.29603
$946$	−1.40023	−0.0455254
$947$	41.4758	1.34778	0.673891	−	0.738831i	$-0.264622\pi$
$947$	41.4758	1.34778	0.673891	+	0.738831i	$0.264622\pi$
$948$	15.3459	0.498411
$949$	43.5665	1.41423
$950$	0	0
$951$	−9.56629	−0.310208
$952$	−13.0158	−0.421846
$953$	−40.1632	−1.30101	−0.650507	−	0.759501i	$-0.725443\pi$
$953$	−40.1632	−1.30101	−0.650507	+	0.759501i	$0.725443\pi$
$954$	−14.7686	−0.478151
$955$	−65.9110	−2.13283
$956$	−3.48819	−0.112816
$957$	−6.11016	−0.197513
$958$	−35.8149	−1.15713
$959$	43.1681	1.39397
$960$	3.55892	0.114864
$961$	19.7267	0.636345
$962$	−31.4016	−1.01243
$963$	9.69143	0.312302
$964$	15.5710	0.501509
$965$	−8.04792	−0.259072
$966$	7.50627	0.241510
$967$	−51.6611	−1.66131	−0.830654	−	0.556789i	$-0.812033\pi$
$967$	−51.6611	−1.66131	−0.830654	+	0.556789i	$0.812033\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	36.8425	1.18294
$971$	−2.05926	−0.0660848	−0.0330424	−	0.999454i	$-0.510520\pi$
$971$	−2.05926	−0.0660848	−0.0330424	+	0.999454i	$0.510520\pi$
$972$	−13.8429	−0.444012
$973$	1.06142	0.0340275
$974$	7.78969	0.249598
$975$	14.5767	0.466827
$976$	−12.9497	−0.414509
$977$	−44.9550	−1.43824	−0.719119	−	0.694887i	$-0.755455\pi$
$977$	−44.9550	−1.43824	−0.719119	+	0.694887i	$0.755455\pi$
$978$	−1.33295	−0.0426231
$979$	6.64714	0.212443
$980$	2.44604	0.0781357
$981$	23.8007	0.759899
$982$	23.4941	0.749726
$983$	−22.7880	−0.726823	−0.363412	−	0.931629i	$-0.618388\pi$
$983$	−22.7880	−0.726823	−0.363412	+	0.931629i	$0.618388\pi$
$984$	−11.1625	−0.355846
$985$	−25.9089	−0.825528
$986$	26.2492	0.835945
$987$	25.5425	0.813026
$988$	0	0
$989$	−3.46908	−0.110310
$990$	4.40292	0.139934
$991$	−8.71855	−0.276954	−0.138477	−	0.990366i	$-0.544221\pi$
$991$	−8.71855	−0.276954	−0.138477	+	0.990366i	$0.544221\pi$
$992$	−7.12227	−0.226132
$993$	10.4134	0.330460
$994$	4.76230	0.151051
$995$	−11.6918	−0.370654
$996$	−15.6056	−0.494484
$997$	35.8247	1.13458	0.567289	−	0.823519i	$-0.307992\pi$
$997$	35.8247	1.13458	0.567289	+	0.823519i	$0.307992\pi$
$998$	43.2397	1.36873
$999$	50.6813	1.60348

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.bv.1.5	yes	12
19.18	odd	2		7942.2.a.bu.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7942.2.a.bu.1.8	✓	12	19.18	odd	2
7942.2.a.bv.1.5	yes	12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.22062 q^{3} +1.00000 q^{4} -2.91567 q^{5} -1.22062 q^{6} -2.48215 q^{7} +1.00000 q^{8} -1.51009 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.22062 q^{3} +1.00000 q^{4} -2.91567 q^{5} -1.22062 q^{6} -2.48215 q^{7} +1.00000 q^{8} -1.51009 q^{9} -2.91567 q^{10} +1.00000 q^{11} -1.22062 q^{12} -3.41090 q^{13} -2.48215 q^{14} +3.55892 q^{15} +1.00000 q^{16} +5.24378 q^{17} -1.51009 q^{18} -2.91567 q^{20} +3.02976 q^{21} +1.00000 q^{22} +2.47751 q^{23} -1.22062 q^{24} +3.50113 q^{25} -3.41090 q^{26} +5.50510 q^{27} -2.48215 q^{28} +5.00578 q^{29} +3.55892 q^{30} -7.12227 q^{31} +1.00000 q^{32} -1.22062 q^{33} +5.24378 q^{34} +7.23713 q^{35} -1.51009 q^{36} +9.20624 q^{37} +4.16341 q^{39} -2.91567 q^{40} +9.14491 q^{41} +3.02976 q^{42} -1.40023 q^{43} +1.00000 q^{44} +4.40292 q^{45} +2.47751 q^{46} +8.43053 q^{47} -1.22062 q^{48} -0.838927 q^{49} +3.50113 q^{50} -6.40066 q^{51} -3.41090 q^{52} +9.77997 q^{53} +5.50510 q^{54} -2.91567 q^{55} -2.48215 q^{56} +5.00578 q^{58} -9.98989 q^{59} +3.55892 q^{60} -12.9497 q^{61} -7.12227 q^{62} +3.74827 q^{63} +1.00000 q^{64} +9.94506 q^{65} -1.22062 q^{66} +2.54199 q^{67} +5.24378 q^{68} -3.02410 q^{69} +7.23713 q^{70} -1.91862 q^{71} -1.51009 q^{72} -12.7727 q^{73} +9.20624 q^{74} -4.27355 q^{75} -2.48215 q^{77} +4.16341 q^{78} -12.5722 q^{79} -2.91567 q^{80} -2.18937 q^{81} +9.14491 q^{82} +12.7850 q^{83} +3.02976 q^{84} -15.2891 q^{85} -1.40023 q^{86} -6.11016 q^{87} +1.00000 q^{88} +6.64714 q^{89} +4.40292 q^{90} +8.46637 q^{91} +2.47751 q^{92} +8.69358 q^{93} +8.43053 q^{94} -1.22062 q^{96} -12.6360 q^{97} -0.838927 q^{98} -1.51009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} - 6 q^{3} + 12 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 12 q^{8} - 2 q^{9}+O(q^{10}) \) 12 * q + 12 * q^2 - 6 * q^3 + 12 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^7 + 12 * q^8 - 2 * q^9	\( 12 q + 12 q^{2} - 6 q^{3} + 12 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 12 q^{8} - 2 q^{9} - 6 q^{10} + 12 q^{11} - 6 q^{12} - 26 q^{13} - 6 q^{14} + 16 q^{15} + 12 q^{16} - 2 q^{18} - 6 q^{20} - 10 q^{21} + 12 q^{22} + 4 q^{23} - 6 q^{24} - 26 q^{26} - 6 q^{27} - 6 q^{28} - 24 q^{29} + 16 q^{30} - 16 q^{31} + 12 q^{32} - 6 q^{33} + 6 q^{35} - 2 q^{36} - 8 q^{37} - 6 q^{40} - 16 q^{41} - 10 q^{42} - 32 q^{43} + 12 q^{44} - 2 q^{45} + 4 q^{46} - 2 q^{47} - 6 q^{48} - 4 q^{49} + 2 q^{51} - 26 q^{52} - 28 q^{53} - 6 q^{54} - 6 q^{55} - 6 q^{56} - 24 q^{58} - 6 q^{59} + 16 q^{60} - 16 q^{62} + 4 q^{63} + 12 q^{64} - 14 q^{65} - 6 q^{66} - 12 q^{67} - 16 q^{69} + 6 q^{70} - 4 q^{71} - 2 q^{72} - 22 q^{73} - 8 q^{74} - 20 q^{75} - 6 q^{77} - 6 q^{80} - 32 q^{81} - 16 q^{82} + 40 q^{83} - 10 q^{84} - 16 q^{85} - 32 q^{86} + 12 q^{88} - 16 q^{89} - 2 q^{90} + 60 q^{91} + 4 q^{92} - 2 q^{94} - 6 q^{96} - 20 q^{97} - 4 q^{98} - 2 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 - 6 * q^3 + 12 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^7 + 12 * q^8 - 2 * q^9 - 6 * q^10 + 12 * q^11 - 6 * q^12 - 26 * q^13 - 6 * q^14 + 16 * q^15 + 12 * q^16 - 2 * q^18 - 6 * q^20 - 10 * q^21 + 12 * q^22 + 4 * q^23 - 6 * q^24 - 26 * q^26 - 6 * q^27 - 6 * q^28 - 24 * q^29 + 16 * q^30 - 16 * q^31 + 12 * q^32 - 6 * q^33 + 6 * q^35 - 2 * q^36 - 8 * q^37 - 6 * q^40 - 16 * q^41 - 10 * q^42 - 32 * q^43 + 12 * q^44 - 2 * q^45 + 4 * q^46 - 2 * q^47 - 6 * q^48 - 4 * q^49 + 2 * q^51 - 26 * q^52 - 28 * q^53 - 6 * q^54 - 6 * q^55 - 6 * q^56 - 24 * q^58 - 6 * q^59 + 16 * q^60 - 16 * q^62 + 4 * q^63 + 12 * q^64 - 14 * q^65 - 6 * q^66 - 12 * q^67 - 16 * q^69 + 6 * q^70 - 4 * q^71 - 2 * q^72 - 22 * q^73 - 8 * q^74 - 20 * q^75 - 6 * q^77 - 6 * q^80 - 32 * q^81 - 16 * q^82 + 40 * q^83 - 10 * q^84 - 16 * q^85 - 32 * q^86 + 12 * q^88 - 16 * q^89 - 2 * q^90 + 60 * q^91 + 4 * q^92 - 2 * q^94 - 6 * q^96 - 20 * q^97 - 4 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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