LMFDB - Embedded newform 8013.2.a.a.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8013,2,Mod(1,8013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8013, 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("8013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8013 = 3 \cdot 2671 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8013.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:	$63.9841271397$
Analytic rank:	$1$
Dimension:	$94$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.03943 q^{2} +1.00000 q^{3} +2.15929 q^{4} -1.46688 q^{5} -2.03943 q^{6} +4.57052 q^{7} -0.324864 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.03943 q^{2} +1.00000 q^{3} +2.15929 q^{4} -1.46688 q^{5} -2.03943 q^{6} +4.57052 q^{7} -0.324864 q^{8} +1.00000 q^{9} +2.99161 q^{10} +1.53473 q^{11} +2.15929 q^{12} -1.89102 q^{13} -9.32128 q^{14} -1.46688 q^{15} -3.65604 q^{16} +1.42499 q^{17} -2.03943 q^{18} -3.67035 q^{19} -3.16742 q^{20} +4.57052 q^{21} -3.12999 q^{22} +3.61777 q^{23} -0.324864 q^{24} -2.84826 q^{25} +3.85661 q^{26} +1.00000 q^{27} +9.86908 q^{28} +2.29707 q^{29} +2.99161 q^{30} -2.96651 q^{31} +8.10599 q^{32} +1.53473 q^{33} -2.90618 q^{34} -6.70441 q^{35} +2.15929 q^{36} -2.50348 q^{37} +7.48544 q^{38} -1.89102 q^{39} +0.476536 q^{40} -7.38270 q^{41} -9.32128 q^{42} -4.77223 q^{43} +3.31394 q^{44} -1.46688 q^{45} -7.37820 q^{46} -0.870813 q^{47} -3.65604 q^{48} +13.8897 q^{49} +5.80884 q^{50} +1.42499 q^{51} -4.08326 q^{52} +6.81935 q^{53} -2.03943 q^{54} -2.25127 q^{55} -1.48480 q^{56} -3.67035 q^{57} -4.68472 q^{58} -11.4974 q^{59} -3.16742 q^{60} -8.61816 q^{61} +6.05001 q^{62} +4.57052 q^{63} -9.21954 q^{64} +2.77390 q^{65} -3.12999 q^{66} -6.89098 q^{67} +3.07698 q^{68} +3.61777 q^{69} +13.6732 q^{70} -5.61929 q^{71} -0.324864 q^{72} -4.52928 q^{73} +5.10568 q^{74} -2.84826 q^{75} -7.92536 q^{76} +7.01454 q^{77} +3.85661 q^{78} -13.2817 q^{79} +5.36298 q^{80} +1.00000 q^{81} +15.0565 q^{82} +8.88396 q^{83} +9.86908 q^{84} -2.09030 q^{85} +9.73265 q^{86} +2.29707 q^{87} -0.498579 q^{88} +5.99448 q^{89} +2.99161 q^{90} -8.64294 q^{91} +7.81181 q^{92} -2.96651 q^{93} +1.77597 q^{94} +5.38397 q^{95} +8.10599 q^{96} +9.05604 q^{97} -28.3270 q^{98} +1.53473 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 94 q - 13 q^{2} + 94 q^{3} + 73 q^{4} - 14 q^{5} - 13 q^{6} - 55 q^{7} - 36 q^{8} + 94 q^{9}+O(q^{10}) $ 94 * q - 13 * q^2 + 94 * q^3 + 73 * q^4 - 14 * q^5 - 13 * q^6 - 55 * q^7 - 36 * q^8 + 94 * q^9	\( 94 q - 13 q^{2} + 94 q^{3} + 73 q^{4} - 14 q^{5} - 13 q^{6} - 55 q^{7} - 36 q^{8} + 94 q^{9} - 39 q^{10} - 49 q^{11} + 73 q^{12} - 52 q^{13} - 7 q^{14} - 14 q^{15} + 43 q^{16} - 22 q^{17} - 13 q^{18} - 89 q^{19} - 22 q^{20} - 55 q^{21} - 36 q^{22} - 46 q^{23} - 36 q^{24} + 18 q^{25} + q^{26} + 94 q^{27} - 123 q^{28} - 20 q^{29} - 39 q^{30} - 61 q^{31} - 65 q^{32} - 49 q^{33} - 67 q^{34} - 40 q^{35} + 73 q^{36} - 83 q^{37} - 19 q^{38} - 52 q^{39} - 101 q^{40} - 25 q^{41} - 7 q^{42} - 150 q^{43} - 71 q^{44} - 14 q^{45} - 72 q^{46} - 39 q^{47} + 43 q^{48} - q^{49} - 45 q^{50} - 22 q^{51} - 110 q^{52} - 30 q^{53} - 13 q^{54} - 54 q^{55} - 5 q^{56} - 89 q^{57} - 77 q^{58} - 43 q^{59} - 22 q^{60} - 109 q^{61} - 33 q^{62} - 55 q^{63} + 10 q^{64} - 66 q^{65} - 36 q^{66} - 155 q^{67} - 46 q^{68} - 46 q^{69} - 43 q^{70} - 27 q^{71} - 36 q^{72} - 157 q^{73} - 29 q^{74} + 18 q^{75} - 176 q^{76} - 9 q^{77} + q^{78} - 99 q^{79} - 18 q^{80} + 94 q^{81} - 53 q^{82} - 144 q^{83} - 123 q^{84} - 105 q^{85} + 23 q^{86} - 20 q^{87} - 88 q^{88} - 4 q^{89} - 39 q^{90} - 99 q^{91} - 76 q^{92} - 61 q^{93} - 65 q^{94} - 49 q^{95} - 65 q^{96} - 139 q^{97} - 6 q^{98} - 49 q^{99}+O(q^{100}) \) 94 * q - 13 * q^2 + 94 * q^3 + 73 * q^4 - 14 * q^5 - 13 * q^6 - 55 * q^7 - 36 * q^8 + 94 * q^9 - 39 * q^10 - 49 * q^11 + 73 * q^12 - 52 * q^13 - 7 * q^14 - 14 * q^15 + 43 * q^16 - 22 * q^17 - 13 * q^18 - 89 * q^19 - 22 * q^20 - 55 * q^21 - 36 * q^22 - 46 * q^23 - 36 * q^24 + 18 * q^25 + q^26 + 94 * q^27 - 123 * q^28 - 20 * q^29 - 39 * q^30 - 61 * q^31 - 65 * q^32 - 49 * q^33 - 67 * q^34 - 40 * q^35 + 73 * q^36 - 83 * q^37 - 19 * q^38 - 52 * q^39 - 101 * q^40 - 25 * q^41 - 7 * q^42 - 150 * q^43 - 71 * q^44 - 14 * q^45 - 72 * q^46 - 39 * q^47 + 43 * q^48 - q^49 - 45 * q^50 - 22 * q^51 - 110 * q^52 - 30 * q^53 - 13 * q^54 - 54 * q^55 - 5 * q^56 - 89 * q^57 - 77 * q^58 - 43 * q^59 - 22 * q^60 - 109 * q^61 - 33 * q^62 - 55 * q^63 + 10 * q^64 - 66 * q^65 - 36 * q^66 - 155 * q^67 - 46 * q^68 - 46 * q^69 - 43 * q^70 - 27 * q^71 - 36 * q^72 - 157 * q^73 - 29 * q^74 + 18 * q^75 - 176 * q^76 - 9 * q^77 + q^78 - 99 * q^79 - 18 * q^80 + 94 * q^81 - 53 * q^82 - 144 * q^83 - 123 * q^84 - 105 * q^85 + 23 * q^86 - 20 * q^87 - 88 * q^88 - 4 * q^89 - 39 * q^90 - 99 * q^91 - 76 * q^92 - 61 * q^93 - 65 * q^94 - 49 * q^95 - 65 * q^96 - 139 * q^97 - 6 * q^98 - 49 * 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 $

$2$	−2.03943	−1.44210	−0.721049	−	0.692884i	$-0.756340\pi$
$2$	−2.03943	−1.44210	−0.721049	+	0.692884i	$0.756340\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.15929	1.07965
$5$	−1.46688	−0.656009	−0.328005	−	0.944676i	$-0.606376\pi$
$5$	−1.46688	−0.656009	−0.328005	+	0.944676i	$0.606376\pi$
$6$	−2.03943	−0.832595
$7$	4.57052	1.72749	0.863747	−	0.503925i	$-0.168111\pi$
$7$	4.57052	1.72749	0.863747	+	0.503925i	$0.168111\pi$
$8$	−0.324864	−0.114857
$9$	1.00000	0.333333
$10$	2.99161	0.946029
$11$	1.53473	0.462740	0.231370	−	0.972866i	$-0.425679\pi$
$11$	1.53473	0.462740	0.231370	+	0.972866i	$0.425679\pi$
$12$	2.15929	0.623334
$13$	−1.89102	−0.524474	−0.262237	−	0.965003i	$-0.584460\pi$
$13$	−1.89102	−0.524474	−0.262237	+	0.965003i	$0.584460\pi$
$14$	−9.32128	−2.49122
$15$	−1.46688	−0.378747
$16$	−3.65604	−0.914011
$17$	1.42499	0.345612	0.172806	−	0.984956i	$-0.444717\pi$
$17$	1.42499	0.345612	0.172806	+	0.984956i	$0.444717\pi$
$18$	−2.03943	−0.480699
$19$	−3.67035	−0.842036	−0.421018	−	0.907052i	$-0.638327\pi$
$19$	−3.67035	−0.842036	−0.421018	+	0.907052i	$0.638327\pi$
$20$	−3.16742	−0.708258
$21$	4.57052	0.997369
$22$	−3.12999	−0.667316
$23$	3.61777	0.754356	0.377178	−	0.926141i	$-0.376894\pi$
$23$	3.61777	0.754356	0.377178	+	0.926141i	$0.376894\pi$
$24$	−0.324864	−0.0663125
$25$	−2.84826	−0.569652
$26$	3.85661	0.756343
$27$	1.00000	0.192450
$28$	9.86908	1.86508
$29$	2.29707	0.426555	0.213278	−	0.976992i	$-0.431586\pi$
$29$	2.29707	0.426555	0.213278	+	0.976992i	$0.431586\pi$
$30$	2.99161	0.546190
$31$	−2.96651	−0.532801	−0.266401	−	0.963862i	$-0.585834\pi$
$31$	−2.96651	−0.532801	−0.266401	+	0.963862i	$0.585834\pi$
$32$	8.10599	1.43295
$33$	1.53473	0.267163
$34$	−2.90618	−0.498406
$35$	−6.70441	−1.13325
$36$	2.15929	0.359882
$37$	−2.50348	−0.411569	−0.205785	−	0.978597i	$-0.565975\pi$
$37$	−2.50348	−0.411569	−0.205785	+	0.978597i	$0.565975\pi$
$38$	7.48544	1.21430
$39$	−1.89102	−0.302805
$40$	0.476536	0.0753470
$41$	−7.38270	−1.15298	−0.576492	−	0.817103i	$-0.695579\pi$
$41$	−7.38270	−1.15298	−0.576492	+	0.817103i	$0.695579\pi$
$42$	−9.32128	−1.43830
$43$	−4.77223	−0.727759	−0.363879	−	0.931446i	$-0.618548\pi$
$43$	−4.77223	−0.727759	−0.363879	+	0.931446i	$0.618548\pi$
$44$	3.31394	0.499595
$45$	−1.46688	−0.218670
$46$	−7.37820	−1.08786
$47$	−0.870813	−0.127021	−0.0635106	−	0.997981i	$-0.520230\pi$
$47$	−0.870813	−0.127021	−0.0635106	+	0.997981i	$0.520230\pi$
$48$	−3.65604	−0.527705
$49$	13.8897	1.98424
$50$	5.80884	0.821493
$51$	1.42499	0.199539
$52$	−4.08326	−0.566247
$53$	6.81935	0.936709	0.468355	−	0.883541i	$-0.344847\pi$
$53$	6.81935	0.936709	0.468355	+	0.883541i	$0.344847\pi$
$54$	−2.03943	−0.277532
$55$	−2.25127	−0.303562
$56$	−1.48480	−0.198414
$57$	−3.67035	−0.486150
$58$	−4.68472	−0.615134
$59$	−11.4974	−1.49684	−0.748420	−	0.663225i	$-0.769187\pi$
$59$	−11.4974	−1.49684	−0.748420	+	0.663225i	$0.769187\pi$
$60$	−3.16742	−0.408913
$61$	−8.61816	−1.10344	−0.551721	−	0.834029i	$-0.686029\pi$
$61$	−8.61816	−1.10344	−0.551721	+	0.834029i	$0.686029\pi$
$62$	6.05001	0.768352
$63$	4.57052	0.575831
$64$	−9.21954	−1.15244
$65$	2.77390	0.344060
$66$	−3.12999	−0.385275
$67$	−6.89098	−0.841867	−0.420934	−	0.907091i	$-0.638297\pi$
$67$	−6.89098	−0.841867	−0.420934	+	0.907091i	$0.638297\pi$
$68$	3.07698	0.373138
$69$	3.61777	0.435528
$70$	13.6732	1.63426
$71$	−5.61929	−0.666887	−0.333444	−	0.942770i	$-0.608211\pi$
$71$	−5.61929	−0.666887	−0.333444	+	0.942770i	$0.608211\pi$
$72$	−0.324864	−0.0382855
$73$	−4.52928	−0.530112	−0.265056	−	0.964233i	$-0.585390\pi$
$73$	−4.52928	−0.530112	−0.265056	+	0.964233i	$0.585390\pi$
$74$	5.10568	0.593523
$75$	−2.84826	−0.328889
$76$	−7.92536	−0.909101
$77$	7.01454	0.799381
$78$	3.85661	0.436675
$79$	−13.2817	−1.49431	−0.747155	−	0.664650i	$-0.768581\pi$
$79$	−13.2817	−1.49431	−0.747155	+	0.664650i	$0.768581\pi$
$80$	5.36298	0.599600
$81$	1.00000	0.111111
$82$	15.0565	1.66272
$83$	8.88396	0.975142	0.487571	−	0.873083i	$-0.337883\pi$
$83$	8.88396	0.975142	0.487571	+	0.873083i	$0.337883\pi$
$84$	9.86908	1.07681
$85$	−2.09030	−0.226725
$86$	9.73265	1.04950
$87$	2.29707	0.246272
$88$	−0.498579	−0.0531487
$89$	5.99448	0.635413	0.317707	−	0.948189i	$-0.397087\pi$
$89$	5.99448	0.635413	0.317707	+	0.948189i	$0.397087\pi$
$90$	2.99161	0.315343
$91$	−8.64294	−0.906027
$92$	7.81181	0.814438
$93$	−2.96651	−0.307613
$94$	1.77597	0.183177
$95$	5.38397	0.552384
$96$	8.10599	0.827314
$97$	9.05604	0.919502	0.459751	−	0.888048i	$-0.347939\pi$
$97$	9.05604	0.919502	0.459751	+	0.888048i	$0.347939\pi$
$98$	−28.3270	−2.86146
$99$	1.53473	0.154247
$100$	−6.15022	−0.615022
$101$	−16.4032	−1.63218	−0.816090	−	0.577924i	$-0.803863\pi$
$101$	−16.4032	−1.63218	−0.816090	+	0.577924i	$0.803863\pi$
$102$	−2.90618	−0.287755
$103$	13.7465	1.35448	0.677241	−	0.735762i	$-0.263176\pi$
$103$	13.7465	1.35448	0.677241	+	0.735762i	$0.263176\pi$
$104$	0.614323	0.0602394
$105$	−6.70441	−0.654284
$106$	−13.9076	−1.35083
$107$	−15.6643	−1.51433	−0.757164	−	0.653224i	$-0.773416\pi$
$107$	−15.6643	−1.51433	−0.757164	+	0.653224i	$0.773416\pi$
$108$	2.15929	0.207778
$109$	−18.9052	−1.81079	−0.905393	−	0.424574i	$-0.860424\pi$
$109$	−18.9052	−1.81079	−0.905393	+	0.424574i	$0.860424\pi$
$110$	4.59132	0.437766
$111$	−2.50348	−0.237619
$112$	−16.7100	−1.57895
$113$	−10.0829	−0.948522	−0.474261	−	0.880384i	$-0.657285\pi$
$113$	−10.0829	−0.948522	−0.474261	+	0.880384i	$0.657285\pi$
$114$	7.48544	0.701076
$115$	−5.30683	−0.494865
$116$	4.96004	0.460528
$117$	−1.89102	−0.174825
$118$	23.4483	2.15859
$119$	6.51297	0.597043
$120$	0.476536	0.0435016
$121$	−8.64459	−0.785872
$122$	17.5762	1.59127
$123$	−7.38270	−0.665676
$124$	−6.40556	−0.575237
$125$	11.5125	1.02971
$126$	−9.32128	−0.830405
$127$	−17.0457	−1.51256	−0.756281	−	0.654247i	$-0.772986\pi$
$127$	−17.0457	−1.51256	−0.756281	+	0.654247i	$0.772986\pi$
$128$	2.59067	0.228985
$129$	−4.77223	−0.420172
$130$	−5.65719	−0.496168
$131$	−7.73192	−0.675541	−0.337770	−	0.941229i	$-0.609673\pi$
$131$	−7.73192	−0.675541	−0.337770	+	0.941229i	$0.609673\pi$
$132$	3.31394	0.288441
$133$	−16.7754	−1.45461
$134$	14.0537	1.21405
$135$	−1.46688	−0.126249
$136$	−0.462929	−0.0396958
$137$	10.7195	0.915830	0.457915	−	0.888996i	$-0.348596\pi$
$137$	10.7195	0.915830	0.457915	+	0.888996i	$0.348596\pi$
$138$	−7.37820	−0.628074
$139$	−2.65086	−0.224843	−0.112422	−	0.993661i	$-0.535861\pi$
$139$	−2.65086	−0.224843	−0.112422	+	0.993661i	$0.535861\pi$
$140$	−14.4768	−1.22351
$141$	−0.870813	−0.0733357
$142$	11.4602	0.961716
$143$	−2.90221	−0.242695
$144$	−3.65604	−0.304670
$145$	−3.36953	−0.279824
$146$	9.23717	0.764474
$147$	13.8897	1.14560
$148$	−5.40573	−0.444349
$149$	−10.7212	−0.878312	−0.439156	−	0.898411i	$-0.644722\pi$
$149$	−10.7212	−0.878312	−0.439156	+	0.898411i	$0.644722\pi$
$150$	5.80884	0.474289
$151$	−17.3360	−1.41078	−0.705390	−	0.708819i	$-0.749228\pi$
$151$	−17.3360	−1.41078	−0.705390	+	0.708819i	$0.749228\pi$
$152$	1.19236	0.0967134
$153$	1.42499	0.115204
$154$	−14.3057	−1.15279
$155$	4.35152	0.349523
$156$	−4.08326	−0.326923
$157$	3.70302	0.295533	0.147766	−	0.989022i	$-0.452792\pi$
$157$	3.70302	0.295533	0.147766	+	0.989022i	$0.452792\pi$
$158$	27.0872	2.15494
$159$	6.81935	0.540809
$160$	−11.8905	−0.940028
$161$	16.5351	1.30315
$162$	−2.03943	−0.160233
$163$	−16.1864	−1.26782	−0.633910	−	0.773407i	$-0.718551\pi$
$163$	−16.1864	−1.26782	−0.633910	+	0.773407i	$0.718551\pi$
$164$	−15.9414	−1.24481
$165$	−2.25127	−0.175261
$166$	−18.1183	−1.40625
$167$	14.9860	1.15965	0.579825	−	0.814741i	$-0.303121\pi$
$167$	14.9860	1.15965	0.579825	+	0.814741i	$0.303121\pi$
$168$	−1.48480	−0.114554
$169$	−9.42404	−0.724927
$170$	4.26302	0.326959
$171$	−3.67035	−0.280679
$172$	−10.3046	−0.785721
$173$	8.24566	0.626906	0.313453	−	0.949604i	$-0.398514\pi$
$173$	8.24566	0.626906	0.313453	+	0.949604i	$0.398514\pi$
$174$	−4.68472	−0.355148
$175$	−13.0180	−0.984070
$176$	−5.61106	−0.422949
$177$	−11.4974	−0.864201
$178$	−12.2253	−0.916328
$179$	8.34703	0.623886	0.311943	−	0.950101i	$-0.399020\pi$
$179$	8.34703	0.623886	0.311943	+	0.950101i	$0.399020\pi$
$180$	−3.16742	−0.236086
$181$	12.2941	0.913810	0.456905	−	0.889516i	$-0.348958\pi$
$181$	12.2941	0.913810	0.456905	+	0.889516i	$0.348958\pi$
$182$	17.6267	1.30658
$183$	−8.61816	−0.637073
$184$	−1.17528	−0.0866428
$185$	3.67230	0.269993
$186$	6.05001	0.443608
$187$	2.18699	0.159928
$188$	−1.88034	−0.137138
$189$	4.57052	0.332456
$190$	−10.9803	−0.796591
$191$	12.5406	0.907405	0.453703	−	0.891153i	$-0.350103\pi$
$191$	12.5406	0.907405	0.453703	+	0.891153i	$0.350103\pi$
$192$	−9.21954	−0.665363
$193$	11.8153	0.850487	0.425244	−	0.905079i	$-0.360188\pi$
$193$	11.8153	0.850487	0.425244	+	0.905079i	$0.360188\pi$
$194$	−18.4692	−1.32601
$195$	2.77390	0.198643
$196$	29.9918	2.14227
$197$	6.29435	0.448454	0.224227	−	0.974537i	$-0.428014\pi$
$197$	6.29435	0.448454	0.224227	+	0.974537i	$0.428014\pi$
$198$	−3.12999	−0.222439
$199$	2.85944	0.202701	0.101350	−	0.994851i	$-0.467684\pi$
$199$	2.85944	0.202701	0.101350	+	0.994851i	$0.467684\pi$
$200$	0.925295	0.0654283
$201$	−6.89098	−0.486052
$202$	33.4533	2.35376
$203$	10.4988	0.736871
$204$	3.07698	0.215432
$205$	10.8295	0.756368
$206$	−28.0350	−1.95329
$207$	3.61777	0.251452
$208$	6.91365	0.479376
$209$	−5.63302	−0.389644
$210$	13.6732	0.943541
$211$	23.8815	1.64407	0.822033	−	0.569439i	$-0.192840\pi$
$211$	23.8815	1.64407	0.822033	+	0.569439i	$0.192840\pi$
$212$	14.7250	1.01131
$213$	−5.61929	−0.385027
$214$	31.9464	2.18381
$215$	7.00030	0.477417
$216$	−0.324864	−0.0221042
$217$	−13.5585	−0.920411
$218$	38.5558	2.61133
$219$	−4.52928	−0.306060
$220$	−4.86116	−0.327739
$221$	−2.69469	−0.181265
$222$	5.10568	0.342670
$223$	−19.7500	−1.32256	−0.661278	−	0.750141i	$-0.729986\pi$
$223$	−19.7500	−1.32256	−0.661278	+	0.750141i	$0.729986\pi$
$224$	37.0486	2.47541
$225$	−2.84826	−0.189884
$226$	20.5635	1.36786
$227$	−8.62650	−0.572561	−0.286280	−	0.958146i	$-0.592419\pi$
$227$	−8.62650	−0.572561	−0.286280	+	0.958146i	$0.592419\pi$
$228$	−7.92536	−0.524870
$229$	7.33196	0.484510	0.242255	−	0.970213i	$-0.422113\pi$
$229$	7.33196	0.484510	0.242255	+	0.970213i	$0.422113\pi$
$230$	10.8229	0.713643
$231$	7.01454	0.461523
$232$	−0.746234	−0.0489927
$233$	22.5006	1.47406	0.737031	−	0.675859i	$-0.236227\pi$
$233$	22.5006	1.47406	0.737031	+	0.675859i	$0.236227\pi$
$234$	3.85661	0.252114
$235$	1.27738	0.0833270
$236$	−24.8263	−1.61606
$237$	−13.2817	−0.862741
$238$	−13.2828	−0.860994
$239$	−2.02579	−0.131037	−0.0655186	−	0.997851i	$-0.520870\pi$
$239$	−2.02579	−0.131037	−0.0655186	+	0.997851i	$0.520870\pi$
$240$	5.36298	0.346179
$241$	13.9514	0.898688	0.449344	−	0.893359i	$-0.351658\pi$
$241$	13.9514	0.898688	0.449344	+	0.893359i	$0.351658\pi$
$242$	17.6301	1.13330
$243$	1.00000	0.0641500
$244$	−18.6091	−1.19133
$245$	−20.3745	−1.30168
$246$	15.0565	0.959969
$247$	6.94071	0.441627
$248$	0.963712	0.0611958
$249$	8.88396	0.562998
$250$	−23.4789	−1.48494
$251$	−20.2182	−1.27616	−0.638081	−	0.769969i	$-0.720272\pi$
$251$	−20.2182	−1.27616	−0.638081	+	0.769969i	$0.720272\pi$
$252$	9.86908	0.621694
$253$	5.55231	0.349071
$254$	34.7636	2.18126
$255$	−2.09030	−0.130900
$256$	13.1556	0.822224
$257$	5.59867	0.349236	0.174618	−	0.984636i	$-0.444131\pi$
$257$	5.59867	0.349236	0.174618	+	0.984636i	$0.444131\pi$
$258$	9.73265	0.605929
$259$	−11.4422	−0.710983
$260$	5.98966	0.371463
$261$	2.29707	0.142185
$262$	15.7687	0.974196
$263$	−6.71436	−0.414025	−0.207013	−	0.978338i	$-0.566374\pi$
$263$	−6.71436	−0.414025	−0.207013	+	0.978338i	$0.566374\pi$
$264$	−0.498579	−0.0306854
$265$	−10.0032	−0.614490
$266$	34.2124	2.09769
$267$	5.99448	0.366856
$268$	−14.8796	−0.908918
$269$	16.8210	1.02560	0.512798	−	0.858509i	$-0.328609\pi$
$269$	16.8210	1.02560	0.512798	+	0.858509i	$0.328609\pi$
$270$	2.99161	0.182063
$271$	−1.06576	−0.0647406	−0.0323703	−	0.999476i	$-0.510306\pi$
$271$	−1.06576	−0.0647406	−0.0323703	+	0.999476i	$0.510306\pi$
$272$	−5.20984	−0.315893
$273$	−8.64294	−0.523095
$274$	−21.8618	−1.32072
$275$	−4.37132	−0.263601
$276$	7.81181	0.470216
$277$	−5.69050	−0.341909	−0.170955	−	0.985279i	$-0.554685\pi$
$277$	−5.69050	−0.341909	−0.170955	+	0.985279i	$0.554685\pi$
$278$	5.40626	0.324246
$279$	−2.96651	−0.177600
$280$	2.17802	0.130162
$281$	−0.238859	−0.0142491	−0.00712457	−	0.999975i	$-0.502268\pi$
$281$	−0.238859	−0.0142491	−0.00712457	+	0.999975i	$0.502268\pi$
$282$	1.77597	0.105757
$283$	9.85609	0.585884	0.292942	−	0.956130i	$-0.405366\pi$
$283$	9.85609	0.585884	0.292942	+	0.956130i	$0.405366\pi$
$284$	−12.1337	−0.720002
$285$	5.38397	0.318919
$286$	5.91887	0.349990
$287$	−33.7428	−1.99177
$288$	8.10599	0.477650
$289$	−14.9694	−0.880552
$290$	6.87193	0.403534
$291$	9.05604	0.530875
$292$	−9.78004	−0.572333
$293$	−29.4490	−1.72043	−0.860215	−	0.509931i	$-0.829671\pi$
$293$	−29.4490	−1.72043	−0.860215	+	0.509931i	$0.829671\pi$
$294$	−28.3270	−1.65207
$295$	16.8654	0.981941
$296$	0.813288	0.0472714
$297$	1.53473	0.0890543
$298$	21.8651	1.26661
$299$	−6.84127	−0.395641
$300$	−6.15022	−0.355083
$301$	−21.8116	−1.25720
$302$	35.3555	2.03448
$303$	−16.4032	−0.942340
$304$	13.4190	0.769631
$305$	12.6418	0.723868
$306$	−2.90618	−0.166135
$307$	−4.00541	−0.228601	−0.114300	−	0.993446i	$-0.536463\pi$
$307$	−4.00541	−0.228601	−0.114300	+	0.993446i	$0.536463\pi$
$308$	15.1464	0.863048
$309$	13.7465	0.782010
$310$	−8.87464	−0.504046
$311$	−5.53723	−0.313988	−0.156994	−	0.987600i	$-0.550180\pi$
$311$	−5.53723	−0.313988	−0.156994	+	0.987600i	$0.550180\pi$
$312$	0.614323	0.0347792
$313$	2.37925	0.134483	0.0672416	−	0.997737i	$-0.478580\pi$
$313$	2.37925	0.134483	0.0672416	+	0.997737i	$0.478580\pi$
$314$	−7.55206	−0.426187
$315$	−6.70441	−0.377751
$316$	−28.6791	−1.61333
$317$	13.0339	0.732058	0.366029	−	0.930604i	$-0.380717\pi$
$317$	13.0339	0.732058	0.366029	+	0.930604i	$0.380717\pi$
$318$	−13.9076	−0.779900
$319$	3.52539	0.197384
$320$	13.5240	0.756013
$321$	−15.6643	−0.874298
$322$	−33.7222	−1.87926
$323$	−5.23023	−0.291018
$324$	2.15929	0.119961
$325$	5.38611	0.298768
$326$	33.0112	1.82832
$327$	−18.9052	−1.04546
$328$	2.39837	0.132428
$329$	−3.98007	−0.219428
$330$	4.59132	0.252744
$331$	21.9346	1.20563	0.602817	−	0.797879i	$-0.294045\pi$
$331$	21.9346	1.20563	0.602817	+	0.797879i	$0.294045\pi$
$332$	19.1831	1.05281
$333$	−2.50348	−0.137190
$334$	−30.5629	−1.67233
$335$	10.1082	0.552273
$336$	−16.7100	−0.911607
$337$	24.1875	1.31757	0.658787	−	0.752329i	$-0.271070\pi$
$337$	24.1875	1.31757	0.658787	+	0.752329i	$0.271070\pi$
$338$	19.2197	1.04541
$339$	−10.0829	−0.547630
$340$	−4.51356	−0.244782
$341$	−4.55281	−0.246548
$342$	7.48544	0.404766
$343$	31.4893	1.70026
$344$	1.55032	0.0835879
$345$	−5.30683	−0.285710
$346$	−16.8165	−0.904059
$347$	13.3458	0.716438	0.358219	−	0.933638i	$-0.383384\pi$
$347$	13.3458	0.716438	0.358219	+	0.933638i	$0.383384\pi$
$348$	4.96004	0.265886
$349$	−35.1694	−1.88257	−0.941287	−	0.337608i	$-0.890382\pi$
$349$	−35.1694	−1.88257	−0.941287	+	0.337608i	$0.890382\pi$
$350$	26.5494	1.41913
$351$	−1.89102	−0.100935
$352$	12.4405	0.663083
$353$	24.5092	1.30449	0.652245	−	0.758008i	$-0.273827\pi$
$353$	24.5092	1.30449	0.652245	+	0.758008i	$0.273827\pi$
$354$	23.4483	1.24626
$355$	8.24283	0.437484
$356$	12.9438	0.686021
$357$	6.51297	0.344703
$358$	−17.0232	−0.899705
$359$	25.5238	1.34709	0.673547	−	0.739145i	$-0.264770\pi$
$359$	25.5238	1.34709	0.673547	+	0.739145i	$0.264770\pi$
$360$	0.476536	0.0251157
$361$	−5.52852	−0.290975
$362$	−25.0729	−1.31780
$363$	−8.64459	−0.453723
$364$	−18.6626	−0.978188
$365$	6.64392	0.347759
$366$	17.5762	0.918721
$367$	25.1884	1.31482	0.657411	−	0.753532i	$-0.271651\pi$
$367$	25.1884	1.31482	0.657411	+	0.753532i	$0.271651\pi$
$368$	−13.2267	−0.689490
$369$	−7.38270	−0.384328
$370$	−7.48942	−0.389356
$371$	31.1680	1.61816
$372$	−6.40556	−0.332113
$373$	24.4701	1.26701	0.633506	−	0.773738i	$-0.281615\pi$
$373$	24.4701	1.26701	0.633506	+	0.773738i	$0.281615\pi$
$374$	−4.46022	−0.230632
$375$	11.5125	0.594501
$376$	0.282895	0.0145892
$377$	−4.34380	−0.223717
$378$	−9.32128	−0.479435
$379$	−32.7878	−1.68420	−0.842098	−	0.539325i	$-0.818680\pi$
$379$	−32.7878	−1.68420	−0.842098	+	0.539325i	$0.818680\pi$
$380$	11.6256	0.596379
$381$	−17.0457	−0.873278
$382$	−25.5757	−1.30857
$383$	−28.0038	−1.43093	−0.715464	−	0.698649i	$-0.753785\pi$
$383$	−28.0038	−1.43093	−0.715464	+	0.698649i	$0.753785\pi$
$384$	2.59067	0.132204
$385$	−10.2895	−0.524401
$386$	−24.0966	−1.22649
$387$	−4.77223	−0.242586
$388$	19.5546	0.992736
$389$	−4.31934	−0.218999	−0.109500	−	0.993987i	$-0.534925\pi$
$389$	−4.31934	−0.218999	−0.109500	+	0.993987i	$0.534925\pi$
$390$	−5.65719	−0.286463
$391$	5.15530	0.260715
$392$	−4.51224	−0.227903
$393$	−7.73192	−0.390024
$394$	−12.8369	−0.646714
$395$	19.4827	0.980282
$396$	3.31394	0.166532
$397$	−30.1790	−1.51464	−0.757319	−	0.653045i	$-0.773491\pi$
$397$	−30.1790	−1.51464	−0.757319	+	0.653045i	$0.773491\pi$
$398$	−5.83164	−0.292314
$399$	−16.7754	−0.839821
$400$	10.4134	0.520668
$401$	29.7457	1.48543	0.742715	−	0.669607i	$-0.233538\pi$
$401$	29.7457	1.48543	0.742715	+	0.669607i	$0.233538\pi$
$402$	14.0537	0.700935
$403$	5.60973	0.279441
$404$	−35.4193	−1.76218
$405$	−1.46688	−0.0728899
$406$	−21.4116	−1.06264
$407$	−3.84217	−0.190449
$408$	−0.462929	−0.0229184
$409$	17.3543	0.858113	0.429056	−	0.903278i	$-0.358846\pi$
$409$	17.3543	0.858113	0.429056	+	0.903278i	$0.358846\pi$
$410$	−22.0861	−1.09076
$411$	10.7195	0.528755
$412$	29.6827	1.46236
$413$	−52.5493	−2.58578
$414$	−7.37820	−0.362619
$415$	−13.0317	−0.639702
$416$	−15.3286	−0.751546
$417$	−2.65086	−0.129813
$418$	11.4882	0.561905
$419$	−5.64773	−0.275910	−0.137955	−	0.990439i	$-0.544053\pi$
$419$	−5.64773	−0.275910	−0.137955	+	0.990439i	$0.544053\pi$
$420$	−14.4768	−0.706394
$421$	2.32366	0.113248	0.0566240	−	0.998396i	$-0.481966\pi$
$421$	2.32366	0.113248	0.0566240	+	0.998396i	$0.481966\pi$
$422$	−48.7046	−2.37091
$423$	−0.870813	−0.0423404
$424$	−2.21536	−0.107587
$425$	−4.05875	−0.196878
$426$	11.4602	0.555247
$427$	−39.3895	−1.90619
$428$	−33.8239	−1.63494
$429$	−2.90221	−0.140120
$430$	−14.2767	−0.688481
$431$	26.5590	1.27930	0.639651	−	0.768666i	$-0.279079\pi$
$431$	26.5590	1.27930	0.639651	+	0.768666i	$0.279079\pi$
$432$	−3.65604	−0.175902
$433$	−8.17559	−0.392894	−0.196447	−	0.980514i	$-0.562940\pi$
$433$	−8.17559	−0.392894	−0.196447	+	0.980514i	$0.562940\pi$
$434$	27.6517	1.32732
$435$	−3.36953	−0.161557
$436$	−40.8218	−1.95501
$437$	−13.2785	−0.635196
$438$	9.23717	0.441369
$439$	8.32123	0.397150	0.198575	−	0.980086i	$-0.436369\pi$
$439$	8.32123	0.397150	0.198575	+	0.980086i	$0.436369\pi$
$440$	0.731357	0.0348661
$441$	13.8897	0.661412
$442$	5.49565	0.261401
$443$	32.0522	1.52285	0.761424	−	0.648254i	$-0.224501\pi$
$443$	32.0522	1.52285	0.761424	+	0.648254i	$0.224501\pi$
$444$	−5.40573	−0.256545
$445$	−8.79319	−0.416837
$446$	40.2788	1.90726
$447$	−10.7212	−0.507094
$448$	−42.1381	−1.99084
$449$	−36.8057	−1.73697	−0.868485	−	0.495715i	$-0.834906\pi$
$449$	−36.8057	−1.73697	−0.868485	+	0.495715i	$0.834906\pi$
$450$	5.80884	0.273831
$451$	−11.3305	−0.533532
$452$	−21.7720	−1.02407
$453$	−17.3360	−0.814514
$454$	17.5932	0.825689
$455$	12.6782	0.594362
$456$	1.19236	0.0558375
$457$	32.6589	1.52772	0.763861	−	0.645381i	$-0.223302\pi$
$457$	32.6589	1.52772	0.763861	+	0.645381i	$0.223302\pi$
$458$	−14.9531	−0.698710
$459$	1.42499	0.0665130
$460$	−11.4590	−0.534279
$461$	−37.0695	−1.72650	−0.863250	−	0.504777i	$-0.831575\pi$
$461$	−37.0695	−1.72650	−0.863250	+	0.504777i	$0.831575\pi$
$462$	−14.3057	−0.665561
$463$	15.4018	0.715784	0.357892	−	0.933763i	$-0.383496\pi$
$463$	15.4018	0.715784	0.357892	+	0.933763i	$0.383496\pi$
$464$	−8.39819	−0.389876
$465$	4.35152	0.201797
$466$	−45.8884	−2.12574
$467$	−4.68913	−0.216987	−0.108494	−	0.994097i	$-0.534603\pi$
$467$	−4.68913	−0.216987	−0.108494	+	0.994097i	$0.534603\pi$
$468$	−4.08326	−0.188749
$469$	−31.4954	−1.45432
$470$	−2.60513	−0.120166
$471$	3.70302	0.170626
$472$	3.73510	0.171922
$473$	−7.32411	−0.336763
$474$	27.0872	1.24416
$475$	10.4541	0.479668
$476$	14.0634	0.644594
$477$	6.81935	0.312236
$478$	4.13146	0.188968
$479$	−40.2966	−1.84120	−0.920600	−	0.390507i	$-0.872300\pi$
$479$	−40.2966	−1.84120	−0.920600	+	0.390507i	$0.872300\pi$
$480$	−11.8905	−0.542726
$481$	4.73412	0.215857
$482$	−28.4529	−1.29600
$483$	16.5351	0.752372
$484$	−18.6662	−0.848463
$485$	−13.2841	−0.603202
$486$	−2.03943	−0.0925106
$487$	3.11757	0.141271	0.0706353	−	0.997502i	$-0.477497\pi$
$487$	3.11757	0.141271	0.0706353	+	0.997502i	$0.477497\pi$
$488$	2.79973	0.126738
$489$	−16.1864	−0.731977
$490$	41.5524	1.87715
$491$	−33.2762	−1.50173	−0.750867	−	0.660454i	$-0.770364\pi$
$491$	−33.2762	−1.50173	−0.750867	+	0.660454i	$0.770364\pi$
$492$	−15.9414	−0.718694
$493$	3.27331	0.147423
$494$	−14.1551	−0.636869
$495$	−2.25127	−0.101187
$496$	10.8457	0.486986
$497$	−25.6831	−1.15204
$498$	−18.1183	−0.811898
$499$	8.95649	0.400948	0.200474	−	0.979699i	$-0.435752\pi$
$499$	8.95649	0.400948	0.200474	+	0.979699i	$0.435752\pi$
$500$	24.8588	1.11172
$501$	14.9860	0.669525
$502$	41.2337	1.84035
$503$	10.6211	0.473572	0.236786	−	0.971562i	$-0.423906\pi$
$503$	10.6211	0.473572	0.236786	+	0.971562i	$0.423906\pi$
$504$	−1.48480	−0.0661380
$505$	24.0616	1.07073
$506$	−11.3236	−0.503394
$507$	−9.42404	−0.418537
$508$	−36.8066	−1.63303
$509$	−0.961213	−0.0426050	−0.0213025	−	0.999773i	$-0.506781\pi$
$509$	−0.961213	−0.0426050	−0.0213025	+	0.999773i	$0.506781\pi$
$510$	4.26302	0.188770
$511$	−20.7012	−0.915766
$512$	−32.0113	−1.41471
$513$	−3.67035	−0.162050
$514$	−11.4181	−0.503632
$515$	−20.1645	−0.888552
$516$	−10.3046	−0.453636
$517$	−1.33647	−0.0587777
$518$	23.3356	1.02531
$519$	8.24566	0.361944
$520$	−0.901140	−0.0395176
$521$	−41.7460	−1.82893	−0.914463	−	0.404669i	$-0.867387\pi$
$521$	−41.7460	−1.82893	−0.914463	+	0.404669i	$0.867387\pi$
$522$	−4.68472	−0.205045
$523$	14.7915	0.646787	0.323393	−	0.946265i	$-0.395176\pi$
$523$	14.7915	0.646787	0.323393	+	0.946265i	$0.395176\pi$
$524$	−16.6955	−0.729345
$525$	−13.0180	−0.568153
$526$	13.6935	0.597065
$527$	−4.22726	−0.184142
$528$	−5.61106	−0.244190
$529$	−9.91177	−0.430946
$530$	20.4008	0.886155
$531$	−11.4974	−0.498947
$532$	−36.2230	−1.57047
$533$	13.9608	0.604711
$534$	−12.2253	−0.529042
$535$	22.9777	0.993414
$536$	2.23863	0.0966940
$537$	8.34703	0.360201
$538$	−34.3054	−1.47901
$539$	21.3169	0.918186
$540$	−3.16742	−0.136304
$541$	2.13747	0.0918972	0.0459486	−	0.998944i	$-0.485369\pi$
$541$	2.13747	0.0918972	0.0459486	+	0.998944i	$0.485369\pi$
$542$	2.17356	0.0933622
$543$	12.2941	0.527588
$544$	11.5510	0.495245
$545$	27.7316	1.18789
$546$	17.6267	0.754354
$547$	8.58062	0.366881	0.183441	−	0.983031i	$-0.441277\pi$
$547$	8.58062	0.366881	0.183441	+	0.983031i	$0.441277\pi$
$548$	23.1466	0.988772
$549$	−8.61816	−0.367814
$550$	8.91502	0.380138
$551$	−8.43105	−0.359175
$552$	−1.17528	−0.0500233
$553$	−60.7044	−2.58141
$554$	11.6054	0.493066
$555$	3.67230	0.155881
$556$	−5.72398	−0.242751
$557$	−42.5777	−1.80407	−0.902037	−	0.431660i	$-0.857928\pi$
$557$	−42.5777	−1.80407	−0.902037	+	0.431660i	$0.857928\pi$
$558$	6.05001	0.256117
$559$	9.02439	0.381691
$560$	24.5116	1.03581
$561$	2.18699	0.0923347
$562$	0.487137	0.0205486
$563$	44.0641	1.85708	0.928540	−	0.371233i	$-0.121065\pi$
$563$	44.0641	1.85708	0.928540	+	0.371233i	$0.121065\pi$
$564$	−1.88034	−0.0791765
$565$	14.7905	0.622240
$566$	−20.1008	−0.844902
$567$	4.57052	0.191944
$568$	1.82550	0.0765964
$569$	9.01769	0.378041	0.189021	−	0.981973i	$-0.439469\pi$
$569$	9.01769	0.378041	0.189021	+	0.981973i	$0.439469\pi$
$570$	−10.9803	−0.459912
$571$	−37.6660	−1.57627	−0.788137	−	0.615500i	$-0.788954\pi$
$571$	−37.6660	−1.57627	−0.788137	+	0.615500i	$0.788954\pi$
$572$	−6.26672	−0.262025
$573$	12.5406	0.523891
$574$	68.8162	2.87233
$575$	−10.3043	−0.429720
$576$	−9.21954	−0.384147
$577$	−11.3255	−0.471487	−0.235744	−	0.971815i	$-0.575753\pi$
$577$	−11.3255	−0.471487	−0.235744	+	0.971815i	$0.575753\pi$
$578$	30.5291	1.26984
$579$	11.8153	0.491029
$580$	−7.27579	−0.302111
$581$	40.6043	1.68455
$582$	−18.4692	−0.765573
$583$	10.4659	0.433453
$584$	1.47140	0.0608869
$585$	2.77390	0.114687
$586$	60.0594	2.48103
$587$	20.9068	0.862915	0.431458	−	0.902133i	$-0.357999\pi$
$587$	20.9068	0.862915	0.431458	+	0.902133i	$0.357999\pi$
$588$	29.9918	1.23684
$589$	10.8881	0.448638
$590$	−34.3959	−1.41605
$591$	6.29435	0.258915
$592$	9.15282	0.376179
$593$	−3.83701	−0.157567	−0.0787837	−	0.996892i	$-0.525104\pi$
$593$	−3.83701	−0.157567	−0.0787837	+	0.996892i	$0.525104\pi$
$594$	−3.12999	−0.128425
$595$	−9.55375	−0.391666
$596$	−23.1501	−0.948266
$597$	2.85944	0.117029
$598$	13.9523	0.570553
$599$	11.2934	0.461436	0.230718	−	0.973021i	$-0.425893\pi$
$599$	11.2934	0.461436	0.230718	+	0.973021i	$0.425893\pi$
$600$	0.925295	0.0377750
$601$	6.52030	0.265968	0.132984	−	0.991118i	$-0.457544\pi$
$601$	6.52030	0.265968	0.132984	+	0.991118i	$0.457544\pi$
$602$	44.4833	1.81300
$603$	−6.89098	−0.280622
$604$	−37.4334	−1.52314
$605$	12.6806	0.515539
$606$	33.4533	1.35895
$607$	32.2238	1.30792	0.653961	−	0.756528i	$-0.273106\pi$
$607$	32.2238	1.30792	0.653961	+	0.756528i	$0.273106\pi$
$608$	−29.7518	−1.20660
$609$	10.4988	0.425433
$610$	−25.7821	−1.04389
$611$	1.64672	0.0666193
$612$	3.07698	0.124379
$613$	−17.6403	−0.712486	−0.356243	−	0.934393i	$-0.615942\pi$
$613$	−17.6403	−0.712486	−0.356243	+	0.934393i	$0.615942\pi$
$614$	8.16877	0.329665
$615$	10.8295	0.436689
$616$	−2.27877	−0.0918142
$617$	36.3533	1.46353	0.731764	−	0.681558i	$-0.238697\pi$
$617$	36.3533	1.46353	0.731764	+	0.681558i	$0.238697\pi$
$618$	−28.0350	−1.12774
$619$	−9.95628	−0.400177	−0.200088	−	0.979778i	$-0.564123\pi$
$619$	−9.95628	−0.400177	−0.200088	+	0.979778i	$0.564123\pi$
$620$	9.39620	0.377361
$621$	3.61777	0.145176
$622$	11.2928	0.452801
$623$	27.3979	1.09767
$624$	6.91365	0.276768
$625$	−2.64613	−0.105845
$626$	−4.85232	−0.193938
$627$	−5.63302	−0.224961
$628$	7.99589	0.319071
$629$	−3.56744	−0.142243
$630$	13.6732	0.544754
$631$	−19.0094	−0.756752	−0.378376	−	0.925652i	$-0.623517\pi$
$631$	−19.0094	−0.756752	−0.378376	+	0.925652i	$0.623517\pi$
$632$	4.31475	0.171631
$633$	23.8815	0.949202
$634$	−26.5818	−1.05570
$635$	25.0040	0.992255
$636$	14.7250	0.583882
$637$	−26.2656	−1.04068
$638$	−7.18980	−0.284647
$639$	−5.61929	−0.222296
$640$	−3.80020	−0.150216
$641$	−9.22706	−0.364447	−0.182223	−	0.983257i	$-0.558329\pi$
$641$	−9.22706	−0.364447	−0.182223	+	0.983257i	$0.558329\pi$
$642$	31.9464	1.26082
$643$	−21.7496	−0.857721	−0.428861	−	0.903371i	$-0.641085\pi$
$643$	−21.7496	−0.857721	−0.428861	+	0.903371i	$0.641085\pi$
$644$	35.7040	1.40694
$645$	7.00030	0.275637
$646$	10.6667	0.419676
$647$	−3.69723	−0.145353	−0.0726766	−	0.997356i	$-0.523154\pi$
$647$	−3.69723	−0.145353	−0.0726766	+	0.997356i	$0.523154\pi$
$648$	−0.324864	−0.0127618
$649$	−17.6455	−0.692648
$650$	−10.9846	−0.430852
$651$	−13.5585	−0.531400
$652$	−34.9513	−1.36880
$653$	−33.9609	−1.32899	−0.664496	−	0.747292i	$-0.731354\pi$
$653$	−33.9609	−1.32899	−0.664496	+	0.747292i	$0.731354\pi$
$654$	38.5558	1.50765
$655$	11.3418	0.443161
$656$	26.9915	1.05384
$657$	−4.52928	−0.176704
$658$	8.11709	0.316437
$659$	16.7449	0.652287	0.326144	−	0.945320i	$-0.394251\pi$
$659$	16.7449	0.652287	0.326144	+	0.945320i	$0.394251\pi$
$660$	−4.86116	−0.189220
$661$	27.8049	1.08148	0.540742	−	0.841189i	$-0.318144\pi$
$661$	27.8049	1.08148	0.540742	+	0.841189i	$0.318144\pi$
$662$	−44.7342	−1.73864
$663$	−2.69469	−0.104653
$664$	−2.88608	−0.112001
$665$	24.6075	0.954240
$666$	5.10568	0.197841
$667$	8.31026	0.321775
$668$	32.3591	1.25201
$669$	−19.7500	−0.763578
$670$	−20.6151	−0.796431
$671$	−13.2266	−0.510607
$672$	37.0486	1.42918
$673$	−30.6048	−1.17973	−0.589865	−	0.807502i	$-0.700819\pi$
$673$	−30.6048	−1.17973	−0.589865	+	0.807502i	$0.700819\pi$
$674$	−49.3287	−1.90007
$675$	−2.84826	−0.109630
$676$	−20.3493	−0.782664
$677$	5.60320	0.215348	0.107674	−	0.994186i	$-0.465660\pi$
$677$	5.60320	0.215348	0.107674	+	0.994186i	$0.465660\pi$
$678$	20.5635	0.789735
$679$	41.3908	1.58843
$680$	0.679062	0.0260408
$681$	−8.62650	−0.330568
$682$	9.28515	0.355547
$683$	−5.49312	−0.210188	−0.105094	−	0.994462i	$-0.533514\pi$
$683$	−5.49312	−0.210188	−0.105094	+	0.994462i	$0.533514\pi$
$684$	−7.92536	−0.303034
$685$	−15.7243	−0.600793
$686$	−64.2204	−2.45195
$687$	7.33196	0.279732
$688$	17.4475	0.665180
$689$	−12.8955	−0.491280
$690$	10.8229	0.412022
$691$	32.0842	1.22054	0.610270	−	0.792193i	$-0.291061\pi$
$691$	32.0842	1.22054	0.610270	+	0.792193i	$0.291061\pi$
$692$	17.8048	0.676836
$693$	7.01454	0.266460
$694$	−27.2178	−1.03317
$695$	3.88850	0.147499
$696$	−0.746234	−0.0282859
$697$	−10.5203	−0.398485
$698$	71.7256	2.71485
$699$	22.5006	0.851050
$700$	−28.1097	−1.06245
$701$	7.89090	0.298035	0.149018	−	0.988835i	$-0.452389\pi$
$701$	7.89090	0.298035	0.149018	+	0.988835i	$0.452389\pi$
$702$	3.85661	0.145558
$703$	9.18864	0.346556
$704$	−14.1495	−0.533281
$705$	1.27738	0.0481089
$706$	−49.9848	−1.88120
$707$	−74.9712	−2.81958
$708$	−24.8263	−0.933031
$709$	10.4173	0.391229	0.195615	−	0.980681i	$-0.437330\pi$
$709$	10.4173	0.391229	0.195615	+	0.980681i	$0.437330\pi$
$710$	−16.8107	−0.630895
$711$	−13.2817	−0.498103
$712$	−1.94739	−0.0729814
$713$	−10.7321	−0.401922
$714$	−13.2828	−0.497095
$715$	4.25720	0.159210
$716$	18.0237	0.673576
$717$	−2.02579	−0.0756543
$718$	−52.0541	−1.94264
$719$	−49.9147	−1.86151	−0.930753	−	0.365649i	$-0.880847\pi$
$719$	−49.9147	−1.86151	−0.930753	+	0.365649i	$0.880847\pi$
$720$	5.36298	0.199867
$721$	62.8286	2.33986
$722$	11.2750	0.419614
$723$	13.9514	0.518858
$724$	26.5464	0.986590
$725$	−6.54265	−0.242988
$726$	17.6301	0.654313
$727$	0.776318	0.0287921	0.0143960	−	0.999896i	$-0.495417\pi$
$727$	0.776318	0.0287921	0.0143960	+	0.999896i	$0.495417\pi$
$728$	2.80778	0.104063
$729$	1.00000	0.0370370
$730$	−13.5498	−0.501502
$731$	−6.80041	−0.251522
$732$	−18.6091	−0.687813
$733$	41.9515	1.54951	0.774757	−	0.632259i	$-0.217872\pi$
$733$	41.9515	1.54951	0.774757	+	0.632259i	$0.217872\pi$
$734$	−51.3700	−1.89610
$735$	−20.3745	−0.751524
$736$	29.3256	1.08095
$737$	−10.5758	−0.389566
$738$	15.0565	0.554238
$739$	5.63357	0.207234	0.103617	−	0.994617i	$-0.466958\pi$
$739$	5.63357	0.207234	0.103617	+	0.994617i	$0.466958\pi$
$740$	7.92957	0.291497
$741$	6.94071	0.254973
$742$	−63.5650	−2.33354
$743$	29.6591	1.08809	0.544044	−	0.839057i	$-0.316893\pi$
$743$	29.6591	1.08809	0.544044	+	0.839057i	$0.316893\pi$
$744$	0.963712	0.0353314
$745$	15.7267	0.576181
$746$	−49.9051	−1.82715
$747$	8.88396	0.325047
$748$	4.72235	0.172666
$749$	−71.5942	−2.61599
$750$	−23.4789	−0.857329
$751$	46.8152	1.70831	0.854156	−	0.520017i	$-0.174074\pi$
$751$	46.8152	1.70831	0.854156	+	0.520017i	$0.174074\pi$
$752$	3.18373	0.116099
$753$	−20.2182	−0.736793
$754$	8.85890	0.322622
$755$	25.4298	0.925485
$756$	9.86908	0.358935
$757$	8.97205	0.326095	0.163047	−	0.986618i	$-0.447868\pi$
$757$	8.97205	0.326095	0.163047	+	0.986618i	$0.447868\pi$
$758$	66.8685	2.42878
$759$	5.55231	0.201536
$760$	−1.74906	−0.0634449
$761$	−30.9950	−1.12357	−0.561784	−	0.827284i	$-0.689885\pi$
$761$	−30.9950	−1.12357	−0.561784	+	0.827284i	$0.689885\pi$
$762$	34.7636	1.25935
$763$	−86.4065	−3.12812
$764$	27.0788	0.979676
$765$	−2.09030	−0.0755749
$766$	57.1119	2.06354
$767$	21.7419	0.785054
$768$	13.1556	0.474711
$769$	−43.9719	−1.58567	−0.792834	−	0.609437i	$-0.791395\pi$
$769$	−43.9719	−1.58567	−0.792834	+	0.609437i	$0.791395\pi$
$770$	20.9847	0.756238
$771$	5.59867	0.201631
$772$	25.5128	0.918225
$773$	45.3007	1.62935	0.814676	−	0.579916i	$-0.196915\pi$
$773$	45.3007	1.62935	0.814676	+	0.579916i	$0.196915\pi$
$774$	9.73265	0.349833
$775$	8.44939	0.303511
$776$	−2.94198	−0.105611
$777$	−11.4422	−0.410486
$778$	8.80900	0.315818
$779$	27.0971	0.970855
$780$	5.98966	0.214464
$781$	−8.62412	−0.308595
$782$	−10.5139	−0.375976
$783$	2.29707	0.0820906
$784$	−50.7812	−1.81361
$785$	−5.43189	−0.193872
$786$	15.7687	0.562452
$787$	38.9434	1.38818	0.694092	−	0.719887i	$-0.255806\pi$
$787$	38.9434	1.38818	0.694092	+	0.719887i	$0.255806\pi$
$788$	13.5913	0.484171
$789$	−6.71436	−0.239038
$790$	−39.7337	−1.41366
$791$	−46.0842	−1.63857
$792$	−0.498579	−0.0177162
$793$	16.2971	0.578727
$794$	61.5480	2.18426
$795$	−10.0032	−0.354776
$796$	6.17437	0.218845
$797$	−12.6935	−0.449627	−0.224814	−	0.974402i	$-0.572177\pi$
$797$	−12.6935	−0.449627	−0.224814	+	0.974402i	$0.572177\pi$
$798$	34.2124	1.21110
$799$	−1.24090	−0.0439000
$800$	−23.0880	−0.816282
$801$	5.99448	0.211804
$802$	−60.6644	−2.14214
$803$	−6.95124	−0.245304
$804$	−14.8796	−0.524764
$805$	−24.2550	−0.854876
$806$	−11.4407	−0.402981
$807$	16.8210	0.592128
$808$	5.32881	0.187467
$809$	9.89626	0.347934	0.173967	−	0.984751i	$-0.444341\pi$
$809$	9.89626	0.347934	0.173967	+	0.984751i	$0.444341\pi$
$810$	2.99161	0.105114
$811$	−5.82584	−0.204573	−0.102287	−	0.994755i	$-0.532616\pi$
$811$	−5.82584	−0.204573	−0.102287	+	0.994755i	$0.532616\pi$
$812$	22.6700	0.795560
$813$	−1.06576	−0.0373780
$814$	7.83586	0.274647
$815$	23.7436	0.831702
$816$	−5.20984	−0.182381
$817$	17.5158	0.612799
$818$	−35.3929	−1.23748
$819$	−8.64294	−0.302009
$820$	23.3841	0.816610
$821$	−32.7796	−1.14402	−0.572008	−	0.820248i	$-0.693835\pi$
$821$	−32.7796	−1.14402	−0.572008	+	0.820248i	$0.693835\pi$
$822$	−21.8618	−0.762516
$823$	−19.5902	−0.682873	−0.341437	−	0.939905i	$-0.610913\pi$
$823$	−19.5902	−0.682873	−0.341437	+	0.939905i	$0.610913\pi$
$824$	−4.46573	−0.155571
$825$	−4.37132	−0.152190
$826$	107.171	3.72895
$827$	0.822046	0.0285853	0.0142927	−	0.999898i	$-0.495450\pi$
$827$	0.822046	0.0285853	0.0142927	+	0.999898i	$0.495450\pi$
$828$	7.81181	0.271479
$829$	−27.1048	−0.941388	−0.470694	−	0.882297i	$-0.655996\pi$
$829$	−27.1048	−0.941388	−0.470694	+	0.882297i	$0.655996\pi$
$830$	26.5773	0.922513
$831$	−5.69050	−0.197401
$832$	17.4343	0.604427
$833$	19.7927	0.685776
$834$	5.40626	0.187203
$835$	−21.9827	−0.760742
$836$	−12.1633	−0.420677
$837$	−2.96651	−0.102538
$838$	11.5182	0.397889
$839$	−9.30482	−0.321238	−0.160619	−	0.987016i	$-0.551349\pi$
$839$	−9.30482	−0.321238	−0.160619	+	0.987016i	$0.551349\pi$
$840$	2.17802	0.0751488
$841$	−23.7235	−0.818051
$842$	−4.73894	−0.163315
$843$	−0.238859	−0.00822674
$844$	51.5670	1.77501
$845$	13.8240	0.475559
$846$	1.77597	0.0610589
$847$	−39.5103	−1.35759
$848$	−24.9318	−0.856163
$849$	9.85609	0.338260
$850$	8.27756	0.283918
$851$	−9.05699	−0.310470
$852$	−12.1337	−0.415693
$853$	−45.4046	−1.55462	−0.777312	−	0.629115i	$-0.783417\pi$
$853$	−45.4046	−1.55462	−0.777312	+	0.629115i	$0.783417\pi$
$854$	80.3322	2.74891
$855$	5.38397	0.184128
$856$	5.08877	0.173931
$857$	−19.9907	−0.682870	−0.341435	−	0.939905i	$-0.610913\pi$
$857$	−19.9907	−0.682870	−0.341435	+	0.939905i	$0.610913\pi$
$858$	5.91887	0.202067
$859$	−37.9417	−1.29455	−0.647276	−	0.762255i	$-0.724092\pi$
$859$	−37.9417	−1.29455	−0.647276	+	0.762255i	$0.724092\pi$
$860$	15.1157	0.515441
$861$	−33.7428	−1.14995
$862$	−54.1653	−1.84488
$863$	53.7612	1.83005	0.915027	−	0.403393i	$-0.132169\pi$
$863$	53.7612	1.83005	0.915027	+	0.403393i	$0.132169\pi$
$864$	8.10599	0.275771
$865$	−12.0954	−0.411256
$866$	16.6736	0.566591
$867$	−14.9694	−0.508387
$868$	−29.2768	−0.993718
$869$	−20.3839	−0.691477
$870$	6.87193	0.232980
$871$	13.0310	0.441538
$872$	6.14160	0.207981
$873$	9.05604	0.306501
$874$	27.0806	0.916014
$875$	52.6180	1.77881
$876$	−9.78004	−0.330437
$877$	35.3584	1.19397	0.596983	−	0.802254i	$-0.296366\pi$
$877$	35.3584	1.19397	0.596983	+	0.802254i	$0.296366\pi$
$878$	−16.9706	−0.572730
$879$	−29.4490	−0.993291
$880$	8.23076	0.277459
$881$	44.7814	1.50872	0.754362	−	0.656459i	$-0.227947\pi$
$881$	44.7814	1.50872	0.754362	+	0.656459i	$0.227947\pi$
$882$	−28.3270	−0.953821
$883$	−5.85591	−0.197067	−0.0985335	−	0.995134i	$-0.531415\pi$
$883$	−5.85591	−0.197067	−0.0985335	+	0.995134i	$0.531415\pi$
$884$	−5.81863	−0.195702
$885$	16.8654	0.566924
$886$	−65.3684	−2.19609
$887$	−33.7534	−1.13333	−0.566663	−	0.823949i	$-0.691766\pi$
$887$	−33.7534	−1.13333	−0.566663	+	0.823949i	$0.691766\pi$
$888$	0.813288	0.0272922
$889$	−77.9078	−2.61294
$890$	17.9331	0.601120
$891$	1.53473	0.0514156
$892$	−42.6460	−1.42789
$893$	3.19619	0.106956
$894$	21.8651	0.731279
$895$	−12.2441	−0.409275
$896$	11.8407	0.395570
$897$	−6.84127	−0.228423
$898$	75.0629	2.50488
$899$	−6.81428	−0.227269
$900$	−6.15022	−0.205007
$901$	9.71753	0.323738
$902$	23.1078	0.769405
$903$	−21.8116	−0.725844
$904$	3.27558	0.108944
$905$	−18.0339	−0.599468
$906$	35.3555	1.17461
$907$	16.9399	0.562481	0.281241	−	0.959637i	$-0.409254\pi$
$907$	16.9399	0.562481	0.281241	+	0.959637i	$0.409254\pi$
$908$	−18.6271	−0.618163
$909$	−16.4032	−0.544060
$910$	−25.8563	−0.857128
$911$	−22.0564	−0.730762	−0.365381	−	0.930858i	$-0.619061\pi$
$911$	−22.0564	−0.730762	−0.365381	+	0.930858i	$0.619061\pi$
$912$	13.4190	0.444346
$913$	13.6345	0.451237
$914$	−66.6058	−2.20312
$915$	12.6418	0.417926
$916$	15.8318	0.523099
$917$	−35.3389	−1.16699
$918$	−2.90618	−0.0959183
$919$	−54.1924	−1.78764	−0.893821	−	0.448424i	$-0.851985\pi$
$919$	−54.1924	−1.78764	−0.893821	+	0.448424i	$0.851985\pi$
$920$	1.72400	0.0568385
$921$	−4.00541	−0.131983
$922$	75.6008	2.48978
$923$	10.6262	0.349765
$924$	15.1464	0.498281
$925$	7.13055	0.234451
$926$	−31.4110	−1.03223
$927$	13.7465	0.451494
$928$	18.6200	0.611232
$929$	21.0606	0.690976	0.345488	−	0.938423i	$-0.387713\pi$
$929$	21.0606	0.690976	0.345488	+	0.938423i	$0.387713\pi$
$930$	−8.87464	−0.291011
$931$	−50.9799	−1.67080
$932$	48.5853	1.59146
$933$	−5.53723	−0.181281
$934$	9.56318	0.312917
$935$	−3.20805	−0.104915
$936$	0.614323	0.0200798
$937$	−2.74422	−0.0896496	−0.0448248	−	0.998995i	$-0.514273\pi$
$937$	−2.74422	−0.0896496	−0.0448248	+	0.998995i	$0.514273\pi$
$938$	64.2327	2.09727
$939$	2.37925	0.0776439
$940$	2.75823	0.0899637
$941$	−19.3153	−0.629660	−0.314830	−	0.949148i	$-0.601948\pi$
$941$	−19.3153	−0.629660	−0.314830	+	0.949148i	$0.601948\pi$
$942$	−7.55206	−0.246059
$943$	−26.7089	−0.869761
$944$	42.0352	1.36813
$945$	−6.70441	−0.218095
$946$	14.9370	0.485645
$947$	−3.20981	−0.104305	−0.0521524	−	0.998639i	$-0.516608\pi$
$947$	−3.20981	−0.104305	−0.0521524	+	0.998639i	$0.516608\pi$
$948$	−28.6791	−0.931454
$949$	8.56496	0.278030
$950$	−21.3205	−0.691727
$951$	13.0339	0.422654
$952$	−2.11583	−0.0685743
$953$	−40.7966	−1.32153	−0.660766	−	0.750592i	$-0.729768\pi$
$953$	−40.7966	−1.32153	−0.660766	+	0.750592i	$0.729768\pi$
$954$	−13.9076	−0.450275
$955$	−18.3956	−0.595266
$956$	−4.37426	−0.141474
$957$	3.52539	0.113960
$958$	82.1823	2.65519
$959$	48.9938	1.58209
$960$	13.5240	0.436484
$961$	−22.1998	−0.716123
$962$	−9.65493	−0.311288
$963$	−15.6643	−0.504776
$964$	30.1251	0.970264
$965$	−17.3317	−0.557928
$966$	−33.7222	−1.08499
$967$	−38.1781	−1.22772	−0.613862	−	0.789413i	$-0.710385\pi$
$967$	−38.1781	−1.22772	−0.613862	+	0.789413i	$0.710385\pi$
$968$	2.80831	0.0902626
$969$	−5.23023	−0.168019
$970$	27.0921	0.869876
$971$	−39.4996	−1.26760	−0.633801	−	0.773496i	$-0.718506\pi$
$971$	−39.4996	−1.26760	−0.633801	+	0.773496i	$0.718506\pi$
$972$	2.15929	0.0692593
$973$	−12.1158	−0.388415
$974$	−6.35808	−0.203726
$975$	5.38611	0.172494
$976$	31.5084	1.00856
$977$	−37.8799	−1.21189	−0.605943	−	0.795508i	$-0.707204\pi$
$977$	−37.8799	−1.21189	−0.605943	+	0.795508i	$0.707204\pi$
$978$	33.0112	1.05558
$979$	9.19993	0.294031
$980$	−43.9944	−1.40535
$981$	−18.9052	−0.603596
$982$	67.8646	2.16565
$983$	20.7250	0.661024	0.330512	−	0.943802i	$-0.392779\pi$
$983$	20.7250	0.661024	0.330512	+	0.943802i	$0.392779\pi$
$984$	2.39837	0.0764572
$985$	−9.23306	−0.294190
$986$	−6.67570	−0.212598
$987$	−3.98007	−0.126687
$988$	14.9870	0.476800
$989$	−17.2648	−0.548989
$990$	4.59132	0.145922
$991$	−42.1840	−1.34002	−0.670009	−	0.742353i	$-0.733710\pi$
$991$	−42.1840	−1.34002	−0.670009	+	0.742353i	$0.733710\pi$
$992$	−24.0465	−0.763478
$993$	21.9346	0.696074
$994$	52.3790	1.66136
$995$	−4.19446	−0.132973
$996$	19.1831	0.607839
$997$	27.4401	0.869035	0.434518	−	0.900663i	$-0.356919\pi$
$997$	27.4401	0.869035	0.434518	+	0.900663i	$0.356919\pi$
$998$	−18.2662	−0.578206
$999$	−2.50348	−0.0792065

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8013.2.a.a.1.18	✓	94

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8013.2.a.a.1.18	✓	94	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:	\( N \)	\(=\)	\( 8013 = 3 \cdot 2671 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.03943 q^{2} +1.00000 q^{3} +2.15929 q^{4} -1.46688 q^{5} -2.03943 q^{6} +4.57052 q^{7} -0.324864 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.03943 q^{2} +1.00000 q^{3} +2.15929 q^{4} -1.46688 q^{5} -2.03943 q^{6} +4.57052 q^{7} -0.324864 q^{8} +1.00000 q^{9} +2.99161 q^{10} +1.53473 q^{11} +2.15929 q^{12} -1.89102 q^{13} -9.32128 q^{14} -1.46688 q^{15} -3.65604 q^{16} +1.42499 q^{17} -2.03943 q^{18} -3.67035 q^{19} -3.16742 q^{20} +4.57052 q^{21} -3.12999 q^{22} +3.61777 q^{23} -0.324864 q^{24} -2.84826 q^{25} +3.85661 q^{26} +1.00000 q^{27} +9.86908 q^{28} +2.29707 q^{29} +2.99161 q^{30} -2.96651 q^{31} +8.10599 q^{32} +1.53473 q^{33} -2.90618 q^{34} -6.70441 q^{35} +2.15929 q^{36} -2.50348 q^{37} +7.48544 q^{38} -1.89102 q^{39} +0.476536 q^{40} -7.38270 q^{41} -9.32128 q^{42} -4.77223 q^{43} +3.31394 q^{44} -1.46688 q^{45} -7.37820 q^{46} -0.870813 q^{47} -3.65604 q^{48} +13.8897 q^{49} +5.80884 q^{50} +1.42499 q^{51} -4.08326 q^{52} +6.81935 q^{53} -2.03943 q^{54} -2.25127 q^{55} -1.48480 q^{56} -3.67035 q^{57} -4.68472 q^{58} -11.4974 q^{59} -3.16742 q^{60} -8.61816 q^{61} +6.05001 q^{62} +4.57052 q^{63} -9.21954 q^{64} +2.77390 q^{65} -3.12999 q^{66} -6.89098 q^{67} +3.07698 q^{68} +3.61777 q^{69} +13.6732 q^{70} -5.61929 q^{71} -0.324864 q^{72} -4.52928 q^{73} +5.10568 q^{74} -2.84826 q^{75} -7.92536 q^{76} +7.01454 q^{77} +3.85661 q^{78} -13.2817 q^{79} +5.36298 q^{80} +1.00000 q^{81} +15.0565 q^{82} +8.88396 q^{83} +9.86908 q^{84} -2.09030 q^{85} +9.73265 q^{86} +2.29707 q^{87} -0.498579 q^{88} +5.99448 q^{89} +2.99161 q^{90} -8.64294 q^{91} +7.81181 q^{92} -2.96651 q^{93} +1.77597 q^{94} +5.38397 q^{95} +8.10599 q^{96} +9.05604 q^{97} -28.3270 q^{98} +1.53473 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 94 q - 13 q^{2} + 94 q^{3} + 73 q^{4} - 14 q^{5} - 13 q^{6} - 55 q^{7} - 36 q^{8} + 94 q^{9}+O(q^{10}) \) 94 * q - 13 * q^2 + 94 * q^3 + 73 * q^4 - 14 * q^5 - 13 * q^6 - 55 * q^7 - 36 * q^8 + 94 * q^9	\( 94 q - 13 q^{2} + 94 q^{3} + 73 q^{4} - 14 q^{5} - 13 q^{6} - 55 q^{7} - 36 q^{8} + 94 q^{9} - 39 q^{10} - 49 q^{11} + 73 q^{12} - 52 q^{13} - 7 q^{14} - 14 q^{15} + 43 q^{16} - 22 q^{17} - 13 q^{18} - 89 q^{19} - 22 q^{20} - 55 q^{21} - 36 q^{22} - 46 q^{23} - 36 q^{24} + 18 q^{25} + q^{26} + 94 q^{27} - 123 q^{28} - 20 q^{29} - 39 q^{30} - 61 q^{31} - 65 q^{32} - 49 q^{33} - 67 q^{34} - 40 q^{35} + 73 q^{36} - 83 q^{37} - 19 q^{38} - 52 q^{39} - 101 q^{40} - 25 q^{41} - 7 q^{42} - 150 q^{43} - 71 q^{44} - 14 q^{45} - 72 q^{46} - 39 q^{47} + 43 q^{48} - q^{49} - 45 q^{50} - 22 q^{51} - 110 q^{52} - 30 q^{53} - 13 q^{54} - 54 q^{55} - 5 q^{56} - 89 q^{57} - 77 q^{58} - 43 q^{59} - 22 q^{60} - 109 q^{61} - 33 q^{62} - 55 q^{63} + 10 q^{64} - 66 q^{65} - 36 q^{66} - 155 q^{67} - 46 q^{68} - 46 q^{69} - 43 q^{70} - 27 q^{71} - 36 q^{72} - 157 q^{73} - 29 q^{74} + 18 q^{75} - 176 q^{76} - 9 q^{77} + q^{78} - 99 q^{79} - 18 q^{80} + 94 q^{81} - 53 q^{82} - 144 q^{83} - 123 q^{84} - 105 q^{85} + 23 q^{86} - 20 q^{87} - 88 q^{88} - 4 q^{89} - 39 q^{90} - 99 q^{91} - 76 q^{92} - 61 q^{93} - 65 q^{94} - 49 q^{95} - 65 q^{96} - 139 q^{97} - 6 q^{98} - 49 q^{99}+O(q^{100}) \) 94 * q - 13 * q^2 + 94 * q^3 + 73 * q^4 - 14 * q^5 - 13 * q^6 - 55 * q^7 - 36 * q^8 + 94 * q^9 - 39 * q^10 - 49 * q^11 + 73 * q^12 - 52 * q^13 - 7 * q^14 - 14 * q^15 + 43 * q^16 - 22 * q^17 - 13 * q^18 - 89 * q^19 - 22 * q^20 - 55 * q^21 - 36 * q^22 - 46 * q^23 - 36 * q^24 + 18 * q^25 + q^26 + 94 * q^27 - 123 * q^28 - 20 * q^29 - 39 * q^30 - 61 * q^31 - 65 * q^32 - 49 * q^33 - 67 * q^34 - 40 * q^35 + 73 * q^36 - 83 * q^37 - 19 * q^38 - 52 * q^39 - 101 * q^40 - 25 * q^41 - 7 * q^42 - 150 * q^43 - 71 * q^44 - 14 * q^45 - 72 * q^46 - 39 * q^47 + 43 * q^48 - q^49 - 45 * q^50 - 22 * q^51 - 110 * q^52 - 30 * q^53 - 13 * q^54 - 54 * q^55 - 5 * q^56 - 89 * q^57 - 77 * q^58 - 43 * q^59 - 22 * q^60 - 109 * q^61 - 33 * q^62 - 55 * q^63 + 10 * q^64 - 66 * q^65 - 36 * q^66 - 155 * q^67 - 46 * q^68 - 46 * q^69 - 43 * q^70 - 27 * q^71 - 36 * q^72 - 157 * q^73 - 29 * q^74 + 18 * q^75 - 176 * q^76 - 9 * q^77 + q^78 - 99 * q^79 - 18 * q^80 + 94 * q^81 - 53 * q^82 - 144 * q^83 - 123 * q^84 - 105 * q^85 + 23 * q^86 - 20 * q^87 - 88 * q^88 - 4 * q^89 - 39 * q^90 - 99 * q^91 - 76 * q^92 - 61 * q^93 - 65 * q^94 - 49 * q^95 - 65 * q^96 - 139 * q^97 - 6 * q^98 - 49 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more