LMFDB - Embedded newform 6031.2.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6031,2,Mod(1,6031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6031, 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("6031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6031 = 37 \cdot 163 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6031.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:	$48.1577774590$
Analytic rank:	$0$
Dimension:	$134$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	6031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56681 q^{2} +3.22691 q^{3} +4.58852 q^{4} +1.32695 q^{5} -8.28287 q^{6} +4.31369 q^{7} -6.64424 q^{8} +7.41296 q^{9} +O(q^{10})$	\(q-2.56681 q^{2} +3.22691 q^{3} +4.58852 q^{4} +1.32695 q^{5} -8.28287 q^{6} +4.31369 q^{7} -6.64424 q^{8} +7.41296 q^{9} -3.40604 q^{10} +5.32174 q^{11} +14.8068 q^{12} -3.79279 q^{13} -11.0724 q^{14} +4.28196 q^{15} +7.87748 q^{16} +0.303616 q^{17} -19.0277 q^{18} +4.34131 q^{19} +6.08875 q^{20} +13.9199 q^{21} -13.6599 q^{22} -0.255081 q^{23} -21.4404 q^{24} -3.23920 q^{25} +9.73536 q^{26} +14.2402 q^{27} +19.7935 q^{28} +5.14825 q^{29} -10.9910 q^{30} -3.14750 q^{31} -6.93152 q^{32} +17.1728 q^{33} -0.779324 q^{34} +5.72406 q^{35} +34.0145 q^{36} +1.00000 q^{37} -11.1433 q^{38} -12.2390 q^{39} -8.81660 q^{40} -4.41400 q^{41} -35.7298 q^{42} -7.37098 q^{43} +24.4189 q^{44} +9.83664 q^{45} +0.654745 q^{46} -5.42960 q^{47} +25.4199 q^{48} +11.6079 q^{49} +8.31441 q^{50} +0.979741 q^{51} -17.4033 q^{52} -6.99688 q^{53} -36.5520 q^{54} +7.06170 q^{55} -28.6612 q^{56} +14.0090 q^{57} -13.2146 q^{58} -4.89294 q^{59} +19.6479 q^{60} +15.5322 q^{61} +8.07904 q^{62} +31.9772 q^{63} +2.03694 q^{64} -5.03285 q^{65} -44.0793 q^{66} +0.480091 q^{67} +1.39315 q^{68} -0.823124 q^{69} -14.6926 q^{70} +4.37475 q^{71} -49.2535 q^{72} -1.00131 q^{73} -2.56681 q^{74} -10.4526 q^{75} +19.9202 q^{76} +22.9563 q^{77} +31.4152 q^{78} -16.2079 q^{79} +10.4530 q^{80} +23.7131 q^{81} +11.3299 q^{82} +3.86973 q^{83} +63.8717 q^{84} +0.402883 q^{85} +18.9199 q^{86} +16.6129 q^{87} -35.3589 q^{88} -1.57119 q^{89} -25.2488 q^{90} -16.3609 q^{91} -1.17044 q^{92} -10.1567 q^{93} +13.9368 q^{94} +5.76072 q^{95} -22.3674 q^{96} -11.8085 q^{97} -29.7954 q^{98} +39.4498 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 134 q + 9 q^{2} + 7 q^{3} + 149 q^{4} + 22 q^{5} + 20 q^{6} + 11 q^{7} + 27 q^{8} + 181 q^{9}+O(q^{10}) $ 134 * q + 9 * q^2 + 7 * q^3 + 149 * q^4 + 22 * q^5 + 20 * q^6 + 11 * q^7 + 27 * q^8 + 181 * q^9	\( 134 q + 9 q^{2} + 7 q^{3} + 149 q^{4} + 22 q^{5} + 20 q^{6} + 11 q^{7} + 27 q^{8} + 181 q^{9} + 15 q^{10} + 20 q^{11} + 28 q^{12} + 11 q^{13} + 17 q^{14} - 13 q^{15} + 143 q^{16} + 76 q^{17} + 23 q^{18} + 15 q^{19} + 67 q^{20} + 63 q^{21} + 2 q^{22} + 22 q^{23} + 33 q^{24} + 160 q^{25} + 65 q^{26} + 31 q^{27} + 10 q^{28} + 73 q^{29} + 20 q^{30} + 10 q^{31} + 53 q^{32} + 72 q^{33} - 7 q^{34} + 52 q^{35} + 201 q^{36} + 134 q^{37} + 70 q^{38} + 6 q^{39} + 11 q^{40} + 182 q^{41} - 15 q^{42} + 12 q^{43} + 33 q^{44} + 29 q^{45} + 24 q^{46} + 80 q^{47} + 21 q^{48} + 229 q^{49} + 37 q^{50} + 57 q^{51} - 15 q^{52} + 75 q^{53} + 95 q^{54} - 9 q^{55} + 39 q^{56} + 19 q^{57} - 21 q^{58} + 91 q^{59} + 62 q^{60} + 58 q^{61} + 108 q^{62} + 9 q^{63} + 167 q^{64} + 76 q^{65} + 105 q^{66} - 17 q^{67} + 109 q^{68} + 48 q^{69} - 55 q^{70} + 56 q^{71} + 48 q^{72} + 54 q^{73} + 9 q^{74} + 28 q^{75} + 82 q^{76} + 156 q^{77} + 16 q^{78} - 2 q^{79} + 98 q^{80} + 270 q^{81} - 42 q^{82} + 130 q^{83} + 229 q^{84} + 22 q^{85} + 72 q^{86} + 22 q^{87} + 61 q^{88} + 157 q^{89} + 176 q^{90} + 31 q^{91} - 18 q^{92} + 36 q^{93} + 83 q^{94} + 98 q^{95} + 111 q^{96} + 35 q^{97} + 53 q^{98} + 55 q^{99}+O(q^{100}) \) 134 * q + 9 * q^2 + 7 * q^3 + 149 * q^4 + 22 * q^5 + 20 * q^6 + 11 * q^7 + 27 * q^8 + 181 * q^9 + 15 * q^10 + 20 * q^11 + 28 * q^12 + 11 * q^13 + 17 * q^14 - 13 * q^15 + 143 * q^16 + 76 * q^17 + 23 * q^18 + 15 * q^19 + 67 * q^20 + 63 * q^21 + 2 * q^22 + 22 * q^23 + 33 * q^24 + 160 * q^25 + 65 * q^26 + 31 * q^27 + 10 * q^28 + 73 * q^29 + 20 * q^30 + 10 * q^31 + 53 * q^32 + 72 * q^33 - 7 * q^34 + 52 * q^35 + 201 * q^36 + 134 * q^37 + 70 * q^38 + 6 * q^39 + 11 * q^40 + 182 * q^41 - 15 * q^42 + 12 * q^43 + 33 * q^44 + 29 * q^45 + 24 * q^46 + 80 * q^47 + 21 * q^48 + 229 * q^49 + 37 * q^50 + 57 * q^51 - 15 * q^52 + 75 * q^53 + 95 * q^54 - 9 * q^55 + 39 * q^56 + 19 * q^57 - 21 * q^58 + 91 * q^59 + 62 * q^60 + 58 * q^61 + 108 * q^62 + 9 * q^63 + 167 * q^64 + 76 * q^65 + 105 * q^66 - 17 * q^67 + 109 * q^68 + 48 * q^69 - 55 * q^70 + 56 * q^71 + 48 * q^72 + 54 * q^73 + 9 * q^74 + 28 * q^75 + 82 * q^76 + 156 * q^77 + 16 * q^78 - 2 * q^79 + 98 * q^80 + 270 * q^81 - 42 * q^82 + 130 * q^83 + 229 * q^84 + 22 * q^85 + 72 * q^86 + 22 * q^87 + 61 * q^88 + 157 * q^89 + 176 * q^90 + 31 * q^91 - 18 * q^92 + 36 * q^93 + 83 * q^94 + 98 * q^95 + 111 * q^96 + 35 * q^97 + 53 * q^98 + 55 * 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 $

$2$	−2.56681	−1.81501	−0.907505	−	0.420042i	$-0.862015\pi$
$2$	−2.56681	−1.81501	−0.907505	+	0.420042i	$0.862015\pi$
$3$	3.22691	1.86306	0.931529	−	0.363667i	$-0.118475\pi$
$3$	3.22691	1.86306	0.931529	+	0.363667i	$0.118475\pi$
$4$	4.58852	2.29426
$5$	1.32695	0.593431	0.296716	−	0.954966i	$-0.404109\pi$
$5$	1.32695	0.593431	0.296716	+	0.954966i	$0.404109\pi$
$6$	−8.28287	−3.38147
$7$	4.31369	1.63042	0.815211	−	0.579164i	$-0.196621\pi$
$7$	4.31369	1.63042	0.815211	+	0.579164i	$0.196621\pi$
$8$	−6.64424	−2.34910
$9$	7.41296	2.47099
$10$	−3.40604	−1.07708
$11$	5.32174	1.60457	0.802283	−	0.596944i	$-0.203619\pi$
$11$	5.32174	1.60457	0.802283	+	0.596944i	$0.203619\pi$
$12$	14.8068	4.27434
$13$	−3.79279	−1.05193	−0.525965	−	0.850506i	$-0.676296\pi$
$13$	−3.79279	−1.05193	−0.525965	+	0.850506i	$0.676296\pi$
$14$	−11.0724	−2.95923
$15$	4.28196	1.10560
$16$	7.87748	1.96937
$17$	0.303616	0.0736376	0.0368188	−	0.999322i	$-0.488278\pi$
$17$	0.303616	0.0736376	0.0368188	+	0.999322i	$0.488278\pi$
$18$	−19.0277	−4.48486
$19$	4.34131	0.995966	0.497983	−	0.867187i	$-0.334074\pi$
$19$	4.34131	0.995966	0.497983	+	0.867187i	$0.334074\pi$
$20$	6.08875	1.36149
$21$	13.9199	3.03757
$22$	−13.6599	−2.91230
$23$	−0.255081	−0.0531880	−0.0265940	−	0.999646i	$-0.508466\pi$
$23$	−0.255081	−0.0531880	−0.0265940	+	0.999646i	$0.508466\pi$
$24$	−21.4404	−4.37650
$25$	−3.23920	−0.647839
$26$	9.73536	1.90926
$27$	14.2402	2.74053
$28$	19.7935	3.74061
$29$	5.14825	0.956006	0.478003	−	0.878358i	$-0.341361\pi$
$29$	5.14825	0.956006	0.478003	+	0.878358i	$0.341361\pi$
$30$	−10.9910	−2.00667
$31$	−3.14750	−0.565308	−0.282654	−	0.959222i	$-0.591215\pi$
$31$	−3.14750	−0.565308	−0.282654	+	0.959222i	$0.591215\pi$
$32$	−6.93152	−1.22533
$33$	17.1728	2.98940
$34$	−0.779324	−0.133653
$35$	5.72406	0.967543
$36$	34.0145	5.66909
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−11.1433	−1.80769
$39$	−12.2390	−1.95981
$40$	−8.81660	−1.39403
$41$	−4.41400	−0.689351	−0.344676	−	0.938722i	$-0.612011\pi$
$41$	−4.41400	−0.689351	−0.344676	+	0.938722i	$0.612011\pi$
$42$	−35.7298	−5.51322
$43$	−7.37098	−1.12406	−0.562032	−	0.827115i	$-0.689980\pi$
$43$	−7.37098	−1.12406	−0.562032	+	0.827115i	$0.689980\pi$
$44$	24.4189	3.68129
$45$	9.83664	1.46636
$46$	0.654745	0.0965368
$47$	−5.42960	−0.791989	−0.395994	−	0.918253i	$-0.629600\pi$
$47$	−5.42960	−0.791989	−0.395994	+	0.918253i	$0.629600\pi$
$48$	25.4199	3.66905
$49$	11.6079	1.65828
$50$	8.31441	1.17583
$51$	0.979741	0.137191
$52$	−17.4033	−2.41340
$53$	−6.99688	−0.961095	−0.480547	−	0.876969i	$-0.659562\pi$
$53$	−6.99688	−0.961095	−0.480547	+	0.876969i	$0.659562\pi$
$54$	−36.5520	−4.97409
$55$	7.06170	0.952199
$56$	−28.6612	−3.83002
$57$	14.0090	1.85554
$58$	−13.2146	−1.73516
$59$	−4.89294	−0.637007	−0.318504	−	0.947922i	$-0.603180\pi$
$59$	−4.89294	−0.637007	−0.318504	+	0.947922i	$0.603180\pi$
$60$	19.6479	2.53653
$61$	15.5322	1.98870	0.994348	−	0.106172i	$-0.0338595\pi$
$61$	15.5322	1.98870	0.994348	+	0.106172i	$0.0338595\pi$
$62$	8.07904	1.02604
$63$	31.9772	4.02875
$64$	2.03694	0.254617
$65$	−5.03285	−0.624248
$66$	−44.0793	−5.42579
$67$	0.480091	0.0586525	0.0293262	−	0.999570i	$-0.490664\pi$
$67$	0.480091	0.0586525	0.0293262	+	0.999570i	$0.490664\pi$
$68$	1.39315	0.168944
$69$	−0.823124	−0.0990924
$70$	−14.6926	−1.75610
$71$	4.37475	0.519187	0.259593	−	0.965718i	$-0.416411\pi$
$71$	4.37475	0.519187	0.259593	+	0.965718i	$0.416411\pi$
$72$	−49.2535	−5.80458
$73$	−1.00131	−0.117195	−0.0585973	−	0.998282i	$-0.518663\pi$
$73$	−1.00131	−0.117195	−0.0585973	+	0.998282i	$0.518663\pi$
$74$	−2.56681	−0.298386
$75$	−10.4526	−1.20696
$76$	19.9202	2.28500
$77$	22.9563	2.61612
$78$	31.4152	3.55707
$79$	−16.2079	−1.82353	−0.911767	−	0.410709i	$-0.865281\pi$
$79$	−16.2079	−1.82353	−0.911767	+	0.410709i	$0.865281\pi$
$80$	10.4530	1.16869
$81$	23.7131	2.63479
$82$	11.3299	1.25118
$83$	3.86973	0.424758	0.212379	−	0.977187i	$-0.431879\pi$
$83$	3.86973	0.424758	0.212379	+	0.977187i	$0.431879\pi$
$84$	63.8717	6.96898
$85$	0.402883	0.0436988
$86$	18.9199	2.04019
$87$	16.6129	1.78109
$88$	−35.3589	−3.76928
$89$	−1.57119	−0.166545	−0.0832727	−	0.996527i	$-0.526537\pi$
$89$	−1.57119	−0.166545	−0.0832727	+	0.996527i	$0.526537\pi$
$90$	−25.2488	−2.66146
$91$	−16.3609	−1.71509
$92$	−1.17044	−0.122027
$93$	−10.1567	−1.05320
$94$	13.9368	1.43747
$95$	5.76072	0.591037
$96$	−22.3674	−2.28286
$97$	−11.8085	−1.19897	−0.599483	−	0.800387i	$-0.704627\pi$
$97$	−11.8085	−1.19897	−0.599483	+	0.800387i	$0.704627\pi$
$98$	−29.7954	−3.00979
$99$	39.4498	3.96486
$100$	−14.8631	−1.48631
$101$	1.92876	0.191919	0.0959594	−	0.995385i	$-0.469408\pi$
$101$	1.92876	0.191919	0.0959594	+	0.995385i	$0.469408\pi$
$102$	−2.51481	−0.249003
$103$	−19.1745	−1.88932	−0.944658	−	0.328058i	$-0.893606\pi$
$103$	−19.1745	−1.88932	−0.944658	+	0.328058i	$0.893606\pi$
$104$	25.2002	2.47108
$105$	18.4710	1.80259
$106$	17.9597	1.74440
$107$	−1.41095	−0.136402	−0.0682010	−	0.997672i	$-0.521726\pi$
$107$	−1.41095	−0.136402	−0.0682010	+	0.997672i	$0.521726\pi$
$108$	65.3416	6.28750
$109$	1.28686	0.123259	0.0616294	−	0.998099i	$-0.480370\pi$
$109$	1.28686	0.123259	0.0616294	+	0.998099i	$0.480370\pi$
$110$	−18.1260	−1.72825
$111$	3.22691	0.306285
$112$	33.9810	3.21090
$113$	17.2554	1.62325	0.811625	−	0.584179i	$-0.198583\pi$
$113$	17.2554	1.62325	0.811625	+	0.584179i	$0.198583\pi$
$114$	−35.9586	−3.36783
$115$	−0.338480	−0.0315634
$116$	23.6228	2.19333
$117$	−28.1158	−2.59930
$118$	12.5593	1.15617
$119$	1.30970	0.120060
$120$	−28.4504	−2.59715
$121$	17.3209	1.57463
$122$	−39.8682	−3.60950
$123$	−14.2436	−1.28430
$124$	−14.4424	−1.29696
$125$	−10.9330	−0.977879
$126$	−82.0795	−7.31222
$127$	−18.4630	−1.63833	−0.819164	−	0.573560i	$-0.805562\pi$
$127$	−18.4630	−1.63833	−0.819164	+	0.573560i	$0.805562\pi$
$128$	8.63460	0.763198
$129$	−23.7855	−2.09420
$130$	12.9184	1.13302
$131$	9.28089	0.810875	0.405438	−	0.914123i	$-0.367119\pi$
$131$	9.28089	0.810875	0.405438	+	0.914123i	$0.367119\pi$
$132$	78.7977	6.85846
$133$	18.7271	1.62384
$134$	−1.23230	−0.106455
$135$	18.8961	1.62632
$136$	−2.01730	−0.172982
$137$	7.97006	0.680928	0.340464	−	0.940257i	$-0.389416\pi$
$137$	7.97006	0.680928	0.340464	+	0.940257i	$0.389416\pi$
$138$	2.11280	0.179854
$139$	3.17430	0.269240	0.134620	−	0.990897i	$-0.457019\pi$
$139$	3.17430	0.269240	0.134620	+	0.990897i	$0.457019\pi$
$140$	26.2650	2.21980
$141$	−17.5208	−1.47552
$142$	−11.2292	−0.942329
$143$	−20.1842	−1.68789
$144$	58.3954	4.86629
$145$	6.83148	0.567324
$146$	2.57018	0.212709
$147$	37.4578	3.08946
$148$	4.58852	0.377174
$149$	−10.5489	−0.864198	−0.432099	−	0.901826i	$-0.642227\pi$
$149$	−10.5489	−0.864198	−0.432099	+	0.901826i	$0.642227\pi$
$150$	26.8299	2.19065
$151$	13.9053	1.13160	0.565798	−	0.824544i	$-0.308568\pi$
$151$	13.9053	1.13160	0.565798	+	0.824544i	$0.308568\pi$
$152$	−28.8448	−2.33962
$153$	2.25069	0.181957
$154$	−58.9246	−4.74828
$155$	−4.17658	−0.335471
$156$	−56.1588	−4.49630
$157$	−15.0720	−1.20288	−0.601438	−	0.798919i	$-0.705405\pi$
$157$	−15.0720	−1.20288	−0.601438	+	0.798919i	$0.705405\pi$
$158$	41.6027	3.30973
$159$	−22.5783	−1.79058
$160$	−9.19780	−0.727150
$161$	−1.10034	−0.0867190
$162$	−60.8670	−4.78216
$163$	−1.00000	−0.0783260
$163$	−1.00000	−0.0783260
$164$	−20.2537	−1.58155
$165$	22.7875	1.77400
$166$	−9.93287	−0.770940
$167$	−18.0619	−1.39767	−0.698837	−	0.715281i	$-0.746299\pi$
$167$	−18.0619	−1.39767	−0.698837	+	0.715281i	$0.746299\pi$
$168$	−92.4872	−7.13554
$169$	1.38522	0.106555
$170$	−1.03413	−0.0793138
$171$	32.1820	2.46102
$172$	−33.8219	−2.57890
$173$	9.39970	0.714646	0.357323	−	0.933981i	$-0.383690\pi$
$173$	9.39970	0.714646	0.357323	+	0.933981i	$0.383690\pi$
$174$	−42.6423	−3.23270
$175$	−13.9729	−1.05625
$176$	41.9219	3.15998
$177$	−15.7891	−1.18678
$178$	4.03294	0.302282
$179$	−15.3660	−1.14851	−0.574255	−	0.818677i	$-0.694708\pi$
$179$	−15.3660	−1.14851	−0.574255	+	0.818677i	$0.694708\pi$
$180$	45.1356	3.36421
$181$	2.43933	0.181314	0.0906568	−	0.995882i	$-0.471103\pi$
$181$	2.43933	0.181314	0.0906568	+	0.995882i	$0.471103\pi$
$182$	41.9954	3.11290
$183$	50.1211	3.70506
$184$	1.69482	0.124944
$185$	1.32695	0.0975595
$186$	26.0704	1.91157
$187$	1.61576	0.118156
$188$	−24.9138	−1.81703
$189$	61.4279	4.46823
$190$	−14.7867	−1.07274
$191$	−3.06229	−0.221580	−0.110790	−	0.993844i	$-0.535338\pi$
$191$	−3.06229	−0.221580	−0.110790	+	0.993844i	$0.535338\pi$
$192$	6.57302	0.474367
$193$	8.64374	0.622190	0.311095	−	0.950379i	$-0.399304\pi$
$193$	8.64374	0.622190	0.311095	+	0.950379i	$0.399304\pi$
$194$	30.3101	2.17614
$195$	−16.2405	−1.16301
$196$	53.2632	3.80452
$197$	15.1766	1.08129	0.540645	−	0.841251i	$-0.318180\pi$
$197$	15.1766	1.08129	0.540645	+	0.841251i	$0.318180\pi$
$198$	−101.260	−7.19626
$199$	25.6870	1.82090	0.910451	−	0.413618i	$-0.135735\pi$
$199$	25.6870	1.82090	0.910451	+	0.413618i	$0.135735\pi$
$200$	21.5220	1.52184
$201$	1.54921	0.109273
$202$	−4.95076	−0.348334
$203$	22.2080	1.55869
$204$	4.49556	0.314752
$205$	−5.85717	−0.409083
$206$	49.2172	3.42913
$207$	−1.89090	−0.131427
$208$	−29.8776	−2.07164
$209$	23.1033	1.59809
$210$	−47.4117	−3.27172
$211$	−22.0413	−1.51738	−0.758692	−	0.651449i	$-0.774161\pi$
$211$	−22.0413	−1.51738	−0.758692	+	0.651449i	$0.774161\pi$
$212$	−32.1053	−2.20500
$213$	14.1169	0.967276
$214$	3.62165	0.247571
$215$	−9.78094	−0.667055
$216$	−94.6156	−6.43777
$217$	−13.5773	−0.921690
$218$	−3.30313	−0.223716
$219$	−3.23114	−0.218341
$220$	32.4027	2.18459
$221$	−1.15155	−0.0774615
$222$	−8.28287	−0.555910
$223$	3.92755	0.263008	0.131504	−	0.991316i	$-0.458019\pi$
$223$	3.92755	0.263008	0.131504	+	0.991316i	$0.458019\pi$
$224$	−29.9004	−1.99781
$225$	−24.0120	−1.60080
$226$	−44.2913	−2.94621
$227$	20.7042	1.37419	0.687094	−	0.726569i	$-0.258886\pi$
$227$	20.7042	1.37419	0.687094	+	0.726569i	$0.258886\pi$
$228$	64.2808	4.25710
$229$	2.96623	0.196014	0.0980068	−	0.995186i	$-0.468753\pi$
$229$	2.96623	0.196014	0.0980068	+	0.995186i	$0.468753\pi$
$230$	0.868815	0.0572880
$231$	74.0781	4.87398
$232$	−34.2062	−2.24575
$233$	−19.6357	−1.28638	−0.643189	−	0.765707i	$-0.722389\pi$
$233$	−19.6357	−1.28638	−0.643189	+	0.765707i	$0.722389\pi$
$234$	72.1679	4.71776
$235$	−7.20482	−0.469991
$236$	−22.4514	−1.46146
$237$	−52.3015	−3.39735
$238$	−3.36176	−0.217911
$239$	5.99087	0.387517	0.193759	−	0.981049i	$-0.437932\pi$
$239$	5.99087	0.387517	0.193759	+	0.981049i	$0.437932\pi$
$240$	33.7310	2.17733
$241$	13.3576	0.860438	0.430219	−	0.902725i	$-0.358436\pi$
$241$	13.3576	0.860438	0.430219	+	0.902725i	$0.358436\pi$
$242$	−44.4595	−2.85797
$243$	33.7993	2.16823
$244$	71.2699	4.56259
$245$	15.4032	0.984073
$246$	36.5606	2.33102
$247$	−16.4657	−1.04769
$248$	20.9128	1.32796
$249$	12.4873	0.791349
$250$	28.0630	1.77486
$251$	−4.04531	−0.255337	−0.127669	−	0.991817i	$-0.540749\pi$
$251$	−4.04531	−0.255337	−0.127669	+	0.991817i	$0.540749\pi$
$252$	146.728	9.24300
$253$	−1.35747	−0.0853437
$254$	47.3911	2.97358
$255$	1.30007	0.0814135
$256$	−26.2373	−1.63983
$257$	5.49045	0.342485	0.171243	−	0.985229i	$-0.445222\pi$
$257$	5.49045	0.342485	0.171243	+	0.985229i	$0.445222\pi$
$258$	61.0529	3.80099
$259$	4.31369	0.268040
$260$	−23.0933	−1.43219
$261$	38.1638	2.36228
$262$	−23.8223	−1.47175
$263$	13.1980	0.813822	0.406911	−	0.913468i	$-0.366606\pi$
$263$	13.1980	0.813822	0.406911	+	0.913468i	$0.366606\pi$
$264$	−114.100	−7.02238
$265$	−9.28452	−0.570344
$266$	−48.0689	−2.94729
$267$	−5.07008	−0.310284
$268$	2.20291	0.134564
$269$	19.6261	1.19663	0.598314	−	0.801262i	$-0.295838\pi$
$269$	19.6261	1.19663	0.598314	+	0.801262i	$0.295838\pi$
$270$	−48.5027	−2.95178
$271$	−0.225110	−0.0136744	−0.00683722	−	0.999977i	$-0.502176\pi$
$271$	−0.225110	−0.0136744	−0.00683722	+	0.999977i	$0.502176\pi$
$272$	2.39173	0.145020
$273$	−52.7952	−3.19531
$274$	−20.4576	−1.23589
$275$	−17.2382	−1.03950
$276$	−3.77692	−0.227344
$277$	−22.1504	−1.33089	−0.665443	−	0.746448i	$-0.731757\pi$
$277$	−22.1504	−1.33089	−0.665443	+	0.746448i	$0.731757\pi$
$278$	−8.14783	−0.488674
$279$	−23.3323	−1.39687
$280$	−38.0321	−2.27285
$281$	18.8480	1.12438	0.562189	−	0.827009i	$-0.309959\pi$
$281$	18.8480	1.12438	0.562189	+	0.827009i	$0.309959\pi$
$282$	44.9727	2.67809
$283$	−31.5909	−1.87788	−0.938942	−	0.344075i	$-0.888193\pi$
$283$	−31.5909	−1.87788	−0.938942	+	0.344075i	$0.888193\pi$
$284$	20.0736	1.19115
$285$	18.5893	1.10114
$286$	51.8091	3.06354
$287$	−19.0406	−1.12393
$288$	−51.3831	−3.02778
$289$	−16.9078	−0.994578
$290$	−17.5351	−1.02970
$291$	−38.1048	−2.23375
$292$	−4.59454	−0.268875
$293$	−3.62288	−0.211651	−0.105826	−	0.994385i	$-0.533749\pi$
$293$	−3.62288	−0.211651	−0.105826	+	0.994385i	$0.533749\pi$
$294$	−96.1470	−5.60741
$295$	−6.49270	−0.378020
$296$	−6.64424	−0.386189
$297$	75.7828	4.39736
$298$	27.0770	1.56853
$299$	0.967467	0.0559501
$300$	−47.9620	−2.76909
$301$	−31.7961	−1.83270
$302$	−35.6922	−2.05386
$303$	6.22394	0.357556
$304$	34.1986	1.96143
$305$	20.6105	1.18015
$306$	−5.77710	−0.330255
$307$	15.1040	0.862030	0.431015	−	0.902345i	$-0.358156\pi$
$307$	15.1040	0.862030	0.431015	+	0.902345i	$0.358156\pi$
$308$	105.336	6.00206
$309$	−61.8743	−3.51990
$310$	10.7205	0.608884
$311$	20.5024	1.16258	0.581292	−	0.813695i	$-0.302547\pi$
$311$	20.5024	1.16258	0.581292	+	0.813695i	$0.302547\pi$
$312$	81.3188	4.60377
$313$	−24.3603	−1.37693	−0.688464	−	0.725271i	$-0.741715\pi$
$313$	−24.3603	−1.37693	−0.688464	+	0.725271i	$0.741715\pi$
$314$	38.6870	2.18323
$315$	42.4322	2.39079
$316$	−74.3704	−4.18366
$317$	5.03600	0.282850	0.141425	−	0.989949i	$-0.454832\pi$
$317$	5.03600	0.282850	0.141425	+	0.989949i	$0.454832\pi$
$318$	57.9542	3.24991
$319$	27.3976	1.53397
$320$	2.70292	0.151098
$321$	−4.55302	−0.254125
$322$	2.82437	0.157396
$323$	1.31809	0.0733405
$324$	108.808	6.04489
$325$	12.2856	0.681481
$326$	2.56681	0.142163
$327$	4.15258	0.229638
$328$	29.3277	1.61935
$329$	−23.4216	−1.29128
$330$	−58.4911	−3.21983
$331$	−9.66676	−0.531333	−0.265667	−	0.964065i	$-0.585592\pi$
$331$	−9.66676	−0.531333	−0.265667	+	0.964065i	$0.585592\pi$
$332$	17.7563	0.974505
$333$	7.41296	0.406228
$334$	46.3616	2.53679
$335$	0.637058	0.0348062
$336$	109.654	5.98210
$337$	−21.1371	−1.15141	−0.575704	−	0.817658i	$-0.695272\pi$
$337$	−21.1371	−1.15141	−0.575704	+	0.817658i	$0.695272\pi$
$338$	−3.55560	−0.193399
$339$	55.6816	3.02421
$340$	1.84864	0.100257
$341$	−16.7502	−0.907073
$342$	−82.6051	−4.46677
$343$	19.8772	1.07327
$344$	48.9746	2.64053
$345$	−1.09225	−0.0588045
$346$	−24.1273	−1.29709
$347$	21.1011	1.13277	0.566383	−	0.824142i	$-0.308342\pi$
$347$	21.1011	1.13277	0.566383	+	0.824142i	$0.308342\pi$
$348$	76.2288	4.08629
$349$	−11.9728	−0.640889	−0.320445	−	0.947267i	$-0.603832\pi$
$349$	−11.9728	−0.640889	−0.320445	+	0.947267i	$0.603832\pi$
$350$	35.8658	1.91711
$351$	−54.0101	−2.88285
$352$	−36.8877	−1.96612
$353$	−5.28058	−0.281057	−0.140528	−	0.990077i	$-0.544880\pi$
$353$	−5.28058	−0.281057	−0.140528	+	0.990077i	$0.544880\pi$
$354$	40.5276	2.15402
$355$	5.80508	0.308102
$356$	−7.20942	−0.382099
$357$	4.22630	0.223679
$358$	39.4416	2.08456
$359$	−5.99284	−0.316290	−0.158145	−	0.987416i	$-0.550551\pi$
$359$	−5.99284	−0.316290	−0.158145	+	0.987416i	$0.550551\pi$
$360$	−65.3571	−3.44462
$361$	−0.152994	−0.00805229
$362$	−6.26129	−0.329086
$363$	55.8931	2.93363
$364$	−75.0723	−3.93486
$365$	−1.32869	−0.0695470
$366$	−128.651	−6.72471
$367$	−6.28430	−0.328038	−0.164019	−	0.986457i	$-0.552446\pi$
$367$	−6.28430	−0.328038	−0.164019	+	0.986457i	$0.552446\pi$
$368$	−2.00939	−0.104747
$369$	−32.7208	−1.70338
$370$	−3.40604	−0.177071
$371$	−30.1824	−1.56699
$372$	−46.6043	−2.41632
$373$	−20.1660	−1.04416	−0.522078	−	0.852898i	$-0.674843\pi$
$373$	−20.1660	−1.04416	−0.522078	+	0.852898i	$0.674843\pi$
$374$	−4.14736	−0.214455
$375$	−35.2799	−1.82185
$376$	36.0756	1.86046
$377$	−19.5262	−1.00565
$378$	−157.674	−8.10987
$379$	6.75008	0.346729	0.173364	−	0.984858i	$-0.444536\pi$
$379$	6.75008	0.346729	0.173364	+	0.984858i	$0.444536\pi$
$380$	26.4332	1.35599
$381$	−59.5785	−3.05230
$382$	7.86033	0.402169
$383$	−19.7201	−1.00765	−0.503824	−	0.863806i	$-0.668074\pi$
$383$	−19.7201	−1.00765	−0.503824	+	0.863806i	$0.668074\pi$
$384$	27.8631	1.42188
$385$	30.4620	1.55249
$386$	−22.1868	−1.12928
$387$	−54.6408	−2.77755
$388$	−54.1833	−2.75074
$389$	28.8855	1.46455	0.732275	−	0.681009i	$-0.238458\pi$
$389$	28.8855	1.46455	0.732275	+	0.681009i	$0.238458\pi$
$390$	41.6864	2.11087
$391$	−0.0774465	−0.00391664
$392$	−77.1259	−3.89545
$393$	29.9486	1.51071
$394$	−38.9555	−1.96255
$395$	−21.5071	−1.08214
$396$	181.016	9.09642
$397$	−34.1951	−1.71620	−0.858102	−	0.513480i	$-0.828356\pi$
$397$	−34.1951	−1.71620	−0.858102	+	0.513480i	$0.828356\pi$
$398$	−65.9336	−3.30495
$399$	60.4307	3.02532
$400$	−25.5167	−1.27584
$401$	−31.6363	−1.57984	−0.789920	−	0.613210i	$-0.789878\pi$
$401$	−31.6363	−1.57984	−0.789920	+	0.613210i	$0.789878\pi$
$402$	−3.97653	−0.198331
$403$	11.9378	0.594664
$404$	8.85015	0.440312
$405$	31.4661	1.56356
$406$	−57.0036	−2.82904
$407$	5.32174	0.263789
$408$	−6.50964	−0.322275
$409$	−12.9836	−0.641996	−0.320998	−	0.947080i	$-0.604018\pi$
$409$	−12.9836	−0.641996	−0.320998	+	0.947080i	$0.604018\pi$
$410$	15.0343	0.742489
$411$	25.7187	1.26861
$412$	−87.9824	−4.33458
$413$	−21.1067	−1.03859
$414$	4.85359	0.238541
$415$	5.13495	0.252065
$416$	26.2898	1.28896
$417$	10.2432	0.501611
$418$	−59.3019	−2.90055
$419$	−21.8272	−1.06633	−0.533165	−	0.846011i	$-0.678998\pi$
$419$	−21.8272	−1.06633	−0.533165	+	0.846011i	$0.678998\pi$
$420$	84.7548	4.13561
$421$	−25.1587	−1.22616	−0.613080	−	0.790021i	$-0.710070\pi$
$421$	−25.1587	−1.22616	−0.613080	+	0.790021i	$0.710070\pi$
$422$	56.5758	2.75407
$423$	−40.2494	−1.95699
$424$	46.4890	2.25770
$425$	−0.983471	−0.0477053
$426$	−36.2355	−1.75561
$427$	67.0011	3.24241
$428$	−6.47419	−0.312942
$429$	−65.1327	−3.14464
$430$	25.1058	1.21071
$431$	2.10560	0.101423	0.0507116	−	0.998713i	$-0.483851\pi$
$431$	2.10560	0.101423	0.0507116	+	0.998713i	$0.483851\pi$
$432$	112.177	5.39712
$433$	32.4481	1.55936	0.779679	−	0.626180i	$-0.215382\pi$
$433$	32.4481	1.55936	0.779679	+	0.626180i	$0.215382\pi$
$434$	34.8505	1.67288
$435$	22.0446	1.05696
$436$	5.90478	0.282788
$437$	−1.10739	−0.0529735
$438$	8.29374	0.396290
$439$	−22.9040	−1.09315	−0.546574	−	0.837411i	$-0.684068\pi$
$439$	−22.9040	−1.09315	−0.546574	+	0.837411i	$0.684068\pi$
$440$	−46.9196	−2.23681
$441$	86.0491	4.09758
$442$	2.95581	0.140593
$443$	14.6665	0.696827	0.348414	−	0.937341i	$-0.386720\pi$
$443$	14.6665	0.696827	0.348414	+	0.937341i	$0.386720\pi$
$444$	14.8068	0.702697
$445$	−2.08489	−0.0988332
$446$	−10.0813	−0.477362
$447$	−34.0403	−1.61005
$448$	8.78673	0.415134
$449$	−36.1429	−1.70569	−0.852846	−	0.522163i	$-0.825125\pi$
$449$	−36.1429	−1.70569	−0.852846	+	0.522163i	$0.825125\pi$
$450$	61.6344	2.90547
$451$	−23.4902	−1.10611
$452$	79.1767	3.72416
$453$	44.8711	2.10823
$454$	−53.1438	−2.49416
$455$	−21.7101	−1.01779
$456$	−93.0795	−4.35885
$457$	−11.0892	−0.518733	−0.259367	−	0.965779i	$-0.583514\pi$
$457$	−11.0892	−0.518733	−0.259367	+	0.965779i	$0.583514\pi$
$458$	−7.61374	−0.355767
$459$	4.32355	0.201806
$460$	−1.55312	−0.0724148
$461$	−6.56477	−0.305752	−0.152876	−	0.988245i	$-0.548853\pi$
$461$	−6.56477	−0.305752	−0.152876	+	0.988245i	$0.548853\pi$
$462$	−190.145	−8.84632
$463$	5.20399	0.241850	0.120925	−	0.992662i	$-0.461414\pi$
$463$	5.20399	0.241850	0.120925	+	0.992662i	$0.461414\pi$
$464$	40.5552	1.88273
$465$	−13.4775	−0.625003
$466$	50.4012	2.33479
$467$	−23.2986	−1.07813	−0.539066	−	0.842264i	$-0.681223\pi$
$467$	−23.2986	−1.07813	−0.539066	+	0.842264i	$0.681223\pi$
$468$	−129.010	−5.96348
$469$	2.07096	0.0956283
$470$	18.4934	0.853038
$471$	−48.6360	−2.24103
$472$	32.5099	1.49639
$473$	−39.2265	−1.80363
$474$	134.248	6.16622
$475$	−14.0624	−0.645226
$476$	6.00960	0.275450
$477$	−51.8675	−2.37485
$478$	−15.3774	−0.703348
$479$	7.12978	0.325768	0.162884	−	0.986645i	$-0.447920\pi$
$479$	7.12978	0.325768	0.162884	+	0.986645i	$0.447920\pi$
$480$	−29.6805	−1.35472
$481$	−3.79279	−0.172936
$482$	−34.2864	−1.56170
$483$	−3.55070	−0.161562
$484$	79.4774	3.61261
$485$	−15.6693	−0.711504
$486$	−86.7565	−3.93535
$487$	−32.8434	−1.48827	−0.744137	−	0.668027i	$-0.767139\pi$
$487$	−32.8434	−1.48827	−0.744137	+	0.668027i	$0.767139\pi$
$488$	−103.200	−4.67164
$489$	−3.22691	−0.145926
$490$	−39.5370	−1.78610
$491$	38.9936	1.75976	0.879878	−	0.475200i	$-0.157624\pi$
$491$	38.9936	1.75976	0.879878	+	0.475200i	$0.157624\pi$
$492$	−65.3570	−2.94652
$493$	1.56309	0.0703980
$494$	42.2643	1.90156
$495$	52.3481	2.35287
$496$	−24.7944	−1.11330
$497$	18.8713	0.846494
$498$	−32.0525	−1.43631
$499$	26.4139	1.18245	0.591223	−	0.806508i	$-0.298645\pi$
$499$	26.4139	1.18245	0.591223	+	0.806508i	$0.298645\pi$
$500$	−50.1664	−2.24351
$501$	−58.2842	−2.60395
$502$	10.3835	0.463440
$503$	−4.77520	−0.212916	−0.106458	−	0.994317i	$-0.533951\pi$
$503$	−4.77520	−0.212916	−0.106458	+	0.994317i	$0.533951\pi$
$504$	−212.464	−9.46392
$505$	2.55937	0.113891
$506$	3.48438	0.154900
$507$	4.46998	0.198519
$508$	−84.7179	−3.75875
$509$	22.0800	0.978679	0.489340	−	0.872093i	$-0.337238\pi$
$509$	22.0800	0.978679	0.489340	+	0.872093i	$0.337238\pi$
$510$	−3.33703	−0.147766
$511$	−4.31935	−0.191077
$512$	50.0769	2.21311
$513$	61.8213	2.72948
$514$	−14.0930	−0.621614
$515$	−25.4436	−1.12118
$516$	−109.140	−4.80463
$517$	−28.8949	−1.27080
$518$	−11.0724	−0.486495
$519$	30.3320	1.33143
$520$	33.4395	1.46642
$521$	37.9878	1.66427	0.832137	−	0.554570i	$-0.187117\pi$
$521$	37.9878	1.66427	0.832137	+	0.554570i	$0.187117\pi$
$522$	−97.9592	−4.28756
$523$	27.4901	1.20206	0.601030	−	0.799226i	$-0.294757\pi$
$523$	27.4901	1.20206	0.601030	+	0.799226i	$0.294757\pi$
$524$	42.5856	1.86036
$525$	−45.0893	−1.96786
$526$	−33.8767	−1.47709
$527$	−0.955630	−0.0416279
$528$	135.278	5.88723
$529$	−22.9349	−0.997171
$530$	23.8316	1.03518
$531$	−36.2712	−1.57404
$532$	85.9296	3.72552
$533$	16.7414	0.725149
$534$	13.0139	0.563168
$535$	−1.87227	−0.0809452
$536$	−3.18984	−0.137780
$537$	−49.5847	−2.13974
$538$	−50.3766	−2.17189
$539$	61.7744	2.66081
$540$	86.7052	3.73120
$541$	−6.34647	−0.272856	−0.136428	−	0.990650i	$-0.543562\pi$
$541$	−6.34647	−0.272856	−0.136428	+	0.990650i	$0.543562\pi$
$542$	0.577814	0.0248192
$543$	7.87149	0.337798
$544$	−2.10452	−0.0902304
$545$	1.70760	0.0731456
$546$	135.515	5.79952
$547$	43.5338	1.86137	0.930686	−	0.365820i	$-0.119211\pi$
$547$	43.5338	1.86137	0.930686	+	0.365820i	$0.119211\pi$
$548$	36.5708	1.56223
$549$	115.140	4.91404
$550$	44.2471	1.88670
$551$	22.3502	0.952149
$552$	5.46903	0.232778
$553$	−69.9159	−2.97313
$554$	56.8558	2.41557
$555$	4.28196	0.181759
$556$	14.5653	0.617708
$557$	−20.5913	−0.872481	−0.436241	−	0.899830i	$-0.643690\pi$
$557$	−20.5913	−0.872481	−0.436241	+	0.899830i	$0.643690\pi$
$558$	59.8896	2.53533
$559$	27.9566	1.18244
$560$	45.0912	1.90545
$561$	5.21393	0.220132
$562$	−48.3793	−2.04076
$563$	−39.3244	−1.65733	−0.828664	−	0.559746i	$-0.810899\pi$
$563$	−39.3244	−1.65733	−0.828664	+	0.559746i	$0.810899\pi$
$564$	−80.3948	−3.38523
$565$	22.8971	0.963287
$566$	81.0879	3.40838
$567$	102.291	4.29581
$568$	−29.0669	−1.21962
$569$	35.7379	1.49821	0.749106	−	0.662450i	$-0.230483\pi$
$569$	35.7379	1.49821	0.749106	+	0.662450i	$0.230483\pi$
$570$	−47.7153	−1.99857
$571$	27.3832	1.14595	0.572975	−	0.819573i	$-0.305789\pi$
$571$	27.3832	1.14595	0.572975	+	0.819573i	$0.305789\pi$
$572$	−92.6157	−3.87246
$573$	−9.88175	−0.412816
$574$	48.8737	2.03995
$575$	0.826257	0.0344573
$576$	15.0997	0.629156
$577$	−19.8282	−0.825460	−0.412730	−	0.910853i	$-0.635425\pi$
$577$	−19.8282	−0.825460	−0.412730	+	0.910853i	$0.635425\pi$
$578$	43.3992	1.80517
$579$	27.8926	1.15918
$580$	31.3464	1.30159
$581$	16.6928	0.692535
$582$	97.8079	4.05427
$583$	−37.2356	−1.54214
$584$	6.65296	0.275301
$585$	−37.3083	−1.54251
$586$	9.29926	0.384149
$587$	31.8677	1.31532	0.657660	−	0.753315i	$-0.271546\pi$
$587$	31.8677	1.31532	0.657660	+	0.753315i	$0.271546\pi$
$588$	171.876	7.08804
$589$	−13.6643	−0.563027
$590$	16.6655	0.686110
$591$	48.9736	2.01451
$592$	7.87748	0.323762
$593$	28.2077	1.15835	0.579176	−	0.815202i	$-0.303374\pi$
$593$	28.2077	1.15835	0.579176	+	0.815202i	$0.303374\pi$
$594$	−194.520	−7.98126
$595$	1.73791	0.0712476
$596$	−48.4038	−1.98270
$597$	82.8896	3.39245
$598$	−2.48331	−0.101550
$599$	−25.9533	−1.06042	−0.530211	−	0.847865i	$-0.677887\pi$
$599$	−25.9533	−1.06042	−0.530211	+	0.847865i	$0.677887\pi$
$600$	69.4496	2.83527
$601$	16.9963	0.693293	0.346646	−	0.937996i	$-0.387320\pi$
$601$	16.9963	0.693293	0.346646	+	0.937996i	$0.387320\pi$
$602$	81.6147	3.32637
$603$	3.55890	0.144929
$604$	63.8047	2.59618
$605$	22.9840	0.934434
$606$	−15.9757	−0.648967
$607$	34.3909	1.39589	0.697943	−	0.716154i	$-0.254099\pi$
$607$	34.3909	1.39589	0.697943	+	0.716154i	$0.254099\pi$
$608$	−30.0919	−1.22039
$609$	71.6631	2.90394
$610$	−52.9033	−2.14199
$611$	20.5933	0.833116
$612$	10.3273	0.417458
$613$	27.8027	1.12294	0.561471	−	0.827496i	$-0.310236\pi$
$613$	27.8027	1.12294	0.561471	+	0.827496i	$0.310236\pi$
$614$	−38.7691	−1.56459
$615$	−18.9006	−0.762145
$616$	−152.528	−6.14551
$617$	−7.30529	−0.294100	−0.147050	−	0.989129i	$-0.546978\pi$
$617$	−7.30529	−0.294100	−0.147050	+	0.989129i	$0.546978\pi$
$618$	158.820	6.38866
$619$	−33.5389	−1.34804	−0.674021	−	0.738713i	$-0.735434\pi$
$619$	−33.5389	−1.34804	−0.674021	+	0.738713i	$0.735434\pi$
$620$	−19.1643	−0.769659
$621$	−3.63241	−0.145764
$622$	−52.6258	−2.11010
$623$	−6.77761	−0.271539
$624$	−96.4124	−3.85958
$625$	1.68838	0.0675354
$626$	62.5284	2.49914
$627$	74.5525	2.97734
$628$	−69.1582	−2.75971
$629$	0.303616	0.0121059
$630$	−108.916	−4.33930
$631$	−34.0796	−1.35669	−0.678343	−	0.734745i	$-0.737302\pi$
$631$	−34.0796	−1.35669	−0.678343	+	0.734745i	$0.737302\pi$
$632$	107.689	4.28365
$633$	−71.1253	−2.82698
$634$	−12.9265	−0.513375
$635$	−24.4995	−0.972234
$636$	−103.601	−4.10805
$637$	−44.0264	−1.74439
$638$	−70.3246	−2.78418
$639$	32.4298	1.28290
$640$	11.4577	0.452905
$641$	28.9087	1.14182	0.570912	−	0.821011i	$-0.306590\pi$
$641$	28.9087	1.14182	0.570912	+	0.821011i	$0.306590\pi$
$642$	11.6867	0.461239
$643$	27.5409	1.08611	0.543054	−	0.839698i	$-0.317268\pi$
$643$	27.5409	1.08611	0.543054	+	0.839698i	$0.317268\pi$
$644$	−5.04893	−0.198956
$645$	−31.5622	−1.24276
$646$	−3.38329	−0.133114
$647$	33.6683	1.32364	0.661818	−	0.749664i	$-0.269785\pi$
$647$	33.6683	1.32364	0.661818	+	0.749664i	$0.269785\pi$
$648$	−157.555	−6.18936
$649$	−26.0390	−1.02212
$650$	−31.5348	−1.23690
$651$	−43.8129	−1.71716
$652$	−4.58852	−0.179700
$653$	−40.6614	−1.59120	−0.795602	−	0.605820i	$-0.792845\pi$
$653$	−40.6614	−1.59120	−0.795602	+	0.605820i	$0.792845\pi$
$654$	−10.6589	−0.416796
$655$	12.3153	0.481199
$656$	−34.7712	−1.35759
$657$	−7.42268	−0.289586
$658$	60.1189	2.34368
$659$	−6.45690	−0.251525	−0.125763	−	0.992060i	$-0.540138\pi$
$659$	−6.45690	−0.251525	−0.125763	+	0.992060i	$0.540138\pi$
$660$	104.561	4.07002
$661$	3.06895	0.119368	0.0596841	−	0.998217i	$-0.480991\pi$
$661$	3.06895	0.119368	0.0596841	+	0.998217i	$0.480991\pi$
$662$	24.8128	0.964375
$663$	−3.71595	−0.144315
$664$	−25.7114	−0.997797
$665$	24.8500	0.963640
$666$	−19.0277	−0.737307
$667$	−1.31322	−0.0508481
$668$	−82.8775	−3.20663
$669$	12.6738	0.489999
$670$	−1.63521	−0.0631736
$671$	82.6584	3.19099
$672$	−96.4860	−3.72203
$673$	51.2755	1.97653	0.988263	−	0.152760i	$-0.0488160\pi$
$673$	51.2755	1.97653	0.988263	+	0.152760i	$0.0488160\pi$
$674$	54.2548	2.08982
$675$	−46.1269	−1.77543
$676$	6.35611	0.244466
$677$	18.3635	0.705766	0.352883	−	0.935668i	$-0.385201\pi$
$677$	18.3635	0.705766	0.352883	+	0.935668i	$0.385201\pi$
$678$	−142.924	−5.48897
$679$	−50.9380	−1.95482
$680$	−2.67686	−0.102653
$681$	66.8107	2.56019
$682$	42.9946	1.64635
$683$	20.1189	0.769829	0.384915	−	0.922952i	$-0.374231\pi$
$683$	20.1189	0.769829	0.384915	+	0.922952i	$0.374231\pi$
$684$	147.668	5.64621
$685$	10.5759	0.404084
$686$	−51.0210	−1.94799
$687$	9.57175	0.365185
$688$	−58.0648	−2.21370
$689$	26.5376	1.01100
$690$	2.80359	0.106731
$691$	2.57728	0.0980444	0.0490222	−	0.998798i	$-0.484389\pi$
$691$	2.57728	0.0980444	0.0490222	+	0.998798i	$0.484389\pi$
$692$	43.1307	1.63958
$693$	170.174	6.46439
$694$	−54.1626	−2.05598
$695$	4.21214	0.159776
$696$	−110.380	−4.18396
$697$	−1.34016	−0.0507622
$698$	30.7319	1.16322
$699$	−63.3627	−2.39660
$700$	−64.1149	−2.42332
$701$	9.52207	0.359644	0.179822	−	0.983699i	$-0.442448\pi$
$701$	9.52207	0.359644	0.179822	+	0.983699i	$0.442448\pi$
$702$	138.634	5.23240
$703$	4.34131	0.163736
$704$	10.8401	0.408550
$705$	−23.2493	−0.875620
$706$	13.5543	0.510121
$707$	8.32007	0.312908
$708$	−72.4486	−2.72279
$709$	14.0473	0.527556	0.263778	−	0.964583i	$-0.415031\pi$
$709$	14.0473	0.527556	0.263778	+	0.964583i	$0.415031\pi$
$710$	−14.9005	−0.559208
$711$	−120.149	−4.50593
$712$	10.4393	0.391231
$713$	0.802867	0.0300676
$714$	−10.8481	−0.405980
$715$	−26.7835	−1.00165
$716$	−70.5072	−2.63498
$717$	19.3320	0.721967
$718$	15.3825	0.574069
$719$	−47.1421	−1.75810	−0.879052	−	0.476725i	$-0.841824\pi$
$719$	−47.1421	−1.75810	−0.879052	+	0.476725i	$0.841824\pi$
$720$	77.4880	2.88781
$721$	−82.7127	−3.08038
$722$	0.392706	0.0146150
$723$	43.1037	1.60305
$724$	11.1929	0.415981
$725$	−16.6762	−0.619338
$726$	−143.467	−5.32456
$727$	−24.7081	−0.916371	−0.458186	−	0.888857i	$-0.651501\pi$
$727$	−24.7081	−0.916371	−0.458186	+	0.888857i	$0.651501\pi$
$728$	108.706	4.02891
$729$	37.9282	1.40475
$730$	3.41050	0.126228
$731$	−2.23795	−0.0827734
$732$	229.982	8.50036
$733$	−16.3897	−0.605369	−0.302684	−	0.953091i	$-0.597883\pi$
$733$	−16.3897	−0.605369	−0.302684	+	0.953091i	$0.597883\pi$
$734$	16.1306	0.595392
$735$	49.7047	1.83338
$736$	1.76810	0.0651730
$737$	2.55492	0.0941117
$738$	83.9882	3.09165
$739$	27.7893	1.02224	0.511122	−	0.859508i	$-0.329230\pi$
$739$	27.7893	1.02224	0.511122	+	0.859508i	$0.329230\pi$
$740$	6.08875	0.223827
$741$	−53.1333	−1.95190
$742$	77.4724	2.84410
$743$	−42.1320	−1.54567	−0.772837	−	0.634604i	$-0.781163\pi$
$743$	−42.1320	−1.54567	−0.772837	+	0.634604i	$0.781163\pi$
$744$	67.4836	2.47407
$745$	−13.9979	−0.512842
$746$	51.7623	1.89515
$747$	28.6861	1.04957
$748$	7.41396	0.271081
$749$	−6.08642	−0.222393
$750$	90.5568	3.30667
$751$	−44.7778	−1.63396	−0.816982	−	0.576663i	$-0.804355\pi$
$751$	−44.7778	−1.63396	−0.816982	+	0.576663i	$0.804355\pi$
$752$	−42.7716	−1.55972
$753$	−13.0538	−0.475709
$754$	50.1201	1.82527
$755$	18.4516	0.671524
$756$	281.863	10.2513
$757$	38.4835	1.39871	0.699353	−	0.714777i	$-0.253472\pi$
$757$	38.4835	1.39871	0.699353	+	0.714777i	$0.253472\pi$
$758$	−17.3262	−0.629316
$759$	−4.38045	−0.159000
$760$	−38.2756	−1.38840
$761$	16.1119	0.584056	0.292028	−	0.956410i	$-0.405670\pi$
$761$	16.1119	0.584056	0.292028	+	0.956410i	$0.405670\pi$
$762$	152.927	5.53995
$763$	5.55111	0.200964
$764$	−14.0514	−0.508362
$765$	2.98656	0.107979
$766$	50.6177	1.82889
$767$	18.5579	0.670086
$768$	−84.6653	−3.05510
$769$	26.7715	0.965404	0.482702	−	0.875785i	$-0.339655\pi$
$769$	26.7715	0.965404	0.482702	+	0.875785i	$0.339655\pi$
$770$	−78.1902	−2.81778
$771$	17.7172	0.638070
$772$	39.6620	1.42747
$773$	−11.9225	−0.428821	−0.214411	−	0.976744i	$-0.568783\pi$
$773$	−11.9225	−0.428821	−0.214411	+	0.976744i	$0.568783\pi$
$774$	140.253	5.04128
$775$	10.1954	0.366229
$776$	78.4583	2.81649
$777$	13.9199	0.499374
$778$	−74.1435	−2.65817
$779$	−19.1626	−0.686570
$780$	−74.5201	−2.66825
$781$	23.2813	0.833069
$782$	0.198791	0.00710874
$783$	73.3122	2.61997
$784$	91.4412	3.26576
$785$	−19.9998	−0.713825
$786$	−76.8724	−2.74195
$787$	18.5566	0.661471	0.330735	−	0.943724i	$-0.392703\pi$
$787$	18.5566	0.661471	0.330735	+	0.943724i	$0.392703\pi$
$788$	69.6383	2.48076
$789$	42.5887	1.51620
$790$	55.2048	1.96410
$791$	74.4344	2.64658
$792$	−262.114	−9.31383
$793$	−58.9103	−2.09197
$794$	87.7724	3.11493
$795$	−29.9603	−1.06258
$796$	117.865	4.17762
$797$	−26.2463	−0.929691	−0.464845	−	0.885392i	$-0.653890\pi$
$797$	−26.2463	−0.929691	−0.464845	+	0.885392i	$0.653890\pi$
$798$	−155.114	−5.49098
$799$	−1.64851	−0.0583201
$800$	22.4526	0.793818
$801$	−11.6471	−0.411531
$802$	81.2044	2.86743
$803$	−5.32872	−0.188046
$804$	7.10859	0.250701
$805$	−1.46010	−0.0514617
$806$	−30.6421	−1.07932
$807$	63.3318	2.22939
$808$	−12.8151	−0.450835
$809$	41.2408	1.44995	0.724975	−	0.688775i	$-0.241851\pi$
$809$	41.2408	1.44995	0.724975	+	0.688775i	$0.241851\pi$
$810$	−80.7676	−2.83788
$811$	−18.5095	−0.649955	−0.324977	−	0.945722i	$-0.605357\pi$
$811$	−18.5095	−0.649955	−0.324977	+	0.945722i	$0.605357\pi$
$812$	101.902	3.57605
$813$	−0.726409	−0.0254763
$814$	−13.6599	−0.478779
$815$	−1.32695	−0.0464811
$816$	7.71789	0.270180
$817$	−31.9997	−1.11953
$818$	33.3264	1.16523
$819$	−121.283	−4.23796
$820$	−26.8757	−0.938542
$821$	−38.2014	−1.33324	−0.666619	−	0.745399i	$-0.732259\pi$
$821$	−38.2014	−1.33324	−0.666619	+	0.745399i	$0.732259\pi$
$822$	−66.0150	−2.30254
$823$	30.9920	1.08031	0.540157	−	0.841565i	$-0.318365\pi$
$823$	30.9920	1.08031	0.540157	+	0.841565i	$0.318365\pi$
$824$	127.400	4.43818
$825$	−55.6260	−1.93665
$826$	54.1768	1.88505
$827$	31.3493	1.09012	0.545061	−	0.838397i	$-0.316507\pi$
$827$	31.3493	1.09012	0.545061	+	0.838397i	$0.316507\pi$
$828$	−8.67645	−0.301528
$829$	47.4556	1.64820	0.824101	−	0.566444i	$-0.191681\pi$
$829$	47.4556	1.64820	0.824101	+	0.566444i	$0.191681\pi$
$830$	−13.1804	−0.457500
$831$	−71.4773	−2.47952
$832$	−7.72567	−0.267840
$833$	3.52435	0.122111
$834$	−26.2923	−0.910428
$835$	−23.9673	−0.829423
$836$	106.010	3.66644
$837$	−44.8211	−1.54924
$838$	56.0264	1.93540
$839$	−50.7392	−1.75171	−0.875855	−	0.482574i	$-0.839702\pi$
$839$	−50.7392	−1.75171	−0.875855	+	0.482574i	$0.839702\pi$
$840$	−122.726	−4.23445
$841$	−2.49553	−0.0860528
$842$	64.5777	2.22549
$843$	60.8209	2.09478
$844$	−101.137	−3.48128
$845$	1.83812	0.0632333
$846$	103.313	3.55196
$847$	74.7171	2.56731
$848$	−55.1178	−1.89275
$849$	−101.941	−3.49861
$850$	2.52438	0.0865857
$851$	−0.255081	−0.00874406
$852$	64.7758	2.21918
$853$	48.6567	1.66597	0.832987	−	0.553293i	$-0.186629\pi$
$853$	48.6567	1.66597	0.832987	+	0.553293i	$0.186629\pi$
$854$	−171.979	−5.88501
$855$	42.7040	1.46044
$856$	9.37472	0.320421
$857$	9.44893	0.322769	0.161385	−	0.986892i	$-0.448404\pi$
$857$	9.44893	0.322769	0.161385	+	0.986892i	$0.448404\pi$
$858$	167.183	5.70755
$859$	51.1064	1.74373	0.871864	−	0.489748i	$-0.162911\pi$
$859$	51.1064	1.74373	0.871864	+	0.489748i	$0.162911\pi$
$860$	−44.8801	−1.53040
$861$	−61.4425	−2.09395
$862$	−5.40468	−0.184084
$863$	−4.19316	−0.142737	−0.0713683	−	0.997450i	$-0.522737\pi$
$863$	−4.19316	−0.142737	−0.0713683	+	0.997450i	$0.522737\pi$
$864$	−98.7064	−3.35806
$865$	12.4730	0.424093
$866$	−83.2882	−2.83025
$867$	−54.5600	−1.85296
$868$	−62.2999	−2.11460
$869$	−86.2543	−2.92598
$870$	−56.5843	−1.91839
$871$	−1.82088	−0.0616982
$872$	−8.55021	−0.289547
$873$	−87.5356	−2.96263
$874$	2.84245	0.0961474
$875$	−47.1617	−1.59436
$876$	−14.8262	−0.500930
$877$	36.2997	1.22575	0.612877	−	0.790178i	$-0.290012\pi$
$877$	36.2997	1.22575	0.612877	+	0.790178i	$0.290012\pi$
$878$	58.7902	1.98407
$879$	−11.6907	−0.394318
$880$	55.6284	1.87523
$881$	−28.8631	−0.972421	−0.486211	−	0.873842i	$-0.661621\pi$
$881$	−28.8631	−0.972421	−0.486211	+	0.873842i	$0.661621\pi$
$882$	−220.872	−7.43714
$883$	10.4855	0.352866	0.176433	−	0.984313i	$-0.443544\pi$
$883$	10.4855	0.352866	0.176433	+	0.984313i	$0.443544\pi$
$884$	−5.28391	−0.177717
$885$	−20.9514	−0.704273
$886$	−37.6462	−1.26475
$887$	31.4912	1.05737	0.528686	−	0.848818i	$-0.322685\pi$
$887$	31.4912	1.05737	0.528686	+	0.848818i	$0.322685\pi$
$888$	−21.4404	−0.719492
$889$	−79.6437	−2.67116
$890$	5.35152	0.179383
$891$	126.195	4.22769
$892$	18.0216	0.603409
$893$	−23.5716	−0.788794
$894$	87.3751	2.92226
$895$	−20.3900	−0.681561
$896$	37.2470	1.24433
$897$	3.12193	0.104238
$898$	92.7721	3.09585
$899$	−16.2041	−0.540438
$900$	−110.180	−3.67266
$901$	−2.12436	−0.0707727
$902$	60.2948	2.00760
$903$	−102.603	−3.41443
$904$	−114.649	−3.81317
$905$	3.23687	0.107597
$906$	−115.176	−3.82646
$907$	13.1707	0.437327	0.218663	−	0.975800i	$-0.429830\pi$
$907$	13.1707	0.437327	0.218663	+	0.975800i	$0.429830\pi$
$908$	95.0018	3.15274
$909$	14.2978	0.474228
$910$	55.7258	1.84729
$911$	−24.3190	−0.805724	−0.402862	−	0.915261i	$-0.631985\pi$
$911$	−24.3190	−0.805724	−0.402862	+	0.915261i	$0.631985\pi$
$912$	110.356	3.65425
$913$	20.5937	0.681552
$914$	28.4640	0.941506
$915$	66.5083	2.19870
$916$	13.6106	0.449706
$917$	40.0349	1.32207
$918$	−11.0978	−0.366280
$919$	24.2640	0.800395	0.400197	−	0.916429i	$-0.368942\pi$
$919$	24.2640	0.800395	0.400197	+	0.916429i	$0.368942\pi$
$920$	2.24895	0.0741455
$921$	48.7392	1.60601
$922$	16.8505	0.554942
$923$	−16.5925	−0.546148
$924$	339.909	11.1822
$925$	−3.23920	−0.106504
$926$	−13.3577	−0.438960
$927$	−142.139	−4.66847
$928$	−35.6852	−1.17142
$929$	1.54277	0.0506166	0.0253083	−	0.999680i	$-0.491943\pi$
$929$	1.54277	0.0506166	0.0253083	+	0.999680i	$0.491943\pi$
$930$	34.5941	1.13439
$931$	50.3937	1.65159
$932$	−90.0989	−2.95129
$933$	66.1594	2.16596
$934$	59.8032	1.95682
$935$	2.14404	0.0701176
$936$	186.808	6.10601
$937$	30.6511	1.00133	0.500663	−	0.865642i	$-0.333090\pi$
$937$	30.6511	1.00133	0.500663	+	0.865642i	$0.333090\pi$
$938$	−5.31578	−0.173566
$939$	−78.6087	−2.56530
$940$	−33.0595	−1.07828
$941$	−18.2229	−0.594049	−0.297024	−	0.954870i	$-0.595994\pi$
$941$	−18.2229	−0.594049	−0.297024	+	0.954870i	$0.595994\pi$
$942$	124.839	4.06749
$943$	1.12593	0.0366652
$944$	−38.5441	−1.25450
$945$	81.5120	2.65158
$946$	100.687	3.27361
$947$	17.2966	0.562064	0.281032	−	0.959698i	$-0.409323\pi$
$947$	17.2966	0.562064	0.281032	+	0.959698i	$0.409323\pi$
$948$	−239.987	−7.79440
$949$	3.79776	0.123281
$950$	36.0955	1.17109
$951$	16.2507	0.526966
$952$	−8.70199	−0.282033
$953$	12.8951	0.417712	0.208856	−	0.977946i	$-0.433026\pi$
$953$	12.8951	0.417712	0.208856	+	0.977946i	$0.433026\pi$
$954$	133.134	4.31038
$955$	−4.06352	−0.131492
$956$	27.4892	0.889066
$957$	88.4098	2.85788
$958$	−18.3008	−0.591272
$959$	34.3804	1.11020
$960$	8.72209	0.281504
$961$	−21.0932	−0.680427
$962$	9.73536	0.313881
$963$	−10.4593	−0.337047
$964$	61.2915	1.97407
$965$	11.4698	0.369227
$966$	9.11398	0.293237
$967$	47.5480	1.52904	0.764520	−	0.644601i	$-0.222976\pi$
$967$	47.5480	1.52904	0.764520	+	0.644601i	$0.222976\pi$
$968$	−115.084	−3.69895
$969$	4.25336	0.136638
$970$	40.2200	1.29139
$971$	−0.211828	−0.00679788	−0.00339894	−	0.999994i	$-0.501082\pi$
$971$	−0.211828	−0.00679788	−0.00339894	+	0.999994i	$0.501082\pi$
$972$	155.089	4.97448
$973$	13.6929	0.438976
$974$	84.3027	2.70123
$975$	39.6445	1.26964
$976$	122.355	3.91648
$977$	−20.5554	−0.657627	−0.328813	−	0.944395i	$-0.606649\pi$
$977$	−20.5554	−0.657627	−0.328813	+	0.944395i	$0.606649\pi$
$978$	8.28287	0.264857
$979$	−8.36145	−0.267233
$980$	70.6778	2.25772
$981$	9.53944	0.304571
$982$	−100.089	−3.19397
$983$	−42.3658	−1.35126	−0.675630	−	0.737241i	$-0.736128\pi$
$983$	−42.3658	−1.35126	−0.675630	+	0.737241i	$0.736128\pi$
$984$	94.6379	3.01695
$985$	20.1387	0.641671
$986$	−4.01215	−0.127773
$987$	−75.5795	−2.40572
$988$	−75.5531	−2.40366
$989$	1.88020	0.0597868
$990$	−134.368	−4.27048
$991$	−7.89982	−0.250946	−0.125473	−	0.992097i	$-0.540045\pi$
$991$	−7.89982	−0.250946	−0.125473	+	0.992097i	$0.540045\pi$
$992$	21.8170	0.692689
$993$	−31.1938	−0.989905
$994$	−48.4391	−1.53639
$995$	34.0854	1.08058
$996$	57.2981	1.81556
$997$	53.1684	1.68386	0.841930	−	0.539587i	$-0.181420\pi$
$997$	53.1684	1.68386	0.841930	+	0.539587i	$0.181420\pi$
$998$	−67.7994	−2.14615
$999$	14.2402	0.450541

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6031.2.a.e.1.6	✓	134

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6031.2.a.e.1.6	✓	134	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:	\( N \)	\(=\)	\( 6031 = 37 \cdot 163 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56681 q^{2} +3.22691 q^{3} +4.58852 q^{4} +1.32695 q^{5} -8.28287 q^{6} +4.31369 q^{7} -6.64424 q^{8} +7.41296 q^{9} +O(q^{10})\)	\(q-2.56681 q^{2} +3.22691 q^{3} +4.58852 q^{4} +1.32695 q^{5} -8.28287 q^{6} +4.31369 q^{7} -6.64424 q^{8} +7.41296 q^{9} -3.40604 q^{10} +5.32174 q^{11} +14.8068 q^{12} -3.79279 q^{13} -11.0724 q^{14} +4.28196 q^{15} +7.87748 q^{16} +0.303616 q^{17} -19.0277 q^{18} +4.34131 q^{19} +6.08875 q^{20} +13.9199 q^{21} -13.6599 q^{22} -0.255081 q^{23} -21.4404 q^{24} -3.23920 q^{25} +9.73536 q^{26} +14.2402 q^{27} +19.7935 q^{28} +5.14825 q^{29} -10.9910 q^{30} -3.14750 q^{31} -6.93152 q^{32} +17.1728 q^{33} -0.779324 q^{34} +5.72406 q^{35} +34.0145 q^{36} +1.00000 q^{37} -11.1433 q^{38} -12.2390 q^{39} -8.81660 q^{40} -4.41400 q^{41} -35.7298 q^{42} -7.37098 q^{43} +24.4189 q^{44} +9.83664 q^{45} +0.654745 q^{46} -5.42960 q^{47} +25.4199 q^{48} +11.6079 q^{49} +8.31441 q^{50} +0.979741 q^{51} -17.4033 q^{52} -6.99688 q^{53} -36.5520 q^{54} +7.06170 q^{55} -28.6612 q^{56} +14.0090 q^{57} -13.2146 q^{58} -4.89294 q^{59} +19.6479 q^{60} +15.5322 q^{61} +8.07904 q^{62} +31.9772 q^{63} +2.03694 q^{64} -5.03285 q^{65} -44.0793 q^{66} +0.480091 q^{67} +1.39315 q^{68} -0.823124 q^{69} -14.6926 q^{70} +4.37475 q^{71} -49.2535 q^{72} -1.00131 q^{73} -2.56681 q^{74} -10.4526 q^{75} +19.9202 q^{76} +22.9563 q^{77} +31.4152 q^{78} -16.2079 q^{79} +10.4530 q^{80} +23.7131 q^{81} +11.3299 q^{82} +3.86973 q^{83} +63.8717 q^{84} +0.402883 q^{85} +18.9199 q^{86} +16.6129 q^{87} -35.3589 q^{88} -1.57119 q^{89} -25.2488 q^{90} -16.3609 q^{91} -1.17044 q^{92} -10.1567 q^{93} +13.9368 q^{94} +5.76072 q^{95} -22.3674 q^{96} -11.8085 q^{97} -29.7954 q^{98} +39.4498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 134 q + 9 q^{2} + 7 q^{3} + 149 q^{4} + 22 q^{5} + 20 q^{6} + 11 q^{7} + 27 q^{8} + 181 q^{9}+O(q^{10}) \) 134 * q + 9 * q^2 + 7 * q^3 + 149 * q^4 + 22 * q^5 + 20 * q^6 + 11 * q^7 + 27 * q^8 + 181 * q^9	\( 134 q + 9 q^{2} + 7 q^{3} + 149 q^{4} + 22 q^{5} + 20 q^{6} + 11 q^{7} + 27 q^{8} + 181 q^{9} + 15 q^{10} + 20 q^{11} + 28 q^{12} + 11 q^{13} + 17 q^{14} - 13 q^{15} + 143 q^{16} + 76 q^{17} + 23 q^{18} + 15 q^{19} + 67 q^{20} + 63 q^{21} + 2 q^{22} + 22 q^{23} + 33 q^{24} + 160 q^{25} + 65 q^{26} + 31 q^{27} + 10 q^{28} + 73 q^{29} + 20 q^{30} + 10 q^{31} + 53 q^{32} + 72 q^{33} - 7 q^{34} + 52 q^{35} + 201 q^{36} + 134 q^{37} + 70 q^{38} + 6 q^{39} + 11 q^{40} + 182 q^{41} - 15 q^{42} + 12 q^{43} + 33 q^{44} + 29 q^{45} + 24 q^{46} + 80 q^{47} + 21 q^{48} + 229 q^{49} + 37 q^{50} + 57 q^{51} - 15 q^{52} + 75 q^{53} + 95 q^{54} - 9 q^{55} + 39 q^{56} + 19 q^{57} - 21 q^{58} + 91 q^{59} + 62 q^{60} + 58 q^{61} + 108 q^{62} + 9 q^{63} + 167 q^{64} + 76 q^{65} + 105 q^{66} - 17 q^{67} + 109 q^{68} + 48 q^{69} - 55 q^{70} + 56 q^{71} + 48 q^{72} + 54 q^{73} + 9 q^{74} + 28 q^{75} + 82 q^{76} + 156 q^{77} + 16 q^{78} - 2 q^{79} + 98 q^{80} + 270 q^{81} - 42 q^{82} + 130 q^{83} + 229 q^{84} + 22 q^{85} + 72 q^{86} + 22 q^{87} + 61 q^{88} + 157 q^{89} + 176 q^{90} + 31 q^{91} - 18 q^{92} + 36 q^{93} + 83 q^{94} + 98 q^{95} + 111 q^{96} + 35 q^{97} + 53 q^{98} + 55 q^{99}+O(q^{100}) \) 134 * q + 9 * q^2 + 7 * q^3 + 149 * q^4 + 22 * q^5 + 20 * q^6 + 11 * q^7 + 27 * q^8 + 181 * q^9 + 15 * q^10 + 20 * q^11 + 28 * q^12 + 11 * q^13 + 17 * q^14 - 13 * q^15 + 143 * q^16 + 76 * q^17 + 23 * q^18 + 15 * q^19 + 67 * q^20 + 63 * q^21 + 2 * q^22 + 22 * q^23 + 33 * q^24 + 160 * q^25 + 65 * q^26 + 31 * q^27 + 10 * q^28 + 73 * q^29 + 20 * q^30 + 10 * q^31 + 53 * q^32 + 72 * q^33 - 7 * q^34 + 52 * q^35 + 201 * q^36 + 134 * q^37 + 70 * q^38 + 6 * q^39 + 11 * q^40 + 182 * q^41 - 15 * q^42 + 12 * q^43 + 33 * q^44 + 29 * q^45 + 24 * q^46 + 80 * q^47 + 21 * q^48 + 229 * q^49 + 37 * q^50 + 57 * q^51 - 15 * q^52 + 75 * q^53 + 95 * q^54 - 9 * q^55 + 39 * q^56 + 19 * q^57 - 21 * q^58 + 91 * q^59 + 62 * q^60 + 58 * q^61 + 108 * q^62 + 9 * q^63 + 167 * q^64 + 76 * q^65 + 105 * q^66 - 17 * q^67 + 109 * q^68 + 48 * q^69 - 55 * q^70 + 56 * q^71 + 48 * q^72 + 54 * q^73 + 9 * q^74 + 28 * q^75 + 82 * q^76 + 156 * q^77 + 16 * q^78 - 2 * q^79 + 98 * q^80 + 270 * q^81 - 42 * q^82 + 130 * q^83 + 229 * q^84 + 22 * q^85 + 72 * q^86 + 22 * q^87 + 61 * q^88 + 157 * q^89 + 176 * q^90 + 31 * q^91 - 18 * q^92 + 36 * q^93 + 83 * q^94 + 98 * q^95 + 111 * q^96 + 35 * q^97 + 53 * q^98 + 55 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more