LMFDB - Embedded newform 1849.2.a.q.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,2,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1849.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:	$14.7643393337$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 27 x^{18} + 300 x^{16} - 1775 x^{14} + 6037 x^{12} - 11859 x^{10} + 12809 x^{8} - 6823 x^{6} + \cdots + 1 $ x^20 - 27x^18 + 300x^16 - 1775x^14 + 6037x^12 - 11859x^10 + 12809x^8 - 6823x^6 + 1500x^4 - 75*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Root			$2.64610$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64610 q^{2} -0.666460 q^{3} +5.00186 q^{4} -2.47875 q^{5} -1.76352 q^{6} -1.01239 q^{7} +7.94323 q^{8} -2.55583 q^{9} +O(q^{10})$	\(q+2.64610 q^{2} -0.666460 q^{3} +5.00186 q^{4} -2.47875 q^{5} -1.76352 q^{6} -1.01239 q^{7} +7.94323 q^{8} -2.55583 q^{9} -6.55903 q^{10} -5.17875 q^{11} -3.33354 q^{12} -1.29153 q^{13} -2.67889 q^{14} +1.65199 q^{15} +11.0149 q^{16} -0.735191 q^{17} -6.76299 q^{18} -5.77571 q^{19} -12.3984 q^{20} +0.674717 q^{21} -13.7035 q^{22} -0.896622 q^{23} -5.29384 q^{24} +1.14421 q^{25} -3.41752 q^{26} +3.70274 q^{27} -5.06383 q^{28} +3.42395 q^{29} +4.37133 q^{30} +3.87131 q^{31} +13.2600 q^{32} +3.45143 q^{33} -1.94539 q^{34} +2.50946 q^{35} -12.7839 q^{36} +4.52623 q^{37} -15.2831 q^{38} +0.860753 q^{39} -19.6893 q^{40} -10.4191 q^{41} +1.78537 q^{42} -25.9034 q^{44} +6.33527 q^{45} -2.37255 q^{46} -0.494696 q^{47} -7.34097 q^{48} -5.97507 q^{49} +3.02771 q^{50} +0.489975 q^{51} -6.46005 q^{52} +10.3172 q^{53} +9.79783 q^{54} +12.8368 q^{55} -8.04164 q^{56} +3.84928 q^{57} +9.06012 q^{58} -1.40654 q^{59} +8.26302 q^{60} +2.40989 q^{61} +10.2439 q^{62} +2.58750 q^{63} +13.0577 q^{64} +3.20138 q^{65} +9.13284 q^{66} -5.71755 q^{67} -3.67732 q^{68} +0.597563 q^{69} +6.64030 q^{70} -0.277410 q^{71} -20.3016 q^{72} +7.80557 q^{73} +11.9769 q^{74} -0.762573 q^{75} -28.8893 q^{76} +5.24292 q^{77} +2.27764 q^{78} +9.25023 q^{79} -27.3032 q^{80} +5.19977 q^{81} -27.5701 q^{82} -6.32179 q^{83} +3.37484 q^{84} +1.82236 q^{85} -2.28193 q^{87} -41.1360 q^{88} +5.33258 q^{89} +16.7638 q^{90} +1.30753 q^{91} -4.48478 q^{92} -2.58007 q^{93} -1.30902 q^{94} +14.3166 q^{95} -8.83729 q^{96} -6.49285 q^{97} -15.8106 q^{98} +13.2360 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 14 q^{4} - 32 q^{6} - 6 q^{9}+O(q^{10}) $ 20 * q + 14 * q^4 - 32 * q^6 - 6 * q^9	$ 20 q + 14 q^{4} - 32 q^{6} - 6 q^{9} + 12 q^{10} - 20 q^{11} + 6 q^{13} - 34 q^{14} - 34 q^{15} + 14 q^{16} - 8 q^{17} + 12 q^{21} - 46 q^{23} + 2 q^{24} + 20 q^{25} - 38 q^{31} - 76 q^{35} - 46 q^{36} - 68 q^{38} - 64 q^{40} - 56 q^{41} - 58 q^{44} - 24 q^{47} - 20 q^{49} - 20 q^{52} - 42 q^{53} + 66 q^{54} - 46 q^{56} + 2 q^{57} + 12 q^{58} - 90 q^{59} + 18 q^{60} - 28 q^{64} + 76 q^{66} - 62 q^{67} - 54 q^{68} + 24 q^{74} - 36 q^{78} - 10 q^{79} - 48 q^{81} - 68 q^{83} + 70 q^{84} - 12 q^{87} + 58 q^{90} - 8 q^{92} - 42 q^{95} - 22 q^{96} - 44 q^{97} - 38 q^{99}+O(q^{100}) $ 20 * q + 14 * q^4 - 32 * q^6 - 6 * q^9 + 12 * q^10 - 20 * q^11 + 6 * q^13 - 34 * q^14 - 34 * q^15 + 14 * q^16 - 8 * q^17 + 12 * q^21 - 46 * q^23 + 2 * q^24 + 20 * q^25 - 38 * q^31 - 76 * q^35 - 46 * q^36 - 68 * q^38 - 64 * q^40 - 56 * q^41 - 58 * q^44 - 24 * q^47 - 20 * q^49 - 20 * q^52 - 42 * q^53 + 66 * q^54 - 46 * q^56 + 2 * q^57 + 12 * q^58 - 90 * q^59 + 18 * q^60 - 28 * q^64 + 76 * q^66 - 62 * q^67 - 54 * q^68 + 24 * q^74 - 36 * q^78 - 10 * q^79 - 48 * q^81 - 68 * q^83 + 70 * q^84 - 12 * q^87 + 58 * q^90 - 8 * q^92 - 42 * q^95 - 22 * q^96 - 44 * q^97 - 38 * 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 $

$2$	2.64610	1.87108	0.935539	−	0.353225i	$-0.114915\pi$
$2$	2.64610	1.87108	0.935539	+	0.353225i	$0.114915\pi$
$3$	−0.666460	−0.384781	−0.192390	−	0.981318i	$-0.561624\pi$
$3$	−0.666460	−0.384781	−0.192390	+	0.981318i	$0.561624\pi$
$4$	5.00186	2.50093
$5$	−2.47875	−1.10853	−0.554266	−	0.832340i	$-0.687001\pi$
$5$	−2.47875	−1.10853	−0.554266	+	0.832340i	$0.687001\pi$
$6$	−1.76352	−0.719954
$7$	−1.01239	−0.382647	−0.191324	−	0.981527i	$-0.561278\pi$
$7$	−1.01239	−0.382647	−0.191324	+	0.981527i	$0.561278\pi$
$8$	7.94323	2.80836
$9$	−2.55583	−0.851944
$10$	−6.55903	−2.07415
$11$	−5.17875	−1.56145	−0.780727	−	0.624873i	$-0.785151\pi$
$11$	−5.17875	−1.56145	−0.780727	+	0.624873i	$0.785151\pi$
$12$	−3.33354	−0.962310
$13$	−1.29153	−0.358206	−0.179103	−	0.983830i	$-0.557320\pi$
$13$	−1.29153	−0.358206	−0.179103	+	0.983830i	$0.557320\pi$
$14$	−2.67889	−0.715963
$15$	1.65199	0.426542
$16$	11.0149	2.75372
$17$	−0.735191	−0.178310	−0.0891550	−	0.996018i	$-0.528417\pi$
$17$	−0.735191	−0.178310	−0.0891550	+	0.996018i	$0.528417\pi$
$18$	−6.76299	−1.59405
$19$	−5.77571	−1.32504	−0.662520	−	0.749044i	$-0.730513\pi$
$19$	−5.77571	−1.32504	−0.662520	+	0.749044i	$0.730513\pi$
$20$	−12.3984	−2.77236
$21$	0.674717	0.147235
$22$	−13.7035	−2.92160
$23$	−0.896622	−0.186959	−0.0934793	−	0.995621i	$-0.529799\pi$
$23$	−0.896622	−0.186959	−0.0934793	+	0.995621i	$0.529799\pi$
$24$	−5.29384	−1.08060
$25$	1.14421	0.228843
$26$	−3.41752	−0.670231
$27$	3.70274	0.712592
$28$	−5.06383	−0.956974
$29$	3.42395	0.635812	0.317906	−	0.948122i	$-0.397020\pi$
$29$	3.42395	0.635812	0.317906	+	0.948122i	$0.397020\pi$
$30$	4.37133	0.798092
$31$	3.87131	0.695307	0.347654	−	0.937623i	$-0.386979\pi$
$31$	3.87131	0.695307	0.347654	+	0.937623i	$0.386979\pi$
$32$	13.2600	2.34407
$33$	3.45143	0.600817
$34$	−1.94539	−0.333632
$35$	2.50946	0.424177
$36$	−12.7839	−2.13065
$37$	4.52623	0.744108	0.372054	−	0.928211i	$-0.378654\pi$
$37$	4.52623	0.744108	0.372054	+	0.928211i	$0.378654\pi$
$38$	−15.2831	−2.47925
$39$	0.860753	0.137831
$40$	−19.6893	−3.11315
$41$	−10.4191	−1.62720	−0.813598	−	0.581428i	$-0.802494\pi$
$41$	−10.4191	−1.62720	−0.813598	+	0.581428i	$0.802494\pi$
$42$	1.78537	0.275489
$43$	0	0
$43$	0	0
$44$	−25.9034	−3.90508
$45$	6.33527	0.944407
$46$	−2.37255	−0.349814
$47$	−0.494696	−0.0721588	−0.0360794	−	0.999349i	$-0.511487\pi$
$47$	−0.494696	−0.0721588	−0.0360794	+	0.999349i	$0.511487\pi$
$48$	−7.34097	−1.05958
$49$	−5.97507	−0.853581
$50$	3.02771	0.428183
$51$	0.489975	0.0686103
$52$	−6.46005	−0.895848
$53$	10.3172	1.41718	0.708589	−	0.705621i	$-0.249332\pi$
$53$	10.3172	1.41718	0.708589	+	0.705621i	$0.249332\pi$
$54$	9.79783	1.33332
$55$	12.8368	1.73092
$56$	−8.04164	−1.07461
$57$	3.84928	0.509850
$58$	9.06012	1.18965
$59$	−1.40654	−0.183116	−0.0915581	−	0.995800i	$-0.529185\pi$
$59$	−1.40654	−0.183116	−0.0915581	+	0.995800i	$0.529185\pi$
$60$	8.26302	1.06675
$61$	2.40989	0.308555	0.154277	−	0.988028i	$-0.450695\pi$
$61$	2.40989	0.308555	0.154277	+	0.988028i	$0.450695\pi$
$62$	10.2439	1.30097
$63$	2.58750	0.325994
$64$	13.0577	1.63221
$65$	3.20138	0.397083
$66$	9.13284	1.12418
$67$	−5.71755	−0.698510	−0.349255	−	0.937028i	$-0.613565\pi$
$67$	−5.71755	−0.698510	−0.349255	+	0.937028i	$0.613565\pi$
$68$	−3.67732	−0.445941
$69$	0.597563	0.0719381
$70$	6.64030	0.793667
$71$	−0.277410	−0.0329225	−0.0164612	−	0.999865i	$-0.505240\pi$
$71$	−0.277410	−0.0329225	−0.0164612	+	0.999865i	$0.505240\pi$
$72$	−20.3016	−2.39256
$73$	7.80557	0.913573	0.456786	−	0.889576i	$-0.349000\pi$
$73$	7.80557	0.913573	0.456786	+	0.889576i	$0.349000\pi$
$74$	11.9769	1.39228
$75$	−0.762573	−0.0880543
$76$	−28.8893	−3.31383
$77$	5.24292	0.597486
$78$	2.27764	0.257892
$79$	9.25023	1.04073	0.520366	−	0.853943i	$-0.325796\pi$
$79$	9.25023	1.04073	0.520366	+	0.853943i	$0.325796\pi$
$80$	−27.3032	−3.05259
$81$	5.19977	0.577752
$82$	−27.5701	−3.04461
$83$	−6.32179	−0.693906	−0.346953	−	0.937882i	$-0.612784\pi$
$83$	−6.32179	−0.693906	−0.346953	+	0.937882i	$0.612784\pi$
$84$	3.37484	0.368225
$85$	1.82236	0.197662
$86$	0	0
$87$	−2.28193	−0.244648
$88$	−41.1360	−4.38512
$89$	5.33258	0.565253	0.282626	−	0.959230i	$-0.408794\pi$
$89$	5.33258	0.565253	0.282626	+	0.959230i	$0.408794\pi$
$90$	16.7638	1.76706
$91$	1.30753	0.137067
$92$	−4.48478	−0.467571
$93$	−2.58007	−0.267541
$94$	−1.30902	−0.135015
$95$	14.3166	1.46885
$96$	−8.83729	−0.901952
$97$	−6.49285	−0.659249	−0.329624	−	0.944112i	$-0.606922\pi$
$97$	−6.49285	−0.659249	−0.329624	+	0.944112i	$0.606922\pi$
$98$	−15.8106	−1.59712
$99$	13.2360	1.33027
$100$	5.72320	0.572320
$101$	−16.2616	−1.61809	−0.809046	−	0.587745i	$-0.800016\pi$
$101$	−16.2616	−1.61809	−0.809046	+	0.587745i	$0.800016\pi$
$102$	1.29653	0.128375
$103$	−18.6787	−1.84047	−0.920233	−	0.391371i	$-0.872001\pi$
$103$	−18.6787	−1.84047	−0.920233	+	0.391371i	$0.872001\pi$
$104$	−10.2589	−1.00597
$105$	−1.67246	−0.163215
$106$	27.3004	2.65165
$107$	−11.6906	−1.13017	−0.565085	−	0.825033i	$-0.691157\pi$
$107$	−11.6906	−1.13017	−0.565085	+	0.825033i	$0.691157\pi$
$108$	18.5206	1.78214
$109$	10.6717	1.02216	0.511081	−	0.859533i	$-0.329245\pi$
$109$	10.6717	1.02216	0.511081	+	0.859533i	$0.329245\pi$
$110$	33.9676	3.23869
$111$	−3.01655	−0.286319
$112$	−11.1513	−1.05370
$113$	−9.94181	−0.935246	−0.467623	−	0.883928i	$-0.654890\pi$
$113$	−9.94181	−0.935246	−0.467623	+	0.883928i	$0.654890\pi$
$114$	10.1856	0.953968
$115$	2.22251	0.207250
$116$	17.1261	1.59012
$117$	3.30093	0.305171
$118$	−3.72185	−0.342624
$119$	0.744300	0.0682299
$120$	13.1221	1.19788
$121$	15.8195	1.43814
$122$	6.37681	0.577330
$123$	6.94394	0.626114
$124$	19.3637	1.73891
$125$	9.55754	0.854852
$126$	6.84678	0.609960
$127$	10.4512	0.927394	0.463697	−	0.885994i	$-0.346523\pi$
$127$	10.4512	0.927394	0.463697	+	0.885994i	$0.346523\pi$
$128$	8.03188	0.709924
$129$	0	0
$130$	8.47119	0.742972
$131$	17.9873	1.57156	0.785778	−	0.618509i	$-0.212263\pi$
$131$	17.9873	1.57156	0.785778	+	0.618509i	$0.212263\pi$
$132$	17.2636	1.50260
$133$	5.84727	0.507023
$134$	−15.1292	−1.30697
$135$	−9.17817	−0.789931
$136$	−5.83979	−0.500758
$137$	−21.6306	−1.84803	−0.924013	−	0.382362i	$-0.875111\pi$
$137$	−21.6306	−1.84803	−0.924013	+	0.382362i	$0.875111\pi$
$138$	1.58121	0.134602
$139$	−7.78711	−0.660494	−0.330247	−	0.943894i	$-0.607132\pi$
$139$	−7.78711	−0.660494	−0.330247	+	0.943894i	$0.607132\pi$
$140$	12.5520	1.06084
$141$	0.329695	0.0277653
$142$	−0.734054	−0.0616005
$143$	6.68852	0.559322
$144$	−28.1522	−2.34601
$145$	−8.48713	−0.704817
$146$	20.6543	1.70937
$147$	3.98214	0.328442
$148$	22.6396	1.86096
$149$	11.1359	0.912291	0.456145	−	0.889905i	$-0.349230\pi$
$149$	11.1359	0.912291	0.456145	+	0.889905i	$0.349230\pi$
$150$	−2.01785	−0.164756
$151$	13.9494	1.13519	0.567594	−	0.823309i	$-0.307875\pi$
$151$	13.9494	1.13519	0.567594	+	0.823309i	$0.307875\pi$
$152$	−45.8778	−3.72118
$153$	1.87902	0.151910
$154$	13.8733	1.11794
$155$	−9.59601	−0.770770
$156$	4.30536	0.344705
$157$	−17.6545	−1.40898	−0.704489	−	0.709715i	$-0.748824\pi$
$157$	−17.6545	−1.40898	−0.704489	+	0.709715i	$0.748824\pi$
$158$	24.4771	1.94729
$159$	−6.87601	−0.545303
$160$	−32.8684	−2.59847
$161$	0.907731	0.0715392
$162$	13.7591	1.08102
$163$	−15.7369	−1.23261	−0.616303	−	0.787509i	$-0.711370\pi$
$163$	−15.7369	−1.23261	−0.616303	+	0.787509i	$0.711370\pi$
$164$	−52.1151	−4.06950
$165$	−8.55524	−0.666025
$166$	−16.7281	−1.29835
$167$	−20.7271	−1.60391	−0.801955	−	0.597384i	$-0.796207\pi$
$167$	−20.7271	−1.60391	−0.801955	+	0.597384i	$0.796207\pi$
$168$	5.35943	0.413489
$169$	−11.3320	−0.871689
$170$	4.82214	0.369842
$171$	14.7617	1.12886
$172$	0	0
$173$	7.58651	0.576792	0.288396	−	0.957511i	$-0.406878\pi$
$173$	7.58651	0.576792	0.288396	+	0.957511i	$0.406878\pi$
$174$	−6.03821	−0.457755
$175$	−1.15839	−0.0875661
$176$	−57.0434	−4.29980
$177$	0.937404	0.0704596
$178$	14.1106	1.05763
$179$	−3.77331	−0.282030	−0.141015	−	0.990007i	$-0.545037\pi$
$179$	−3.77331	−0.282030	−0.141015	+	0.990007i	$0.545037\pi$
$180$	31.6881	2.36189
$181$	4.12671	0.306736	0.153368	−	0.988169i	$-0.450988\pi$
$181$	4.12671	0.306736	0.153368	+	0.988169i	$0.450988\pi$
$182$	3.45986	0.256462
$183$	−1.60609	−0.118726
$184$	−7.12208	−0.525046
$185$	−11.2194	−0.824868
$186$	−6.82713	−0.500590
$187$	3.80737	0.278423
$188$	−2.47440	−0.180464
$189$	−3.74861	−0.272672
$190$	37.8831	2.74833
$191$	−7.18164	−0.519645	−0.259823	−	0.965656i	$-0.583664\pi$
$191$	−7.18164	−0.519645	−0.259823	+	0.965656i	$0.583664\pi$
$192$	−8.70242	−0.628043
$193$	1.61440	0.116207	0.0581035	−	0.998311i	$-0.481495\pi$
$193$	1.61440	0.116207	0.0581035	+	0.998311i	$0.481495\pi$
$194$	−17.1807	−1.23351
$195$	−2.13359	−0.152790
$196$	−29.8864	−2.13475
$197$	20.6185	1.46901	0.734504	−	0.678605i	$-0.237415\pi$
$197$	20.6185	1.46901	0.734504	+	0.678605i	$0.237415\pi$
$198$	35.0239	2.48904
$199$	2.09208	0.148303	0.0741516	−	0.997247i	$-0.476375\pi$
$199$	2.09208	0.148303	0.0741516	+	0.997247i	$0.476375\pi$
$200$	9.08875	0.642672
$201$	3.81052	0.268773
$202$	−43.0299	−3.02758
$203$	−3.46637	−0.243292
$204$	2.45079	0.171589
$205$	25.8265	1.80380
$206$	−49.4257	−3.44365
$207$	2.29162	0.159278
$208$	−14.2260	−0.986399
$209$	29.9110	2.06899
$210$	−4.42549	−0.305388
$211$	14.9191	1.02708	0.513538	−	0.858067i	$-0.328334\pi$
$211$	14.9191	1.02708	0.513538	+	0.858067i	$0.328334\pi$
$212$	51.6053	3.54426
$213$	0.184882	0.0126679
$214$	−30.9345	−2.11464
$215$	0	0
$216$	29.4117	2.00121
$217$	−3.91927	−0.266057
$218$	28.2384	1.91254
$219$	−5.20210	−0.351525
$220$	64.2081	4.32891
$221$	0.949521	0.0638717
$222$	−7.98211	−0.535724
$223$	−8.99839	−0.602577	−0.301288	−	0.953533i	$-0.597417\pi$
$223$	−8.99839	−0.602577	−0.301288	+	0.953533i	$0.597417\pi$
$224$	−13.4243	−0.896951
$225$	−2.92442	−0.194961
$226$	−26.3070	−1.74992
$227$	14.7727	0.980497	0.490249	−	0.871583i	$-0.336906\pi$
$227$	14.7727	0.980497	0.490249	+	0.871583i	$0.336906\pi$
$228$	19.2536	1.27510
$229$	17.3557	1.14690	0.573448	−	0.819242i	$-0.305605\pi$
$229$	17.3557	1.14690	0.573448	+	0.819242i	$0.305605\pi$
$230$	5.88098	0.387780
$231$	−3.49419	−0.229901
$232$	27.1972	1.78558
$233$	−5.17803	−0.339224	−0.169612	−	0.985511i	$-0.554251\pi$
$233$	−5.17803	−0.339224	−0.169612	+	0.985511i	$0.554251\pi$
$234$	8.73461	0.570999
$235$	1.22623	0.0799903
$236$	−7.03533	−0.457961
$237$	−6.16491	−0.400454
$238$	1.96949	0.127663
$239$	−1.52959	−0.0989407	−0.0494704	−	0.998776i	$-0.515753\pi$
$239$	−1.52959	−0.0989407	−0.0494704	+	0.998776i	$0.515753\pi$
$240$	18.1965	1.17458
$241$	3.55448	0.228964	0.114482	−	0.993425i	$-0.463479\pi$
$241$	3.55448	0.228964	0.114482	+	0.993425i	$0.463479\pi$
$242$	41.8600	2.69086
$243$	−14.5737	−0.934900
$244$	12.0539	0.771674
$245$	14.8107	0.946222
$246$	18.3744	1.17151
$247$	7.45951	0.474637
$248$	30.7507	1.95267
$249$	4.21322	0.267002
$250$	25.2902	1.59949
$251$	−24.5195	−1.54766	−0.773829	−	0.633395i	$-0.781661\pi$
$251$	−24.5195	−1.54766	−0.773829	+	0.633395i	$0.781661\pi$
$252$	12.9423	0.815288
$253$	4.64339	0.291927
$254$	27.6549	1.73523
$255$	−1.21453	−0.0760567
$256$	−4.86219	−0.303887
$257$	−9.68556	−0.604169	−0.302084	−	0.953281i	$-0.597682\pi$
$257$	−9.68556	−0.604169	−0.302084	+	0.953281i	$0.597682\pi$
$258$	0	0
$259$	−4.58231	−0.284731
$260$	16.0129	0.993076
$261$	−8.75104	−0.541676
$262$	47.5962	2.94050
$263$	5.19348	0.320244	0.160122	−	0.987097i	$-0.448811\pi$
$263$	5.19348	0.320244	0.160122	+	0.987097i	$0.448811\pi$
$264$	27.4155	1.68731
$265$	−25.5738	−1.57099
$266$	15.4725	0.948679
$267$	−3.55395	−0.217498
$268$	−28.5984	−1.74692
$269$	−6.37184	−0.388498	−0.194249	−	0.980952i	$-0.562227\pi$
$269$	−6.37184	−0.388498	−0.194249	+	0.980952i	$0.562227\pi$
$270$	−24.2864	−1.47802
$271$	28.8525	1.75266	0.876332	−	0.481709i	$-0.159984\pi$
$271$	28.8525	1.75266	0.876332	+	0.481709i	$0.159984\pi$
$272$	−8.09804	−0.491016
$273$	−0.871417	−0.0527406
$274$	−57.2367	−3.45780
$275$	−5.92560	−0.357327
$276$	2.98893	0.179912
$277$	−6.33573	−0.380677	−0.190338	−	0.981719i	$-0.560959\pi$
$277$	−6.33573	−0.380677	−0.190338	+	0.981719i	$0.560959\pi$
$278$	−20.6055	−1.23584
$279$	−9.89441	−0.592363
$280$	19.9332	1.19124
$281$	−14.0084	−0.835669	−0.417835	−	0.908523i	$-0.637211\pi$
$281$	−14.0084	−0.835669	−0.417835	+	0.908523i	$0.637211\pi$
$282$	0.872407	0.0519511
$283$	−12.8475	−0.763702	−0.381851	−	0.924224i	$-0.624713\pi$
$283$	−12.8475	−0.763702	−0.381851	+	0.924224i	$0.624713\pi$
$284$	−1.38756	−0.0823368
$285$	−9.54141	−0.565185
$286$	17.6985	1.04653
$287$	10.5482	0.622642
$288$	−33.8904	−1.99701
$289$	−16.4595	−0.968206
$290$	−22.4578	−1.31877
$291$	4.32722	0.253666
$292$	39.0424	2.28478
$293$	−20.7598	−1.21280	−0.606401	−	0.795159i	$-0.707387\pi$
$293$	−20.7598	−1.21280	−0.606401	+	0.795159i	$0.707387\pi$
$294$	10.5372	0.614540
$295$	3.48647	0.202990
$296$	35.9529	2.08972
$297$	−19.1756	−1.11268
$298$	29.4668	1.70697
$299$	1.15801	0.0669697
$300$	−3.81428	−0.220218
$301$	0	0
$302$	36.9116	2.12402
$303$	10.8377	0.622611
$304$	−63.6188	−3.64879
$305$	−5.97352	−0.342043
$306$	4.97209	0.284236
$307$	−10.8928	−0.621683	−0.310842	−	0.950462i	$-0.600611\pi$
$307$	−10.8928	−0.621683	−0.310842	+	0.950462i	$0.600611\pi$
$308$	26.2243	1.49427
$309$	12.4486	0.708176
$310$	−25.3920	−1.44217
$311$	0.518250	0.0293873	0.0146936	−	0.999892i	$-0.495323\pi$
$311$	0.518250	0.0293873	0.0146936	+	0.999892i	$0.495323\pi$
$312$	6.83716	0.387078
$313$	9.36639	0.529419	0.264710	−	0.964328i	$-0.414724\pi$
$313$	9.36639	0.529419	0.264710	+	0.964328i	$0.414724\pi$
$314$	−46.7155	−2.63631
$315$	−6.41376	−0.361375
$316$	46.2683	2.60280
$317$	22.7639	1.27855	0.639273	−	0.768980i	$-0.279236\pi$
$317$	22.7639	1.27855	0.639273	+	0.768980i	$0.279236\pi$
$318$	−18.1946	−1.02030
$319$	−17.7318	−0.992790
$320$	−32.3668	−1.80936
$321$	7.79130	0.434868
$322$	2.40195	0.133855
$323$	4.24625	0.236268
$324$	26.0085	1.44492
$325$	−1.47779	−0.0819728
$326$	−41.6413	−2.30630
$327$	−7.11225	−0.393308
$328$	−82.7616	−4.56974
$329$	0.500825	0.0276114
$330$	−22.6381	−1.24618
$331$	25.2966	1.39043	0.695213	−	0.718804i	$-0.255310\pi$
$331$	25.2966	1.39043	0.695213	+	0.718804i	$0.255310\pi$
$332$	−31.6207	−1.73541
$333$	−11.5683	−0.633938
$334$	−54.8460	−3.00104
$335$	14.1724	0.774321
$336$	7.43193	0.405445
$337$	28.3161	1.54248	0.771239	−	0.636546i	$-0.219637\pi$
$337$	28.3161	1.54248	0.771239	+	0.636546i	$0.219637\pi$
$338$	−29.9855	−1.63100
$339$	6.62581	0.359865
$340$	9.11517	0.494340
$341$	−20.0485	−1.08569
$342$	39.0611	2.11218
$343$	13.1358	0.709268
$344$	0	0
$345$	−1.48121	−0.0797457
$346$	20.0747	1.07922
$347$	17.4247	0.935406	0.467703	−	0.883886i	$-0.345082\pi$
$347$	17.4247	0.935406	0.467703	+	0.883886i	$0.345082\pi$
$348$	−11.4139	−0.611848
$349$	−27.3650	−1.46482	−0.732408	−	0.680866i	$-0.761604\pi$
$349$	−27.3650	−1.46482	−0.732408	+	0.680866i	$0.761604\pi$
$350$	−3.06522	−0.163843
$351$	−4.78220	−0.255255
$352$	−68.6705	−3.66015
$353$	−1.05837	−0.0563312	−0.0281656	−	0.999603i	$-0.508967\pi$
$353$	−1.05837	−0.0563312	−0.0281656	+	0.999603i	$0.508967\pi$
$354$	2.48047	0.131835
$355$	0.687630	0.0364956
$356$	26.6728	1.41366
$357$	−0.496046	−0.0262535
$358$	−9.98456	−0.527700
$359$	−2.33274	−0.123117	−0.0615587	−	0.998103i	$-0.519607\pi$
$359$	−2.33274	−0.123117	−0.0615587	+	0.998103i	$0.519607\pi$
$360$	50.3225	2.65223
$361$	14.3589	0.755730
$362$	10.9197	0.573926
$363$	−10.5431	−0.553367
$364$	6.54009	0.342794
$365$	−19.3481	−1.01272
$366$	−4.24989	−0.222145
$367$	−10.4810	−0.547106	−0.273553	−	0.961857i	$-0.588199\pi$
$367$	−10.4810	−0.547106	−0.273553	+	0.961857i	$0.588199\pi$
$368$	−9.87619	−0.514832
$369$	26.6296	1.38628
$370$	−29.6877	−1.54339
$371$	−10.4450	−0.542280
$372$	−12.9052	−0.669101
$373$	6.30435	0.326427	0.163213	−	0.986591i	$-0.447814\pi$
$373$	6.30435	0.326427	0.163213	+	0.986591i	$0.447814\pi$
$374$	10.0747	0.520951
$375$	−6.36972	−0.328931
$376$	−3.92948	−0.202648
$377$	−4.42213	−0.227751
$378$	−9.91922	−0.510189
$379$	1.42939	0.0734226	0.0367113	−	0.999326i	$-0.488312\pi$
$379$	1.42939	0.0734226	0.0367113	+	0.999326i	$0.488312\pi$
$380$	71.6094	3.67349
$381$	−6.96530	−0.356843
$382$	−19.0034	−0.972296
$383$	−15.2608	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$383$	−15.2608	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$384$	−5.35292	−0.273165
$385$	−12.9959	−0.662332
$386$	4.27187	0.217432
$387$	0	0
$388$	−32.4763	−1.64873
$389$	−22.8598	−1.15904	−0.579519	−	0.814959i	$-0.696760\pi$
$389$	−22.8598	−1.15904	−0.579519	+	0.814959i	$0.696760\pi$
$390$	−5.64571	−0.285881
$391$	0.659189	0.0333366
$392$	−47.4613	−2.39716
$393$	−11.9878	−0.604704
$394$	54.5587	2.74863
$395$	−22.9290	−1.15368
$396$	66.2047	3.32691
$397$	36.7056	1.84220	0.921101	−	0.389323i	$-0.127291\pi$
$397$	36.7056	1.84220	0.921101	+	0.389323i	$0.127291\pi$
$398$	5.53585	0.277487
$399$	−3.89697	−0.195093
$400$	12.6034	0.630169
$401$	−22.7121	−1.13419	−0.567095	−	0.823653i	$-0.691933\pi$
$401$	−22.7121	−1.13419	−0.567095	+	0.823653i	$0.691933\pi$
$402$	10.0830	0.502895
$403$	−4.99991	−0.249063
$404$	−81.3384	−4.04674
$405$	−12.8889	−0.640456
$406$	−9.17238	−0.455217
$407$	−23.4403	−1.16189
$408$	3.89199	0.192682
$409$	21.2962	1.05303	0.526515	−	0.850166i	$-0.323498\pi$
$409$	21.2962	1.05303	0.526515	+	0.850166i	$0.323498\pi$
$410$	68.3395	3.37505
$411$	14.4159	0.711084
$412$	−93.4282	−4.60288
$413$	1.42397	0.0700689
$414$	6.06385	0.298022
$415$	15.6701	0.769217
$416$	−17.1257	−0.839659
$417$	5.18980	0.254146
$418$	79.1476	3.87123
$419$	−5.26938	−0.257426	−0.128713	−	0.991682i	$-0.541085\pi$
$419$	−5.26938	−0.257426	−0.128713	+	0.991682i	$0.541085\pi$
$420$	−8.36539	−0.408189
$421$	10.7715	0.524972	0.262486	−	0.964936i	$-0.415458\pi$
$421$	10.7715	0.524972	0.262486	+	0.964936i	$0.415458\pi$
$422$	39.4776	1.92174
$423$	1.26436	0.0614752
$424$	81.9520	3.97994
$425$	−0.841216	−0.0408050
$426$	0.489218	0.0237027
$427$	−2.43975	−0.118068
$428$	−58.4746	−2.82648
$429$	−4.45763	−0.215216
$430$	0	0
$431$	−21.8440	−1.05219	−0.526094	−	0.850427i	$-0.676344\pi$
$431$	−21.8440	−1.05219	−0.526094	+	0.850427i	$0.676344\pi$
$432$	40.7852	1.96228
$433$	−22.1447	−1.06421	−0.532104	−	0.846679i	$-0.678599\pi$
$433$	−22.1447	−1.06421	−0.532104	+	0.846679i	$0.678599\pi$
$434$	−10.3708	−0.497814
$435$	5.65633	0.271200
$436$	53.3782	2.55635
$437$	5.17863	0.247728
$438$	−13.7653	−0.657731
$439$	−19.6948	−0.939983	−0.469992	−	0.882671i	$-0.655743\pi$
$439$	−19.6948	−0.939983	−0.469992	+	0.882671i	$0.655743\pi$
$440$	101.966	4.86104
$441$	15.2713	0.727203
$442$	2.51253	0.119509
$443$	14.1271	0.671198	0.335599	−	0.942005i	$-0.391061\pi$
$443$	14.1271	0.671198	0.335599	+	0.942005i	$0.391061\pi$
$444$	−15.0884	−0.716063
$445$	−13.2182	−0.626600
$446$	−23.8107	−1.12747
$447$	−7.42165	−0.351032
$448$	−13.2195	−0.624561
$449$	−19.9690	−0.942394	−0.471197	−	0.882028i	$-0.656178\pi$
$449$	−19.9690	−0.942394	−0.471197	+	0.882028i	$0.656178\pi$
$450$	−7.73831	−0.364787
$451$	53.9581	2.54079
$452$	−49.7275	−2.33899
$453$	−9.29672	−0.436798
$454$	39.0900	1.83459
$455$	−3.24105	−0.151943
$456$	30.5757	1.43184
$457$	19.4197	0.908414	0.454207	−	0.890896i	$-0.349923\pi$
$457$	19.4197	0.908414	0.454207	+	0.890896i	$0.349923\pi$
$458$	45.9250	2.14593
$459$	−2.72222	−0.127062
$460$	11.1167	0.518317
$461$	9.78257	0.455620	0.227810	−	0.973706i	$-0.426844\pi$
$461$	9.78257	0.455620	0.227810	+	0.973706i	$0.426844\pi$
$462$	−9.24599	−0.430163
$463$	−6.53117	−0.303529	−0.151765	−	0.988417i	$-0.548496\pi$
$463$	−6.53117	−0.303529	−0.151765	+	0.988417i	$0.548496\pi$
$464$	37.7144	1.75085
$465$	6.39536	0.296578
$466$	−13.7016	−0.634715
$467$	−22.8687	−1.05823	−0.529117	−	0.848549i	$-0.677477\pi$
$467$	−22.8687	−1.05823	−0.529117	+	0.848549i	$0.677477\pi$
$468$	16.5108	0.763212
$469$	5.78839	0.267283
$470$	3.24473	0.149668
$471$	11.7660	0.542148
$472$	−11.1725	−0.514255
$473$	0	0
$474$	−16.3130	−0.749280
$475$	−6.60865	−0.303226
$476$	3.72288	0.170638
$477$	−26.3691	−1.20736
$478$	−4.04744	−0.185126
$479$	−8.59652	−0.392785	−0.196392	−	0.980525i	$-0.562923\pi$
$479$	−8.59652	−0.392785	−0.196392	+	0.980525i	$0.562923\pi$
$480$	21.9054	0.999842
$481$	−5.84577	−0.266544
$482$	9.40551	0.428409
$483$	−0.604966	−0.0275269
$484$	79.1269	3.59668
$485$	16.0942	0.730798
$486$	−38.5634	−1.74927
$487$	34.6207	1.56882	0.784408	−	0.620245i	$-0.212967\pi$
$487$	34.6207	1.56882	0.784408	+	0.620245i	$0.212967\pi$
$488$	19.1423	0.866531
$489$	10.4880	0.474283
$490$	39.1907	1.77045
$491$	−15.4789	−0.698552	−0.349276	−	0.937020i	$-0.613572\pi$
$491$	−15.4789	−0.698552	−0.349276	+	0.937020i	$0.613572\pi$
$492$	34.7326	1.56587
$493$	−2.51726	−0.113372
$494$	19.7386	0.888083
$495$	−32.8088	−1.47465
$496$	42.6420	1.91468
$497$	0.280847	0.0125977
$498$	11.1486	0.499581
$499$	5.42032	0.242647	0.121324	−	0.992613i	$-0.461286\pi$
$499$	5.42032	0.242647	0.121324	+	0.992613i	$0.461286\pi$
$500$	47.8055	2.13793
$501$	13.8138	0.617154
$502$	−64.8811	−2.89579
$503$	−42.7129	−1.90447	−0.952237	−	0.305360i	$-0.901223\pi$
$503$	−42.7129	−1.90447	−0.952237	+	0.305360i	$0.901223\pi$
$504$	20.5531	0.915507
$505$	40.3086	1.79371
$506$	12.2869	0.546218
$507$	7.55229	0.335409
$508$	52.2754	2.31935
$509$	−11.2661	−0.499361	−0.249681	−	0.968328i	$-0.580326\pi$
$509$	−11.2661	−0.499361	−0.249681	+	0.968328i	$0.580326\pi$
$510$	−3.21376	−0.142308
$511$	−7.90228	−0.349576
$512$	−28.9296	−1.27852
$513$	−21.3860	−0.944213
$514$	−25.6290	−1.13045
$515$	46.2998	2.04021
$516$	0	0
$517$	2.56191	0.112673
$518$	−12.1253	−0.532754
$519$	−5.05611	−0.221938
$520$	25.4293	1.11515
$521$	−3.83959	−0.168216	−0.0841078	−	0.996457i	$-0.526804\pi$
$521$	−3.83959	−0.168216	−0.0841078	+	0.996457i	$0.526804\pi$
$522$	−23.1561	−1.01352
$523$	29.4776	1.28896	0.644482	−	0.764619i	$-0.277073\pi$
$523$	29.4776	1.28896	0.644482	+	0.764619i	$0.277073\pi$
$524$	89.9698	3.93035
$525$	0.772021	0.0336937
$526$	13.7425	0.599201
$527$	−2.84615	−0.123980
$528$	38.0171	1.65448
$529$	−22.1961	−0.965046
$530$	−67.6710	−2.93944
$531$	3.59488	0.156005
$532$	29.2472	1.26803
$533$	13.4566	0.582871
$534$	−9.40412	−0.406956
$535$	28.9780	1.25283
$536$	−45.4158	−1.96166
$537$	2.51476	0.108520
$538$	−16.8605	−0.726909
$539$	30.9434	1.33283
$540$	−45.9079	−1.97556
$541$	−2.55702	−0.109935	−0.0549675	−	0.998488i	$-0.517506\pi$
$541$	−2.55702	−0.109935	−0.0549675	+	0.998488i	$0.517506\pi$
$542$	76.3466	3.27937
$543$	−2.75028	−0.118026
$544$	−9.74867	−0.417971
$545$	−26.4525	−1.13310
$546$	−2.30586	−0.0986817
$547$	−21.3945	−0.914761	−0.457381	−	0.889271i	$-0.651212\pi$
$547$	−21.3945	−0.914761	−0.457381	+	0.889271i	$0.651212\pi$
$548$	−108.193	−4.62178
$549$	−6.15927	−0.262871
$550$	−15.6798	−0.668587
$551$	−19.7758	−0.842476
$552$	4.74658	0.202028
$553$	−9.36483	−0.398233
$554$	−16.7650	−0.712276
$555$	7.47729	0.317393
$556$	−38.9500	−1.65185
$557$	27.2745	1.15566	0.577829	−	0.816158i	$-0.303900\pi$
$557$	27.2745	1.15566	0.577829	+	0.816158i	$0.303900\pi$
$558$	−26.1816	−1.10836
$559$	0	0
$560$	27.6414	1.16806
$561$	−2.53746	−0.107132
$562$	−37.0676	−1.56360
$563$	−8.66764	−0.365298	−0.182649	−	0.983178i	$-0.558467\pi$
$563$	−8.66764	−0.365298	−0.182649	+	0.983178i	$0.558467\pi$
$564$	1.64909	0.0694391
$565$	24.6433	1.03675
$566$	−33.9957	−1.42895
$567$	−5.26419	−0.221075
$568$	−2.20353	−0.0924580
$569$	−26.2869	−1.10200	−0.551001	−	0.834504i	$-0.685754\pi$
$569$	−26.2869	−1.10200	−0.551001	+	0.834504i	$0.685754\pi$
$570$	−25.2476	−1.05750
$571$	−24.5424	−1.02707	−0.513533	−	0.858070i	$-0.671664\pi$
$571$	−24.5424	−1.02707	−0.513533	+	0.858070i	$0.671664\pi$
$572$	33.4550	1.39882
$573$	4.78628	0.199950
$574$	27.9117	1.16501
$575$	−1.02593	−0.0427841
$576$	−33.3732	−1.39055
$577$	23.1670	0.964456	0.482228	−	0.876046i	$-0.339828\pi$
$577$	23.1670	0.964456	0.482228	+	0.876046i	$0.339828\pi$
$578$	−43.5535	−1.81159
$579$	−1.07593	−0.0447142
$580$	−42.4514	−1.76270
$581$	6.40011	0.265521
$582$	11.4503	0.474629
$583$	−53.4303	−2.21286
$584$	62.0014	2.56564
$585$	−8.18219	−0.338292
$586$	−54.9327	−2.26925
$587$	−37.2311	−1.53669	−0.768346	−	0.640035i	$-0.778920\pi$
$587$	−37.2311	−1.53669	−0.768346	+	0.640035i	$0.778920\pi$
$588$	19.9181	0.821409
$589$	−22.3596	−0.921310
$590$	9.22556	0.379810
$591$	−13.7414	−0.565246
$592$	49.8559	2.04907
$593$	14.0271	0.576022	0.288011	−	0.957627i	$-0.407006\pi$
$593$	14.0271	0.576022	0.288011	+	0.957627i	$0.407006\pi$
$594$	−50.7405	−2.08191
$595$	−1.84494	−0.0756350
$596$	55.7003	2.28157
$597$	−1.39428	−0.0570643
$598$	3.06423	0.125306
$599$	10.0367	0.410087	0.205043	−	0.978753i	$-0.434266\pi$
$599$	10.0367	0.410087	0.205043	+	0.978753i	$0.434266\pi$
$600$	−6.05729	−0.247288
$601$	−19.8537	−0.809849	−0.404924	−	0.914350i	$-0.632702\pi$
$601$	−19.8537	−0.809849	−0.404924	+	0.914350i	$0.632702\pi$
$602$	0	0
$603$	14.6131	0.595091
$604$	69.7730	2.83902
$605$	−39.2126	−1.59422
$606$	28.6777	1.16495
$607$	4.70014	0.190773	0.0953863	−	0.995440i	$-0.469591\pi$
$607$	4.70014	0.190773	0.0953863	+	0.995440i	$0.469591\pi$
$608$	−76.5862	−3.10598
$609$	2.31020	0.0936139
$610$	−15.8065	−0.639988
$611$	0.638914	0.0258477
$612$	9.39862	0.379917
$613$	−14.8092	−0.598137	−0.299069	−	0.954232i	$-0.596676\pi$
$613$	−14.8092	−0.598137	−0.299069	+	0.954232i	$0.596676\pi$
$614$	−28.8234	−1.16322
$615$	−17.2123	−0.694067
$616$	41.6457	1.67795
$617$	11.6585	0.469353	0.234676	−	0.972074i	$-0.424597\pi$
$617$	11.6585	0.469353	0.234676	+	0.972074i	$0.424597\pi$
$618$	32.9403	1.32505
$619$	−28.2375	−1.13496	−0.567480	−	0.823387i	$-0.692081\pi$
$619$	−28.2375	−1.13496	−0.567480	+	0.823387i	$0.692081\pi$
$620$	−47.9979	−1.92764
$621$	−3.31996	−0.133225
$622$	1.37134	0.0549858
$623$	−5.39865	−0.216292
$624$	9.48109	0.379547
$625$	−29.4118	−1.17647
$626$	24.7844	0.990585
$627$	−19.9345	−0.796106
$628$	−88.3051	−3.52376
$629$	−3.32765	−0.132682
$630$	−16.9715	−0.676160
$631$	−15.5979	−0.620944	−0.310472	−	0.950583i	$-0.600487\pi$
$631$	−15.5979	−0.620944	−0.310472	+	0.950583i	$0.600487\pi$
$632$	73.4767	2.92275
$633$	−9.94301	−0.395199
$634$	60.2355	2.39226
$635$	−25.9059	−1.02805
$636$	−34.3928	−1.36376
$637$	7.71698	0.305758
$638$	−46.9202	−1.85759
$639$	0.709012	0.0280481
$640$	−19.9090	−0.786974
$641$	2.57760	0.101809	0.0509046	−	0.998704i	$-0.483790\pi$
$641$	2.57760	0.101809	0.0509046	+	0.998704i	$0.483790\pi$
$642$	20.6166	0.813671
$643$	−27.9723	−1.10312	−0.551559	−	0.834136i	$-0.685967\pi$
$643$	−27.9723	−1.10312	−0.551559	+	0.834136i	$0.685967\pi$
$644$	4.54034	0.178915
$645$	0	0
$646$	11.2360	0.442075
$647$	5.47509	0.215248	0.107624	−	0.994192i	$-0.465676\pi$
$647$	5.47509	0.215248	0.107624	+	0.994192i	$0.465676\pi$
$648$	41.3029	1.62253
$649$	7.28414	0.285927
$650$	−3.91037	−0.153378
$651$	2.61204	0.102374
$652$	−78.7135	−3.08266
$653$	25.8330	1.01092	0.505462	−	0.862849i	$-0.331322\pi$
$653$	25.8330	1.01092	0.505462	+	0.862849i	$0.331322\pi$
$654$	−18.8197	−0.735910
$655$	−44.5860	−1.74212
$656$	−114.766	−4.48084
$657$	−19.9497	−0.778313
$658$	1.32523	0.0516630
$659$	−3.43526	−0.133819	−0.0669093	−	0.997759i	$-0.521314\pi$
$659$	−3.43526	−0.133819	−0.0669093	+	0.997759i	$0.521314\pi$
$660$	−42.7921	−1.66568
$661$	24.5731	0.955782	0.477891	−	0.878419i	$-0.341401\pi$
$661$	24.5731	0.955782	0.477891	+	0.878419i	$0.341401\pi$
$662$	66.9373	2.60159
$663$	−0.632818	−0.0245766
$664$	−50.2154	−1.94874
$665$	−14.4939	−0.562051
$666$	−30.6109	−1.18615
$667$	−3.06999	−0.118871
$668$	−103.674	−4.01127
$669$	5.99706	0.231860
$670$	37.5016	1.44881
$671$	−12.4802	−0.481794
$672$	8.94678	0.345129
$673$	19.3560	0.746120	0.373060	−	0.927807i	$-0.378309\pi$
$673$	19.3560	0.746120	0.373060	+	0.927807i	$0.378309\pi$
$674$	74.9274	2.88610
$675$	4.23672	0.163072
$676$	−56.6808	−2.18003
$677$	35.7269	1.37310	0.686548	−	0.727084i	$-0.259125\pi$
$677$	35.7269	1.37310	0.686548	+	0.727084i	$0.259125\pi$
$678$	17.5326	0.673335
$679$	6.57329	0.252260
$680$	14.4754	0.555106
$681$	−9.84540	−0.377277
$682$	−53.0505	−2.03141
$683$	28.7683	1.10079	0.550394	−	0.834905i	$-0.314478\pi$
$683$	28.7683	1.10079	0.550394	+	0.834905i	$0.314478\pi$
$684$	73.8362	2.82320
$685$	53.6169	2.04859
$686$	34.7587	1.32709
$687$	−11.5669	−0.441304
$688$	0	0
$689$	−13.3250	−0.507642
$690$	−3.91943	−0.149210
$691$	49.8613	1.89681	0.948406	−	0.317058i	$-0.102695\pi$
$691$	49.8613	1.89681	0.948406	+	0.317058i	$0.102695\pi$
$692$	37.9467	1.44252
$693$	−13.4000	−0.509024
$694$	46.1075	1.75022
$695$	19.3023	0.732179
$696$	−18.1259	−0.687059
$697$	7.66006	0.290145
$698$	−72.4107	−2.74079
$699$	3.45095	0.130527
$700$	−5.79411	−0.218997
$701$	27.7330	1.04746	0.523731	−	0.851884i	$-0.324540\pi$
$701$	27.7330	1.04746	0.523731	+	0.851884i	$0.324540\pi$
$702$	−12.6542	−0.477601
$703$	−26.1422	−0.985973
$704$	−67.6225	−2.54862
$705$	−0.817232	−0.0307787
$706$	−2.80055	−0.105400
$707$	16.4631	0.619159
$708$	4.68876	0.176214
$709$	18.3416	0.688833	0.344416	−	0.938817i	$-0.388077\pi$
$709$	18.3416	0.688833	0.344416	+	0.938817i	$0.388077\pi$
$710$	1.81954	0.0682861
$711$	−23.6420	−0.886645
$712$	42.3579	1.58743
$713$	−3.47110	−0.129994
$714$	−1.31259	−0.0491224
$715$	−16.5792	−0.620026
$716$	−18.8736	−0.705338
$717$	1.01941	0.0380705
$718$	−6.17267	−0.230362
$719$	−31.7888	−1.18552	−0.592762	−	0.805378i	$-0.701962\pi$
$719$	−31.7888	−1.18552	−0.592762	+	0.805378i	$0.701962\pi$
$720$	69.7823	2.60063
$721$	18.9101	0.704249
$722$	37.9950	1.41403
$723$	−2.36892	−0.0881009
$724$	20.6412	0.767125
$725$	3.91773	0.145501
$726$	−27.8980	−1.03539
$727$	30.7127	1.13907	0.569535	−	0.821967i	$-0.307123\pi$
$727$	30.7127	1.13907	0.569535	+	0.821967i	$0.307123\pi$
$728$	10.3860	0.384932
$729$	−5.88655	−0.218020
$730$	−51.1970	−1.89489
$731$	0	0
$732$	−8.03346	−0.296925
$733$	−27.7556	−1.02518	−0.512588	−	0.858635i	$-0.671313\pi$
$733$	−27.7556	−1.02518	−0.512588	+	0.858635i	$0.671313\pi$
$734$	−27.7339	−1.02368
$735$	−9.87075	−0.364088
$736$	−11.8893	−0.438244
$737$	29.6098	1.09069
$738$	70.4645	2.59384
$739$	29.8662	1.09865	0.549324	−	0.835610i	$-0.314885\pi$
$739$	29.8662	1.09865	0.549324	+	0.835610i	$0.314885\pi$
$740$	−56.1179	−2.06294
$741$	−4.97146	−0.182631
$742$	−27.6387	−1.01465
$743$	−34.7067	−1.27326	−0.636632	−	0.771167i	$-0.719673\pi$
$743$	−34.7067	−1.27326	−0.636632	+	0.771167i	$0.719673\pi$
$744$	−20.4941	−0.751350
$745$	−27.6032	−1.01130
$746$	16.6820	0.610770
$747$	16.1574	0.591169
$748$	19.0440	0.696316
$749$	11.8354	0.432457
$750$	−16.8549	−0.615455
$751$	−48.0491	−1.75334	−0.876668	−	0.481096i	$-0.840239\pi$
$751$	−48.0491	−1.75334	−0.876668	+	0.481096i	$0.840239\pi$
$752$	−5.44901	−0.198705
$753$	16.3413	0.595509
$754$	−11.7014	−0.426141
$755$	−34.5771	−1.25839
$756$	−18.7500	−0.681932
$757$	−26.9502	−0.979520	−0.489760	−	0.871857i	$-0.662916\pi$
$757$	−26.9502	−0.979520	−0.489760	+	0.871857i	$0.662916\pi$
$758$	3.78230	0.137379
$759$	−3.09463	−0.112328
$760$	113.720	4.12505
$761$	−31.3355	−1.13591	−0.567956	−	0.823059i	$-0.692266\pi$
$761$	−31.3355	−1.13591	−0.567956	+	0.823059i	$0.692266\pi$
$762$	−18.4309	−0.667681
$763$	−10.8039	−0.391127
$764$	−35.9216	−1.29960
$765$	−4.65764	−0.168397
$766$	−40.3817	−1.45905
$767$	1.81659	0.0655933
$768$	3.24045	0.116930
$769$	46.6635	1.68273	0.841365	−	0.540467i	$-0.181753\pi$
$769$	46.6635	1.68273	0.841365	+	0.540467i	$0.181753\pi$
$770$	−34.3885	−1.23927
$771$	6.45504	0.232473
$772$	8.07500	0.290626
$773$	−11.4797	−0.412895	−0.206447	−	0.978458i	$-0.566190\pi$
$773$	−11.4797	−0.412895	−0.206447	+	0.978458i	$0.566190\pi$
$774$	0	0
$775$	4.42960	0.159116
$776$	−51.5742	−1.85140
$777$	3.05393	0.109559
$778$	−60.4894	−2.16865
$779$	60.1779	2.15610
$780$	−10.6719	−0.382116
$781$	1.43664	0.0514069
$782$	1.74428	0.0623754
$783$	12.6780	0.453074
$784$	−65.8146	−2.35052
$785$	43.7610	1.56190
$786$	−31.7209	−1.13145
$787$	−53.7215	−1.91496	−0.957482	−	0.288493i	$-0.906846\pi$
$787$	−53.7215	−1.91496	−0.957482	+	0.288493i	$0.906846\pi$
$788$	103.131	3.67388
$789$	−3.46124	−0.123224
$790$	−60.6726	−2.15863
$791$	10.0650	0.357870
$792$	105.137	3.73587
$793$	−3.11244	−0.110526
$794$	97.1269	3.44690
$795$	17.0439	0.604486
$796$	10.4643	0.370896
$797$	−8.94720	−0.316926	−0.158463	−	0.987365i	$-0.550654\pi$
$797$	−8.94720	−0.316926	−0.158463	+	0.987365i	$0.550654\pi$
$798$	−10.3118	−0.365033
$799$	0.363696	0.0128666
$800$	15.1723	0.536423
$801$	−13.6292	−0.481563
$802$	−60.0986	−2.12216
$803$	−40.4231	−1.42650
$804$	19.0597	0.672183
$805$	−2.25004	−0.0793035
$806$	−13.2303	−0.466016
$807$	4.24657	0.149486
$808$	−129.170	−4.54418
$809$	−23.9640	−0.842530	−0.421265	−	0.906938i	$-0.638414\pi$
$809$	−23.9640	−0.842530	−0.421265	+	0.906938i	$0.638414\pi$
$810$	−34.1055	−1.19834
$811$	−36.0546	−1.26605	−0.633025	−	0.774132i	$-0.718187\pi$
$811$	−36.0546	−1.26605	−0.633025	+	0.774132i	$0.718187\pi$
$812$	−17.3383	−0.608455
$813$	−19.2290	−0.674391
$814$	−62.0253	−2.17399
$815$	39.0078	1.36638
$816$	5.39702	0.188933
$817$	0	0
$818$	56.3520	1.97030
$819$	−3.34183	−0.116773
$820$	129.180	4.51117
$821$	9.66165	0.337194	0.168597	−	0.985685i	$-0.446076\pi$
$821$	9.66165	0.337194	0.168597	+	0.985685i	$0.446076\pi$
$822$	38.1460	1.33049
$823$	−6.96400	−0.242750	−0.121375	−	0.992607i	$-0.538730\pi$
$823$	−6.96400	−0.242750	−0.121375	+	0.992607i	$0.538730\pi$
$824$	−148.369	−5.16868
$825$	3.94918	0.137493
$826$	3.76797	0.131104
$827$	3.49683	0.121597	0.0607983	−	0.998150i	$-0.480635\pi$
$827$	3.49683	0.121597	0.0607983	+	0.998150i	$0.480635\pi$
$828$	11.4623	0.398344
$829$	37.9814	1.31915	0.659574	−	0.751639i	$-0.270737\pi$
$829$	37.9814	1.31915	0.659574	+	0.751639i	$0.270737\pi$
$830$	41.4648	1.43926
$831$	4.22251	0.146477
$832$	−16.8644	−0.584667
$833$	4.39282	0.152202
$834$	13.7327	0.475526
$835$	51.3773	1.77799
$836$	149.611	5.17439
$837$	14.3344	0.495471
$838$	−13.9433	−0.481664
$839$	−7.56918	−0.261317	−0.130659	−	0.991427i	$-0.541709\pi$
$839$	−7.56918	−0.261317	−0.130659	+	0.991427i	$0.541709\pi$
$840$	−13.2847	−0.458366
$841$	−17.2766	−0.595744
$842$	28.5026	0.982263
$843$	9.33602	0.321550
$844$	74.6235	2.56865
$845$	28.0891	0.966294
$846$	3.34562	0.115025
$847$	−16.0155	−0.550299
$848$	113.643	3.90251
$849$	8.56231	0.293858
$850$	−2.22594	−0.0763492
$851$	−4.05832	−0.139118
$852$	0.924755	0.0316816
$853$	15.2674	0.522747	0.261374	−	0.965238i	$-0.415825\pi$
$853$	15.2674	0.522747	0.261374	+	0.965238i	$0.415825\pi$
$854$	−6.45582	−0.220914
$855$	−36.5907	−1.25138
$856$	−92.8609	−3.17392
$857$	5.41977	0.185136	0.0925679	−	0.995706i	$-0.470492\pi$
$857$	5.41977	0.185136	0.0925679	+	0.995706i	$0.470492\pi$
$858$	−11.7953	−0.402686
$859$	11.0657	0.377557	0.188779	−	0.982020i	$-0.439547\pi$
$859$	11.0657	0.377557	0.188779	+	0.982020i	$0.439547\pi$
$860$	0	0
$861$	−7.02997	−0.239581
$862$	−57.8014	−1.96872
$863$	−22.9067	−0.779752	−0.389876	−	0.920867i	$-0.627482\pi$
$863$	−22.9067	−0.779752	−0.389876	+	0.920867i	$0.627482\pi$
$864$	49.0985	1.67036
$865$	−18.8051	−0.639392
$866$	−58.5973	−1.99122
$867$	10.9696	0.372547
$868$	−19.6036	−0.665391
$869$	−47.9047	−1.62505
$870$	14.9672	0.507436
$871$	7.38439	0.250210
$872$	84.7676	2.87059
$873$	16.5946	0.561643
$874$	13.7032	0.463518
$875$	−9.67595	−0.327107
$876$	−26.0202	−0.879140
$877$	21.2708	0.718263	0.359131	−	0.933287i	$-0.383073\pi$
$877$	21.2708	0.718263	0.359131	+	0.933287i	$0.383073\pi$
$878$	−52.1146	−1.75878
$879$	13.8356	0.466663
$880$	141.396	4.76647
$881$	6.98959	0.235485	0.117743	−	0.993044i	$-0.462434\pi$
$881$	6.98959	0.235485	0.117743	+	0.993044i	$0.462434\pi$
$882$	40.4093	1.36065
$883$	48.0090	1.61563	0.807816	−	0.589435i	$-0.200650\pi$
$883$	48.0090	1.61563	0.807816	+	0.589435i	$0.200650\pi$
$884$	4.74937	0.159739
$885$	−2.32359	−0.0781067
$886$	37.3817	1.25586
$887$	14.5124	0.487279	0.243639	−	0.969866i	$-0.421659\pi$
$887$	14.5124	0.487279	0.243639	+	0.969866i	$0.421659\pi$
$888$	−23.9612	−0.804084
$889$	−10.5807	−0.354865
$890$	−34.9766	−1.17242
$891$	−26.9283	−0.902133
$892$	−45.0087	−1.50700
$893$	2.85722	0.0956133
$894$	−19.6384	−0.656808
$895$	9.35310	0.312640
$896$	−8.13139	−0.271651
$897$	−0.771770	−0.0257687
$898$	−52.8400	−1.76329
$899$	13.2552	0.442084
$900$	−14.6275	−0.487584
$901$	−7.58513	−0.252697
$902$	142.779	4.75401
$903$	0	0
$904$	−78.9700	−2.62650
$905$	−10.2291	−0.340026
$906$	−24.6001	−0.817283
$907$	−44.9578	−1.49280	−0.746400	−	0.665498i	$-0.768219\pi$
$907$	−44.9578	−1.49280	−0.746400	+	0.665498i	$0.768219\pi$
$908$	73.8909	2.45216
$909$	41.5620	1.37852
$910$	−8.57614	−0.284296
$911$	−9.30118	−0.308162	−0.154081	−	0.988058i	$-0.549242\pi$
$911$	−9.30118	−0.308162	−0.154081	+	0.988058i	$0.549242\pi$
$912$	42.3994	1.40398
$913$	32.7390	1.08350
$914$	51.3864	1.69971
$915$	3.98111	0.131611
$916$	86.8108	2.86831
$917$	−18.2101	−0.601351
$918$	−7.20328	−0.237744
$919$	−33.7150	−1.11215	−0.556077	−	0.831131i	$-0.687694\pi$
$919$	−33.7150	−1.11215	−0.556077	+	0.831131i	$0.687694\pi$
$920$	17.6539	0.582031
$921$	7.25959	0.239212
$922$	25.8857	0.852500
$923$	0.358283	0.0117930
$924$	−17.4775	−0.574966
$925$	5.17898	0.170284
$926$	−17.2821	−0.567926
$927$	47.7396	1.56797
$928$	45.4017	1.49039
$929$	54.1378	1.77620	0.888102	−	0.459647i	$-0.152024\pi$
$929$	54.1378	1.77620	0.888102	+	0.459647i	$0.152024\pi$
$930$	16.9228	0.554919
$931$	34.5103	1.13103
$932$	−25.8998	−0.848376
$933$	−0.345393	−0.0113076
$934$	−60.5128	−1.98004
$935$	−9.43754	−0.308641
$936$	26.2201	0.857030
$937$	−27.4944	−0.898204	−0.449102	−	0.893480i	$-0.648256\pi$
$937$	−27.4944	−0.898204	−0.449102	+	0.893480i	$0.648256\pi$
$938$	15.3167	0.500107
$939$	−6.24232	−0.203710
$940$	6.13342	0.200050
$941$	47.6720	1.55406	0.777031	−	0.629462i	$-0.216725\pi$
$941$	47.6720	1.55406	0.777031	+	0.629462i	$0.216725\pi$
$942$	31.1340	1.01440
$943$	9.34203	0.304218
$944$	−15.4929	−0.504251
$945$	9.29189	0.302265
$946$	0	0
$947$	35.0300	1.13832	0.569162	−	0.822226i	$-0.307268\pi$
$947$	35.0300	1.13832	0.569162	+	0.822226i	$0.307268\pi$
$948$	−30.8360	−1.00151
$949$	−10.0811	−0.327247
$950$	−17.4872	−0.567359
$951$	−15.1712	−0.491960
$952$	5.91214	0.191614
$953$	−19.4195	−0.629060	−0.314530	−	0.949248i	$-0.601847\pi$
$953$	−19.4195	−0.629060	−0.314530	+	0.949248i	$0.601847\pi$
$954$	−69.7753	−2.25906
$955$	17.8015	0.576043
$956$	−7.65078	−0.247444
$957$	11.8175	0.382006
$958$	−22.7473	−0.734931
$959$	21.8986	0.707142
$960$	21.5711	0.696206
$961$	−16.0130	−0.516548
$962$	−15.4685	−0.498724
$963$	29.8791	0.962842
$964$	17.7790	0.572623
$965$	−4.00170	−0.128819
$966$	−1.60080	−0.0515050
$967$	19.4564	0.625674	0.312837	−	0.949807i	$-0.398721\pi$
$967$	19.4564	0.625674	0.312837	+	0.949807i	$0.398721\pi$
$968$	125.658	4.03880
$969$	−2.82996	−0.0909113
$970$	42.5868	1.36738
$971$	4.06617	0.130490	0.0652448	−	0.997869i	$-0.479217\pi$
$971$	4.06617	0.130490	0.0652448	+	0.997869i	$0.479217\pi$
$972$	−72.8954	−2.33812
$973$	7.88359	0.252736
$974$	91.6100	2.93538
$975$	0.984885	0.0315416
$976$	26.5446	0.849673
$977$	44.3882	1.42010	0.710052	−	0.704149i	$-0.248671\pi$
$977$	44.3882	1.42010	0.710052	+	0.704149i	$0.248671\pi$
$978$	27.7523	0.887420
$979$	−27.6161	−0.882615
$980$	74.0811	2.36643
$981$	−27.2750	−0.870824
$982$	−40.9587	−1.30704
$983$	−38.3739	−1.22394	−0.611969	−	0.790882i	$-0.709622\pi$
$983$	−38.3739	−1.22394	−0.611969	+	0.790882i	$0.709622\pi$
$984$	55.1573	1.75835
$985$	−51.1082	−1.62844
$986$	−6.66092	−0.212127
$987$	−0.333780	−0.0106243
$988$	37.3114	1.18703
$989$	0	0
$990$	−86.8155	−2.75918
$991$	−56.8096	−1.80462	−0.902309	−	0.431090i	$-0.858129\pi$
$991$	−56.8096	−1.80462	−0.902309	+	0.431090i	$0.858129\pi$
$992$	51.3337	1.62985
$993$	−16.8591	−0.535009
$994$	0.743149	0.0235712
$995$	−5.18574	−0.164399
$996$	21.0739	0.667753
$997$	6.95198	0.220171	0.110086	−	0.993922i	$-0.464887\pi$
$997$	6.95198	0.220171	0.110086	+	0.993922i	$0.464887\pi$
$998$	14.3427	0.454011
$999$	16.7595	0.530246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.q.1.20	yes	20
43.42	odd	2	inner	1849.2.a.q.1.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.q.1.1	✓	20	43.42	odd	2	inner
1849.2.a.q.1.20	yes	20	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64610 q^{2} -0.666460 q^{3} +5.00186 q^{4} -2.47875 q^{5} -1.76352 q^{6} -1.01239 q^{7} +7.94323 q^{8} -2.55583 q^{9} +O(q^{10})\)	\(q+2.64610 q^{2} -0.666460 q^{3} +5.00186 q^{4} -2.47875 q^{5} -1.76352 q^{6} -1.01239 q^{7} +7.94323 q^{8} -2.55583 q^{9} -6.55903 q^{10} -5.17875 q^{11} -3.33354 q^{12} -1.29153 q^{13} -2.67889 q^{14} +1.65199 q^{15} +11.0149 q^{16} -0.735191 q^{17} -6.76299 q^{18} -5.77571 q^{19} -12.3984 q^{20} +0.674717 q^{21} -13.7035 q^{22} -0.896622 q^{23} -5.29384 q^{24} +1.14421 q^{25} -3.41752 q^{26} +3.70274 q^{27} -5.06383 q^{28} +3.42395 q^{29} +4.37133 q^{30} +3.87131 q^{31} +13.2600 q^{32} +3.45143 q^{33} -1.94539 q^{34} +2.50946 q^{35} -12.7839 q^{36} +4.52623 q^{37} -15.2831 q^{38} +0.860753 q^{39} -19.6893 q^{40} -10.4191 q^{41} +1.78537 q^{42} -25.9034 q^{44} +6.33527 q^{45} -2.37255 q^{46} -0.494696 q^{47} -7.34097 q^{48} -5.97507 q^{49} +3.02771 q^{50} +0.489975 q^{51} -6.46005 q^{52} +10.3172 q^{53} +9.79783 q^{54} +12.8368 q^{55} -8.04164 q^{56} +3.84928 q^{57} +9.06012 q^{58} -1.40654 q^{59} +8.26302 q^{60} +2.40989 q^{61} +10.2439 q^{62} +2.58750 q^{63} +13.0577 q^{64} +3.20138 q^{65} +9.13284 q^{66} -5.71755 q^{67} -3.67732 q^{68} +0.597563 q^{69} +6.64030 q^{70} -0.277410 q^{71} -20.3016 q^{72} +7.80557 q^{73} +11.9769 q^{74} -0.762573 q^{75} -28.8893 q^{76} +5.24292 q^{77} +2.27764 q^{78} +9.25023 q^{79} -27.3032 q^{80} +5.19977 q^{81} -27.5701 q^{82} -6.32179 q^{83} +3.37484 q^{84} +1.82236 q^{85} -2.28193 q^{87} -41.1360 q^{88} +5.33258 q^{89} +16.7638 q^{90} +1.30753 q^{91} -4.48478 q^{92} -2.58007 q^{93} -1.30902 q^{94} +14.3166 q^{95} -8.83729 q^{96} -6.49285 q^{97} -15.8106 q^{98} +13.2360 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 14 q^{4} - 32 q^{6} - 6 q^{9}+O(q^{10}) \) 20 * q + 14 * q^4 - 32 * q^6 - 6 * q^9	\( 20 q + 14 q^{4} - 32 q^{6} - 6 q^{9} + 12 q^{10} - 20 q^{11} + 6 q^{13} - 34 q^{14} - 34 q^{15} + 14 q^{16} - 8 q^{17} + 12 q^{21} - 46 q^{23} + 2 q^{24} + 20 q^{25} - 38 q^{31} - 76 q^{35} - 46 q^{36} - 68 q^{38} - 64 q^{40} - 56 q^{41} - 58 q^{44} - 24 q^{47} - 20 q^{49} - 20 q^{52} - 42 q^{53} + 66 q^{54} - 46 q^{56} + 2 q^{57} + 12 q^{58} - 90 q^{59} + 18 q^{60} - 28 q^{64} + 76 q^{66} - 62 q^{67} - 54 q^{68} + 24 q^{74} - 36 q^{78} - 10 q^{79} - 48 q^{81} - 68 q^{83} + 70 q^{84} - 12 q^{87} + 58 q^{90} - 8 q^{92} - 42 q^{95} - 22 q^{96} - 44 q^{97} - 38 q^{99}+O(q^{100}) \) 20 * q + 14 * q^4 - 32 * q^6 - 6 * q^9 + 12 * q^10 - 20 * q^11 + 6 * q^13 - 34 * q^14 - 34 * q^15 + 14 * q^16 - 8 * q^17 + 12 * q^21 - 46 * q^23 + 2 * q^24 + 20 * q^25 - 38 * q^31 - 76 * q^35 - 46 * q^36 - 68 * q^38 - 64 * q^40 - 56 * q^41 - 58 * q^44 - 24 * q^47 - 20 * q^49 - 20 * q^52 - 42 * q^53 + 66 * q^54 - 46 * q^56 + 2 * q^57 + 12 * q^58 - 90 * q^59 + 18 * q^60 - 28 * q^64 + 76 * q^66 - 62 * q^67 - 54 * q^68 + 24 * q^74 - 36 * q^78 - 10 * q^79 - 48 * q^81 - 68 * q^83 + 70 * q^84 - 12 * q^87 + 58 * q^90 - 8 * q^92 - 42 * q^95 - 22 * q^96 - 44 * q^97 - 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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