LMFDB - Embedded newform 409.2.a.b.1.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [409,2,Mod(1,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("409.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 409 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	409.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.26588144267$
Analytic rank:	$0$
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} - 5 x^{19} - 19 x^{18} + 126 x^{17} + 100 x^{16} - 1283 x^{15} + 247 x^{14} + 6767 x^{13} + \cdots - 4 $ x^20 - 5x^19 - 19x^18 + 126x^17 + 100x^16 - 1283x^15 + 247x^14 + 6767x^13 - 4554x^12 - 19689x^11 + 18771x^10 + 31011x^9 - 35515x^8 - 23548x^7 + 31466x^6 + 5354x^5 - 10552x^4 + 1129x^3 + 523x^2 - 54*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$1.89605$ of defining polynomial
Character	$\chi$	$=$	409.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.89605 q^{2} +0.972004 q^{3} +1.59500 q^{4} +1.17356 q^{5} +1.84297 q^{6} +1.87820 q^{7} -0.767899 q^{8} -2.05521 q^{9} +2.22513 q^{10} -0.660558 q^{11} +1.55035 q^{12} +3.13946 q^{13} +3.56116 q^{14} +1.14071 q^{15} -4.64597 q^{16} +3.53185 q^{17} -3.89678 q^{18} -1.60271 q^{19} +1.87183 q^{20} +1.82562 q^{21} -1.25245 q^{22} -1.52749 q^{23} -0.746400 q^{24} -3.62275 q^{25} +5.95256 q^{26} -4.91368 q^{27} +2.99573 q^{28} -6.19279 q^{29} +2.16284 q^{30} +2.78733 q^{31} -7.27320 q^{32} -0.642065 q^{33} +6.69656 q^{34} +2.20419 q^{35} -3.27806 q^{36} -11.4685 q^{37} -3.03881 q^{38} +3.05156 q^{39} -0.901178 q^{40} +7.45968 q^{41} +3.46146 q^{42} +8.74141 q^{43} -1.05359 q^{44} -2.41192 q^{45} -2.89619 q^{46} +6.10570 q^{47} -4.51590 q^{48} -3.47237 q^{49} -6.86891 q^{50} +3.43297 q^{51} +5.00743 q^{52} +12.1790 q^{53} -9.31658 q^{54} -0.775207 q^{55} -1.44227 q^{56} -1.55784 q^{57} -11.7418 q^{58} -3.55992 q^{59} +1.81943 q^{60} -8.45118 q^{61} +5.28492 q^{62} -3.86009 q^{63} -4.49839 q^{64} +3.68435 q^{65} -1.21739 q^{66} -5.59502 q^{67} +5.63331 q^{68} -1.48472 q^{69} +4.17924 q^{70} -9.21827 q^{71} +1.57819 q^{72} +12.4803 q^{73} -21.7448 q^{74} -3.52132 q^{75} -2.55632 q^{76} -1.24066 q^{77} +5.78591 q^{78} -12.0097 q^{79} -5.45235 q^{80} +1.38951 q^{81} +14.1439 q^{82} -2.59714 q^{83} +2.91186 q^{84} +4.14485 q^{85} +16.5741 q^{86} -6.01941 q^{87} +0.507241 q^{88} -4.80605 q^{89} -4.57312 q^{90} +5.89652 q^{91} -2.43634 q^{92} +2.70930 q^{93} +11.5767 q^{94} -1.88088 q^{95} -7.06957 q^{96} +4.44768 q^{97} -6.58379 q^{98} +1.35758 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) $ 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.89605	1.34071	0.670354	−	0.742041i	$-0.266142\pi$
$2$	1.89605	1.34071	0.670354	+	0.742041i	$0.266142\pi$
$3$	0.972004	0.561187	0.280593	−	0.959827i	$-0.409469\pi$
$3$	0.972004	0.561187	0.280593	+	0.959827i	$0.409469\pi$
$4$	1.59500	0.797500
$5$	1.17356	0.524834	0.262417	−	0.964955i	$-0.415480\pi$
$5$	1.17356	0.524834	0.262417	+	0.964955i	$0.415480\pi$
$6$	1.84297	0.752388
$7$	1.87820	0.709892	0.354946	−	0.934887i	$-0.384499\pi$
$7$	1.87820	0.709892	0.354946	+	0.934887i	$0.384499\pi$
$8$	−0.767899	−0.271493
$9$	−2.05521	−0.685070
$10$	2.22513	0.703649
$11$	−0.660558	−0.199166	−0.0995828	−	0.995029i	$-0.531751\pi$
$11$	−0.660558	−0.199166	−0.0995828	+	0.995029i	$0.531751\pi$
$12$	1.55035	0.447546
$13$	3.13946	0.870728	0.435364	−	0.900254i	$-0.356620\pi$
$13$	3.13946	0.870728	0.435364	+	0.900254i	$0.356620\pi$
$14$	3.56116	0.951759
$15$	1.14071	0.294530
$16$	−4.64597	−1.16149
$17$	3.53185	0.856600	0.428300	−	0.903637i	$-0.359113\pi$
$17$	3.53185	0.856600	0.428300	+	0.903637i	$0.359113\pi$
$18$	−3.89678	−0.918479
$19$	−1.60271	−0.367687	−0.183843	−	0.982956i	$-0.558854\pi$
$19$	−1.60271	−0.367687	−0.183843	+	0.982956i	$0.558854\pi$
$20$	1.87183	0.418555
$21$	1.82562	0.398382
$22$	−1.25245	−0.267023
$23$	−1.52749	−0.318503	−0.159251	−	0.987238i	$-0.550908\pi$
$23$	−1.52749	−0.318503	−0.159251	+	0.987238i	$0.550908\pi$
$24$	−0.746400	−0.152358
$25$	−3.62275	−0.724550
$26$	5.95256	1.16739
$27$	−4.91368	−0.945638
$28$	2.99573	0.566139
$29$	−6.19279	−1.14997	−0.574986	−	0.818163i	$-0.694992\pi$
$29$	−6.19279	−1.14997	−0.574986	+	0.818163i	$0.694992\pi$
$30$	2.16284	0.394878
$31$	2.78733	0.500620	0.250310	−	0.968166i	$-0.419467\pi$
$31$	2.78733	0.500620	0.250310	+	0.968166i	$0.419467\pi$
$32$	−7.27320	−1.28573
$33$	−0.642065	−0.111769
$34$	6.69656	1.14845
$35$	2.20419	0.372575
$36$	−3.27806	−0.546343
$37$	−11.4685	−1.88540	−0.942701	−	0.333637i	$-0.891724\pi$
$37$	−11.4685	−1.88540	−0.942701	+	0.333637i	$0.891724\pi$
$38$	−3.03881	−0.492961
$39$	3.05156	0.488641
$40$	−0.901178	−0.142489
$41$	7.45968	1.16501	0.582503	−	0.812829i	$-0.302073\pi$
$41$	7.45968	1.16501	0.582503	+	0.812829i	$0.302073\pi$
$42$	3.46146	0.534114
$43$	8.74141	1.33305	0.666526	−	0.745481i	$-0.267780\pi$
$43$	8.74141	1.33305	0.666526	+	0.745481i	$0.267780\pi$
$44$	−1.05359	−0.158835
$45$	−2.41192	−0.359548
$46$	−2.89619	−0.427019
$47$	6.10570	0.890608	0.445304	−	0.895379i	$-0.353096\pi$
$47$	6.10570	0.890608	0.445304	+	0.895379i	$0.353096\pi$
$48$	−4.51590	−0.651815
$49$	−3.47237	−0.496053
$50$	−6.86891	−0.971410
$51$	3.43297	0.480712
$52$	5.00743	0.694406
$53$	12.1790	1.67291	0.836454	−	0.548037i	$-0.184625\pi$
$53$	12.1790	1.67291	0.836454	+	0.548037i	$0.184625\pi$
$54$	−9.31658	−1.26783
$55$	−0.775207	−0.104529
$56$	−1.44227	−0.192731
$57$	−1.55784	−0.206341
$58$	−11.7418	−1.54178
$59$	−3.55992	−0.463462	−0.231731	−	0.972780i	$-0.574439\pi$
$59$	−3.55992	−0.463462	−0.231731	+	0.972780i	$0.574439\pi$
$60$	1.81943	0.234887
$61$	−8.45118	−1.08206	−0.541031	−	0.841002i	$-0.681966\pi$
$61$	−8.45118	−1.08206	−0.541031	+	0.841002i	$0.681966\pi$
$62$	5.28492	0.671186
$63$	−3.86009	−0.486326
$64$	−4.49839	−0.562298
$65$	3.68435	0.456988
$66$	−1.21739	−0.149850
$67$	−5.59502	−0.683541	−0.341770	−	0.939783i	$-0.611027\pi$
$67$	−5.59502	−0.683541	−0.341770	+	0.939783i	$0.611027\pi$
$68$	5.63331	0.683139
$69$	−1.48472	−0.178739
$70$	4.17924	0.499515
$71$	−9.21827	−1.09401	−0.547004	−	0.837130i	$-0.684232\pi$
$71$	−9.21827	−1.09401	−0.547004	+	0.837130i	$0.684232\pi$
$72$	1.57819	0.185992
$73$	12.4803	1.46071	0.730353	−	0.683070i	$-0.239356\pi$
$73$	12.4803	1.46071	0.730353	+	0.683070i	$0.239356\pi$
$74$	−21.7448	−2.52778
$75$	−3.52132	−0.406608
$76$	−2.55632	−0.293230
$77$	−1.24066	−0.141386
$78$	5.78591	0.655125
$79$	−12.0097	−1.35119	−0.675597	−	0.737271i	$-0.736114\pi$
$79$	−12.0097	−1.35119	−0.675597	+	0.737271i	$0.736114\pi$
$80$	−5.45235	−0.609591
$81$	1.38951	0.154390
$82$	14.1439	1.56193
$83$	−2.59714	−0.285074	−0.142537	−	0.989790i	$-0.545526\pi$
$83$	−2.59714	−0.285074	−0.142537	+	0.989790i	$0.545526\pi$
$84$	2.91186	0.317710
$85$	4.14485	0.449572
$86$	16.5741	1.78724
$87$	−6.01941	−0.645348
$88$	0.507241	0.0540721
$89$	−4.80605	−0.509440	−0.254720	−	0.967015i	$-0.581983\pi$
$89$	−4.80605	−0.509440	−0.254720	+	0.967015i	$0.581983\pi$
$90$	−4.57312	−0.482049
$91$	5.89652	0.618123
$92$	−2.43634	−0.254006
$93$	2.70930	0.280941
$94$	11.5767	1.19405
$95$	−1.88088	−0.192974
$96$	−7.06957	−0.721535
$97$	4.44768	0.451593	0.225797	−	0.974174i	$-0.427501\pi$
$97$	4.44768	0.451593	0.225797	+	0.974174i	$0.427501\pi$
$98$	−6.58379	−0.665063
$99$	1.35758	0.136442
$100$	−5.77829	−0.577829
$101$	3.69821	0.367986	0.183993	−	0.982928i	$-0.441098\pi$
$101$	3.69821	0.367986	0.183993	+	0.982928i	$0.441098\pi$
$102$	6.50908	0.644495
$103$	9.02564	0.889323	0.444661	−	0.895699i	$-0.353324\pi$
$103$	9.02564	0.889323	0.444661	+	0.895699i	$0.353324\pi$
$104$	−2.41078	−0.236397
$105$	2.14248	0.209084
$106$	23.0919	2.24288
$107$	8.91245	0.861599	0.430800	−	0.902448i	$-0.358232\pi$
$107$	8.91245	0.861599	0.430800	+	0.902448i	$0.358232\pi$
$108$	−7.83732	−0.754147
$109$	6.41961	0.614887	0.307444	−	0.951566i	$-0.400526\pi$
$109$	6.41961	0.614887	0.307444	+	0.951566i	$0.400526\pi$
$110$	−1.46983	−0.140143
$111$	−11.1474	−1.05806
$112$	−8.72606	−0.824535
$113$	−2.79405	−0.262842	−0.131421	−	0.991327i	$-0.541954\pi$
$113$	−2.79405	−0.262842	−0.131421	+	0.991327i	$0.541954\pi$
$114$	−2.95374	−0.276643
$115$	−1.79260	−0.167161
$116$	−9.87750	−0.917102
$117$	−6.45224	−0.596510
$118$	−6.74979	−0.621368
$119$	6.63352	0.608094
$120$	−0.875948	−0.0799628
$121$	−10.5637	−0.960333
$122$	−16.0238	−1.45073
$123$	7.25083	0.653786
$124$	4.44580	0.399244
$125$	−10.1193	−0.905102
$126$	−7.31892	−0.652021
$127$	15.5725	1.38184	0.690919	−	0.722932i	$-0.257206\pi$
$127$	15.5725	1.38184	0.690919	+	0.722932i	$0.257206\pi$
$128$	6.01723	0.531853
$129$	8.49668	0.748091
$130$	6.98571	0.612687
$131$	13.6245	1.19038	0.595189	−	0.803586i	$-0.297077\pi$
$131$	13.6245	1.19038	0.595189	+	0.803586i	$0.297077\pi$
$132$	−1.02409	−0.0891359
$133$	−3.01020	−0.261018
$134$	−10.6084	−0.916429
$135$	−5.76652	−0.496303
$136$	−2.71210	−0.232561
$137$	10.0052	0.854803	0.427401	−	0.904062i	$-0.359429\pi$
$137$	10.0052	0.854803	0.427401	+	0.904062i	$0.359429\pi$
$138$	−2.81510	−0.239638
$139$	7.04859	0.597854	0.298927	−	0.954276i	$-0.403371\pi$
$139$	7.04859	0.597854	0.298927	+	0.954276i	$0.403371\pi$
$140$	3.51568	0.297129
$141$	5.93476	0.499797
$142$	−17.4783	−1.46674
$143$	−2.07379	−0.173419
$144$	9.54845	0.795704
$145$	−7.26763	−0.603544
$146$	23.6632	1.95838
$147$	−3.37516	−0.278378
$148$	−18.2922	−1.50361
$149$	4.26885	0.349718	0.174859	−	0.984593i	$-0.444053\pi$
$149$	4.26885	0.349718	0.174859	+	0.984593i	$0.444053\pi$
$150$	−6.67660	−0.545142
$151$	6.87874	0.559784	0.279892	−	0.960031i	$-0.409701\pi$
$151$	6.87874	0.559784	0.279892	+	0.960031i	$0.409701\pi$
$152$	1.23072	0.0998244
$153$	−7.25869	−0.586831
$154$	−2.35235	−0.189558
$155$	3.27111	0.262742
$156$	4.86724	0.389691
$157$	−12.3880	−0.988669	−0.494334	−	0.869272i	$-0.664588\pi$
$157$	−12.3880	−0.988669	−0.494334	+	0.869272i	$0.664588\pi$
$158$	−22.7709	−1.81156
$159$	11.8380	0.938814
$160$	−8.53556	−0.674795
$161$	−2.86892	−0.226103
$162$	2.63458	0.206992
$163$	20.3543	1.59427	0.797136	−	0.603799i	$-0.206347\pi$
$163$	20.3543	1.59427	0.797136	+	0.603799i	$0.206347\pi$
$164$	11.8982	0.929092
$165$	−0.753504	−0.0586602
$166$	−4.92431	−0.382201
$167$	18.4501	1.42772	0.713858	−	0.700291i	$-0.246946\pi$
$167$	18.4501	1.42772	0.713858	+	0.700291i	$0.246946\pi$
$168$	−1.40189	−0.108158
$169$	−3.14382	−0.241832
$170$	7.85884	0.602746
$171$	3.29390	0.251891
$172$	13.9426	1.06311
$173$	3.92588	0.298479	0.149240	−	0.988801i	$-0.452317\pi$
$173$	3.92588	0.298479	0.149240	+	0.988801i	$0.452317\pi$
$174$	−11.4131	−0.865224
$175$	−6.80424	−0.514352
$176$	3.06893	0.231330
$177$	−3.46026	−0.260089
$178$	−9.11251	−0.683011
$179$	−2.51299	−0.187830	−0.0939150	−	0.995580i	$-0.529938\pi$
$179$	−2.51299	−0.187830	−0.0939150	+	0.995580i	$0.529938\pi$
$180$	−3.84701	−0.286739
$181$	12.4802	0.927646	0.463823	−	0.885928i	$-0.346477\pi$
$181$	12.4802	0.927646	0.463823	+	0.885928i	$0.346477\pi$
$182$	11.1801	0.828723
$183$	−8.21458	−0.607239
$184$	1.17295	0.0864713
$185$	−13.4590	−0.989523
$186$	5.13696	0.376660
$187$	−2.33299	−0.170605
$188$	9.73859	0.710260
$189$	−9.22887	−0.671301
$190$	−3.56624	−0.258722
$191$	−9.07445	−0.656604	−0.328302	−	0.944573i	$-0.606476\pi$
$191$	−9.07445	−0.656604	−0.328302	+	0.944573i	$0.606476\pi$
$192$	−4.37245	−0.315554
$193$	−15.5956	−1.12260	−0.561300	−	0.827613i	$-0.689698\pi$
$193$	−15.5956	−1.12260	−0.561300	+	0.827613i	$0.689698\pi$
$194$	8.43302	0.605455
$195$	3.58120	0.256455
$196$	−5.53843	−0.395602
$197$	2.35090	0.167495	0.0837474	−	0.996487i	$-0.473311\pi$
$197$	2.35090	0.167495	0.0837474	+	0.996487i	$0.473311\pi$
$198$	2.57405	0.182929
$199$	0.675453	0.0478816	0.0239408	−	0.999713i	$-0.492379\pi$
$199$	0.675453	0.0478816	0.0239408	+	0.999713i	$0.492379\pi$
$200$	2.78190	0.196710
$201$	−5.43838	−0.383594
$202$	7.01199	0.493362
$203$	−11.6313	−0.816356
$204$	5.47559	0.383368
$205$	8.75441	0.611434
$206$	17.1131	1.19232
$207$	3.13930	0.218197
$208$	−14.5858	−1.01135
$209$	1.05868	0.0732305
$210$	4.06224	0.280321
$211$	−4.62446	−0.318361	−0.159181	−	0.987249i	$-0.550885\pi$
$211$	−4.62446	−0.318361	−0.159181	+	0.987249i	$0.550885\pi$
$212$	19.4254	1.33415
$213$	−8.96019	−0.613942
$214$	16.8984	1.15515
$215$	10.2586	0.699631
$216$	3.77321	0.256734
$217$	5.23516	0.355386
$218$	12.1719	0.824385
$219$	12.1309	0.819728
$220$	−1.23646	−0.0833618
$221$	11.0881	0.745866
$222$	−21.1360	−1.41855
$223$	−7.41478	−0.496531	−0.248265	−	0.968692i	$-0.579860\pi$
$223$	−7.41478	−0.496531	−0.248265	+	0.968692i	$0.579860\pi$
$224$	−13.6605	−0.912731
$225$	7.44550	0.496367
$226$	−5.29766	−0.352395
$227$	22.1135	1.46772	0.733862	−	0.679298i	$-0.237716\pi$
$227$	22.1135	1.46772	0.733862	+	0.679298i	$0.237716\pi$
$228$	−2.48475	−0.164557
$229$	−17.5974	−1.16287	−0.581434	−	0.813593i	$-0.697508\pi$
$229$	−17.5974	−1.16287	−0.581434	+	0.813593i	$0.697508\pi$
$230$	−3.39886	−0.224114
$231$	−1.20592	−0.0793440
$232$	4.75543	0.312209
$233$	10.5325	0.690007	0.345003	−	0.938601i	$-0.387878\pi$
$233$	10.5325	0.690007	0.345003	+	0.938601i	$0.387878\pi$
$234$	−12.2338	−0.799746
$235$	7.16543	0.467421
$236$	−5.67808	−0.369611
$237$	−11.6734	−0.758272
$238$	12.5775	0.815276
$239$	−7.61277	−0.492429	−0.246215	−	0.969215i	$-0.579187\pi$
$239$	−7.61277	−0.492429	−0.246215	+	0.969215i	$0.579187\pi$
$240$	−5.29970	−0.342094
$241$	−21.2758	−1.37050	−0.685248	−	0.728310i	$-0.740306\pi$
$241$	−21.2758	−1.37050	−0.685248	+	0.728310i	$0.740306\pi$
$242$	−20.0292	−1.28753
$243$	16.0917	1.03228
$244$	−13.4796	−0.862945
$245$	−4.07505	−0.260345
$246$	13.7479	0.876536
$247$	−5.03163	−0.320155
$248$	−2.14039	−0.135915
$249$	−2.52443	−0.159979
$250$	−19.1868	−1.21348
$251$	−6.96387	−0.439555	−0.219778	−	0.975550i	$-0.570533\pi$
$251$	−6.96387	−0.439555	−0.219778	+	0.975550i	$0.570533\pi$
$252$	−6.15685	−0.387845
$253$	1.00899	0.0634348
$254$	29.5263	1.85264
$255$	4.02881	0.252294
$256$	20.4057	1.27536
$257$	−7.68381	−0.479303	−0.239651	−	0.970859i	$-0.577033\pi$
$257$	−7.68381	−0.479303	−0.239651	+	0.970859i	$0.577033\pi$
$258$	16.1101	1.00297
$259$	−21.5400	−1.33843
$260$	5.87654	0.364448
$261$	12.7275	0.787810
$262$	25.8327	1.59595
$263$	22.5399	1.38987	0.694934	−	0.719074i	$-0.255434\pi$
$263$	22.5399	1.38987	0.694934	+	0.719074i	$0.255434\pi$
$264$	0.493041	0.0303445
$265$	14.2928	0.877999
$266$	−5.70749	−0.349949
$267$	−4.67150	−0.285891
$268$	−8.92406	−0.545124
$269$	−20.6354	−1.25816	−0.629082	−	0.777339i	$-0.716569\pi$
$269$	−20.6354	−1.25816	−0.629082	+	0.777339i	$0.716569\pi$
$270$	−10.9336	−0.665398
$271$	9.53213	0.579035	0.289518	−	0.957173i	$-0.406505\pi$
$271$	9.53213	0.579035	0.289518	+	0.957173i	$0.406505\pi$
$272$	−16.4089	−0.994935
$273$	5.73144	0.346882
$274$	18.9704	1.14604
$275$	2.39303	0.144305
$276$	−2.36813	−0.142545
$277$	11.3525	0.682106	0.341053	−	0.940044i	$-0.389217\pi$
$277$	11.3525	0.682106	0.341053	+	0.940044i	$0.389217\pi$
$278$	13.3645	0.801548
$279$	−5.72855	−0.342959
$280$	−1.69259	−0.101152
$281$	−0.243462	−0.0145237	−0.00726185	−	0.999974i	$-0.502312\pi$
$281$	−0.243462	−0.0145237	−0.00726185	+	0.999974i	$0.502312\pi$
$282$	11.2526	0.670082
$283$	−20.8454	−1.23913	−0.619565	−	0.784945i	$-0.712691\pi$
$283$	−20.8454	−1.23913	−0.619565	+	0.784945i	$0.712691\pi$
$284$	−14.7031	−0.872471
$285$	−1.82822	−0.108295
$286$	−3.93201	−0.232505
$287$	14.0108	0.827028
$288$	14.9479	0.880816
$289$	−4.52603	−0.266237
$290$	−13.7798	−0.809176
$291$	4.32316	0.253428
$292$	19.9060	1.16491
$293$	10.3636	0.605447	0.302723	−	0.953078i	$-0.402104\pi$
$293$	10.3636	0.605447	0.302723	+	0.953078i	$0.402104\pi$
$294$	−6.39946	−0.373224
$295$	−4.17780	−0.243241
$296$	8.80661	0.511874
$297$	3.24577	0.188339
$298$	8.09395	0.468870
$299$	−4.79547	−0.277329
$300$	−5.61651	−0.324270
$301$	16.4181	0.946324
$302$	13.0424	0.750507
$303$	3.59467	0.206509
$304$	7.44614	0.427066
$305$	−9.91800	−0.567903
$306$	−13.7628	−0.786769
$307$	−14.2586	−0.813783	−0.406891	−	0.913477i	$-0.633387\pi$
$307$	−14.2586	−0.813783	−0.406891	+	0.913477i	$0.633387\pi$
$308$	−1.97885	−0.112756
$309$	8.77295	0.499076
$310$	6.20219	0.352261
$311$	32.9594	1.86895	0.934477	−	0.356023i	$-0.115868\pi$
$311$	32.9594	1.86895	0.934477	+	0.356023i	$0.115868\pi$
$312$	−2.34329	−0.132663
$313$	8.96988	0.507008	0.253504	−	0.967334i	$-0.418417\pi$
$313$	8.96988	0.507008	0.253504	+	0.967334i	$0.418417\pi$
$314$	−23.4882	−1.32552
$315$	−4.53006	−0.255240
$316$	−19.1554	−1.07758
$317$	−31.6969	−1.78028	−0.890138	−	0.455691i	$-0.849392\pi$
$317$	−31.6969	−1.78028	−0.890138	+	0.455691i	$0.849392\pi$
$318$	22.4454	1.25868
$319$	4.09069	0.229035
$320$	−5.27914	−0.295113
$321$	8.66294	0.483518
$322$	−5.43961	−0.303138
$323$	−5.66053	−0.314960
$324$	2.21627	0.123126
$325$	−11.3735	−0.630886
$326$	38.5928	2.13746
$327$	6.23989	0.345066
$328$	−5.72828	−0.316291
$329$	11.4677	0.632236
$330$	−1.42868	−0.0786462
$331$	6.11301	0.336002	0.168001	−	0.985787i	$-0.446269\pi$
$331$	6.11301	0.336002	0.168001	+	0.985787i	$0.446269\pi$
$332$	−4.14245	−0.227346
$333$	23.5701	1.29163
$334$	34.9824	1.91415
$335$	−6.56612	−0.358745
$336$	−8.48176	−0.462718
$337$	−27.7611	−1.51224	−0.756121	−	0.654432i	$-0.772908\pi$
$337$	−27.7611	−1.51224	−0.756121	+	0.654432i	$0.772908\pi$
$338$	−5.96083	−0.324227
$339$	−2.71583	−0.147504
$340$	6.61104	0.358534
$341$	−1.84120	−0.0997063
$342$	6.24540	0.337712
$343$	−19.6692	−1.06204
$344$	−6.71252	−0.361915
$345$	−1.74242	−0.0938085
$346$	7.44366	0.400174
$347$	14.3085	0.768121	0.384061	−	0.923308i	$-0.374525\pi$
$347$	14.3085	0.768121	0.384061	+	0.923308i	$0.374525\pi$
$348$	−9.60096	−0.514666
$349$	−25.9079	−1.38682	−0.693409	−	0.720544i	$-0.743892\pi$
$349$	−25.9079	−1.38682	−0.693409	+	0.720544i	$0.743892\pi$
$350$	−12.9012	−0.689596
$351$	−15.4263	−0.823394
$352$	4.80437	0.256074
$353$	−16.2569	−0.865265	−0.432633	−	0.901570i	$-0.642415\pi$
$353$	−16.2569	−0.865265	−0.432633	+	0.901570i	$0.642415\pi$
$354$	−6.56082	−0.348703
$355$	−10.8182	−0.574172
$356$	−7.66566	−0.406279
$357$	6.44780	0.341254
$358$	−4.76476	−0.251825
$359$	−2.01625	−0.106413	−0.0532067	−	0.998584i	$-0.516944\pi$
$359$	−2.01625	−0.106413	−0.0532067	+	0.998584i	$0.516944\pi$
$360$	1.85211	0.0976147
$361$	−16.4313	−0.864807
$362$	23.6631	1.24370
$363$	−10.2679	−0.538926
$364$	9.40495	0.492953
$365$	14.6464	0.766627
$366$	−15.5752	−0.814131
$367$	2.99174	0.156168	0.0780839	−	0.996947i	$-0.475120\pi$
$367$	2.99174	0.156168	0.0780839	+	0.996947i	$0.475120\pi$
$368$	7.09666	0.369939
$369$	−15.3312	−0.798110
$370$	−25.5189	−1.32666
$371$	22.8745	1.18758
$372$	4.32133	0.224051
$373$	−1.59062	−0.0823591	−0.0411795	−	0.999152i	$-0.513112\pi$
$373$	−1.59062	−0.0823591	−0.0411795	+	0.999152i	$0.513112\pi$
$374$	−4.42347	−0.228732
$375$	−9.83604	−0.507931
$376$	−4.68856	−0.241794
$377$	−19.4420	−1.00131
$378$	−17.4984	−0.900020
$379$	−32.2797	−1.65810	−0.829049	−	0.559176i	$-0.811118\pi$
$379$	−32.2797	−1.65810	−0.829049	+	0.559176i	$0.811118\pi$
$380$	−3.00001	−0.153897
$381$	15.1365	0.775469
$382$	−17.2056	−0.880315
$383$	25.0354	1.27925	0.639625	−	0.768687i	$-0.279090\pi$
$383$	25.0354	1.27925	0.639625	+	0.768687i	$0.279090\pi$
$384$	5.84877	0.298469
$385$	−1.45599	−0.0742042
$386$	−29.5701	−1.50508
$387$	−17.9654	−0.913234
$388$	7.09405	0.360146
$389$	12.3661	0.626986	0.313493	−	0.949590i	$-0.398501\pi$
$389$	12.3661	0.626986	0.313493	+	0.949590i	$0.398501\pi$
$390$	6.79013	0.343832
$391$	−5.39485	−0.272829
$392$	2.66643	0.134675
$393$	13.2431	0.668024
$394$	4.45742	0.224562
$395$	−14.0941	−0.709152
$396$	2.16535	0.108813
$397$	−15.4081	−0.773309	−0.386654	−	0.922225i	$-0.626369\pi$
$397$	−15.4081	−0.773309	−0.386654	+	0.922225i	$0.626369\pi$
$398$	1.28069	0.0641953
$399$	−2.92593	−0.146480
$400$	16.8312	0.841560
$401$	−0.347141	−0.0173354	−0.00866770	−	0.999962i	$-0.502759\pi$
$401$	−0.347141	−0.0173354	−0.00866770	+	0.999962i	$0.502759\pi$
$402$	−10.3114	−0.514288
$403$	8.75071	0.435904
$404$	5.89865	0.293469
$405$	1.63068	0.0810291
$406$	−22.0535	−1.09450
$407$	7.57558	0.375508
$408$	−2.63617	−0.130510
$409$	1.00000	0.0494468
$409$	1.00000	0.0494468
$410$	16.5988	0.819755
$411$	9.72510	0.479704
$412$	14.3959	0.709235
$413$	−6.68624	−0.329008
$414$	5.95227	0.292538
$415$	−3.04791	−0.149616
$416$	−22.8339	−1.11952
$417$	6.85126	0.335507
$418$	2.00731	0.0981808
$419$	−10.1111	−0.493959	−0.246979	−	0.969021i	$-0.579438\pi$
$419$	−10.1111	−0.493959	−0.246979	+	0.969021i	$0.579438\pi$
$420$	3.41725	0.166745
$421$	−29.1186	−1.41915	−0.709576	−	0.704629i	$-0.751113\pi$
$421$	−29.1186	−1.41915	−0.709576	+	0.704629i	$0.751113\pi$
$422$	−8.76821	−0.426830
$423$	−12.5485	−0.610128
$424$	−9.35221	−0.454183
$425$	−12.7950	−0.620649
$426$	−16.9890	−0.823118
$427$	−15.8730	−0.768148
$428$	14.2154	0.687126
$429$	−2.01573	−0.0973205
$430$	19.4508	0.938001
$431$	−15.0403	−0.724466	−0.362233	−	0.932088i	$-0.617986\pi$
$431$	−15.0403	−0.724466	−0.362233	+	0.932088i	$0.617986\pi$
$432$	22.8288	1.09835
$433$	−29.2100	−1.40374	−0.701872	−	0.712303i	$-0.747652\pi$
$433$	−29.2100	−1.40374	−0.701872	+	0.712303i	$0.747652\pi$
$434$	9.92613	0.476469
$435$	−7.06416	−0.338701
$436$	10.2393	0.490373
$437$	2.44811	0.117109
$438$	23.0007	1.09902
$439$	33.0594	1.57784	0.788920	−	0.614496i	$-0.210641\pi$
$439$	33.0594	1.57784	0.788920	+	0.614496i	$0.210641\pi$
$440$	0.595280	0.0283789
$441$	7.13645	0.339831
$442$	21.0236	0.999989
$443$	−19.7670	−0.939157	−0.469578	−	0.882891i	$-0.655594\pi$
$443$	−19.7670	−0.939157	−0.469578	+	0.882891i	$0.655594\pi$
$444$	−17.7801	−0.843805
$445$	−5.64021	−0.267371
$446$	−14.0588	−0.665703
$447$	4.14934	0.196257
$448$	−8.44886	−0.399171
$449$	−8.94927	−0.422342	−0.211171	−	0.977449i	$-0.567728\pi$
$449$	−8.94927	−0.422342	−0.211171	+	0.977449i	$0.567728\pi$
$450$	14.1170	0.665484
$451$	−4.92755	−0.232029
$452$	−4.45651	−0.209617
$453$	6.68616	0.314143
$454$	41.9283	1.96779
$455$	6.91994	0.324412
$456$	1.19626	0.0560201
$457$	15.2739	0.714484	0.357242	−	0.934012i	$-0.383717\pi$
$457$	15.2739	0.714484	0.357242	+	0.934012i	$0.383717\pi$
$458$	−33.3655	−1.55907
$459$	−17.3544	−0.810034
$460$	−2.85920	−0.133311
$461$	−37.9187	−1.76605	−0.883026	−	0.469325i	$-0.844497\pi$
$461$	−37.9187	−1.76605	−0.883026	+	0.469325i	$0.844497\pi$
$462$	−2.28649	−0.106377
$463$	13.6247	0.633194	0.316597	−	0.948560i	$-0.397460\pi$
$463$	13.6247	0.633194	0.316597	+	0.948560i	$0.397460\pi$
$464$	28.7715	1.33568
$465$	3.17953	0.147447
$466$	19.9701	0.925098
$467$	4.17503	0.193197	0.0965987	−	0.995323i	$-0.469204\pi$
$467$	4.17503	0.193197	0.0965987	+	0.995323i	$0.469204\pi$
$468$	−10.2913	−0.475717
$469$	−10.5086	−0.485240
$470$	13.5860	0.626675
$471$	−12.0412	−0.554827
$472$	2.73366	0.125827
$473$	−5.77421	−0.265498
$474$	−22.1334	−1.01662
$475$	5.80621	0.266407
$476$	10.5805	0.484955
$477$	−25.0303	−1.14606
$478$	−14.4342	−0.660204
$479$	−3.57205	−0.163211	−0.0816056	−	0.996665i	$-0.526005\pi$
$479$	−3.57205	−0.163211	−0.0816056	+	0.996665i	$0.526005\pi$
$480$	−8.29659	−0.378686
$481$	−36.0047	−1.64167
$482$	−40.3400	−1.83744
$483$	−2.78860	−0.126886
$484$	−16.8490	−0.765866
$485$	5.21963	0.237011
$486$	30.5106	1.38399
$487$	−21.2793	−0.964255	−0.482128	−	0.876101i	$-0.660136\pi$
$487$	−21.2793	−0.964255	−0.482128	+	0.876101i	$0.660136\pi$
$488$	6.48965	0.293773
$489$	19.7845	0.894684
$490$	−7.72649	−0.349047
$491$	28.4185	1.28251	0.641255	−	0.767328i	$-0.278414\pi$
$491$	28.4185	1.28251	0.641255	+	0.767328i	$0.278414\pi$
$492$	11.5651	0.521394
$493$	−21.8720	−0.985065
$494$	−9.54022	−0.429235
$495$	1.59321	0.0716095
$496$	−12.9499	−0.581467
$497$	−17.3137	−0.776627
$498$	−4.78645	−0.214486
$499$	−35.0686	−1.56988	−0.784942	−	0.619569i	$-0.787308\pi$
$499$	−35.0686	−1.56988	−0.784942	+	0.619569i	$0.787308\pi$
$500$	−16.1404	−0.721819
$501$	17.9336	0.801215
$502$	−13.2038	−0.589316
$503$	0.262159	0.0116891	0.00584454	−	0.999983i	$-0.498140\pi$
$503$	0.262159	0.0116891	0.00584454	+	0.999983i	$0.498140\pi$
$504$	2.96416	0.132034
$505$	4.34009	0.193131
$506$	1.91310	0.0850476
$507$	−3.05580	−0.135713
$508$	24.8382	1.10202
$509$	27.3021	1.21014	0.605072	−	0.796171i	$-0.293144\pi$
$509$	27.3021	1.21014	0.605072	+	0.796171i	$0.293144\pi$
$510$	7.63882	0.338253
$511$	23.4404	1.03694
$512$	26.6558	1.17803
$513$	7.87520	0.347699
$514$	−14.5689	−0.642605
$515$	10.5922	0.466746
$516$	13.5522	0.596603
$517$	−4.03317	−0.177379
$518$	−40.8410	−1.79445
$519$	3.81597	0.167502
$520$	−2.82921	−0.124069
$521$	18.0665	0.791509	0.395754	−	0.918356i	$-0.370483\pi$
$521$	18.0665	0.791509	0.395754	+	0.918356i	$0.370483\pi$
$522$	24.1319	1.05622
$523$	−20.6909	−0.904752	−0.452376	−	0.891827i	$-0.649423\pi$
$523$	−20.6909	−0.904752	−0.452376	+	0.891827i	$0.649423\pi$
$524$	21.7311	0.949327
$525$	−6.61375	−0.288648
$526$	42.7367	1.86341
$527$	9.84445	0.428831
$528$	2.98302	0.129819
$529$	−20.6668	−0.898556
$530$	27.0998	1.17714
$531$	7.31638	0.317504
$532$	−4.80128	−0.208162
$533$	23.4193	1.01440
$534$	−8.85739	−0.383297
$535$	10.4593	0.452196
$536$	4.29641	0.185577
$537$	−2.44264	−0.105408
$538$	−39.1257	−1.68683
$539$	2.29370	0.0987967
$540$	−9.19760	−0.395802
$541$	30.5888	1.31511	0.657557	−	0.753405i	$-0.271590\pi$
$541$	30.5888	1.31511	0.657557	+	0.753405i	$0.271590\pi$
$542$	18.0734	0.776318
$543$	12.1308	0.520582
$544$	−25.6878	−1.10136
$545$	7.53382	0.322714
$546$	10.8671	0.465068
$547$	14.7666	0.631372	0.315686	−	0.948864i	$-0.397765\pi$
$547$	14.7666	0.631372	0.315686	+	0.948864i	$0.397765\pi$
$548$	15.9583	0.681706
$549$	17.3689	0.741288
$550$	4.53731	0.193472
$551$	9.92523	0.422829
$552$	1.14012	0.0485265
$553$	−22.5566	−0.959202
$554$	21.5249	0.914505
$555$	−13.0822	−0.555307
$556$	11.2425	0.476788
$557$	32.1543	1.36242	0.681211	−	0.732087i	$-0.261454\pi$
$557$	32.1543	1.36242	0.681211	+	0.732087i	$0.261454\pi$
$558$	−10.8616	−0.459809
$559$	27.4433	1.16073
$560$	−10.2406	−0.432744
$561$	−2.26768	−0.0957414
$562$	−0.461615	−0.0194721
$563$	−39.4405	−1.66222	−0.831110	−	0.556109i	$-0.812294\pi$
$563$	−39.4405	−1.66222	−0.831110	+	0.556109i	$0.812294\pi$
$564$	9.46595	0.398588
$565$	−3.27900	−0.137948
$566$	−39.5239	−1.66131
$567$	2.60978	0.109600
$568$	7.07869	0.297015
$569$	−0.771919	−0.0323605	−0.0161803	−	0.999869i	$-0.505151\pi$
$569$	−0.771919	−0.0323605	−0.0161803	+	0.999869i	$0.505151\pi$
$570$	−3.46640	−0.145191
$571$	−6.66763	−0.279032	−0.139516	−	0.990220i	$-0.544555\pi$
$571$	−6.66763	−0.279032	−0.139516	+	0.990220i	$0.544555\pi$
$572$	−3.30770	−0.138302
$573$	−8.82039	−0.368477
$574$	26.5651	1.10880
$575$	5.53370	0.230771
$576$	9.24512	0.385213
$577$	31.2723	1.30188	0.650942	−	0.759128i	$-0.274374\pi$
$577$	31.2723	1.30188	0.650942	+	0.759128i	$0.274374\pi$
$578$	−8.58156	−0.356946
$579$	−15.1590	−0.629988
$580$	−11.5919	−0.481326
$581$	−4.87795	−0.202372
$582$	8.19692	0.339773
$583$	−8.04491	−0.333186
$584$	−9.58358	−0.396571
$585$	−7.57211	−0.313068
$586$	19.6499	0.811728
$587$	4.05302	0.167286	0.0836431	−	0.996496i	$-0.473344\pi$
$587$	4.05302	0.167286	0.0836431	+	0.996496i	$0.473344\pi$
$588$	−5.38338	−0.222007
$589$	−4.46728	−0.184071
$590$	−7.92130	−0.326115
$591$	2.28508	0.0939958
$592$	53.2822	2.18988
$593$	6.60358	0.271177	0.135588	−	0.990765i	$-0.456708\pi$
$593$	6.60358	0.271177	0.135588	+	0.990765i	$0.456708\pi$
$594$	6.15414	0.252507
$595$	7.78486	0.319148
$596$	6.80882	0.278900
$597$	0.656543	0.0268705
$598$	−9.09245	−0.371818
$599$	28.1040	1.14830	0.574149	−	0.818750i	$-0.305333\pi$
$599$	28.1040	1.14830	0.574149	+	0.818750i	$0.305333\pi$
$600$	2.70402	0.110391
$601$	−32.2145	−1.31406	−0.657029	−	0.753866i	$-0.728187\pi$
$601$	−32.2145	−1.31406	−0.657029	+	0.753866i	$0.728187\pi$
$602$	31.1295	1.26874
$603$	11.4989	0.468273
$604$	10.9716	0.446428
$605$	−12.3971	−0.504015
$606$	6.81568	0.276868
$607$	24.6563	1.00077	0.500384	−	0.865804i	$-0.333192\pi$
$607$	24.6563	1.00077	0.500384	+	0.865804i	$0.333192\pi$
$608$	11.6568	0.472746
$609$	−11.3056	−0.458128
$610$	−18.8050	−0.761392
$611$	19.1686	0.775477
$612$	−11.5776	−0.467998
$613$	19.0444	0.769195	0.384597	−	0.923084i	$-0.374340\pi$
$613$	19.0444	0.769195	0.384597	+	0.923084i	$0.374340\pi$
$614$	−27.0350	−1.09105
$615$	8.50931	0.343129
$616$	0.952700	0.0383854
$617$	26.1846	1.05415	0.527076	−	0.849818i	$-0.323288\pi$
$617$	26.1846	1.05415	0.527076	+	0.849818i	$0.323288\pi$
$618$	16.6339	0.669116
$619$	−31.8280	−1.27928	−0.639638	−	0.768677i	$-0.720916\pi$
$619$	−31.8280	−1.27928	−0.639638	+	0.768677i	$0.720916\pi$
$620$	5.21743	0.209537
$621$	7.50558	0.301188
$622$	62.4926	2.50572
$623$	−9.02672	−0.361648
$624$	−14.1775	−0.567553
$625$	6.23805	0.249522
$626$	17.0073	0.679750
$627$	1.02904	0.0410960
$628$	−19.7588	−0.788463
$629$	−40.5049	−1.61504
$630$	−8.58922	−0.342203
$631$	25.5972	1.01901	0.509504	−	0.860468i	$-0.329829\pi$
$631$	25.5972	1.01901	0.509504	+	0.860468i	$0.329829\pi$
$632$	9.22221	0.366840
$633$	−4.49499	−0.178660
$634$	−60.0989	−2.38683
$635$	18.2753	0.725235
$636$	18.8816	0.748704
$637$	−10.9014	−0.431927
$638$	7.75615	0.307069
$639$	18.9455	0.749471
$640$	7.06161	0.279135
$641$	17.8941	0.706774	0.353387	−	0.935477i	$-0.385030\pi$
$641$	17.8941	0.706774	0.353387	+	0.935477i	$0.385030\pi$
$642$	16.4253	0.648257
$643$	24.8619	0.980459	0.490229	−	0.871593i	$-0.336913\pi$
$643$	24.8619	0.980459	0.490229	+	0.871593i	$0.336913\pi$
$644$	−4.57593	−0.180317
$645$	9.97140	0.392623
$646$	−10.7326	−0.422270
$647$	−38.6520	−1.51957	−0.759783	−	0.650177i	$-0.774695\pi$
$647$	−38.6520	−1.51957	−0.759783	+	0.650177i	$0.774695\pi$
$648$	−1.06700	−0.0419158
$649$	2.35153	0.0923058
$650$	−21.5646	−0.845834
$651$	5.08860	0.199438
$652$	32.4651	1.27143
$653$	−24.4718	−0.957656	−0.478828	−	0.877909i	$-0.658938\pi$
$653$	−24.4718	−0.957656	−0.478828	+	0.877909i	$0.658938\pi$
$654$	11.8311	0.462634
$655$	15.9892	0.624751
$656$	−34.6575	−1.35315
$657$	−25.6496	−1.00068
$658$	21.7433	0.847644
$659$	26.3373	1.02596	0.512978	−	0.858402i	$-0.328542\pi$
$659$	26.3373	1.02596	0.512978	+	0.858402i	$0.328542\pi$
$660$	−1.20184	−0.0467815
$661$	3.44609	0.134038	0.0670188	−	0.997752i	$-0.478651\pi$
$661$	3.44609	0.134038	0.0670188	+	0.997752i	$0.478651\pi$
$662$	11.5906	0.450480
$663$	10.7777	0.418570
$664$	1.99434	0.0773955
$665$	−3.53267	−0.136991
$666$	44.6900	1.73170
$667$	9.45939	0.366269
$668$	29.4280	1.13860
$669$	−7.20720	−0.278646
$670$	−12.4497	−0.480973
$671$	5.58249	0.215510
$672$	−13.2781	−0.512212
$673$	−17.2653	−0.665527	−0.332764	−	0.943010i	$-0.607981\pi$
$673$	−17.2653	−0.665527	−0.332764	+	0.943010i	$0.607981\pi$
$674$	−52.6363	−2.02748
$675$	17.8010	0.685162
$676$	−5.01439	−0.192861
$677$	2.92530	0.112428	0.0562142	−	0.998419i	$-0.482097\pi$
$677$	2.92530	0.112428	0.0562142	+	0.998419i	$0.482097\pi$
$678$	−5.14934	−0.197759
$679$	8.35362	0.320583
$680$	−3.18283	−0.122056
$681$	21.4944	0.823667
$682$	−3.49100	−0.133677
$683$	−18.2366	−0.697804	−0.348902	−	0.937159i	$-0.613445\pi$
$683$	−18.2366	−0.697804	−0.348902	+	0.937159i	$0.613445\pi$
$684$	5.25377	0.200883
$685$	11.7418	0.448629
$686$	−37.2937	−1.42388
$687$	−17.1047	−0.652586
$688$	−40.6124	−1.54833
$689$	38.2353	1.45665
$690$	−3.30370	−0.125770
$691$	29.4210	1.11923	0.559613	−	0.828754i	$-0.310950\pi$
$691$	29.4210	1.11923	0.559613	+	0.828754i	$0.310950\pi$
$692$	6.26178	0.238037
$693$	2.54981	0.0968594
$694$	27.1296	1.02983
$695$	8.27197	0.313774
$696$	4.62230	0.175208
$697$	26.3465	0.997944
$698$	−49.1227	−1.85932
$699$	10.2376	0.387223
$700$	−10.8528	−0.410196
$701$	18.2846	0.690601	0.345301	−	0.938492i	$-0.387777\pi$
$701$	18.2846	0.690601	0.345301	+	0.938492i	$0.387777\pi$
$702$	−29.2490	−1.10393
$703$	18.3806	0.693237
$704$	2.97144	0.111990
$705$	6.96482	0.262310
$706$	−30.8238	−1.16007
$707$	6.94597	0.261230
$708$	−5.51911	−0.207421
$709$	11.2944	0.424171	0.212085	−	0.977251i	$-0.431975\pi$
$709$	11.2944	0.424171	0.212085	+	0.977251i	$0.431975\pi$
$710$	−20.5119	−0.769797
$711$	24.6824	0.925662
$712$	3.69056	0.138310
$713$	−4.25761	−0.159449
$714$	12.2253	0.457522
$715$	−2.43373	−0.0910162
$716$	−4.00823	−0.149794
$717$	−7.39964	−0.276345
$718$	−3.82290	−0.142669
$719$	−28.4111	−1.05955	−0.529777	−	0.848137i	$-0.677724\pi$
$719$	−28.4111	−1.05955	−0.529777	+	0.848137i	$0.677724\pi$
$720$	11.2057	0.417612
$721$	16.9519	0.631323
$722$	−31.1546	−1.15945
$723$	−20.6802	−0.769104
$724$	19.9059	0.739798
$725$	22.4349	0.833211
$726$	−19.4685	−0.722543
$727$	−11.3690	−0.421651	−0.210826	−	0.977524i	$-0.567615\pi$
$727$	−11.3690	−0.421651	−0.210826	+	0.977524i	$0.567615\pi$
$728$	−4.52793	−0.167816
$729$	11.4726	0.424912
$730$	27.7703	1.02782
$731$	30.8734	1.14189
$732$	−13.1023	−0.484273
$733$	17.7129	0.654242	0.327121	−	0.944982i	$-0.393921\pi$
$733$	17.7129	0.654242	0.327121	+	0.944982i	$0.393921\pi$
$734$	5.67249	0.209376
$735$	−3.96096	−0.146102
$736$	11.1097	0.409509
$737$	3.69584	0.136138
$738$	−29.0687	−1.07003
$739$	−5.53202	−0.203499	−0.101749	−	0.994810i	$-0.532444\pi$
$739$	−5.53202	−0.203499	−0.101749	+	0.994810i	$0.532444\pi$
$740$	−21.4671	−0.789145
$741$	−4.89076	−0.179667
$742$	43.3712	1.59221
$743$	−31.1293	−1.14202	−0.571012	−	0.820942i	$-0.693449\pi$
$743$	−31.1293	−1.14202	−0.571012	+	0.820942i	$0.693449\pi$
$744$	−2.08047	−0.0762736
$745$	5.00977	0.183544
$746$	−3.01589	−0.110420
$747$	5.33767	0.195295
$748$	−3.72112	−0.136058
$749$	16.7394	0.611643
$750$	−18.6496	−0.680987
$751$	−14.5955	−0.532599	−0.266299	−	0.963890i	$-0.585801\pi$
$751$	−14.5955	−0.532599	−0.266299	+	0.963890i	$0.585801\pi$
$752$	−28.3669	−1.03444
$753$	−6.76890	−0.246673
$754$	−36.8629	−1.34247
$755$	8.07264	0.293793
$756$	−14.7200	−0.535363
$757$	−3.08384	−0.112084	−0.0560420	−	0.998428i	$-0.517848\pi$
$757$	−3.08384	−0.112084	−0.0560420	+	0.998428i	$0.517848\pi$
$758$	−61.2039	−2.22303
$759$	0.980744	0.0355988
$760$	1.44433	0.0523912
$761$	3.11254	0.112829	0.0564147	−	0.998407i	$-0.482033\pi$
$761$	3.11254	0.112829	0.0564147	+	0.998407i	$0.482033\pi$
$762$	28.6996	1.03968
$763$	12.0573	0.436504
$764$	−14.4737	−0.523642
$765$	−8.51854	−0.307988
$766$	47.4683	1.71510
$767$	−11.1762	−0.403550
$768$	19.8345	0.715714
$769$	−21.7099	−0.782880	−0.391440	−	0.920204i	$-0.628023\pi$
$769$	−21.7099	−0.782880	−0.391440	+	0.920204i	$0.628023\pi$
$770$	−2.76063	−0.0994863
$771$	−7.46869	−0.268978
$772$	−24.8751	−0.895273
$773$	25.2412	0.907863	0.453931	−	0.891037i	$-0.350021\pi$
$773$	25.2412	0.907863	0.453931	+	0.891037i	$0.350021\pi$
$774$	−34.0633	−1.22438
$775$	−10.0978	−0.362724
$776$	−3.41537	−0.122605
$777$	−20.9370	−0.751111
$778$	23.4467	0.840606
$779$	−11.9557	−0.428357
$780$	5.71202	0.204523
$781$	6.08920	0.217889
$782$	−10.2289	−0.365785
$783$	30.4294	1.08746
$784$	16.1325	0.576162
$785$	−14.5381	−0.518887
$786$	25.1095	0.895626
$787$	18.4022	0.655969	0.327984	−	0.944683i	$-0.393631\pi$
$787$	18.4022	0.655969	0.327984	+	0.944683i	$0.393631\pi$
$788$	3.74969	0.133577
$789$	21.9088	0.779975
$790$	−26.7231	−0.950767
$791$	−5.24778	−0.186590
$792$	−1.04249	−0.0370432
$793$	−26.5321	−0.942182
$794$	−29.2144	−1.03678
$795$	13.8926	0.492721
$796$	1.07735	0.0381856
$797$	−29.8970	−1.05900	−0.529502	−	0.848308i	$-0.677621\pi$
$797$	−29.8970	−1.05900	−0.529502	+	0.848308i	$0.677621\pi$
$798$	−5.54770	−0.196387
$799$	21.5644	0.762894
$800$	26.3490	0.931576
$801$	9.87744	0.349002
$802$	−0.658197	−0.0232417
$803$	−8.24394	−0.290922
$804$	−8.67422	−0.305916
$805$	−3.36686	−0.118666
$806$	16.5918	0.584420
$807$	−20.0577	−0.706064
$808$	−2.83985	−0.0999056
$809$	−13.7840	−0.484621	−0.242310	−	0.970199i	$-0.577905\pi$
$809$	−13.7840	−0.484621	−0.242310	+	0.970199i	$0.577905\pi$
$810$	3.09185	0.108636
$811$	−54.7860	−1.92380	−0.961898	−	0.273407i	$-0.911849\pi$
$811$	−54.7860	−1.92380	−0.961898	+	0.273407i	$0.911849\pi$
$812$	−18.5519	−0.651044
$813$	9.26526	0.324947
$814$	14.3637	0.503446
$815$	23.8871	0.836728
$816$	−15.9495	−0.558344
$817$	−14.0099	−0.490146
$818$	1.89605	0.0662938
$819$	−12.1186	−0.423457
$820$	13.9633	0.487619
$821$	−3.64212	−0.127111	−0.0635555	−	0.997978i	$-0.520244\pi$
$821$	−3.64212	−0.127111	−0.0635555	+	0.997978i	$0.520244\pi$
$822$	18.4393	0.643143
$823$	12.7841	0.445627	0.222813	−	0.974861i	$-0.428476\pi$
$823$	12.7841	0.445627	0.222813	+	0.974861i	$0.428476\pi$
$824$	−6.93078	−0.241445
$825$	2.32604	0.0809823
$826$	−12.6774	−0.441104
$827$	−42.4885	−1.47747	−0.738735	−	0.673996i	$-0.764577\pi$
$827$	−42.4885	−1.47747	−0.738735	+	0.673996i	$0.764577\pi$
$828$	5.00719	0.174012
$829$	−5.45416	−0.189431	−0.0947154	−	0.995504i	$-0.530194\pi$
$829$	−5.45416	−0.189431	−0.0947154	+	0.995504i	$0.530194\pi$
$830$	−5.77899	−0.200592
$831$	11.0347	0.382788
$832$	−14.1225	−0.489609
$833$	−12.2639	−0.424919
$834$	12.9903	0.449818
$835$	21.6524	0.749313
$836$	1.68860	0.0584014
$837$	−13.6961	−0.473405
$838$	−19.1711	−0.662255
$839$	40.8536	1.41042	0.705211	−	0.708997i	$-0.250852\pi$
$839$	40.8536	1.41042	0.705211	+	0.708997i	$0.250852\pi$
$840$	−1.64520	−0.0567649
$841$	9.35059	0.322434
$842$	−55.2102	−1.90267
$843$	−0.236646	−0.00815050
$844$	−7.37602	−0.253893
$845$	−3.68947	−0.126922
$846$	−23.7925	−0.818005
$847$	−19.8407	−0.681733
$848$	−56.5831	−1.94307
$849$	−20.2618	−0.695383
$850$	−24.2600	−0.832110
$851$	17.5179	0.600506
$852$	−14.2915	−0.489619
$853$	−41.8276	−1.43215	−0.716075	−	0.698023i	$-0.754063\pi$
$853$	−41.8276	−1.43215	−0.716075	+	0.698023i	$0.754063\pi$
$854$	−30.0960	−1.02986
$855$	3.86560	0.132201
$856$	−6.84386	−0.233918
$857$	−52.8929	−1.80679	−0.903393	−	0.428813i	$-0.858932\pi$
$857$	−52.8929	−1.80679	−0.903393	+	0.428813i	$0.858932\pi$
$858$	−3.82193	−0.130478
$859$	−5.22079	−0.178131	−0.0890656	−	0.996026i	$-0.528388\pi$
$859$	−5.22079	−0.178131	−0.0890656	+	0.996026i	$0.528388\pi$
$860$	16.3625	0.557956
$861$	13.6185	0.464117
$862$	−28.5172	−0.971298
$863$	−54.2898	−1.84805	−0.924024	−	0.382335i	$-0.875120\pi$
$863$	−54.2898	−1.84805	−0.924024	+	0.382335i	$0.875120\pi$
$864$	35.7382	1.21584
$865$	4.60727	0.156652
$866$	−55.3836	−1.88201
$867$	−4.39931	−0.149408
$868$	8.35009	0.283421
$869$	7.93309	0.269112
$870$	−13.3940	−0.454099
$871$	−17.5653	−0.595178
$872$	−4.92961	−0.166938
$873$	−9.14091	−0.309373
$874$	4.64174	0.157009
$875$	−19.0061	−0.642525
$876$	19.3487	0.653733
$877$	−19.3162	−0.652263	−0.326131	−	0.945324i	$-0.605745\pi$
$877$	−19.3162	−0.652263	−0.326131	+	0.945324i	$0.605745\pi$
$878$	62.6823	2.11542
$879$	10.0734	0.339769
$880$	3.60159	0.121410
$881$	10.2758	0.346200	0.173100	−	0.984904i	$-0.444622\pi$
$881$	10.2758	0.346200	0.173100	+	0.984904i	$0.444622\pi$
$882$	13.5311	0.455614
$883$	−0.673102	−0.0226517	−0.0113258	−	0.999936i	$-0.503605\pi$
$883$	−0.673102	−0.0226517	−0.0113258	+	0.999936i	$0.503605\pi$
$884$	17.6855	0.594828
$885$	−4.06083	−0.136503
$886$	−37.4791	−1.25914
$887$	46.6681	1.56696	0.783481	−	0.621416i	$-0.213442\pi$
$887$	46.6681	1.56696	0.783481	+	0.621416i	$0.213442\pi$
$888$	8.56006	0.287257
$889$	29.2483	0.980956
$890$	−10.6941	−0.358467
$891$	−0.917852	−0.0307492
$892$	−11.8266	−0.395983
$893$	−9.78566	−0.327464
$894$	7.86735	0.263123
$895$	−2.94916	−0.0985795
$896$	11.3016	0.377559
$897$	−4.66122	−0.155634
$898$	−16.9683	−0.566238
$899$	−17.2614	−0.575699
$900$	11.8756	0.395853
$901$	43.0143	1.43301
$902$	−9.34287	−0.311084
$903$	15.9585	0.531064
$904$	2.14555	0.0713599
$905$	14.6463	0.486860
$906$	12.6773	0.421175
$907$	2.11538	0.0702401	0.0351200	−	0.999383i	$-0.488819\pi$
$907$	2.11538	0.0702401	0.0351200	+	0.999383i	$0.488819\pi$
$908$	35.2710	1.17051
$909$	−7.60059	−0.252096
$910$	13.1205	0.434942
$911$	56.4199	1.86927	0.934637	−	0.355603i	$-0.115725\pi$
$911$	56.4199	1.86927	0.934637	+	0.355603i	$0.115725\pi$
$912$	7.23768	0.239663
$913$	1.71556	0.0567769
$914$	28.9601	0.957915
$915$	−9.64033	−0.318699
$916$	−28.0679	−0.927388
$917$	25.5895	0.845040
$918$	−32.9048	−1.08602
$919$	−23.4794	−0.774513	−0.387256	−	0.921972i	$-0.626577\pi$
$919$	−23.4794	−0.774513	−0.387256	+	0.921972i	$0.626577\pi$
$920$	1.37654	0.0453831
$921$	−13.8594	−0.456684
$922$	−71.8957	−2.36776
$923$	−28.9403	−0.952583
$924$	−1.92345	−0.0632769
$925$	41.5473	1.36607
$926$	25.8331	0.848929
$927$	−18.5496	−0.609248
$928$	45.0413	1.47855
$929$	46.7568	1.53404	0.767020	−	0.641623i	$-0.221738\pi$
$929$	46.7568	1.53404	0.767020	+	0.641623i	$0.221738\pi$
$930$	6.02855	0.197684
$931$	5.56520	0.182392
$932$	16.7993	0.550281
$933$	32.0366	1.04883
$934$	7.91607	0.259022
$935$	−2.73792	−0.0895394
$936$	4.95466	0.161948
$937$	54.1331	1.76845	0.884226	−	0.467060i	$-0.154687\pi$
$937$	54.1331	1.76845	0.884226	+	0.467060i	$0.154687\pi$
$938$	−19.9247	−0.650566
$939$	8.71876	0.284526
$940$	11.4289	0.372768
$941$	−21.2548	−0.692888	−0.346444	−	0.938071i	$-0.612611\pi$
$941$	−21.2548	−0.692888	−0.346444	+	0.938071i	$0.612611\pi$
$942$	−22.8306	−0.743862
$943$	−11.3945	−0.371057
$944$	16.5393	0.538309
$945$	−10.8307	−0.352322
$946$	−10.9482	−0.355956
$947$	−34.7588	−1.12951	−0.564755	−	0.825259i	$-0.691029\pi$
$947$	−34.7588	−1.12951	−0.564755	+	0.825259i	$0.691029\pi$
$948$	−18.6192	−0.604722
$949$	39.1813	1.27188
$950$	11.0089	0.357174
$951$	−30.8095	−0.999067
$952$	−5.09387	−0.165093
$953$	−33.2464	−1.07696	−0.538479	−	0.842639i	$-0.681001\pi$
$953$	−33.2464	−1.07696	−0.538479	+	0.842639i	$0.681001\pi$
$954$	−47.4587	−1.53653
$955$	−10.6494	−0.344608
$956$	−12.1424	−0.392712
$957$	3.97617	0.128531
$958$	−6.77278	−0.218819
$959$	18.7918	0.606818
$960$	−5.13134	−0.165613
$961$	−23.2308	−0.749380
$962$	−68.2667	−2.20101
$963$	−18.3170	−0.590256
$964$	−33.9349	−1.09297
$965$	−18.3025	−0.589178
$966$	−5.28732	−0.170117
$967$	57.3850	1.84538	0.922689	−	0.385545i	$-0.125987\pi$
$967$	57.3850	1.84538	0.922689	+	0.385545i	$0.125987\pi$
$968$	8.11182	0.260724
$969$	−5.50205	−0.176751
$970$	9.89668	0.317763
$971$	12.2948	0.394560	0.197280	−	0.980347i	$-0.436789\pi$
$971$	12.2948	0.394560	0.197280	+	0.980347i	$0.436789\pi$
$972$	25.6662	0.823244
$973$	13.2387	0.424412
$974$	−40.3465	−1.29279
$975$	−11.0550	−0.354045
$976$	39.2640	1.25681
$977$	42.3917	1.35623	0.678115	−	0.734956i	$-0.262797\pi$
$977$	42.3917	1.35623	0.678115	+	0.734956i	$0.262797\pi$
$978$	37.5123	1.19951
$979$	3.17468	0.101463
$980$	−6.49971	−0.207625
$981$	−13.1936	−0.421241
$982$	53.8829	1.71947
$983$	18.9499	0.604407	0.302204	−	0.953243i	$-0.402278\pi$
$983$	18.9499	0.604407	0.302204	+	0.953243i	$0.402278\pi$
$984$	−5.56790	−0.177498
$985$	2.75893	0.0879069
$986$	−41.4704	−1.32069
$987$	11.1467	0.354802
$988$	−8.02546	−0.255324
$989$	−13.3524	−0.424581
$990$	3.02081	0.0960076
$991$	57.1892	1.81667	0.908337	−	0.418239i	$-0.137353\pi$
$991$	57.1892	1.81667	0.908337	+	0.418239i	$0.137353\pi$
$992$	−20.2728	−0.643663
$993$	5.94187	0.188560
$994$	−32.8277	−1.04123
$995$	0.792687	0.0251299
$996$	−4.02647	−0.127584
$997$	46.9941	1.48832	0.744160	−	0.668002i	$-0.232850\pi$
$997$	46.9941	1.48832	0.744160	+	0.668002i	$0.232850\pi$
$998$	−66.4917	−2.10476
$999$	56.3524	1.78291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	409.2.a.b.1.16	✓	20
3.2	odd	2		3681.2.a.i.1.5		20
4.3	odd	2		6544.2.a.i.1.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
409.2.a.b.1.16	✓	20	1.1	even	1	trivial
3681.2.a.i.1.5		20	3.2	odd	2
6544.2.a.i.1.9		20	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 409 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	409.a (trivial)

\(f(q)\)	\(=\)	\(q+1.89605 q^{2} +0.972004 q^{3} +1.59500 q^{4} +1.17356 q^{5} +1.84297 q^{6} +1.87820 q^{7} -0.767899 q^{8} -2.05521 q^{9} +2.22513 q^{10} -0.660558 q^{11} +1.55035 q^{12} +3.13946 q^{13} +3.56116 q^{14} +1.14071 q^{15} -4.64597 q^{16} +3.53185 q^{17} -3.89678 q^{18} -1.60271 q^{19} +1.87183 q^{20} +1.82562 q^{21} -1.25245 q^{22} -1.52749 q^{23} -0.746400 q^{24} -3.62275 q^{25} +5.95256 q^{26} -4.91368 q^{27} +2.99573 q^{28} -6.19279 q^{29} +2.16284 q^{30} +2.78733 q^{31} -7.27320 q^{32} -0.642065 q^{33} +6.69656 q^{34} +2.20419 q^{35} -3.27806 q^{36} -11.4685 q^{37} -3.03881 q^{38} +3.05156 q^{39} -0.901178 q^{40} +7.45968 q^{41} +3.46146 q^{42} +8.74141 q^{43} -1.05359 q^{44} -2.41192 q^{45} -2.89619 q^{46} +6.10570 q^{47} -4.51590 q^{48} -3.47237 q^{49} -6.86891 q^{50} +3.43297 q^{51} +5.00743 q^{52} +12.1790 q^{53} -9.31658 q^{54} -0.775207 q^{55} -1.44227 q^{56} -1.55784 q^{57} -11.7418 q^{58} -3.55992 q^{59} +1.81943 q^{60} -8.45118 q^{61} +5.28492 q^{62} -3.86009 q^{63} -4.49839 q^{64} +3.68435 q^{65} -1.21739 q^{66} -5.59502 q^{67} +5.63331 q^{68} -1.48472 q^{69} +4.17924 q^{70} -9.21827 q^{71} +1.57819 q^{72} +12.4803 q^{73} -21.7448 q^{74} -3.52132 q^{75} -2.55632 q^{76} -1.24066 q^{77} +5.78591 q^{78} -12.0097 q^{79} -5.45235 q^{80} +1.38951 q^{81} +14.1439 q^{82} -2.59714 q^{83} +2.91186 q^{84} +4.14485 q^{85} +16.5741 q^{86} -6.01941 q^{87} +0.507241 q^{88} -4.80605 q^{89} -4.57312 q^{90} +5.89652 q^{91} -2.43634 q^{92} +2.70930 q^{93} +11.5767 q^{94} -1.88088 q^{95} -7.06957 q^{96} +4.44768 q^{97} -6.58379 q^{98} +1.35758 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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