LMFDB - Embedded newform 409.2.a.b.1.5

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			1.5
Root			$-1.38030$ 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)

$f(q)$	$=$	\(q-1.38030 q^{2} -1.32806 q^{3} -0.0947758 q^{4} -2.04589 q^{5} +1.83312 q^{6} -0.300238 q^{7} +2.89142 q^{8} -1.23626 q^{9} +2.82394 q^{10} -3.45738 q^{11} +0.125868 q^{12} -3.89705 q^{13} +0.414418 q^{14} +2.71707 q^{15} -3.80147 q^{16} +5.59382 q^{17} +1.70640 q^{18} -0.669118 q^{19} +0.193901 q^{20} +0.398734 q^{21} +4.77222 q^{22} +3.49818 q^{23} -3.83997 q^{24} -0.814317 q^{25} +5.37910 q^{26} +5.62600 q^{27} +0.0284553 q^{28} +8.88357 q^{29} -3.75037 q^{30} +7.37344 q^{31} -0.535674 q^{32} +4.59161 q^{33} -7.72114 q^{34} +0.614254 q^{35} +0.117167 q^{36} -3.54541 q^{37} +0.923582 q^{38} +5.17552 q^{39} -5.91553 q^{40} +7.92837 q^{41} -0.550371 q^{42} +0.985097 q^{43} +0.327676 q^{44} +2.52925 q^{45} -4.82853 q^{46} -2.95435 q^{47} +5.04858 q^{48} -6.90986 q^{49} +1.12400 q^{50} -7.42893 q^{51} +0.369346 q^{52} -6.43289 q^{53} -7.76556 q^{54} +7.07343 q^{55} -0.868112 q^{56} +0.888629 q^{57} -12.2620 q^{58} +4.42090 q^{59} -0.257513 q^{60} -0.236790 q^{61} -10.1775 q^{62} +0.371170 q^{63} +8.34232 q^{64} +7.97296 q^{65} -6.33779 q^{66} +2.33250 q^{67} -0.530159 q^{68} -4.64579 q^{69} -0.847854 q^{70} +6.91848 q^{71} -3.57453 q^{72} +4.76591 q^{73} +4.89373 q^{74} +1.08146 q^{75} +0.0634161 q^{76} +1.03804 q^{77} -7.14377 q^{78} +3.80018 q^{79} +7.77740 q^{80} -3.76290 q^{81} -10.9435 q^{82} -7.24637 q^{83} -0.0377903 q^{84} -11.4444 q^{85} -1.35973 q^{86} -11.7979 q^{87} -9.99672 q^{88} +14.2851 q^{89} -3.49112 q^{90} +1.17004 q^{91} -0.331542 q^{92} -9.79237 q^{93} +4.07789 q^{94} +1.36894 q^{95} +0.711407 q^{96} -10.8161 q^{97} +9.53767 q^{98} +4.27421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.38030	−0.976018	−0.488009	−	0.872838i	$-0.662277\pi$
$2$	−1.38030	−0.976018	−0.488009	+	0.872838i	$0.662277\pi$
$3$	−1.32806	−0.766756	−0.383378	−	0.923592i	$-0.625239\pi$
$3$	−1.32806	−0.766756	−0.383378	+	0.923592i	$0.625239\pi$
$4$	−0.0947758	−0.0473879
$5$	−2.04589	−0.914952	−0.457476	−	0.889222i	$-0.651246\pi$
$5$	−2.04589	−0.914952	−0.457476	+	0.889222i	$0.651246\pi$
$6$	1.83312	0.748368
$7$	−0.300238	−0.113479	−0.0567396	−	0.998389i	$-0.518070\pi$
$7$	−0.300238	−0.113479	−0.0567396	+	0.998389i	$0.518070\pi$
$8$	2.89142	1.02227
$9$	−1.23626	−0.412085
$10$	2.82394	0.893010
$11$	−3.45738	−1.04244	−0.521220	−	0.853423i	$-0.674523\pi$
$11$	−3.45738	−1.04244	−0.521220	+	0.853423i	$0.674523\pi$
$12$	0.125868	0.0363350
$13$	−3.89705	−1.08085	−0.540424	−	0.841393i	$-0.681736\pi$
$13$	−3.89705	−1.08085	−0.540424	+	0.841393i	$0.681736\pi$
$14$	0.414418	0.110758
$15$	2.71707	0.701545
$16$	−3.80147	−0.950366
$17$	5.59382	1.35670	0.678350	−	0.734739i	$-0.262695\pi$
$17$	5.59382	1.35670	0.678350	+	0.734739i	$0.262695\pi$
$18$	1.70640	0.402203
$19$	−0.669118	−0.153506	−0.0767531	−	0.997050i	$-0.524455\pi$
$19$	−0.669118	−0.153506	−0.0767531	+	0.997050i	$0.524455\pi$
$20$	0.193901	0.0433576
$21$	0.398734	0.0870108
$22$	4.77222	1.01744
$23$	3.49818	0.729420	0.364710	−	0.931121i	$-0.381168\pi$
$23$	3.49818	0.729420	0.364710	+	0.931121i	$0.381168\pi$
$24$	−3.83997	−0.783832
$25$	−0.814317	−0.162863
$26$	5.37910	1.05493
$27$	5.62600	1.08272
$28$	0.0284553	0.00537754
$29$	8.88357	1.64964	0.824819	−	0.565397i	$-0.191277\pi$
$29$	8.88357	1.64964	0.824819	+	0.565397i	$0.191277\pi$
$30$	−3.75037	−0.684721
$31$	7.37344	1.32431	0.662154	−	0.749368i	$-0.269642\pi$
$31$	7.37344	1.32431	0.662154	+	0.749368i	$0.269642\pi$
$32$	−0.535674	−0.0946947
$33$	4.59161	0.799297
$34$	−7.72114	−1.32416
$35$	0.614254	0.103828
$36$	0.117167	0.0195279
$37$	−3.54541	−0.582862	−0.291431	−	0.956592i	$-0.594131\pi$
$37$	−3.54541	−0.582862	−0.291431	+	0.956592i	$0.594131\pi$
$38$	0.923582	0.149825
$39$	5.17552	0.828747
$40$	−5.91553	−0.935328
$41$	7.92837	1.23820	0.619102	−	0.785311i	$-0.287497\pi$
$41$	7.92837	1.23820	0.619102	+	0.785311i	$0.287497\pi$
$42$	−0.550371	−0.0849242
$43$	0.985097	0.150226	0.0751130	−	0.997175i	$-0.476068\pi$
$43$	0.985097	0.150226	0.0751130	+	0.997175i	$0.476068\pi$
$44$	0.327676	0.0493990
$45$	2.52925	0.377038
$46$	−4.82853	−0.711927
$47$	−2.95435	−0.430936	−0.215468	−	0.976511i	$-0.569128\pi$
$47$	−2.95435	−0.430936	−0.215468	+	0.976511i	$0.569128\pi$
$48$	5.04858	0.728699
$49$	−6.90986	−0.987122
$50$	1.12400	0.158958
$51$	−7.42893	−1.04026
$52$	0.369346	0.0512191
$53$	−6.43289	−0.883625	−0.441813	−	0.897107i	$-0.645664\pi$
$53$	−6.43289	−0.883625	−0.441813	+	0.897107i	$0.645664\pi$
$54$	−7.76556	−1.05676
$55$	7.07343	0.953782
$56$	−0.868112	−0.116006
$57$	0.888629	0.117702
$58$	−12.2620	−1.61008
$59$	4.42090	0.575552	0.287776	−	0.957698i	$-0.407084\pi$
$59$	4.42090	0.575552	0.287776	+	0.957698i	$0.407084\pi$
$60$	−0.257513	−0.0332447
$61$	−0.236790	−0.0303179	−0.0151590	−	0.999885i	$-0.504825\pi$
$61$	−0.236790	−0.0303179	−0.0151590	+	0.999885i	$0.504825\pi$
$62$	−10.1775	−1.29255
$63$	0.371170	0.0467631
$64$	8.34232	1.04279
$65$	7.97296	0.988924
$66$	−6.33779	−0.780128
$67$	2.33250	0.284960	0.142480	−	0.989798i	$-0.454492\pi$
$67$	2.33250	0.284960	0.142480	+	0.989798i	$0.454492\pi$
$68$	−0.530159	−0.0642912
$69$	−4.64579	−0.559287
$70$	−0.847854	−0.101338
$71$	6.91848	0.821073	0.410536	−	0.911844i	$-0.365341\pi$
$71$	6.91848	0.821073	0.410536	+	0.911844i	$0.365341\pi$
$72$	−3.57453	−0.421262
$73$	4.76591	0.557807	0.278904	−	0.960319i	$-0.410029\pi$
$73$	4.76591	0.557807	0.278904	+	0.960319i	$0.410029\pi$
$74$	4.89373	0.568884
$75$	1.08146	0.124876
$76$	0.0634161	0.00727433
$77$	1.03804	0.118295
$78$	−7.14377	−0.808872
$79$	3.80018	0.427553	0.213777	−	0.976883i	$-0.431423\pi$
$79$	3.80018	0.427553	0.213777	+	0.976883i	$0.431423\pi$
$80$	7.77740	0.869539
$81$	−3.76290	−0.418100
$82$	−10.9435	−1.20851
$83$	−7.24637	−0.795392	−0.397696	−	0.917517i	$-0.630190\pi$
$83$	−7.24637	−0.795392	−0.397696	+	0.917517i	$0.630190\pi$
$84$	−0.0377903	−0.00412326
$85$	−11.4444	−1.24132
$86$	−1.35973	−0.146623
$87$	−11.7979	−1.26487
$88$	−9.99672	−1.06565
$89$	14.2851	1.51421	0.757107	−	0.653291i	$-0.226612\pi$
$89$	14.2851	1.51421	0.757107	+	0.653291i	$0.226612\pi$
$90$	−3.49112	−0.367996
$91$	1.17004	0.122654
$92$	−0.331542	−0.0345657
$93$	−9.79237	−1.01542
$94$	4.07789	0.420602
$95$	1.36894	0.140451
$96$	0.711407	0.0726077
$97$	−10.8161	−1.09820	−0.549102	−	0.835755i	$-0.685030\pi$
$97$	−10.8161	−1.09820	−0.549102	+	0.835755i	$0.685030\pi$
$98$	9.53767	0.963450
$99$	4.27421	0.429574
$100$	0.0771775	0.00771775
$101$	−14.6903	−1.46174	−0.730872	−	0.682515i	$-0.760886\pi$
$101$	−14.6903	−1.46174	−0.730872	+	0.682515i	$0.760886\pi$
$102$	10.2541	1.01531
$103$	−13.1277	−1.29351	−0.646753	−	0.762699i	$-0.723874\pi$
$103$	−13.1277	−1.29351	−0.646753	+	0.762699i	$0.723874\pi$
$104$	−11.2680	−1.10492
$105$	−0.815767	−0.0796107
$106$	8.87931	0.862435
$107$	11.7666	1.13752	0.568760	−	0.822504i	$-0.307423\pi$
$107$	11.7666	1.13752	0.568760	+	0.822504i	$0.307423\pi$
$108$	−0.533209	−0.0513080
$109$	−2.17412	−0.208243	−0.104122	−	0.994565i	$-0.533203\pi$
$109$	−2.17412	−0.208243	−0.104122	+	0.994565i	$0.533203\pi$
$110$	−9.76345	−0.930908
$111$	4.70852	0.446913
$112$	1.14134	0.107847
$113$	−12.7111	−1.19576	−0.597881	−	0.801585i	$-0.703991\pi$
$113$	−12.7111	−1.19576	−0.597881	+	0.801585i	$0.703991\pi$
$114$	−1.22657	−0.114879
$115$	−7.15690	−0.667384
$116$	−0.841948	−0.0781729
$117$	4.81776	0.445402
$118$	−6.10216	−0.561750
$119$	−1.67947	−0.153957
$120$	7.85618	0.717168
$121$	0.953476	0.0866796
$122$	0.326841	0.0295908
$123$	−10.5294	−0.949400
$124$	−0.698823	−0.0627562
$125$	11.8955	1.06396
$126$	−0.512326	−0.0456416
$127$	16.2137	1.43873	0.719365	−	0.694633i	$-0.244433\pi$
$127$	16.2137	1.43873	0.719365	+	0.694633i	$0.244433\pi$
$128$	−10.4435	−0.923088
$129$	−1.30827	−0.115187
$130$	−11.0051	−0.965208
$131$	−1.83753	−0.160546	−0.0802729	−	0.996773i	$-0.525579\pi$
$131$	−1.83753	−0.160546	−0.0802729	+	0.996773i	$0.525579\pi$
$132$	−0.435173	−0.0378770
$133$	0.200894	0.0174197
$134$	−3.21954	−0.278126
$135$	−11.5102	−0.990641
$136$	16.1741	1.38691
$137$	5.89871	0.503961	0.251980	−	0.967732i	$-0.418918\pi$
$137$	5.89871	0.503961	0.251980	+	0.967732i	$0.418918\pi$
$138$	6.41257	0.545875
$139$	8.57012	0.726908	0.363454	−	0.931612i	$-0.381597\pi$
$139$	8.57012	0.726908	0.363454	+	0.931612i	$0.381597\pi$
$140$	−0.0582164	−0.00492019
$141$	3.92356	0.330423
$142$	−9.54957	−0.801382
$143$	13.4736	1.12672
$144$	4.69958	0.391632
$145$	−18.1749	−1.50934
$146$	−6.57837	−0.544430
$147$	9.17671	0.756882
$148$	0.336019	0.0276206
$149$	20.0774	1.64481	0.822403	−	0.568905i	$-0.192633\pi$
$149$	20.0774	1.64481	0.822403	+	0.568905i	$0.192633\pi$
$150$	−1.49274	−0.121882
$151$	21.5706	1.75539	0.877696	−	0.479218i	$-0.159080\pi$
$151$	21.5706	1.75539	0.877696	+	0.479218i	$0.159080\pi$
$152$	−1.93470	−0.156925
$153$	−6.91539	−0.559076
$154$	−1.43280	−0.115458
$155$	−15.0853	−1.21168
$156$	−0.490514	−0.0392726
$157$	−4.99937	−0.398993	−0.199497	−	0.979899i	$-0.563931\pi$
$157$	−4.99937	−0.398993	−0.199497	+	0.979899i	$0.563931\pi$
$158$	−5.24538	−0.417300
$159$	8.54327	0.677525
$160$	1.09593	0.0866411
$161$	−1.05028	−0.0827740
$162$	5.19393	0.408074
$163$	−17.7843	−1.39298	−0.696488	−	0.717568i	$-0.745255\pi$
$163$	−17.7843	−1.39298	−0.696488	+	0.717568i	$0.745255\pi$
$164$	−0.751418	−0.0586759
$165$	−9.39395	−0.731318
$166$	10.0022	0.776318
$167$	23.1226	1.78928	0.894640	−	0.446787i	$-0.147432\pi$
$167$	23.1226	1.78928	0.894640	+	0.446787i	$0.147432\pi$
$168$	1.15290	0.0889485
$169$	2.18703	0.168233
$170$	15.7966	1.21155
$171$	0.827201	0.0632576
$172$	−0.0933634	−0.00711889
$173$	−16.8117	−1.27817	−0.639087	−	0.769135i	$-0.720687\pi$
$173$	−16.8117	−1.27817	−0.639087	+	0.769135i	$0.720687\pi$
$174$	16.2847	1.23454
$175$	0.244489	0.0184816
$176$	13.1431	0.990699
$177$	−5.87122	−0.441308
$178$	−19.7176	−1.47790
$179$	18.9790	1.41856	0.709280	−	0.704927i	$-0.249020\pi$
$179$	18.9790	1.41856	0.709280	+	0.704927i	$0.249020\pi$
$180$	−0.239712	−0.0178670
$181$	8.22419	0.611299	0.305649	−	0.952144i	$-0.401126\pi$
$181$	8.22419	0.611299	0.305649	+	0.952144i	$0.401126\pi$
$182$	−1.61501	−0.119712
$183$	0.314472	0.0232464
$184$	10.1147	0.745664
$185$	7.25354	0.533291
$186$	13.5164	0.991070
$187$	−19.3400	−1.41428
$188$	0.280001	0.0204212
$189$	−1.68914	−0.122867
$190$	−1.88955	−0.137082
$191$	3.31876	0.240137	0.120069	−	0.992766i	$-0.461689\pi$
$191$	3.31876	0.240137	0.120069	+	0.992766i	$0.461689\pi$
$192$	−11.0791	−0.799566
$193$	22.2933	1.60471	0.802355	−	0.596848i	$-0.203580\pi$
$193$	22.2933	1.60471	0.802355	+	0.596848i	$0.203580\pi$
$194$	14.9294	1.07187
$195$	−10.5886	−0.758263
$196$	0.654887	0.0467777
$197$	12.2417	0.872182	0.436091	−	0.899903i	$-0.356363\pi$
$197$	12.2417	0.872182	0.436091	+	0.899903i	$0.356363\pi$
$198$	−5.89968	−0.419272
$199$	18.2365	1.29275	0.646377	−	0.763018i	$-0.276283\pi$
$199$	18.2365	1.29275	0.646377	+	0.763018i	$0.276283\pi$
$200$	−2.35453	−0.166490
$201$	−3.09770	−0.218495
$202$	20.2771	1.42669
$203$	−2.66718	−0.187200
$204$	0.704083	0.0492956
$205$	−16.2206	−1.13290
$206$	18.1201	1.26249
$207$	−4.32464	−0.300583
$208$	14.8145	1.02720
$209$	2.31339	0.160021
$210$	1.12600	0.0777015
$211$	−1.49736	−0.103082	−0.0515411	−	0.998671i	$-0.516413\pi$
$211$	−1.49736	−0.103082	−0.0515411	+	0.998671i	$0.516413\pi$
$212$	0.609682	0.0418731
$213$	−9.18816	−0.629563
$214$	−16.2414	−1.11024
$215$	−2.01540	−0.137449
$216$	16.2671	1.10684
$217$	−2.21378	−0.150281
$218$	3.00094	0.203249
$219$	−6.32941	−0.427702
$220$	−0.670390	−0.0451977
$221$	−21.7994	−1.46639
$222$	−6.49916	−0.436195
$223$	0.368166	0.0246542	0.0123271	−	0.999924i	$-0.496076\pi$
$223$	0.368166	0.0246542	0.0123271	+	0.999924i	$0.496076\pi$
$224$	0.160829	0.0107459
$225$	1.00670	0.0671136
$226$	17.5452	1.16709
$227$	20.1086	1.33465	0.667326	−	0.744765i	$-0.267439\pi$
$227$	20.1086	1.33465	0.667326	+	0.744765i	$0.267439\pi$
$228$	−0.0842205	−0.00557764
$229$	−19.5150	−1.28958	−0.644792	−	0.764358i	$-0.723056\pi$
$229$	−19.5150	−1.28958	−0.644792	+	0.764358i	$0.723056\pi$
$230$	9.87865	0.651379
$231$	−1.37857	−0.0907035
$232$	25.6861	1.68638
$233$	−10.3574	−0.678537	−0.339269	−	0.940690i	$-0.610180\pi$
$233$	−10.3574	−0.678537	−0.339269	+	0.940690i	$0.610180\pi$
$234$	−6.64994	−0.434720
$235$	6.04429	0.394286
$236$	−0.418994	−0.0272742
$237$	−5.04687	−0.327829
$238$	2.31818	0.150265
$239$	−9.67826	−0.626034	−0.313017	−	0.949747i	$-0.601340\pi$
$239$	−9.67826	−0.626034	−0.313017	+	0.949747i	$0.601340\pi$
$240$	−10.3289	−0.666725
$241$	−23.4147	−1.50827	−0.754137	−	0.656718i	$-0.771944\pi$
$241$	−23.4147	−1.50827	−0.754137	+	0.656718i	$0.771944\pi$
$242$	−1.31608	−0.0846009
$243$	−11.8806	−0.762144
$244$	0.0224420	0.00143670
$245$	14.1368	0.903169
$246$	14.5337	0.926632
$247$	2.60759	0.165917
$248$	21.3197	1.35380
$249$	9.62362	0.609872
$250$	−16.4193	−1.03845
$251$	23.9836	1.51383	0.756917	−	0.653511i	$-0.226705\pi$
$251$	23.9836	1.51383	0.756917	+	0.653511i	$0.226705\pi$
$252$	−0.0351780	−0.00221600
$253$	−12.0945	−0.760376
$254$	−22.3797	−1.40423
$255$	15.1988	0.951786
$256$	−2.26943	−0.141839
$257$	−19.8386	−1.23750	−0.618750	−	0.785588i	$-0.712361\pi$
$257$	−19.8386	−1.23750	−0.618750	+	0.785588i	$0.712361\pi$
$258$	1.80580	0.112424
$259$	1.06447	0.0661427
$260$	−0.755644	−0.0468630
$261$	−10.9824	−0.679792
$262$	2.53634	0.156696
$263$	5.57483	0.343759	0.171879	−	0.985118i	$-0.445016\pi$
$263$	5.57483	0.343759	0.171879	+	0.985118i	$0.445016\pi$
$264$	13.2763	0.817097
$265$	13.1610	0.808475
$266$	−0.277294	−0.0170020
$267$	−18.9714	−1.16103
$268$	−0.221064	−0.0135036
$269$	−17.9676	−1.09550	−0.547752	−	0.836641i	$-0.684516\pi$
$269$	−17.9676	−1.09550	−0.547752	+	0.836641i	$0.684516\pi$
$270$	15.8875	0.966884
$271$	4.64456	0.282137	0.141068	−	0.990000i	$-0.454946\pi$
$271$	4.64456	0.282137	0.141068	+	0.990000i	$0.454946\pi$
$272$	−21.2647	−1.28936
$273$	−1.55389	−0.0940455
$274$	−8.14198	−0.491875
$275$	2.81540	0.169775
$276$	0.440308	0.0265034
$277$	19.2078	1.15409	0.577043	−	0.816714i	$-0.304206\pi$
$277$	19.2078	1.15409	0.577043	+	0.816714i	$0.304206\pi$
$278$	−11.8293	−0.709476
$279$	−9.11545	−0.545728
$280$	1.77606	0.106140
$281$	17.7157	1.05683	0.528416	−	0.848986i	$-0.322786\pi$
$281$	17.7157	1.05683	0.528416	+	0.848986i	$0.322786\pi$
$282$	−5.41568	−0.322499
$283$	17.7025	1.05231	0.526153	−	0.850390i	$-0.323634\pi$
$283$	17.7025	1.05231	0.526153	+	0.850390i	$0.323634\pi$
$284$	−0.655705	−0.0389089
$285$	−1.81804	−0.107691
$286$	−18.5976	−1.09970
$287$	−2.38040	−0.140510
$288$	0.662230	0.0390223
$289$	14.2908	0.840636
$290$	25.0867	1.47314
$291$	14.3644	0.842054
$292$	−0.451693	−0.0264333
$293$	−3.13261	−0.183009	−0.0915047	−	0.995805i	$-0.529168\pi$
$293$	−3.13261	−0.183009	−0.0915047	+	0.995805i	$0.529168\pi$
$294$	−12.6666	−0.738731
$295$	−9.04469	−0.526602
$296$	−10.2513	−0.595842
$297$	−19.4512	−1.12867
$298$	−27.7128	−1.60536
$299$	−13.6326	−0.788392
$300$	−0.102496	−0.00591763
$301$	−0.295763	−0.0170475
$302$	−29.7739	−1.71329
$303$	19.5097	1.12080
$304$	2.54363	0.145887
$305$	0.484448	0.0277394
$306$	9.54530	0.545669
$307$	0.217861	0.0124340	0.00621700	−	0.999981i	$-0.498021\pi$
$307$	0.217861	0.0124340	0.00621700	+	0.999981i	$0.498021\pi$
$308$	−0.0983806	−0.00560576
$309$	17.4343	0.991804
$310$	20.8222	1.18262
$311$	−17.2626	−0.978872	−0.489436	−	0.872039i	$-0.662797\pi$
$311$	−17.2626	−0.978872	−0.489436	+	0.872039i	$0.662797\pi$
$312$	14.9646	0.847203
$313$	11.6304	0.657388	0.328694	−	0.944437i	$-0.393392\pi$
$313$	11.6304	0.657388	0.328694	+	0.944437i	$0.393392\pi$
$314$	6.90062	0.389425
$315$	−0.759376	−0.0427860
$316$	−0.360165	−0.0202609
$317$	5.02237	0.282085	0.141042	−	0.990004i	$-0.454955\pi$
$317$	5.02237	0.282085	0.141042	+	0.990004i	$0.454955\pi$
$318$	−11.7923	−0.661277
$319$	−30.7139	−1.71965
$320$	−17.0675	−0.954103
$321$	−15.6267	−0.872200
$322$	1.44971	0.0807889
$323$	−3.74292	−0.208262
$324$	0.356632	0.0198129
$325$	3.17344	0.176031
$326$	24.5477	1.35957
$327$	2.88736	0.159672
$328$	22.9242	1.26578
$329$	0.887007	0.0489023
$330$	12.9665	0.713780
$331$	7.47113	0.410651	0.205325	−	0.978694i	$-0.434175\pi$
$331$	7.47113	0.410651	0.205325	+	0.978694i	$0.434175\pi$
$332$	0.686780	0.0376920
$333$	4.38303	0.240189
$334$	−31.9161	−1.74637
$335$	−4.77204	−0.260724
$336$	−1.51577	−0.0826922
$337$	14.1266	0.769525	0.384763	−	0.923016i	$-0.374283\pi$
$337$	14.1266	0.769525	0.384763	+	0.923016i	$0.374283\pi$
$338$	−3.01875	−0.164198
$339$	16.8811	0.916858
$340$	1.08465	0.0588233
$341$	−25.4928	−1.38051
$342$	−1.14178	−0.0617406
$343$	4.17626	0.225497
$344$	2.84833	0.153571
$345$	9.50479	0.511721
$346$	23.2052	1.24752
$347$	−17.3816	−0.933091	−0.466546	−	0.884497i	$-0.654502\pi$
$347$	−17.3816	−0.933091	−0.466546	+	0.884497i	$0.654502\pi$
$348$	1.11816	0.0599395
$349$	−11.3743	−0.608850	−0.304425	−	0.952536i	$-0.598464\pi$
$349$	−11.3743	−0.608850	−0.304425	+	0.952536i	$0.598464\pi$
$350$	−0.337467	−0.0180384
$351$	−21.9248	−1.17026
$352$	1.85203	0.0987135
$353$	−27.8215	−1.48079	−0.740395	−	0.672172i	$-0.765362\pi$
$353$	−27.8215	−1.48079	−0.740395	+	0.672172i	$0.765362\pi$
$354$	8.10404	0.430725
$355$	−14.1545	−0.751242
$356$	−1.35388	−0.0717554
$357$	2.23044	0.118048
$358$	−26.1967	−1.38454
$359$	31.8150	1.67913	0.839567	−	0.543257i	$-0.182809\pi$
$359$	31.8150	1.67913	0.839567	+	0.543257i	$0.182809\pi$
$360$	7.31311	0.385435
$361$	−18.5523	−0.976436
$362$	−11.3518	−0.596639
$363$	−1.26627	−0.0664621
$364$	−0.110892	−0.00581230
$365$	−9.75054	−0.510367
$366$	−0.434065	−0.0226890
$367$	−3.39169	−0.177045	−0.0885224	−	0.996074i	$-0.528214\pi$
$367$	−3.39169	−0.177045	−0.0885224	+	0.996074i	$0.528214\pi$
$368$	−13.2982	−0.693216
$369$	−9.80150	−0.510246
$370$	−10.0120	−0.520501
$371$	1.93140	0.100273
$372$	0.928079	0.0481187
$373$	16.6667	0.862970	0.431485	−	0.902120i	$-0.357990\pi$
$373$	16.6667	0.862970	0.431485	+	0.902120i	$0.357990\pi$
$374$	26.6949	1.38036
$375$	−15.7979	−0.815801
$376$	−8.54226	−0.440533
$377$	−34.6198	−1.78301
$378$	2.33151	0.119920
$379$	6.52444	0.335138	0.167569	−	0.985860i	$-0.446408\pi$
$379$	6.52444	0.335138	0.167569	+	0.985860i	$0.446408\pi$
$380$	−0.129743	−0.00665566
$381$	−21.5327	−1.10315
$382$	−4.58089	−0.234378
$383$	8.66460	0.442740	0.221370	−	0.975190i	$-0.428947\pi$
$383$	8.66460	0.442740	0.221370	+	0.975190i	$0.428947\pi$
$384$	13.8697	0.707783
$385$	−2.12371	−0.108234
$386$	−30.7715	−1.56623
$387$	−1.21783	−0.0619059
$388$	1.02510	0.0520416
$389$	25.3450	1.28504	0.642522	−	0.766267i	$-0.277888\pi$
$389$	25.3450	1.28504	0.642522	+	0.766267i	$0.277888\pi$
$390$	14.6154	0.740079
$391$	19.5682	0.989604
$392$	−19.9793	−1.00911
$393$	2.44035	0.123099
$394$	−16.8971	−0.851266
$395$	−7.77476	−0.391191
$396$	−0.405091	−0.0203566
$397$	17.7609	0.891394	0.445697	−	0.895184i	$-0.352956\pi$
$397$	17.7609	0.891394	0.445697	+	0.895184i	$0.352956\pi$
$398$	−25.1719	−1.26175
$399$	−0.266800	−0.0133567
$400$	3.09560	0.154780
$401$	6.25522	0.312371	0.156185	−	0.987728i	$-0.450080\pi$
$401$	6.25522	0.312371	0.156185	+	0.987728i	$0.450080\pi$
$402$	4.27574	0.213255
$403$	−28.7347	−1.43138
$404$	1.39229	0.0692689
$405$	7.69850	0.382542
$406$	3.68151	0.182710
$407$	12.2578	0.607598
$408$	−21.4801	−1.06342
$409$	1.00000	0.0494468
$409$	1.00000	0.0494468
$410$	22.3893	1.10573
$411$	−7.83384	−0.386415
$412$	1.24418	0.0612965
$413$	−1.32732	−0.0653132
$414$	5.96929	0.293375
$415$	14.8253	0.727746
$416$	2.08755	0.102351
$417$	−11.3816	−0.557361
$418$	−3.19317	−0.156183
$419$	−17.4774	−0.853825	−0.426913	−	0.904293i	$-0.640399\pi$
$419$	−17.4774	−0.853825	−0.426913	+	0.904293i	$0.640399\pi$
$420$	0.0773149	0.00377258
$421$	23.7864	1.15928	0.579640	−	0.814872i	$-0.303193\pi$
$421$	23.7864	1.15928	0.579640	+	0.814872i	$0.303193\pi$
$422$	2.06680	0.100610
$423$	3.65233	0.177583
$424$	−18.6002	−0.903304
$425$	−4.55514	−0.220957
$426$	12.6824	0.614465
$427$	0.0710934	0.00344045
$428$	−1.11519	−0.0539047
$429$	−17.8937	−0.863918
$430$	2.78186	0.134153
$431$	9.72228	0.468306	0.234153	−	0.972200i	$-0.424768\pi$
$431$	9.72228	0.468306	0.234153	+	0.972200i	$0.424768\pi$
$432$	−21.3871	−1.02899
$433$	12.8718	0.618582	0.309291	−	0.950968i	$-0.399908\pi$
$433$	12.8718	0.618582	0.309291	+	0.950968i	$0.399908\pi$
$434$	3.05568	0.146677
$435$	24.1373	1.15729
$436$	0.206054	0.00986820
$437$	−2.34069	−0.111970
$438$	8.73648	0.417445
$439$	16.3107	0.778466	0.389233	−	0.921139i	$-0.372740\pi$
$439$	16.3107	0.778466	0.389233	+	0.921139i	$0.372740\pi$
$440$	20.4522	0.975022
$441$	8.54235	0.406779
$442$	30.0897	1.43122
$443$	−20.7520	−0.985957	−0.492978	−	0.870042i	$-0.664092\pi$
$443$	−20.7520	−0.985957	−0.492978	+	0.870042i	$0.664092\pi$
$444$	−0.446254	−0.0211783
$445$	−29.2257	−1.38543
$446$	−0.508179	−0.0240630
$447$	−26.6640	−1.26116
$448$	−2.50468	−0.118335
$449$	−14.7710	−0.697086	−0.348543	−	0.937293i	$-0.613324\pi$
$449$	−14.7710	−0.697086	−0.348543	+	0.937293i	$0.613324\pi$
$450$	−1.38955	−0.0655041
$451$	−27.4114	−1.29075
$452$	1.20471	0.0566647
$453$	−28.6471	−1.34596
$454$	−27.7558	−1.30265
$455$	−2.39378	−0.112222
$456$	2.56939	0.120323
$457$	−15.0736	−0.705113	−0.352557	−	0.935790i	$-0.614688\pi$
$457$	−15.0736	−0.705113	−0.352557	+	0.935790i	$0.614688\pi$
$458$	26.9365	1.25866
$459$	31.4708	1.46893
$460$	0.678301	0.0316259
$461$	−19.8978	−0.926733	−0.463366	−	0.886167i	$-0.653359\pi$
$461$	−19.8978	−0.926733	−0.463366	+	0.886167i	$0.653359\pi$
$462$	1.90284	0.0885283
$463$	−27.4958	−1.27784	−0.638919	−	0.769274i	$-0.720618\pi$
$463$	−27.4958	−1.27784	−0.638919	+	0.769274i	$0.720618\pi$
$464$	−33.7706	−1.56776
$465$	20.0342	0.929061
$466$	14.2963	0.662265
$467$	−18.1186	−0.838429	−0.419215	−	0.907887i	$-0.637695\pi$
$467$	−18.1186	−0.838429	−0.419215	+	0.907887i	$0.637695\pi$
$468$	−0.456607	−0.0211066
$469$	−0.700303	−0.0323370
$470$	−8.34292	−0.384830
$471$	6.63946	0.305930
$472$	12.7827	0.588370
$473$	−3.40586	−0.156601
$474$	6.96618	0.319967
$475$	0.544874	0.0250005
$476$	0.159174	0.00729571
$477$	7.95270	0.364129
$478$	13.3589	0.611021
$479$	1.62305	0.0741590	0.0370795	−	0.999312i	$-0.488195\pi$
$479$	1.62305	0.0741590	0.0370795	+	0.999312i	$0.488195\pi$
$480$	−1.45546	−0.0664325
$481$	13.8167	0.629985
$482$	32.3193	1.47210
$483$	1.39484	0.0634674
$484$	−0.0903664	−0.00410757
$485$	22.1285	1.00480
$486$	16.3988	0.743866
$487$	17.6722	0.800805	0.400403	−	0.916339i	$-0.368870\pi$
$487$	17.6722	0.800805	0.400403	+	0.916339i	$0.368870\pi$
$488$	−0.684660	−0.0309931
$489$	23.6187	1.06807
$490$	−19.5131	−0.881510
$491$	40.4253	1.82437	0.912183	−	0.409782i	$-0.134395\pi$
$491$	40.4253	1.82437	0.912183	+	0.409782i	$0.134395\pi$
$492$	0.997928	0.0449901
$493$	49.6931	2.23806
$494$	−3.59925	−0.161938
$495$	−8.74457	−0.393039
$496$	−28.0299	−1.25858
$497$	−2.07719	−0.0931747
$498$	−13.2835	−0.595246
$499$	−2.67593	−0.119791	−0.0598955	−	0.998205i	$-0.519077\pi$
$499$	−2.67593	−0.119791	−0.0598955	+	0.998205i	$0.519077\pi$
$500$	−1.12740	−0.0504190
$501$	−30.7082	−1.37194
$502$	−33.1046	−1.47753
$503$	13.3355	0.594599	0.297300	−	0.954784i	$-0.403914\pi$
$503$	13.3355	0.594599	0.297300	+	0.954784i	$0.403914\pi$
$504$	1.07321	0.0478045
$505$	30.0549	1.33742
$506$	16.6941	0.742141
$507$	−2.90450	−0.128994
$508$	−1.53666	−0.0681783
$509$	32.0232	1.41940	0.709702	−	0.704502i	$-0.248830\pi$
$509$	32.0232	1.41940	0.709702	+	0.704502i	$0.248830\pi$
$510$	−20.9789	−0.928961
$511$	−1.43090	−0.0632995
$512$	24.0196	1.06153
$513$	−3.76446	−0.166205
$514$	27.3833	1.20782
$515$	26.8578	1.18350
$516$	0.123992	0.00545845
$517$	10.2143	0.449225
$518$	−1.46928	−0.0645565
$519$	22.3270	0.980047
$520$	23.0531	1.01095
$521$	24.3133	1.06519	0.532594	−	0.846371i	$-0.321217\pi$
$521$	24.3133	1.06519	0.532594	+	0.846371i	$0.321217\pi$
$522$	15.1589	0.663489
$523$	−3.22360	−0.140958	−0.0704791	−	0.997513i	$-0.522453\pi$
$523$	−3.22360	−0.140958	−0.0704791	+	0.997513i	$0.522453\pi$
$524$	0.174153	0.00760793
$525$	−0.324696	−0.0141709
$526$	−7.69493	−0.335515
$527$	41.2457	1.79669
$528$	−17.4548	−0.759625
$529$	−10.7628	−0.467946
$530$	−18.1661	−0.789086
$531$	−5.46536	−0.237177
$532$	−0.0190399	−0.000825485 0
$533$	−30.8973	−1.33831
$534$	26.1862	1.13319
$535$	−24.0732	−1.04078
$536$	6.74422	0.291306
$537$	−25.2053	−1.08769
$538$	24.8007	1.06923
$539$	23.8900	1.02902
$540$	1.09089	0.0469444
$541$	−26.5420	−1.14113	−0.570564	−	0.821253i	$-0.693275\pi$
$541$	−26.5420	−1.14113	−0.570564	+	0.821253i	$0.693275\pi$
$542$	−6.41087	−0.275371
$543$	−10.9222	−0.468717
$544$	−2.99646	−0.128472
$545$	4.44802	0.190532
$546$	2.14483	0.0917901
$547$	20.5342	0.877981	0.438990	−	0.898492i	$-0.355336\pi$
$547$	20.5342	0.877981	0.438990	+	0.898492i	$0.355336\pi$
$548$	−0.559055	−0.0238816
$549$	0.292734	0.0124936
$550$	−3.88610	−0.165704
$551$	−5.94416	−0.253230
$552$	−13.4329	−0.571742
$553$	−1.14096	−0.0485184
$554$	−26.5125	−1.12641
$555$	−9.63313	−0.408904
$556$	−0.812240	−0.0344466
$557$	33.5780	1.42275	0.711373	−	0.702814i	$-0.248073\pi$
$557$	33.5780	1.42275	0.711373	+	0.702814i	$0.248073\pi$
$558$	12.5820	0.532641
$559$	−3.83898	−0.162371
$560$	−2.33507	−0.0986746
$561$	25.6846	1.08441
$562$	−24.4530	−1.03149
$563$	−34.2480	−1.44338	−0.721692	−	0.692215i	$-0.756635\pi$
$563$	−34.2480	−1.44338	−0.721692	+	0.692215i	$0.756635\pi$
$564$	−0.371858	−0.0156580
$565$	26.0056	1.09406
$566$	−24.4348	−1.02707
$567$	1.12977	0.0474457
$568$	20.0042	0.839358
$569$	7.92621	0.332284	0.166142	−	0.986102i	$-0.446869\pi$
$569$	7.92621	0.332284	0.166142	+	0.986102i	$0.446869\pi$
$570$	2.50944	0.105109
$571$	16.8742	0.706165	0.353082	−	0.935592i	$-0.385134\pi$
$571$	16.8742	0.706165	0.353082	+	0.935592i	$0.385134\pi$
$572$	−1.27697	−0.0533928
$573$	−4.40752	−0.184127
$574$	3.28566	0.137141
$575$	−2.84862	−0.118796
$576$	−10.3132	−0.429718
$577$	−4.20001	−0.174849	−0.0874244	−	0.996171i	$-0.527864\pi$
$577$	−4.20001	−0.174849	−0.0874244	+	0.996171i	$0.527864\pi$
$578$	−19.7256	−0.820476
$579$	−29.6069	−1.23042
$580$	1.72254	0.0715244
$581$	2.17563	0.0902605
$582$	−19.8271	−0.821861
$583$	22.2409	0.921126
$584$	13.7802	0.570230
$585$	−9.85662	−0.407521
$586$	4.32394	0.178620
$587$	26.0714	1.07608	0.538042	−	0.842918i	$-0.319164\pi$
$587$	26.0714	1.07608	0.538042	+	0.842918i	$0.319164\pi$
$588$	−0.869730	−0.0358670
$589$	−4.93370	−0.203289
$590$	12.4844	0.513974
$591$	−16.2577	−0.668751
$592$	13.4778	0.553932
$593$	−30.8085	−1.26515	−0.632577	−	0.774497i	$-0.718003\pi$
$593$	−30.8085	−1.26515	−0.632577	+	0.774497i	$0.718003\pi$
$594$	26.8485	1.10161
$595$	3.43603	0.140863
$596$	−1.90285	−0.0779439
$597$	−24.2192	−0.991227
$598$	18.8170	0.769485
$599$	−9.25916	−0.378319	−0.189159	−	0.981946i	$-0.560576\pi$
$599$	−9.25916	−0.378319	−0.189159	+	0.981946i	$0.560576\pi$
$600$	3.12696	0.127657
$601$	6.56273	0.267699	0.133850	−	0.991002i	$-0.457266\pi$
$601$	6.56273	0.267699	0.133850	+	0.991002i	$0.457266\pi$
$602$	0.408242	0.0166387
$603$	−2.88356	−0.117428
$604$	−2.04437	−0.0831843
$605$	−1.95071	−0.0793077
$606$	−26.9291	−1.09392
$607$	−14.5775	−0.591682	−0.295841	−	0.955237i	$-0.595600\pi$
$607$	−14.5775	−0.591682	−0.295841	+	0.955237i	$0.595600\pi$
$608$	0.358429	0.0145362
$609$	3.54218	0.143536
$610$	−0.668683	−0.0270742
$611$	11.5133	0.465777
$612$	0.655412	0.0264934
$613$	−6.37447	−0.257462	−0.128731	−	0.991680i	$-0.541090\pi$
$613$	−6.37447	−0.257462	−0.128731	+	0.991680i	$0.541090\pi$
$614$	−0.300714	−0.0121358
$615$	21.5420	0.868655
$616$	3.00139	0.120930
$617$	6.10386	0.245732	0.122866	−	0.992423i	$-0.460791\pi$
$617$	6.10386	0.245732	0.122866	+	0.992423i	$0.460791\pi$
$618$	−24.0646	−0.968019
$619$	−20.0277	−0.804981	−0.402491	−	0.915424i	$-0.631855\pi$
$619$	−20.0277	−0.804981	−0.402491	+	0.915424i	$0.631855\pi$
$620$	1.42972	0.0574189
$621$	19.6807	0.789761
$622$	23.8275	0.955397
$623$	−4.28891	−0.171832
$624$	−19.6746	−0.787613
$625$	−20.2653	−0.810612
$626$	−16.0534	−0.641622
$627$	−3.07233	−0.122697
$628$	0.473819	0.0189074
$629$	−19.8324	−0.790769
$630$	1.04816	0.0417599
$631$	−4.19378	−0.166952	−0.0834758	−	0.996510i	$-0.526602\pi$
$631$	−4.19378	−0.166952	−0.0834758	+	0.996510i	$0.526602\pi$
$632$	10.9879	0.437075
$633$	1.98858	0.0790389
$634$	−6.93237	−0.275320
$635$	−33.1714	−1.31637
$636$	−0.809695	−0.0321065
$637$	26.9281	1.06693
$638$	42.3943	1.67841
$639$	−8.55302	−0.338352
$640$	21.3664	0.844581
$641$	18.0977	0.714815	0.357407	−	0.933949i	$-0.383661\pi$
$641$	18.0977	0.714815	0.357407	+	0.933949i	$0.383661\pi$
$642$	21.5696	0.851283
$643$	−34.3118	−1.35313	−0.676563	−	0.736384i	$-0.736532\pi$
$643$	−34.3118	−1.35313	−0.676563	+	0.736384i	$0.736532\pi$
$644$	0.0995415	0.00392248
$645$	2.67658	0.105390
$646$	5.16635	0.203267
$647$	−42.9999	−1.69050	−0.845250	−	0.534372i	$-0.820548\pi$
$647$	−42.9999	−1.69050	−0.845250	+	0.534372i	$0.820548\pi$
$648$	−10.8801	−0.427412
$649$	−15.2847	−0.599978
$650$	−4.38029	−0.171809
$651$	2.94004	0.115229
$652$	1.68552	0.0660102
$653$	−25.7091	−1.00607	−0.503037	−	0.864265i	$-0.667784\pi$
$653$	−25.7091	−1.00607	−0.503037	+	0.864265i	$0.667784\pi$
$654$	−3.98543	−0.155842
$655$	3.75939	0.146892
$656$	−30.1394	−1.17675
$657$	−5.89188	−0.229864
$658$	−1.22433	−0.0477295
$659$	−21.5003	−0.837534	−0.418767	−	0.908094i	$-0.637538\pi$
$659$	−21.5003	−0.837534	−0.418767	+	0.908094i	$0.637538\pi$
$660$	0.890319	0.0346556
$661$	43.9110	1.70794	0.853970	−	0.520322i	$-0.174188\pi$
$661$	43.9110	1.70794	0.853970	+	0.520322i	$0.174188\pi$
$662$	−10.3124	−0.400803
$663$	28.9509	1.12436
$664$	−20.9523	−0.813106
$665$	−0.411008	−0.0159382
$666$	−6.04990	−0.234429
$667$	31.0763	1.20328
$668$	−2.19146	−0.0847902
$669$	−0.488946	−0.0189038
$670$	6.58684	0.254472
$671$	0.818674	0.0316046
$672$	−0.213591	−0.00823946
$673$	49.1559	1.89482	0.947410	−	0.320022i	$-0.103690\pi$
$673$	49.1559	1.89482	0.947410	+	0.320022i	$0.103690\pi$
$674$	−19.4989	−0.751071
$675$	−4.58135	−0.176336
$676$	−0.207277	−0.00797220
$677$	23.1357	0.889178	0.444589	−	0.895735i	$-0.353350\pi$
$677$	23.1357	0.889178	0.444589	+	0.895735i	$0.353350\pi$
$678$	−23.3010	−0.894870
$679$	3.24739	0.124623
$680$	−33.0904	−1.26896
$681$	−26.7054	−1.02335
$682$	35.1876	1.34740
$683$	−44.0487	−1.68548	−0.842739	−	0.538323i	$-0.819058\pi$
$683$	−44.0487	−1.68548	−0.842739	+	0.538323i	$0.819058\pi$
$684$	−0.0783986	−0.00299764
$685$	−12.0681	−0.461100
$686$	−5.76449	−0.220089
$687$	25.9170	0.988797
$688$	−3.74481	−0.142770
$689$	25.0693	0.955065
$690$	−13.1194	−0.499449
$691$	16.0482	0.610504	0.305252	−	0.952272i	$-0.401259\pi$
$691$	16.0482	0.610504	0.305252	+	0.952272i	$0.401259\pi$
$692$	1.59335	0.0605699
$693$	−1.28328	−0.0487477
$694$	23.9918	0.910714
$695$	−17.5336	−0.665086
$696$	−34.1127	−1.29304
$697$	44.3499	1.67987
$698$	15.6999	0.594249
$699$	13.7553	0.520273
$700$	−0.0231716	−0.000875804 0
$701$	−50.2864	−1.89929	−0.949646	−	0.313324i	$-0.898557\pi$
$701$	−50.2864	−1.89929	−0.949646	+	0.313324i	$0.898557\pi$
$702$	30.2628	1.14220
$703$	2.37230	0.0894729
$704$	−28.8426	−1.08705
$705$	−8.02718	−0.302321
$706$	38.4020	1.44528
$707$	4.41059	0.165877
$708$	0.556450	0.0209127
$709$	10.6030	0.398203	0.199102	−	0.979979i	$-0.436198\pi$
$709$	10.6030	0.398203	0.199102	+	0.979979i	$0.436198\pi$
$710$	19.5374	0.733226
$711$	−4.69799	−0.176188
$712$	41.3040	1.54793
$713$	25.7936	0.965977
$714$	−3.07868	−0.115217
$715$	−27.5656	−1.03089
$716$	−1.79875	−0.0672225
$717$	12.8533	0.480016
$718$	−43.9142	−1.63887
$719$	50.7851	1.89397	0.946983	−	0.321284i	$-0.104114\pi$
$719$	50.7851	1.89397	0.946983	+	0.321284i	$0.104114\pi$
$720$	−9.61485	−0.358324
$721$	3.94142	0.146786
$722$	25.6077	0.953019
$723$	31.0961	1.15648
$724$	−0.779454	−0.0289682
$725$	−7.23404	−0.268666
$726$	1.74784	0.0648683
$727$	−17.9301	−0.664990	−0.332495	−	0.943105i	$-0.607890\pi$
$727$	−17.9301	−0.664990	−0.332495	+	0.943105i	$0.607890\pi$
$728$	3.38308	0.125385
$729$	27.0669	1.00248
$730$	13.4587	0.498127
$731$	5.51046	0.203812
$732$	−0.0298043	−0.00110160
$733$	26.8303	0.990999	0.495500	−	0.868608i	$-0.334985\pi$
$733$	26.8303	0.990999	0.495500	+	0.868608i	$0.334985\pi$
$734$	4.68155	0.172799
$735$	−18.7746	−0.692511
$736$	−1.87388	−0.0690722
$737$	−8.06433	−0.297053
$738$	13.5290	0.498009
$739$	25.1889	0.926587	0.463294	−	0.886205i	$-0.346668\pi$
$739$	25.1889	0.926587	0.463294	+	0.886205i	$0.346668\pi$
$740$	−0.687460	−0.0252715
$741$	−3.46303	−0.127218
$742$	−2.66590	−0.0978683
$743$	22.3360	0.819430	0.409715	−	0.912214i	$-0.365628\pi$
$743$	22.3360	0.819430	0.409715	+	0.912214i	$0.365628\pi$
$744$	−28.3138	−1.03803
$745$	−41.0763	−1.50492
$746$	−23.0051	−0.842275
$747$	8.95837	0.327770
$748$	1.83296	0.0670196
$749$	−3.53277	−0.129085
$750$	21.8058	0.796237
$751$	23.4082	0.854176	0.427088	−	0.904210i	$-0.359539\pi$
$751$	23.4082	0.854176	0.427088	+	0.904210i	$0.359539\pi$
$752$	11.2309	0.409547
$753$	−31.8517	−1.16074
$754$	47.7856	1.74025
$755$	−44.1312	−1.60610
$756$	0.160089	0.00582239
$757$	15.8828	0.577271	0.288635	−	0.957439i	$-0.406798\pi$
$757$	15.8828	0.577271	0.288635	+	0.957439i	$0.406798\pi$
$758$	−9.00567	−0.327101
$759$	16.0623	0.583023
$760$	3.95819	0.143579
$761$	11.3967	0.413129	0.206564	−	0.978433i	$-0.433772\pi$
$761$	11.3967	0.413129	0.206564	+	0.978433i	$0.433772\pi$
$762$	29.7216	1.07670
$763$	0.652753	0.0236312
$764$	−0.314538	−0.0113796
$765$	14.1482	0.511528
$766$	−11.9597	−0.432123
$767$	−17.2285	−0.622085
$768$	3.01394	0.108756
$769$	28.4670	1.02655	0.513274	−	0.858225i	$-0.328432\pi$
$769$	28.4670	1.02655	0.513274	+	0.858225i	$0.328432\pi$
$770$	2.93135	0.105639
$771$	26.3469	0.948861
$772$	−2.11287	−0.0760438
$773$	22.8660	0.822432	0.411216	−	0.911538i	$-0.365104\pi$
$773$	22.8660	0.822432	0.411216	+	0.911538i	$0.365104\pi$
$774$	1.68097	0.0604213
$775$	−6.00431	−0.215681
$776$	−31.2737	−1.12266
$777$	−1.41367	−0.0507153
$778$	−34.9837	−1.25423
$779$	−5.30501	−0.190072
$780$	1.00354	0.0359325
$781$	−23.9198	−0.855919
$782$	−27.0099	−0.965872
$783$	49.9790	1.78610
$784$	26.2676	0.938128
$785$	10.2282	0.365059
$786$	−3.36841	−0.120147
$787$	−0.582587	−0.0207670	−0.0103835	−	0.999946i	$-0.503305\pi$
$787$	−0.582587	−0.0207670	−0.0103835	+	0.999946i	$0.503305\pi$
$788$	−1.16021	−0.0413309
$789$	−7.40371	−0.263579
$790$	10.7315	0.381809
$791$	3.81636	0.135694
$792$	12.3585	0.439140
$793$	0.922785	0.0327691
$794$	−24.5153	−0.870017
$795$	−17.4786	−0.619903
$796$	−1.72838	−0.0612609
$797$	−4.59138	−0.162635	−0.0813176	−	0.996688i	$-0.525913\pi$
$797$	−4.59138	−0.162635	−0.0813176	+	0.996688i	$0.525913\pi$
$798$	0.368263	0.0130364
$799$	−16.5261	−0.584651
$800$	0.436208	0.0154223
$801$	−17.6600	−0.623985
$802$	−8.63407	−0.304880
$803$	−16.4776	−0.581480
$804$	0.293587	0.0103540
$805$	2.14877	0.0757342
$806$	39.6624	1.39705
$807$	23.8621	0.839985
$808$	−42.4759	−1.49430
$809$	−18.1700	−0.638824	−0.319412	−	0.947616i	$-0.603485\pi$
$809$	−18.1700	−0.638824	−0.319412	+	0.947616i	$0.603485\pi$
$810$	−10.6262	−0.373368
$811$	−53.0220	−1.86185	−0.930926	−	0.365207i	$-0.880998\pi$
$811$	−53.0220	−1.86185	−0.930926	+	0.365207i	$0.880998\pi$
$812$	0.252784	0.00887099
$813$	−6.16825	−0.216330
$814$	−16.9195	−0.593027
$815$	36.3849	1.27451
$816$	28.2408	0.988626
$817$	−0.659146	−0.0230606
$818$	−1.38030	−0.0482610
$819$	−1.44647	−0.0505438
$820$	1.53732	0.0536856
$821$	36.7124	1.28127	0.640635	−	0.767845i	$-0.278671\pi$
$821$	36.7124	1.28127	0.640635	+	0.767845i	$0.278671\pi$
$822$	10.8130	0.377148
$823$	−49.3411	−1.71992	−0.859961	−	0.510360i	$-0.829512\pi$
$823$	−49.3411	−1.71992	−0.859961	+	0.510360i	$0.829512\pi$
$824$	−37.9575	−1.32231
$825$	−3.73902	−0.130176
$826$	1.83210	0.0637469
$827$	−34.0759	−1.18494	−0.592468	−	0.805594i	$-0.701846\pi$
$827$	−34.0759	−1.18494	−0.592468	+	0.805594i	$0.701846\pi$
$828$	0.409871	0.0142440
$829$	−16.5307	−0.574134	−0.287067	−	0.957911i	$-0.592680\pi$
$829$	−16.5307	−0.574134	−0.287067	+	0.957911i	$0.592680\pi$
$830$	−20.4633	−0.710293
$831$	−25.5092	−0.884903
$832$	−32.5105	−1.12710
$833$	−38.6525	−1.33923
$834$	15.7101	0.543995
$835$	−47.3064	−1.63711
$836$	−0.219254	−0.00758305
$837$	41.4830	1.43386
$838$	24.1240	0.833349
$839$	−13.6583	−0.471538	−0.235769	−	0.971809i	$-0.575761\pi$
$839$	−13.6583	−0.471538	−0.235769	+	0.971809i	$0.575761\pi$
$840$	−2.35872	−0.0813836
$841$	49.9179	1.72131
$842$	−32.8324	−1.13148
$843$	−23.5276	−0.810332
$844$	0.141913	0.00488485
$845$	−4.47443	−0.153925
$846$	−5.04131	−0.173324
$847$	−0.286269	−0.00983633
$848$	24.4544	0.839768
$849$	−23.5100	−0.806862
$850$	6.28745	0.215658
$851$	−12.4025	−0.425151
$852$	0.870815	0.0298336
$853$	−28.4297	−0.973414	−0.486707	−	0.873565i	$-0.661802\pi$
$853$	−28.4297	−0.973414	−0.486707	+	0.873565i	$0.661802\pi$
$854$	−0.0981301	−0.00335794
$855$	−1.69236	−0.0578777
$856$	34.0221	1.16285
$857$	27.5759	0.941975	0.470988	−	0.882140i	$-0.343898\pi$
$857$	27.5759	0.941975	0.470988	+	0.882140i	$0.343898\pi$
$858$	24.6987	0.843200
$859$	−28.7472	−0.980843	−0.490422	−	0.871485i	$-0.663157\pi$
$859$	−28.7472	−0.980843	−0.490422	+	0.871485i	$0.663157\pi$
$860$	0.191012	0.00651344
$861$	3.16131	0.107737
$862$	−13.4197	−0.457075
$863$	28.4792	0.969444	0.484722	−	0.874668i	$-0.338921\pi$
$863$	28.4792	0.969444	0.484722	+	0.874668i	$0.338921\pi$
$864$	−3.01370	−0.102528
$865$	34.3950	1.16947
$866$	−17.7670	−0.603747
$867$	−18.9791	−0.644562
$868$	0.209813	0.00712152
$869$	−13.1387	−0.445699
$870$	−33.3167	−1.12954
$871$	−9.08986	−0.307998
$872$	−6.28629	−0.212881
$873$	13.3714	0.452554
$874$	3.23085	0.109285
$875$	−3.57147	−0.120738
$876$	0.599875	0.0202679
$877$	−38.9562	−1.31546	−0.657729	−	0.753255i	$-0.728483\pi$
$877$	−38.9562	−1.31546	−0.657729	+	0.753255i	$0.728483\pi$
$878$	−22.5136	−0.759797
$879$	4.16030	0.140323
$880$	−26.8894	−0.906442
$881$	−47.3697	−1.59593	−0.797963	−	0.602707i	$-0.794089\pi$
$881$	−47.3697	−1.59593	−0.797963	+	0.602707i	$0.794089\pi$
$882$	−11.7910	−0.397023
$883$	30.7696	1.03548	0.517739	−	0.855539i	$-0.326774\pi$
$883$	30.7696	1.03548	0.517739	+	0.855539i	$0.326774\pi$
$884$	2.06606	0.0694890
$885$	12.0119	0.403776
$886$	28.6439	0.962312
$887$	−10.3300	−0.346846	−0.173423	−	0.984847i	$-0.555483\pi$
$887$	−10.3300	−0.346846	−0.173423	+	0.984847i	$0.555483\pi$
$888$	13.6143	0.456866
$889$	−4.86795	−0.163266
$890$	40.3402	1.35221
$891$	13.0098	0.435844
$892$	−0.0348932	−0.00116831
$893$	1.97681	0.0661514
$894$	36.8043	1.23092
$895$	−38.8291	−1.29791
$896$	3.13555	0.104751
$897$	18.1049	0.604505
$898$	20.3884	0.680369
$899$	65.5025	2.18463
$900$	−0.0954112	−0.00318037
$901$	−35.9844	−1.19881
$902$	37.8359	1.25980
$903$	0.392791	0.0130713
$904$	−36.7532	−1.22239
$905$	−16.8258	−0.559309
$906$	39.5415	1.31368
$907$	−48.0569	−1.59570	−0.797851	−	0.602854i	$-0.794030\pi$
$907$	−48.0569	−1.59570	−0.797851	+	0.602854i	$0.794030\pi$
$908$	−1.90581	−0.0632464
$909$	18.1610	0.602363
$910$	3.30413	0.109531
$911$	32.9823	1.09275	0.546376	−	0.837540i	$-0.316007\pi$
$911$	32.9823	1.09275	0.546376	+	0.837540i	$0.316007\pi$
$912$	−3.37809	−0.111860
$913$	25.0535	0.829148
$914$	20.8061	0.688204
$915$	−0.643376	−0.0212694
$916$	1.84955	0.0611107
$917$	0.551696	0.0182186
$918$	−43.4392	−1.43371
$919$	−22.9775	−0.757959	−0.378979	−	0.925405i	$-0.623725\pi$
$919$	−22.9775	−0.757959	−0.378979	+	0.925405i	$0.623725\pi$
$920$	−20.6936	−0.682247
$921$	−0.289333	−0.00953384
$922$	27.4649	0.904508
$923$	−26.9617	−0.887455
$924$	0.130655	0.00429825
$925$	2.88709	0.0949269
$926$	37.9524	1.24719
$927$	16.2291	0.533035
$928$	−4.75870	−0.156212
$929$	33.4643	1.09793	0.548964	−	0.835846i	$-0.315022\pi$
$929$	33.4643	1.09793	0.548964	+	0.835846i	$0.315022\pi$
$930$	−27.6531	−0.906781
$931$	4.62351	0.151529
$932$	0.981633	0.0321545
$933$	22.9258	0.750556
$934$	25.0091	0.818323
$935$	39.5675	1.29400
$936$	13.9301	0.455321
$937$	−12.3697	−0.404101	−0.202050	−	0.979375i	$-0.564760\pi$
$937$	−12.3697	−0.404101	−0.202050	+	0.979375i	$0.564760\pi$
$938$	0.966627	0.0315615
$939$	−15.4458	−0.504056
$940$	−0.572852	−0.0186844
$941$	−14.7118	−0.479590	−0.239795	−	0.970824i	$-0.577080\pi$
$941$	−14.7118	−0.479590	−0.239795	+	0.970824i	$0.577080\pi$
$942$	−9.16444	−0.298594
$943$	27.7348	0.903171
$944$	−16.8059	−0.546985
$945$	3.45580	0.112417
$946$	4.70110	0.152846
$947$	49.1993	1.59876	0.799381	−	0.600824i	$-0.205161\pi$
$947$	49.1993	1.59876	0.799381	+	0.600824i	$0.205161\pi$
$948$	0.478321	0.0155351
$949$	−18.5730	−0.602905
$950$	−0.752088	−0.0244010
$951$	−6.67001	−0.216290
$952$	−4.85606	−0.157386
$953$	2.42440	0.0785342	0.0392671	−	0.999229i	$-0.487498\pi$
$953$	2.42440	0.0785342	0.0392671	+	0.999229i	$0.487498\pi$
$954$	−10.9771	−0.355397
$955$	−6.78984	−0.219714
$956$	0.917264	0.0296665
$957$	40.7899	1.31855
$958$	−2.24029	−0.0723805
$959$	−1.77101	−0.0571890
$960$	22.6667	0.731564
$961$	23.3676	0.753792
$962$	−19.0711	−0.614877
$963$	−14.5465	−0.468755
$964$	2.21915	0.0714739
$965$	−45.6098	−1.46823
$966$	−1.92530	−0.0619454
$967$	37.1967	1.19617	0.598083	−	0.801434i	$-0.295929\pi$
$967$	37.1967	1.19617	0.598083	+	0.801434i	$0.295929\pi$
$968$	2.75690	0.0886100
$969$	4.97083	0.159686
$970$	−30.5439	−0.980707
$971$	42.2843	1.35697	0.678484	−	0.734615i	$-0.262637\pi$
$971$	42.2843	1.35697	0.678484	+	0.734615i	$0.262637\pi$
$972$	1.12600	0.0361164
$973$	−2.57307	−0.0824889
$974$	−24.3930	−0.781601
$975$	−4.21451	−0.134973
$976$	0.900151	0.0288131
$977$	−47.6762	−1.52530	−0.762648	−	0.646813i	$-0.776101\pi$
$977$	−47.6762	−1.52530	−0.762648	+	0.646813i	$0.776101\pi$
$978$	−32.6008	−1.04246
$979$	−49.3889	−1.57848
$980$	−1.33983	−0.0427993
$981$	2.68777	0.0858139
$982$	−55.7989	−1.78062
$983$	26.4277	0.842912	0.421456	−	0.906849i	$-0.361519\pi$
$983$	26.4277	0.842912	0.421456	+	0.906849i	$0.361519\pi$
$984$	−30.4448	−0.970543
$985$	−25.0451	−0.798004
$986$	−68.5913	−2.18439
$987$	−1.17800	−0.0374961
$988$	−0.247136	−0.00786245
$989$	3.44604	0.109578
$990$	12.0701	0.383614
$991$	−20.0705	−0.637561	−0.318780	−	0.947829i	$-0.603273\pi$
$991$	−20.0705	−0.637561	−0.318780	+	0.947829i	$0.603273\pi$
$992$	−3.94976	−0.125405
$993$	−9.92211	−0.314869
$994$	2.86714	0.0909402
$995$	−37.3100	−1.18281
$996$	−0.912086	−0.0289005
$997$	−42.8122	−1.35587	−0.677937	−	0.735120i	$-0.737126\pi$
$997$	−42.8122	−1.35587	−0.677937	+	0.735120i	$0.737126\pi$
$998$	3.69358	0.116918
$999$	−19.9465	−0.631079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	409.2.a.b.1.5	✓	20
3.2	odd	2		3681.2.a.i.1.16		20
4.3	odd	2		6544.2.a.i.1.14		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
409.2.a.b.1.5	✓	20	1.1	even	1	trivial
3681.2.a.i.1.16		20	3.2	odd	2
6544.2.a.i.1.14		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)

\(f(q)\)	\(=\)	\(q-1.38030 q^{2} -1.32806 q^{3} -0.0947758 q^{4} -2.04589 q^{5} +1.83312 q^{6} -0.300238 q^{7} +2.89142 q^{8} -1.23626 q^{9} +2.82394 q^{10} -3.45738 q^{11} +0.125868 q^{12} -3.89705 q^{13} +0.414418 q^{14} +2.71707 q^{15} -3.80147 q^{16} +5.59382 q^{17} +1.70640 q^{18} -0.669118 q^{19} +0.193901 q^{20} +0.398734 q^{21} +4.77222 q^{22} +3.49818 q^{23} -3.83997 q^{24} -0.814317 q^{25} +5.37910 q^{26} +5.62600 q^{27} +0.0284553 q^{28} +8.88357 q^{29} -3.75037 q^{30} +7.37344 q^{31} -0.535674 q^{32} +4.59161 q^{33} -7.72114 q^{34} +0.614254 q^{35} +0.117167 q^{36} -3.54541 q^{37} +0.923582 q^{38} +5.17552 q^{39} -5.91553 q^{40} +7.92837 q^{41} -0.550371 q^{42} +0.985097 q^{43} +0.327676 q^{44} +2.52925 q^{45} -4.82853 q^{46} -2.95435 q^{47} +5.04858 q^{48} -6.90986 q^{49} +1.12400 q^{50} -7.42893 q^{51} +0.369346 q^{52} -6.43289 q^{53} -7.76556 q^{54} +7.07343 q^{55} -0.868112 q^{56} +0.888629 q^{57} -12.2620 q^{58} +4.42090 q^{59} -0.257513 q^{60} -0.236790 q^{61} -10.1775 q^{62} +0.371170 q^{63} +8.34232 q^{64} +7.97296 q^{65} -6.33779 q^{66} +2.33250 q^{67} -0.530159 q^{68} -4.64579 q^{69} -0.847854 q^{70} +6.91848 q^{71} -3.57453 q^{72} +4.76591 q^{73} +4.89373 q^{74} +1.08146 q^{75} +0.0634161 q^{76} +1.03804 q^{77} -7.14377 q^{78} +3.80018 q^{79} +7.77740 q^{80} -3.76290 q^{81} -10.9435 q^{82} -7.24637 q^{83} -0.0377903 q^{84} -11.4444 q^{85} -1.35973 q^{86} -11.7979 q^{87} -9.99672 q^{88} +14.2851 q^{89} -3.49112 q^{90} +1.17004 q^{91} -0.331542 q^{92} -9.79237 q^{93} +4.07789 q^{94} +1.36894 q^{95} +0.711407 q^{96} -10.8161 q^{97} +9.53767 q^{98} +4.27421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) \) 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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