LMFDB - Embedded newform 4489.2.a.h.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4489,2,Mod(1,4489)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4489, 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("4489.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4489 = 67^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4489.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.8448454674$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 67)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.68251$ of defining polynomial
Character	$\chi$	$=$	4489.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.478891 q^{2} +1.83083 q^{3} -1.77066 q^{4} +0.830830 q^{5} -0.876769 q^{6} +1.08816 q^{7} +1.80574 q^{8} +0.351939 q^{9} -0.397877 q^{10} -2.87677 q^{11} -3.24178 q^{12} +4.39011 q^{13} -0.521109 q^{14} +1.52111 q^{15} +2.67657 q^{16} -3.28463 q^{17} -0.168540 q^{18} +3.55131 q^{19} -1.47112 q^{20} +1.99223 q^{21} +1.37766 q^{22} +6.41437 q^{23} +3.30600 q^{24} -4.30972 q^{25} -2.10238 q^{26} -4.84815 q^{27} -1.92676 q^{28} -1.97270 q^{29} -0.728446 q^{30} -2.68807 q^{31} -4.89326 q^{32} -5.26687 q^{33} +1.57298 q^{34} +0.904073 q^{35} -0.623165 q^{36} +8.07686 q^{37} -1.70069 q^{38} +8.03754 q^{39} +1.50026 q^{40} +2.71113 q^{41} -0.954061 q^{42} +7.74408 q^{43} +5.09379 q^{44} +0.292401 q^{45} -3.07179 q^{46} +10.9626 q^{47} +4.90035 q^{48} -5.81592 q^{49} +2.06389 q^{50} -6.01360 q^{51} -7.77340 q^{52} -7.49900 q^{53} +2.32174 q^{54} -2.39011 q^{55} +1.96492 q^{56} +6.50184 q^{57} +0.944707 q^{58} +10.6543 q^{59} -2.69337 q^{60} +9.25103 q^{61} +1.28729 q^{62} +0.382964 q^{63} -3.00980 q^{64} +3.64743 q^{65} +2.52226 q^{66} +5.81597 q^{68} +11.7436 q^{69} -0.432953 q^{70} -1.73418 q^{71} +0.635509 q^{72} -1.46398 q^{73} -3.86794 q^{74} -7.89037 q^{75} -6.28817 q^{76} -3.13037 q^{77} -3.84911 q^{78} -15.4857 q^{79} +2.22378 q^{80} -9.93195 q^{81} -1.29833 q^{82} -8.08733 q^{83} -3.52757 q^{84} -2.72897 q^{85} -3.70857 q^{86} -3.61167 q^{87} -5.19469 q^{88} +15.4408 q^{89} -0.140028 q^{90} +4.77712 q^{91} -11.3577 q^{92} -4.92139 q^{93} -5.24990 q^{94} +2.95053 q^{95} -8.95874 q^{96} -1.03304 q^{97} +2.78519 q^{98} -1.01245 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 5 q^{6} + 2 q^{7} + 9 q^{8} - 3 q^{9} + 7 q^{10} - 5 q^{11} + 12 q^{12} - 2 q^{13} - 3 q^{14} + 8 q^{15} + 6 q^{16} - 16 q^{17} + 21 q^{18} + 10 q^{19} + 8 q^{20}+ \cdots + 36 q^{99}+O(q^{100}) $ 5 * q - 2 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 5 * q^6 + 2 * q^7 + 9 * q^8 - 3 * q^9 + 7 * q^10 - 5 * q^11 + 12 * q^12 - 2 * q^13 - 3 * q^14 + 8 * q^15 + 6 * q^16 - 16 * q^17 + 21 * q^18 + 10 * q^19 + 8 * q^20 - 5 * q^21 + 24 * q^22 + 2 * q^23 + 16 * q^24 - 16 * q^25 - 19 * q^26 + q^27 - 16 * q^28 - 12 * q^29 + 10 * q^30 - 16 * q^31 + 6 * q^32 + 7 * q^33 + 2 * q^34 - 7 * q^35 + 13 * q^36 + q^37 + 18 * q^38 - 6 * q^39 + 7 * q^40 + 20 * q^41 - 9 * q^42 + q^43 + 18 * q^44 + 16 * q^45 + 19 * q^46 + 11 * q^47 + 18 * q^48 - 21 * q^49 + 13 * q^50 - 15 * q^51 - 28 * q^52 + 17 * q^53 + 15 * q^54 + 12 * q^55 - 3 * q^56 + 8 * q^57 + 18 * q^58 + 14 * q^59 + 13 * q^60 - 19 * q^61 + 24 * q^62 - 10 * q^63 - 21 * q^64 - 4 * q^65 + 39 * q^66 - 15 * q^68 + 6 * q^69 - 6 * q^70 - 19 * q^71 + 21 * q^72 - 4 * q^73 - 18 * q^74 - 15 * q^75 - 14 * q^76 - 13 * q^77 - 46 * q^78 + 4 * q^79 + 12 * q^80 + q^81 - 19 * q^82 - 36 * q^83 - 15 * q^84 + q^85 - 18 * q^86 + 8 * q^87 + 24 * q^88 + 42 * q^89 + 9 * q^90 + 8 * q^91 - 38 * q^92 + 7 * q^93 - 2 * q^95 + 7 * q^96 - 27 * q^97 + 15 * q^98 + 36 * 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 $

$2$	−0.478891	−0.338627	−0.169314	−	0.985562i	$-0.554155\pi$
$2$	−0.478891	−0.338627	−0.169314	+	0.985562i	$0.554155\pi$
$3$	1.83083	1.05703	0.528515	−	0.848924i	$-0.322749\pi$
$3$	1.83083	1.05703	0.528515	+	0.848924i	$0.322749\pi$
$4$	−1.77066	−0.885331
$5$	0.830830	0.371558	0.185779	−	0.982592i	$-0.440519\pi$
$5$	0.830830	0.371558	0.185779	+	0.982592i	$0.440519\pi$
$6$	−0.876769	−0.357939
$7$	1.08816	0.411284	0.205642	−	0.978627i	$-0.434072\pi$
$7$	1.08816	0.411284	0.205642	+	0.978627i	$0.434072\pi$
$8$	1.80574	0.638425
$9$	0.351939	0.117313
$10$	−0.397877	−0.125820
$11$	−2.87677	−0.867378	−0.433689	−	0.901063i	$-0.642788\pi$
$11$	−2.87677	−0.867378	−0.433689	+	0.901063i	$0.642788\pi$
$12$	−3.24178	−0.935822
$13$	4.39011	1.21760	0.608798	−	0.793325i	$-0.291652\pi$
$13$	4.39011	1.21760	0.608798	+	0.793325i	$0.291652\pi$
$14$	−0.521109	−0.139272
$15$	1.52111	0.392749
$16$	2.67657	0.669143
$17$	−3.28463	−0.796640	−0.398320	−	0.917247i	$-0.630407\pi$
$17$	−3.28463	−0.796640	−0.398320	+	0.917247i	$0.630407\pi$
$18$	−0.168540	−0.0397253
$19$	3.55131	0.814726	0.407363	−	0.913266i	$-0.366448\pi$
$19$	3.55131	0.814726	0.407363	+	0.913266i	$0.366448\pi$
$20$	−1.47112	−0.328952
$21$	1.99223	0.434740
$22$	1.37766	0.293718
$23$	6.41437	1.33749	0.668745	−	0.743492i	$-0.266832\pi$
$23$	6.41437	1.33749	0.668745	+	0.743492i	$0.266832\pi$
$24$	3.30600	0.674834
$25$	−4.30972	−0.861944
$26$	−2.10238	−0.412311
$27$	−4.84815	−0.933027
$28$	−1.92676	−0.364123
$29$	−1.97270	−0.366320	−0.183160	−	0.983083i	$-0.558633\pi$
$29$	−1.97270	−0.366320	−0.183160	+	0.983083i	$0.558633\pi$
$30$	−0.728446	−0.132995
$31$	−2.68807	−0.482791	−0.241396	−	0.970427i	$-0.577605\pi$
$31$	−2.68807	−0.482791	−0.241396	+	0.970427i	$0.577605\pi$
$32$	−4.89326	−0.865015
$33$	−5.26687	−0.916845
$34$	1.57298	0.269764
$35$	0.904073	0.152816
$36$	−0.623165	−0.103861
$37$	8.07686	1.32783	0.663914	−	0.747809i	$-0.268894\pi$
$37$	8.07686	1.32783	0.663914	+	0.747809i	$0.268894\pi$
$38$	−1.70069	−0.275889
$39$	8.03754	1.28704
$40$	1.50026	0.237212
$41$	2.71113	0.423407	0.211703	−	0.977334i	$-0.432099\pi$
$41$	2.71113	0.423407	0.211703	+	0.977334i	$0.432099\pi$
$42$	−0.954061	−0.147215
$43$	7.74408	1.18096	0.590480	−	0.807052i	$-0.298938\pi$
$43$	7.74408	1.18096	0.590480	+	0.807052i	$0.298938\pi$
$44$	5.09379	0.767917
$45$	0.292401	0.0435886
$46$	−3.07179	−0.452910
$47$	10.9626	1.59906	0.799530	−	0.600626i	$-0.205082\pi$
$47$	10.9626	1.59906	0.799530	+	0.600626i	$0.205082\pi$
$48$	4.90035	0.707305
$49$	−5.81592	−0.830845
$50$	2.06389	0.291878
$51$	−6.01360	−0.842072
$52$	−7.77340	−1.07798
$53$	−7.49900	−1.03007	−0.515034	−	0.857170i	$-0.672221\pi$
$53$	−7.49900	−1.03007	−0.515034	+	0.857170i	$0.672221\pi$
$54$	2.32174	0.315948
$55$	−2.39011	−0.322282
$56$	1.96492	0.262574
$57$	6.50184	0.861190
$58$	0.944707	0.124046
$59$	10.6543	1.38707	0.693533	−	0.720424i	$-0.256053\pi$
$59$	10.6543	1.38707	0.693533	+	0.720424i	$0.256053\pi$
$60$	−2.69337	−0.347713
$61$	9.25103	1.18447	0.592237	−	0.805764i	$-0.298245\pi$
$61$	9.25103	1.18447	0.592237	+	0.805764i	$0.298245\pi$
$62$	1.28729	0.163486
$63$	0.382964	0.0482489
$64$	−3.00980	−0.376226
$65$	3.64743	0.452408
$66$	2.52226	0.310469
$67$	0	0
$67$	0	0
$68$	5.81597	0.705290
$69$	11.7436	1.41377
$70$	−0.432953	−0.0517477
$71$	−1.73418	−0.205810	−0.102905	−	0.994691i	$-0.532814\pi$
$71$	−1.73418	−0.205810	−0.102905	+	0.994691i	$0.532814\pi$
$72$	0.635509	0.0748954
$73$	−1.46398	−0.171346	−0.0856728	−	0.996323i	$-0.527304\pi$
$73$	−1.46398	−0.171346	−0.0856728	+	0.996323i	$0.527304\pi$
$74$	−3.86794	−0.449639
$75$	−7.89037	−0.911101
$76$	−6.28817	−0.721303
$77$	−3.13037	−0.356739
$78$	−3.84911	−0.435826
$79$	−15.4857	−1.74227	−0.871137	−	0.491040i	$-0.836617\pi$
$79$	−15.4857	−1.74227	−0.871137	+	0.491040i	$0.836617\pi$
$80$	2.22378	0.248626
$81$	−9.93195	−1.10355
$82$	−1.29833	−0.143377
$83$	−8.08733	−0.887700	−0.443850	−	0.896101i	$-0.646388\pi$
$83$	−8.08733	−0.887700	−0.443850	+	0.896101i	$0.646388\pi$
$84$	−3.52757	−0.384889
$85$	−2.72897	−0.295998
$86$	−3.70857	−0.399906
$87$	−3.61167	−0.387212
$88$	−5.19469	−0.553756
$89$	15.4408	1.63672	0.818361	−	0.574704i	$-0.194883\pi$
$89$	15.4408	1.63672	0.818361	+	0.574704i	$0.194883\pi$
$90$	−0.140028	−0.0147603
$91$	4.77712	0.500778
$92$	−11.3577	−1.18412
$93$	−4.92139	−0.510325
$94$	−5.24990	−0.541486
$95$	2.95053	0.302718
$96$	−8.95874	−0.914347
$97$	−1.03304	−0.104890	−0.0524448	−	0.998624i	$-0.516701\pi$
$97$	−1.03304	−0.104890	−0.0524448	+	0.998624i	$0.516701\pi$
$98$	2.78519	0.281347
$99$	−1.01245	−0.101755
$100$	7.63106	0.763106
$101$	15.4230	1.53464	0.767321	−	0.641263i	$-0.221589\pi$
$101$	15.4230	1.53464	0.767321	+	0.641263i	$0.221589\pi$
$102$	2.87986	0.285149
$103$	−4.04222	−0.398291	−0.199146	−	0.979970i	$-0.563817\pi$
$103$	−4.04222	−0.398291	−0.199146	+	0.979970i	$0.563817\pi$
$104$	7.92738	0.777344
$105$	1.65520	0.161531
$106$	3.59121	0.348809
$107$	10.2876	0.994541	0.497271	−	0.867595i	$-0.334336\pi$
$107$	10.2876	0.994541	0.497271	+	0.867595i	$0.334336\pi$
$108$	8.58444	0.826038
$109$	7.91008	0.757648	0.378824	−	0.925469i	$-0.376329\pi$
$109$	7.91008	0.757648	0.378824	+	0.925469i	$0.376329\pi$
$110$	1.14460	0.109133
$111$	14.7874	1.40355
$112$	2.91253	0.275208
$113$	13.1872	1.24055	0.620273	−	0.784386i	$-0.287022\pi$
$113$	13.1872	1.24055	0.620273	+	0.784386i	$0.287022\pi$
$114$	−3.11368	−0.291623
$115$	5.32925	0.496955
$116$	3.49298	0.324315
$117$	1.54505	0.142840
$118$	−5.10224	−0.469699
$119$	−3.57419	−0.327645
$120$	2.74672	0.250740
$121$	−2.72420	−0.247655
$122$	−4.43024	−0.401095
$123$	4.96361	0.447554
$124$	4.75966	0.427430
$125$	−7.73480	−0.691821
$126$	−0.183398	−0.0163384
$127$	1.12811	0.100103	0.0500516	−	0.998747i	$-0.484061\pi$
$127$	1.12811	0.100103	0.0500516	+	0.998747i	$0.484061\pi$
$128$	11.2279	0.992415
$129$	14.1781	1.24831
$130$	−1.74672	−0.153198
$131$	2.13769	0.186771	0.0933856	−	0.995630i	$-0.470231\pi$
$131$	2.13769	0.186771	0.0933856	+	0.995630i	$0.470231\pi$
$132$	9.32586	0.811712
$133$	3.86438	0.335084
$134$	0	0
$135$	−4.02799	−0.346674
$136$	−5.93118	−0.508595
$137$	10.6793	0.912393	0.456197	−	0.889879i	$-0.349211\pi$
$137$	10.6793	0.912393	0.456197	+	0.889879i	$0.349211\pi$
$138$	−5.62392	−0.478740
$139$	−0.839840	−0.0712343	−0.0356171	−	0.999366i	$-0.511340\pi$
$139$	−0.839840	−0.0712343	−0.0356171	+	0.999366i	$0.511340\pi$
$140$	−1.60081	−0.135293
$141$	20.0707	1.69026
$142$	0.830486	0.0696928
$143$	−12.6293	−1.05612
$144$	0.941989	0.0784991
$145$	−1.63898	−0.136109
$146$	0.701087	0.0580223
$147$	−10.6480	−0.878229
$148$	−14.3014	−1.17557
$149$	12.3234	1.00957	0.504785	−	0.863245i	$-0.331572\pi$
$149$	12.3234	1.00957	0.504785	+	0.863245i	$0.331572\pi$
$150$	3.77863	0.308524
$151$	−11.0223	−0.896984	−0.448492	−	0.893787i	$-0.648039\pi$
$151$	−11.0223	−0.896984	−0.448492	+	0.893787i	$0.648039\pi$
$152$	6.41274	0.520142
$153$	−1.15599	−0.0934561
$154$	1.49911	0.120802
$155$	−2.23333	−0.179385
$156$	−14.2318	−1.13945
$157$	8.31265	0.663422	0.331711	−	0.943381i	$-0.392374\pi$
$157$	8.31265	0.663422	0.331711	+	0.943381i	$0.392374\pi$
$158$	7.41596	0.589982
$159$	−13.7294	−1.08881
$160$	−4.06547	−0.321404
$161$	6.97984	0.550088
$162$	4.75633	0.373692
$163$	6.95133	0.544470	0.272235	−	0.962231i	$-0.412237\pi$
$163$	6.95133	0.544470	0.272235	+	0.962231i	$0.412237\pi$
$164$	−4.80049	−0.374855
$165$	−4.37588	−0.340662
$166$	3.87295	0.300600
$167$	−18.9954	−1.46991	−0.734956	−	0.678115i	$-0.762797\pi$
$167$	−18.9954	−1.46991	−0.734956	+	0.678115i	$0.762797\pi$
$168$	3.59744	0.277549
$169$	6.27303	0.482541
$170$	1.30688	0.100233
$171$	1.24984	0.0955779
$172$	−13.7122	−1.04554
$173$	9.82344	0.746862	0.373431	−	0.927658i	$-0.378181\pi$
$173$	9.82344	0.746862	0.373431	+	0.927658i	$0.378181\pi$
$174$	1.72960	0.131121
$175$	−4.68965	−0.354504
$176$	−7.69988	−0.580401
$177$	19.5061	1.46617
$178$	−7.39447	−0.554239
$179$	21.7154	1.62309	0.811544	−	0.584291i	$-0.198627\pi$
$179$	21.7154	1.62309	0.811544	+	0.584291i	$0.198627\pi$
$180$	−0.517744	−0.0385904
$181$	−25.7742	−1.91578	−0.957889	−	0.287138i	$-0.907296\pi$
$181$	−25.7742	−1.91578	−0.957889	+	0.287138i	$0.907296\pi$
$182$	−2.28772	−0.169577
$183$	16.9371	1.25202
$184$	11.5827	0.853886
$185$	6.71050	0.493365
$186$	2.35681	0.172810
$187$	9.44912	0.690988
$188$	−19.4111	−1.41570
$189$	−5.27554	−0.383739
$190$	−1.41299	−0.102509
$191$	−16.0976	−1.16478	−0.582392	−	0.812908i	$-0.697883\pi$
$191$	−16.0976	−1.16478	−0.582392	+	0.812908i	$0.697883\pi$
$192$	−5.51044	−0.397682
$193$	24.9141	1.79335	0.896677	−	0.442686i	$-0.145974\pi$
$193$	24.9141	1.79335	0.896677	+	0.442686i	$0.145974\pi$
$194$	0.494715	0.0355185
$195$	6.67783	0.478209
$196$	10.2980	0.735573
$197$	12.0793	0.860612	0.430306	−	0.902683i	$-0.358406\pi$
$197$	12.0793	0.860612	0.430306	+	0.902683i	$0.358406\pi$
$198$	0.484852	0.0344569
$199$	−2.35813	−0.167163	−0.0835816	−	0.996501i	$-0.526636\pi$
$199$	−2.35813	−0.167163	−0.0835816	+	0.996501i	$0.526636\pi$
$200$	−7.78223	−0.550287
$201$	0	0
$202$	−7.38593	−0.519672
$203$	−2.14660	−0.150662
$204$	10.6481	0.745513
$205$	2.25248	0.157320
$206$	1.93578	0.134872
$207$	2.25747	0.156905
$208$	11.7504	0.814746
$209$	−10.2163	−0.706676
$210$	−0.792663	−0.0546989
$211$	−0.703159	−0.0484075	−0.0242037	−	0.999707i	$-0.507705\pi$
$211$	−0.703159	−0.0484075	−0.0242037	+	0.999707i	$0.507705\pi$
$212$	13.2782	0.911951
$213$	−3.17500	−0.217547
$214$	−4.92665	−0.336779
$215$	6.43401	0.438796
$216$	−8.75449	−0.595668
$217$	−2.92504	−0.198564
$218$	−3.78807	−0.256560
$219$	−2.68030	−0.181118
$220$	4.23207	0.285326
$221$	−14.4199	−0.969986
$222$	−7.08154	−0.475282
$223$	7.64266	0.511791	0.255895	−	0.966704i	$-0.417630\pi$
$223$	7.64266	0.511791	0.255895	+	0.966704i	$0.417630\pi$
$224$	−5.32463	−0.355767
$225$	−1.51676	−0.101117
$226$	−6.31523	−0.420083
$227$	10.9267	0.725229	0.362615	−	0.931939i	$-0.381884\pi$
$227$	10.9267	0.725229	0.362615	+	0.931939i	$0.381884\pi$
$228$	−11.5126	−0.762439
$229$	26.5908	1.75717	0.878585	−	0.477586i	$-0.158488\pi$
$229$	26.5908	1.75717	0.878585	+	0.477586i	$0.158488\pi$
$230$	−2.55213	−0.168283
$231$	−5.73118	−0.377084
$232$	−3.56217	−0.233868
$233$	2.47015	0.161825	0.0809123	−	0.996721i	$-0.474217\pi$
$233$	2.47015	0.161825	0.0809123	+	0.996721i	$0.474217\pi$
$234$	−0.739910	−0.0483694
$235$	9.10806	0.594144
$236$	−18.8651	−1.22801
$237$	−28.3516	−1.84164
$238$	1.71165	0.110950
$239$	15.7639	1.01968	0.509839	−	0.860270i	$-0.329705\pi$
$239$	15.7639	1.01968	0.509839	+	0.860270i	$0.329705\pi$
$240$	4.07136	0.262805
$241$	−18.5929	−1.19768	−0.598838	−	0.800870i	$-0.704371\pi$
$241$	−18.5929	−1.19768	−0.598838	+	0.800870i	$0.704371\pi$
$242$	1.30460	0.0838626
$243$	−3.63927	−0.233459
$244$	−16.3805	−1.04865
$245$	−4.83204	−0.308708
$246$	−2.37703	−0.151554
$247$	15.5906	0.992008
$248$	−4.85394	−0.308226
$249$	−14.8065	−0.938326
$250$	3.70413	0.234270
$251$	13.6325	0.860477	0.430238	−	0.902715i	$-0.358429\pi$
$251$	13.6325	0.860477	0.430238	+	0.902715i	$0.358429\pi$
$252$	−0.678100	−0.0427163
$253$	−18.4527	−1.16011
$254$	−0.540240	−0.0338977
$255$	−4.99628	−0.312879
$256$	0.642664	0.0401665
$257$	−4.85968	−0.303138	−0.151569	−	0.988447i	$-0.548433\pi$
$257$	−4.85968	−0.303138	−0.151569	+	0.988447i	$0.548433\pi$
$258$	−6.78977	−0.422712
$259$	8.78888	0.546114
$260$	−6.45837	−0.400531
$261$	−0.694268	−0.0429741
$262$	−1.02372	−0.0632458
$263$	12.4893	0.770125	0.385063	−	0.922890i	$-0.374180\pi$
$263$	12.4893	0.770125	0.385063	+	0.922890i	$0.374180\pi$
$264$	−9.51060	−0.585337
$265$	−6.23040	−0.382730
$266$	−1.85062	−0.113469
$267$	28.2695	1.73007
$268$	0	0
$269$	−10.3046	−0.628282	−0.314141	−	0.949376i	$-0.601716\pi$
$269$	−10.3046	−0.628282	−0.314141	+	0.949376i	$0.601716\pi$
$270$	1.92897	0.117393
$271$	4.31277	0.261982	0.130991	−	0.991384i	$-0.458184\pi$
$271$	4.31277	0.261982	0.130991	+	0.991384i	$0.458184\pi$
$272$	−8.79155	−0.533066
$273$	8.74609	0.529338
$274$	−5.11422	−0.308961
$275$	12.3981	0.747632
$276$	−20.7940	−1.25165
$277$	−16.4124	−0.986128	−0.493064	−	0.869993i	$-0.664123\pi$
$277$	−16.4124	−0.986128	−0.493064	+	0.869993i	$0.664123\pi$
$278$	0.402192	0.0241219
$279$	−0.946034	−0.0566376
$280$	1.63252	0.0975616
$281$	−24.1040	−1.43793	−0.718963	−	0.695048i	$-0.755383\pi$
$281$	−24.1040	−1.43793	−0.718963	+	0.695048i	$0.755383\pi$
$282$	−9.61167	−0.572367
$283$	−26.2489	−1.56034	−0.780169	−	0.625569i	$-0.784867\pi$
$283$	−26.2489	−1.56034	−0.780169	+	0.625569i	$0.784867\pi$
$284$	3.07066	0.182210
$285$	5.40193	0.319983
$286$	6.04807	0.357630
$287$	2.95013	0.174141
$288$	−1.72213	−0.101477
$289$	−6.21121	−0.365365
$290$	0.784891	0.0460904
$291$	−1.89133	−0.110871
$292$	2.59221	0.151698
$293$	17.1283	1.00065	0.500323	−	0.865839i	$-0.333215\pi$
$293$	17.1283	1.00065	0.500323	+	0.865839i	$0.333215\pi$
$294$	5.09921	0.297392
$295$	8.85188	0.515376
$296$	14.5847	0.847718
$297$	13.9470	0.809287
$298$	−5.90156	−0.341868
$299$	28.1598	1.62852
$300$	13.9712	0.806627
$301$	8.42676	0.485711
$302$	5.27850	0.303743
$303$	28.2368	1.62216
$304$	9.50534	0.545169
$305$	7.68603	0.440101
$306$	0.553593	0.0316468
$307$	8.83455	0.504214	0.252107	−	0.967699i	$-0.418876\pi$
$307$	8.83455	0.504214	0.252107	+	0.967699i	$0.418876\pi$
$308$	5.54284	0.315832
$309$	−7.40061	−0.421006
$310$	1.06952	0.0607447
$311$	−21.3828	−1.21251	−0.606254	−	0.795271i	$-0.707328\pi$
$311$	−21.3828	−1.21251	−0.606254	+	0.795271i	$0.707328\pi$
$312$	14.5137	0.821676
$313$	−2.92283	−0.165208	−0.0826040	−	0.996582i	$-0.526324\pi$
$313$	−2.92283	−0.165208	−0.0826040	+	0.996582i	$0.526324\pi$
$314$	−3.98086	−0.224653
$315$	0.318178	0.0179273
$316$	27.4199	1.54249
$317$	−7.77890	−0.436906	−0.218453	−	0.975847i	$-0.570101\pi$
$317$	−7.77890	−0.436906	−0.218453	+	0.975847i	$0.570101\pi$
$318$	6.57489	0.368702
$319$	5.67499	0.317738
$320$	−2.50064	−0.139790
$321$	18.8349	1.05126
$322$	−3.34258	−0.186275
$323$	−11.6647	−0.649043
$324$	17.5861	0.977008
$325$	−18.9201	−1.04950
$326$	−3.32893	−0.184372
$327$	14.4820	0.800857
$328$	4.89558	0.270313
$329$	11.9290	0.657668
$330$	2.09557	0.115357
$331$	32.9661	1.81198	0.905992	−	0.423296i	$-0.139127\pi$
$331$	32.9661	1.81198	0.905992	+	0.423296i	$0.139127\pi$
$332$	14.3199	0.785909
$333$	2.84256	0.155771
$334$	9.09676	0.497752
$335$	0	0
$336$	5.33235	0.290903
$337$	27.5402	1.50021	0.750104	−	0.661320i	$-0.230003\pi$
$337$	27.5402	1.50021	0.750104	+	0.661320i	$0.230003\pi$
$338$	−3.00410	−0.163402
$339$	24.1435	1.31129
$340$	4.83208	0.262057
$341$	7.73295	0.418763
$342$	−0.598539	−0.0323653
$343$	−13.9457	−0.752998
$344$	13.9838	0.753955
$345$	9.75696	0.525297
$346$	−4.70436	−0.252908
$347$	−8.46173	−0.454250	−0.227125	−	0.973866i	$-0.572933\pi$
$347$	−8.46173	−0.454250	−0.227125	+	0.973866i	$0.572933\pi$
$348$	6.39505	0.342811
$349$	28.7613	1.53956	0.769778	−	0.638311i	$-0.220367\pi$
$349$	28.7613	1.53956	0.769778	+	0.638311i	$0.220367\pi$
$350$	2.24583	0.120045
$351$	−21.2839	−1.13605
$352$	14.0768	0.750295
$353$	22.1501	1.17893	0.589465	−	0.807794i	$-0.299339\pi$
$353$	22.1501	1.17893	0.589465	+	0.807794i	$0.299339\pi$
$354$	−9.34133	−0.496486
$355$	−1.44081	−0.0764704
$356$	−27.3405	−1.44904
$357$	−6.54373	−0.346331
$358$	−10.3993	−0.549622
$359$	8.40059	0.443366	0.221683	−	0.975119i	$-0.428845\pi$
$359$	8.40059	0.443366	0.221683	+	0.975119i	$0.428845\pi$
$360$	0.528000	0.0278280
$361$	−6.38820	−0.336221
$362$	12.3430	0.648735
$363$	−4.98755	−0.261778
$364$	−8.45867	−0.443355
$365$	−1.21632	−0.0636649
$366$	−8.11102	−0.423970
$367$	3.70888	0.193602	0.0968010	−	0.995304i	$-0.469139\pi$
$367$	3.70888	0.193602	0.0968010	+	0.995304i	$0.469139\pi$
$368$	17.1685	0.894972
$369$	0.954150	0.0496710
$370$	−3.21360	−0.167067
$371$	−8.16008	−0.423650
$372$	8.71413	0.451807
$373$	−11.2150	−0.580689	−0.290345	−	0.956922i	$-0.593770\pi$
$373$	−11.2150	−0.580689	−0.290345	+	0.956922i	$0.593770\pi$
$374$	−4.52510	−0.233987
$375$	−14.1611	−0.731276
$376$	19.7956	1.02088
$377$	−8.66035	−0.446030
$378$	2.52641	0.129945
$379$	−5.95507	−0.305891	−0.152946	−	0.988235i	$-0.548876\pi$
$379$	−5.95507	−0.305891	−0.152946	+	0.988235i	$0.548876\pi$
$380$	−5.22440	−0.268006
$381$	2.06537	0.105812
$382$	7.70902	0.394428
$383$	14.2175	0.726482	0.363241	−	0.931695i	$-0.381670\pi$
$383$	14.2175	0.726482	0.363241	+	0.931695i	$0.381670\pi$
$384$	20.5564	1.04901
$385$	−2.60081	−0.132549
$386$	−11.9311	−0.607279
$387$	2.72544	0.138542
$388$	1.82917	0.0928620
$389$	−14.1432	−0.717091	−0.358545	−	0.933512i	$-0.616727\pi$
$389$	−14.1432	−0.717091	−0.358545	+	0.933512i	$0.616727\pi$
$390$	−3.19795	−0.161935
$391$	−21.0688	−1.06550
$392$	−10.5020	−0.530432
$393$	3.91375	0.197423
$394$	−5.78466	−0.291427
$395$	−12.8660	−0.647357
$396$	1.79270	0.0900866
$397$	−0.316050	−0.0158621	−0.00793105	−	0.999969i	$-0.502525\pi$
$397$	−0.316050	−0.0158621	−0.00793105	+	0.999969i	$0.502525\pi$
$398$	1.12929	0.0566061
$399$	7.07502	0.354194
$400$	−11.5353	−0.576764
$401$	−30.4915	−1.52267	−0.761336	−	0.648358i	$-0.775456\pi$
$401$	−30.4915	−1.52267	−0.761336	+	0.648358i	$0.775456\pi$
$402$	0	0
$403$	−11.8009	−0.587845
$404$	−27.3089	−1.35867
$405$	−8.25177	−0.410034
$406$	1.02799	0.0510182
$407$	−23.2353	−1.15173
$408$	−10.8590	−0.537600
$409$	10.7150	0.529825	0.264912	−	0.964272i	$-0.414657\pi$
$409$	10.7150	0.529825	0.264912	+	0.964272i	$0.414657\pi$
$410$	−1.07870	−0.0532730
$411$	19.5520	0.964427
$412$	7.15740	0.352620
$413$	11.5935	0.570479
$414$	−1.08108	−0.0531322
$415$	−6.71920	−0.329832
$416$	−21.4819	−1.05324
$417$	−1.53760	−0.0752968
$418$	4.89250	0.239300
$419$	−4.40477	−0.215187	−0.107594	−	0.994195i	$-0.534315\pi$
$419$	−4.40477	−0.215187	−0.107594	+	0.994195i	$0.534315\pi$
$420$	−2.93081	−0.143009
$421$	−2.66691	−0.129977	−0.0649886	−	0.997886i	$-0.520701\pi$
$421$	−2.66691	−0.129977	−0.0649886	+	0.997886i	$0.520701\pi$
$422$	0.336737	0.0163921
$423$	3.85816	0.187590
$424$	−13.5412	−0.657621
$425$	14.1558	0.686659
$426$	1.52048	0.0736674
$427$	10.0666	0.487155
$428$	−18.2159	−0.880499
$429$	−23.1221	−1.11635
$430$	−3.08119	−0.148588
$431$	−10.2150	−0.492037	−0.246019	−	0.969265i	$-0.579122\pi$
$431$	−10.2150	−0.492037	−0.246019	+	0.969265i	$0.579122\pi$
$432$	−12.9764	−0.624329
$433$	0.208104	0.0100009	0.00500043	−	0.999987i	$-0.498408\pi$
$433$	0.208104	0.0100009	0.00500043	+	0.999987i	$0.498408\pi$
$434$	1.40077	0.0672393
$435$	−3.00069	−0.143872
$436$	−14.0061	−0.670770
$437$	22.7794	1.08969
$438$	1.28357	0.0613314
$439$	−4.44240	−0.212024	−0.106012	−	0.994365i	$-0.533808\pi$
$439$	−4.44240	−0.212024	−0.106012	+	0.994365i	$0.533808\pi$
$440$	−4.31591	−0.205753
$441$	−2.04685	−0.0974688
$442$	6.90555	0.328464
$443$	−28.0190	−1.33122	−0.665611	−	0.746299i	$-0.731829\pi$
$443$	−28.0190	−1.33122	−0.665611	+	0.746299i	$0.731829\pi$
$444$	−26.1834	−1.24261
$445$	12.8287	0.608138
$446$	−3.66001	−0.173306
$447$	22.5620	1.06715
$448$	−3.27514	−0.154736
$449$	0.295186	0.0139307	0.00696534	−	0.999976i	$-0.497783\pi$
$449$	0.295186	0.0139307	0.00696534	+	0.999976i	$0.497783\pi$
$450$	0.726362	0.0342410
$451$	−7.79928	−0.367254
$452$	−23.3500	−1.09829
$453$	−20.1800	−0.948139
$454$	−5.23269	−0.245583
$455$	3.96897	0.186068
$456$	11.7406	0.549805
$457$	17.7444	0.830047	0.415024	−	0.909811i	$-0.363773\pi$
$457$	17.7444	0.830047	0.415024	+	0.909811i	$0.363773\pi$
$458$	−12.7341	−0.595026
$459$	15.9244	0.743286
$460$	−9.43631	−0.439970
$461$	−22.4541	−1.04579	−0.522896	−	0.852396i	$-0.675149\pi$
$461$	−22.4541	−1.04579	−0.522896	+	0.852396i	$0.675149\pi$
$462$	2.74461	0.127691
$463$	−19.7880	−0.919626	−0.459813	−	0.888016i	$-0.652084\pi$
$463$	−19.7880	−0.919626	−0.459813	+	0.888016i	$0.652084\pi$
$464$	−5.28007	−0.245121
$465$	−4.08884	−0.189615
$466$	−1.18293	−0.0547983
$467$	22.1724	1.02602	0.513008	−	0.858384i	$-0.328531\pi$
$467$	22.1724	1.02602	0.513008	+	0.858384i	$0.328531\pi$
$468$	−2.73576	−0.126460
$469$	0	0
$470$	−4.36177	−0.201194
$471$	15.2191	0.701257
$472$	19.2388	0.885538
$473$	−22.2779	−1.02434
$474$	13.5774	0.623629
$475$	−15.3052	−0.702249
$476$	6.32868	0.290075
$477$	−2.63919	−0.120840
$478$	−7.54917	−0.345291
$479$	−3.90594	−0.178467	−0.0892336	−	0.996011i	$-0.528442\pi$
$479$	−3.90594	−0.178467	−0.0892336	+	0.996011i	$0.528442\pi$
$480$	−7.44319	−0.339733
$481$	35.4583	1.61676
$482$	8.90400	0.405566
$483$	12.7789	0.581460
$484$	4.82364	0.219256
$485$	−0.858283	−0.0389726
$486$	1.74282	0.0790557
$487$	−26.8235	−1.21549	−0.607745	−	0.794132i	$-0.707926\pi$
$487$	−26.8235	−1.21549	−0.607745	+	0.794132i	$0.707926\pi$
$488$	16.7049	0.756197
$489$	12.7267	0.575521
$490$	2.31402	0.104537
$491$	36.8371	1.66244	0.831218	−	0.555946i	$-0.187644\pi$
$491$	36.8371	1.66244	0.831218	+	0.555946i	$0.187644\pi$
$492$	−8.78888	−0.396233
$493$	6.47958	0.291825
$494$	−7.46622	−0.335921
$495$	−0.841170	−0.0378078
$496$	−7.19481	−0.323056
$497$	−1.88706	−0.0846464
$498$	7.09072	0.317743
$499$	−3.06877	−0.137377	−0.0686885	−	0.997638i	$-0.521881\pi$
$499$	−3.06877	−0.137377	−0.0686885	+	0.997638i	$0.521881\pi$
$500$	13.6957	0.612491
$501$	−34.7774	−1.55374
$502$	−6.52850	−0.291381
$503$	−3.67220	−0.163735	−0.0818676	−	0.996643i	$-0.526088\pi$
$503$	−3.67220	−0.163735	−0.0818676	+	0.996643i	$0.526088\pi$
$504$	0.691533	0.0308033
$505$	12.8139	0.570210
$506$	8.83683	0.392845
$507$	11.4849	0.510060
$508$	−1.99749	−0.0886245
$509$	15.5016	0.687095	0.343548	−	0.939135i	$-0.388371\pi$
$509$	15.5016	0.687095	0.343548	+	0.939135i	$0.388371\pi$
$510$	2.39267	0.105949
$511$	−1.59304	−0.0704718
$512$	−22.7636	−1.00602
$513$	−17.2173	−0.760162
$514$	2.32726	0.102651
$515$	−3.35840	−0.147989
$516$	−25.1046	−1.10517
$517$	−31.5369	−1.38699
$518$	−4.20892	−0.184929
$519$	17.9851	0.789456
$520$	6.58631	0.288829
$521$	11.0777	0.485321	0.242661	−	0.970111i	$-0.421980\pi$
$521$	11.0777	0.485321	0.242661	+	0.970111i	$0.421980\pi$
$522$	0.332479	0.0145522
$523$	19.3717	0.847067	0.423533	−	0.905881i	$-0.360790\pi$
$523$	19.3717	0.847067	0.423533	+	0.905881i	$0.360790\pi$
$524$	−3.78514	−0.165354
$525$	−8.58595	−0.374722
$526$	−5.98103	−0.260785
$527$	8.82930	0.384610
$528$	−14.0972	−0.613501
$529$	18.1442	0.788877
$530$	2.98368	0.129603
$531$	3.74965	0.162721
$532$	−6.84251	−0.296661
$533$	11.9021	0.515538
$534$	−13.5380	−0.585848
$535$	8.54726	0.369530
$536$	0	0
$537$	39.7573	1.71565
$538$	4.93478	0.212754
$539$	16.7310	0.720657
$540$	7.13221	0.306921
$541$	6.89287	0.296347	0.148174	−	0.988961i	$-0.452661\pi$
$541$	6.89287	0.296347	0.148174	+	0.988961i	$0.452661\pi$
$542$	−2.06535	−0.0887143
$543$	−47.1881	−2.02504
$544$	16.0726	0.689105
$545$	6.57193	0.281511
$546$	−4.18843	−0.179248
$547$	−9.68042	−0.413905	−0.206952	−	0.978351i	$-0.566355\pi$
$547$	−9.68042	−0.413905	−0.206952	+	0.978351i	$0.566355\pi$
$548$	−18.9094	−0.807771
$549$	3.25579	0.138954
$550$	−5.93733	−0.253169
$551$	−7.00565	−0.298451
$552$	21.2059	0.902584
$553$	−16.8508	−0.716570
$554$	7.85978	0.333930
$555$	12.2858	0.521502
$556$	1.48707	0.0630660
$557$	16.2834	0.689951	0.344976	−	0.938612i	$-0.387887\pi$
$557$	16.2834	0.689951	0.344976	+	0.938612i	$0.387887\pi$
$558$	0.453048	0.0191790
$559$	33.9973	1.43793
$560$	2.41982	0.102256
$561$	17.2997	0.730395
$562$	11.5432	0.486921
$563$	13.2273	0.557463	0.278731	−	0.960369i	$-0.410086\pi$
$563$	13.2273	0.557463	0.278731	+	0.960369i	$0.410086\pi$
$564$	−35.5384	−1.49644
$565$	10.9563	0.460935
$566$	12.5704	0.528373
$567$	−10.8075	−0.453873
$568$	−3.13148	−0.131394
$569$	−45.3651	−1.90180	−0.950902	−	0.309492i	$-0.899841\pi$
$569$	−45.3651	−1.90180	−0.950902	+	0.309492i	$0.899841\pi$
$570$	−2.58694	−0.108355
$571$	−21.1131	−0.883554	−0.441777	−	0.897125i	$-0.645652\pi$
$571$	−21.1131	−0.883554	−0.441777	+	0.897125i	$0.645652\pi$
$572$	22.3623	0.935013
$573$	−29.4720	−1.23121
$574$	−1.41279	−0.0589687
$575$	−27.6442	−1.15284
$576$	−1.05927	−0.0441361
$577$	23.9403	0.996646	0.498323	−	0.866991i	$-0.333949\pi$
$577$	23.9403	0.996646	0.498323	+	0.866991i	$0.333949\pi$
$578$	2.97449	0.123723
$579$	45.6134	1.89563
$580$	2.90207	0.120502
$581$	−8.80028	−0.365097
$582$	0.905739	0.0375441
$583$	21.5729	0.893458
$584$	−2.64356	−0.109391
$585$	1.28367	0.0530733
$586$	−8.20260	−0.338846
$587$	38.1142	1.57314	0.786570	−	0.617501i	$-0.211855\pi$
$587$	38.1142	1.57314	0.786570	+	0.617501i	$0.211855\pi$
$588$	18.8539	0.777523
$589$	−9.54616	−0.393343
$590$	−4.23909	−0.174521
$591$	22.1151	0.909693
$592$	21.6183	0.888507
$593$	21.1357	0.867940	0.433970	−	0.900927i	$-0.357112\pi$
$593$	21.1357	0.867940	0.433970	+	0.900927i	$0.357112\pi$
$594$	−6.67910	−0.274047
$595$	−2.96954	−0.121739
$596$	−21.8205	−0.893804
$597$	−4.31733	−0.176697
$598$	−13.4855	−0.551462
$599$	−11.6156	−0.474600	−0.237300	−	0.971436i	$-0.576263\pi$
$599$	−11.6156	−0.474600	−0.237300	+	0.971436i	$0.576263\pi$
$600$	−14.2479	−0.581670
$601$	−48.4579	−1.97664	−0.988319	−	0.152399i	$-0.951300\pi$
$601$	−48.4579	−1.97664	−0.988319	+	0.152399i	$0.951300\pi$
$602$	−4.03550	−0.164475
$603$	0	0
$604$	19.5168	0.794128
$605$	−2.26335	−0.0920182
$606$	−13.5224	−0.549309
$607$	−22.0351	−0.894376	−0.447188	−	0.894440i	$-0.647575\pi$
$607$	−22.0351	−0.894376	−0.447188	+	0.894440i	$0.647575\pi$
$608$	−17.3775	−0.704751
$609$	−3.93006	−0.159254
$610$	−3.68078	−0.149030
$611$	48.1270	1.94701
$612$	2.04687	0.0827396
$613$	−19.4122	−0.784052	−0.392026	−	0.919954i	$-0.628226\pi$
$613$	−19.4122	−0.784052	−0.392026	+	0.919954i	$0.628226\pi$
$614$	−4.23079	−0.170741
$615$	4.12392	0.166292
$616$	−5.65263	−0.227751
$617$	7.68409	0.309350	0.154675	−	0.987965i	$-0.450567\pi$
$617$	7.68409	0.309350	0.154675	+	0.987965i	$0.450567\pi$
$618$	3.54409	0.142564
$619$	−28.1469	−1.13132	−0.565659	−	0.824639i	$-0.691378\pi$
$619$	−28.1469	−1.13132	−0.565659	+	0.824639i	$0.691378\pi$
$620$	3.95447	0.158815
$621$	−31.0978	−1.24791
$622$	10.2400	0.410588
$623$	16.8020	0.673158
$624$	21.5131	0.861212
$625$	15.1223	0.604892
$626$	1.39972	0.0559439
$627$	−18.7043	−0.746978
$628$	−14.7189	−0.587348
$629$	−26.5295	−1.05780
$630$	−0.152373	−0.00607068
$631$	−7.40083	−0.294622	−0.147311	−	0.989090i	$-0.547062\pi$
$631$	−7.40083	−0.294622	−0.147311	+	0.989090i	$0.547062\pi$
$632$	−27.9631	−1.11231
$633$	−1.28736	−0.0511682
$634$	3.72525	0.147948
$635$	0.937264	0.0371942
$636$	24.3101	0.963960
$637$	−25.5325	−1.01163
$638$	−2.71770	−0.107595
$639$	−0.610326	−0.0241441
$640$	9.32848	0.368740
$641$	6.58756	0.260193	0.130097	−	0.991501i	$-0.458471\pi$
$641$	6.58756	0.260193	0.130097	+	0.991501i	$0.458471\pi$
$642$	−9.01986	−0.355985
$643$	0.345529	0.0136263	0.00681317	−	0.999977i	$-0.497831\pi$
$643$	0.345529	0.0136263	0.00681317	+	0.999977i	$0.497831\pi$
$644$	−12.3589	−0.487010
$645$	11.7796	0.463821
$646$	5.58614	0.219784
$647$	7.87818	0.309723	0.154862	−	0.987936i	$-0.450507\pi$
$647$	7.87818	0.309723	0.154862	+	0.987936i	$0.450507\pi$
$648$	−17.9345	−0.704534
$649$	−30.6499	−1.20311
$650$	9.06069	0.355390
$651$	−5.35524	−0.209889
$652$	−12.3085	−0.482036
$653$	−31.6335	−1.23791	−0.618957	−	0.785425i	$-0.712445\pi$
$653$	−31.6335	−1.23791	−0.618957	+	0.785425i	$0.712445\pi$
$654$	−6.93531	−0.271192
$655$	1.77606	0.0693964
$656$	7.25653	0.283320
$657$	−0.515230	−0.0201011
$658$	−5.71271	−0.222705
$659$	4.33886	0.169018	0.0845091	−	0.996423i	$-0.473068\pi$
$659$	4.33886	0.169018	0.0845091	+	0.996423i	$0.473068\pi$
$660$	7.74820	0.301598
$661$	8.55237	0.332649	0.166324	−	0.986071i	$-0.446810\pi$
$661$	8.55237	0.332649	0.166324	+	0.986071i	$0.446810\pi$
$662$	−15.7872	−0.613587
$663$	−26.4003	−1.02530
$664$	−14.6036	−0.566730
$665$	3.21064	0.124503
$666$	−1.36128	−0.0527484
$667$	−12.6536	−0.489950
$668$	33.6345	1.30136
$669$	13.9924	0.540978
$670$	0	0
$671$	−26.6131	−1.02739
$672$	−9.74850	−0.376057
$673$	−36.6687	−1.41347	−0.706736	−	0.707477i	$-0.749833\pi$
$673$	−36.6687	−1.41347	−0.706736	+	0.707477i	$0.749833\pi$
$674$	−13.1887	−0.508012
$675$	20.8942	0.804217
$676$	−11.1074	−0.427209
$677$	−28.4485	−1.09337	−0.546683	−	0.837340i	$-0.684110\pi$
$677$	−28.4485	−1.09337	−0.546683	+	0.837340i	$0.684110\pi$
$678$	−11.5621	−0.444040
$679$	−1.12411	−0.0431394
$680$	−4.92780	−0.188973
$681$	20.0049	0.766589
$682$	−3.70324	−0.141804
$683$	35.8081	1.37016	0.685080	−	0.728468i	$-0.259767\pi$
$683$	35.8081	1.37016	0.685080	+	0.728468i	$0.259767\pi$
$684$	−2.21305	−0.0846181
$685$	8.87268	0.339008
$686$	6.67848	0.254986
$687$	48.6832	1.85738
$688$	20.7276	0.790232
$689$	−32.9214	−1.25421
$690$	−4.67252	−0.177880
$691$	22.1819	0.843840	0.421920	−	0.906633i	$-0.361356\pi$
$691$	22.1819	0.843840	0.421920	+	0.906633i	$0.361356\pi$
$692$	−17.3940	−0.661221
$693$	−1.10170	−0.0418501
$694$	4.05225	0.153821
$695$	−0.697764	−0.0264677
$696$	−6.52173	−0.247206
$697$	−8.90504	−0.337303
$698$	−13.7735	−0.521336
$699$	4.52242	0.171054
$700$	8.30379	0.313854
$701$	−12.3989	−0.468300	−0.234150	−	0.972200i	$-0.575231\pi$
$701$	−12.3989	−0.468300	−0.234150	+	0.972200i	$0.575231\pi$
$702$	10.1927	0.384698
$703$	28.6834	1.08182
$704$	8.65851	0.326330
$705$	16.6753	0.628029
$706$	−10.6075	−0.399218
$707$	16.7826	0.631174
$708$	−34.5388	−1.29805
$709$	24.9837	0.938284	0.469142	−	0.883123i	$-0.344563\pi$
$709$	24.9837	0.938284	0.469142	+	0.883123i	$0.344563\pi$
$710$	0.689993	0.0258950
$711$	−5.45001	−0.204391
$712$	27.8821	1.04492
$713$	−17.2423	−0.645728
$714$	3.13374	0.117277
$715$	−10.4928	−0.392409
$716$	−38.4507	−1.43697
$717$	28.8609	1.07783
$718$	−4.02297	−0.150136
$719$	−45.1649	−1.68437	−0.842184	−	0.539191i	$-0.818730\pi$
$719$	−45.1649	−1.68437	−0.842184	+	0.539191i	$0.818730\pi$
$720$	0.782633	0.0291670
$721$	−4.39856	−0.163811
$722$	3.05925	0.113854
$723$	−34.0405	−1.26598
$724$	45.6374	1.69610
$725$	8.50177	0.315748
$726$	2.38849	0.0886453
$727$	37.2055	1.37988	0.689938	−	0.723868i	$-0.257638\pi$
$727$	37.2055	1.37988	0.689938	+	0.723868i	$0.257638\pi$
$728$	8.62623	0.319709
$729$	23.1330	0.856777
$730$	0.582484	0.0215587
$731$	−25.4364	−0.940800
$732$	−29.9898	−1.10846
$733$	−21.9918	−0.812287	−0.406144	−	0.913809i	$-0.633127\pi$
$733$	−21.9918	−0.812287	−0.406144	+	0.913809i	$0.633127\pi$
$734$	−1.77615	−0.0655589
$735$	−8.84664	−0.326313
$736$	−31.3872	−1.15695
$737$	0	0
$738$	−0.456934	−0.0168200
$739$	−27.3506	−1.00611	−0.503053	−	0.864255i	$-0.667790\pi$
$739$	−27.3506	−1.00611	−0.503053	+	0.864255i	$0.667790\pi$
$740$	−11.8820	−0.436792
$741$	28.5438	1.04858
$742$	3.90779	0.143460
$743$	34.6339	1.27059	0.635297	−	0.772268i	$-0.280878\pi$
$743$	34.6339	1.27059	0.635297	+	0.772268i	$0.280878\pi$
$744$	−8.88675	−0.325804
$745$	10.2386	0.375114
$746$	5.37075	0.196637
$747$	−2.84624	−0.104139
$748$	−16.7312	−0.611754
$749$	11.1945	0.409039
$750$	6.78163	0.247630
$751$	20.3387	0.742169	0.371085	−	0.928599i	$-0.378986\pi$
$751$	20.3387	0.742169	0.371085	+	0.928599i	$0.378986\pi$
$752$	29.3422	1.07000
$753$	24.9588	0.909550
$754$	4.14737	0.151038
$755$	−9.15768	−0.333282
$756$	9.34121	0.339737
$757$	1.05198	0.0382348	0.0191174	−	0.999817i	$-0.493914\pi$
$757$	1.05198	0.0382348	0.0191174	+	0.999817i	$0.493914\pi$
$758$	2.85183	0.103583
$759$	−33.7837	−1.22627
$760$	5.32789	0.193263
$761$	−42.8478	−1.55323	−0.776615	−	0.629975i	$-0.783065\pi$
$761$	−42.8478	−1.55323	−0.776615	+	0.629975i	$0.783065\pi$
$762$	−0.989088	−0.0358309
$763$	8.60740	0.311609
$764$	28.5035	1.03122
$765$	−0.960429	−0.0347244
$766$	−6.80865	−0.246007
$767$	46.7733	1.68889
$768$	1.17661	0.0424572
$769$	−20.0059	−0.721431	−0.360715	−	0.932676i	$-0.617467\pi$
$769$	−20.0059	−0.721431	−0.360715	+	0.932676i	$0.617467\pi$
$770$	1.24550	0.0448849
$771$	−8.89724	−0.320426
$772$	−44.1144	−1.58771
$773$	6.67646	0.240135	0.120068	−	0.992766i	$-0.461689\pi$
$773$	6.67646	0.240135	0.120068	+	0.992766i	$0.461689\pi$
$774$	−1.30519	−0.0469141
$775$	11.5848	0.416139
$776$	−1.86540	−0.0669641
$777$	16.0909	0.577259
$778$	6.77307	0.242826
$779$	9.62805	0.344961
$780$	−11.8242	−0.423374
$781$	4.98885	0.178515
$782$	10.0897	0.360806
$783$	9.56393	0.341787
$784$	−15.5667	−0.555955
$785$	6.90640	0.246500
$786$	−1.87426	−0.0668528
$787$	−18.3611	−0.654502	−0.327251	−	0.944937i	$-0.606122\pi$
$787$	−18.3611	−0.654502	−0.327251	+	0.944937i	$0.606122\pi$
$788$	−21.3883	−0.761927
$789$	22.8658	0.814045
$790$	6.16140	0.219213
$791$	14.3497	0.510217
$792$	−1.82821	−0.0649627
$793$	40.6130	1.44221
$794$	0.151354	0.00537134
$795$	−11.4068	−0.404557
$796$	4.17545	0.147995
$797$	49.4409	1.75129	0.875644	−	0.482958i	$-0.160438\pi$
$797$	49.4409	1.75129	0.875644	+	0.482958i	$0.160438\pi$
$798$	−3.38817	−0.119940
$799$	−36.0081	−1.27388
$800$	21.0886	0.745595
$801$	5.43422	0.192009
$802$	14.6021	0.515618
$803$	4.21153	0.148622
$804$	0	0
$805$	5.79906	0.204390
$806$	5.65135	0.199060
$807$	−18.8660	−0.664113
$808$	27.8498	0.979754
$809$	−50.4211	−1.77271	−0.886355	−	0.463006i	$-0.846771\pi$
$809$	−50.4211	−1.77271	−0.886355	+	0.463006i	$0.846771\pi$
$810$	3.95170	0.138849
$811$	−37.1590	−1.30483	−0.652415	−	0.757862i	$-0.726244\pi$
$811$	−37.1590	−1.30483	−0.652415	+	0.757862i	$0.726244\pi$
$812$	3.80091	0.133386
$813$	7.89594	0.276923
$814$	11.1272	0.390007
$815$	5.77537	0.202302
$816$	−16.0958	−0.563467
$817$	27.5016	0.962160
$818$	−5.13134	−0.179413
$819$	1.68125	0.0587477
$820$	−3.98839	−0.139281
$821$	−35.3955	−1.23531	−0.617656	−	0.786448i	$-0.711918\pi$
$821$	−35.3955	−1.23531	−0.617656	+	0.786448i	$0.711918\pi$
$822$	−9.36327	−0.326582
$823$	−38.0519	−1.32641	−0.663203	−	0.748440i	$-0.730803\pi$
$823$	−38.0519	−1.32641	−0.663203	+	0.748440i	$0.730803\pi$
$824$	−7.29919	−0.254279
$825$	22.6988	0.790269
$826$	−5.55203	−0.193180
$827$	−52.9396	−1.84089	−0.920444	−	0.390874i	$-0.872173\pi$
$827$	−52.9396	−1.84089	−0.920444	+	0.390874i	$0.872173\pi$
$828$	−3.99721	−0.138913
$829$	31.1563	1.08210	0.541051	−	0.840990i	$-0.318027\pi$
$829$	31.1563	1.08210	0.541051	+	0.840990i	$0.318027\pi$
$830$	3.21777	0.111690
$831$	−30.0484	−1.04237
$832$	−13.2134	−0.458091
$833$	19.1031	0.661884
$834$	0.736346	0.0254976
$835$	−15.7820	−0.546158
$836$	18.0896	0.625643
$837$	13.0322	0.450457
$838$	2.10941	0.0728683
$839$	−32.0479	−1.10642	−0.553209	−	0.833043i	$-0.686597\pi$
$839$	−32.0479	−1.10642	−0.553209	+	0.833043i	$0.686597\pi$
$840$	2.98886	0.103126
$841$	−25.1085	−0.865809
$842$	1.27716	0.0440138
$843$	−44.1304	−1.51993
$844$	1.24506	0.0428567
$845$	5.21182	0.179292
$846$	−1.84764	−0.0635232
$847$	−2.96436	−0.101856
$848$	−20.0716	−0.689263
$849$	−48.0573	−1.64932
$850$	−6.77911	−0.232522
$851$	51.8080	1.77595
$852$	5.62185	0.192601
$853$	−11.7743	−0.403145	−0.201573	−	0.979474i	$-0.564605\pi$
$853$	−11.7743	−0.403145	−0.201573	+	0.979474i	$0.564605\pi$
$854$	−4.82079	−0.164964
$855$	1.03841	0.0355128
$856$	18.5767	0.634940
$857$	−32.8390	−1.12176	−0.560880	−	0.827897i	$-0.689537\pi$
$857$	−32.8390	−1.12176	−0.560880	+	0.827897i	$0.689537\pi$
$858$	11.0730	0.378026
$859$	−23.1472	−0.789773	−0.394886	−	0.918730i	$-0.629216\pi$
$859$	−23.1472	−0.789773	−0.394886	+	0.918730i	$0.629216\pi$
$860$	−11.3925	−0.388480
$861$	5.40118	0.184072
$862$	4.89186	0.166617
$863$	−21.3357	−0.726276	−0.363138	−	0.931735i	$-0.618295\pi$
$863$	−21.3357	−0.726276	−0.363138	+	0.931735i	$0.618295\pi$
$864$	23.7233	0.807082
$865$	8.16161	0.277503
$866$	−0.0996595	−0.00338657
$867$	−11.3717	−0.386202
$868$	5.17925	0.175795
$869$	44.5487	1.51121
$870$	1.43700	0.0487189
$871$	0	0
$872$	14.2835	0.483701
$873$	−0.363568	−0.0123049
$874$	−10.9089	−0.368998
$875$	−8.41666	−0.284535
$876$	4.74590	0.160349
$877$	−3.28825	−0.111036	−0.0555181	−	0.998458i	$-0.517681\pi$
$877$	−3.28825	−0.111036	−0.0555181	+	0.998458i	$0.517681\pi$
$878$	2.12743	0.0717973
$879$	31.3590	1.05771
$880$	−6.39729	−0.215653
$881$	−38.1003	−1.28363	−0.641816	−	0.766858i	$-0.721819\pi$
$881$	−38.1003	−1.28363	−0.641816	+	0.766858i	$0.721819\pi$
$882$	0.980217	0.0330056
$883$	41.0647	1.38194	0.690968	−	0.722885i	$-0.257184\pi$
$883$	41.0647	1.38194	0.690968	+	0.722885i	$0.257184\pi$
$884$	25.5327	0.858759
$885$	16.2063	0.544768
$886$	13.4181	0.450788
$887$	38.1040	1.27941	0.639704	−	0.768622i	$-0.279057\pi$
$887$	38.1040	1.27941	0.639704	+	0.768622i	$0.279057\pi$
$888$	26.7021	0.896063
$889$	1.22755	0.0411709
$890$	−6.14355	−0.205932
$891$	28.5719	0.957196
$892$	−13.5326	−0.453104
$893$	38.9316	1.30280
$894$	−10.8048	−0.361365
$895$	18.0418	0.603072
$896$	12.2177	0.408165
$897$	51.5558	1.72140
$898$	−0.141362	−0.00471731
$899$	5.30274	0.176856
$900$	2.68567	0.0895222
$901$	24.6314	0.820592
$902$	3.73501	0.124362
$903$	15.4280	0.513411
$904$	23.8126	0.791995
$905$	−21.4139	−0.711824
$906$	9.66403	0.321066
$907$	36.6101	1.21562	0.607808	−	0.794084i	$-0.292049\pi$
$907$	36.6101	1.21562	0.607808	+	0.794084i	$0.292049\pi$
$908$	−19.3475	−0.642068
$909$	5.42794	0.180033
$910$	−1.90071	−0.0630079
$911$	−43.3269	−1.43548	−0.717742	−	0.696309i	$-0.754824\pi$
$911$	−43.3269	−1.43548	−0.717742	+	0.696309i	$0.754824\pi$
$912$	17.4027	0.576260
$913$	23.2654	0.769972
$914$	−8.49763	−0.281077
$915$	14.0718	0.465200
$916$	−47.0834	−1.55568
$917$	2.32614	0.0768161
$918$	−7.62605	−0.251697
$919$	−22.9015	−0.755452	−0.377726	−	0.925917i	$-0.623294\pi$
$919$	−22.9015	−0.755452	−0.377726	+	0.925917i	$0.623294\pi$
$920$	9.62324	0.317269
$921$	16.1746	0.532970
$922$	10.7531	0.354134
$923$	−7.61325	−0.250593
$924$	10.1480	0.333844
$925$	−34.8090	−1.14451
$926$	9.47630	0.311411
$927$	−1.42261	−0.0467247
$928$	9.65292	0.316873
$929$	27.5021	0.902316	0.451158	−	0.892444i	$-0.351011\pi$
$929$	27.5021	0.902316	0.451158	+	0.892444i	$0.351011\pi$
$930$	1.95811	0.0642090
$931$	−20.6541	−0.676911
$932$	−4.37380	−0.143268
$933$	−39.1483	−1.28166
$934$	−10.6182	−0.347437
$935$	7.85061	0.256742
$936$	2.78995	0.0911924
$937$	−4.58217	−0.149693	−0.0748464	−	0.997195i	$-0.523847\pi$
$937$	−4.58217	−0.149693	−0.0748464	+	0.997195i	$0.523847\pi$
$938$	0	0
$939$	−5.35120	−0.174630
$940$	−16.1273	−0.526015
$941$	−14.0525	−0.458097	−0.229049	−	0.973415i	$-0.573561\pi$
$941$	−14.0525	−0.458097	−0.229049	+	0.973415i	$0.573561\pi$
$942$	−7.28827	−0.237465
$943$	17.3902	0.566302
$944$	28.5169	0.928147
$945$	−4.38308	−0.142582
$946$	10.6687	0.346870
$947$	−9.79821	−0.318399	−0.159200	−	0.987246i	$-0.550891\pi$
$947$	−9.79821	−0.318399	−0.159200	+	0.987246i	$0.550891\pi$
$948$	50.2012	1.63046
$949$	−6.42702	−0.208630
$950$	7.32951	0.237801
$951$	−14.2418	−0.461823
$952$	−6.45405	−0.209177
$953$	28.6760	0.928908	0.464454	−	0.885597i	$-0.346251\pi$
$953$	28.6760	0.928908	0.464454	+	0.885597i	$0.346251\pi$
$954$	1.26388	0.0409198
$955$	−13.3744	−0.432785
$956$	−27.9125	−0.902754
$957$	10.3899	0.335859
$958$	1.87052	0.0604339
$959$	11.6207	0.375253
$960$	−4.57824	−0.147762
$961$	−23.7743	−0.766913
$962$	−16.9807	−0.547478
$963$	3.62061	0.116672
$964$	32.9218	1.06034
$965$	20.6994	0.666336
$966$	−6.11970	−0.196898
$967$	3.62365	0.116529	0.0582644	−	0.998301i	$-0.481443\pi$
$967$	3.62365	0.116529	0.0582644	+	0.998301i	$0.481443\pi$
$968$	−4.91919	−0.158109
$969$	−21.3561	−0.686058
$970$	0.411024	0.0131972
$971$	−14.0498	−0.450879	−0.225439	−	0.974257i	$-0.572382\pi$
$971$	−14.0498	−0.450879	−0.225439	+	0.974257i	$0.572382\pi$
$972$	6.44392	0.206689
$973$	−0.913877	−0.0292975
$974$	12.8456	0.411598
$975$	−34.6395	−1.10935
$976$	24.7611	0.792582
$977$	13.1192	0.419719	0.209860	−	0.977732i	$-0.432699\pi$
$977$	13.1192	0.419719	0.209860	+	0.977732i	$0.432699\pi$
$978$	−6.09471	−0.194887
$979$	−44.4196	−1.41966
$980$	8.55591	0.273309
$981$	2.78386	0.0888819
$982$	−17.6410	−0.562946
$983$	26.9322	0.859004	0.429502	−	0.903066i	$-0.358689\pi$
$983$	26.9322	0.859004	0.429502	+	0.903066i	$0.358689\pi$
$984$	8.96298	0.285729
$985$	10.0358	0.319768
$986$	−3.10301	−0.0988201
$987$	21.8400	0.695175
$988$	−27.6057	−0.878256
$989$	49.6734	1.57952
$990$	0.402829	0.0128028
$991$	−32.6333	−1.03663	−0.518315	−	0.855190i	$-0.673441\pi$
$991$	−32.6333	−1.03663	−0.518315	+	0.855190i	$0.673441\pi$
$992$	13.1534	0.417622
$993$	60.3554	1.91532
$994$	0.903698	0.0286636
$995$	−1.95920	−0.0621109
$996$	26.2174	0.830729
$997$	−39.5207	−1.25163	−0.625816	−	0.779971i	$-0.715234\pi$
$997$	−39.5207	−1.25163	−0.625816	+	0.779971i	$0.715234\pi$
$998$	1.46961	0.0465196
$999$	−39.1578	−1.23890

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4489.2.a.h.1.3		5
67.8	odd	22		67.2.e.b.64.1	yes	10
67.42	odd	22		67.2.e.b.22.1	✓	10
67.66	odd	2		4489.2.a.i.1.3		5
201.8	even	22		603.2.u.a.64.1		10
201.176	even	22		603.2.u.a.424.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.e.b.22.1	✓	10	67.42	odd	22
67.2.e.b.64.1	yes	10	67.8	odd	22
603.2.u.a.64.1		10	201.8	even	22
603.2.u.a.424.1		10	201.176	even	22
4489.2.a.h.1.3		5	1.1	even	1	trivial
4489.2.a.i.1.3		5	67.66	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:	\( N \)	\(=\)	\( 4489 = 67^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4489.a (trivial)

\(f(q)\)	\(=\)	\(q-0.478891 q^{2} +1.83083 q^{3} -1.77066 q^{4} +0.830830 q^{5} -0.876769 q^{6} +1.08816 q^{7} +1.80574 q^{8} +0.351939 q^{9} -0.397877 q^{10} -2.87677 q^{11} -3.24178 q^{12} +4.39011 q^{13} -0.521109 q^{14} +1.52111 q^{15} +2.67657 q^{16} -3.28463 q^{17} -0.168540 q^{18} +3.55131 q^{19} -1.47112 q^{20} +1.99223 q^{21} +1.37766 q^{22} +6.41437 q^{23} +3.30600 q^{24} -4.30972 q^{25} -2.10238 q^{26} -4.84815 q^{27} -1.92676 q^{28} -1.97270 q^{29} -0.728446 q^{30} -2.68807 q^{31} -4.89326 q^{32} -5.26687 q^{33} +1.57298 q^{34} +0.904073 q^{35} -0.623165 q^{36} +8.07686 q^{37} -1.70069 q^{38} +8.03754 q^{39} +1.50026 q^{40} +2.71113 q^{41} -0.954061 q^{42} +7.74408 q^{43} +5.09379 q^{44} +0.292401 q^{45} -3.07179 q^{46} +10.9626 q^{47} +4.90035 q^{48} -5.81592 q^{49} +2.06389 q^{50} -6.01360 q^{51} -7.77340 q^{52} -7.49900 q^{53} +2.32174 q^{54} -2.39011 q^{55} +1.96492 q^{56} +6.50184 q^{57} +0.944707 q^{58} +10.6543 q^{59} -2.69337 q^{60} +9.25103 q^{61} +1.28729 q^{62} +0.382964 q^{63} -3.00980 q^{64} +3.64743 q^{65} +2.52226 q^{66} +5.81597 q^{68} +11.7436 q^{69} -0.432953 q^{70} -1.73418 q^{71} +0.635509 q^{72} -1.46398 q^{73} -3.86794 q^{74} -7.89037 q^{75} -6.28817 q^{76} -3.13037 q^{77} -3.84911 q^{78} -15.4857 q^{79} +2.22378 q^{80} -9.93195 q^{81} -1.29833 q^{82} -8.08733 q^{83} -3.52757 q^{84} -2.72897 q^{85} -3.70857 q^{86} -3.61167 q^{87} -5.19469 q^{88} +15.4408 q^{89} -0.140028 q^{90} +4.77712 q^{91} -11.3577 q^{92} -4.92139 q^{93} -5.24990 q^{94} +2.95053 q^{95} -8.95874 q^{96} -1.03304 q^{97} +2.78519 q^{98} -1.01245 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 5 q^{6} + 2 q^{7} + 9 q^{8} - 3 q^{9} + 7 q^{10} - 5 q^{11} + 12 q^{12} - 2 q^{13} - 3 q^{14} + 8 q^{15} + 6 q^{16} - 16 q^{17} + 21 q^{18} + 10 q^{19} + 8 q^{20}+ \cdots + 36 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 5 * q^6 + 2 * q^7 + 9 * q^8 - 3 * q^9 + 7 * q^10 - 5 * q^11 + 12 * q^12 - 2 * q^13 - 3 * q^14 + 8 * q^15 + 6 * q^16 - 16 * q^17 + 21 * q^18 + 10 * q^19 + 8 * q^20 - 5 * q^21 + 24 * q^22 + 2 * q^23 + 16 * q^24 - 16 * q^25 - 19 * q^26 + q^27 - 16 * q^28 - 12 * q^29 + 10 * q^30 - 16 * q^31 + 6 * q^32 + 7 * q^33 + 2 * q^34 - 7 * q^35 + 13 * q^36 + q^37 + 18 * q^38 - 6 * q^39 + 7 * q^40 + 20 * q^41 - 9 * q^42 + q^43 + 18 * q^44 + 16 * q^45 + 19 * q^46 + 11 * q^47 + 18 * q^48 - 21 * q^49 + 13 * q^50 - 15 * q^51 - 28 * q^52 + 17 * q^53 + 15 * q^54 + 12 * q^55 - 3 * q^56 + 8 * q^57 + 18 * q^58 + 14 * q^59 + 13 * q^60 - 19 * q^61 + 24 * q^62 - 10 * q^63 - 21 * q^64 - 4 * q^65 + 39 * q^66 - 15 * q^68 + 6 * q^69 - 6 * q^70 - 19 * q^71 + 21 * q^72 - 4 * q^73 - 18 * q^74 - 15 * q^75 - 14 * q^76 - 13 * q^77 - 46 * q^78 + 4 * q^79 + 12 * q^80 + q^81 - 19 * q^82 - 36 * q^83 - 15 * q^84 + q^85 - 18 * q^86 + 8 * q^87 + 24 * q^88 + 42 * q^89 + 9 * q^90 + 8 * q^91 - 38 * q^92 + 7 * q^93 - 2 * q^95 + 7 * q^96 - 27 * q^97 + 15 * q^98 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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