LMFDB - Embedded newform 4489.2.a.i.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:	$1$
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			$1.91899$ 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	$-0.445845\pi$
$2$	0.478891	0.338627	0.169314	+	0.985562i	$0.445845\pi$
$3$	−1.83083	−1.05703	−0.528515	−	0.848924i	$-0.677251\pi$
$3$	−1.83083	−1.05703	−0.528515	+	0.848924i	$0.677251\pi$
$4$	−1.77066	−0.885331
$5$	−0.830830	−0.371558	−0.185779	−	0.982592i	$-0.559481\pi$
$5$	−0.830830	−0.371558	−0.185779	+	0.982592i	$0.559481\pi$
$6$	−0.876769	−0.357939
$7$	−1.08816	−0.411284	−0.205642	−	0.978627i	$-0.565928\pi$
$7$	−1.08816	−0.411284	−0.205642	+	0.978627i	$0.565928\pi$
$8$	−1.80574	−0.638425
$9$	0.351939	0.117313
$10$	−0.397877	−0.125820
$11$	2.87677	0.867378	0.433689	−	0.901063i	$-0.357212\pi$
$11$	2.87677	0.867378	0.433689	+	0.901063i	$0.357212\pi$
$12$	3.24178	0.935822
$13$	−4.39011	−1.21760	−0.608798	−	0.793325i	$-0.708348\pi$
$13$	−4.39011	−1.21760	−0.608798	+	0.793325i	$0.708348\pi$
$14$	−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	$-0.422395\pi$
$31$	2.68807	0.482791	0.241396	+	0.970427i	$0.422395\pi$
$32$	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	$-0.567901\pi$
$41$	−2.71113	−0.423407	−0.211703	+	0.977334i	$0.567901\pi$
$42$	0.954061	0.147215
$43$	−7.74408	−1.18096	−0.590480	−	0.807052i	$-0.701062\pi$
$43$	−7.74408	−1.18096	−0.590480	+	0.807052i	$0.701062\pi$
$44$	−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	$-0.327779\pi$
$53$	7.49900	1.03007	0.515034	+	0.857170i	$0.327779\pi$
$54$	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	$-0.701755\pi$
$61$	−9.25103	−1.18447	−0.592237	+	0.805764i	$0.701755\pi$
$62$	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	$-0.163383\pi$
$79$	15.4857	1.74227	0.871137	+	0.491040i	$0.163383\pi$
$80$	−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	$-0.483299\pi$
$97$	1.03304	0.104890	0.0524448	+	0.998624i	$0.483299\pi$
$98$	−2.78519	−0.281347
$99$	1.01245	0.101755
$100$	7.63106	0.763106
$101$	−15.4230	−1.53464	−0.767321	−	0.641263i	$-0.778411\pi$
$101$	−15.4230	−1.53464	−0.767321	+	0.641263i	$0.778411\pi$
$102$	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	$-0.623671\pi$
$109$	−7.91008	−0.757648	−0.378824	+	0.925469i	$0.623671\pi$
$110$	−1.14460	−0.109133
$111$	−14.7874	−1.40355
$112$	−2.91253	−0.275208
$113$	−13.1872	−1.24055	−0.620273	−	0.784386i	$-0.712978\pi$
$113$	−13.1872	−1.24055	−0.620273	+	0.784386i	$0.712978\pi$
$114$	−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	$-0.650789\pi$
$137$	−10.6793	−0.912393	−0.456197	+	0.889879i	$0.650789\pi$
$138$	−5.62392	−0.478740
$139$	0.839840	0.0712343	0.0356171	−	0.999366i	$-0.488660\pi$
$139$	0.839840	0.0712343	0.0356171	+	0.999366i	$0.488660\pi$
$140$	−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	$-0.801373\pi$
$179$	−21.7154	−1.62309	−0.811544	+	0.584291i	$0.801373\pi$
$180$	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	$-0.302117\pi$
$191$	16.0976	1.16478	0.582392	+	0.812908i	$0.302117\pi$
$192$	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	$-0.641594\pi$
$197$	−12.0793	−0.860612	−0.430306	+	0.902683i	$0.641594\pi$
$198$	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	$-0.841512\pi$
$229$	−26.5908	−1.75717	−0.878585	+	0.477586i	$0.841512\pi$
$230$	−2.55213	−0.168283
$231$	5.73118	0.377084
$232$	3.56217	0.233868
$233$	−2.47015	−0.161825	−0.0809123	−	0.996721i	$-0.525783\pi$
$233$	−2.47015	−0.161825	−0.0809123	+	0.996721i	$0.525783\pi$
$234$	−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	$-0.670295\pi$
$239$	−15.7639	−1.01968	−0.509839	+	0.860270i	$0.670295\pi$
$240$	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	$-0.641571\pi$
$251$	−13.6325	−0.860477	−0.430238	+	0.902715i	$0.641571\pi$
$252$	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	$-0.541816\pi$
$271$	−4.31277	−0.261982	−0.130991	+	0.991384i	$0.541816\pi$
$272$	−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	$-0.244617\pi$
$281$	24.1040	1.43793	0.718963	+	0.695048i	$0.244617\pi$
$282$	−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	$-0.292672\pi$
$311$	21.3828	1.21251	0.606254	+	0.795271i	$0.292672\pi$
$312$	−14.5137	−0.821676
$313$	2.92283	0.165208	0.0826040	−	0.996582i	$-0.473676\pi$
$313$	2.92283	0.165208	0.0826040	+	0.996582i	$0.473676\pi$
$314$	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	$-0.860873\pi$
$331$	−32.9661	−1.81198	−0.905992	+	0.423296i	$0.860873\pi$
$332$	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	$-0.769997\pi$
$337$	−27.5402	−1.50021	−0.750104	+	0.661320i	$0.769997\pi$
$338$	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	$-0.427067\pi$
$347$	8.46173	0.454250	0.227125	+	0.973866i	$0.427067\pi$
$348$	−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	$-0.700661\pi$
$353$	−22.1501	−1.17893	−0.589465	+	0.807794i	$0.700661\pi$
$354$	−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	$-0.530861\pi$
$367$	−3.70888	−0.193602	−0.0968010	+	0.995304i	$0.530861\pi$
$368$	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	$-0.406230\pi$
$373$	11.2150	0.580689	0.290345	+	0.956922i	$0.406230\pi$
$374$	−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	$-0.451124\pi$
$379$	5.95507	0.305891	0.152946	+	0.988235i	$0.451124\pi$
$380$	5.22440	0.268006
$381$	−2.06537	−0.105812
$382$	7.70902	0.394428
$383$	−14.2175	−0.726482	−0.363241	−	0.931695i	$-0.618330\pi$
$383$	−14.2175	−0.726482	−0.363241	+	0.931695i	$0.618330\pi$
$384$	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	$-0.224544\pi$
$401$	30.4915	1.52267	0.761336	+	0.648358i	$0.224544\pi$
$402$	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	$-0.585343\pi$
$409$	−10.7150	−0.529825	−0.264912	+	0.964272i	$0.585343\pi$
$410$	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	$-0.501592\pi$
$433$	−0.208104	−0.0100009	−0.00500043	+	0.999987i	$0.501592\pi$
$434$	−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	$-0.268171\pi$
$443$	28.0190	1.33122	0.665611	+	0.746299i	$0.268171\pi$
$444$	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	$-0.347916\pi$
$463$	19.7880	0.919626	0.459813	+	0.888016i	$0.347916\pi$
$464$	−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	$-0.292074\pi$
$487$	26.8235	1.21549	0.607745	+	0.794132i	$0.292074\pi$
$488$	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	$-0.478119\pi$
$499$	3.06877	0.137377	0.0686885	+	0.997638i	$0.478119\pi$
$500$	−13.6957	−0.612491
$501$	34.7774	1.55374
$502$	−6.52850	−0.291381
$503$	3.67220	0.163735	0.0818676	−	0.996643i	$-0.473912\pi$
$503$	3.67220	0.163735	0.0818676	+	0.996643i	$0.473912\pi$
$504$	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	$-0.578020\pi$
$521$	−11.0777	−0.485321	−0.242661	+	0.970111i	$0.578020\pi$
$522$	−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	$-0.547339\pi$
$541$	−6.89287	−0.296347	−0.148174	+	0.988961i	$0.547339\pi$
$542$	−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	$-0.433645\pi$
$547$	9.68042	0.413905	0.206952	+	0.978351i	$0.433645\pi$
$548$	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	$-0.589914\pi$
$563$	−13.2273	−0.557463	−0.278731	+	0.960369i	$0.589914\pi$
$564$	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	$-0.666051\pi$
$577$	−23.9403	−0.996646	−0.498323	+	0.866991i	$0.666051\pi$
$578$	−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	$-0.788145\pi$
$587$	−38.1142	−1.57314	−0.786570	+	0.617501i	$0.788145\pi$
$588$	−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	$-0.642888\pi$
$593$	−21.1357	−0.867940	−0.433970	+	0.900927i	$0.642888\pi$
$594$	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	$-0.423737\pi$
$599$	11.6156	0.474600	0.237300	+	0.971436i	$0.423737\pi$
$600$	−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	$-0.452938\pi$
$631$	7.40083	0.294622	0.147311	+	0.989090i	$0.452938\pi$
$632$	−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	$-0.541529\pi$
$641$	−6.58756	−0.260193	−0.130097	+	0.991501i	$0.541529\pi$
$642$	−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	$-0.549493\pi$
$647$	−7.87818	−0.309723	−0.154862	+	0.987936i	$0.549493\pi$
$648$	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	$-0.287555\pi$
$653$	31.6335	1.23791	0.618957	+	0.785425i	$0.287555\pi$
$654$	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	$-0.553190\pi$
$661$	−8.55237	−0.332649	−0.166324	+	0.986071i	$0.553190\pi$
$662$	−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	$-0.250167\pi$
$673$	36.6687	1.41347	0.706736	+	0.707477i	$0.250167\pi$
$674$	−13.1887	−0.508012
$675$	−20.8942	−0.804217
$676$	−11.1074	−0.427209
$677$	28.4485	1.09337	0.546683	−	0.837340i	$-0.315890\pi$
$677$	28.4485	1.09337	0.546683	+	0.837340i	$0.315890\pi$
$678$	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	$-0.740233\pi$
$683$	−35.8081	−1.37016	−0.685080	+	0.728468i	$0.740233\pi$
$684$	−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	$-0.424769\pi$
$701$	12.3989	0.468300	0.234150	+	0.972200i	$0.424769\pi$
$702$	−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	$-0.742362\pi$
$727$	−37.2055	−1.37988	−0.689938	+	0.723868i	$0.742362\pi$
$728$	−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	$-0.366873\pi$
$733$	21.9918	0.812287	0.406144	+	0.913809i	$0.366873\pi$
$734$	−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	$-0.332210\pi$
$739$	27.3506	1.00611	0.503053	+	0.864255i	$0.332210\pi$
$740$	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	$-0.506086\pi$
$757$	−1.05198	−0.0382348	−0.0191174	+	0.999817i	$0.506086\pi$
$758$	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	$-0.382533\pi$
$769$	20.0059	0.721431	0.360715	+	0.932676i	$0.382533\pi$
$770$	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	$-0.393878\pi$
$787$	18.3611	0.654502	0.327251	+	0.944937i	$0.393878\pi$
$788$	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	$-0.153229\pi$
$809$	50.4211	1.77271	0.886355	+	0.463006i	$0.153229\pi$
$810$	3.95170	0.138849
$811$	37.1590	1.30483	0.652415	−	0.757862i	$-0.273756\pi$
$811$	37.1590	1.30483	0.652415	+	0.757862i	$0.273756\pi$
$812$	−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	$-0.310463\pi$
$857$	32.8390	1.12176	0.560880	+	0.827897i	$0.310463\pi$
$858$	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	$-0.742816\pi$
$883$	−41.0647	−1.38194	−0.690968	+	0.722885i	$0.742816\pi$
$884$	−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	$-0.376706\pi$
$919$	22.9015	0.755452	0.377726	+	0.925917i	$0.376706\pi$
$920$	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	$-0.648989\pi$
$929$	−27.5021	−0.902316	−0.451158	+	0.892444i	$0.648989\pi$
$930$	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	$-0.476153\pi$
$937$	4.58217	0.149693	0.0748464	+	0.997195i	$0.476153\pi$
$938$	0	0
$939$	−5.35120	−0.174630
$940$	16.1273	0.526015
$941$	14.0525	0.458097	0.229049	−	0.973415i	$-0.426439\pi$
$941$	14.0525	0.458097	0.229049	+	0.973415i	$0.426439\pi$
$942$	−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	$-0.641311\pi$
$983$	−26.9322	−0.859004	−0.429502	+	0.903066i	$0.641311\pi$
$984$	−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	$-0.326559\pi$
$991$	32.6333	1.03663	0.518315	+	0.855190i	$0.326559\pi$
$992$	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.i.1.3		5
67.25	even	11		67.2.e.b.22.1	✓	10
67.59	even	11		67.2.e.b.64.1	yes	10
67.66	odd	2		4489.2.a.h.1.3		5
201.59	odd	22		603.2.u.a.64.1		10
201.92	odd	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	67.25	even	11
67.2.e.b.64.1	yes	10	67.59	even	11
603.2.u.a.64.1		10	201.59	odd	22
603.2.u.a.424.1		10	201.92	odd	22
4489.2.a.h.1.3		5	67.66	odd	2
4489.2.a.i.1.3		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Properties

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