LMFDB - Embedded newform 38.4.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [38,4,Mod(1,38)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(38, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("38.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	38.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$2.24207258022$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.15207$ of defining polynomial
Character	$\chi$	$=$	38.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -6.15207 q^{3} +4.00000 q^{4} +18.3041 q^{5} +12.3041 q^{6} +21.8479 q^{7} -8.00000 q^{8} +10.8479 q^{9} -36.6083 q^{10} -8.30413 q^{11} -24.6083 q^{12} +53.0645 q^{13} -43.6959 q^{14} -112.608 q^{15} +16.0000 q^{16} +74.2810 q^{17} -21.6959 q^{18} -19.0000 q^{19} +73.2165 q^{20} -134.410 q^{21} +16.6083 q^{22} -163.977 q^{23} +49.2165 q^{24} +210.041 q^{25} -106.129 q^{26} +99.3686 q^{27} +87.3917 q^{28} -232.410 q^{29} +225.217 q^{30} +98.4331 q^{31} -32.0000 q^{32} +51.0876 q^{33} -148.562 q^{34} +399.908 q^{35} +43.3917 q^{36} +296.433 q^{37} +38.0000 q^{38} -326.456 q^{39} -146.433 q^{40} -434.912 q^{41} +268.820 q^{42} -171.299 q^{43} -33.2165 q^{44} +198.562 q^{45} +327.954 q^{46} -366.083 q^{47} -98.4331 q^{48} +134.332 q^{49} -420.083 q^{50} -456.982 q^{51} +212.258 q^{52} +138.631 q^{53} -198.737 q^{54} -152.000 q^{55} -174.783 q^{56} +116.889 q^{57} +464.820 q^{58} +572.797 q^{59} -450.433 q^{60} -632.691 q^{61} -196.866 q^{62} +237.005 q^{63} +64.0000 q^{64} +971.299 q^{65} -102.175 q^{66} -183.461 q^{67} +297.124 q^{68} +1008.80 q^{69} -799.815 q^{70} +56.6545 q^{71} -86.7835 q^{72} +68.1521 q^{73} -592.866 q^{74} -1292.19 q^{75} -76.0000 q^{76} -181.428 q^{77} +652.912 q^{78} -332.820 q^{79} +292.866 q^{80} -904.217 q^{81} +869.825 q^{82} +1152.91 q^{83} -537.640 q^{84} +1359.65 q^{85} +342.598 q^{86} +1429.80 q^{87} +66.4331 q^{88} -368.479 q^{89} -397.124 q^{90} +1159.35 q^{91} -655.908 q^{92} -605.567 q^{93} +732.165 q^{94} -347.779 q^{95} +196.866 q^{96} -426.443 q^{97} -268.664 q^{98} -90.0827 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + q^{3} + 8 q^{4} + 10 q^{5} - 2 q^{6} + 57 q^{7} - 16 q^{8} + 35 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} + 13 q^{13} - 114 q^{14} - 172 q^{15} + 32 q^{16} - 51 q^{17} - 70 q^{18} - 38 q^{19}+ \cdots + 352 q^{99}+O(q^{100}) $ 2 * q - 4 * q^2 + q^3 + 8 * q^4 + 10 * q^5 - 2 * q^6 + 57 * q^7 - 16 * q^8 + 35 * q^9 - 20 * q^10 + 10 * q^11 + 4 * q^12 + 13 * q^13 - 114 * q^14 - 172 * q^15 + 32 * q^16 - 51 * q^17 - 70 * q^18 - 38 * q^19 + 40 * q^20 + 117 * q^21 - 20 * q^22 - 155 * q^23 - 8 * q^24 + 154 * q^25 - 26 * q^26 + 79 * q^27 + 228 * q^28 - 79 * q^29 + 344 * q^30 - 16 * q^31 - 64 * q^32 + 182 * q^33 + 102 * q^34 + 108 * q^35 + 140 * q^36 + 380 * q^37 + 76 * q^38 - 613 * q^39 - 80 * q^40 - 790 * q^41 - 234 * q^42 + 296 * q^43 + 40 * q^44 - 2 * q^45 + 310 * q^46 - 200 * q^47 + 16 * q^48 + 1027 * q^49 - 308 * q^50 - 1353 * q^51 + 52 * q^52 + 397 * q^53 - 158 * q^54 - 304 * q^55 - 456 * q^56 - 19 * q^57 + 158 * q^58 + 201 * q^59 - 688 * q^60 - 680 * q^61 + 32 * q^62 + 1086 * q^63 + 128 * q^64 + 1304 * q^65 - 364 * q^66 - 939 * q^67 - 204 * q^68 + 1073 * q^69 - 216 * q^70 + 406 * q^71 - 280 * q^72 + 123 * q^73 - 760 * q^74 - 1693 * q^75 - 152 * q^76 + 462 * q^77 + 1226 * q^78 + 106 * q^79 + 160 * q^80 - 1702 * q^81 + 1580 * q^82 + 2226 * q^83 + 468 * q^84 + 2400 * q^85 - 592 * q^86 + 2527 * q^87 - 80 * q^88 - 870 * q^89 + 4 * q^90 - 249 * q^91 - 620 * q^92 - 1424 * q^93 + 400 * q^94 - 190 * q^95 - 32 * q^96 - 1864 * q^97 - 2054 * q^98 + 352 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−6.15207	−1.18397	−0.591983	−	0.805950i	$-0.701655\pi$
$3$	−6.15207	−1.18397	−0.591983	+	0.805950i	$0.701655\pi$
$4$	4.00000	0.500000
$5$	18.3041	1.63717	0.818586	−	0.574384i	$-0.194758\pi$
$5$	18.3041	1.63717	0.818586	+	0.574384i	$0.194758\pi$
$6$	12.3041	0.837190
$7$	21.8479	1.17968	0.589839	−	0.807521i	$-0.299191\pi$
$7$	21.8479	1.17968	0.589839	+	0.807521i	$0.299191\pi$
$8$	−8.00000	−0.353553
$9$	10.8479	0.401775
$10$	−36.6083	−1.15766
$11$	−8.30413	−0.227617	−0.113809	−	0.993503i	$-0.536305\pi$
$11$	−8.30413	−0.227617	−0.113809	+	0.993503i	$0.536305\pi$
$12$	−24.6083	−0.591983
$13$	53.0645	1.13211	0.566055	−	0.824367i	$-0.308469\pi$
$13$	53.0645	1.13211	0.566055	+	0.824367i	$0.308469\pi$
$14$	−43.6959	−0.834158
$15$	−112.608	−1.93836
$16$	16.0000	0.250000
$17$	74.2810	1.05975	0.529876	−	0.848075i	$-0.322238\pi$
$17$	74.2810	1.05975	0.529876	+	0.848075i	$0.322238\pi$
$18$	−21.6959	−0.284098
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	73.2165	0.818586
$21$	−134.410	−1.39670
$22$	16.6083	0.160950
$23$	−163.977	−1.48659	−0.743294	−	0.668964i	$-0.766738\pi$
$23$	−163.977	−1.48659	−0.743294	+	0.668964i	$0.766738\pi$
$24$	49.2165	0.418595
$25$	210.041	1.68033
$26$	−106.129	−0.800523
$27$	99.3686	0.708278
$28$	87.3917	0.589839
$29$	−232.410	−1.48819	−0.744094	−	0.668075i	$-0.767118\pi$
$29$	−232.410	−1.48819	−0.744094	+	0.668075i	$0.767118\pi$
$30$	225.217	1.37062
$31$	98.4331	0.570294	0.285147	−	0.958484i	$-0.407958\pi$
$31$	98.4331	0.570294	0.285147	+	0.958484i	$0.407958\pi$
$32$	−32.0000	−0.176777
$33$	51.0876	0.269491
$34$	−148.562	−0.749358
$35$	399.908	1.93133
$36$	43.3917	0.200888
$37$	296.433	1.31712	0.658558	−	0.752530i	$-0.271167\pi$
$37$	296.433	1.31712	0.658558	+	0.752530i	$0.271167\pi$
$38$	38.0000	0.162221
$39$	−326.456	−1.34038
$40$	−146.433	−0.578828
$41$	−434.912	−1.65663	−0.828316	−	0.560261i	$-0.810701\pi$
$41$	−434.912	−1.65663	−0.828316	+	0.560261i	$0.810701\pi$
$42$	268.820	0.987615
$43$	−171.299	−0.607509	−0.303755	−	0.952750i	$-0.598240\pi$
$43$	−171.299	−0.607509	−0.303755	+	0.952750i	$0.598240\pi$
$44$	−33.2165	−0.113809
$45$	198.562	0.657775
$46$	327.954	1.05118
$47$	−366.083	−1.13614	−0.568071	−	0.822980i	$-0.692310\pi$
$47$	−366.083	−1.13614	−0.568071	+	0.822980i	$0.692310\pi$
$48$	−98.4331	−0.295991
$49$	134.332	0.391639
$50$	−420.083	−1.18817
$51$	−456.982	−1.25471
$52$	212.258	0.566055
$53$	138.631	0.359292	0.179646	−	0.983731i	$-0.442505\pi$
$53$	138.631	0.359292	0.179646	+	0.983731i	$0.442505\pi$
$54$	−198.737	−0.500828
$55$	−152.000	−0.372649
$56$	−174.783	−0.417079
$57$	116.889	0.271620
$58$	464.820	1.05231
$59$	572.797	1.26393	0.631964	−	0.774997i	$-0.282249\pi$
$59$	572.797	1.26393	0.631964	+	0.774997i	$0.282249\pi$
$60$	−450.433	−0.969178
$61$	−632.691	−1.32800	−0.663998	−	0.747734i	$-0.731142\pi$
$61$	−632.691	−1.32800	−0.663998	+	0.747734i	$0.731142\pi$
$62$	−196.866	−0.403258
$63$	237.005	0.473965
$64$	64.0000	0.125000
$65$	971.299	1.85346
$66$	−102.175	−0.190559
$67$	−183.461	−0.334527	−0.167264	−	0.985912i	$-0.553493\pi$
$67$	−183.461	−0.334527	−0.167264	+	0.985912i	$0.553493\pi$
$68$	297.124	0.529876
$69$	1008.80	1.76007
$70$	−799.815	−1.36566
$71$	56.6545	0.0946994	0.0473497	−	0.998878i	$-0.484922\pi$
$71$	56.6545	0.0946994	0.0473497	+	0.998878i	$0.484922\pi$
$72$	−86.7835	−0.142049
$73$	68.1521	0.109268	0.0546342	−	0.998506i	$-0.482601\pi$
$73$	68.1521	0.109268	0.0546342	+	0.998506i	$0.482601\pi$
$74$	−592.866	−0.931342
$75$	−1292.19	−1.98945
$76$	−76.0000	−0.114708
$77$	−181.428	−0.268515
$78$	652.912	0.947792
$79$	−332.820	−0.473989	−0.236995	−	0.971511i	$-0.576162\pi$
$79$	−332.820	−0.473989	−0.236995	+	0.971511i	$0.576162\pi$
$80$	292.866	0.409293
$81$	−904.217	−1.24035
$82$	869.825	1.17142
$83$	1152.91	1.52468	0.762341	−	0.647176i	$-0.224050\pi$
$83$	1152.91	1.52468	0.762341	+	0.647176i	$0.224050\pi$
$84$	−537.640	−0.698349
$85$	1359.65	1.73500
$86$	342.598	0.429574
$87$	1429.80	1.76196
$88$	66.4331	0.0804749
$89$	−368.479	−0.438862	−0.219431	−	0.975628i	$-0.570420\pi$
$89$	−368.479	−0.438862	−0.219431	+	0.975628i	$0.570420\pi$
$90$	−397.124	−0.465117
$91$	1159.35	1.33553
$92$	−655.908	−0.743294
$93$	−605.567	−0.675208
$94$	732.165	0.803373
$95$	−347.779	−0.375593
$96$	196.866	0.209298
$97$	−426.443	−0.446378	−0.223189	−	0.974775i	$-0.571647\pi$
$97$	−426.443	−0.446378	−0.223189	+	0.974775i	$0.571647\pi$
$98$	−268.664	−0.276931
$99$	−90.0827	−0.0914510
$100$	840.165	0.840165
$101$	−403.124	−0.397152	−0.198576	−	0.980086i	$-0.563632\pi$
$101$	−403.124	−0.397152	−0.198576	+	0.980086i	$0.563632\pi$
$102$	913.964	0.887214
$103$	−1135.68	−1.08642	−0.543211	−	0.839596i	$-0.682792\pi$
$103$	−1135.68	−1.08642	−0.543211	+	0.839596i	$0.682792\pi$
$104$	−424.516	−0.400262
$105$	−2460.26	−2.28663
$106$	−277.263	−0.254058
$107$	−380.096	−0.343414	−0.171707	−	0.985148i	$-0.554928\pi$
$107$	−380.096	−0.343414	−0.171707	+	0.985148i	$0.554928\pi$
$108$	397.474	0.354139
$109$	1180.74	1.03756	0.518782	−	0.854907i	$-0.326386\pi$
$109$	1180.74	1.03756	0.518782	+	0.854907i	$0.326386\pi$
$110$	304.000	0.263502
$111$	−1823.68	−1.55942
$112$	349.567	0.294919
$113$	−1132.51	−0.942807	−0.471404	−	0.881918i	$-0.656252\pi$
$113$	−1132.51	−0.942807	−0.471404	+	0.881918i	$0.656252\pi$
$114$	−233.779	−0.192065
$115$	−3001.45	−2.43380
$116$	−929.640	−0.744094
$117$	575.640	0.454854
$118$	−1145.59	−0.893733
$119$	1622.89	1.25017
$120$	900.866	0.685312
$121$	−1262.04	−0.948190
$122$	1265.38	0.939035
$123$	2675.61	1.96140
$124$	393.732	0.285147
$125$	1556.61	1.11382
$126$	−474.010	−0.335144
$127$	−40.0462	−0.0279806	−0.0139903	−	0.999902i	$-0.504453\pi$
$127$	−40.0462	−0.0279806	−0.0139903	+	0.999902i	$0.504453\pi$
$128$	−128.000	−0.0883883
$129$	1053.84	0.719270
$130$	−1942.60	−1.31059
$131$	−177.898	−0.118649	−0.0593244	−	0.998239i	$-0.518895\pi$
$131$	−177.898	−0.118649	−0.0593244	+	0.998239i	$0.518895\pi$
$132$	204.350	0.134746
$133$	−415.111	−0.270637
$134$	366.922	0.236547
$135$	1818.86	1.15957
$136$	−594.248	−0.374679
$137$	−24.7603	−0.0154410	−0.00772050	−	0.999970i	$-0.502458\pi$
$137$	−24.7603	−0.0154410	−0.00772050	+	0.999970i	$0.502458\pi$
$138$	−2017.59	−1.24456
$139$	2867.21	1.74959	0.874796	−	0.484491i	$-0.160995\pi$
$139$	2867.21	1.74959	0.874796	+	0.484491i	$0.160995\pi$
$140$	1599.63	0.965667
$141$	2252.17	1.34515
$142$	−113.309	−0.0669626
$143$	−440.655	−0.257688
$144$	173.567	0.100444
$145$	−4254.06	−2.43642
$146$	−136.304	−0.0772645
$147$	−826.421	−0.463687
$148$	1185.73	0.658558
$149$	1949.35	1.07179	0.535895	−	0.844285i	$-0.319974\pi$
$149$	1949.35	1.07179	0.535895	+	0.844285i	$0.319974\pi$
$150$	2584.38	1.40676
$151$	1120.99	0.604136	0.302068	−	0.953286i	$-0.402323\pi$
$151$	1120.99	0.604136	0.302068	+	0.953286i	$0.402323\pi$
$152$	152.000	0.0811107
$153$	805.795	0.425782
$154$	362.856	0.189869
$155$	1801.73	0.933669
$156$	−1305.82	−0.670190
$157$	−2360.23	−1.19979	−0.599894	−	0.800079i	$-0.704791\pi$
$157$	−2360.23	−1.19979	−0.599894	+	0.800079i	$0.704791\pi$
$158$	665.640	0.335161
$159$	−852.870	−0.425390
$160$	−585.732	−0.289414
$161$	−3582.56	−1.75370
$162$	1808.43	0.877061
$163$	861.825	0.414131	0.207065	−	0.978327i	$-0.433609\pi$
$163$	861.825	0.414131	0.207065	+	0.978327i	$0.433609\pi$
$164$	−1739.65	−0.828316
$165$	935.114	0.441203
$166$	−2305.82	−1.07811
$167$	1686.51	0.781472	0.390736	−	0.920503i	$-0.372221\pi$
$167$	1686.51	0.781472	0.390736	+	0.920503i	$0.372221\pi$
$168$	1075.28	0.493807
$169$	618.838	0.281674
$170$	−2719.30	−1.22683
$171$	−206.111	−0.0921736
$172$	−685.197	−0.303755
$173$	−3191.44	−1.40255	−0.701273	−	0.712893i	$-0.747384\pi$
$173$	−3191.44	−1.40255	−0.701273	+	0.712893i	$0.747384\pi$
$174$	−2859.60	−1.24590
$175$	4588.97	1.98225
$176$	−132.866	−0.0569043
$177$	−3523.88	−1.49645
$178$	736.959	0.310322
$179$	−1229.49	−0.513389	−0.256695	−	0.966493i	$-0.582633\pi$
$179$	−1229.49	−0.513389	−0.256695	+	0.966493i	$0.582633\pi$
$180$	794.248	0.328888
$181$	−3108.95	−1.27672	−0.638360	−	0.769738i	$-0.720387\pi$
$181$	−3108.95	−1.27672	−0.638360	+	0.769738i	$0.720387\pi$
$182$	−2318.70	−0.944359
$183$	3892.36	1.57230
$184$	1311.82	0.525589
$185$	5425.95	2.15635
$186$	1211.13	0.477444
$187$	−616.840	−0.241218
$188$	−1464.33	−0.568071
$189$	2171.00	0.835539
$190$	695.557	0.265584
$191$	−1415.48	−0.536233	−0.268117	−	0.963386i	$-0.586401\pi$
$191$	−1415.48	−0.536233	−0.268117	+	0.963386i	$0.586401\pi$
$192$	−393.732	−0.147996
$193$	1443.40	0.538333	0.269167	−	0.963094i	$-0.413252\pi$
$193$	1443.40	0.538333	0.269167	+	0.963094i	$0.413252\pi$
$194$	852.886	0.315637
$195$	−5975.50	−2.19443
$196$	537.329	0.195819
$197$	5271.92	1.90664	0.953322	−	0.301954i	$-0.0976389\pi$
$197$	5271.92	1.90664	0.953322	+	0.301954i	$0.0976389\pi$
$198$	180.165	0.0646656
$199$	−2510.19	−0.894183	−0.447091	−	0.894488i	$-0.647540\pi$
$199$	−2510.19	−0.894183	−0.447091	+	0.894488i	$0.647540\pi$
$200$	−1680.33	−0.594087
$201$	1128.67	0.396069
$202$	806.248	0.280829
$203$	−5077.68	−1.75558
$204$	−1827.93	−0.627355
$205$	−7960.70	−2.71219
$206$	2271.35	0.768217
$207$	−1778.81	−0.597275
$208$	849.032	0.283028
$209$	157.779	0.0522190
$210$	4920.52	1.61689
$211$	1854.44	0.605046	0.302523	−	0.953142i	$-0.402171\pi$
$211$	1854.44	0.605046	0.302523	+	0.953142i	$0.402171\pi$
$212$	554.526	0.179646
$213$	−348.542	−0.112121
$214$	760.192	0.242830
$215$	−3135.48	−0.994597
$216$	−794.949	−0.250414
$217$	2150.56	0.672763
$218$	−2361.48	−0.733668
$219$	−419.276	−0.129370
$220$	−608.000	−0.186324
$221$	3941.68	1.19976
$222$	3647.35	1.10268
$223$	1880.34	0.564649	0.282325	−	0.959319i	$-0.408895\pi$
$223$	1880.34	0.564649	0.282325	+	0.959319i	$0.408895\pi$
$224$	−699.134	−0.208539
$225$	2278.51	0.675115
$226$	2265.01	0.666665
$227$	1799.23	0.526075	0.263038	−	0.964786i	$-0.415276\pi$
$227$	1799.23	0.526075	0.263038	+	0.964786i	$0.415276\pi$
$228$	467.557	0.135810
$229$	4835.34	1.39532	0.697660	−	0.716429i	$-0.254224\pi$
$229$	4835.34	1.39532	0.697660	+	0.716429i	$0.254224\pi$
$230$	6002.91	1.72096
$231$	1116.16	0.317913
$232$	1859.28	0.526154
$233$	865.299	0.243295	0.121647	−	0.992573i	$-0.461182\pi$
$233$	865.299	0.243295	0.121647	+	0.992573i	$0.461182\pi$
$234$	−1151.28	−0.321630
$235$	−6700.83	−1.86006
$236$	2291.19	0.631964
$237$	2047.53	0.561187
$238$	−3245.77	−0.884001
$239$	−4764.27	−1.28943	−0.644717	−	0.764421i	$-0.723025\pi$
$239$	−4764.27	−1.28943	−0.644717	+	0.764421i	$0.723025\pi$
$240$	−1801.73	−0.484589
$241$	615.336	0.164470	0.0822350	−	0.996613i	$-0.473794\pi$
$241$	615.336	0.164470	0.0822350	+	0.996613i	$0.473794\pi$
$242$	2524.08	0.670472
$243$	2879.85	0.760257
$244$	−2530.76	−0.663998
$245$	2458.83	0.641180
$246$	−5351.22	−1.38692
$247$	−1008.22	−0.259724
$248$	−787.465	−0.201629
$249$	−7092.79	−1.80517
$250$	−3113.22	−0.787588
$251$	−1658.08	−0.416959	−0.208480	−	0.978027i	$-0.566852\pi$
$251$	−1658.08	−0.416959	−0.208480	+	0.978027i	$0.566852\pi$
$252$	948.020	0.236983
$253$	1361.69	0.338373
$254$	80.0925	0.0197852
$255$	−8364.66	−2.05418
$256$	256.000	0.0625000
$257$	3446.12	0.836432	0.418216	−	0.908348i	$-0.362656\pi$
$257$	3446.12	0.836432	0.418216	+	0.908348i	$0.362656\pi$
$258$	−2107.69	−0.508601
$259$	6476.45	1.55377
$260$	3885.20	0.926730
$261$	−2521.17	−0.597917
$262$	355.795	0.0838974
$263$	5755.80	1.34950	0.674748	−	0.738048i	$-0.264252\pi$
$263$	5755.80	1.34950	0.674748	+	0.738048i	$0.264252\pi$
$264$	−408.701	−0.0952795
$265$	2537.53	0.588223
$266$	830.221	0.191369
$267$	2266.91	0.519598
$268$	−733.844	−0.167264
$269$	2257.28	0.511631	0.255816	−	0.966726i	$-0.417656\pi$
$269$	2257.28	0.511631	0.255816	+	0.966726i	$0.417656\pi$
$270$	−3637.71	−0.819941
$271$	7012.13	1.57180	0.785898	−	0.618357i	$-0.212201\pi$
$271$	7012.13	1.57180	0.785898	+	0.618357i	$0.212201\pi$
$272$	1188.50	0.264938
$273$	−7132.39	−1.58122
$274$	49.5207	0.0109184
$275$	−1744.21	−0.382472
$276$	4035.19	0.880035
$277$	−372.810	−0.0808664	−0.0404332	−	0.999182i	$-0.512874\pi$
$277$	−372.810	−0.0808664	−0.0404332	+	0.999182i	$0.512874\pi$
$278$	−5734.41	−1.23715
$279$	1067.80	0.229130
$280$	−3199.26	−0.682830
$281$	−1888.96	−0.401017	−0.200508	−	0.979692i	$-0.564259\pi$
$281$	−1888.96	−0.401017	−0.200508	+	0.979692i	$0.564259\pi$
$282$	−4504.33	−0.951167
$283$	−3884.43	−0.815920	−0.407960	−	0.913000i	$-0.633760\pi$
$283$	−3884.43	−0.815920	−0.407960	+	0.913000i	$0.633760\pi$
$284$	226.618	0.0473497
$285$	2139.56	0.444689
$286$	881.309	0.182213
$287$	−9501.94	−1.95429
$288$	−347.134	−0.0710245
$289$	604.668	0.123075
$290$	8508.13	1.72281
$291$	2623.51	0.528497
$292$	272.608	0.0546342
$293$	−1273.01	−0.253823	−0.126911	−	0.991914i	$-0.540506\pi$
$293$	−1273.01	−0.253823	−0.126911	+	0.991914i	$0.540506\pi$
$294$	1652.84	0.327876
$295$	10484.5	2.06927
$296$	−2371.46	−0.465671
$297$	−825.170	−0.161216
$298$	−3898.69	−0.757869
$299$	−8701.35	−1.68298
$300$	−5168.75	−0.994727
$301$	−3742.53	−0.716665
$302$	−2241.97	−0.427188
$303$	2480.05	0.470214
$304$	−304.000	−0.0573539
$305$	−11580.9	−2.17416
$306$	−1611.59	−0.301074
$307$	819.153	0.152285	0.0761426	−	0.997097i	$-0.475740\pi$
$307$	819.153	0.152285	0.0761426	+	0.997097i	$0.475740\pi$
$308$	−725.713	−0.134258
$309$	6986.76	1.28629
$310$	−3603.46	−0.660203
$311$	2104.67	0.383745	0.191872	−	0.981420i	$-0.438544\pi$
$311$	2104.67	0.383745	0.191872	+	0.981420i	$0.438544\pi$
$312$	2611.65	0.473896
$313$	2395.47	0.432587	0.216293	−	0.976328i	$-0.430603\pi$
$313$	2395.47	0.432587	0.216293	+	0.976328i	$0.430603\pi$
$314$	4720.46	0.848378
$315$	4338.17	0.775962
$316$	−1331.28	−0.236995
$317$	2158.23	0.382393	0.191196	−	0.981552i	$-0.438763\pi$
$317$	2158.23	0.382393	0.191196	+	0.981552i	$0.438763\pi$
$318$	1705.74	0.300796
$319$	1929.96	0.338737
$320$	1171.46	0.204646
$321$	2338.38	0.406590
$322$	7165.11	1.24005
$323$	−1411.34	−0.243124
$324$	−3616.87	−0.620176
$325$	11145.7	1.90232
$326$	−1723.65	−0.292835
$327$	−7264.00	−1.22844
$328$	3479.30	0.585708
$329$	−7998.15	−1.34028
$330$	−1870.23	−0.311978
$331$	517.782	0.0859815	0.0429908	−	0.999075i	$-0.486311\pi$
$331$	517.782	0.0859815	0.0429908	+	0.999075i	$0.486311\pi$
$332$	4611.65	0.762341
$333$	3215.69	0.529185
$334$	−3373.01	−0.552584
$335$	−3358.10	−0.547679
$336$	−2150.56	−0.349174
$337$	6503.63	1.05126	0.525631	−	0.850713i	$-0.323829\pi$
$337$	6503.63	1.05126	0.525631	+	0.850713i	$0.323829\pi$
$338$	−1237.68	−0.199174
$339$	6967.25	1.11625
$340$	5438.60	0.867498
$341$	−817.402	−0.129809
$342$	412.221	0.0651766
$343$	−4558.96	−0.717670
$344$	1370.39	0.214787
$345$	18465.2	2.88154
$346$	6382.87	0.991749
$347$	6058.33	0.937257	0.468628	−	0.883395i	$-0.344748\pi$
$347$	6058.33	0.937257	0.468628	+	0.883395i	$0.344748\pi$
$348$	5719.21	0.880982
$349$	−10955.1	−1.68027	−0.840135	−	0.542377i	$-0.817524\pi$
$349$	−10955.1	−1.68027	−0.840135	+	0.542377i	$0.817524\pi$
$350$	−9177.94	−1.40166
$351$	5272.94	0.801849
$352$	265.732	0.0402374
$353$	−1806.43	−0.272369	−0.136185	−	0.990683i	$-0.543484\pi$
$353$	−1806.43	−0.272369	−0.136185	+	0.990683i	$0.543484\pi$
$354$	7047.77	1.05815
$355$	1037.01	0.155039
$356$	−1473.92	−0.219431
$357$	−9984.11	−1.48015
$358$	2458.99	0.363021
$359$	−7964.50	−1.17089	−0.585446	−	0.810711i	$-0.699081\pi$
$359$	−7964.50	−1.17089	−0.585446	+	0.810711i	$0.699081\pi$
$360$	−1588.50	−0.232559
$361$	361.000	0.0526316
$362$	6217.90	0.902777
$363$	7764.16	1.12263
$364$	4637.40	0.667763
$365$	1247.46	0.178891
$366$	−7784.71	−1.11179
$367$	−7311.58	−1.03995	−0.519974	−	0.854182i	$-0.674059\pi$
$367$	−7311.58	−1.03995	−0.519974	+	0.854182i	$0.674059\pi$
$368$	−2623.63	−0.371647
$369$	−4717.90	−0.665594
$370$	−10851.9	−1.52477
$371$	3028.81	0.423849
$372$	−2422.27	−0.337604
$373$	5518.38	0.766035	0.383017	−	0.923741i	$-0.374885\pi$
$373$	5518.38	0.766035	0.383017	+	0.923741i	$0.374885\pi$
$374$	1233.68	0.170567
$375$	−9576.36	−1.31872
$376$	2928.66	0.401687
$377$	−12332.7	−1.68479
$378$	−4342.00	−0.590815
$379$	−1139.97	−0.154502	−0.0772512	−	0.997012i	$-0.524614\pi$
$379$	−1139.97	−0.154502	−0.0772512	+	0.997012i	$0.524614\pi$
$380$	−1391.11	−0.187796
$381$	246.367	0.0331280
$382$	2830.96	0.379174
$383$	−10409.5	−1.38877	−0.694385	−	0.719604i	$-0.744323\pi$
$383$	−10409.5	−1.38877	−0.694385	+	0.719604i	$0.744323\pi$
$384$	787.465	0.104649
$385$	−3320.89	−0.439605
$386$	−2886.80	−0.380659
$387$	−1858.24	−0.244082
$388$	−1705.77	−0.223189
$389$	10471.2	1.36481	0.682404	−	0.730975i	$-0.260934\pi$
$389$	10471.2	1.36481	0.682404	+	0.730975i	$0.260934\pi$
$390$	11951.0	1.55170
$391$	−12180.4	−1.57542
$392$	−1074.66	−0.138465
$393$	1094.44	0.140476
$394$	−10543.8	−1.34820
$395$	−6091.98	−0.776002
$396$	−360.331	−0.0457255
$397$	−9588.68	−1.21220	−0.606098	−	0.795390i	$-0.707266\pi$
$397$	−9588.68	−1.21220	−0.606098	+	0.795390i	$0.707266\pi$
$398$	5020.37	0.632283
$399$	2553.79	0.320424
$400$	3360.66	0.420083
$401$	8549.30	1.06467	0.532334	−	0.846535i	$-0.321315\pi$
$401$	8549.30	1.06467	0.532334	+	0.846535i	$0.321315\pi$
$402$	−2257.33	−0.280063
$403$	5223.30	0.645635
$404$	−1612.50	−0.198576
$405$	−16550.9	−2.03067
$406$	10155.4	1.24138
$407$	−2461.62	−0.299798
$408$	3655.85	0.443607
$409$	266.960	0.0322746	0.0161373	−	0.999870i	$-0.494863\pi$
$409$	266.960	0.0322746	0.0161373	+	0.999870i	$0.494863\pi$
$410$	15921.4	1.91781
$411$	152.327	0.0182816
$412$	−4542.71	−0.543211
$413$	12514.4	1.49103
$414$	3557.62	0.422337
$415$	21103.1	2.49617
$416$	−1698.06	−0.200131
$417$	−17639.2	−2.07146
$418$	−315.557	−0.0369244
$419$	8643.83	1.00783	0.503913	−	0.863755i	$-0.331893\pi$
$419$	8643.83	1.00783	0.503913	+	0.863755i	$0.331893\pi$
$420$	−9841.03	−1.14332
$421$	−16801.3	−1.94500	−0.972499	−	0.232905i	$-0.925177\pi$
$421$	−16801.3	−1.94500	−0.972499	+	0.232905i	$0.925177\pi$
$422$	−3708.87	−0.427832
$423$	−3971.24	−0.456474
$424$	−1109.05	−0.127029
$425$	15602.1	1.78073
$426$	697.085	0.0792814
$427$	−13823.0	−1.56661
$428$	−1520.38	−0.171707
$429$	2710.94	0.305094
$430$	6270.97	0.703286
$431$	12053.5	1.34710	0.673548	−	0.739144i	$-0.264769\pi$
$431$	12053.5	1.34710	0.673548	+	0.739144i	$0.264769\pi$
$432$	1589.90	0.177069
$433$	9034.61	1.00271	0.501357	−	0.865240i	$-0.332834\pi$
$433$	9034.61	1.00271	0.501357	+	0.865240i	$0.332834\pi$
$434$	−4301.12	−0.475715
$435$	26171.3	2.88464
$436$	4722.96	0.518782
$437$	3115.56	0.341047
$438$	838.552	0.0914785
$439$	3008.87	0.327120	0.163560	−	0.986533i	$-0.447702\pi$
$439$	3008.87	0.327120	0.163560	+	0.986533i	$0.447702\pi$
$440$	1216.00	0.131751
$441$	1457.23	0.157351
$442$	−7883.37	−0.848356
$443$	−229.594	−0.0246237	−0.0123119	−	0.999924i	$-0.503919\pi$
$443$	−229.594	−0.0246237	−0.0123119	+	0.999924i	$0.503919\pi$
$444$	−7294.71	−0.779710
$445$	−6744.70	−0.718493
$446$	−3760.68	−0.399267
$447$	−11992.5	−1.26896
$448$	1398.27	0.147460
$449$	−7559.44	−0.794548	−0.397274	−	0.917700i	$-0.630044\pi$
$449$	−7559.44	−0.794548	−0.397274	+	0.917700i	$0.630044\pi$
$450$	−4557.03	−0.477379
$451$	3611.57	0.377078
$452$	−4530.02	−0.471404
$453$	−6896.38	−0.715276
$454$	−3598.46	−0.371991
$455$	21220.9	2.18648
$456$	−935.114	−0.0960323
$457$	11556.4	1.18290	0.591449	−	0.806343i	$-0.298556\pi$
$457$	11556.4	1.18290	0.591449	+	0.806343i	$0.298556\pi$
$458$	−9670.69	−0.986641
$459$	7381.20	0.750599
$460$	−12005.8	−1.21690
$461$	9191.58	0.928622	0.464311	−	0.885672i	$-0.346302\pi$
$461$	9191.58	0.928622	0.464311	+	0.885672i	$0.346302\pi$
$462$	−2232.32	−0.224798
$463$	1356.03	0.136113	0.0680564	−	0.997681i	$-0.478320\pi$
$463$	1356.03	0.136113	0.0680564	+	0.997681i	$0.478320\pi$
$464$	−3718.56	−0.372047
$465$	−11084.4	−1.10543
$466$	−1730.60	−0.172035
$467$	−14808.9	−1.46740	−0.733700	−	0.679473i	$-0.762208\pi$
$467$	−14808.9	−1.46740	−0.733700	+	0.679473i	$0.762208\pi$
$468$	2302.56	0.227427
$469$	−4008.25	−0.394635
$470$	13401.7	1.31526
$471$	14520.3	1.42051
$472$	−4582.37	−0.446866
$473$	1422.49	0.138280
$474$	−4095.06	−0.396819
$475$	−3990.79	−0.385494
$476$	6491.55	0.625083
$477$	1503.86	0.144355
$478$	9528.54	0.911768
$479$	9834.66	0.938115	0.469058	−	0.883168i	$-0.344594\pi$
$479$	9834.66	0.938115	0.469058	+	0.883168i	$0.344594\pi$
$480$	3603.46	0.342656
$481$	15730.1	1.49112
$482$	−1230.67	−0.116298
$483$	22040.1	2.07632
$484$	−5048.17	−0.474095
$485$	−7805.67	−0.730798
$486$	−5759.70	−0.537583
$487$	3687.82	0.343144	0.171572	−	0.985172i	$-0.445115\pi$
$487$	3687.82	0.343144	0.171572	+	0.985172i	$0.445115\pi$
$488$	5061.53	0.469518
$489$	−5302.00	−0.490317
$490$	−4917.67	−0.453383
$491$	−11197.4	−1.02919	−0.514593	−	0.857435i	$-0.672057\pi$
$491$	−11197.4	−1.02919	−0.514593	+	0.857435i	$0.672057\pi$
$492$	10702.4	0.980698
$493$	−17263.6	−1.57711
$494$	2016.45	0.183653
$495$	−1648.89	−0.149721
$496$	1574.93	0.142573
$497$	1237.78	0.111715
$498$	14185.6	1.27645
$499$	−12101.6	−1.08566	−0.542829	−	0.839843i	$-0.682647\pi$
$499$	−12101.6	−1.08566	−0.542829	+	0.839843i	$0.682647\pi$
$500$	6226.43	0.556909
$501$	−10375.5	−0.925236
$502$	3316.15	0.294835
$503$	−7266.58	−0.644136	−0.322068	−	0.946716i	$-0.604378\pi$
$503$	−7266.58	−0.644136	−0.322068	+	0.946716i	$0.604378\pi$
$504$	−1896.04	−0.167572
$505$	−7378.84	−0.650206
$506$	−2723.37	−0.239266
$507$	−3807.13	−0.333493
$508$	−160.185	−0.0139903
$509$	2564.99	0.223362	0.111681	−	0.993744i	$-0.464376\pi$
$509$	2564.99	0.223362	0.111681	+	0.993744i	$0.464376\pi$
$510$	16729.3	1.45252
$511$	1488.98	0.128902
$512$	−512.000	−0.0441942
$513$	−1888.00	−0.162490
$514$	−6892.24	−0.591447
$515$	−20787.6	−1.77866
$516$	4215.38	0.359635
$517$	3040.00	0.258606
$518$	−12952.9	−1.09868
$519$	19633.9	1.66057
$520$	−7770.39	−0.655297
$521$	11254.6	0.946401	0.473201	−	0.880955i	$-0.343099\pi$
$521$	11254.6	0.946401	0.473201	+	0.880955i	$0.343099\pi$
$522$	5042.34	0.422791
$523$	−18357.1	−1.53480	−0.767401	−	0.641168i	$-0.778450\pi$
$523$	−18357.1	−1.53480	−0.767401	+	0.641168i	$0.778450\pi$
$524$	−711.591	−0.0593244
$525$	−28231.6	−2.34691
$526$	−11511.6	−0.954238
$527$	7311.71	0.604370
$528$	817.402	0.0673728
$529$	14721.4	1.20995
$530$	−5075.06	−0.415936
$531$	6213.66	0.507815
$532$	−1660.44	−0.135318
$533$	−23078.4	−1.87549
$534$	−4533.82	−0.367411
$535$	−6957.33	−0.562227
$536$	1467.69	0.118273
$537$	7563.93	0.607836
$538$	−4514.56	−0.361778
$539$	−1115.51	−0.0891438
$540$	7275.43	0.579786
$541$	11381.8	0.904510	0.452255	−	0.891889i	$-0.350620\pi$
$541$	11381.8	0.904510	0.452255	+	0.891889i	$0.350620\pi$
$542$	−14024.3	−1.11143
$543$	19126.5	1.51159
$544$	−2376.99	−0.187340
$545$	21612.4	1.69867
$546$	14264.8	1.11809
$547$	−8996.84	−0.703248	−0.351624	−	0.936141i	$-0.614371\pi$
$547$	−8996.84	−0.703248	−0.351624	+	0.936141i	$0.614371\pi$
$548$	−99.0413	−0.00772050
$549$	−6863.39	−0.533556
$550$	3488.42	0.270449
$551$	4415.79	0.341414
$552$	−8070.37	−0.622279
$553$	−7271.43	−0.559155
$554$	745.620	0.0571812
$555$	−33380.8	−2.55304
$556$	11468.8	0.874796
$557$	10759.8	0.818507	0.409253	−	0.912421i	$-0.365789\pi$
$557$	10759.8	0.818507	0.409253	+	0.912421i	$0.365789\pi$
$558$	−2135.59	−0.162019
$559$	−9089.90	−0.687767
$560$	6398.52	0.482834
$561$	3794.84	0.285594
$562$	3777.91	0.283562
$563$	25381.8	1.90003	0.950015	−	0.312205i	$-0.101068\pi$
$563$	25381.8	1.90003	0.950015	+	0.312205i	$0.101068\pi$
$564$	9008.66	0.672576
$565$	−20729.5	−1.54354
$566$	7768.86	0.576943
$567$	−19755.3	−1.46322
$568$	−453.236	−0.0334813
$569$	−5546.00	−0.408612	−0.204306	−	0.978907i	$-0.565494\pi$
$569$	−5546.00	−0.408612	−0.204306	+	0.978907i	$0.565494\pi$
$570$	−4279.11	−0.314443
$571$	7714.96	0.565431	0.282715	−	0.959204i	$-0.408765\pi$
$571$	7714.96	0.565431	0.282715	+	0.959204i	$0.408765\pi$
$572$	−1762.62	−0.128844
$573$	8708.13	0.634882
$574$	19003.9	1.38189
$575$	−34441.9	−2.49796
$576$	694.268	0.0502219
$577$	21335.1	1.53933	0.769665	−	0.638448i	$-0.220423\pi$
$577$	21335.1	1.53933	0.769665	+	0.638448i	$0.220423\pi$
$578$	−1209.34	−0.0870273
$579$	−8879.90	−0.637368
$580$	−17016.3	−1.21821
$581$	25188.8	1.79863
$582$	−5247.01	−0.373704
$583$	−1151.21	−0.0817811
$584$	−545.217	−0.0386322
$585$	10536.6	0.744674
$586$	2546.02	0.179480
$587$	12370.9	0.869846	0.434923	−	0.900468i	$-0.356776\pi$
$587$	12370.9	0.869846	0.434923	+	0.900468i	$0.356776\pi$
$588$	−3305.68	−0.231844
$589$	−1870.23	−0.130834
$590$	−20969.1	−1.46319
$591$	−32433.2	−2.25740
$592$	4742.93	0.329279
$593$	12982.5	0.899034	0.449517	−	0.893272i	$-0.351596\pi$
$593$	12982.5	0.899034	0.449517	+	0.893272i	$0.351596\pi$
$594$	1650.34	0.113997
$595$	29705.5	2.04674
$596$	7797.38	0.535895
$597$	15442.8	1.05868
$598$	17402.7	1.19005
$599$	13163.7	0.897921	0.448960	−	0.893552i	$-0.351794\pi$
$599$	13163.7	0.897921	0.448960	+	0.893552i	$0.351794\pi$
$600$	10337.5	0.703378
$601$	−29325.4	−1.99036	−0.995180	−	0.0980640i	$-0.968735\pi$
$601$	−29325.4	−1.99036	−0.995180	+	0.0980640i	$0.968735\pi$
$602$	7485.07	0.506758
$603$	−1990.17	−0.134405
$604$	4483.94	0.302068
$605$	−23100.6	−1.55235
$606$	−4960.09	−0.332492
$607$	−2170.11	−0.145110	−0.0725552	−	0.997364i	$-0.523115\pi$
$607$	−2170.11	−0.145110	−0.0725552	+	0.997364i	$0.523115\pi$
$608$	608.000	0.0405554
$609$	31238.2	2.07855
$610$	23161.7	1.53736
$611$	−19426.0	−1.28624
$612$	3223.18	0.212891
$613$	−4917.96	−0.324036	−0.162018	−	0.986788i	$-0.551800\pi$
$613$	−4917.96	−0.324036	−0.162018	+	0.986788i	$0.551800\pi$
$614$	−1638.31	−0.107682
$615$	48974.7	3.21114
$616$	1451.43	0.0949344
$617$	24763.5	1.61579	0.807894	−	0.589328i	$-0.200607\pi$
$617$	24763.5	1.61579	0.807894	+	0.589328i	$0.200607\pi$
$618$	−13973.5	−0.909542
$619$	−27552.0	−1.78902	−0.894512	−	0.447043i	$-0.852477\pi$
$619$	−27552.0	−1.78902	−0.894512	+	0.447043i	$0.852477\pi$
$620$	7206.93	0.466834
$621$	−16294.2	−1.05292
$622$	−4209.33	−0.271348
$623$	−8050.51	−0.517716
$624$	−5223.30	−0.335095
$625$	2237.20	0.143181
$626$	−4790.93	−0.305885
$627$	−970.664	−0.0618255
$628$	−9440.91	−0.599894
$629$	22019.3	1.39582
$630$	−8676.34	−0.548688
$631$	377.839	0.0238376	0.0119188	−	0.999929i	$-0.496206\pi$
$631$	377.839	0.0238376	0.0119188	+	0.999929i	$0.496206\pi$
$632$	2662.56	0.167581
$633$	−11408.6	−0.716354
$634$	−4316.47	−0.270393
$635$	−733.012	−0.0458090
$636$	−3411.48	−0.212695
$637$	7128.27	0.443379
$638$	−3859.93	−0.239523
$639$	614.584	0.0380479
$640$	−2342.93	−0.144707
$641$	−14047.3	−0.865576	−0.432788	−	0.901496i	$-0.642470\pi$
$641$	−14047.3	−0.865576	−0.432788	+	0.901496i	$0.642470\pi$
$642$	−4676.75	−0.287503
$643$	5768.94	0.353818	0.176909	−	0.984227i	$-0.443390\pi$
$643$	5768.94	0.353818	0.176909	+	0.984227i	$0.443390\pi$
$644$	−14330.2	−0.876848
$645$	19289.7	1.17757
$646$	2822.68	0.171915
$647$	4568.59	0.277604	0.138802	−	0.990320i	$-0.455675\pi$
$647$	4568.59	0.277604	0.138802	+	0.990320i	$0.455675\pi$
$648$	7233.73	0.438531
$649$	−4756.58	−0.287692
$650$	−22291.5	−1.34514
$651$	−13230.4	−0.796528
$652$	3447.30	0.207065
$653$	−16532.7	−0.990774	−0.495387	−	0.868672i	$-0.664974\pi$
$653$	−16532.7	−0.990774	−0.495387	+	0.868672i	$0.664974\pi$
$654$	14528.0	0.868638
$655$	−3256.26	−0.194248
$656$	−6958.60	−0.414158
$657$	739.309	0.0439014
$658$	15996.3	0.947721
$659$	−18630.0	−1.10125	−0.550624	−	0.834753i	$-0.685610\pi$
$659$	−18630.0	−1.10125	−0.550624	+	0.834753i	$0.685610\pi$
$660$	3740.46	0.220602
$661$	28851.3	1.69771	0.848854	−	0.528627i	$-0.177293\pi$
$661$	28851.3	1.69771	0.848854	+	0.528627i	$0.177293\pi$
$662$	−1035.56	−0.0607981
$663$	−24249.5	−1.42047
$664$	−9223.30	−0.539056
$665$	−7598.24	−0.443079
$666$	−6431.37	−0.374190
$667$	38109.9	2.21232
$668$	6746.02	0.390736
$669$	−11568.0	−0.668525
$670$	6716.19	0.387267
$671$	5253.95	0.302275
$672$	4301.12	0.246904
$673$	−3873.41	−0.221856	−0.110928	−	0.993828i	$-0.535382\pi$
$673$	−3873.41	−0.221856	−0.110928	+	0.993828i	$0.535382\pi$
$674$	−13007.3	−0.743354
$675$	20871.5	1.19014
$676$	2475.35	0.140837
$677$	5025.24	0.285282	0.142641	−	0.989775i	$-0.454441\pi$
$677$	5025.24	0.285282	0.142641	+	0.989775i	$0.454441\pi$
$678$	−13934.5	−0.789309
$679$	−9316.90	−0.526583
$680$	−10877.2	−0.613414
$681$	−11069.0	−0.622855
$682$	1634.80	0.0917886
$683$	−14157.0	−0.793122	−0.396561	−	0.918008i	$-0.629796\pi$
$683$	−14157.0	−0.793122	−0.396561	+	0.918008i	$0.629796\pi$
$684$	−824.443	−0.0460868
$685$	−453.217	−0.0252796
$686$	9117.92	0.507469
$687$	−29747.4	−1.65201
$688$	−2740.79	−0.151877
$689$	7356.40	0.406758
$690$	−36930.3	−2.03755
$691$	−2437.65	−0.134200	−0.0671002	−	0.997746i	$-0.521375\pi$
$691$	−2437.65	−0.134200	−0.0671002	+	0.997746i	$0.521375\pi$
$692$	−12765.7	−0.701273
$693$	−1968.12	−0.107883
$694$	−12116.7	−0.662741
$695$	52481.7	2.86438
$696$	−11438.4	−0.622948
$697$	−32305.7	−1.75562
$698$	21910.2	1.18813
$699$	−5323.38	−0.288052
$700$	18355.9	0.991124
$701$	9496.96	0.511691	0.255845	−	0.966718i	$-0.417646\pi$
$701$	9496.96	0.511691	0.255845	+	0.966718i	$0.417646\pi$
$702$	−10545.9	−0.566993
$703$	−5632.23	−0.302167
$704$	−531.465	−0.0284522
$705$	41223.9	2.20225
$706$	3612.85	0.192594
$707$	−8807.43	−0.468511
$708$	−14095.5	−0.748224
$709$	6350.33	0.336377	0.168189	−	0.985755i	$-0.446208\pi$
$709$	6350.33	0.336377	0.168189	+	0.985755i	$0.446208\pi$
$710$	−2074.02	−0.109629
$711$	−3610.41	−0.190437
$712$	2947.83	0.155161
$713$	−16140.7	−0.847792
$714$	19968.2	1.04663
$715$	−8065.80	−0.421879
$716$	−4917.98	−0.256695
$717$	29310.1	1.52665
$718$	15929.0	0.827946
$719$	−19722.6	−1.02299	−0.511496	−	0.859286i	$-0.670908\pi$
$719$	−19722.6	−1.02299	−0.511496	+	0.859286i	$0.670908\pi$
$720$	3176.99	0.164444
$721$	−24812.2	−1.28163
$722$	−722.000	−0.0372161
$723$	−3785.59	−0.194727
$724$	−12435.8	−0.638360
$725$	−48815.7	−2.50065
$726$	−15528.3	−0.793816
$727$	28325.8	1.44504	0.722520	−	0.691350i	$-0.242984\pi$
$727$	28325.8	1.44504	0.722520	+	0.691350i	$0.242984\pi$
$728$	−9274.79	−0.472179
$729$	6696.82	0.340234
$730$	−2494.93	−0.126495
$731$	−12724.3	−0.643809
$732$	15569.4	0.786151
$733$	33981.8	1.71234	0.856170	−	0.516694i	$-0.172837\pi$
$733$	33981.8	1.71234	0.856170	+	0.516694i	$0.172837\pi$
$734$	14623.2	0.735355
$735$	−15126.9	−0.759135
$736$	5247.26	0.262794
$737$	1523.49	0.0761443
$738$	9435.80	0.470646
$739$	9147.10	0.455320	0.227660	−	0.973741i	$-0.426893\pi$
$739$	9147.10	0.455320	0.227660	+	0.973741i	$0.426893\pi$
$740$	21703.8	1.07817
$741$	6202.67	0.307504
$742$	−6057.62	−0.299706
$743$	−34159.2	−1.68665	−0.843324	−	0.537405i	$-0.819404\pi$
$743$	−34159.2	−1.68665	−0.843324	+	0.537405i	$0.819404\pi$
$744$	4844.54	0.238722
$745$	35681.1	1.75470
$746$	−11036.8	−0.541668
$747$	12506.7	0.612579
$748$	−2467.36	−0.120609
$749$	−8304.31	−0.405117
$750$	19152.7	0.932478
$751$	28361.8	1.37808	0.689038	−	0.724725i	$-0.258033\pi$
$751$	28361.8	1.37808	0.689038	+	0.724725i	$0.258033\pi$
$752$	−5857.32	−0.284035
$753$	10200.6	0.493666
$754$	24665.4	1.19133
$755$	20518.7	0.989074
$756$	8683.99	0.417770
$757$	11464.9	0.550462	0.275231	−	0.961378i	$-0.411246\pi$
$757$	11464.9	0.550462	0.275231	+	0.961378i	$0.411246\pi$
$758$	2279.94	0.109250
$759$	−8377.18	−0.400623
$760$	2782.23	0.132792
$761$	14289.3	0.680666	0.340333	−	0.940305i	$-0.389460\pi$
$761$	14289.3	0.680666	0.340333	+	0.940305i	$0.389460\pi$
$762$	−492.734	−0.0234250
$763$	25796.7	1.22399
$764$	−5661.92	−0.268117
$765$	14749.4	0.697079
$766$	20818.9	0.982008
$767$	30395.2	1.43091
$768$	−1574.93	−0.0739979
$769$	100.811	0.00472734	0.00236367	−	0.999997i	$-0.499248\pi$
$769$	100.811	0.00472734	0.00236367	+	0.999997i	$0.499248\pi$
$770$	6641.77	0.310848
$771$	−21200.8	−0.990307
$772$	5773.61	0.269167
$773$	52.1073	0.00242454	0.00121227	−	0.999999i	$-0.499614\pi$
$773$	52.1073	0.00242454	0.00121227	+	0.999999i	$0.499614\pi$
$774$	3716.49	0.172592
$775$	20675.0	0.958282
$776$	3411.54	0.157819
$777$	−39843.6	−1.83961
$778$	−20942.4	−0.965065
$779$	8263.34	0.380057
$780$	−23902.0	−1.09722
$781$	−470.467	−0.0215552
$782$	24360.7	1.11399
$783$	−23094.3	−1.05405
$784$	2149.31	0.0979097
$785$	−43201.9	−1.96426
$786$	−2188.88	−0.0993316
$787$	15261.8	0.691266	0.345633	−	0.938370i	$-0.387664\pi$
$787$	15261.8	0.691266	0.345633	+	0.938370i	$0.387664\pi$
$788$	21087.7	0.953322
$789$	−35410.0	−1.59776
$790$	12184.0	0.548716
$791$	−24742.9	−1.11221
$792$	720.662	0.0323328
$793$	−33573.4	−1.50344
$794$	19177.4	0.857152
$795$	−15611.0	−0.696436
$796$	−10040.7	−0.447091
$797$	24432.3	1.08587	0.542934	−	0.839775i	$-0.317313\pi$
$797$	24432.3	1.08587	0.542934	+	0.839775i	$0.317313\pi$
$798$	−5107.58	−0.226574
$799$	−27193.0	−1.20403
$800$	−6721.32	−0.297043
$801$	−3997.24	−0.176324
$802$	−17098.6	−0.752834
$803$	−565.944	−0.0248714
$804$	4514.66	0.198035
$805$	−65575.6	−2.87110
$806$	−10446.6	−0.456533
$807$	−13886.9	−0.605754
$808$	3224.99	0.140414
$809$	3635.86	0.158010	0.0790050	−	0.996874i	$-0.474826\pi$
$809$	3635.86	0.158010	0.0790050	+	0.996874i	$0.474826\pi$
$810$	33101.8	1.43590
$811$	1087.93	0.0471051	0.0235526	−	0.999723i	$-0.492502\pi$
$811$	1087.93	0.0471051	0.0235526	+	0.999723i	$0.492502\pi$
$812$	−20310.7	−0.877791
$813$	−43139.1	−1.86095
$814$	4923.24	0.211990
$815$	15775.0	0.678003
$816$	−7311.71	−0.313678
$817$	3254.69	0.139372
$818$	−533.920	−0.0228216
$819$	12576.5	0.536581
$820$	−31842.8	−1.35610
$821$	10953.7	0.465636	0.232818	−	0.972520i	$-0.425205\pi$
$821$	10953.7	0.465636	0.232818	+	0.972520i	$0.425205\pi$
$822$	−304.655	−0.0129271
$823$	13502.5	0.571893	0.285947	−	0.958246i	$-0.407692\pi$
$823$	13502.5	0.571893	0.285947	+	0.958246i	$0.407692\pi$
$824$	9085.41	0.384108
$825$	10730.5	0.452834
$826$	−25028.9	−1.05432
$827$	26812.4	1.12740	0.563699	−	0.825980i	$-0.309378\pi$
$827$	26812.4	1.12740	0.563699	+	0.825980i	$0.309378\pi$
$828$	−7115.24	−0.298637
$829$	−4497.94	−0.188444	−0.0942218	−	0.995551i	$-0.530036\pi$
$829$	−4497.94	−0.188444	−0.0942218	+	0.995551i	$0.530036\pi$
$830$	−42206.1	−1.76506
$831$	2293.55	0.0957430
$832$	3396.13	0.141514
$833$	9978.33	0.415040
$834$	35278.5	1.46474
$835$	30870.0	1.27940
$836$	631.114	0.0261095
$837$	9781.16	0.403926
$838$	−17287.7	−0.712640
$839$	30324.9	1.24783	0.623917	−	0.781491i	$-0.285540\pi$
$839$	30324.9	1.24783	0.623917	+	0.781491i	$0.285540\pi$
$840$	19682.1	0.808447
$841$	29625.4	1.21470
$842$	33602.6	1.37532
$843$	11621.0	0.474790
$844$	7417.75	0.302523
$845$	11327.3	0.461149
$846$	7942.48	0.322776
$847$	−27573.0	−1.11856
$848$	2218.10	0.0898230
$849$	23897.3	0.966022
$850$	−31204.2	−1.25917
$851$	−48608.2	−1.95801
$852$	−1394.17	−0.0560604
$853$	17230.8	0.691642	0.345821	−	0.938300i	$-0.387600\pi$
$853$	17230.8	0.691642	0.345821	+	0.938300i	$0.387600\pi$
$854$	27646.0	1.10776
$855$	−3772.68	−0.150904
$856$	3040.77	0.121415
$857$	−6724.12	−0.268018	−0.134009	−	0.990980i	$-0.542785\pi$
$857$	−6724.12	−0.268018	−0.134009	+	0.990980i	$0.542785\pi$
$858$	−5421.87	−0.215734
$859$	−36183.5	−1.43721	−0.718606	−	0.695417i	$-0.755219\pi$
$859$	−36183.5	−1.43721	−0.718606	+	0.695417i	$0.755219\pi$
$860$	−12541.9	−0.497298
$861$	58456.6	2.31381
$862$	−24107.1	−0.952541
$863$	14350.8	0.566056	0.283028	−	0.959112i	$-0.408661\pi$
$863$	14350.8	0.566056	0.283028	+	0.959112i	$0.408661\pi$
$864$	−3179.80	−0.125207
$865$	−58416.5	−2.29621
$866$	−18069.2	−0.709027
$867$	−3719.96	−0.145717
$868$	8602.24	0.336381
$869$	2763.78	0.107888
$870$	−52342.6	−2.03975
$871$	−9735.27	−0.378722
$872$	−9445.93	−0.366834
$873$	−4626.02	−0.179344
$874$	−6231.12	−0.241157
$875$	34008.7	1.31395
$876$	−1677.10	−0.0646851
$877$	−42978.6	−1.65483	−0.827414	−	0.561592i	$-0.810189\pi$
$877$	−42978.6	−1.65483	−0.827414	+	0.561592i	$0.810189\pi$
$878$	−6017.74	−0.231309
$879$	7831.65	0.300518
$880$	−2432.00	−0.0931622
$881$	−7298.75	−0.279116	−0.139558	−	0.990214i	$-0.544568\pi$
$881$	−7298.75	−0.279116	−0.139558	+	0.990214i	$0.544568\pi$
$882$	−2914.45	−0.111264
$883$	46722.1	1.78066	0.890331	−	0.455314i	$-0.150473\pi$
$883$	46722.1	1.78066	0.890331	+	0.455314i	$0.150473\pi$
$884$	15766.7	0.599878
$885$	−64501.7	−2.44994
$886$	459.187	0.0174116
$887$	35268.7	1.33507	0.667535	−	0.744578i	$-0.267349\pi$
$887$	35268.7	1.33507	0.667535	+	0.744578i	$0.267349\pi$
$888$	14589.4	0.551338
$889$	−874.928	−0.0330080
$890$	13489.4	0.508051
$891$	7508.74	0.282326
$892$	7521.35	0.282325
$893$	6955.57	0.260649
$894$	23985.0	0.897292
$895$	−22504.8	−0.840506
$896$	−2796.54	−0.104270
$897$	53531.3	1.99259
$898$	15118.9	0.561830
$899$	−22876.8	−0.848704
$900$	9114.06	0.337558
$901$	10297.7	0.380761
$902$	−7223.14	−0.266635
$903$	23024.3	0.848507
$904$	9060.05	0.333333
$905$	−56906.6	−2.09021
$906$	13792.8	0.505777
$907$	−46292.7	−1.69474	−0.847368	−	0.531007i	$-0.821814\pi$
$907$	−46292.7	−1.69474	−0.847368	+	0.531007i	$0.821814\pi$
$908$	7196.92	0.263038
$909$	−4373.06	−0.159566
$910$	−42441.8	−1.54608
$911$	−42085.5	−1.53058	−0.765288	−	0.643688i	$-0.777403\pi$
$911$	−42085.5	−1.53058	−0.765288	+	0.643688i	$0.777403\pi$
$912$	1870.23	0.0679051
$913$	−9573.94	−0.347044
$914$	−23112.7	−0.836435
$915$	71246.2	2.57413
$916$	19341.4	0.697660
$917$	−3886.70	−0.139967
$918$	−14762.4	−0.530754
$919$	21331.8	0.765691	0.382845	−	0.923812i	$-0.374944\pi$
$919$	21331.8	0.765691	0.382845	+	0.923812i	$0.374944\pi$
$920$	24011.6	0.860479
$921$	−5039.49	−0.180300
$922$	−18383.2	−0.656635
$923$	3006.34	0.107210
$924$	4464.63	0.158956
$925$	62263.2	2.21319
$926$	−2712.07	−0.0962463
$927$	−12319.7	−0.436498
$928$	7437.12	0.263077
$929$	−34331.5	−1.21247	−0.606233	−	0.795287i	$-0.707320\pi$
$929$	−34331.5	−1.21247	−0.606233	+	0.795287i	$0.707320\pi$
$930$	22168.8	0.781658
$931$	−2552.31	−0.0898481
$932$	3461.20	0.121647
$933$	−12948.0	−0.454341
$934$	29617.9	1.03761
$935$	−11290.7	−0.394915
$936$	−4605.12	−0.160815
$937$	13625.1	0.475041	0.237521	−	0.971383i	$-0.423665\pi$
$937$	13625.1	0.475041	0.237521	+	0.971383i	$0.423665\pi$
$938$	8016.49	0.279049
$939$	−14737.1	−0.512168
$940$	−26803.3	−0.930029
$941$	7086.48	0.245497	0.122748	−	0.992438i	$-0.460829\pi$
$941$	7086.48	0.245497	0.122748	+	0.992438i	$0.460829\pi$
$942$	−29040.6	−1.00445
$943$	71315.6	2.46273
$944$	9164.75	0.315982
$945$	39738.3	1.36792
$946$	−2844.98	−0.0977784
$947$	38735.9	1.32920	0.664598	−	0.747202i	$-0.268603\pi$
$947$	38735.9	1.32920	0.664598	+	0.747202i	$0.268603\pi$
$948$	8190.12	0.280594
$949$	3616.45	0.123704
$950$	7981.57	0.272586
$951$	−13277.6	−0.452740
$952$	−12983.1	−0.442000
$953$	30964.3	1.05250	0.526249	−	0.850330i	$-0.323598\pi$
$953$	30964.3	1.05250	0.526249	+	0.850330i	$0.323598\pi$
$954$	−3007.73	−0.102074
$955$	−25909.1	−0.877906
$956$	−19057.1	−0.644717
$957$	−11873.3	−0.401053
$958$	−19669.3	−0.663347
$959$	−540.962	−0.0182154
$960$	−7206.93	−0.242294
$961$	−20101.9	−0.674765
$962$	−31460.1	−1.05438
$963$	−4123.26	−0.137975
$964$	2461.34	0.0822350
$965$	26420.2	0.881344
$966$	−44080.2	−1.46818
$967$	10968.0	0.364743	0.182371	−	0.983230i	$-0.441623\pi$
$967$	10968.0	0.364743	0.182371	+	0.983230i	$0.441623\pi$
$968$	10096.3	0.335236
$969$	8682.65	0.287850
$970$	15611.3	0.516752
$971$	−30081.2	−0.994183	−0.497091	−	0.867698i	$-0.665599\pi$
$971$	−30081.2	−0.994183	−0.497091	+	0.867698i	$0.665599\pi$
$972$	11519.4	0.380128
$973$	62642.5	2.06395
$974$	−7375.64	−0.242640
$975$	−68569.3	−2.25228
$976$	−10123.1	−0.331999
$977$	26628.1	0.871962	0.435981	−	0.899956i	$-0.356401\pi$
$977$	26628.1	0.871962	0.435981	+	0.899956i	$0.356401\pi$
$978$	10604.0	0.346706
$979$	3059.90	0.0998926
$980$	9835.34	0.320590
$981$	12808.6	0.416867
$982$	22394.7	0.727744
$983$	25495.3	0.827236	0.413618	−	0.910450i	$-0.364265\pi$
$983$	25495.3	0.827236	0.413618	+	0.910450i	$0.364265\pi$
$984$	−21404.9	−0.693458
$985$	96498.0	3.12150
$986$	34527.3	1.11519
$987$	49205.2	1.58685
$988$	−4032.90	−0.129862
$989$	28089.1	0.903116
$990$	3297.77	0.105869
$991$	−54385.4	−1.74330	−0.871649	−	0.490130i	$-0.836949\pi$
$991$	−54385.4	−1.74330	−0.871649	+	0.490130i	$0.836949\pi$
$992$	−3149.86	−0.100815
$993$	−3185.43	−0.101799
$994$	−2475.57	−0.0789942
$995$	−45946.8	−1.46393
$996$	−28371.2	−0.902586
$997$	−36846.5	−1.17045	−0.585227	−	0.810870i	$-0.698994\pi$
$997$	−36846.5	−1.17045	−0.585227	+	0.810870i	$0.698994\pi$
$998$	24203.3	0.767677
$999$	29456.1	0.932884

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.4.a.b.1.1	✓	2
3.2	odd	2		342.4.a.k.1.1		2
4.3	odd	2		304.4.a.d.1.2		2
5.2	odd	4		950.4.b.g.799.2		4
5.3	odd	4		950.4.b.g.799.3		4
5.4	even	2		950.4.a.h.1.2		2
7.6	odd	2		1862.4.a.b.1.2		2
8.3	odd	2		1216.4.a.l.1.1		2
8.5	even	2		1216.4.a.j.1.2		2
19.18	odd	2		722.4.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.b.1.1	✓	2	1.1	even	1	trivial
304.4.a.d.1.2		2	4.3	odd	2
342.4.a.k.1.1		2	3.2	odd	2
722.4.a.i.1.2		2	19.18	odd	2
950.4.a.h.1.2		2	5.4	even	2
950.4.b.g.799.2		4	5.2	odd	4
950.4.b.g.799.3		4	5.3	odd	4
1216.4.a.j.1.2		2	8.5	even	2
1216.4.a.l.1.1		2	8.3	odd	2
1862.4.a.b.1.2		2	7.6	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:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -6.15207 q^{3} +4.00000 q^{4} +18.3041 q^{5} +12.3041 q^{6} +21.8479 q^{7} -8.00000 q^{8} +10.8479 q^{9} -36.6083 q^{10} -8.30413 q^{11} -24.6083 q^{12} +53.0645 q^{13} -43.6959 q^{14} -112.608 q^{15} +16.0000 q^{16} +74.2810 q^{17} -21.6959 q^{18} -19.0000 q^{19} +73.2165 q^{20} -134.410 q^{21} +16.6083 q^{22} -163.977 q^{23} +49.2165 q^{24} +210.041 q^{25} -106.129 q^{26} +99.3686 q^{27} +87.3917 q^{28} -232.410 q^{29} +225.217 q^{30} +98.4331 q^{31} -32.0000 q^{32} +51.0876 q^{33} -148.562 q^{34} +399.908 q^{35} +43.3917 q^{36} +296.433 q^{37} +38.0000 q^{38} -326.456 q^{39} -146.433 q^{40} -434.912 q^{41} +268.820 q^{42} -171.299 q^{43} -33.2165 q^{44} +198.562 q^{45} +327.954 q^{46} -366.083 q^{47} -98.4331 q^{48} +134.332 q^{49} -420.083 q^{50} -456.982 q^{51} +212.258 q^{52} +138.631 q^{53} -198.737 q^{54} -152.000 q^{55} -174.783 q^{56} +116.889 q^{57} +464.820 q^{58} +572.797 q^{59} -450.433 q^{60} -632.691 q^{61} -196.866 q^{62} +237.005 q^{63} +64.0000 q^{64} +971.299 q^{65} -102.175 q^{66} -183.461 q^{67} +297.124 q^{68} +1008.80 q^{69} -799.815 q^{70} +56.6545 q^{71} -86.7835 q^{72} +68.1521 q^{73} -592.866 q^{74} -1292.19 q^{75} -76.0000 q^{76} -181.428 q^{77} +652.912 q^{78} -332.820 q^{79} +292.866 q^{80} -904.217 q^{81} +869.825 q^{82} +1152.91 q^{83} -537.640 q^{84} +1359.65 q^{85} +342.598 q^{86} +1429.80 q^{87} +66.4331 q^{88} -368.479 q^{89} -397.124 q^{90} +1159.35 q^{91} -655.908 q^{92} -605.567 q^{93} +732.165 q^{94} -347.779 q^{95} +196.866 q^{96} -426.443 q^{97} -268.664 q^{98} -90.0827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + q^{3} + 8 q^{4} + 10 q^{5} - 2 q^{6} + 57 q^{7} - 16 q^{8} + 35 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} + 13 q^{13} - 114 q^{14} - 172 q^{15} + 32 q^{16} - 51 q^{17} - 70 q^{18} - 38 q^{19}+ \cdots + 352 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + q^3 + 8 * q^4 + 10 * q^5 - 2 * q^6 + 57 * q^7 - 16 * q^8 + 35 * q^9 - 20 * q^10 + 10 * q^11 + 4 * q^12 + 13 * q^13 - 114 * q^14 - 172 * q^15 + 32 * q^16 - 51 * q^17 - 70 * q^18 - 38 * q^19 + 40 * q^20 + 117 * q^21 - 20 * q^22 - 155 * q^23 - 8 * q^24 + 154 * q^25 - 26 * q^26 + 79 * q^27 + 228 * q^28 - 79 * q^29 + 344 * q^30 - 16 * q^31 - 64 * q^32 + 182 * q^33 + 102 * q^34 + 108 * q^35 + 140 * q^36 + 380 * q^37 + 76 * q^38 - 613 * q^39 - 80 * q^40 - 790 * q^41 - 234 * q^42 + 296 * q^43 + 40 * q^44 - 2 * q^45 + 310 * q^46 - 200 * q^47 + 16 * q^48 + 1027 * q^49 - 308 * q^50 - 1353 * q^51 + 52 * q^52 + 397 * q^53 - 158 * q^54 - 304 * q^55 - 456 * q^56 - 19 * q^57 + 158 * q^58 + 201 * q^59 - 688 * q^60 - 680 * q^61 + 32 * q^62 + 1086 * q^63 + 128 * q^64 + 1304 * q^65 - 364 * q^66 - 939 * q^67 - 204 * q^68 + 1073 * q^69 - 216 * q^70 + 406 * q^71 - 280 * q^72 + 123 * q^73 - 760 * q^74 - 1693 * q^75 - 152 * q^76 + 462 * q^77 + 1226 * q^78 + 106 * q^79 + 160 * q^80 - 1702 * q^81 + 1580 * q^82 + 2226 * q^83 + 468 * q^84 + 2400 * q^85 - 592 * q^86 + 2527 * q^87 - 80 * q^88 - 870 * q^89 + 4 * q^90 - 249 * q^91 - 620 * q^92 - 1424 * q^93 + 400 * q^94 - 190 * q^95 - 32 * q^96 - 1864 * q^97 - 2054 * q^98 + 352 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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