LMFDB - Embedded newform 409.2.a.b.1.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			1.14
Root			$1.59576$ of defining polynomial
Character	$\chi$	$=$	409.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.59576 q^{2} -2.30987 q^{3} +0.546465 q^{4} +2.27401 q^{5} -3.68601 q^{6} +1.11928 q^{7} -2.31950 q^{8} +2.33551 q^{9} +3.62878 q^{10} +5.64263 q^{11} -1.26226 q^{12} +4.63250 q^{13} +1.78611 q^{14} -5.25267 q^{15} -4.79431 q^{16} -5.13442 q^{17} +3.72692 q^{18} +3.82544 q^{19} +1.24267 q^{20} -2.58540 q^{21} +9.00431 q^{22} +8.64693 q^{23} +5.35775 q^{24} +0.171118 q^{25} +7.39237 q^{26} +1.53489 q^{27} +0.611649 q^{28} +3.47670 q^{29} -8.38203 q^{30} -3.14509 q^{31} -3.01158 q^{32} -13.0338 q^{33} -8.19333 q^{34} +2.54526 q^{35} +1.27627 q^{36} -7.83219 q^{37} +6.10450 q^{38} -10.7005 q^{39} -5.27456 q^{40} -7.22331 q^{41} -4.12569 q^{42} +6.31621 q^{43} +3.08350 q^{44} +5.31097 q^{45} +13.7985 q^{46} -6.96264 q^{47} +11.0742 q^{48} -5.74720 q^{49} +0.273064 q^{50} +11.8599 q^{51} +2.53150 q^{52} -11.5281 q^{53} +2.44932 q^{54} +12.8314 q^{55} -2.59618 q^{56} -8.83628 q^{57} +5.54799 q^{58} -4.04329 q^{59} -2.87040 q^{60} +6.10123 q^{61} -5.01882 q^{62} +2.61410 q^{63} +4.78283 q^{64} +10.5343 q^{65} -20.7988 q^{66} -12.9796 q^{67} -2.80578 q^{68} -19.9733 q^{69} +4.06164 q^{70} -0.129308 q^{71} -5.41721 q^{72} +6.82272 q^{73} -12.4983 q^{74} -0.395261 q^{75} +2.09047 q^{76} +6.31571 q^{77} -17.0754 q^{78} +14.7727 q^{79} -10.9023 q^{80} -10.5519 q^{81} -11.5267 q^{82} +6.48568 q^{83} -1.41283 q^{84} -11.6757 q^{85} +10.0792 q^{86} -8.03073 q^{87} -13.0881 q^{88} +4.34780 q^{89} +8.47506 q^{90} +5.18508 q^{91} +4.72524 q^{92} +7.26475 q^{93} -11.1107 q^{94} +8.69909 q^{95} +6.95637 q^{96} -1.38126 q^{97} -9.17118 q^{98} +13.1784 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) $ 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

Coefficient data

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

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

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

$2$	1.59576	1.12838	0.564188	−	0.825646i	$-0.309189\pi$
$2$	1.59576	1.12838	0.564188	+	0.825646i	$0.309189\pi$
$3$	−2.30987	−1.33361	−0.666803	−	0.745234i	$-0.732338\pi$
$3$	−2.30987	−1.33361	−0.666803	+	0.745234i	$0.732338\pi$
$4$	0.546465	0.273232
$5$	2.27401	1.01697	0.508484	−	0.861071i	$-0.330206\pi$
$5$	2.27401	1.01697	0.508484	+	0.861071i	$0.330206\pi$
$6$	−3.68601	−1.50481
$7$	1.11928	0.423050	0.211525	−	0.977373i	$-0.432157\pi$
$7$	1.11928	0.423050	0.211525	+	0.977373i	$0.432157\pi$
$8$	−2.31950	−0.820067
$9$	2.33551	0.778503
$10$	3.62878	1.14752
$11$	5.64263	1.70132	0.850659	−	0.525718i	$-0.176203\pi$
$11$	5.64263	1.70132	0.850659	+	0.525718i	$0.176203\pi$
$12$	−1.26226	−0.364384
$13$	4.63250	1.28482	0.642412	−	0.766360i	$-0.277934\pi$
$13$	4.63250	1.28482	0.642412	+	0.766360i	$0.277934\pi$
$14$	1.78611	0.477359
$15$	−5.25267	−1.35623
$16$	−4.79431	−1.19858
$17$	−5.13442	−1.24528	−0.622640	−	0.782508i	$-0.713940\pi$
$17$	−5.13442	−1.24528	−0.622640	+	0.782508i	$0.713940\pi$
$18$	3.72692	0.878444
$19$	3.82544	0.877616	0.438808	−	0.898581i	$-0.355401\pi$
$19$	3.82544	0.877616	0.438808	+	0.898581i	$0.355401\pi$
$20$	1.24267	0.277869
$21$	−2.58540	−0.564181
$22$	9.00431	1.91973
$23$	8.64693	1.80301	0.901505	−	0.432769i	$-0.142463\pi$
$23$	8.64693	1.80301	0.901505	+	0.432769i	$0.142463\pi$
$24$	5.35775	1.09365
$25$	0.171118	0.0342236
$26$	7.39237	1.44976
$27$	1.53489	0.295389
$28$	0.611649	0.115591
$29$	3.47670	0.645607	0.322803	−	0.946466i	$-0.395375\pi$
$29$	3.47670	0.645607	0.322803	+	0.946466i	$0.395375\pi$
$30$	−8.38203	−1.53034
$31$	−3.14509	−0.564874	−0.282437	−	0.959286i	$-0.591143\pi$
$31$	−3.14509	−0.564874	−0.282437	+	0.959286i	$0.591143\pi$
$32$	−3.01158	−0.532378
$33$	−13.0338	−2.26889
$34$	−8.19333	−1.40514
$35$	2.54526	0.430228
$36$	1.27627	0.212712
$37$	−7.83219	−1.28760	−0.643802	−	0.765192i	$-0.722644\pi$
$37$	−7.83219	−1.28760	−0.643802	+	0.765192i	$0.722644\pi$
$38$	6.10450	0.990281
$39$	−10.7005	−1.71345
$40$	−5.27456	−0.833982
$41$	−7.22331	−1.12809	−0.564046	−	0.825744i	$-0.690756\pi$
$41$	−7.22331	−1.12809	−0.564046	+	0.825744i	$0.690756\pi$
$42$	−4.12569	−0.636608
$43$	6.31621	0.963213	0.481607	−	0.876387i	$-0.340053\pi$
$43$	6.31621	0.963213	0.481607	+	0.876387i	$0.340053\pi$
$44$	3.08350	0.464855
$45$	5.31097	0.791713
$46$	13.7985	2.03447
$47$	−6.96264	−1.01561	−0.507803	−	0.861473i	$-0.669542\pi$
$47$	−6.96264	−1.01561	−0.507803	+	0.861473i	$0.669542\pi$
$48$	11.0742	1.59843
$49$	−5.74720	−0.821029
$50$	0.273064	0.0386171
$51$	11.8599	1.66071
$52$	2.53150	0.351055
$53$	−11.5281	−1.58350	−0.791752	−	0.610843i	$-0.790831\pi$
$53$	−11.5281	−1.58350	−0.791752	+	0.610843i	$0.790831\pi$
$54$	2.44932	0.333310
$55$	12.8314	1.73019
$56$	−2.59618	−0.346929
$57$	−8.83628	−1.17039
$58$	5.54799	0.728487
$59$	−4.04329	−0.526391	−0.263196	−	0.964742i	$-0.584776\pi$
$59$	−4.04329	−0.526391	−0.263196	+	0.964742i	$0.584776\pi$
$60$	−2.87040	−0.370567
$61$	6.10123	0.781183	0.390591	−	0.920564i	$-0.372271\pi$
$61$	6.10123	0.781183	0.390591	+	0.920564i	$0.372271\pi$
$62$	−5.01882	−0.637390
$63$	2.61410	0.329345
$64$	4.78283	0.597854
$65$	10.5343	1.30662
$66$	−20.7988	−2.56016
$67$	−12.9796	−1.58571	−0.792853	−	0.609413i	$-0.791405\pi$
$67$	−12.9796	−1.58571	−0.792853	+	0.609413i	$0.791405\pi$
$68$	−2.80578	−0.340251
$69$	−19.9733	−2.40450
$70$	4.06164	0.485459
$71$	−0.129308	−0.0153461	−0.00767303	−	0.999971i	$-0.502442\pi$
$71$	−0.129308	−0.0153461	−0.00767303	+	0.999971i	$0.502442\pi$
$72$	−5.41721	−0.638425
$73$	6.82272	0.798540	0.399270	−	0.916833i	$-0.369264\pi$
$73$	6.82272	0.798540	0.399270	+	0.916833i	$0.369264\pi$
$74$	−12.4983	−1.45290
$75$	−0.395261	−0.0456408
$76$	2.09047	0.239793
$77$	6.31571	0.719742
$78$	−17.0754	−1.93341
$79$	14.7727	1.66205	0.831027	−	0.556232i	$-0.187753\pi$
$79$	14.7727	1.66205	0.831027	+	0.556232i	$0.187753\pi$
$80$	−10.9023	−1.21891
$81$	−10.5519	−1.17244
$82$	−11.5267	−1.27291
$83$	6.48568	0.711896	0.355948	−	0.934506i	$-0.384158\pi$
$83$	6.48568	0.711896	0.355948	+	0.934506i	$0.384158\pi$
$84$	−1.41283	−0.154153
$85$	−11.6757	−1.26641
$86$	10.0792	1.08687
$87$	−8.03073	−0.860984
$88$	−13.0881	−1.39519
$89$	4.34780	0.460866	0.230433	−	0.973088i	$-0.425986\pi$
$89$	4.34780	0.460866	0.230433	+	0.973088i	$0.425986\pi$
$90$	8.47506	0.893350
$91$	5.18508	0.543544
$92$	4.72524	0.492641
$93$	7.26475	0.753319
$94$	−11.1107	−1.14598
$95$	8.69909	0.892508
$96$	6.95637	0.709982
$97$	−1.38126	−0.140246	−0.0701229	−	0.997538i	$-0.522339\pi$
$97$	−1.38126	−0.140246	−0.0701229	+	0.997538i	$0.522339\pi$
$98$	−9.17118	−0.926430
$99$	13.1784	1.32448
$100$	0.0935101	0.00935101
$101$	0.945729	0.0941035	0.0470518	−	0.998892i	$-0.485017\pi$
$101$	0.945729	0.0941035	0.0470518	+	0.998892i	$0.485017\pi$
$102$	18.9255	1.87391
$103$	−18.7810	−1.85055	−0.925273	−	0.379302i	$-0.876164\pi$
$103$	−18.7810	−1.85055	−0.925273	+	0.379302i	$0.876164\pi$
$104$	−10.7451	−1.05364
$105$	−5.87923	−0.573754
$106$	−18.3961	−1.78679
$107$	6.38130	0.616904	0.308452	−	0.951240i	$-0.400189\pi$
$107$	6.38130	0.616904	0.308452	+	0.951240i	$0.400189\pi$
$108$	0.838762	0.0807099
$109$	11.9434	1.14397	0.571984	−	0.820265i	$-0.306174\pi$
$109$	11.9434	1.14397	0.571984	+	0.820265i	$0.306174\pi$
$110$	20.4759	1.95230
$111$	18.0914	1.71716
$112$	−5.36619	−0.507057
$113$	−8.83064	−0.830717	−0.415359	−	0.909658i	$-0.636344\pi$
$113$	−8.83064	−0.830717	−0.415359	+	0.909658i	$0.636344\pi$
$114$	−14.1006	−1.32064
$115$	19.6632	1.83360
$116$	1.89989	0.176401
$117$	10.8192	1.00024
$118$	−6.45214	−0.593967
$119$	−5.74687	−0.526815
$120$	12.1836	1.11220
$121$	20.8393	1.89448
$122$	9.73613	0.881468
$123$	16.6849	1.50443
$124$	−1.71868	−0.154342
$125$	−10.9809	−0.982164
$126$	4.17149	0.371625
$127$	−17.6388	−1.56519	−0.782597	−	0.622528i	$-0.786106\pi$
$127$	−17.6388	−1.56519	−0.782597	+	0.622528i	$0.786106\pi$
$128$	13.6554	1.20698
$129$	−14.5896	−1.28455
$130$	16.8103	1.47436
$131$	−4.30376	−0.376021	−0.188011	−	0.982167i	$-0.560204\pi$
$131$	−4.30376	−0.376021	−0.188011	+	0.982167i	$0.560204\pi$
$132$	−7.12249	−0.619933
$133$	4.28175	0.371275
$134$	−20.7123	−1.78927
$135$	3.49035	0.300401
$136$	11.9093	1.02121
$137$	7.74286	0.661518	0.330759	−	0.943715i	$-0.392695\pi$
$137$	7.74286	0.661518	0.330759	+	0.943715i	$0.392695\pi$
$138$	−31.8727	−2.71318
$139$	−18.7780	−1.59273	−0.796365	−	0.604816i	$-0.793247\pi$
$139$	−18.7780	−1.59273	−0.796365	+	0.604816i	$0.793247\pi$
$140$	1.39090	0.117552
$141$	16.0828	1.35442
$142$	−0.206345	−0.0173161
$143$	26.1395	2.18589
$144$	−11.1971	−0.933096
$145$	7.90604	0.656561
$146$	10.8875	0.901053
$147$	13.2753	1.09493
$148$	−4.28002	−0.351815
$149$	5.83633	0.478131	0.239065	−	0.971003i	$-0.423159\pi$
$149$	5.83633	0.478131	0.239065	+	0.971003i	$0.423159\pi$
$150$	−0.630744	−0.0515000
$151$	12.2018	0.992970	0.496485	−	0.868045i	$-0.334624\pi$
$151$	12.2018	0.992970	0.496485	+	0.868045i	$0.334624\pi$
$152$	−8.87311	−0.719704
$153$	−11.9915	−0.969454
$154$	10.0784	0.812139
$155$	−7.15195	−0.574459
$156$	−5.84743	−0.468169
$157$	−12.0530	−0.961936	−0.480968	−	0.876738i	$-0.659715\pi$
$157$	−12.0530	−0.961936	−0.480968	+	0.876738i	$0.659715\pi$
$158$	23.5737	1.87542
$159$	26.6284	2.11177
$160$	−6.84837	−0.541411
$161$	9.67837	0.762762
$162$	−16.8384	−1.32295
$163$	5.27366	0.413065	0.206532	−	0.978440i	$-0.433782\pi$
$163$	5.27366	0.413065	0.206532	+	0.978440i	$0.433782\pi$
$164$	−3.94728	−0.308231
$165$	−29.6389	−2.30738
$166$	10.3496	0.803287
$167$	−0.638378	−0.0493992	−0.0246996	−	0.999695i	$-0.507863\pi$
$167$	−0.638378	−0.0493992	−0.0246996	+	0.999695i	$0.507863\pi$
$168$	5.99684	0.462666
$169$	8.46001	0.650770
$170$	−18.6317	−1.42899
$171$	8.93435	0.683227
$172$	3.45159	0.263181
$173$	4.88337	0.371276	0.185638	−	0.982618i	$-0.440565\pi$
$173$	4.88337	0.371276	0.185638	+	0.982618i	$0.440565\pi$
$174$	−12.8152	−0.971514
$175$	0.191530	0.0144783
$176$	−27.0525	−2.03916
$177$	9.33948	0.701998
$178$	6.93807	0.520030
$179$	−13.1127	−0.980091	−0.490046	−	0.871697i	$-0.663020\pi$
$179$	−13.1127	−0.980091	−0.490046	+	0.871697i	$0.663020\pi$
$180$	2.90226	0.216322
$181$	−3.22990	−0.240076	−0.120038	−	0.992769i	$-0.538302\pi$
$181$	−3.22990	−0.240076	−0.120038	+	0.992769i	$0.538302\pi$
$182$	8.27416	0.613322
$183$	−14.0931	−1.04179
$184$	−20.0566	−1.47859
$185$	−17.8105	−1.30945
$186$	11.5928	0.850027
$187$	−28.9716	−2.11862
$188$	−3.80484	−0.277496
$189$	1.71798	0.124964
$190$	13.8817	1.00708
$191$	22.1721	1.60431	0.802157	−	0.597114i	$-0.203686\pi$
$191$	22.1721	1.60431	0.802157	+	0.597114i	$0.203686\pi$
$192$	−11.0477	−0.797302
$193$	4.82665	0.347430	0.173715	−	0.984796i	$-0.444423\pi$
$193$	4.82665	0.347430	0.173715	+	0.984796i	$0.444423\pi$
$194$	−2.20417	−0.158250
$195$	−24.3330	−1.74252
$196$	−3.14064	−0.224332
$197$	−3.05779	−0.217858	−0.108929	−	0.994050i	$-0.534742\pi$
$197$	−3.05779	−0.217858	−0.108929	+	0.994050i	$0.534742\pi$
$198$	21.0297	1.49451
$199$	−22.8974	−1.62315	−0.811575	−	0.584248i	$-0.801390\pi$
$199$	−22.8974	−1.62315	−0.811575	+	0.584248i	$0.801390\pi$
$200$	−0.396909	−0.0280657
$201$	29.9811	2.11471
$202$	1.50916	0.106184
$203$	3.89141	0.273124
$204$	6.48099	0.453760
$205$	−16.4259	−1.14723
$206$	−29.9700	−2.08811
$207$	20.1950	1.40365
$208$	−22.2096	−1.53996
$209$	21.5856	1.49310
$210$	−9.38187	−0.647410
$211$	16.2204	1.11666	0.558329	−	0.829620i	$-0.311443\pi$
$211$	16.2204	1.11666	0.558329	+	0.829620i	$0.311443\pi$
$212$	−6.29969	−0.432665
$213$	0.298685	0.0204656
$214$	10.1831	0.696099
$215$	14.3631	0.979557
$216$	−3.56017	−0.242239
$217$	−3.52024	−0.238970
$218$	19.0588	1.29083
$219$	−15.7596	−1.06494
$220$	7.01191	0.472743
$221$	−23.7852	−1.59996
$222$	28.8696	1.93760
$223$	−17.9300	−1.20068	−0.600342	−	0.799743i	$-0.704969\pi$
$223$	−17.9300	−1.20068	−0.600342	+	0.799743i	$0.704969\pi$
$224$	−3.37082	−0.225222
$225$	0.399648	0.0266432
$226$	−14.0916	−0.937361
$227$	1.13782	0.0755196	0.0377598	−	0.999287i	$-0.487978\pi$
$227$	1.13782	0.0755196	0.0377598	+	0.999287i	$0.487978\pi$
$228$	−4.82871	−0.319789
$229$	−17.8832	−1.18175	−0.590877	−	0.806762i	$-0.701218\pi$
$229$	−17.8832	−1.18175	−0.590877	+	0.806762i	$0.701218\pi$
$230$	31.3778	2.06899
$231$	−14.5885	−0.959851
$232$	−8.06420	−0.529441
$233$	12.0658	0.790458	0.395229	−	0.918583i	$-0.370665\pi$
$233$	12.0658	0.790458	0.395229	+	0.918583i	$0.370665\pi$
$234$	17.2650	1.12865
$235$	−15.8331	−1.03284
$236$	−2.20951	−0.143827
$237$	−34.1230	−2.21652
$238$	−9.17066	−0.594445
$239$	20.4634	1.32367	0.661835	−	0.749650i	$-0.269778\pi$
$239$	20.4634	1.32367	0.661835	+	0.749650i	$0.269778\pi$
$240$	25.1829	1.62555
$241$	−2.48714	−0.160211	−0.0801053	−	0.996786i	$-0.525526\pi$
$241$	−2.48714	−0.160211	−0.0801053	+	0.996786i	$0.525526\pi$
$242$	33.2546	2.13769
$243$	19.7689	1.26818
$244$	3.33411	0.213444
$245$	−13.0692	−0.834960
$246$	26.6252	1.69756
$247$	17.7213	1.12758
$248$	7.29503	0.463235
$249$	−14.9811	−0.949389
$250$	−17.5230	−1.10825
$251$	8.23208	0.519605	0.259802	−	0.965662i	$-0.416343\pi$
$251$	8.23208	0.519605	0.259802	+	0.965662i	$0.416343\pi$
$252$	1.42851	0.0899878
$253$	48.7915	3.06749
$254$	−28.1474	−1.76613
$255$	26.9694	1.68889
$256$	12.2252	0.764075
$257$	4.41696	0.275522	0.137761	−	0.990465i	$-0.456009\pi$
$257$	4.41696	0.275522	0.137761	+	0.990465i	$0.456009\pi$
$258$	−23.2816	−1.44945
$259$	−8.76644	−0.544720
$260$	5.75664	0.357012
$261$	8.11986	0.502607
$262$	−6.86778	−0.424293
$263$	−4.26365	−0.262908	−0.131454	−	0.991322i	$-0.541965\pi$
$263$	−4.26365	−0.262908	−0.131454	+	0.991322i	$0.541965\pi$
$264$	30.2318	1.86064
$265$	−26.2150	−1.61037
$266$	6.83267	0.418938
$267$	−10.0429	−0.614613
$268$	−7.09287	−0.433266
$269$	−23.2292	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$269$	−23.2292	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$270$	5.56978	0.338966
$271$	−18.4623	−1.12150	−0.560752	−	0.827984i	$-0.689488\pi$
$271$	−18.4623	−1.12150	−0.560752	+	0.827984i	$0.689488\pi$
$272$	24.6160	1.49256
$273$	−11.9769	−0.724873
$274$	12.3558	0.746441
$275$	0.965557	0.0582253
$276$	−10.9147	−0.656988
$277$	−8.70780	−0.523201	−0.261601	−	0.965176i	$-0.584250\pi$
$277$	−8.70780	−0.523201	−0.261601	+	0.965176i	$0.584250\pi$
$278$	−29.9653	−1.79720
$279$	−7.34538	−0.439756
$280$	−5.90373	−0.352816
$281$	7.43068	0.443277	0.221639	−	0.975129i	$-0.428859\pi$
$281$	7.43068	0.443277	0.221639	+	0.975129i	$0.428859\pi$
$282$	25.6644	1.52829
$283$	28.8258	1.71351	0.856757	−	0.515721i	$-0.172476\pi$
$283$	28.8258	1.71351	0.856757	+	0.515721i	$0.172476\pi$
$284$	−0.0706624	−0.00419304
$285$	−20.0938	−1.19025
$286$	41.7124	2.46651
$287$	−8.08493	−0.477238
$288$	−7.03358	−0.414458
$289$	9.36227	0.550722
$290$	12.6162	0.740848
$291$	3.19054	0.187033
$292$	3.72838	0.218187
$293$	−5.44959	−0.318368	−0.159184	−	0.987249i	$-0.550886\pi$
$293$	−5.44959	−0.318368	−0.159184	+	0.987249i	$0.550886\pi$
$294$	21.1843	1.23549
$295$	−9.19448	−0.535323
$296$	18.1668	1.05592
$297$	8.66081	0.502551
$298$	9.31341	0.539511
$299$	40.0569	2.31655
$300$	−0.215996	−0.0124706
$301$	7.06963	0.407487
$302$	19.4712	1.12044
$303$	−2.18451	−0.125497
$304$	−18.3403	−1.05189
$305$	13.8743	0.794438
$306$	−19.1356	−1.09391
$307$	−20.9184	−1.19388	−0.596939	−	0.802287i	$-0.703617\pi$
$307$	−20.9184	−1.19388	−0.596939	+	0.802287i	$0.703617\pi$
$308$	3.45131	0.196657
$309$	43.3817	2.46790
$310$	−11.4128	−0.648206
$311$	−4.98628	−0.282746	−0.141373	−	0.989956i	$-0.545152\pi$
$311$	−4.98628	−0.282746	−0.141373	+	0.989956i	$0.545152\pi$
$312$	24.8197	1.40514
$313$	−17.8811	−1.01070	−0.505350	−	0.862914i	$-0.668637\pi$
$313$	−17.8811	−1.01070	−0.505350	+	0.862914i	$0.668637\pi$
$314$	−19.2338	−1.08543
$315$	5.94448	0.334934
$316$	8.07274	0.454127
$317$	2.44848	0.137521	0.0687603	−	0.997633i	$-0.478096\pi$
$317$	2.44848	0.137521	0.0687603	+	0.997633i	$0.478096\pi$
$318$	42.4927	2.38287
$319$	19.6177	1.09838
$320$	10.8762	0.607999
$321$	−14.7400	−0.822706
$322$	15.4444	0.860683
$323$	−19.6414	−1.09288
$324$	−5.76625	−0.320347
$325$	0.792704	0.0439713
$326$	8.41551	0.466092
$327$	−27.5877	−1.52560
$328$	16.7545	0.925111
$329$	−7.79317	−0.429651
$330$	−47.2967	−2.60360
$331$	−26.2243	−1.44142	−0.720709	−	0.693238i	$-0.756183\pi$
$331$	−26.2243	−1.44142	−0.720709	+	0.693238i	$0.756183\pi$
$332$	3.54420	0.194513
$333$	−18.2922	−1.00240
$334$	−1.01870	−0.0557408
$335$	−29.5157	−1.61261
$336$	12.3952	0.676214
$337$	1.66023	0.0904382	0.0452191	−	0.998977i	$-0.485601\pi$
$337$	1.66023	0.0904382	0.0452191	+	0.998977i	$0.485601\pi$
$338$	13.5002	0.734313
$339$	20.3977	1.10785
$340$	−6.38037	−0.346024
$341$	−17.7466	−0.961030
$342$	14.2571	0.770937
$343$	−14.2677	−0.770385
$344$	−14.6505	−0.789900
$345$	−45.4195	−2.44530
$346$	7.79271	0.418939
$347$	−19.9063	−1.06863	−0.534313	−	0.845287i	$-0.679430\pi$
$347$	−19.9063	−1.06863	−0.534313	+	0.845287i	$0.679430\pi$
$348$	−4.38851	−0.235249
$349$	−15.8591	−0.848919	−0.424459	−	0.905447i	$-0.639536\pi$
$349$	−15.8591	−0.848919	−0.424459	+	0.905447i	$0.639536\pi$
$350$	0.305637	0.0163370
$351$	7.11036	0.379523
$352$	−16.9933	−0.905744
$353$	−3.28287	−0.174729	−0.0873647	−	0.996176i	$-0.527845\pi$
$353$	−3.28287	−0.174729	−0.0873647	+	0.996176i	$0.527845\pi$
$354$	14.9036	0.792118
$355$	−0.294048	−0.0156064
$356$	2.37592	0.125923
$357$	13.2745	0.702563
$358$	−20.9248	−1.10591
$359$	32.9164	1.73726	0.868630	−	0.495461i	$-0.165001\pi$
$359$	32.9164	1.73726	0.868630	+	0.495461i	$0.165001\pi$
$360$	−12.3188	−0.649258
$361$	−4.36600	−0.229790
$362$	−5.15416	−0.270896
$363$	−48.1361	−2.52649
$364$	2.83346	0.148514
$365$	15.5149	0.812089
$366$	−22.4892	−1.17553
$367$	−14.5464	−0.759314	−0.379657	−	0.925127i	$-0.623958\pi$
$367$	−14.5464	−0.759314	−0.379657	+	0.925127i	$0.623958\pi$
$368$	−41.4560	−2.16105
$369$	−16.8701	−0.878223
$370$	−28.4213	−1.47755
$371$	−12.9032	−0.669901
$372$	3.96993	0.205831
$373$	−30.2342	−1.56547	−0.782735	−	0.622355i	$-0.786176\pi$
$373$	−30.2342	−1.56547	−0.782735	+	0.622355i	$0.786176\pi$
$374$	−46.2319	−2.39060
$375$	25.3645	1.30982
$376$	16.1498	0.832865
$377$	16.1058	0.829490
$378$	2.74148	0.141007
$379$	11.5374	0.592638	0.296319	−	0.955089i	$-0.404241\pi$
$379$	11.5374	0.592638	0.296319	+	0.955089i	$0.404241\pi$
$380$	4.75374	0.243862
$381$	40.7435	2.08735
$382$	35.3814	1.81027
$383$	8.30939	0.424590	0.212295	−	0.977206i	$-0.431906\pi$
$383$	8.30939	0.424590	0.212295	+	0.977206i	$0.431906\pi$
$384$	−31.5423	−1.60964
$385$	14.3620	0.731954
$386$	7.70220	0.392031
$387$	14.7516	0.749865
$388$	−0.754811	−0.0383197
$389$	19.3524	0.981208	0.490604	−	0.871383i	$-0.336776\pi$
$389$	19.3524	0.981208	0.490604	+	0.871383i	$0.336776\pi$
$390$	−38.8297	−1.96622
$391$	−44.3970	−2.24525
$392$	13.3306	0.673299
$393$	9.94113	0.501464
$394$	−4.87951	−0.245826
$395$	33.5932	1.69026
$396$	7.20154	0.361891
$397$	12.4940	0.627055	0.313527	−	0.949579i	$-0.398489\pi$
$397$	12.4940	0.627055	0.313527	+	0.949579i	$0.398489\pi$
$398$	−36.5388	−1.83152
$399$	−9.89030	−0.495134
$400$	−0.820393	−0.0410196
$401$	21.8124	1.08926	0.544630	−	0.838677i	$-0.316670\pi$
$401$	21.8124	1.08926	0.544630	+	0.838677i	$0.316670\pi$
$402$	47.8428	2.38618
$403$	−14.5696	−0.725763
$404$	0.516808	0.0257121
$405$	−23.9952	−1.19233
$406$	6.20978	0.308186
$407$	−44.1942	−2.19062
$408$	−27.5089	−1.36190
$409$	1.00000	0.0494468
$409$	1.00000	0.0494468
$410$	−26.2118	−1.29451
$411$	−17.8850	−0.882203
$412$	−10.2631	−0.505629
$413$	−4.52559	−0.222690
$414$	32.2265	1.58384
$415$	14.7485	0.723976
$416$	−13.9511	−0.684011
$417$	43.3748	2.12407
$418$	34.4455	1.68478
$419$	13.1765	0.643714	0.321857	−	0.946788i	$-0.395693\pi$
$419$	13.1765	0.643714	0.321857	+	0.946788i	$0.395693\pi$
$420$	−3.21279	−0.156768
$421$	8.91268	0.434377	0.217189	−	0.976130i	$-0.430311\pi$
$421$	8.91268	0.434377	0.217189	+	0.976130i	$0.430311\pi$
$422$	25.8839	1.26001
$423$	−16.2613	−0.790652
$424$	26.7394	1.29858
$425$	−0.878593	−0.0426180
$426$	0.476632	0.0230929
$427$	6.82901	0.330479
$428$	3.48716	0.168558
$429$	−60.3788	−2.91512
$430$	22.9202	1.10531
$431$	−11.3904	−0.548658	−0.274329	−	0.961636i	$-0.588456\pi$
$431$	−11.3904	−0.548658	−0.274329	+	0.961636i	$0.588456\pi$
$432$	−7.35872	−0.354047
$433$	25.4354	1.22235	0.611174	−	0.791496i	$-0.290698\pi$
$433$	25.4354	1.22235	0.611174	+	0.791496i	$0.290698\pi$
$434$	−5.61748	−0.269648
$435$	−18.2620	−0.875593
$436$	6.52664	0.312569
$437$	33.0783	1.58235
$438$	−25.1486	−1.20165
$439$	26.7471	1.27657	0.638286	−	0.769799i	$-0.279644\pi$
$439$	26.7471	1.27657	0.638286	+	0.769799i	$0.279644\pi$
$440$	−29.7624	−1.41887
$441$	−13.4226	−0.639174
$442$	−37.9555	−1.80536
$443$	−4.09607	−0.194610	−0.0973051	−	0.995255i	$-0.531022\pi$
$443$	−4.09607	−0.194610	−0.0973051	+	0.995255i	$0.531022\pi$
$444$	9.88629	0.469183
$445$	9.88694	0.468686
$446$	−28.6121	−1.35482
$447$	−13.4812	−0.637638
$448$	5.35335	0.252922
$449$	2.67865	0.126413	0.0632065	−	0.998000i	$-0.479867\pi$
$449$	2.67865	0.126413	0.0632065	+	0.998000i	$0.479867\pi$
$450$	0.637744	0.0300636
$451$	−40.7585	−1.91924
$452$	−4.82564	−0.226979
$453$	−28.1847	−1.32423
$454$	1.81569	0.0852144
$455$	11.7909	0.552767
$456$	20.4957	0.959801
$457$	−35.5703	−1.66391	−0.831953	−	0.554846i	$-0.812777\pi$
$457$	−35.5703	−1.66391	−0.831953	+	0.554846i	$0.812777\pi$
$458$	−28.5373	−1.33346
$459$	−7.88076	−0.367842
$460$	10.7452	0.501000
$461$	11.1466	0.519148	0.259574	−	0.965723i	$-0.416418\pi$
$461$	11.1466	0.519148	0.259574	+	0.965723i	$0.416418\pi$
$462$	−23.2798	−1.08307
$463$	13.6820	0.635857	0.317928	−	0.948115i	$-0.397013\pi$
$463$	13.6820	0.635857	0.317928	+	0.948115i	$0.397013\pi$
$464$	−16.6684	−0.773809
$465$	16.5201	0.766101
$466$	19.2542	0.891934
$467$	6.21481	0.287587	0.143793	−	0.989608i	$-0.454070\pi$
$467$	6.21481	0.287587	0.143793	+	0.989608i	$0.454070\pi$
$468$	5.91233	0.273298
$469$	−14.5278	−0.670832
$470$	−25.2659	−1.16543
$471$	27.8409	1.28284
$472$	9.37841	0.431676
$473$	35.6401	1.63873
$474$	−54.4522	−2.50107
$475$	0.654603	0.0300352
$476$	−3.14046	−0.143943
$477$	−26.9239	−1.23276
$478$	32.6548	1.49360
$479$	8.30122	0.379293	0.189646	−	0.981852i	$-0.439266\pi$
$479$	8.30122	0.379293	0.189646	+	0.981852i	$0.439266\pi$
$480$	15.8189	0.722029
$481$	−36.2826	−1.65434
$482$	−3.96889	−0.180778
$483$	−22.3558	−1.01722
$484$	11.3879	0.517634
$485$	−3.14100	−0.142626
$486$	31.5466	1.43098
$487$	−13.7269	−0.622025	−0.311013	−	0.950406i	$-0.600668\pi$
$487$	−13.7269	−0.622025	−0.311013	+	0.950406i	$0.600668\pi$
$488$	−14.1518	−0.640622
$489$	−12.1815	−0.550865
$490$	−20.8554	−0.942149
$491$	29.8108	1.34534	0.672670	−	0.739942i	$-0.265147\pi$
$491$	29.8108	1.34534	0.672670	+	0.739942i	$0.265147\pi$
$492$	9.11772	0.411059
$493$	−17.8508	−0.803961
$494$	28.2791	1.27234
$495$	29.9679	1.34695
$496$	15.0785	0.677045
$497$	−0.144733	−0.00649214
$498$	−23.9063	−1.07127
$499$	21.3462	0.955589	0.477795	−	0.878472i	$-0.341436\pi$
$499$	21.3462	0.955589	0.477795	+	0.878472i	$0.341436\pi$
$500$	−6.00069	−0.268359
$501$	1.47457	0.0658790
$502$	13.1365	0.586309
$503$	−13.4129	−0.598053	−0.299027	−	0.954245i	$-0.596662\pi$
$503$	−13.4129	−0.598053	−0.299027	+	0.954245i	$0.596662\pi$
$504$	−6.06340	−0.270085
$505$	2.15060	0.0957003
$506$	77.8597	3.46128
$507$	−19.5415	−0.867871
$508$	−9.63901	−0.427662
$509$	−6.97595	−0.309203	−0.154602	−	0.987977i	$-0.549409\pi$
$509$	−6.97595	−0.309203	−0.154602	+	0.987977i	$0.549409\pi$
$510$	43.0368	1.90570
$511$	7.63656	0.337822
$512$	−7.80233	−0.344818
$513$	5.87162	0.259238
$514$	7.04843	0.310893
$515$	−42.7081	−1.88195
$516$	−7.97273	−0.350980
$517$	−39.2876	−1.72787
$518$	−13.9892	−0.614649
$519$	−11.2800	−0.495135
$520$	−24.4344	−1.07152
$521$	31.9655	1.40043	0.700216	−	0.713931i	$-0.253087\pi$
$521$	31.9655	1.40043	0.700216	+	0.713931i	$0.253087\pi$
$522$	12.9574	0.567129
$523$	−12.6093	−0.551365	−0.275682	−	0.961249i	$-0.588904\pi$
$523$	−12.6093	−0.551365	−0.275682	+	0.961249i	$0.588904\pi$
$524$	−2.35185	−0.102741
$525$	−0.442409	−0.0193083
$526$	−6.80379	−0.296659
$527$	16.1482	0.703426
$528$	62.4878	2.71943
$529$	51.7694	2.25084
$530$	−41.8329	−1.81711
$531$	−9.44314	−0.409797
$532$	2.33983	0.101444
$533$	−33.4619	−1.44940
$534$	−16.0260	−0.693515
$535$	14.5111	0.627371
$536$	30.1061	1.30039
$537$	30.2887	1.30705
$538$	−37.0684	−1.59813
$539$	−32.4294	−1.39683
$540$	1.90735	0.0820794
$541$	−25.5509	−1.09852	−0.549259	−	0.835652i	$-0.685090\pi$
$541$	−25.5509	−1.09852	−0.549259	+	0.835652i	$0.685090\pi$
$542$	−29.4614	−1.26548
$543$	7.46065	0.320167
$544$	15.4627	0.662959
$545$	27.1594	1.16338
$546$	−19.1123	−0.817929
$547$	15.3477	0.656218	0.328109	−	0.944640i	$-0.393589\pi$
$547$	15.3477	0.656218	0.328109	+	0.944640i	$0.393589\pi$
$548$	4.23120	0.180748
$549$	14.2495	0.608153
$550$	1.54080	0.0657000
$551$	13.2999	0.566595
$552$	46.3281	1.97185
$553$	16.5348	0.703131
$554$	−13.8956	−0.590368
$555$	41.1399	1.74629
$556$	−10.2615	−0.435185
$557$	0.0909675	0.00385442	0.00192721	−	0.999998i	$-0.499387\pi$
$557$	0.0909675	0.00385442	0.00192721	+	0.999998i	$0.499387\pi$
$558$	−11.7215	−0.496210
$559$	29.2598	1.23756
$560$	−12.2028	−0.515661
$561$	66.9208	2.82540
$562$	11.8576	0.500183
$563$	0.631492	0.0266142	0.0133071	−	0.999911i	$-0.495764\pi$
$563$	0.631492	0.0266142	0.0133071	+	0.999911i	$0.495764\pi$
$564$	8.78869	0.370070
$565$	−20.0810	−0.844813
$566$	45.9991	1.93349
$567$	−11.8106	−0.495998
$568$	0.299930	0.0125848
$569$	15.6213	0.654879	0.327440	−	0.944872i	$-0.393814\pi$
$569$	15.6213	0.654879	0.327440	+	0.944872i	$0.393814\pi$
$570$	−32.0649	−1.34305
$571$	16.4264	0.687421	0.343711	−	0.939076i	$-0.388316\pi$
$571$	16.4264	0.687421	0.343711	+	0.939076i	$0.388316\pi$
$572$	14.2843	0.597256
$573$	−51.2146	−2.13952
$574$	−12.9016	−0.538504
$575$	1.47965	0.0617056
$576$	11.1704	0.465431
$577$	16.0284	0.667271	0.333636	−	0.942702i	$-0.391724\pi$
$577$	16.0284	0.667271	0.333636	+	0.942702i	$0.391724\pi$
$578$	14.9400	0.621421
$579$	−11.1489	−0.463334
$580$	4.32037	0.179394
$581$	7.25932	0.301167
$582$	5.09135	0.211043
$583$	−65.0487	−2.69404
$584$	−15.8253	−0.654856
$585$	24.6030	1.01721
$586$	−8.69626	−0.359239
$587$	41.8813	1.72863	0.864313	−	0.502954i	$-0.167753\pi$
$587$	41.8813	1.72863	0.864313	+	0.502954i	$0.167753\pi$
$588$	7.25449	0.299170
$589$	−12.0313	−0.495743
$590$	−14.6722	−0.604046
$591$	7.06310	0.290537
$592$	37.5499	1.54329
$593$	29.8719	1.22669	0.613345	−	0.789815i	$-0.289823\pi$
$593$	29.8719	1.22669	0.613345	+	0.789815i	$0.289823\pi$
$594$	13.8206	0.567067
$595$	−13.0684	−0.535754
$596$	3.18935	0.130641
$597$	52.8900	2.16464
$598$	63.9213	2.61394
$599$	24.6241	1.00611	0.503057	−	0.864253i	$-0.332208\pi$
$599$	24.6241	1.00611	0.503057	+	0.864253i	$0.332208\pi$
$600$	0.916808	0.0374285
$601$	36.8165	1.50178	0.750888	−	0.660430i	$-0.229626\pi$
$601$	36.8165	1.50178	0.750888	+	0.660430i	$0.229626\pi$
$602$	11.2815	0.459799
$603$	−30.3139	−1.23448
$604$	6.66787	0.271312
$605$	47.3888	1.92663
$606$	−3.48597	−0.141608
$607$	15.5623	0.631653	0.315826	−	0.948817i	$-0.397718\pi$
$607$	15.5623	0.631653	0.315826	+	0.948817i	$0.397718\pi$
$608$	−11.5206	−0.467223
$609$	−8.98866	−0.364239
$610$	22.1400	0.896424
$611$	−32.2544	−1.30487
$612$	−6.55293	−0.264886
$613$	−10.3255	−0.417042	−0.208521	−	0.978018i	$-0.566865\pi$
$613$	−10.3255	−0.417042	−0.208521	+	0.978018i	$0.566865\pi$
$614$	−33.3809	−1.34714
$615$	37.9417	1.52996
$616$	−14.6493	−0.590236
$617$	−39.8929	−1.60603	−0.803014	−	0.595960i	$-0.796772\pi$
$617$	−39.8929	−1.60603	−0.803014	+	0.595960i	$0.796772\pi$
$618$	69.2270	2.78472
$619$	37.1188	1.49193	0.745966	−	0.665984i	$-0.231988\pi$
$619$	37.1188	1.49193	0.745966	+	0.665984i	$0.231988\pi$
$620$	−3.90829	−0.156961
$621$	13.2721	0.532590
$622$	−7.95692	−0.319044
$623$	4.86642	0.194969
$624$	51.3013	2.05370
$625$	−25.8263	−1.03305
$626$	−28.5341	−1.14045
$627$	−49.8599	−1.99121
$628$	−6.58655	−0.262832
$629$	40.2138	1.60343
$630$	9.48600	0.377931
$631$	5.80965	0.231278	0.115639	−	0.993291i	$-0.463108\pi$
$631$	5.80965	0.231278	0.115639	+	0.993291i	$0.463108\pi$
$632$	−34.2652	−1.36300
$633$	−37.4670	−1.48918
$634$	3.90721	0.155175
$635$	−40.1109	−1.59175
$636$	14.5515	0.577004
$637$	−26.6239	−1.05488
$638$	31.3053	1.23939
$639$	−0.302000	−0.0119470
$640$	31.0526	1.22746
$641$	−0.528917	−0.0208910	−0.0104455	−	0.999945i	$-0.503325\pi$
$641$	−0.528917	−0.0208910	−0.0104455	+	0.999945i	$0.503325\pi$
$642$	−23.5216	−0.928322
$643$	−7.32405	−0.288833	−0.144416	−	0.989517i	$-0.546130\pi$
$643$	−7.32405	−0.288833	−0.144416	+	0.989517i	$0.546130\pi$
$644$	5.28889	0.208411
$645$	−33.1770	−1.30634
$646$	−31.3431	−1.23318
$647$	45.7871	1.80007	0.900037	−	0.435813i	$-0.143539\pi$
$647$	45.7871	1.80007	0.900037	+	0.435813i	$0.143539\pi$
$648$	24.4752	0.961476
$649$	−22.8148	−0.895559
$650$	1.26497	0.0496162
$651$	8.13131	0.318691
$652$	2.88187	0.112863
$653$	−17.8131	−0.697079	−0.348539	−	0.937294i	$-0.613322\pi$
$653$	−17.8131	−0.697079	−0.348539	+	0.937294i	$0.613322\pi$
$654$	−44.0234	−1.72145
$655$	−9.78678	−0.382401
$656$	34.6307	1.35210
$657$	15.9345	0.621666
$658$	−12.4361	−0.484808
$659$	34.7524	1.35376	0.676880	−	0.736093i	$-0.263331\pi$
$659$	34.7524	1.35376	0.676880	+	0.736093i	$0.263331\pi$
$660$	−16.1966	−0.630452
$661$	−27.8081	−1.08161	−0.540804	−	0.841149i	$-0.681880\pi$
$661$	−27.8081	−1.08161	−0.540804	+	0.841149i	$0.681880\pi$
$662$	−41.8478	−1.62646
$663$	54.9407	2.13372
$664$	−15.0435	−0.583803
$665$	9.73675	0.377575
$666$	−29.1900	−1.13109
$667$	30.0628	1.16404
$668$	−0.348851	−0.0134975
$669$	41.4161	1.60124
$670$	−47.1000	−1.81963
$671$	34.4270	1.32904
$672$	7.78615	0.300357
$673$	−24.2109	−0.933262	−0.466631	−	0.884452i	$-0.654532\pi$
$673$	−24.2109	−0.933262	−0.466631	+	0.884452i	$0.654532\pi$
$674$	2.64933	0.102048
$675$	0.262647	0.0101093
$676$	4.62310	0.177811
$677$	−3.64316	−0.140018	−0.0700090	−	0.997546i	$-0.522303\pi$
$677$	−3.64316	−0.140018	−0.0700090	+	0.997546i	$0.522303\pi$
$678$	32.5499	1.25007
$679$	−1.54602	−0.0593309
$680$	27.0818	1.03854
$681$	−2.62821	−0.100713
$682$	−28.3193	−1.08440
$683$	35.1573	1.34526	0.672628	−	0.739981i	$-0.265166\pi$
$683$	35.1573	1.34526	0.672628	+	0.739981i	$0.265166\pi$
$684$	4.88231	0.186680
$685$	17.6073	0.672742
$686$	−22.7680	−0.869284
$687$	41.3078	1.57599
$688$	−30.2819	−1.15448
$689$	−53.4038	−2.03452
$690$	−72.4788	−2.75922
$691$	−6.49256	−0.246989	−0.123494	−	0.992345i	$-0.539410\pi$
$691$	−6.49256	−0.246989	−0.123494	+	0.992345i	$0.539410\pi$
$692$	2.66859	0.101445
$693$	14.7504	0.560321
$694$	−31.7658	−1.20581
$695$	−42.7014	−1.61976
$696$	18.6273	0.706065
$697$	37.0875	1.40479
$698$	−25.3074	−0.957899
$699$	−27.8705	−1.05416
$700$	0.104664	0.00395594
$701$	−37.3339	−1.41008	−0.705040	−	0.709167i	$-0.749071\pi$
$701$	−37.3339	−1.41008	−0.705040	+	0.709167i	$0.749071\pi$
$702$	11.3465	0.428245
$703$	−29.9616	−1.13002
$704$	26.9878	1.01714
$705$	36.5725	1.37740
$706$	−5.23869	−0.197161
$707$	1.05854	0.0398105
$708$	5.10370	0.191809
$709$	12.3533	0.463938	0.231969	−	0.972723i	$-0.425483\pi$
$709$	12.3533	0.463938	0.231969	+	0.972723i	$0.425483\pi$
$710$	−0.469231	−0.0176099
$711$	34.5017	1.29391
$712$	−10.0847	−0.377941
$713$	−27.1953	−1.01847
$714$	21.1830	0.792756
$715$	59.4414	2.22298
$716$	−7.16564	−0.267793
$717$	−47.2679	−1.76525
$718$	52.5268	1.96028
$719$	−30.9709	−1.15502	−0.577510	−	0.816383i	$-0.695976\pi$
$719$	−30.9709	−1.15502	−0.577510	+	0.816383i	$0.695976\pi$
$720$	−25.4624	−0.948928
$721$	−21.0213	−0.782873
$722$	−6.96712	−0.259289
$723$	5.74497	0.213658
$724$	−1.76503	−0.0655966
$725$	0.594926	0.0220950
$726$	−76.8139	−2.85083
$727$	41.1577	1.52646	0.763228	−	0.646129i	$-0.223613\pi$
$727$	41.1577	1.52646	0.763228	+	0.646129i	$0.223613\pi$
$728$	−12.0268	−0.445742
$729$	−14.0079	−0.518812
$730$	24.7582	0.916342
$731$	−32.4301	−1.19947
$732$	−7.70136	−0.284651
$733$	15.0958	0.557575	0.278787	−	0.960353i	$-0.410067\pi$
$733$	15.0958	0.557575	0.278787	+	0.960353i	$0.410067\pi$
$734$	−23.2126	−0.856791
$735$	30.1882	1.11351
$736$	−26.0410	−0.959882
$737$	−73.2389	−2.69779
$738$	−26.9207	−0.990965
$739$	13.4496	0.494753	0.247377	−	0.968919i	$-0.420431\pi$
$739$	13.4496	0.494753	0.247377	+	0.968919i	$0.420431\pi$
$740$	−9.73280	−0.357785
$741$	−40.9340	−1.50375
$742$	−20.5905	−0.755900
$743$	−1.89225	−0.0694198	−0.0347099	−	0.999397i	$-0.511051\pi$
$743$	−1.89225	−0.0694198	−0.0347099	+	0.999397i	$0.511051\pi$
$744$	−16.8506	−0.617772
$745$	13.2719	0.486244
$746$	−48.2467	−1.76644
$747$	15.1474	0.554213
$748$	−15.8320	−0.578875
$749$	7.14249	0.260981
$750$	40.4758	1.47797
$751$	−25.7910	−0.941128	−0.470564	−	0.882366i	$-0.655950\pi$
$751$	−25.7910	−0.941128	−0.470564	+	0.882366i	$0.655950\pi$
$752$	33.3810	1.21728
$753$	−19.0151	−0.692947
$754$	25.7010	0.935977
$755$	27.7471	1.00982
$756$	0.938813	0.0341443
$757$	9.60626	0.349146	0.174573	−	0.984644i	$-0.444146\pi$
$757$	9.60626	0.349146	0.174573	+	0.984644i	$0.444146\pi$
$758$	18.4110	0.668718
$759$	−112.702	−4.09082
$760$	−20.1775	−0.731916
$761$	3.35648	0.121672	0.0608362	−	0.998148i	$-0.480623\pi$
$761$	3.35648	0.121672	0.0608362	+	0.998148i	$0.480623\pi$
$762$	65.0170	2.35532
$763$	13.3680	0.483955
$764$	12.1162	0.438350
$765$	−27.2688	−0.985904
$766$	13.2598	0.479097
$767$	−18.7305	−0.676320
$768$	−28.2387	−1.01897
$769$	−39.6230	−1.42884	−0.714420	−	0.699717i	$-0.753310\pi$
$769$	−39.6230	−1.42884	−0.714420	+	0.699717i	$0.753310\pi$
$770$	22.9183	0.825919
$771$	−10.2026	−0.367438
$772$	2.63759	0.0949291
$773$	−5.24230	−0.188552	−0.0942762	−	0.995546i	$-0.530054\pi$
$773$	−5.24230	−0.188552	−0.0942762	+	0.995546i	$0.530054\pi$
$774$	23.5400	0.846129
$775$	−0.538181	−0.0193320
$776$	3.20384	0.115011
$777$	20.2494	0.726442
$778$	30.8819	1.10717
$779$	−27.6323	−0.990031
$780$	−13.2971	−0.476113
$781$	−0.729638	−0.0261085
$782$	−70.8471	−2.53349
$783$	5.33634	0.190705
$784$	27.5539	0.984066
$785$	−27.4087	−0.978258
$786$	15.8637	0.565840
$787$	−2.54440	−0.0906980	−0.0453490	−	0.998971i	$-0.514440\pi$
$787$	−2.54440	−0.0906980	−0.0453490	+	0.998971i	$0.514440\pi$
$788$	−1.67097	−0.0595259
$789$	9.84850	0.350616
$790$	53.6068	1.90724
$791$	−9.88400	−0.351434
$792$	−30.5674	−1.08616
$793$	28.2639	1.00368
$794$	19.9375	0.707554
$795$	60.5532	2.14760
$796$	−12.5126	−0.443497
$797$	−16.8841	−0.598065	−0.299033	−	0.954243i	$-0.596664\pi$
$797$	−16.8841	−0.598065	−0.299033	+	0.954243i	$0.596664\pi$
$798$	−15.7826	−0.558698
$799$	35.7491	1.26471
$800$	−0.515337	−0.0182199
$801$	10.1543	0.358786
$802$	34.8075	1.22909
$803$	38.4981	1.35857
$804$	16.3836	0.577806
$805$	22.0087	0.775705
$806$	−23.2496	−0.818934
$807$	53.6565	1.88880
$808$	−2.19362	−0.0771712
$809$	−3.36769	−0.118402	−0.0592008	−	0.998246i	$-0.518855\pi$
$809$	−3.36769	−0.118402	−0.0592008	+	0.998246i	$0.518855\pi$
$810$	−38.2906	−1.34540
$811$	−2.53677	−0.0890779	−0.0445390	−	0.999008i	$-0.514182\pi$
$811$	−2.53677	−0.0890779	−0.0445390	+	0.999008i	$0.514182\pi$
$812$	2.12652	0.0746262
$813$	42.6455	1.49564
$814$	−70.5235	−2.47185
$815$	11.9923	0.420073
$816$	−56.8598	−1.99049
$817$	24.1623	0.845332
$818$	1.59576	0.0557946
$819$	12.1098	0.423151
$820$	−8.97616	−0.313461
$821$	15.2527	0.532324	0.266162	−	0.963928i	$-0.414244\pi$
$821$	15.2527	0.532324	0.266162	+	0.963928i	$0.414244\pi$
$822$	−28.5403	−0.995457
$823$	−15.2327	−0.530978	−0.265489	−	0.964114i	$-0.585533\pi$
$823$	−15.2327	−0.530978	−0.265489	+	0.964114i	$0.585533\pi$
$824$	43.5625	1.51757
$825$	−2.23031	−0.0776495
$826$	−7.22177	−0.251278
$827$	−41.2946	−1.43595	−0.717977	−	0.696066i	$-0.754932\pi$
$827$	−41.2946	−1.43595	−0.717977	+	0.696066i	$0.754932\pi$
$828$	11.0359	0.383522
$829$	−0.908418	−0.0315507	−0.0157753	−	0.999876i	$-0.505022\pi$
$829$	−0.908418	−0.0315507	−0.0157753	+	0.999876i	$0.505022\pi$
$830$	23.5351	0.816917
$831$	20.1139	0.697744
$832$	22.1565	0.768137
$833$	29.5086	1.02241
$834$	69.2160	2.39675
$835$	−1.45168	−0.0502374
$836$	11.7957	0.407964
$837$	−4.82735	−0.166858
$838$	21.0266	0.726352
$839$	−24.0199	−0.829261	−0.414630	−	0.909990i	$-0.636089\pi$
$839$	−24.0199	−0.829261	−0.414630	+	0.909990i	$0.636089\pi$
$840$	13.6369	0.470517
$841$	−16.9126	−0.583192
$842$	14.2225	0.490141
$843$	−17.1639	−0.591157
$844$	8.86387	0.305107
$845$	19.2381	0.661812
$846$	−25.9492	−0.892153
$847$	23.3251	0.801460
$848$	55.2692	1.89795
$849$	−66.5838	−2.28515
$850$	−1.40203	−0.0480891
$851$	−67.7244	−2.32156
$852$	0.163221	0.00559186
$853$	35.2482	1.20687	0.603437	−	0.797411i	$-0.293797\pi$
$853$	35.2482	1.20687	0.603437	+	0.797411i	$0.293797\pi$
$854$	10.8975	0.372904
$855$	20.3168	0.694820
$856$	−14.8014	−0.505902
$857$	53.8046	1.83793	0.918965	−	0.394338i	$-0.129026\pi$
$857$	53.8046	1.83793	0.918965	+	0.394338i	$0.129026\pi$
$858$	−96.3504	−3.28935
$859$	42.6015	1.45355	0.726773	−	0.686878i	$-0.241019\pi$
$859$	42.6015	1.45355	0.726773	+	0.686878i	$0.241019\pi$
$860$	7.84894	0.267647
$861$	18.6752	0.636448
$862$	−18.1764	−0.619092
$863$	18.3270	0.623860	0.311930	−	0.950105i	$-0.399025\pi$
$863$	18.3270	0.623860	0.311930	+	0.950105i	$0.399025\pi$
$864$	−4.62244	−0.157259
$865$	11.1048	0.377576
$866$	40.5889	1.37927
$867$	−21.6257	−0.734446
$868$	−1.92369	−0.0652943
$869$	83.3567	2.82768
$870$	−29.1418	−0.987999
$871$	−60.1278	−2.03735
$872$	−27.7027	−0.938131
$873$	−3.22595	−0.109182
$874$	52.7852	1.78549
$875$	−12.2908	−0.415504
$876$	−8.61208	−0.290975
$877$	9.22122	0.311378	0.155689	−	0.987806i	$-0.450240\pi$
$877$	9.22122	0.311378	0.155689	+	0.987806i	$0.450240\pi$
$878$	42.6822	1.44045
$879$	12.5878	0.424577
$880$	−61.5176	−2.07376
$881$	9.85588	0.332053	0.166027	−	0.986121i	$-0.446906\pi$
$881$	9.85588	0.332053	0.166027	+	0.986121i	$0.446906\pi$
$882$	−21.4194	−0.721228
$883$	−28.6958	−0.965691	−0.482845	−	0.875706i	$-0.660397\pi$
$883$	−28.6958	−0.965691	−0.482845	+	0.875706i	$0.660397\pi$
$884$	−12.9978	−0.437162
$885$	21.2381	0.713910
$886$	−6.53636	−0.219593
$887$	41.0932	1.37978	0.689888	−	0.723916i	$-0.257660\pi$
$887$	41.0932	1.37978	0.689888	+	0.723916i	$0.257660\pi$
$888$	−41.9629	−1.40818
$889$	−19.7429	−0.662155
$890$	15.7772	0.528854
$891$	−59.5406	−1.99469
$892$	−9.79814	−0.328066
$893$	−26.6352	−0.891312
$894$	−21.5128	−0.719495
$895$	−29.8185	−0.996721
$896$	15.2843	0.510613
$897$	−92.5262	−3.08936
$898$	4.27449	0.142641
$899$	−10.9345	−0.364686
$900$	0.218394	0.00727979
$901$	59.1900	1.97191
$902$	−65.0409	−2.16563
$903$	−16.3300	−0.543427
$904$	20.4827	0.681244
$905$	−7.34482	−0.244150
$906$	−44.9761	−1.49423
$907$	−28.8950	−0.959443	−0.479722	−	0.877421i	$-0.659262\pi$
$907$	−28.8950	−0.959443	−0.479722	+	0.877421i	$0.659262\pi$
$908$	0.621777	0.0206344
$909$	2.20876	0.0732599
$910$	18.8155	0.623728
$911$	5.65818	0.187464	0.0937319	−	0.995597i	$-0.470120\pi$
$911$	5.65818	0.187464	0.0937319	+	0.995597i	$0.470120\pi$
$912$	42.3638	1.40281
$913$	36.5963	1.21116
$914$	−56.7618	−1.87751
$915$	−32.0478	−1.05947
$916$	−9.77252	−0.322893
$917$	−4.81713	−0.159076
$918$	−12.5758	−0.415065
$919$	20.2347	0.667483	0.333741	−	0.942665i	$-0.391689\pi$
$919$	20.2347	0.667483	0.333741	+	0.942665i	$0.391689\pi$
$920$	−45.6088	−1.50368
$921$	48.3189	1.59216
$922$	17.7873	0.585794
$923$	−0.599020	−0.0197170
$924$	−7.97209	−0.262262
$925$	−1.34023	−0.0440665
$926$	21.8333	0.717486
$927$	−43.8632	−1.44066
$928$	−10.4704	−0.343707
$929$	−21.6182	−0.709269	−0.354634	−	0.935005i	$-0.615395\pi$
$929$	−21.6182	−0.709269	−0.354634	+	0.935005i	$0.615395\pi$
$930$	26.3622	0.864450
$931$	−21.9856	−0.720548
$932$	6.59355	0.215979
$933$	11.5177	0.377071
$934$	9.91737	0.324506
$935$	−65.8818	−2.15456
$936$	−25.0952	−0.820263
$937$	−37.1276	−1.21291	−0.606453	−	0.795119i	$-0.707408\pi$
$937$	−37.1276	−1.21291	−0.606453	+	0.795119i	$0.707408\pi$
$938$	−23.1830	−0.756951
$939$	41.3031	1.34788
$940$	−8.65223	−0.282205
$941$	29.7695	0.970456	0.485228	−	0.874388i	$-0.338737\pi$
$941$	29.7695	0.970456	0.485228	+	0.874388i	$0.338737\pi$
$942$	44.4276	1.44753
$943$	−62.4594	−2.03396
$944$	19.3848	0.630920
$945$	3.90669	0.127085
$946$	56.8732	1.84911
$947$	−35.7140	−1.16055	−0.580274	−	0.814421i	$-0.697054\pi$
$947$	−35.7140	−1.16055	−0.580274	+	0.814421i	$0.697054\pi$
$948$	−18.6470	−0.605626
$949$	31.6062	1.02598
$950$	1.04459	0.0338910
$951$	−5.65569	−0.183398
$952$	13.3299	0.432024
$953$	−32.0434	−1.03799	−0.518993	−	0.854778i	$-0.673693\pi$
$953$	−32.0434	−1.03799	−0.518993	+	0.854778i	$0.673693\pi$
$954$	−42.9643	−1.39102
$955$	50.4195	1.63154
$956$	11.1825	0.361669
$957$	−45.3144	−1.46481
$958$	13.2468	0.427985
$959$	8.66646	0.279855
$960$	−25.1227	−0.810830
$961$	−21.1084	−0.680917
$962$	−57.8985	−1.86672
$963$	14.9036	0.480261
$964$	−1.35913	−0.0437747
$965$	10.9758	0.353325
$966$	−35.6746	−1.14781
$967$	35.2531	1.13366	0.566832	−	0.823834i	$-0.308169\pi$
$967$	35.2531	1.13366	0.566832	+	0.823834i	$0.308169\pi$
$968$	−48.3368	−1.55360
$969$	45.3692	1.45747
$970$	−5.01230	−0.160935
$971$	−14.6476	−0.470065	−0.235032	−	0.971988i	$-0.575520\pi$
$971$	−14.6476	−0.470065	−0.235032	+	0.971988i	$0.575520\pi$
$972$	10.8030	0.346507
$973$	−21.0179	−0.673804
$974$	−21.9049	−0.701878
$975$	−1.83105	−0.0586404
$976$	−29.2512	−0.936307
$977$	−56.6508	−1.81242	−0.906210	−	0.422828i	$-0.861037\pi$
$977$	−56.6508	−1.81242	−0.906210	+	0.422828i	$0.861037\pi$
$978$	−19.4388	−0.621583
$979$	24.5330	0.784079
$980$	−7.14185	−0.228138
$981$	27.8939	0.890583
$982$	47.5709	1.51805
$983$	58.3738	1.86184	0.930918	−	0.365228i	$-0.119009\pi$
$983$	58.3738	1.86184	0.930918	+	0.365228i	$0.119009\pi$
$984$	−38.7007	−1.23373
$985$	−6.95344	−0.221555
$986$	−28.4857	−0.907170
$987$	18.0012	0.572985
$988$	9.68408	0.308092
$989$	54.6159	1.73668
$990$	47.8216	1.51987
$991$	35.8945	1.14023	0.570114	−	0.821566i	$-0.306899\pi$
$991$	35.8945	1.14023	0.570114	+	0.821566i	$0.306899\pi$
$992$	9.47169	0.300726
$993$	60.5748	1.92228
$994$	−0.230959	−0.00732558
$995$	−52.0688	−1.65069
$996$	−8.18664	−0.259404
$997$	−17.8457	−0.565180	−0.282590	−	0.959241i	$-0.591194\pi$
$997$	−17.8457	−0.565180	−0.282590	+	0.959241i	$0.591194\pi$
$998$	34.0636	1.07826
$999$	−12.0215	−0.380345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	409.2.a.b.1.14	✓	20
3.2	odd	2		3681.2.a.i.1.7		20
4.3	odd	2		6544.2.a.i.1.17		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
409.2.a.b.1.14	✓	20	1.1	even	1	trivial
3681.2.a.i.1.7		20	3.2	odd	2
6544.2.a.i.1.17		20	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.59576 q^{2} -2.30987 q^{3} +0.546465 q^{4} +2.27401 q^{5} -3.68601 q^{6} +1.11928 q^{7} -2.31950 q^{8} +2.33551 q^{9} +3.62878 q^{10} +5.64263 q^{11} -1.26226 q^{12} +4.63250 q^{13} +1.78611 q^{14} -5.25267 q^{15} -4.79431 q^{16} -5.13442 q^{17} +3.72692 q^{18} +3.82544 q^{19} +1.24267 q^{20} -2.58540 q^{21} +9.00431 q^{22} +8.64693 q^{23} +5.35775 q^{24} +0.171118 q^{25} +7.39237 q^{26} +1.53489 q^{27} +0.611649 q^{28} +3.47670 q^{29} -8.38203 q^{30} -3.14509 q^{31} -3.01158 q^{32} -13.0338 q^{33} -8.19333 q^{34} +2.54526 q^{35} +1.27627 q^{36} -7.83219 q^{37} +6.10450 q^{38} -10.7005 q^{39} -5.27456 q^{40} -7.22331 q^{41} -4.12569 q^{42} +6.31621 q^{43} +3.08350 q^{44} +5.31097 q^{45} +13.7985 q^{46} -6.96264 q^{47} +11.0742 q^{48} -5.74720 q^{49} +0.273064 q^{50} +11.8599 q^{51} +2.53150 q^{52} -11.5281 q^{53} +2.44932 q^{54} +12.8314 q^{55} -2.59618 q^{56} -8.83628 q^{57} +5.54799 q^{58} -4.04329 q^{59} -2.87040 q^{60} +6.10123 q^{61} -5.01882 q^{62} +2.61410 q^{63} +4.78283 q^{64} +10.5343 q^{65} -20.7988 q^{66} -12.9796 q^{67} -2.80578 q^{68} -19.9733 q^{69} +4.06164 q^{70} -0.129308 q^{71} -5.41721 q^{72} +6.82272 q^{73} -12.4983 q^{74} -0.395261 q^{75} +2.09047 q^{76} +6.31571 q^{77} -17.0754 q^{78} +14.7727 q^{79} -10.9023 q^{80} -10.5519 q^{81} -11.5267 q^{82} +6.48568 q^{83} -1.41283 q^{84} -11.6757 q^{85} +10.0792 q^{86} -8.03073 q^{87} -13.0881 q^{88} +4.34780 q^{89} +8.47506 q^{90} +5.18508 q^{91} +4.72524 q^{92} +7.26475 q^{93} -11.1107 q^{94} +8.69909 q^{95} +6.95637 q^{96} -1.38126 q^{97} -9.17118 q^{98} +13.1784 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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