LMFDB - Embedded newform 755.2.a.j.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [755,2,Mod(1,755)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(755, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("755.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 755 = 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	755.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-2,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.02870535261$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 2 x^{14} - 22 x^{13} + 48 x^{12} + 171 x^{11} - 423 x^{10} - 527 x^{9} + 1641 x^{8} + 400 x^{7} + \cdots + 8 $ x^15 - 2x^14 - 22x^13 + 48x^12 + 171x^11 - 423x^10 - 527x^9 + 1641x^8 + 400x^7 - 2651x^6 + 421x^5 + 1591x^4 - 396x^3 - 288x^2 + 60x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.0966636$ of defining polynomial
Character	$\chi$	$=$	755.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.0966636 q^{2} -1.19614 q^{3} -1.99066 q^{4} +1.00000 q^{5} -0.115623 q^{6} +2.27227 q^{7} -0.385751 q^{8} -1.56924 q^{9} +0.0966636 q^{10} +1.59018 q^{11} +2.38111 q^{12} +0.511125 q^{13} +0.219645 q^{14} -1.19614 q^{15} +3.94402 q^{16} -4.87901 q^{17} -0.151689 q^{18} -2.76804 q^{19} -1.99066 q^{20} -2.71795 q^{21} +0.153712 q^{22} +0.727885 q^{23} +0.461413 q^{24} +1.00000 q^{25} +0.0494071 q^{26} +5.46547 q^{27} -4.52330 q^{28} +7.79700 q^{29} -0.115623 q^{30} +10.2709 q^{31} +1.15275 q^{32} -1.90208 q^{33} -0.471623 q^{34} +2.27227 q^{35} +3.12383 q^{36} +6.19960 q^{37} -0.267568 q^{38} -0.611378 q^{39} -0.385751 q^{40} +7.54761 q^{41} -0.262727 q^{42} +6.41231 q^{43} -3.16550 q^{44} -1.56924 q^{45} +0.0703600 q^{46} +1.87248 q^{47} -4.71761 q^{48} -1.83681 q^{49} +0.0966636 q^{50} +5.83599 q^{51} -1.01747 q^{52} +6.64777 q^{53} +0.528311 q^{54} +1.59018 q^{55} -0.876529 q^{56} +3.31096 q^{57} +0.753686 q^{58} -9.62671 q^{59} +2.38111 q^{60} -10.7110 q^{61} +0.992824 q^{62} -3.56574 q^{63} -7.77662 q^{64} +0.511125 q^{65} -0.183862 q^{66} +7.88901 q^{67} +9.71244 q^{68} -0.870654 q^{69} +0.219645 q^{70} -0.985313 q^{71} +0.605338 q^{72} +14.1335 q^{73} +0.599275 q^{74} -1.19614 q^{75} +5.51021 q^{76} +3.61331 q^{77} -0.0590980 q^{78} -5.80342 q^{79} +3.94402 q^{80} -1.82974 q^{81} +0.729579 q^{82} -12.6138 q^{83} +5.41051 q^{84} -4.87901 q^{85} +0.619837 q^{86} -9.32632 q^{87} -0.613413 q^{88} +16.8464 q^{89} -0.151689 q^{90} +1.16141 q^{91} -1.44897 q^{92} -12.2855 q^{93} +0.181000 q^{94} -2.76804 q^{95} -1.37885 q^{96} +1.32171 q^{97} -0.177553 q^{98} -2.49538 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 2 q^{2} + 5 q^{3} + 18 q^{4} + 15 q^{5} - 4 q^{6} + 11 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 11 q^{13} - 9 q^{14} + 5 q^{15} + 40 q^{16} + 25 q^{17} - 15 q^{18} - 3 q^{19}+ \cdots - 11 q^{99}+O(q^{100}) $ 15 * q - 2 * q^2 + 5 * q^3 + 18 * q^4 + 15 * q^5 - 4 * q^6 + 11 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^10 + 2 * q^11 + 4 * q^12 + 11 * q^13 - 9 * q^14 + 5 * q^15 + 40 * q^16 + 25 * q^17 - 15 * q^18 - 3 * q^19 + 18 * q^20 - 11 * q^21 + 13 * q^22 + 8 * q^23 - 21 * q^24 + 15 * q^25 - 45 * q^26 + 20 * q^27 + 29 * q^28 + 24 * q^29 - 4 * q^30 - 11 * q^31 + 27 * q^32 + 26 * q^33 - 11 * q^34 + 11 * q^35 - 22 * q^36 + 35 * q^37 + 2 * q^38 + 11 * q^39 + 12 * q^40 - 20 * q^41 - 17 * q^42 - 13 * q^43 - 32 * q^44 + 10 * q^45 - 5 * q^46 + 6 * q^47 - 11 * q^48 + 40 * q^49 - 2 * q^50 + 23 * q^51 + 27 * q^52 + 38 * q^53 - 30 * q^54 + 2 * q^55 - 36 * q^56 + 4 * q^57 + 6 * q^58 - 15 * q^59 + 4 * q^60 + 33 * q^61 - 13 * q^62 + 2 * q^63 + 74 * q^64 + 11 * q^65 + 8 * q^66 + 24 * q^67 + 8 * q^68 + 13 * q^69 - 9 * q^70 + 19 * q^71 + 56 * q^72 + 4 * q^73 - 28 * q^74 + 5 * q^75 + 19 * q^76 + 14 * q^77 - 15 * q^78 - 4 * q^79 + 40 * q^80 - 13 * q^81 + 33 * q^82 + 19 * q^83 - 71 * q^84 + 25 * q^85 - 59 * q^86 + 49 * q^87 - 48 * q^88 - 35 * q^89 - 15 * q^90 - q^91 + 18 * q^92 + 2 * q^93 - 70 * q^94 - 3 * q^95 - 35 * q^96 + 55 * q^97 - 64 * q^98 - 11 * q^99

Coefficient data

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

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

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

$2$	0.0966636	0.0683515	0.0341757	−	0.999416i	$-0.489119\pi$
$2$	0.0966636	0.0683515	0.0341757	+	0.999416i	$0.489119\pi$
$3$	−1.19614	−0.690593	−0.345296	−	0.938494i	$-0.612222\pi$
$3$	−1.19614	−0.690593	−0.345296	+	0.938494i	$0.612222\pi$
$4$	−1.99066	−0.995328
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.115623	−0.0472030
$7$	2.27227	0.858835	0.429418	−	0.903106i	$-0.358719\pi$
$7$	2.27227	0.858835	0.429418	+	0.903106i	$0.358719\pi$
$8$	−0.385751	−0.136384
$9$	−1.56924	−0.523081
$10$	0.0966636	0.0305677
$11$	1.59018	0.479457	0.239729	−	0.970840i	$-0.422942\pi$
$11$	1.59018	0.479457	0.239729	+	0.970840i	$0.422942\pi$
$12$	2.38111	0.687367
$13$	0.511125	0.141760	0.0708802	−	0.997485i	$-0.477419\pi$
$13$	0.511125	0.141760	0.0708802	+	0.997485i	$0.477419\pi$
$14$	0.219645	0.0587027
$15$	−1.19614	−0.308843
$16$	3.94402	0.986006
$17$	−4.87901	−1.18333	−0.591667	−	0.806182i	$-0.701530\pi$
$17$	−4.87901	−1.18333	−0.591667	+	0.806182i	$0.701530\pi$
$18$	−0.151689	−0.0357534
$19$	−2.76804	−0.635031	−0.317516	−	0.948253i	$-0.602849\pi$
$19$	−2.76804	−0.635031	−0.317516	+	0.948253i	$0.602849\pi$
$20$	−1.99066	−0.445124
$21$	−2.71795	−0.593106
$22$	0.153712	0.0327716
$23$	0.727885	0.151774	0.0758872	−	0.997116i	$-0.475821\pi$
$23$	0.727885	0.151774	0.0758872	+	0.997116i	$0.475821\pi$
$24$	0.461413	0.0941856
$25$	1.00000	0.200000
$26$	0.0494071	0.00968954
$27$	5.46547	1.05183
$28$	−4.52330	−0.854823
$29$	7.79700	1.44787	0.723933	−	0.689870i	$-0.242332\pi$
$29$	7.79700	1.44787	0.723933	+	0.689870i	$0.242332\pi$
$30$	−0.115623	−0.0211098
$31$	10.2709	1.84471	0.922356	−	0.386342i	$-0.126262\pi$
$31$	10.2709	1.84471	0.922356	+	0.386342i	$0.126262\pi$
$32$	1.15275	0.203779
$33$	−1.90208	−0.331110
$34$	−0.471623	−0.0808827
$35$	2.27227	0.384083
$36$	3.12383	0.520638
$37$	6.19960	1.01921	0.509604	−	0.860409i	$-0.329792\pi$
$37$	6.19960	1.01921	0.509604	+	0.860409i	$0.329792\pi$
$38$	−0.267568	−0.0434053
$39$	−0.611378	−0.0978988
$40$	−0.385751	−0.0609926
$41$	7.54761	1.17874	0.589369	−	0.807864i	$-0.299376\pi$
$41$	7.54761	1.17874	0.589369	+	0.807864i	$0.299376\pi$
$42$	−0.262727	−0.0405397
$43$	6.41231	0.977868	0.488934	−	0.872321i	$-0.337386\pi$
$43$	6.41231	0.977868	0.488934	+	0.872321i	$0.337386\pi$
$44$	−3.16550	−0.477217
$45$	−1.56924	−0.233929
$46$	0.0703600	0.0103740
$47$	1.87248	0.273129	0.136564	−	0.990631i	$-0.456394\pi$
$47$	1.87248	0.273129	0.136564	+	0.990631i	$0.456394\pi$
$48$	−4.71761	−0.680929
$49$	−1.83681	−0.262402
$50$	0.0966636	0.0136703
$51$	5.83599	0.817203
$52$	−1.01747	−0.141098
$53$	6.64777	0.913141	0.456571	−	0.889687i	$-0.349078\pi$
$53$	6.64777	0.913141	0.456571	+	0.889687i	$0.349078\pi$
$54$	0.528311	0.0718941
$55$	1.59018	0.214420
$56$	−0.876529	−0.117131
$57$	3.31096	0.438548
$58$	0.753686	0.0989638
$59$	−9.62671	−1.25329	−0.626645	−	0.779305i	$-0.715572\pi$
$59$	−9.62671	−1.25329	−0.626645	+	0.779305i	$0.715572\pi$
$60$	2.38111	0.307400
$61$	−10.7110	−1.37140	−0.685698	−	0.727886i	$-0.740503\pi$
$61$	−10.7110	−1.37140	−0.685698	+	0.727886i	$0.740503\pi$
$62$	0.992824	0.126089
$63$	−3.56574	−0.449241
$64$	−7.77662	−0.972077
$65$	0.511125	0.0633972
$66$	−0.183862	−0.0226318
$67$	7.88901	0.963796	0.481898	−	0.876227i	$-0.339948\pi$
$67$	7.88901	0.963796	0.481898	+	0.876227i	$0.339948\pi$
$68$	9.71244	1.17781
$69$	−0.870654	−0.104814
$70$	0.219645	0.0262526
$71$	−0.985313	−0.116935	−0.0584675	−	0.998289i	$-0.518621\pi$
$71$	−0.985313	−0.116935	−0.0584675	+	0.998289i	$0.518621\pi$
$72$	0.605338	0.0713397
$73$	14.1335	1.65421	0.827103	−	0.562050i	$-0.189987\pi$
$73$	14.1335	1.65421	0.827103	+	0.562050i	$0.189987\pi$
$74$	0.599275	0.0696643
$75$	−1.19614	−0.138119
$76$	5.51021	0.632064
$77$	3.61331	0.411775
$78$	−0.0590980	−0.00669153
$79$	−5.80342	−0.652935	−0.326468	−	0.945208i	$-0.605858\pi$
$79$	−5.80342	−0.652935	−0.326468	+	0.945208i	$0.605858\pi$
$80$	3.94402	0.440955
$81$	−1.82974	−0.203305
$82$	0.729579	0.0805685
$83$	−12.6138	−1.38455	−0.692273	−	0.721635i	$-0.743391\pi$
$83$	−12.6138	−1.38455	−0.692273	+	0.721635i	$0.743391\pi$
$84$	5.41051	0.590335
$85$	−4.87901	−0.529203
$86$	0.619837	0.0668387
$87$	−9.32632	−0.999886
$88$	−0.613413	−0.0653901
$89$	16.8464	1.78571	0.892855	−	0.450344i	$-0.148699\pi$
$89$	16.8464	1.78571	0.892855	+	0.450344i	$0.148699\pi$
$90$	−0.151689	−0.0159894
$91$	1.16141	0.121749
$92$	−1.44897	−0.151065
$93$	−12.2855	−1.27394
$94$	0.181000	0.0186687
$95$	−2.76804	−0.283995
$96$	−1.37885	−0.140728
$97$	1.32171	0.134199	0.0670997	−	0.997746i	$-0.478625\pi$
$97$	1.32171	0.134199	0.0670997	+	0.997746i	$0.478625\pi$
$98$	−0.177553	−0.0179355
$99$	−2.49538	−0.250795
$100$	−1.99066	−0.199066
$101$	−8.82633	−0.878253	−0.439126	−	0.898425i	$-0.644712\pi$
$101$	−8.82633	−0.878253	−0.439126	+	0.898425i	$0.644712\pi$
$102$	0.564128	0.0558570
$103$	−6.47124	−0.637631	−0.318815	−	0.947817i	$-0.603285\pi$
$103$	−6.47124	−0.637631	−0.318815	+	0.947817i	$0.603285\pi$
$104$	−0.197167	−0.0193338
$105$	−2.71795	−0.265245
$106$	0.642597	0.0624146
$107$	5.79841	0.560553	0.280277	−	0.959919i	$-0.409574\pi$
$107$	5.79841	0.560553	0.280277	+	0.959919i	$0.409574\pi$
$108$	−10.8799	−1.04692
$109$	−11.8131	−1.13149	−0.565743	−	0.824582i	$-0.691411\pi$
$109$	−11.8131	−1.13149	−0.565743	+	0.824582i	$0.691411\pi$
$110$	0.153712	0.0146559
$111$	−7.41560	−0.703857
$112$	8.96187	0.846817
$113$	17.9838	1.69177	0.845885	−	0.533365i	$-0.179073\pi$
$113$	17.9838	1.69177	0.845885	+	0.533365i	$0.179073\pi$
$114$	0.320050	0.0299754
$115$	0.727885	0.0678756
$116$	−15.5211	−1.44110
$117$	−0.802079	−0.0741523
$118$	−0.930552	−0.0856643
$119$	−11.0864	−1.01629
$120$	0.461413	0.0421211
$121$	−8.47133	−0.770121
$122$	−1.03536	−0.0937370
$123$	−9.02801	−0.814029
$124$	−20.4459	−1.83609
$125$	1.00000	0.0894427
$126$	−0.344677	−0.0307063
$127$	6.49484	0.576324	0.288162	−	0.957582i	$-0.406956\pi$
$127$	6.49484	0.576324	0.288162	+	0.957582i	$0.406956\pi$
$128$	−3.05721	−0.270222
$129$	−7.67003	−0.675308
$130$	0.0494071	0.00433329
$131$	−17.7180	−1.54803	−0.774015	−	0.633168i	$-0.781754\pi$
$131$	−17.7180	−1.54803	−0.774015	+	0.633168i	$0.781754\pi$
$132$	3.78639	0.329563
$133$	−6.28971	−0.545387
$134$	0.762580	0.0658769
$135$	5.46547	0.470392
$136$	1.88209	0.161387
$137$	2.15258	0.183907	0.0919536	−	0.995763i	$-0.470689\pi$
$137$	2.15258	0.183907	0.0919536	+	0.995763i	$0.470689\pi$
$138$	−0.0841605	−0.00716422
$139$	18.6282	1.58002	0.790011	−	0.613092i	$-0.210075\pi$
$139$	18.6282	1.58002	0.790011	+	0.613092i	$0.210075\pi$
$140$	−4.52330	−0.382289
$141$	−2.23975	−0.188621
$142$	−0.0952438	−0.00799269
$143$	0.812780	0.0679681
$144$	−6.18914	−0.515761
$145$	7.79700	0.647506
$146$	1.36620	0.113067
$147$	2.19709	0.181213
$148$	−12.3413	−1.01445
$149$	0.719970	0.0589822	0.0294911	−	0.999565i	$-0.490611\pi$
$149$	0.719970	0.0589822	0.0294911	+	0.999565i	$0.490611\pi$
$150$	−0.115623	−0.00944061
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	1.06777	0.0866078
$153$	7.65636	0.618980
$154$	0.349275	0.0281454
$155$	10.2709	0.824980
$156$	1.21704	0.0974414
$157$	20.0909	1.60343	0.801715	−	0.597707i	$-0.203921\pi$
$157$	20.0909	1.60343	0.801715	+	0.597707i	$0.203921\pi$
$158$	−0.560979	−0.0446291
$159$	−7.95168	−0.630609
$160$	1.15275	0.0911326
$161$	1.65395	0.130349
$162$	−0.176869	−0.0138962
$163$	−11.7359	−0.919224	−0.459612	−	0.888120i	$-0.652012\pi$
$163$	−11.7359	−0.919224	−0.459612	+	0.888120i	$0.652012\pi$
$164$	−15.0247	−1.17323
$165$	−1.90208	−0.148077
$166$	−1.21930	−0.0946358
$167$	−11.8796	−0.919267	−0.459634	−	0.888109i	$-0.652019\pi$
$167$	−11.8796	−0.919267	−0.459634	+	0.888109i	$0.652019\pi$
$168$	1.04845	0.0808899
$169$	−12.7388	−0.979904
$170$	−0.471623	−0.0361718
$171$	4.34372	0.332173
$172$	−12.7647	−0.973299
$173$	−11.9977	−0.912169	−0.456084	−	0.889937i	$-0.650749\pi$
$173$	−11.9977	−0.912169	−0.456084	+	0.889937i	$0.650749\pi$
$174$	−0.901515	−0.0683437
$175$	2.27227	0.171767
$176$	6.27171	0.472748
$177$	11.5149	0.865514
$178$	1.62843	0.122056
$179$	13.7491	1.02766	0.513828	−	0.857893i	$-0.328227\pi$
$179$	13.7491	1.02766	0.513828	+	0.857893i	$0.328227\pi$
$180$	3.12383	0.232836
$181$	18.4750	1.37324	0.686619	−	0.727017i	$-0.259094\pi$
$181$	18.4750	1.37324	0.686619	+	0.727017i	$0.259094\pi$
$182$	0.112266	0.00832172
$183$	12.8118	0.947077
$184$	−0.280782	−0.0206996
$185$	6.19960	0.455803
$186$	−1.18756	−0.0870760
$187$	−7.75851	−0.567358
$188$	−3.72745	−0.271853
$189$	12.4190	0.903348
$190$	−0.267568	−0.0194114
$191$	−7.50204	−0.542829	−0.271414	−	0.962463i	$-0.587491\pi$
$191$	−7.50204	−0.542829	−0.271414	+	0.962463i	$0.587491\pi$
$192$	9.30194	0.671310
$193$	24.8318	1.78743	0.893715	−	0.448635i	$-0.148090\pi$
$193$	24.8318	1.78743	0.893715	+	0.448635i	$0.148090\pi$
$194$	0.127761	0.00917272
$195$	−0.611378	−0.0437817
$196$	3.65646	0.261176
$197$	−5.22821	−0.372495	−0.186247	−	0.982503i	$-0.559633\pi$
$197$	−5.22821	−0.372495	−0.186247	+	0.982503i	$0.559633\pi$
$198$	−0.241212	−0.0171422
$199$	−18.5508	−1.31503	−0.657514	−	0.753442i	$-0.728392\pi$
$199$	−18.5508	−1.31503	−0.657514	+	0.753442i	$0.728392\pi$
$200$	−0.385751	−0.0272767
$201$	−9.43638	−0.665591
$202$	−0.853185	−0.0600299
$203$	17.7168	1.24348
$204$	−11.6175	−0.813385
$205$	7.54761	0.527148
$206$	−0.625534	−0.0435830
$207$	−1.14223	−0.0793904
$208$	2.01589	0.139777
$209$	−4.40167	−0.304470
$210$	−0.262727	−0.0181299
$211$	−14.6039	−1.00537	−0.502686	−	0.864469i	$-0.667655\pi$
$211$	−14.6039	−1.00537	−0.502686	+	0.864469i	$0.667655\pi$
$212$	−13.2334	−0.908875
$213$	1.17857	0.0807546
$214$	0.560495	0.0383147
$215$	6.41231	0.437316
$216$	−2.10831	−0.143452
$217$	23.3382	1.58430
$218$	−1.14189	−0.0773387
$219$	−16.9057	−1.14238
$220$	−3.16550	−0.213418
$221$	−2.49378	−0.167750
$222$	−0.716818	−0.0481097
$223$	16.1119	1.07893	0.539465	−	0.842008i	$-0.318626\pi$
$223$	16.1119	1.07893	0.539465	+	0.842008i	$0.318626\pi$
$224$	2.61934	0.175012
$225$	−1.56924	−0.104616
$226$	1.73838	0.115635
$227$	−17.7995	−1.18139	−0.590696	−	0.806894i	$-0.701147\pi$
$227$	−17.7995	−1.18139	−0.590696	+	0.806894i	$0.701147\pi$
$228$	−6.59099	−0.436499
$229$	15.0937	0.997417	0.498708	−	0.866770i	$-0.333808\pi$
$229$	15.0937	0.997417	0.498708	+	0.866770i	$0.333808\pi$
$230$	0.0703600	0.00463940
$231$	−4.32203	−0.284369
$232$	−3.00770	−0.197465
$233$	−25.9661	−1.70110	−0.850549	−	0.525895i	$-0.823730\pi$
$233$	−25.9661	−1.70110	−0.850549	+	0.525895i	$0.823730\pi$
$234$	−0.0775319	−0.00506842
$235$	1.87248	0.122147
$236$	19.1635	1.24744
$237$	6.94171	0.450913
$238$	−1.07165	−0.0694649
$239$	−12.1956	−0.788870	−0.394435	−	0.918924i	$-0.629060\pi$
$239$	−12.1956	−0.788870	−0.394435	+	0.918924i	$0.629060\pi$
$240$	−4.71761	−0.304521
$241$	−10.6823	−0.688109	−0.344055	−	0.938950i	$-0.611801\pi$
$241$	−10.6823	−0.688109	−0.344055	+	0.938950i	$0.611801\pi$
$242$	−0.818869	−0.0526389
$243$	−14.2078	−0.911429
$244$	21.3218	1.36499
$245$	−1.83681	−0.117350
$246$	−0.872680	−0.0556401
$247$	−1.41481	−0.0900223
$248$	−3.96202	−0.251588
$249$	15.0879	0.956158
$250$	0.0966636	0.00611354
$251$	20.2113	1.27572	0.637862	−	0.770151i	$-0.279819\pi$
$251$	20.2113	1.27572	0.637862	+	0.770151i	$0.279819\pi$
$252$	7.09816	0.447142
$253$	1.15747	0.0727694
$254$	0.627814	0.0393926
$255$	5.83599	0.365464
$256$	15.2577	0.953607
$257$	−7.75398	−0.483680	−0.241840	−	0.970316i	$-0.577751\pi$
$257$	−7.75398	−0.483680	−0.241840	+	0.970316i	$0.577751\pi$
$258$	−0.741413	−0.0461583
$259$	14.0871	0.875331
$260$	−1.01747	−0.0631010
$261$	−12.2354	−0.757352
$262$	−1.71269	−0.105810
$263$	−12.7590	−0.786756	−0.393378	−	0.919377i	$-0.628694\pi$
$263$	−12.7590	−0.786756	−0.393378	+	0.919377i	$0.628694\pi$
$264$	0.733730	0.0451579
$265$	6.64777	0.408369
$266$	−0.607986	−0.0372780
$267$	−20.1506	−1.23320
$268$	−15.7043	−0.959293
$269$	3.70890	0.226136	0.113068	−	0.993587i	$-0.463932\pi$
$269$	3.70890	0.226136	0.113068	+	0.993587i	$0.463932\pi$
$270$	0.528311	0.0321520
$271$	−18.5078	−1.12427	−0.562134	−	0.827047i	$-0.690019\pi$
$271$	−18.5078	−1.12427	−0.562134	+	0.827047i	$0.690019\pi$
$272$	−19.2430	−1.16678
$273$	−1.38921	−0.0840789
$274$	0.208076	0.0125703
$275$	1.59018	0.0958914
$276$	1.73317	0.104325
$277$	32.2642	1.93857	0.969284	−	0.245944i	$-0.0790980\pi$
$277$	32.2642	1.93857	0.969284	+	0.245944i	$0.0790980\pi$
$278$	1.80067	0.107997
$279$	−16.1176	−0.964934
$280$	−0.876529	−0.0523826
$281$	−9.20114	−0.548894	−0.274447	−	0.961602i	$-0.588495\pi$
$281$	−9.20114	−0.548894	−0.274447	+	0.961602i	$0.588495\pi$
$282$	−0.216502	−0.0128925
$283$	12.3214	0.732432	0.366216	−	0.930530i	$-0.380653\pi$
$283$	12.3214	0.732432	0.366216	+	0.930530i	$0.380653\pi$
$284$	1.96142	0.116389
$285$	3.31096	0.196125
$286$	0.0785662	0.00464572
$287$	17.1502	1.01234
$288$	−1.80894	−0.106593
$289$	6.80478	0.400281
$290$	0.753686	0.0442580
$291$	−1.58095	−0.0926771
$292$	−28.1350	−1.64648
$293$	−9.07694	−0.530280	−0.265140	−	0.964210i	$-0.585418\pi$
$293$	−9.07694	−0.530280	−0.265140	+	0.964210i	$0.585418\pi$
$294$	0.212378	0.0123862
$295$	−9.62671	−0.560489
$296$	−2.39150	−0.139003
$297$	8.69107	0.504307
$298$	0.0695949	0.00403152
$299$	0.372040	0.0215156
$300$	2.38111	0.137473
$301$	14.5705	0.839827
$302$	−0.0966636	−0.00556236
$303$	10.5575	0.606515
$304$	−10.9172	−0.626145
$305$	−10.7110	−0.613307
$306$	0.740092	0.0423082
$307$	11.6591	0.665420	0.332710	−	0.943029i	$-0.392037\pi$
$307$	11.6591	0.665420	0.332710	+	0.943029i	$0.392037\pi$
$308$	−7.19286	−0.409851
$309$	7.74053	0.440343
$310$	0.992824	0.0563886
$311$	−7.33067	−0.415684	−0.207842	−	0.978162i	$-0.566644\pi$
$311$	−7.33067	−0.415684	−0.207842	+	0.978162i	$0.566644\pi$
$312$	0.235840	0.0133518
$313$	6.71834	0.379743	0.189872	−	0.981809i	$-0.439193\pi$
$313$	6.71834	0.379743	0.189872	+	0.981809i	$0.439193\pi$
$314$	1.94206	0.109597
$315$	−3.56574	−0.200907
$316$	11.5526	0.649885
$317$	14.3752	0.807393	0.403696	−	0.914893i	$-0.367725\pi$
$317$	14.3752	0.807393	0.403696	+	0.914893i	$0.367725\pi$
$318$	−0.768638	−0.0431031
$319$	12.3986	0.694190
$320$	−7.77662	−0.434726
$321$	−6.93572	−0.387114
$322$	0.159876	0.00890957
$323$	13.5053	0.751454
$324$	3.64238	0.202355
$325$	0.511125	0.0283521
$326$	−1.13443	−0.0628303
$327$	14.1301	0.781396
$328$	−2.91150	−0.160761
$329$	4.25476	0.234572
$330$	−0.183862	−0.0101213
$331$	15.5871	0.856747	0.428374	−	0.903602i	$-0.359087\pi$
$331$	15.5871	0.856747	0.428374	+	0.903602i	$0.359087\pi$
$332$	25.1098	1.37808
$333$	−9.72868	−0.533128
$334$	−1.14832	−0.0628333
$335$	7.88901	0.431023
$336$	−10.7197	−0.584806
$337$	−7.86878	−0.428640	−0.214320	−	0.976764i	$-0.568753\pi$
$337$	−7.86878	−0.428640	−0.214320	+	0.976764i	$0.568753\pi$
$338$	−1.23137	−0.0669779
$339$	−21.5111	−1.16832
$340$	9.71244	0.526731
$341$	16.3326	0.884460
$342$	0.419880	0.0227045
$343$	−20.0796	−1.08420
$344$	−2.47355	−0.133365
$345$	−0.870654	−0.0468744
$346$	−1.15974	−0.0623481
$347$	−9.31806	−0.500220	−0.250110	−	0.968217i	$-0.580467\pi$
$347$	−9.31806	−0.500220	−0.250110	+	0.968217i	$0.580467\pi$
$348$	18.5655	0.995215
$349$	25.0571	1.34128	0.670639	−	0.741784i	$-0.266020\pi$
$349$	25.0571	1.34128	0.670639	+	0.741784i	$0.266020\pi$
$350$	0.219645	0.0117405
$351$	2.79353	0.149108
$352$	1.83307	0.0977031
$353$	13.5445	0.720903	0.360452	−	0.932778i	$-0.382623\pi$
$353$	13.5445	0.720903	0.360452	+	0.932778i	$0.382623\pi$
$354$	1.11307	0.0591591
$355$	−0.985313	−0.0522950
$356$	−33.5353	−1.77737
$357$	13.2609	0.701843
$358$	1.32904	0.0702418
$359$	−6.76969	−0.357290	−0.178645	−	0.983914i	$-0.557171\pi$
$359$	−6.76969	−0.357290	−0.178645	+	0.983914i	$0.557171\pi$
$360$	0.605338	0.0319041
$361$	−11.3380	−0.596735
$362$	1.78586	0.0938629
$363$	10.1329	0.531840
$364$	−2.31197	−0.121180
$365$	14.1335	0.739784
$366$	1.23844	0.0647341
$367$	−1.38006	−0.0720384	−0.0360192	−	0.999351i	$-0.511468\pi$
$367$	−1.38006	−0.0720384	−0.0360192	+	0.999351i	$0.511468\pi$
$368$	2.87080	0.149651
$369$	−11.8440	−0.616576
$370$	0.599275	0.0311548
$371$	15.1055	0.784238
$372$	24.4562	1.26799
$373$	11.0544	0.572373	0.286187	−	0.958174i	$-0.407612\pi$
$373$	11.0544	0.572373	0.286187	+	0.958174i	$0.407612\pi$
$374$	−0.749965	−0.0387798
$375$	−1.19614	−0.0617685
$376$	−0.722309	−0.0372503
$377$	3.98524	0.205250
$378$	1.20046	0.0617452
$379$	−2.04480	−0.105034	−0.0525170	−	0.998620i	$-0.516724\pi$
$379$	−2.04480	−0.105034	−0.0525170	+	0.998620i	$0.516724\pi$
$380$	5.51021	0.282668
$381$	−7.76875	−0.398005
$382$	−0.725174	−0.0371031
$383$	−29.5699	−1.51095	−0.755475	−	0.655177i	$-0.772594\pi$
$383$	−29.5699	−1.51095	−0.755475	+	0.655177i	$0.772594\pi$
$384$	3.65685	0.186613
$385$	3.61331	0.184151
$386$	2.40033	0.122173
$387$	−10.0625	−0.511504
$388$	−2.63107	−0.133572
$389$	−20.4711	−1.03792	−0.518962	−	0.854797i	$-0.673682\pi$
$389$	−20.4711	−1.03792	−0.518962	+	0.854797i	$0.673682\pi$
$390$	−0.0590980	−0.00299254
$391$	−3.55136	−0.179600
$392$	0.708552	0.0357873
$393$	21.1933	1.06906
$394$	−0.505378	−0.0254606
$395$	−5.80342	−0.292002
$396$	4.96744	0.249623
$397$	−8.54840	−0.429032	−0.214516	−	0.976720i	$-0.568817\pi$
$397$	−8.54840	−0.429032	−0.214516	+	0.976720i	$0.568817\pi$
$398$	−1.79318	−0.0898841
$399$	7.52339	0.376641
$400$	3.94402	0.197201
$401$	−4.04137	−0.201816	−0.100908	−	0.994896i	$-0.532175\pi$
$401$	−4.04137	−0.201816	−0.100908	+	0.994896i	$0.532175\pi$
$402$	−0.912154	−0.0454941
$403$	5.24972	0.261507
$404$	17.5702	0.874150
$405$	−1.82974	−0.0909206
$406$	1.71257	0.0849936
$407$	9.85847	0.488666
$408$	−2.25124	−0.111453
$409$	−3.23377	−0.159900	−0.0799499	−	0.996799i	$-0.525476\pi$
$409$	−3.23377	−0.159900	−0.0799499	+	0.996799i	$0.525476\pi$
$410$	0.729579	0.0360313
$411$	−2.57479	−0.127005
$412$	12.8820	0.634652
$413$	−21.8744	−1.07637
$414$	−0.110412	−0.00542645
$415$	−12.6138	−0.619188
$416$	0.589197	0.0288877
$417$	−22.2820	−1.09115
$418$	−0.425482	−0.0208110
$419$	−2.26889	−0.110843	−0.0554214	−	0.998463i	$-0.517650\pi$
$419$	−2.26889	−0.110843	−0.0554214	+	0.998463i	$0.517650\pi$
$420$	5.41051	0.264006
$421$	11.4331	0.557215	0.278607	−	0.960405i	$-0.410127\pi$
$421$	11.4331	0.557215	0.278607	+	0.960405i	$0.410127\pi$
$422$	−1.41166	−0.0687186
$423$	−2.93837	−0.142868
$424$	−2.56438	−0.124538
$425$	−4.87901	−0.236667
$426$	0.113925	0.00551969
$427$	−24.3381	−1.17780
$428$	−11.5426	−0.557935
$429$	−0.972200	−0.0469383
$430$	0.619837	0.0298912
$431$	−1.47623	−0.0711073	−0.0355536	−	0.999368i	$-0.511319\pi$
$431$	−1.47623	−0.0711073	−0.0355536	+	0.999368i	$0.511319\pi$
$432$	21.5559	1.03711
$433$	17.6811	0.849701	0.424850	−	0.905264i	$-0.360327\pi$
$433$	17.6811	0.849701	0.424850	+	0.905264i	$0.360327\pi$
$434$	2.25596	0.108289
$435$	−9.32632	−0.447163
$436$	23.5157	1.12620
$437$	−2.01481	−0.0963815
$438$	−1.63417	−0.0780836
$439$	6.07846	0.290109	0.145055	−	0.989424i	$-0.453664\pi$
$439$	6.07846	0.290109	0.145055	+	0.989424i	$0.453664\pi$
$440$	−0.613413	−0.0292433
$441$	2.88241	0.137257
$442$	−0.241058	−0.0114660
$443$	33.7945	1.60563	0.802813	−	0.596231i	$-0.203336\pi$
$443$	33.7945	1.60563	0.802813	+	0.596231i	$0.203336\pi$
$444$	14.7619	0.700569
$445$	16.8464	0.798594
$446$	1.55743	0.0737465
$447$	−0.861186	−0.0407327
$448$	−17.6705	−0.834855
$449$	29.4543	1.39003	0.695017	−	0.718993i	$-0.255397\pi$
$449$	29.4543	1.39003	0.695017	+	0.718993i	$0.255397\pi$
$450$	−0.151689	−0.00715068
$451$	12.0021	0.565155
$452$	−35.7995	−1.68387
$453$	1.19614	0.0561997
$454$	−1.72056	−0.0807499
$455$	1.16141	0.0544478
$456$	−1.27721	−0.0598108
$457$	32.3364	1.51263	0.756316	−	0.654206i	$-0.226997\pi$
$457$	32.3364	1.51263	0.756316	+	0.654206i	$0.226997\pi$
$458$	1.45901	0.0681749
$459$	−26.6661	−1.24467
$460$	−1.44897	−0.0675585
$461$	23.0033	1.07137	0.535686	−	0.844417i	$-0.320053\pi$
$461$	23.0033	1.07137	0.535686	+	0.844417i	$0.320053\pi$
$462$	−0.417783	−0.0194370
$463$	19.3557	0.899535	0.449767	−	0.893146i	$-0.351507\pi$
$463$	19.3557	0.899535	0.449767	+	0.893146i	$0.351507\pi$
$464$	30.7516	1.42761
$465$	−12.2855	−0.569725
$466$	−2.50998	−0.116273
$467$	−19.2813	−0.892231	−0.446115	−	0.894975i	$-0.647193\pi$
$467$	−19.2813	−0.892231	−0.446115	+	0.894975i	$0.647193\pi$
$468$	1.59666	0.0738058
$469$	17.9259	0.827742
$470$	0.181000	0.00834891
$471$	−24.0316	−1.10732
$472$	3.71351	0.170928
$473$	10.1967	0.468846
$474$	0.671011	0.0308205
$475$	−2.76804	−0.127006
$476$	22.0692	1.01154
$477$	−10.4320	−0.477647
$478$	−1.17887	−0.0539204
$479$	−36.0658	−1.64789	−0.823944	−	0.566671i	$-0.808231\pi$
$479$	−36.0658	−1.64789	−0.823944	+	0.566671i	$0.808231\pi$
$480$	−1.37885	−0.0629355
$481$	3.16877	0.144483
$482$	−1.03259	−0.0470333
$483$	−1.97836	−0.0900183
$484$	16.8635	0.766523
$485$	1.32171	0.0600158
$486$	−1.37337	−0.0622975
$487$	−8.25037	−0.373860	−0.186930	−	0.982373i	$-0.559854\pi$
$487$	−8.25037	−0.373860	−0.186930	+	0.982373i	$0.559854\pi$
$488$	4.13176	0.187036
$489$	14.0378	0.634810
$490$	−0.177553	−0.00802102
$491$	−32.4302	−1.46355	−0.731777	−	0.681544i	$-0.761309\pi$
$491$	−32.4302	−1.46355	−0.731777	+	0.681544i	$0.761309\pi$
$492$	17.9717	0.810226
$493$	−38.0417	−1.71331
$494$	−0.136761	−0.00615316
$495$	−2.49538	−0.112159
$496$	40.5087	1.81890
$497$	−2.23889	−0.100428
$498$	1.45845	0.0653548
$499$	30.2011	1.35199	0.675993	−	0.736908i	$-0.263715\pi$
$499$	30.2011	1.35199	0.675993	+	0.736908i	$0.263715\pi$
$500$	−1.99066	−0.0890248
$501$	14.2096	0.634840
$502$	1.95369	0.0871976
$503$	27.7054	1.23532	0.617662	−	0.786443i	$-0.288080\pi$
$503$	27.7054	1.23532	0.617662	+	0.786443i	$0.288080\pi$
$504$	1.37549	0.0612691
$505$	−8.82633	−0.392767
$506$	0.111885	0.00497389
$507$	15.2374	0.676715
$508$	−12.9290	−0.573631
$509$	−35.0506	−1.55359	−0.776795	−	0.629754i	$-0.783156\pi$
$509$	−35.0506	−1.55359	−0.776795	+	0.629754i	$0.783156\pi$
$510$	0.564128	0.0249800
$511$	32.1152	1.42069
$512$	7.58928	0.335402
$513$	−15.1286	−0.667944
$514$	−0.749527	−0.0330602
$515$	−6.47124	−0.285157
$516$	15.2684	0.672154
$517$	2.97757	0.130953
$518$	1.36171	0.0598302
$519$	14.3510	0.629937
$520$	−0.197167	−0.00864634
$521$	−14.7846	−0.647726	−0.323863	−	0.946104i	$-0.604982\pi$
$521$	−14.7846	−0.647726	−0.323863	+	0.946104i	$0.604982\pi$
$522$	−1.18272	−0.0517661
$523$	−11.3221	−0.495082	−0.247541	−	0.968877i	$-0.579623\pi$
$523$	−11.3221	−0.495082	−0.247541	+	0.968877i	$0.579623\pi$
$524$	35.2705	1.54080
$525$	−2.71795	−0.118621
$526$	−1.23333	−0.0537759
$527$	−50.1120	−2.18291
$528$	−7.50185	−0.326476
$529$	−22.4702	−0.976965
$530$	0.642597	0.0279126
$531$	15.1067	0.655573
$532$	12.5207	0.542839
$533$	3.85777	0.167099
$534$	−1.94783	−0.0842910
$535$	5.79841	0.250687
$536$	−3.04319	−0.131446
$537$	−16.4459	−0.709692
$538$	0.358515	0.0154567
$539$	−2.92086	−0.125810
$540$	−10.8799	−0.468195
$541$	15.5789	0.669790	0.334895	−	0.942255i	$-0.391299\pi$
$541$	15.5789	0.669790	0.334895	+	0.942255i	$0.391299\pi$
$542$	−1.78903	−0.0768453
$543$	−22.0988	−0.948349
$544$	−5.62426	−0.241138
$545$	−11.8131	−0.506016
$546$	−0.134286	−0.00574692
$547$	40.8523	1.74672	0.873359	−	0.487077i	$-0.161937\pi$
$547$	40.8523	1.74672	0.873359	+	0.487077i	$0.161937\pi$
$548$	−4.28504	−0.183048
$549$	16.8081	0.717352
$550$	0.153712	0.00655432
$551$	−21.5824	−0.919440
$552$	0.335856	0.0142950
$553$	−13.1869	−0.560764
$554$	3.11877	0.132504
$555$	−7.41560	−0.314775
$556$	−37.0823	−1.57264
$557$	−11.1921	−0.474224	−0.237112	−	0.971482i	$-0.576201\pi$
$557$	−11.1921	−0.474224	−0.237112	+	0.971482i	$0.576201\pi$
$558$	−1.55798	−0.0659547
$559$	3.27749	0.138623
$560$	8.96187	0.378708
$561$	9.28028	0.391814
$562$	−0.889415	−0.0375177
$563$	35.9528	1.51523	0.757615	−	0.652701i	$-0.226364\pi$
$563$	35.9528	1.51523	0.757615	+	0.652701i	$0.226364\pi$
$564$	4.45856	0.187739
$565$	17.9838	0.756583
$566$	1.19103	0.0500628
$567$	−4.15766	−0.174605
$568$	0.380085	0.0159480
$569$	19.2740	0.808009	0.404005	−	0.914757i	$-0.367618\pi$
$569$	19.2740	0.808009	0.404005	+	0.914757i	$0.367618\pi$
$570$	0.320050	0.0134054
$571$	−14.4897	−0.606376	−0.303188	−	0.952931i	$-0.598051\pi$
$571$	−14.4897	−0.606376	−0.303188	+	0.952931i	$0.598051\pi$
$572$	−1.61797	−0.0676505
$573$	8.97351	0.374874
$574$	1.65780	0.0691951
$575$	0.727885	0.0303549
$576$	12.2034	0.508476
$577$	23.8990	0.994929	0.497465	−	0.867484i	$-0.334264\pi$
$577$	23.8990	0.994929	0.497465	+	0.867484i	$0.334264\pi$
$578$	0.657774	0.0273598
$579$	−29.7023	−1.23439
$580$	−15.5211	−0.644480
$581$	−28.6619	−1.18910
$582$	−0.152821	−0.00633462
$583$	10.5711	0.437812
$584$	−5.45203	−0.225607
$585$	−0.802079	−0.0331619
$586$	−0.877409	−0.0362454
$587$	−29.1268	−1.20219	−0.601096	−	0.799177i	$-0.705269\pi$
$587$	−29.1268	−1.20219	−0.601096	+	0.799177i	$0.705269\pi$
$588$	−4.37365	−0.180366
$589$	−28.4303	−1.17145
$590$	−0.930552	−0.0383102
$591$	6.25368	0.257242
$592$	24.4514	1.00494
$593$	−6.03778	−0.247942	−0.123971	−	0.992286i	$-0.539563\pi$
$593$	−6.03778	−0.247942	−0.123971	+	0.992286i	$0.539563\pi$
$594$	0.840110	0.0344701
$595$	−11.0864	−0.454499
$596$	−1.43321	−0.0587067
$597$	22.1893	0.908149
$598$	0.0359627	0.00147062
$599$	−5.02575	−0.205347	−0.102673	−	0.994715i	$-0.532740\pi$
$599$	−5.02575	−0.205347	−0.102673	+	0.994715i	$0.532740\pi$
$600$	0.461413	0.0188371
$601$	−22.4024	−0.913815	−0.456907	−	0.889514i	$-0.651043\pi$
$601$	−22.4024	−0.913815	−0.456907	+	0.889514i	$0.651043\pi$
$602$	1.40843	0.0574034
$603$	−12.3798	−0.504144
$604$	1.99066	0.0809987
$605$	−8.47133	−0.344409
$606$	1.02053	0.0414562
$607$	−31.5451	−1.28038	−0.640188	−	0.768218i	$-0.721144\pi$
$607$	−31.5451	−1.28038	−0.640188	+	0.768218i	$0.721144\pi$
$608$	−3.19084	−0.129406
$609$	−21.1919	−0.858738
$610$	−1.03536	−0.0419205
$611$	0.957068	0.0387188
$612$	−15.2412	−0.616089
$613$	−5.61406	−0.226750	−0.113375	−	0.993552i	$-0.536166\pi$
$613$	−5.61406	−0.226750	−0.113375	+	0.993552i	$0.536166\pi$
$614$	1.12701	0.0454825
$615$	−9.02801	−0.364045
$616$	−1.39384	−0.0561593
$617$	−27.1765	−1.09408	−0.547042	−	0.837105i	$-0.684246\pi$
$617$	−27.1765	−1.09408	−0.547042	+	0.837105i	$0.684246\pi$
$618$	0.748227	0.0300981
$619$	15.7632	0.633576	0.316788	−	0.948496i	$-0.397396\pi$
$619$	15.7632	0.633576	0.316788	+	0.948496i	$0.397396\pi$
$620$	−20.4459	−0.821126
$621$	3.97823	0.159641
$622$	−0.708608	−0.0284126
$623$	38.2794	1.53363
$624$	−2.41129	−0.0965288
$625$	1.00000	0.0400000
$626$	0.649419	0.0259560
$627$	5.26503	0.210265
$628$	−39.9941	−1.59594
$629$	−30.2479	−1.20606
$630$	−0.344677	−0.0137323
$631$	−20.2509	−0.806177	−0.403088	−	0.915161i	$-0.632063\pi$
$631$	−20.2509	−0.806177	−0.403088	+	0.915161i	$0.632063\pi$
$632$	2.23867	0.0890497
$633$	17.4683	0.694303
$634$	1.38956	0.0551865
$635$	6.49484	0.257740
$636$	15.8291	0.627663
$637$	−0.938839	−0.0371982
$638$	1.19850	0.0474489
$639$	1.54620	0.0611666
$640$	−3.05721	−0.120847
$641$	−7.29725	−0.288224	−0.144112	−	0.989561i	$-0.546033\pi$
$641$	−7.29725	−0.288224	−0.144112	+	0.989561i	$0.546033\pi$
$642$	−0.670432	−0.0264598
$643$	26.1640	1.03181	0.515904	−	0.856646i	$-0.327456\pi$
$643$	26.1640	1.03181	0.515904	+	0.856646i	$0.327456\pi$
$644$	−3.29244	−0.129740
$645$	−7.67003	−0.302007
$646$	1.30547	0.0513630
$647$	−10.6515	−0.418753	−0.209377	−	0.977835i	$-0.567143\pi$
$647$	−10.6515	−0.418753	−0.209377	+	0.977835i	$0.567143\pi$
$648$	0.705825	0.0277274
$649$	−15.3082	−0.600899
$650$	0.0494071	0.00193791
$651$	−27.9159	−1.09411
$652$	23.3621	0.914930
$653$	−34.4823	−1.34940	−0.674699	−	0.738093i	$-0.735727\pi$
$653$	−34.4823	−1.34940	−0.674699	+	0.738093i	$0.735727\pi$
$654$	1.36587	0.0534096
$655$	−17.7180	−0.692300
$656$	29.7680	1.16224
$657$	−22.1790	−0.865285
$658$	0.411280	0.0160334
$659$	8.44721	0.329057	0.164528	−	0.986372i	$-0.447390\pi$
$659$	8.44721	0.329057	0.164528	+	0.986372i	$0.447390\pi$
$660$	3.78639	0.147385
$661$	13.6095	0.529350	0.264675	−	0.964338i	$-0.414735\pi$
$661$	13.6095	0.529350	0.264675	+	0.964338i	$0.414735\pi$
$662$	1.50671	0.0585599
$663$	2.98292	0.115847
$664$	4.86580	0.188829
$665$	−6.28971	−0.243905
$666$	−0.940409	−0.0364401
$667$	5.67532	0.219749
$668$	23.6481	0.914973
$669$	−19.2721	−0.745102
$670$	0.762580	0.0294610
$671$	−17.0323	−0.657526
$672$	−3.13311	−0.120862
$673$	−12.0350	−0.463917	−0.231958	−	0.972726i	$-0.574513\pi$
$673$	−12.0350	−0.463917	−0.231958	+	0.972726i	$0.574513\pi$
$674$	−0.760624	−0.0292981
$675$	5.46547	0.210366
$676$	25.3585	0.975326
$677$	19.2527	0.739942	0.369971	−	0.929043i	$-0.379368\pi$
$677$	19.2527	0.739942	0.369971	+	0.929043i	$0.379368\pi$
$678$	−2.07934	−0.0798567
$679$	3.00328	0.115255
$680$	1.88209	0.0721747
$681$	21.2907	0.815862
$682$	1.57877	0.0604541
$683$	3.72985	0.142719	0.0713593	−	0.997451i	$-0.477266\pi$
$683$	3.72985	0.142719	0.0713593	+	0.997451i	$0.477266\pi$
$684$	−8.64686	−0.330621
$685$	2.15258	0.0822458
$686$	−1.94096	−0.0741063
$687$	−18.0542	−0.688809
$688$	25.2903	0.964183
$689$	3.39784	0.129447
$690$	−0.0841605	−0.00320394
$691$	6.13982	0.233570	0.116785	−	0.993157i	$-0.462741\pi$
$691$	6.13982	0.233570	0.116785	+	0.993157i	$0.462741\pi$
$692$	23.8833	0.907907
$693$	−5.67016	−0.215392
$694$	−0.900717	−0.0341907
$695$	18.6282	0.706608
$696$	3.59764	0.136368
$697$	−36.8249	−1.39484
$698$	2.42211	0.0916783
$699$	31.0592	1.17477
$700$	−4.52330	−0.170965
$701$	−28.3951	−1.07247	−0.536233	−	0.844070i	$-0.680153\pi$
$701$	−28.3951	−1.07247	−0.536233	+	0.844070i	$0.680153\pi$
$702$	0.270033	0.0101917
$703$	−17.1607	−0.647228
$704$	−12.3662	−0.466069
$705$	−2.23975	−0.0843537
$706$	1.30926	0.0492748
$707$	−20.0558	−0.754275
$708$	−22.9222	−0.861470
$709$	25.4749	0.956730	0.478365	−	0.878161i	$-0.341230\pi$
$709$	25.4749	0.956730	0.478365	+	0.878161i	$0.341230\pi$
$710$	−0.0952438	−0.00357444
$711$	9.10698	0.341538
$712$	−6.49850	−0.243542
$713$	7.47605	0.279980
$714$	1.28185	0.0479720
$715$	0.812780	0.0303962
$716$	−27.3697	−1.02285
$717$	14.5877	0.544788
$718$	−0.654382	−0.0244213
$719$	13.1315	0.489723	0.244861	−	0.969558i	$-0.421258\pi$
$719$	13.1315	0.489723	0.244861	+	0.969558i	$0.421258\pi$
$720$	−6.18914	−0.230656
$721$	−14.7044	−0.547620
$722$	−1.09597	−0.0407877
$723$	12.7776	0.475203
$724$	−36.7774	−1.36682
$725$	7.79700	0.289573
$726$	0.979484	0.0363521
$727$	−43.0674	−1.59728	−0.798640	−	0.601809i	$-0.794447\pi$
$727$	−43.0674	−1.59728	−0.798640	+	0.601809i	$0.794447\pi$
$728$	−0.448015	−0.0166046
$729$	22.4837	0.832731
$730$	1.36620	0.0505653
$731$	−31.2857	−1.15714
$732$	−25.5039	−0.942652
$733$	20.7114	0.764992	0.382496	−	0.923957i	$-0.375065\pi$
$733$	20.7114	0.764992	0.382496	+	0.923957i	$0.375065\pi$
$734$	−0.133401	−0.00492393
$735$	2.19709	0.0810408
$736$	0.839066	0.0309284
$737$	12.5449	0.462099
$738$	−1.14489	−0.0421439
$739$	−8.52181	−0.313480	−0.156740	−	0.987640i	$-0.550098\pi$
$739$	−8.52181	−0.313480	−0.156740	+	0.987640i	$0.550098\pi$
$740$	−12.3413	−0.453674
$741$	1.69232	0.0621688
$742$	1.46015	0.0536038
$743$	23.3471	0.856520	0.428260	−	0.903656i	$-0.359127\pi$
$743$	23.3471	0.856520	0.428260	+	0.903656i	$0.359127\pi$
$744$	4.73914	0.173745
$745$	0.719970	0.0263777
$746$	1.06855	0.0391225
$747$	19.7942	0.724231
$748$	15.4445	0.564708
$749$	13.1755	0.481423
$750$	−0.115623	−0.00422197
$751$	−19.3870	−0.707441	−0.353721	−	0.935351i	$-0.615084\pi$
$751$	−19.3870	−0.707441	−0.353721	+	0.935351i	$0.615084\pi$
$752$	7.38509	0.269306
$753$	−24.1756	−0.881006
$754$	0.385227	0.0140292
$755$	−1.00000	−0.0363937
$756$	−24.7219	−0.899128
$757$	−18.5063	−0.672624	−0.336312	−	0.941751i	$-0.609180\pi$
$757$	−18.5063	−0.672624	−0.336312	+	0.941751i	$0.609180\pi$
$758$	−0.197657	−0.00717923
$759$	−1.38450	−0.0502540
$760$	1.06777	0.0387322
$761$	18.9370	0.686465	0.343233	−	0.939250i	$-0.388478\pi$
$761$	18.9370	0.686465	0.343233	+	0.939250i	$0.388478\pi$
$762$	−0.750955	−0.0272042
$763$	−26.8424	−0.971760
$764$	14.9340	0.540293
$765$	7.65636	0.276816
$766$	−2.85833	−0.103276
$767$	−4.92045	−0.177667
$768$	−18.2504	−0.658555
$769$	−10.0819	−0.363561	−0.181780	−	0.983339i	$-0.558186\pi$
$769$	−10.0819	−0.363561	−0.181780	+	0.983339i	$0.558186\pi$
$770$	0.349275	0.0125870
$771$	9.27486	0.334026
$772$	−49.4315	−1.77908
$773$	50.3202	1.80989	0.904945	−	0.425528i	$-0.139911\pi$
$773$	50.3202	1.80989	0.904945	+	0.425528i	$0.139911\pi$
$774$	−0.972675	−0.0349621
$775$	10.2709	0.368942
$776$	−0.509851	−0.0183026
$777$	−16.8502	−0.604498
$778$	−1.97881	−0.0709437
$779$	−20.8921	−0.748536
$780$	1.21704	0.0435771
$781$	−1.56682	−0.0560654
$782$	−0.343287	−0.0122759
$783$	42.6142	1.52291
$784$	−7.24443	−0.258730
$785$	20.0909	0.717075
$786$	2.04862	0.0730717
$787$	−7.40414	−0.263929	−0.131965	−	0.991254i	$-0.542129\pi$
$787$	−7.40414	−0.263929	−0.131965	+	0.991254i	$0.542129\pi$
$788$	10.4076	0.370754
$789$	15.2616	0.543328
$790$	−0.560979	−0.0199587
$791$	40.8639	1.45295
$792$	0.962595	0.0342043
$793$	−5.47463	−0.194410
$794$	−0.826319	−0.0293250
$795$	−7.95168	−0.282017
$796$	36.9282	1.30888
$797$	−44.6027	−1.57991	−0.789953	−	0.613167i	$-0.789895\pi$
$797$	−44.6027	−1.57991	−0.789953	+	0.613167i	$0.789895\pi$
$798$	0.727238	0.0257439
$799$	−9.13583	−0.323203
$800$	1.15275	0.0407557
$801$	−26.4360	−0.934072
$802$	−0.390654	−0.0137945
$803$	22.4749	0.793121
$804$	18.7846	0.662481
$805$	1.65395	0.0582940
$806$	0.507457	0.0178744
$807$	−4.43637	−0.156168
$808$	3.40477	0.119779
$809$	−1.69501	−0.0595935	−0.0297968	−	0.999556i	$-0.509486\pi$
$809$	−1.69501	−0.0595935	−0.0297968	+	0.999556i	$0.509486\pi$
$810$	−0.176869	−0.00621455
$811$	19.1531	0.672556	0.336278	−	0.941763i	$-0.390832\pi$
$811$	19.1531	0.672556	0.336278	+	0.941763i	$0.390832\pi$
$812$	−35.2682	−1.23767
$813$	22.1379	0.776411
$814$	0.952955	0.0334011
$815$	−11.7359	−0.411090
$816$	23.0173	0.805767
$817$	−17.7495	−0.620976
$818$	−0.312588	−0.0109294
$819$	−1.82254	−0.0636846
$820$	−15.0247	−0.524685
$821$	20.1450	0.703067	0.351534	−	0.936175i	$-0.385660\pi$
$821$	20.1450	0.703067	0.351534	+	0.936175i	$0.385660\pi$
$822$	−0.248888	−0.00868098
$823$	−45.3823	−1.58193	−0.790964	−	0.611863i	$-0.790421\pi$
$823$	−45.3823	−1.58193	−0.790964	+	0.611863i	$0.790421\pi$
$824$	2.49629	0.0869624
$825$	−1.90208	−0.0662219
$826$	−2.11446	−0.0735715
$827$	13.0922	0.455262	0.227631	−	0.973748i	$-0.426902\pi$
$827$	13.0922	0.455262	0.227631	+	0.973748i	$0.426902\pi$
$828$	2.27379	0.0790195
$829$	−35.9478	−1.24852	−0.624260	−	0.781217i	$-0.714599\pi$
$829$	−35.9478	−1.24852	−0.624260	+	0.781217i	$0.714599\pi$
$830$	−1.21930	−0.0423224
$831$	−38.5926	−1.33876
$832$	−3.97482	−0.137802
$833$	8.96183	0.310509
$834$	−2.15385	−0.0745819
$835$	−11.8796	−0.411109
$836$	8.76222	0.303048
$837$	56.1353	1.94032
$838$	−0.219319	−0.00757626
$839$	36.7395	1.26839	0.634194	−	0.773174i	$-0.281332\pi$
$839$	36.7395	1.26839	0.634194	+	0.773174i	$0.281332\pi$
$840$	1.04845	0.0361751
$841$	31.7932	1.09632
$842$	1.10516	0.0380865
$843$	11.0059	0.379062
$844$	29.0713	1.00067
$845$	−12.7388	−0.438226
$846$	−0.284033	−0.00976527
$847$	−19.2491	−0.661407
$848$	26.2190	0.900363
$849$	−14.7382	−0.505812
$850$	−0.471623	−0.0161765
$851$	4.51259	0.154690
$852$	−2.34614	−0.0803773
$853$	−6.34618	−0.217289	−0.108644	−	0.994081i	$-0.534651\pi$
$853$	−6.34618	−0.217289	−0.108644	+	0.994081i	$0.534651\pi$
$854$	−2.35261	−0.0805046
$855$	4.34372	0.148552
$856$	−2.23674	−0.0764503
$857$	−40.1034	−1.36991	−0.684953	−	0.728587i	$-0.740177\pi$
$857$	−40.1034	−1.36991	−0.684953	+	0.728587i	$0.740177\pi$
$858$	−0.0939763	−0.00320830
$859$	−36.4285	−1.24293	−0.621463	−	0.783444i	$-0.713461\pi$
$859$	−36.4285	−1.24293	−0.621463	+	0.783444i	$0.713461\pi$
$860$	−12.7647	−0.435273
$861$	−20.5140	−0.699117
$862$	−0.142697	−0.00486029
$863$	23.0286	0.783902	0.391951	−	0.919986i	$-0.371800\pi$
$863$	23.0286	0.783902	0.391951	+	0.919986i	$0.371800\pi$
$864$	6.30029	0.214340
$865$	−11.9977	−0.407934
$866$	1.70912	0.0580783
$867$	−8.13948	−0.276431
$868$	−46.4584	−1.57690
$869$	−9.22847	−0.313055
$870$	−0.901515	−0.0305642
$871$	4.03227	0.136628
$872$	4.55690	0.154316
$873$	−2.07409	−0.0701972
$874$	−0.194759	−0.00658782
$875$	2.27227	0.0768166
$876$	33.6535	1.13705
$877$	8.60574	0.290595	0.145298	−	0.989388i	$-0.453586\pi$
$877$	8.60574	0.290595	0.145298	+	0.989388i	$0.453586\pi$
$878$	0.587566	0.0198294
$879$	10.8573	0.366208
$880$	6.27171	0.211419
$881$	−51.2334	−1.72610	−0.863049	−	0.505120i	$-0.831448\pi$
$881$	−51.2334	−1.72610	−0.863049	+	0.505120i	$0.831448\pi$
$882$	0.278624	0.00938174
$883$	−1.97334	−0.0664080	−0.0332040	−	0.999449i	$-0.510571\pi$
$883$	−1.97334	−0.0664080	−0.0332040	+	0.999449i	$0.510571\pi$
$884$	4.96427	0.166966
$885$	11.5149	0.387069
$886$	3.26670	0.109747
$887$	−51.9823	−1.74540	−0.872698	−	0.488261i	$-0.837631\pi$
$887$	−51.9823	−1.74540	−0.872698	+	0.488261i	$0.837631\pi$
$888$	2.86058	0.0959946
$889$	14.7580	0.494967
$890$	1.62843	0.0545851
$891$	−2.90962	−0.0974758
$892$	−32.0732	−1.07389
$893$	−5.18308	−0.173445
$894$	−0.0832454	−0.00278414
$895$	13.7491	0.459582
$896$	−6.94679	−0.232076
$897$	−0.445013	−0.0148585
$898$	2.84716	0.0950109
$899$	80.0823	2.67090
$900$	3.12383	0.104128
$901$	−32.4346	−1.08055
$902$	1.16016	0.0386292
$903$	−17.4283	−0.579979
$904$	−6.93726	−0.230730
$905$	18.4750	0.614131
$906$	0.115623	0.00384133
$907$	−38.1905	−1.26810	−0.634048	−	0.773294i	$-0.718608\pi$
$907$	−38.1905	−1.26810	−0.634048	+	0.773294i	$0.718608\pi$
$908$	35.4326	1.17587
$909$	13.8507	0.459398
$910$	0.112266	0.00372159
$911$	20.6233	0.683281	0.341641	−	0.939831i	$-0.389017\pi$
$911$	20.6233	0.683281	0.341641	+	0.939831i	$0.389017\pi$
$912$	13.0585	0.432411
$913$	−20.0582	−0.663831
$914$	3.12575	0.103391
$915$	12.8118	0.423546
$916$	−30.0463	−0.992757
$917$	−40.2600	−1.32950
$918$	−2.57764	−0.0850748
$919$	31.3088	1.03278	0.516391	−	0.856353i	$-0.327275\pi$
$919$	31.3088	1.03278	0.516391	+	0.856353i	$0.327275\pi$
$920$	−0.280782	−0.00925712
$921$	−13.9459	−0.459535
$922$	2.22358	0.0732298
$923$	−0.503618	−0.0165768
$924$	8.60368	0.283040
$925$	6.19960	0.203841
$926$	1.87099	0.0614845
$927$	10.1550	0.333533
$928$	8.98796	0.295044
$929$	−42.3558	−1.38965	−0.694824	−	0.719180i	$-0.744518\pi$
$929$	−42.3558	−1.38965	−0.694824	+	0.719180i	$0.744518\pi$
$930$	−1.18756	−0.0389416
$931$	5.08436	0.166633
$932$	51.6897	1.69315
$933$	8.76852	0.287068
$934$	−1.86380	−0.0609853
$935$	−7.75851	−0.253730
$936$	0.309403	0.0101132
$937$	−54.0586	−1.76602	−0.883009	−	0.469356i	$-0.844486\pi$
$937$	−54.0586	−1.76602	−0.883009	+	0.469356i	$0.844486\pi$
$938$	1.73278	0.0565774
$939$	−8.03609	−0.262248
$940$	−3.72745	−0.121576
$941$	−22.7532	−0.741733	−0.370866	−	0.928686i	$-0.620939\pi$
$941$	−22.7532	−0.741733	−0.370866	+	0.928686i	$0.620939\pi$
$942$	−2.32298	−0.0756867
$943$	5.49379	0.178902
$944$	−37.9680	−1.23575
$945$	12.4190	0.403990
$946$	0.985651	0.0320463
$947$	−14.9319	−0.485222	−0.242611	−	0.970124i	$-0.578004\pi$
$947$	−14.9319	−0.485222	−0.242611	+	0.970124i	$0.578004\pi$
$948$	−13.8186	−0.448806
$949$	7.22400	0.234501
$950$	−0.267568	−0.00868106
$951$	−17.1948	−0.557580
$952$	4.27660	0.138605
$953$	48.5632	1.57312	0.786558	−	0.617516i	$-0.211861\pi$
$953$	48.5632	1.57312	0.786558	+	0.617516i	$0.211861\pi$
$954$	−1.00839	−0.0326479
$955$	−7.50204	−0.242760
$956$	24.2773	0.785184
$957$	−14.8305	−0.479403
$958$	−3.48625	−0.112636
$959$	4.89123	0.157946
$960$	9.30194	0.300219
$961$	74.4917	2.40296
$962$	0.306304	0.00987565
$963$	−9.09912	−0.293215
$964$	21.2648	0.684895
$965$	24.8318	0.799363
$966$	−0.191235	−0.00615289
$967$	11.6167	0.373567	0.186783	−	0.982401i	$-0.440194\pi$
$967$	11.6167	0.373567	0.186783	+	0.982401i	$0.440194\pi$
$968$	3.26782	0.105032
$969$	−16.1542	−0.518949
$970$	0.127761	0.00410217
$971$	−43.4080	−1.39303	−0.696515	−	0.717543i	$-0.745267\pi$
$971$	−43.4080	−1.39303	−0.696515	+	0.717543i	$0.745267\pi$
$972$	28.2828	0.907170
$973$	42.3282	1.35698
$974$	−0.797511	−0.0255539
$975$	−0.611378	−0.0195798
$976$	−42.2442	−1.35221
$977$	61.8457	1.97862	0.989309	−	0.145835i	$-0.0465870\pi$
$977$	61.8457	1.97862	0.989309	+	0.145835i	$0.0465870\pi$
$978$	1.35694	0.0433902
$979$	26.7887	0.856172
$980$	3.65646	0.116801
$981$	18.5376	0.591859
$982$	−3.13482	−0.100036
$983$	−27.4281	−0.874820	−0.437410	−	0.899262i	$-0.644104\pi$
$983$	−27.4281	−0.874820	−0.437410	+	0.899262i	$0.644104\pi$
$984$	3.48257	0.111020
$985$	−5.22821	−0.166585
$986$	−3.67724	−0.117107
$987$	−5.08930	−0.161994
$988$	2.81640	0.0896017
$989$	4.66742	0.148415
$990$	−0.241212	−0.00766623
$991$	−21.4283	−0.680693	−0.340347	−	0.940300i	$-0.610544\pi$
$991$	−21.4283	−0.680693	−0.340347	+	0.940300i	$0.610544\pi$
$992$	11.8398	0.375913
$993$	−18.6444	−0.591664
$994$	−0.216419	−0.00686440
$995$	−18.5508	−0.588098
$996$	−30.0349	−0.951691
$997$	−42.9189	−1.35926	−0.679628	−	0.733557i	$-0.737859\pi$
$997$	−42.9189	−1.35926	−0.679628	+	0.733557i	$0.737859\pi$
$998$	2.91935	0.0924103
$999$	33.8837	1.07203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	755.2.a.j.1.9	✓	15
3.2	odd	2		6795.2.a.bh.1.7		15
5.4	even	2		3775.2.a.q.1.7		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.a.j.1.9	✓	15	1.1	even	1	trivial
3775.2.a.q.1.7		15	5.4	even	2
6795.2.a.bh.1.7		15	3.2	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:	\( N \)	\(=\)	\( 755 = 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	755.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0966636 q^{2} -1.19614 q^{3} -1.99066 q^{4} +1.00000 q^{5} -0.115623 q^{6} +2.27227 q^{7} -0.385751 q^{8} -1.56924 q^{9} +0.0966636 q^{10} +1.59018 q^{11} +2.38111 q^{12} +0.511125 q^{13} +0.219645 q^{14} -1.19614 q^{15} +3.94402 q^{16} -4.87901 q^{17} -0.151689 q^{18} -2.76804 q^{19} -1.99066 q^{20} -2.71795 q^{21} +0.153712 q^{22} +0.727885 q^{23} +0.461413 q^{24} +1.00000 q^{25} +0.0494071 q^{26} +5.46547 q^{27} -4.52330 q^{28} +7.79700 q^{29} -0.115623 q^{30} +10.2709 q^{31} +1.15275 q^{32} -1.90208 q^{33} -0.471623 q^{34} +2.27227 q^{35} +3.12383 q^{36} +6.19960 q^{37} -0.267568 q^{38} -0.611378 q^{39} -0.385751 q^{40} +7.54761 q^{41} -0.262727 q^{42} +6.41231 q^{43} -3.16550 q^{44} -1.56924 q^{45} +0.0703600 q^{46} +1.87248 q^{47} -4.71761 q^{48} -1.83681 q^{49} +0.0966636 q^{50} +5.83599 q^{51} -1.01747 q^{52} +6.64777 q^{53} +0.528311 q^{54} +1.59018 q^{55} -0.876529 q^{56} +3.31096 q^{57} +0.753686 q^{58} -9.62671 q^{59} +2.38111 q^{60} -10.7110 q^{61} +0.992824 q^{62} -3.56574 q^{63} -7.77662 q^{64} +0.511125 q^{65} -0.183862 q^{66} +7.88901 q^{67} +9.71244 q^{68} -0.870654 q^{69} +0.219645 q^{70} -0.985313 q^{71} +0.605338 q^{72} +14.1335 q^{73} +0.599275 q^{74} -1.19614 q^{75} +5.51021 q^{76} +3.61331 q^{77} -0.0590980 q^{78} -5.80342 q^{79} +3.94402 q^{80} -1.82974 q^{81} +0.729579 q^{82} -12.6138 q^{83} +5.41051 q^{84} -4.87901 q^{85} +0.619837 q^{86} -9.32632 q^{87} -0.613413 q^{88} +16.8464 q^{89} -0.151689 q^{90} +1.16141 q^{91} -1.44897 q^{92} -12.2855 q^{93} +0.181000 q^{94} -2.76804 q^{95} -1.37885 q^{96} +1.32171 q^{97} -0.177553 q^{98} -2.49538 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 2 q^{2} + 5 q^{3} + 18 q^{4} + 15 q^{5} - 4 q^{6} + 11 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 11 q^{13} - 9 q^{14} + 5 q^{15} + 40 q^{16} + 25 q^{17} - 15 q^{18} - 3 q^{19}+ \cdots - 11 q^{99}+O(q^{100}) \) 15 * q - 2 * q^2 + 5 * q^3 + 18 * q^4 + 15 * q^5 - 4 * q^6 + 11 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^10 + 2 * q^11 + 4 * q^12 + 11 * q^13 - 9 * q^14 + 5 * q^15 + 40 * q^16 + 25 * q^17 - 15 * q^18 - 3 * q^19 + 18 * q^20 - 11 * q^21 + 13 * q^22 + 8 * q^23 - 21 * q^24 + 15 * q^25 - 45 * q^26 + 20 * q^27 + 29 * q^28 + 24 * q^29 - 4 * q^30 - 11 * q^31 + 27 * q^32 + 26 * q^33 - 11 * q^34 + 11 * q^35 - 22 * q^36 + 35 * q^37 + 2 * q^38 + 11 * q^39 + 12 * q^40 - 20 * q^41 - 17 * q^42 - 13 * q^43 - 32 * q^44 + 10 * q^45 - 5 * q^46 + 6 * q^47 - 11 * q^48 + 40 * q^49 - 2 * q^50 + 23 * q^51 + 27 * q^52 + 38 * q^53 - 30 * q^54 + 2 * q^55 - 36 * q^56 + 4 * q^57 + 6 * q^58 - 15 * q^59 + 4 * q^60 + 33 * q^61 - 13 * q^62 + 2 * q^63 + 74 * q^64 + 11 * q^65 + 8 * q^66 + 24 * q^67 + 8 * q^68 + 13 * q^69 - 9 * q^70 + 19 * q^71 + 56 * q^72 + 4 * q^73 - 28 * q^74 + 5 * q^75 + 19 * q^76 + 14 * q^77 - 15 * q^78 - 4 * q^79 + 40 * q^80 - 13 * q^81 + 33 * q^82 + 19 * q^83 - 71 * q^84 + 25 * q^85 - 59 * q^86 + 49 * q^87 - 48 * q^88 - 35 * q^89 - 15 * q^90 - q^91 + 18 * q^92 + 2 * q^93 - 70 * q^94 - 3 * q^95 - 35 * q^96 + 55 * q^97 - 64 * q^98 - 11 * q^99

Introduction

L-functions

Modular forms

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