LMFDB - Embedded newform 1875.2.b.e.1249.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 23x^{10} + 199x^{8} + 794x^{6} + 1399x^{4} + 783x^{2} + 81 $ x^12 + 23x^10 + 199x^8 + 794x^6 + 1399x^4 + 783*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.6
Root			$-0.364088i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.e.1249.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.364088i q^{2} +1.00000i q^{3} +1.86744 q^{4} +0.364088 q^{6} +2.24941i q^{7} -1.40809i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-0.364088i q^{2} +1.00000i q^{3} +1.86744 q^{4} +0.364088 q^{6} +2.24941i q^{7} -1.40809i q^{8} -1.00000 q^{9} +4.63962 q^{11} +1.86744i q^{12} -3.75730i q^{13} +0.818981 q^{14} +3.22221 q^{16} -5.36779i q^{17} +0.364088i q^{18} +5.66369 q^{19} -2.24941 q^{21} -1.68923i q^{22} -1.32214i q^{23} +1.40809 q^{24} -1.36799 q^{26} -1.00000i q^{27} +4.20063i q^{28} -8.86744 q^{29} +1.21200 q^{31} -3.98934i q^{32} +4.63962i q^{33} -1.95435 q^{34} -1.86744 q^{36} -2.15975i q^{37} -2.06208i q^{38} +3.75730 q^{39} +3.49965 q^{41} +0.818981i q^{42} -1.16800i q^{43} +8.66420 q^{44} -0.481374 q^{46} -2.36614i q^{47} +3.22221i q^{48} +1.94017 q^{49} +5.36779 q^{51} -7.01653i q^{52} +14.1656i q^{53} -0.364088 q^{54} +3.16736 q^{56} +5.66369i q^{57} +3.22853i q^{58} -0.367089 q^{59} +5.29590 q^{61} -0.441273i q^{62} -2.24941i q^{63} +4.99195 q^{64} +1.68923 q^{66} +8.06723i q^{67} -10.0240i q^{68} +1.32214 q^{69} +11.4185 q^{71} +1.40809i q^{72} +13.7580i q^{73} -0.786340 q^{74} +10.5766 q^{76} +10.4364i q^{77} -1.36799i q^{78} -11.6247 q^{79} +1.00000 q^{81} -1.27418i q^{82} -11.1027i q^{83} -4.20063 q^{84} -0.425253 q^{86} -8.86744i q^{87} -6.53299i q^{88} -16.6526 q^{89} +8.45169 q^{91} -2.46901i q^{92} +1.21200i q^{93} -0.861482 q^{94} +3.98934 q^{96} +14.4790i q^{97} -0.706393i q^{98} -4.63962 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 22 q^{4} - 2 q^{6} - 12 q^{9}+O(q^{10}) $ 12 * q - 22 * q^4 - 2 * q^6 - 12 * q^9	$ 12 q - 22 q^{4} - 2 q^{6} - 12 q^{9} + 8 q^{14} + 34 q^{16} + 4 q^{19} + 4 q^{21} + 12 q^{24} + 74 q^{26} - 62 q^{29} - 4 q^{31} - 74 q^{34} + 22 q^{36} + 66 q^{41} + 22 q^{44} - 24 q^{46} - 8 q^{49} - 4 q^{51} + 2 q^{54} - 60 q^{56} + 16 q^{59} + 68 q^{61} - 24 q^{64} - 18 q^{66} - 2 q^{69} - 6 q^{71} - 72 q^{74} + 54 q^{76} - 50 q^{79} + 12 q^{81} - 88 q^{84} - 60 q^{86} - 36 q^{89} + 56 q^{91} + 100 q^{94} - 66 q^{96}+O(q^{100}) $ 12 * q - 22 * q^4 - 2 * q^6 - 12 * q^9 + 8 * q^14 + 34 * q^16 + 4 * q^19 + 4 * q^21 + 12 * q^24 + 74 * q^26 - 62 * q^29 - 4 * q^31 - 74 * q^34 + 22 * q^36 + 66 * q^41 + 22 * q^44 - 24 * q^46 - 8 * q^49 - 4 * q^51 + 2 * q^54 - 60 * q^56 + 16 * q^59 + 68 * q^61 - 24 * q^64 - 18 * q^66 - 2 * q^69 - 6 * q^71 - 72 * q^74 + 54 * q^76 - 50 * q^79 + 12 * q^81 - 88 * q^84 - 60 * q^86 - 36 * q^89 + 56 * q^91 + 100 * q^94 - 66 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 0.364088i	− 0.257449i	−0.991680	−	0.128724i	$-0.958912\pi$
$2$	− 0.364088i	− 0.257449i	0.991680	−	0.128724i	$-0.0410883\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	1.86744	0.933720
$5$	0	0
$5$	0	0
$6$	0.364088	0.148638
$7$	2.24941i	0.850196i	0.905147	+	0.425098i	$0.139760\pi$
$7$	2.24941i	0.850196i	−0.905147	+	0.425098i	$0.860240\pi$
$8$	− 1.40809i	− 0.497834i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.63962	1.39890	0.699448	−	0.714683i	$-0.253429\pi$
$11$	4.63962	1.39890	0.699448	+	0.714683i	$0.253429\pi$
$12$	1.86744i	0.539084i
$13$	− 3.75730i	− 1.04209i	−0.853530	−	0.521044i	$-0.825543\pi$
$13$	− 3.75730i	− 1.04209i	0.853530	−	0.521044i	$-0.174457\pi$
$14$	0.818981	0.218882
$15$	0	0
$16$	3.22221	0.805553
$17$	− 5.36779i	− 1.30188i	−0.759129	−	0.650940i	$-0.774375\pi$
$17$	− 5.36779i	− 1.30188i	0.759129	−	0.650940i	$-0.225625\pi$
$18$	0.364088i	0.0858163i
$19$	5.66369	1.29934	0.649669	−	0.760217i	$-0.274907\pi$
$19$	5.66369	1.29934	0.649669	+	0.760217i	$0.274907\pi$
$20$	0	0
$21$	−2.24941	−0.490861
$22$	− 1.68923i	− 0.360144i
$23$	− 1.32214i	− 0.275685i	−0.990454	−	0.137842i	$-0.955983\pi$
$23$	− 1.32214i	− 0.275685i	0.990454	−	0.137842i	$-0.0440168\pi$
$24$	1.40809	0.287425
$25$	0	0
$26$	−1.36799	−0.268284
$27$	− 1.00000i	− 0.192450i
$28$	4.20063i	0.793845i
$29$	−8.86744	−1.64664	−0.823321	−	0.567576i	$-0.807881\pi$
$29$	−8.86744	−1.64664	−0.823321	+	0.567576i	$0.807881\pi$
$30$	0	0
$31$	1.21200	0.217681	0.108841	−	0.994059i	$-0.465286\pi$
$31$	1.21200	0.217681	0.108841	+	0.994059i	$0.465286\pi$
$32$	− 3.98934i	− 0.705223i
$33$	4.63962i	0.807653i
$34$	−1.95435	−0.335168
$35$	0	0
$36$	−1.86744	−0.311240
$37$	− 2.15975i	− 0.355061i	−0.984115	−	0.177531i	$-0.943189\pi$
$37$	− 2.15975i	− 0.355061i	0.984115	−	0.177531i	$-0.0568109\pi$
$38$	− 2.06208i	− 0.334513i
$39$	3.75730	0.601649
$40$	0	0
$41$	3.49965	0.546553	0.273277	−	0.961935i	$-0.411893\pi$
$41$	3.49965	0.546553	0.273277	+	0.961935i	$0.411893\pi$
$42$	0.818981i	0.126372i
$43$	− 1.16800i	− 0.178118i	−0.996026	−	0.0890589i	$-0.971614\pi$
$43$	− 1.16800i	− 0.178118i	0.996026	−	0.0890589i	$-0.0283859\pi$
$44$	8.66420	1.30618
$45$	0	0
$46$	−0.481374	−0.0709748
$47$	− 2.36614i	− 0.345137i	−0.984998	−	0.172568i	$-0.944793\pi$
$47$	− 2.36614i	− 0.345137i	0.984998	−	0.172568i	$-0.0552066\pi$
$48$	3.22221i	0.465086i
$49$	1.94017	0.277167
$50$	0	0
$51$	5.36779	0.751641
$52$	− 7.01653i	− 0.973018i
$53$	14.1656i	1.94580i	0.231223	+	0.972901i	$0.425727\pi$
$53$	14.1656i	1.94580i	−0.231223	+	0.972901i	$0.574273\pi$
$54$	−0.364088	−0.0495461
$55$	0	0
$56$	3.16736	0.423256
$57$	5.66369i	0.750174i
$58$	3.22853i	0.423926i
$59$	−0.367089	−0.0477909	−0.0238955	−	0.999714i	$-0.507607\pi$
$59$	−0.367089	−0.0477909	−0.0238955	+	0.999714i	$0.507607\pi$
$60$	0	0
$61$	5.29590	0.678070	0.339035	−	0.940774i	$-0.389899\pi$
$61$	5.29590	0.678070	0.339035	+	0.940774i	$0.389899\pi$
$62$	− 0.441273i	− 0.0560417i
$63$	− 2.24941i	− 0.283399i
$64$	4.99195	0.623994
$65$	0	0
$66$	1.68923	0.207930
$67$	8.06723i	0.985570i	0.870151	+	0.492785i	$0.164021\pi$
$67$	8.06723i	0.985570i	−0.870151	+	0.492785i	$0.835979\pi$
$68$	− 10.0240i	− 1.21559i
$69$	1.32214	0.159167
$70$	0	0
$71$	11.4185	1.35513	0.677563	−	0.735465i	$-0.263036\pi$
$71$	11.4185	1.35513	0.677563	+	0.735465i	$0.263036\pi$
$72$	1.40809i	0.165945i
$73$	13.7580i	1.61025i	0.593104	+	0.805126i	$0.297902\pi$
$73$	13.7580i	1.61025i	−0.593104	+	0.805126i	$0.702098\pi$
$74$	−0.786340	−0.0914101
$75$	0	0
$76$	10.5766	1.21322
$77$	10.4364i	1.18934i
$78$	− 1.36799i	− 0.154894i
$79$	−11.6247	−1.30789	−0.653943	−	0.756544i	$-0.726886\pi$
$79$	−11.6247	−1.30789	−0.653943	+	0.756544i	$0.726886\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 1.27418i	− 0.140710i
$83$	− 11.1027i	− 1.21868i	−0.792910	−	0.609338i	$-0.791435\pi$
$83$	− 11.1027i	− 1.21868i	0.792910	−	0.609338i	$-0.208565\pi$
$84$	−4.20063	−0.458326
$85$	0	0
$86$	−0.425253	−0.0458562
$87$	− 8.86744i	− 0.950689i
$88$	− 6.53299i	− 0.696419i
$89$	−16.6526	−1.76518	−0.882588	−	0.470148i	$-0.844201\pi$
$89$	−16.6526	−1.76518	−0.882588	+	0.470148i	$0.844201\pi$
$90$	0	0
$91$	8.45169	0.885978
$92$	− 2.46901i	− 0.257412i
$93$	1.21200i	0.125678i
$94$	−0.861482	−0.0888551
$95$	0	0
$96$	3.98934	0.407161
$97$	14.4790i	1.47012i	0.678001	+	0.735061i	$0.262847\pi$
$97$	14.4790i	1.47012i	−0.678001	+	0.735061i	$0.737153\pi$
$98$	− 0.706393i	− 0.0713565i
$99$	−4.63962	−0.466299
$100$	0	0
$101$	2.16089	0.215017	0.107508	−	0.994204i	$-0.465713\pi$
$101$	2.16089	0.215017	0.107508	+	0.994204i	$0.465713\pi$
$102$	− 1.95435i	− 0.193509i
$103$	7.13512i	0.703044i	0.936180	+	0.351522i	$0.114336\pi$
$103$	7.13512i	0.703044i	−0.936180	+	0.351522i	$0.885664\pi$
$104$	−5.29061	−0.518787
$105$	0	0
$106$	5.15754	0.500944
$107$	− 14.3985i	− 1.39196i	−0.718062	−	0.695979i	$-0.754971\pi$
$107$	− 14.3985i	− 1.39196i	0.718062	−	0.695979i	$-0.245029\pi$
$108$	− 1.86744i	− 0.179695i
$109$	−16.4693	−1.57748	−0.788738	−	0.614730i	$-0.789265\pi$
$109$	−16.4693	−1.57748	−0.788738	+	0.614730i	$0.789265\pi$
$110$	0	0
$111$	2.15975	0.204995
$112$	7.24806i	0.684878i
$113$	− 16.2639i	− 1.52998i	−0.644042	−	0.764990i	$-0.722744\pi$
$113$	− 16.2639i	− 1.52998i	0.644042	−	0.764990i	$-0.277256\pi$
$114$	2.06208	0.193131
$115$	0	0
$116$	−16.5594	−1.53750
$117$	3.75730i	0.347362i
$118$	0.133653i	0.0123037i
$119$	12.0743	1.10685
$120$	0	0
$121$	10.5260	0.956912
$122$	− 1.92817i	− 0.174568i
$123$	3.49965i	0.315553i
$124$	2.26333	0.203253
$125$	0	0
$126$	−0.818981	−0.0729606
$127$	5.14048i	0.456144i	0.973644	+	0.228072i	$0.0732422\pi$
$127$	5.14048i	0.456144i	−0.973644	+	0.228072i	$0.926758\pi$
$128$	− 9.79620i	− 0.865870i
$129$	1.16800	0.102836
$130$	0	0
$131$	6.52438	0.570038	0.285019	−	0.958522i	$-0.408000\pi$
$131$	6.52438	0.570038	0.285019	+	0.958522i	$0.408000\pi$
$132$	8.66420i	0.754122i
$133$	12.7399i	1.10469i
$134$	2.93718	0.253734
$135$	0	0
$136$	−7.55832	−0.648121
$137$	13.5071i	1.15399i	0.816749	+	0.576993i	$0.195774\pi$
$137$	13.5071i	1.15399i	−0.816749	+	0.576993i	$0.804226\pi$
$138$	− 0.481374i	− 0.0409773i
$139$	21.1376	1.79287	0.896435	−	0.443176i	$-0.146148\pi$
$139$	21.1376	1.79287	0.896435	+	0.443176i	$0.146148\pi$
$140$	0	0
$141$	2.36614	0.199265
$142$	− 4.15733i	− 0.348876i
$143$	− 17.4324i	− 1.45777i
$144$	−3.22221	−0.268518
$145$	0	0
$146$	5.00912	0.414558
$147$	1.94017i	0.160023i
$148$	− 4.03321i	− 0.331528i
$149$	3.22853	0.264491	0.132246	−	0.991217i	$-0.457781\pi$
$149$	3.22853	0.264491	0.132246	+	0.991217i	$0.457781\pi$
$150$	0	0
$151$	−15.9240	−1.29588	−0.647939	−	0.761692i	$-0.724369\pi$
$151$	−15.9240	−1.29588	−0.647939	+	0.761692i	$0.724369\pi$
$152$	− 7.97497i	− 0.646855i
$153$	5.36779i	0.433960i
$154$	3.79976	0.306193
$155$	0	0
$156$	7.01653	0.561772
$157$	3.21251i	0.256387i	0.991749	+	0.128193i	$0.0409178\pi$
$157$	3.21251i	0.256387i	−0.991749	+	0.128193i	$0.959082\pi$
$158$	4.23243i	0.336714i
$159$	−14.1656	−1.12341
$160$	0	0
$161$	2.97403	0.234386
$162$	− 0.364088i	− 0.0286054i
$163$	13.7482i	1.07685i	0.842675	+	0.538423i	$0.180980\pi$
$163$	13.7482i	1.07685i	−0.842675	+	0.538423i	$0.819020\pi$
$164$	6.53538	0.510328
$165$	0	0
$166$	−4.04235	−0.313747
$167$	0.787486i	0.0609375i	0.999536	+	0.0304688i	$0.00970001\pi$
$167$	0.787486i	0.0609375i	−0.999536	+	0.0304688i	$0.990300\pi$
$168$	3.16736i	0.244367i
$169$	−1.11729	−0.0859456
$170$	0	0
$171$	−5.66369	−0.433113
$172$	− 2.18116i	− 0.166312i
$173$	− 2.58986i	− 0.196904i	−0.995142	−	0.0984518i	$-0.968611\pi$
$173$	− 2.58986i	− 0.196904i	0.995142	−	0.0984518i	$-0.0313890\pi$
$174$	−3.22853	−0.244754
$175$	0	0
$176$	14.9498	1.12689
$177$	− 0.367089i	− 0.0275921i
$178$	6.06302i	0.454443i
$179$	−15.0630	−1.12586	−0.562930	−	0.826505i	$-0.690326\pi$
$179$	−15.0630	−1.12586	−0.562930	+	0.826505i	$0.690326\pi$
$180$	0	0
$181$	−12.5492	−0.932771	−0.466386	−	0.884582i	$-0.654444\pi$
$181$	−12.5492	−0.932771	−0.466386	+	0.884582i	$0.654444\pi$
$182$	− 3.07716i	− 0.228094i
$183$	5.29590i	0.391484i
$184$	−1.86169	−0.137245
$185$	0	0
$186$	0.441273	0.0323557
$187$	− 24.9045i	− 1.82120i
$188$	− 4.41862i	− 0.322261i
$189$	2.24941	0.163620
$190$	0	0
$191$	16.0684	1.16267	0.581334	−	0.813665i	$-0.302531\pi$
$191$	16.0684	1.16267	0.581334	+	0.813665i	$0.302531\pi$
$192$	4.99195i	0.360263i
$193$	− 8.48878i	− 0.611036i	−0.952186	−	0.305518i	$-0.901170\pi$
$193$	− 8.48878i	− 0.611036i	0.952186	−	0.305518i	$-0.0988296\pi$
$194$	5.27164	0.378481
$195$	0	0
$196$	3.62316	0.258797
$197$	− 4.68061i	− 0.333479i	−0.986001	−	0.166740i	$-0.946676\pi$
$197$	− 4.68061i	− 0.333479i	0.986001	−	0.166740i	$-0.0533240\pi$
$198$	1.68923i	0.120048i
$199$	−1.45880	−0.103411	−0.0517057	−	0.998662i	$-0.516466\pi$
$199$	−1.45880	−0.103411	−0.0517057	+	0.998662i	$0.516466\pi$
$200$	0	0
$201$	−8.06723	−0.569019
$202$	− 0.786753i	− 0.0553558i
$203$	− 19.9465i	− 1.39997i
$204$	10.0240	0.701822
$205$	0	0
$206$	2.59781	0.180998
$207$	1.32214i	0.0918950i
$208$	− 12.1068i	− 0.839457i
$209$	26.2773	1.81764
$210$	0	0
$211$	12.8107	0.881926	0.440963	−	0.897525i	$-0.354637\pi$
$211$	12.8107	0.881926	0.440963	+	0.897525i	$0.354637\pi$
$212$	26.4535i	1.81683i
$213$	11.4185i	0.782382i
$214$	−5.24233	−0.358358
$215$	0	0
$216$	−1.40809	−0.0958082
$217$	2.72627i	0.185071i
$218$	5.99628i	0.406119i
$219$	−13.7580	−0.929680
$220$	0	0
$221$	−20.1684	−1.35667
$222$	− 0.786340i	− 0.0527757i
$223$	0.223725i	0.0149817i	0.999972	+	0.00749086i	$0.00238444\pi$
$223$	0.223725i	0.0149817i	−0.999972	+	0.00749086i	$0.997616\pi$
$224$	8.97365	0.599577
$225$	0	0
$226$	−5.92149	−0.393892
$227$	28.2672i	1.87616i	0.346416	+	0.938081i	$0.387398\pi$
$227$	28.2672i	1.87616i	−0.346416	+	0.938081i	$0.612602\pi$
$228$	10.5766i	0.700452i
$229$	16.6618	1.10104	0.550521	−	0.834821i	$-0.314429\pi$
$229$	16.6618	1.10104	0.550521	+	0.834821i	$0.314429\pi$
$230$	0	0
$231$	−10.4364	−0.686663
$232$	12.4861i	0.819755i
$233$	− 19.9807i	− 1.30898i	−0.756070	−	0.654491i	$-0.772883\pi$
$233$	− 19.9807i	− 1.30898i	0.756070	−	0.654491i	$-0.227117\pi$
$234$	1.36799	0.0894281
$235$	0	0
$236$	−0.685517	−0.0446233
$237$	− 11.6247i	− 0.755108i
$238$	− 4.39612i	− 0.284958i
$239$	−18.6586	−1.20692	−0.603462	−	0.797392i	$-0.706212\pi$
$239$	−18.6586	−1.20692	−0.603462	+	0.797392i	$0.706212\pi$
$240$	0	0
$241$	7.10919	0.457943	0.228972	−	0.973433i	$-0.426464\pi$
$241$	7.10919	0.457943	0.228972	+	0.973433i	$0.426464\pi$
$242$	− 3.83240i	− 0.246356i
$243$	1.00000i	0.0641500i
$244$	9.88977	0.633128
$245$	0	0
$246$	1.27418	0.0812387
$247$	− 21.2802i	− 1.35402i
$248$	− 1.70660i	− 0.108369i
$249$	11.1027	0.703603
$250$	0	0
$251$	−10.5636	−0.666767	−0.333384	−	0.942791i	$-0.608190\pi$
$251$	−10.5636	−0.666767	−0.333384	+	0.942791i	$0.608190\pi$
$252$	− 4.20063i	− 0.264615i
$253$	− 6.13421i	− 0.385655i
$254$	1.87159	0.117434
$255$	0	0
$256$	6.41723	0.401077
$257$	− 2.94764i	− 0.183869i	−0.995765	−	0.0919344i	$-0.970695\pi$
$257$	− 2.94764i	− 0.183869i	0.995765	−	0.0919344i	$-0.0293050\pi$
$258$	− 0.425253i	− 0.0264751i
$259$	4.85816	0.301872
$260$	0	0
$261$	8.86744	0.548881
$262$	− 2.37545i	− 0.146756i
$263$	0.220418i	0.0135916i	0.999977	+	0.00679578i	$0.00216318\pi$
$263$	0.220418i	0.0135916i	−0.999977	+	0.00679578i	$0.997837\pi$
$264$	6.53299	0.402077
$265$	0	0
$266$	4.63845	0.284402
$267$	− 16.6526i	− 1.01912i
$268$	15.0651i	0.920246i
$269$	−10.4148	−0.634999	−0.317499	−	0.948258i	$-0.602843\pi$
$269$	−10.4148	−0.634999	−0.317499	+	0.948258i	$0.602843\pi$
$270$	0	0
$271$	−23.4416	−1.42397	−0.711987	−	0.702193i	$-0.752205\pi$
$271$	−23.4416	−1.42397	−0.711987	+	0.702193i	$0.752205\pi$
$272$	− 17.2962i	− 1.04873i
$273$	8.45169i	0.511520i
$274$	4.91775	0.297092
$275$	0	0
$276$	2.46901	0.148617
$277$	− 18.4064i	− 1.10593i	−0.833204	−	0.552966i	$-0.813496\pi$
$277$	− 18.4064i	− 1.10593i	0.833204	−	0.552966i	$-0.186504\pi$
$278$	− 7.69595i	− 0.461572i
$279$	−1.21200	−0.0725603
$280$	0	0
$281$	0.120288	0.00717577	0.00358789	−	0.999994i	$-0.498858\pi$
$281$	0.120288	0.00717577	0.00358789	+	0.999994i	$0.498858\pi$
$282$	− 0.861482i	− 0.0513005i
$283$	− 0.0913058i	− 0.00542756i	−0.999996	−	0.00271378i	$-0.999136\pi$
$283$	− 0.0913058i	− 0.00542756i	0.999996	−	0.00271378i	$-0.000863825\pi$
$284$	21.3234	1.26531
$285$	0	0
$286$	−6.34693	−0.375302
$287$	7.87213i	0.464677i
$288$	3.98934i	0.235074i
$289$	−11.8132	−0.694893
$290$	0	0
$291$	−14.4790	−0.848776
$292$	25.6922i	1.50352i
$293$	19.6709i	1.14919i	0.818439	+	0.574593i	$0.194840\pi$
$293$	19.6709i	1.14919i	−0.818439	+	0.574593i	$0.805160\pi$
$294$	0.706393	0.0411977
$295$	0	0
$296$	−3.04112	−0.176762
$297$	− 4.63962i	− 0.269218i
$298$	− 1.17547i	− 0.0680930i
$299$	−4.96767	−0.287288
$300$	0	0
$301$	2.62730	0.151435
$302$	5.79774i	0.333623i
$303$	2.16089i	0.124140i
$304$	18.2496	1.04669
$305$	0	0
$306$	1.95435	0.111723
$307$	9.01250i	0.514371i	0.966362	+	0.257185i	$0.0827951\pi$
$307$	9.01250i	0.514371i	−0.966362	+	0.257185i	$0.917205\pi$
$308$	19.4893i	1.11051i
$309$	−7.13512	−0.405903
$310$	0	0
$311$	−23.0844	−1.30899	−0.654497	−	0.756065i	$-0.727119\pi$
$311$	−23.0844	−1.30899	−0.654497	+	0.756065i	$0.727119\pi$
$312$	− 5.29061i	− 0.299522i
$313$	6.87528i	0.388614i	0.980941	+	0.194307i	$0.0622458\pi$
$313$	6.87528i	0.388614i	−0.980941	+	0.194307i	$0.937754\pi$
$314$	1.16964	0.0660064
$315$	0	0
$316$	−21.7085	−1.22120
$317$	− 12.9741i	− 0.728696i	−0.931263	−	0.364348i	$-0.881292\pi$
$317$	− 12.9741i	− 0.728696i	0.931263	−	0.364348i	$-0.118708\pi$
$318$	5.15754i	0.289220i
$319$	−41.1415	−2.30348
$320$	0	0
$321$	14.3985	0.803647
$322$	− 1.08281i	− 0.0603424i
$323$	− 30.4015i	− 1.69158i
$324$	1.86744	0.103747
$325$	0	0
$326$	5.00557	0.277233
$327$	− 16.4693i	− 0.910756i
$328$	− 4.92781i	− 0.272093i
$329$	5.32241	0.293434
$330$	0	0
$331$	−5.77297	−0.317311	−0.158656	−	0.987334i	$-0.550716\pi$
$331$	−5.77297	−0.317311	−0.158656	+	0.987334i	$0.550716\pi$
$332$	− 20.7336i	− 1.13790i
$333$	2.15975i	0.118354i
$334$	0.286714	0.0156883
$335$	0	0
$336$	−7.24806	−0.395414
$337$	− 13.1047i	− 0.713860i	−0.934131	−	0.356930i	$-0.883823\pi$
$337$	− 13.1047i	− 0.713860i	0.934131	−	0.356930i	$-0.116177\pi$
$338$	0.406792i	0.0221266i
$339$	16.2639	0.883335
$340$	0	0
$341$	5.62320	0.304513
$342$	2.06208i	0.111504i
$343$	20.1101i	1.08584i
$344$	−1.64464	−0.0886731
$345$	0	0
$346$	−0.942937	−0.0506926
$347$	30.8690i	1.65713i	0.559890	+	0.828567i	$0.310843\pi$
$347$	30.8690i	1.65713i	−0.559890	+	0.828567i	$0.689157\pi$
$348$	− 16.5594i	− 0.887678i
$349$	−28.0753	−1.50284	−0.751419	−	0.659825i	$-0.770630\pi$
$349$	−28.0753	−1.50284	−0.751419	+	0.659825i	$0.770630\pi$
$350$	0	0
$351$	−3.75730	−0.200550
$352$	− 18.5090i	− 0.986534i
$353$	− 13.5595i	− 0.721697i	−0.932624	−	0.360849i	$-0.882487\pi$
$353$	− 13.5595i	− 0.721697i	0.932624	−	0.360849i	$-0.117513\pi$
$354$	−0.133653	−0.00710356
$355$	0	0
$356$	−31.0978	−1.64818
$357$	12.0743i	0.639042i
$358$	5.48425i	0.289851i
$359$	1.81642	0.0958672	0.0479336	−	0.998851i	$-0.484736\pi$
$359$	1.81642	0.0958672	0.0479336	+	0.998851i	$0.484736\pi$
$360$	0	0
$361$	13.0773	0.688282
$362$	4.56899i	0.240141i
$363$	10.5260i	0.552473i
$364$	15.7830	0.827255
$365$	0	0
$366$	1.92817	0.100787
$367$	− 31.2892i	− 1.63328i	−0.577147	−	0.816640i	$-0.695834\pi$
$367$	− 31.2892i	− 1.63328i	0.577147	−	0.816640i	$-0.304166\pi$
$368$	− 4.26021i	− 0.222079i
$369$	−3.49965	−0.182184
$370$	0	0
$371$	−31.8643	−1.65431
$372$	2.26333i	0.117348i
$373$	− 24.2034i	− 1.25321i	−0.779338	−	0.626603i	$-0.784445\pi$
$373$	− 24.2034i	− 1.25321i	0.779338	−	0.626603i	$-0.215555\pi$
$374$	−9.06742	−0.468865
$375$	0	0
$376$	−3.33173	−0.171821
$377$	33.3176i	1.71594i
$378$	− 0.818981i	− 0.0421239i
$379$	−0.676661	−0.0347577	−0.0173789	−	0.999849i	$-0.505532\pi$
$379$	−0.676661	−0.0347577	−0.0173789	+	0.999849i	$0.505532\pi$
$380$	0	0
$381$	−5.14048	−0.263355
$382$	− 5.85030i	− 0.299327i
$383$	8.32588i	0.425433i	0.977114	+	0.212716i	$0.0682310\pi$
$383$	8.32588i	0.425433i	−0.977114	+	0.212716i	$0.931769\pi$
$384$	9.79620	0.499910
$385$	0	0
$386$	−3.09066	−0.157311
$387$	1.16800i	0.0593726i
$388$	27.0387i	1.37268i
$389$	−34.8213	−1.76551	−0.882754	−	0.469835i	$-0.844313\pi$
$389$	−34.8213	−1.76551	−0.882754	+	0.469835i	$0.844313\pi$
$390$	0	0
$391$	−7.09696	−0.358909
$392$	− 2.73193i	− 0.137983i
$393$	6.52438i	0.329111i
$394$	−1.70415	−0.0858539
$395$	0	0
$396$	−8.66420	−0.435393
$397$	24.3197i	1.22057i	0.792182	+	0.610286i	$0.208945\pi$
$397$	24.3197i	1.22057i	−0.792182	+	0.610286i	$0.791055\pi$
$398$	0.531130i	0.0266232i
$399$	−12.7399	−0.637794
$400$	0	0
$401$	−21.0996	−1.05367	−0.526833	−	0.849969i	$-0.676621\pi$
$401$	−21.0996	−1.05367	−0.526833	+	0.849969i	$0.676621\pi$
$402$	2.93718i	0.146493i
$403$	− 4.55383i	− 0.226843i
$404$	4.03533	0.200765
$405$	0	0
$406$	−7.26227	−0.360420
$407$	− 10.0204i	− 0.496694i
$408$	− 7.55832i	− 0.374193i
$409$	−1.03583	−0.0512184	−0.0256092	−	0.999672i	$-0.508153\pi$
$409$	−1.03583	−0.0512184	−0.0256092	+	0.999672i	$0.508153\pi$
$410$	0	0
$411$	−13.5071	−0.666254
$412$	13.3244i	0.656447i
$413$	− 0.825732i	− 0.0406316i
$414$	0.481374	0.0236583
$415$	0	0
$416$	−14.9892	−0.734904
$417$	21.1376i	1.03511i
$418$	− 9.56725i	− 0.467950i
$419$	−2.57098	−0.125601	−0.0628004	−	0.998026i	$-0.520003\pi$
$419$	−2.57098	−0.125601	−0.0628004	+	0.998026i	$0.520003\pi$
$420$	0	0
$421$	19.4929	0.950025	0.475012	−	0.879979i	$-0.342444\pi$
$421$	19.4929	0.950025	0.475012	+	0.879979i	$0.342444\pi$
$422$	− 4.66423i	− 0.227051i
$423$	2.36614i	0.115046i
$424$	19.9465	0.968686
$425$	0	0
$426$	4.15733	0.201424
$427$	11.9126i	0.576492i
$428$	− 26.8884i	− 1.29970i
$429$	17.4324	0.841645
$430$	0	0
$431$	−10.8376	−0.522026	−0.261013	−	0.965335i	$-0.584057\pi$
$431$	−10.8376	−0.522026	−0.261013	+	0.965335i	$0.584057\pi$
$432$	− 3.22221i	− 0.155029i
$433$	− 29.8688i	− 1.43540i	−0.696351	−	0.717701i	$-0.745194\pi$
$433$	− 29.8688i	− 1.43540i	0.696351	−	0.717701i	$-0.254806\pi$
$434$	0.992603	0.0476464
$435$	0	0
$436$	−30.7555	−1.47292
$437$	− 7.48818i	− 0.358208i
$438$	5.00912i	0.239345i
$439$	−13.0354	−0.622148	−0.311074	−	0.950386i	$-0.600689\pi$
$439$	−13.0354	−0.622148	−0.311074	+	0.950386i	$0.600689\pi$
$440$	0	0
$441$	−1.94017	−0.0923892
$442$	7.34307i	0.349274i
$443$	− 18.1333i	− 0.861539i	−0.902462	−	0.430770i	$-0.858242\pi$
$443$	− 18.1333i	− 0.861539i	0.902462	−	0.430770i	$-0.141758\pi$
$444$	4.03321	0.191408
$445$	0	0
$446$	0.0814555	0.00385703
$447$	3.22853i	0.152704i
$448$	11.2289i	0.530517i
$449$	−13.7087	−0.646954	−0.323477	−	0.946236i	$-0.604852\pi$
$449$	−13.7087	−0.646954	−0.323477	+	0.946236i	$0.604852\pi$
$450$	0	0
$451$	16.2370	0.764572
$452$	− 30.3719i	− 1.42857i
$453$	− 15.9240i	− 0.748176i
$454$	10.2918	0.483016
$455$	0	0
$456$	7.97497	0.373462
$457$	− 17.8434i	− 0.834680i	−0.908750	−	0.417340i	$-0.862962\pi$
$457$	− 17.8434i	− 0.834680i	0.908750	−	0.417340i	$-0.137038\pi$
$458$	− 6.06635i	− 0.283462i
$459$	−5.36779	−0.250547
$460$	0	0
$461$	1.25912	0.0586430	0.0293215	−	0.999570i	$-0.490665\pi$
$461$	1.25912	0.0586430	0.0293215	+	0.999570i	$0.490665\pi$
$462$	3.79976i	0.176781i
$463$	22.6971i	1.05482i	0.849610	+	0.527412i	$0.176838\pi$
$463$	22.6971i	1.05482i	−0.849610	+	0.527412i	$0.823162\pi$
$464$	−28.5728	−1.32646
$465$	0	0
$466$	−7.27474	−0.336996
$467$	10.1460i	0.469501i	0.972056	+	0.234751i	$0.0754274\pi$
$467$	10.1460i	0.469501i	−0.972056	+	0.234751i	$0.924573\pi$
$468$	7.01653i	0.324339i
$469$	−18.1465	−0.837927
$470$	0	0
$471$	−3.21251	−0.148025
$472$	0.516894i	0.0237920i
$473$	− 5.41906i	− 0.249168i
$474$	−4.23243	−0.194402
$475$	0	0
$476$	22.5481	1.03349
$477$	− 14.1656i	− 0.648600i
$478$	6.79337i	0.310721i
$479$	8.86088	0.404864	0.202432	−	0.979296i	$-0.435116\pi$
$479$	8.86088	0.404864	0.202432	+	0.979296i	$0.435116\pi$
$480$	0	0
$481$	−8.11484	−0.370005
$482$	− 2.58837i	− 0.117897i
$483$	2.97403i	0.135323i
$484$	19.6567	0.893488
$485$	0	0
$486$	0.364088	0.0165154
$487$	8.37280i	0.379408i	0.981841	+	0.189704i	$0.0607528\pi$
$487$	8.37280i	0.379408i	−0.981841	+	0.189704i	$0.939247\pi$
$488$	− 7.45709i	− 0.337566i
$489$	−13.7482	−0.621717
$490$	0	0
$491$	3.44669	0.155547	0.0777734	−	0.996971i	$-0.475219\pi$
$491$	3.44669	0.155547	0.0777734	+	0.996971i	$0.475219\pi$
$492$	6.53538i	0.294638i
$493$	47.5986i	2.14373i
$494$	−7.74785	−0.348592
$495$	0	0
$496$	3.90531	0.175354
$497$	25.6848i	1.15212i
$498$	− 4.04235i	− 0.181142i
$499$	0.476153	0.0213155	0.0106578	−	0.999943i	$-0.496607\pi$
$499$	0.476153	0.0213155	0.0106578	+	0.999943i	$0.496607\pi$
$500$	0	0
$501$	−0.787486	−0.0351823
$502$	3.84607i	0.171658i
$503$	− 9.90779i	− 0.441766i	−0.975300	−	0.220883i	$-0.929106\pi$
$503$	− 9.90779i	− 0.441766i	0.975300	−	0.220883i	$-0.0708940\pi$
$504$	−3.16736	−0.141085
$505$	0	0
$506$	−2.23339	−0.0992864
$507$	− 1.11729i	− 0.0496207i
$508$	9.59955i	0.425911i
$509$	11.8211	0.523963	0.261981	−	0.965073i	$-0.415624\pi$
$509$	11.8211	0.523963	0.261981	+	0.965073i	$0.415624\pi$
$510$	0	0
$511$	−30.9473	−1.36903
$512$	− 21.9288i	− 0.969126i
$513$	− 5.66369i	− 0.250058i
$514$	−1.07320	−0.0473368
$515$	0	0
$516$	2.18116	0.0960204
$517$	− 10.9780i	− 0.482811i
$518$	− 1.76880i	− 0.0777165i
$519$	2.58986	0.113682
$520$	0	0
$521$	25.0561	1.09773	0.548863	−	0.835912i	$-0.315061\pi$
$521$	25.0561	1.09773	0.548863	+	0.835912i	$0.315061\pi$
$522$	− 3.22853i	− 0.141309i
$523$	19.5306i	0.854012i	0.904249	+	0.427006i	$0.140432\pi$
$523$	19.5306i	0.854012i	−0.904249	+	0.427006i	$0.859568\pi$
$524$	12.1839	0.532256
$525$	0	0
$526$	0.0802515	0.00349913
$527$	− 6.50574i	− 0.283395i
$528$	14.9498i	0.650608i
$529$	21.2520	0.923998
$530$	0	0
$531$	0.367089	0.0159303
$532$	23.7911i	1.03147i
$533$	− 13.1492i	− 0.569556i
$534$	−6.06302	−0.262373
$535$	0	0
$536$	11.3594	0.490650
$537$	− 15.0630i	− 0.650015i
$538$	3.79188i	0.163480i
$539$	9.00165	0.387729
$540$	0	0
$541$	−8.96076	−0.385253	−0.192627	−	0.981272i	$-0.561701\pi$
$541$	−8.96076	−0.385253	−0.192627	+	0.981272i	$0.561701\pi$
$542$	8.53479i	0.366601i
$543$	− 12.5492i	− 0.538536i
$544$	−21.4140	−0.918116
$545$	0	0
$546$	3.07716	0.131690
$547$	− 6.82992i	− 0.292026i	−0.989283	−	0.146013i	$-0.953356\pi$
$547$	− 6.82992i	− 0.292026i	0.989283	−	0.146013i	$-0.0466442\pi$
$548$	25.2236i	1.07750i
$549$	−5.29590	−0.226023
$550$	0	0
$551$	−50.2224	−2.13955
$552$	− 1.86169i	− 0.0792386i
$553$	− 26.1488i	− 1.11196i
$554$	−6.70153	−0.284721
$555$	0	0
$556$	39.4732	1.67404
$557$	5.53535i	0.234540i	0.993100	+	0.117270i	$0.0374144\pi$
$557$	5.53535i	0.234540i	−0.993100	+	0.117270i	$0.962586\pi$
$558$	0.441273i	0.0186806i
$559$	−4.38851	−0.185614
$560$	0	0
$561$	24.9045	1.05147
$562$	− 0.0437954i	− 0.00184740i
$563$	39.1974i	1.65198i	0.563688	+	0.825988i	$0.309382\pi$
$563$	39.1974i	1.65198i	−0.563688	+	0.825988i	$0.690618\pi$
$564$	4.41862	0.186058
$565$	0	0
$566$	−0.0332433	−0.00139732
$567$	2.24941i	0.0944662i
$568$	− 16.0782i	− 0.674628i
$569$	−19.8092	−0.830444	−0.415222	−	0.909720i	$-0.636296\pi$
$569$	−19.8092	−0.830444	−0.415222	+	0.909720i	$0.636296\pi$
$570$	0	0
$571$	−22.9768	−0.961548	−0.480774	−	0.876845i	$-0.659644\pi$
$571$	−22.9768	−0.961548	−0.480774	+	0.876845i	$0.659644\pi$
$572$	− 32.5540i	− 1.36115i
$573$	16.0684i	0.671266i
$574$	2.86615	0.119631
$575$	0	0
$576$	−4.99195	−0.207998
$577$	12.1833i	0.507198i	0.967309	+	0.253599i	$0.0816143\pi$
$577$	12.1833i	0.507198i	−0.967309	+	0.253599i	$0.918386\pi$
$578$	4.30103i	0.178899i
$579$	8.48878	0.352782
$580$	0	0
$581$	24.9744	1.03611
$582$	5.27164i	0.218516i
$583$	65.7232i	2.72198i
$584$	19.3725	0.801639
$585$	0	0
$586$	7.16194	0.295857
$587$	12.0991i	0.499384i	0.968325	+	0.249692i	$0.0803294\pi$
$587$	12.0991i	0.499384i	−0.968325	+	0.249692i	$0.919671\pi$
$588$	3.62316i	0.149416i
$589$	6.86437	0.282841
$590$	0	0
$591$	4.68061	0.192534
$592$	− 6.95918i	− 0.286021i
$593$	26.8465i	1.10245i	0.834355	+	0.551227i	$0.185840\pi$
$593$	26.8465i	1.10245i	−0.834355	+	0.551227i	$0.814160\pi$
$594$	−1.68923	−0.0693098
$595$	0	0
$596$	6.02908	0.246961
$597$	− 1.45880i	− 0.0597046i
$598$	1.80867i	0.0739619i
$599$	−5.16011	−0.210836	−0.105418	−	0.994428i	$-0.533618\pi$
$599$	−5.16011	−0.210836	−0.105418	+	0.994428i	$0.533618\pi$
$600$	0	0
$601$	−20.5481	−0.838173	−0.419086	−	0.907946i	$-0.637650\pi$
$601$	−20.5481	−0.838173	−0.419086	+	0.907946i	$0.637650\pi$
$602$	− 0.956567i	− 0.0389868i
$603$	− 8.06723i	− 0.328523i
$604$	−29.7372	−1.20999
$605$	0	0
$606$	0.786753	0.0319597
$607$	7.83380i	0.317964i	0.987281	+	0.158982i	$0.0508212\pi$
$607$	7.83380i	0.317964i	−0.987281	+	0.158982i	$0.949179\pi$
$608$	− 22.5944i	− 0.916324i
$609$	19.9465	0.808272
$610$	0	0
$611$	−8.89029	−0.359663
$612$	10.0240i	0.405197i
$613$	− 21.4304i	− 0.865564i	−0.901499	−	0.432782i	$-0.857532\pi$
$613$	− 21.4304i	− 0.865564i	0.901499	−	0.432782i	$-0.142468\pi$
$614$	3.28134	0.132424
$615$	0	0
$616$	14.6953	0.592092
$617$	− 7.08270i	− 0.285139i	−0.989785	−	0.142569i	$-0.954464\pi$
$617$	− 7.08270i	− 0.285139i	0.989785	−	0.142569i	$-0.0455364\pi$
$618$	2.59781i	0.104499i
$619$	38.5317	1.54872	0.774360	−	0.632745i	$-0.218072\pi$
$619$	38.5317	1.54872	0.774360	+	0.632745i	$0.218072\pi$
$620$	0	0
$621$	−1.32214	−0.0530556
$622$	8.40473i	0.336999i
$623$	− 37.4585i	− 1.50074i
$624$	12.1068	0.484660
$625$	0	0
$626$	2.50321	0.100048
$627$	26.2773i	1.04942i
$628$	5.99918i	0.239393i
$629$	−11.5931	−0.462247
$630$	0	0
$631$	−35.1731	−1.40022	−0.700110	−	0.714035i	$-0.746866\pi$
$631$	−35.1731	−1.40022	−0.700110	+	0.714035i	$0.746866\pi$
$632$	16.3687i	0.651110i
$633$	12.8107i	0.509180i
$634$	−4.72370	−0.187602
$635$	0	0
$636$	−26.4535	−1.04895
$637$	− 7.28981i	− 0.288833i
$638$	14.9791i	0.593029i
$639$	−11.4185	−0.451709
$640$	0	0
$641$	−12.9089	−0.509872	−0.254936	−	0.966958i	$-0.582054\pi$
$641$	−12.9089	−0.509872	−0.254936	+	0.966958i	$0.582054\pi$
$642$	− 5.24233i	− 0.206898i
$643$	− 42.0703i	− 1.65909i	−0.558438	−	0.829546i	$-0.688599\pi$
$643$	− 42.0703i	− 1.65909i	0.558438	−	0.829546i	$-0.311401\pi$
$644$	5.55381	0.218851
$645$	0	0
$646$	−11.0688	−0.435497
$647$	2.45766i	0.0966207i	0.998832	+	0.0483103i	$0.0153836\pi$
$647$	2.45766i	0.0966207i	−0.998832	+	0.0483103i	$0.984616\pi$
$648$	− 1.40809i	− 0.0553149i
$649$	−1.70315	−0.0668546
$650$	0	0
$651$	−2.72627	−0.106851
$652$	25.6740i	1.00547i
$653$	42.7727i	1.67382i	0.547337	+	0.836912i	$0.315641\pi$
$653$	42.7727i	1.67382i	−0.547337	+	0.836912i	$0.684359\pi$
$654$	−5.99628	−0.234473
$655$	0	0
$656$	11.2766	0.440278
$657$	− 13.7580i	− 0.536751i
$658$	− 1.93782i	− 0.0755442i
$659$	−3.41659	−0.133092	−0.0665458	−	0.997783i	$-0.521198\pi$
$659$	−3.41659	−0.133092	−0.0665458	+	0.997783i	$0.521198\pi$
$660$	0	0
$661$	42.6380	1.65843	0.829213	−	0.558933i	$-0.188789\pi$
$661$	42.6380	1.65843	0.829213	+	0.558933i	$0.188789\pi$
$662$	2.10187i	0.0816914i
$663$	− 20.1684i	− 0.783276i
$664$	−15.6335	−0.606699
$665$	0	0
$666$	0.786340	0.0304700
$667$	11.7240i	0.453954i
$668$	1.47058i	0.0568986i
$669$	−0.223725	−0.00864970
$670$	0	0
$671$	24.5709	0.948550
$672$	8.97365i	0.346166i
$673$	7.02233i	0.270691i	0.990798	+	0.135345i	$0.0432144\pi$
$673$	7.02233i	0.270691i	−0.990798	+	0.135345i	$0.956786\pi$
$674$	−4.77127	−0.183782
$675$	0	0
$676$	−2.08648	−0.0802491
$677$	4.96936i	0.190988i	0.995430	+	0.0954941i	$0.0304431\pi$
$677$	4.96936i	0.190988i	−0.995430	+	0.0954941i	$0.969557\pi$
$678$	− 5.92149i	− 0.227414i
$679$	−32.5692	−1.24989
$680$	0	0
$681$	−28.2672	−1.08320
$682$	− 2.04734i	− 0.0783966i
$683$	2.27348i	0.0869922i	0.999054	+	0.0434961i	$0.0138496\pi$
$683$	2.27348i	0.0869922i	−0.999054	+	0.0434961i	$0.986150\pi$
$684$	−10.5766	−0.404406
$685$	0	0
$686$	7.32183	0.279549
$687$	16.6618i	0.635687i
$688$	− 3.76353i	− 0.143483i
$689$	53.2246	2.02769
$690$	0	0
$691$	−16.7141	−0.635836	−0.317918	−	0.948118i	$-0.602984\pi$
$691$	−16.7141	−0.635836	−0.317918	+	0.948118i	$0.602984\pi$
$692$	− 4.83641i	− 0.183853i
$693$	− 10.4364i	− 0.396445i
$694$	11.2390	0.426627
$695$	0	0
$696$	−12.4861	−0.473286
$697$	− 18.7854i	− 0.711547i
$698$	10.2219i	0.386904i
$699$	19.9807	0.755741
$700$	0	0
$701$	10.4650	0.395256	0.197628	−	0.980277i	$-0.436676\pi$
$701$	10.4650	0.395256	0.197628	+	0.980277i	$0.436676\pi$
$702$	1.36799i	0.0516313i
$703$	− 12.2322i	− 0.461345i
$704$	23.1607	0.872904
$705$	0	0
$706$	−4.93683	−0.185800
$707$	4.86072i	0.182806i
$708$	− 0.685517i	− 0.0257633i
$709$	−10.4906	−0.393984	−0.196992	−	0.980405i	$-0.563117\pi$
$709$	−10.4906	−0.393984	−0.196992	+	0.980405i	$0.563117\pi$
$710$	0	0
$711$	11.6247	0.435962
$712$	23.4484i	0.878765i
$713$	− 1.60243i	− 0.0600114i
$714$	4.39612	0.164521
$715$	0	0
$716$	−28.1292	−1.05124
$717$	− 18.6586i	− 0.696818i
$718$	− 0.661338i	− 0.0246809i
$719$	51.8986	1.93549	0.967746	−	0.251929i	$-0.0810648\pi$
$719$	51.8986	1.93549	0.967746	+	0.251929i	$0.0810648\pi$
$720$	0	0
$721$	−16.0498	−0.597725
$722$	− 4.76130i	− 0.177197i
$723$	7.10919i	0.264394i
$724$	−23.4348	−0.870947
$725$	0	0
$726$	3.83240	0.142234
$727$	− 36.1382i	− 1.34029i	−0.742230	−	0.670145i	$-0.766232\pi$
$727$	− 36.1382i	− 1.34029i	0.742230	−	0.670145i	$-0.233768\pi$
$728$	− 11.9007i	− 0.441070i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−6.26956	−0.231888
$732$	9.88977i	0.365536i
$733$	− 49.1978i	− 1.81716i	−0.417709	−	0.908581i	$-0.637167\pi$
$733$	− 49.1978i	− 1.81716i	0.417709	−	0.908581i	$-0.362833\pi$
$734$	−11.3920	−0.420486
$735$	0	0
$736$	−5.27446	−0.194419
$737$	37.4289i	1.37871i
$738$	1.27418i	0.0469032i
$739$	0.931024	0.0342483	0.0171241	−	0.999853i	$-0.494549\pi$
$739$	0.931024	0.0342483	0.0171241	+	0.999853i	$0.494549\pi$
$740$	0	0
$741$	21.2802	0.781746
$742$	11.6014i	0.425901i
$743$	− 7.46765i	− 0.273962i	−0.990574	−	0.136981i	$-0.956260\pi$
$743$	− 7.46765i	− 0.273962i	0.990574	−	0.136981i	$-0.0437399\pi$
$744$	1.70660	0.0625669
$745$	0	0
$746$	−8.81218	−0.322637
$747$	11.1027i	0.406225i
$748$	− 46.5076i	− 1.70049i
$749$	32.3881	1.18344
$750$	0	0
$751$	−11.8157	−0.431160	−0.215580	−	0.976486i	$-0.569164\pi$
$751$	−11.8157	−0.431160	−0.215580	+	0.976486i	$0.569164\pi$
$752$	− 7.62420i	− 0.278026i
$753$	− 10.5636i	− 0.384958i
$754$	12.1305	0.441768
$755$	0	0
$756$	4.20063	0.152775
$757$	36.5733i	1.32928i	0.747163	+	0.664640i	$0.231415\pi$
$757$	36.5733i	1.32928i	−0.747163	+	0.664640i	$0.768585\pi$
$758$	0.246364i	0.00894834i
$759$	6.13421	0.222658
$760$	0	0
$761$	12.9848	0.470700	0.235350	−	0.971911i	$-0.424376\pi$
$761$	12.9848	0.470700	0.235350	+	0.971911i	$0.424376\pi$
$762$	1.87159i	0.0678005i
$763$	− 37.0462i	− 1.34116i
$764$	30.0067	1.08561
$765$	0	0
$766$	3.03135	0.109527
$767$	1.37926i	0.0498023i
$768$	6.41723i	0.231562i
$769$	−0.337323	−0.0121642	−0.00608208	−	0.999982i	$-0.501936\pi$
$769$	−0.337323	−0.0121642	−0.00608208	+	0.999982i	$0.501936\pi$
$770$	0	0
$771$	2.94764	0.106157
$772$	− 15.8523i	− 0.570536i
$773$	32.7655i	1.17849i	0.807953	+	0.589246i	$0.200575\pi$
$773$	32.7655i	1.17849i	−0.807953	+	0.589246i	$0.799425\pi$
$774$	0.425253	0.0152854
$775$	0	0
$776$	20.3877	0.731877
$777$	4.85816i	0.174286i
$778$	12.6780i	0.454528i
$779$	19.8209	0.710158
$780$	0	0
$781$	52.9774	1.89568
$782$	2.58392i	0.0924007i
$783$	8.86744i	0.316896i
$784$	6.25165	0.223273
$785$	0	0
$786$	2.37545	0.0847294
$787$	− 24.6230i	− 0.877715i	−0.898557	−	0.438857i	$-0.855383\pi$
$787$	− 24.6230i	− 0.877715i	0.898557	−	0.438857i	$-0.144617\pi$
$788$	− 8.74075i	− 0.311376i
$789$	−0.220418	−0.00784709
$790$	0	0
$791$	36.5842	1.30078
$792$	6.53299i	0.232140i
$793$	− 19.8983i	− 0.706608i
$794$	8.85451	0.314235
$795$	0	0
$796$	−2.72422	−0.0965573
$797$	− 28.1988i	− 0.998852i	−0.866357	−	0.499426i	$-0.833544\pi$
$797$	− 28.1988i	− 0.998852i	0.866357	−	0.499426i	$-0.166456\pi$
$798$	4.63845i	0.164199i
$799$	−12.7009	−0.449327
$800$	0	0
$801$	16.6526	0.588392
$802$	7.68212i	0.271265i
$803$	63.8318i	2.25258i
$804$	−15.0651	−0.531304
$805$	0	0
$806$	−1.65799	−0.0584004
$807$	− 10.4148i	− 0.366617i
$808$	− 3.04272i	− 0.107043i
$809$	−46.0975	−1.62070	−0.810352	−	0.585944i	$-0.800724\pi$
$809$	−46.0975	−1.62070	−0.810352	+	0.585944i	$0.800724\pi$
$810$	0	0
$811$	19.6551	0.690183	0.345092	−	0.938569i	$-0.387848\pi$
$811$	19.6551	0.690183	0.345092	+	0.938569i	$0.387848\pi$
$812$	− 37.2488i	− 1.30718i
$813$	− 23.4416i	− 0.822132i
$814$	−3.64831	−0.127873
$815$	0	0
$816$	17.2962	0.605487
$817$	− 6.61517i	− 0.231435i
$818$	0.377132i	0.0131861i
$819$	−8.45169	−0.295326
$820$	0	0
$821$	28.8469	1.00676	0.503382	−	0.864064i	$-0.332089\pi$
$821$	28.8469	1.00676	0.503382	+	0.864064i	$0.332089\pi$
$822$	4.91775i	0.171526i
$823$	4.32008i	0.150589i	0.997161	+	0.0752943i	$0.0239896\pi$
$823$	4.32008i	0.150589i	−0.997161	+	0.0752943i	$0.976010\pi$
$824$	10.0469	0.349999
$825$	0	0
$826$	−0.300639	−0.0104606
$827$	− 54.9133i	− 1.90952i	−0.297371	−	0.954762i	$-0.596110\pi$
$827$	− 54.9133i	− 1.90952i	0.297371	−	0.954762i	$-0.403890\pi$
$828$	2.46901i	0.0858042i
$829$	15.8046	0.548916	0.274458	−	0.961599i	$-0.411502\pi$
$829$	15.8046	0.548916	0.274458	+	0.961599i	$0.411502\pi$
$830$	0	0
$831$	18.4064	0.638510
$832$	− 18.7563i	− 0.650256i
$833$	− 10.4144i	− 0.360839i
$834$	7.69595	0.266489
$835$	0	0
$836$	49.0713	1.69717
$837$	− 1.21200i	− 0.0418927i
$838$	0.936064i	0.0323358i
$839$	20.7086	0.714939	0.357469	−	0.933925i	$-0.383640\pi$
$839$	20.7086	0.714939	0.357469	+	0.933925i	$0.383640\pi$
$840$	0	0
$841$	49.6315	1.71143
$842$	− 7.09712i	− 0.244583i
$843$	0.120288i	0.00414294i
$844$	23.9233	0.823472
$845$	0	0
$846$	0.861482	0.0296184
$847$	23.6773i	0.813562i
$848$	45.6447i	1.56745i
$849$	0.0913058	0.00313361
$850$	0	0
$851$	−2.85549	−0.0978850
$852$	21.3234i	0.730526i
$853$	− 8.27523i	− 0.283338i	−0.989914	−	0.141669i	$-0.954753\pi$
$853$	− 8.27523i	− 0.283338i	0.989914	−	0.141669i	$-0.0452469\pi$
$854$	4.33724	0.148417
$855$	0	0
$856$	−20.2744	−0.692964
$857$	− 6.29920i	− 0.215177i	−0.994196	−	0.107588i	$-0.965687\pi$
$857$	− 6.29920i	− 0.215177i	0.994196	−	0.107588i	$-0.0343128\pi$
$858$	− 6.34693i	− 0.216681i
$859$	−26.4130	−0.901200	−0.450600	−	0.892726i	$-0.648790\pi$
$859$	−26.4130	−0.901200	−0.450600	+	0.892726i	$0.648790\pi$
$860$	0	0
$861$	−7.87213	−0.268282
$862$	3.94582i	0.134395i
$863$	− 34.5355i	− 1.17560i	−0.809005	−	0.587802i	$-0.799993\pi$
$863$	− 34.5355i	− 1.17560i	0.809005	−	0.587802i	$-0.200007\pi$
$864$	−3.98934	−0.135720
$865$	0	0
$866$	−10.8749	−0.369543
$867$	− 11.8132i	− 0.401197i
$868$	5.09115i	0.172805i
$869$	−53.9343	−1.82960
$870$	0	0
$871$	30.3110	1.02705
$872$	23.1903i	0.785321i
$873$	− 14.4790i	− 0.490041i
$874$	−2.72635	−0.0922203
$875$	0	0
$876$	−25.6922	−0.868060
$877$	44.1083i	1.48943i	0.667382	+	0.744716i	$0.267415\pi$
$877$	44.1083i	1.48943i	−0.667382	+	0.744716i	$0.732585\pi$
$878$	4.74605i	0.160171i
$879$	−19.6709	−0.663483
$880$	0	0
$881$	34.5724	1.16477	0.582386	−	0.812912i	$-0.302119\pi$
$881$	34.5724	1.16477	0.582386	+	0.812912i	$0.302119\pi$
$882$	0.706393i	0.0237855i
$883$	− 3.56120i	− 0.119844i	−0.998203	−	0.0599220i	$-0.980915\pi$
$883$	− 3.56120i	− 0.119844i	0.998203	−	0.0599220i	$-0.0190852\pi$
$884$	−37.6633	−1.26675
$885$	0	0
$886$	−6.60212	−0.221802
$887$	− 53.6149i	− 1.80021i	−0.435671	−	0.900106i	$-0.643489\pi$
$887$	− 53.6149i	− 1.80021i	0.435671	−	0.900106i	$-0.356511\pi$
$888$	− 3.04112i	− 0.102053i
$889$	−11.5630	−0.387812
$890$	0	0
$891$	4.63962	0.155433
$892$	0.417793i	0.0139887i
$893$	− 13.4011i	− 0.448450i
$894$	1.17547	0.0393135
$895$	0	0
$896$	22.0356	0.736159
$897$	− 4.96767i	− 0.165866i
$898$	4.99117i	0.166558i
$899$	−10.7473	−0.358443
$900$	0	0
$901$	76.0382	2.53320
$902$	− 5.91170i	− 0.196838i
$903$	2.62730i	0.0874310i
$904$	−22.9010	−0.761677
$905$	0	0
$906$	−5.79774	−0.192617
$907$	− 41.7962i	− 1.38782i	−0.720062	−	0.693909i	$-0.755887\pi$
$907$	− 41.7962i	− 1.38782i	0.720062	−	0.693909i	$-0.244113\pi$
$908$	52.7874i	1.75181i
$909$	−2.16089	−0.0716722
$910$	0	0
$911$	−45.8484	−1.51903	−0.759513	−	0.650492i	$-0.774563\pi$
$911$	−45.8484	−1.51903	−0.759513	+	0.650492i	$0.774563\pi$
$912$	18.2496i	0.604305i
$913$	− 51.5121i	− 1.70480i
$914$	−6.49657	−0.214887
$915$	0	0
$916$	31.1149	1.02806
$917$	14.6760i	0.484644i
$918$	1.95435i	0.0645031i
$919$	28.3017	0.933588	0.466794	−	0.884366i	$-0.345409\pi$
$919$	28.3017	0.933588	0.466794	+	0.884366i	$0.345409\pi$
$920$	0	0
$921$	−9.01250	−0.296972
$922$	− 0.458429i	− 0.0150976i
$923$	− 42.9027i	− 1.41216i
$924$	−19.4893	−0.641151
$925$	0	0
$926$	8.26374	0.271563
$927$	− 7.13512i	− 0.234348i
$928$	35.3753i	1.16125i
$929$	−14.0835	−0.462064	−0.231032	−	0.972946i	$-0.574210\pi$
$929$	−14.0835	−0.462064	−0.231032	+	0.972946i	$0.574210\pi$
$930$	0	0
$931$	10.9885	0.360134
$932$	− 37.3128i	− 1.22222i
$933$	− 23.0844i	− 0.755748i
$934$	3.69404	0.120873
$935$	0	0
$936$	5.29061	0.172929
$937$	17.7320i	0.579280i	0.957136	+	0.289640i	$0.0935356\pi$
$937$	17.7320i	0.579280i	−0.957136	+	0.289640i	$0.906464\pi$
$938$	6.60691i	0.215723i
$939$	−6.87528	−0.224366
$940$	0	0
$941$	−13.8982	−0.453068	−0.226534	−	0.974003i	$-0.572739\pi$
$941$	−13.8982	−0.453068	−0.226534	+	0.974003i	$0.572739\pi$
$942$	1.16964i	0.0381088i
$943$	− 4.62702i	− 0.150676i
$944$	−1.18284	−0.0384981
$945$	0	0
$946$	−1.97301	−0.0641482
$947$	− 28.7291i	− 0.933571i	−0.884371	−	0.466786i	$-0.845412\pi$
$947$	− 28.7291i	− 0.933571i	0.884371	−	0.466786i	$-0.154588\pi$
$948$	− 21.7085i	− 0.705059i
$949$	51.6929	1.67802
$950$	0	0
$951$	12.9741	0.420713
$952$	− 17.0017i	− 0.551029i
$953$	40.5303i	1.31291i	0.754367	+	0.656453i	$0.227944\pi$
$953$	40.5303i	1.31291i	−0.754367	+	0.656453i	$0.772056\pi$
$954$	−5.15754	−0.166981
$955$	0	0
$956$	−34.8438	−1.12693
$957$	− 41.1415i	− 1.32992i
$958$	− 3.22614i	− 0.104232i
$959$	−30.3829	−0.981114
$960$	0	0
$961$	−29.5311	−0.952615
$962$	2.95451i	0.0952573i
$963$	14.3985i	0.463986i
$964$	13.2760	0.427591
$965$	0	0
$966$	1.08281	0.0348387
$967$	− 21.1771i	− 0.681010i	−0.940243	−	0.340505i	$-0.889402\pi$
$967$	− 21.1771i	− 0.681010i	0.940243	−	0.340505i	$-0.110598\pi$
$968$	− 14.8216i	− 0.476383i
$969$	30.4015	0.976636
$970$	0	0
$971$	−41.7880	−1.34104	−0.670520	−	0.741891i	$-0.733929\pi$
$971$	−41.7880	−1.34104	−0.670520	+	0.741891i	$0.733929\pi$
$972$	1.86744i	0.0598982i
$973$	47.5471i	1.52429i
$974$	3.04843	0.0976781
$975$	0	0
$976$	17.0645	0.546221
$977$	− 20.1059i	− 0.643245i	−0.946868	−	0.321623i	$-0.895772\pi$
$977$	− 20.1059i	− 0.643245i	0.946868	−	0.321623i	$-0.104228\pi$
$978$	5.00557i	0.160060i
$979$	−77.2618	−2.46930
$980$	0	0
$981$	16.4693	0.525825
$982$	− 1.25490i	− 0.0400454i
$983$	− 1.74028i	− 0.0555063i	−0.999615	−	0.0277531i	$-0.991165\pi$
$983$	− 1.74028i	− 0.0555063i	0.999615	−	0.0277531i	$-0.00883523\pi$
$984$	4.92781	0.157093
$985$	0	0
$986$	17.3301	0.551901
$987$	5.32241i	0.169414i
$988$	− 39.7394i	− 1.26428i
$989$	−1.54425	−0.0491044
$990$	0	0
$991$	−13.8725	−0.440676	−0.220338	−	0.975424i	$-0.570716\pi$
$991$	−13.8725	−0.440676	−0.220338	+	0.975424i	$0.570716\pi$
$992$	− 4.83507i	− 0.153514i
$993$	− 5.77297i	− 0.183200i
$994$	9.35153	0.296613
$995$	0	0
$996$	20.7336	0.656968
$997$	− 23.7256i	− 0.751399i	−0.926742	−	0.375699i	$-0.877403\pi$
$997$	− 23.7256i	− 0.751399i	0.926742	−	0.375699i	$-0.122597\pi$
$998$	− 0.173362i	− 0.00548766i
$999$	−2.15975	−0.0683316

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.e.1249.6		12
5.2	odd	4		1875.2.a.i.1.4	✓	6
5.3	odd	4		1875.2.a.l.1.3	yes	6
5.4	even	2	inner	1875.2.b.e.1249.7		12
15.2	even	4		5625.2.a.r.1.3		6
15.8	even	4		5625.2.a.o.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.i.1.4	✓	6	5.2	odd	4
1875.2.a.l.1.3	yes	6	5.3	odd	4
1875.2.b.e.1249.6		12	1.1	even	1	trivial
1875.2.b.e.1249.7		12	5.4	even	2	inner
5625.2.a.o.1.4		6	15.8	even	4
5625.2.a.r.1.3		6	15.2	even	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.364088i q^{2} +1.00000i q^{3} +1.86744 q^{4} +0.364088 q^{6} +2.24941i q^{7} -1.40809i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-0.364088i q^{2} +1.00000i q^{3} +1.86744 q^{4} +0.364088 q^{6} +2.24941i q^{7} -1.40809i q^{8} -1.00000 q^{9} +4.63962 q^{11} +1.86744i q^{12} -3.75730i q^{13} +0.818981 q^{14} +3.22221 q^{16} -5.36779i q^{17} +0.364088i q^{18} +5.66369 q^{19} -2.24941 q^{21} -1.68923i q^{22} -1.32214i q^{23} +1.40809 q^{24} -1.36799 q^{26} -1.00000i q^{27} +4.20063i q^{28} -8.86744 q^{29} +1.21200 q^{31} -3.98934i q^{32} +4.63962i q^{33} -1.95435 q^{34} -1.86744 q^{36} -2.15975i q^{37} -2.06208i q^{38} +3.75730 q^{39} +3.49965 q^{41} +0.818981i q^{42} -1.16800i q^{43} +8.66420 q^{44} -0.481374 q^{46} -2.36614i q^{47} +3.22221i q^{48} +1.94017 q^{49} +5.36779 q^{51} -7.01653i q^{52} +14.1656i q^{53} -0.364088 q^{54} +3.16736 q^{56} +5.66369i q^{57} +3.22853i q^{58} -0.367089 q^{59} +5.29590 q^{61} -0.441273i q^{62} -2.24941i q^{63} +4.99195 q^{64} +1.68923 q^{66} +8.06723i q^{67} -10.0240i q^{68} +1.32214 q^{69} +11.4185 q^{71} +1.40809i q^{72} +13.7580i q^{73} -0.786340 q^{74} +10.5766 q^{76} +10.4364i q^{77} -1.36799i q^{78} -11.6247 q^{79} +1.00000 q^{81} -1.27418i q^{82} -11.1027i q^{83} -4.20063 q^{84} -0.425253 q^{86} -8.86744i q^{87} -6.53299i q^{88} -16.6526 q^{89} +8.45169 q^{91} -2.46901i q^{92} +1.21200i q^{93} -0.861482 q^{94} +3.98934 q^{96} +14.4790i q^{97} -0.706393i q^{98} -4.63962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 22 q^{4} - 2 q^{6} - 12 q^{9}+O(q^{10}) \) 12 * q - 22 * q^4 - 2 * q^6 - 12 * q^9	\( 12 q - 22 q^{4} - 2 q^{6} - 12 q^{9} + 8 q^{14} + 34 q^{16} + 4 q^{19} + 4 q^{21} + 12 q^{24} + 74 q^{26} - 62 q^{29} - 4 q^{31} - 74 q^{34} + 22 q^{36} + 66 q^{41} + 22 q^{44} - 24 q^{46} - 8 q^{49} - 4 q^{51} + 2 q^{54} - 60 q^{56} + 16 q^{59} + 68 q^{61} - 24 q^{64} - 18 q^{66} - 2 q^{69} - 6 q^{71} - 72 q^{74} + 54 q^{76} - 50 q^{79} + 12 q^{81} - 88 q^{84} - 60 q^{86} - 36 q^{89} + 56 q^{91} + 100 q^{94} - 66 q^{96}+O(q^{100}) \) 12 * q - 22 * q^4 - 2 * q^6 - 12 * q^9 + 8 * q^14 + 34 * q^16 + 4 * q^19 + 4 * q^21 + 12 * q^24 + 74 * q^26 - 62 * q^29 - 4 * q^31 - 74 * q^34 + 22 * q^36 + 66 * q^41 + 22 * q^44 - 24 * q^46 - 8 * q^49 - 4 * q^51 + 2 * q^54 - 60 * q^56 + 16 * q^59 + 68 * q^61 - 24 * q^64 - 18 * q^66 - 2 * q^69 - 6 * q^71 - 72 * q^74 + 54 * q^76 - 50 * q^79 + 12 * q^81 - 88 * q^84 - 60 * q^86 - 36 * q^89 + 56 * q^91 + 100 * q^94 - 66 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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