LMFDB - Embedded newform 3900.2.cd.f.2701.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3900,2,Mod(901,3900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3900, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3900.901"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3900.cd (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-2,0,0,0,0,0,-2,0,12,0,-4,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$31.1416567883$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 13x^{2} + 169 $ x^4 - 13*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2701.1
Root			$-3.12250 - 1.80278i$ of defining polynomial
Character	$\chi$	$=$	3900.2701
Dual form			3900.2.cd.f.901.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{3} +(-3.12250 - 1.80278i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(3.00000 - 1.73205i) q^{11} +(-1.00000 - 3.46410i) q^{13} +(1.00000 - 1.73205i) q^{17} +(6.12250 + 3.53483i) q^{19} +3.60555i q^{21} +(2.62250 + 4.54230i) q^{23} +1.00000 q^{27} +(3.62250 + 6.27435i) q^{29} -3.46410i q^{31} +(-3.00000 - 1.73205i) q^{33} +(6.24500 - 3.60555i) q^{37} +(-2.50000 + 2.59808i) q^{39} +(6.24500 - 3.60555i) q^{41} +(4.12250 - 7.14038i) q^{43} +10.6752i q^{47} +(3.00000 + 5.19615i) q^{49} -2.00000 q^{51} -9.24500 q^{53} -7.06965i q^{57} +(-4.62250 - 2.66880i) q^{59} +(3.12250 - 1.80278i) q^{63} +(-0.122499 + 0.0707248i) q^{67} +(2.62250 - 4.54230i) q^{69} +(-6.24500 - 3.60555i) q^{71} +2.01495i q^{73} -12.4900 q^{77} -4.24500 q^{79} +(-0.500000 - 0.866025i) q^{81} -12.2658i q^{83} +(3.62250 - 6.27435i) q^{87} +(4.37750 - 2.52735i) q^{89} +(-3.12250 + 12.6194i) q^{91} +(-3.00000 + 1.73205i) q^{93} +(0.244998 + 0.141450i) q^{97} +3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 2 q^{9} + 12 q^{11} - 4 q^{13} + 4 q^{17} + 12 q^{19} - 2 q^{23} + 4 q^{27} + 2 q^{29} - 12 q^{33} - 10 q^{39} + 4 q^{43} + 12 q^{49} - 8 q^{51} - 12 q^{53} - 6 q^{59} + 12 q^{67} - 2 q^{69}+ \cdots - 24 q^{97}+O(q^{100}) $ 4 * q - 2 * q^3 - 2 * q^9 + 12 * q^11 - 4 * q^13 + 4 * q^17 + 12 * q^19 - 2 * q^23 + 4 * q^27 + 2 * q^29 - 12 * q^33 - 10 * q^39 + 4 * q^43 + 12 * q^49 - 8 * q^51 - 12 * q^53 - 6 * q^59 + 12 * q^67 - 2 * q^69 + 8 * q^79 - 2 * q^81 + 2 * q^87 + 30 * q^89 - 12 * q^93 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1951$	$3277$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$	$1$

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			0
$2$		0			0
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.12250	−	1.80278i	−1.18019	−	0.681385i	−0.224134	−	0.974558i	$-0.571955\pi$
$7$	−3.12250	−	1.80278i	−1.18019	−	0.681385i	−0.956059	+	0.293173i	$0.905289\pi$
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	3.00000	−	1.73205i	0.904534	−	0.522233i	0.0258656	−	0.999665i	$-0.491766\pi$
$11$	3.00000	−	1.73205i	0.904534	−	0.522233i	0.878668	+	0.477432i	$0.158432\pi$
$12$		0			0
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	−0.718900	−	0.695113i	$-0.755354\pi$
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	0.961436	+	0.275029i	$0.0886875\pi$
$18$		0			0
$19$	6.12250	+	3.53483i	1.40460	+	0.810945i	0.994860	−	0.101259i	$-0.0322870\pi$
$19$	6.12250	+	3.53483i	1.40460	+	0.810945i	0.409737	+	0.912203i	$0.365620\pi$
$20$		0			0
$21$			3.60555i			0.786796i
$22$		0			0
$23$	2.62250	+	4.54230i	0.546829	+	0.947135i	0.998489	+	0.0549455i	$0.0174985\pi$
$23$	2.62250	+	4.54230i	0.546829	+	0.947135i	−0.451661	+	0.892190i	$0.649168\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	3.62250	+	6.27435i	0.672681	+	1.16512i	0.977141	+	0.212593i	$0.0681908\pi$
$29$	3.62250	+	6.27435i	0.672681	+	1.16512i	−0.304460	+	0.952525i	$0.598476\pi$
$30$		0			0
$31$		−	3.46410i		−	0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$		−	3.46410i		−	0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$		0			0
$33$	−3.00000	−	1.73205i	−0.522233	−	0.301511i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.24500	−	3.60555i	1.02667	−	0.592749i	0.110642	−	0.993860i	$-0.464709\pi$
$37$	6.24500	−	3.60555i	1.02667	−	0.592749i	0.916029	+	0.401111i	$0.131376\pi$
$38$		0			0
$39$	−2.50000	+	2.59808i	−0.400320	+	0.416025i
$40$		0			0
$41$	6.24500	−	3.60555i	0.975305	−	0.563093i	0.0744555	−	0.997224i	$-0.476278\pi$
$41$	6.24500	−	3.60555i	0.975305	−	0.563093i	0.900849	+	0.434132i	$0.142945\pi$
$42$		0			0
$43$	4.12250	−	7.14038i	0.628675	−	1.08890i	−0.359143	−	0.933283i	$-0.616931\pi$
$43$	4.12250	−	7.14038i	0.628675	−	1.08890i	0.987818	−	0.155615i	$-0.0497358\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.6752i			1.55714i	0.627559	+	0.778569i	$0.284054\pi$
$47$			10.6752i			1.55714i	−0.627559	+	0.778569i	$0.715946\pi$
$48$		0			0
$49$	3.00000	+	5.19615i	0.428571	+	0.742307i
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$	−9.24500			−1.26990			−0.634949	−	0.772554i	$-0.718979\pi$
$53$	−9.24500			−1.26990			−0.634949	+	0.772554i	$0.718979\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	7.06965i		−	0.936398i
$58$		0			0
$59$	−4.62250	−	2.66880i	−0.601798	−	0.347448i	0.167951	−	0.985795i	$-0.446285\pi$
$59$	−4.62250	−	2.66880i	−0.601798	−	0.347448i	−0.769749	+	0.638347i	$0.779618\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$	3.12250	−	1.80278i	0.393398	−	0.227128i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−0.122499	+	0.0707248i	−0.0149656	+	0.00864041i	−0.507464	−	0.861673i	$-0.669417\pi$
$67$	−0.122499	+	0.0707248i	−0.0149656	+	0.00864041i	0.492499	+	0.870313i	$0.336084\pi$
$68$		0			0
$69$	2.62250	−	4.54230i	0.315712	−	0.546829i
$70$		0			0
$71$	−6.24500	−	3.60555i	−0.741145	−	0.427900i	0.0813405	−	0.996686i	$-0.474080\pi$
$71$	−6.24500	−	3.60555i	−0.741145	−	0.427900i	−0.822485	+	0.568786i	$0.807413\pi$
$72$		0			0
$73$			2.01495i			0.235832i	0.993024	+	0.117916i	$0.0376214\pi$
$73$			2.01495i			0.235832i	−0.993024	+	0.117916i	$0.962379\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.4900			−1.42337
$78$		0			0
$79$	−4.24500			−0.477600			−0.238800	−	0.971069i	$-0.576754\pi$
$79$	−4.24500			−0.477600			−0.238800	+	0.971069i	$0.576754\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		−	12.2658i		−	1.34635i	−0.739485	−	0.673174i	$-0.764931\pi$
$83$		−	12.2658i		−	1.34635i	0.739485	−	0.673174i	$-0.235069\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	3.62250	−	6.27435i	0.388373	−	0.672681i
$88$		0			0
$89$	4.37750	−	2.52735i	0.464014	−	0.267899i	−0.249716	−	0.968319i	$-0.580337\pi$
$89$	4.37750	−	2.52735i	0.464014	−	0.267899i	0.713731	+	0.700420i	$0.247004\pi$
$90$		0			0
$91$	−3.12250	+	12.6194i	−0.327327	+	1.32288i
$92$		0			0
$93$	−3.00000	+	1.73205i	−0.311086	+	0.179605i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.244998	+	0.141450i	0.0248758	+	0.0143620i	0.512386	−	0.858755i	$-0.328762\pi$
$97$	0.244998	+	0.141450i	0.0248758	+	0.0143620i	−0.487511	+	0.873117i	$0.662095\pi$
$98$		0			0
$99$			3.46410i			0.348155i
$100$		0			0
$101$	−5.62250	−	9.73845i	−0.559460	−	0.969012i	−0.997542	−	0.0700774i	$-0.977675\pi$
$101$	−5.62250	−	9.73845i	−0.559460	−	0.969012i	0.438082	−	0.898935i	$-0.355658\pi$
$102$		0			0
$103$	−14.2450			−1.40360			−0.701801	−	0.712373i	$-0.747620\pi$
$103$	−14.2450			−1.40360			−0.701801	+	0.712373i	$0.747620\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.62250	+	2.81025i	0.156853	+	0.271677i	0.933732	−	0.357972i	$-0.116532\pi$
$107$	1.62250	+	2.81025i	0.156853	+	0.271677i	−0.776879	+	0.629650i	$0.783198\pi$
$108$		0			0
$109$		−	10.9581i		−	1.04960i	−0.851227	−	0.524798i	$-0.824141\pi$
$109$		−	10.9581i		−	1.04960i	0.851227	−	0.524798i	$-0.175859\pi$
$110$		0			0
$111$	−6.24500	−	3.60555i	−0.592749	−	0.342224i
$112$		0			0
$113$	6.00000	−	10.3923i	0.564433	−	0.977626i	−0.432670	−	0.901553i	$-0.642428\pi$
$113$	6.00000	−	10.3923i	0.564433	−	0.977626i	0.997102	−	0.0760733i	$-0.0242383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	3.50000	+	0.866025i	0.323575	+	0.0800641i
$118$		0			0
$119$	−6.24500	+	3.60555i	−0.572478	+	0.330520i
$120$		0			0
$121$	0.500000	−	0.866025i	0.0454545	−	0.0787296i
$122$		0			0
$123$	−6.24500	−	3.60555i	−0.563093	−	0.325102i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−8.12250	−	14.0686i	−0.720755	−	1.24838i	−0.960697	−	0.277598i	$-0.910462\pi$
$127$	−8.12250	−	14.0686i	−0.720755	−	1.24838i	0.239942	−	0.970787i	$-0.422871\pi$
$128$		0			0
$129$	−8.24500			−0.725932
$130$		0			0
$131$	11.2450			0.982480			0.491240	−	0.871024i	$-0.336544\pi$
$131$	11.2450			0.982480			0.491240	+	0.871024i	$0.336544\pi$
$132$		0			0
$133$	−12.7450	−	22.0750i	−1.10513	−	1.91414i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.1125	+	8.14785i	1.20571	+	0.696118i	0.961820	−	0.273684i	$-0.0882423\pi$
$137$	14.1125	+	8.14785i	1.20571	+	0.696118i	0.243892	+	0.969802i	$0.421576\pi$
$138$		0			0
$139$	0.122499	−	0.212174i	0.0103902	−	0.0179964i	−0.860784	−	0.508971i	$-0.830026\pi$
$139$	0.122499	−	0.212174i	0.0103902	−	0.0179964i	0.871174	+	0.490975i	$0.163359\pi$
$140$		0			0
$141$	9.24500	−	5.33760i	0.778569	−	0.449507i
$142$		0			0
$143$	−9.00000	−	8.66025i	−0.752618	−	0.724207i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	3.00000	−	5.19615i	0.247436	−	0.428571i
$148$		0			0
$149$	19.6225	+	11.3291i	1.60754	+	0.928112i	0.989919	+	0.141635i	$0.0452358\pi$
$149$	19.6225	+	11.3291i	1.60754	+	0.928112i	0.617619	+	0.786478i	$0.288097\pi$
$150$		0			0
$151$		−	21.2090i		−	1.72596i	−0.505238	−	0.862980i	$-0.668595\pi$
$151$		−	21.2090i		−	1.72596i	0.505238	−	0.862980i	$-0.331405\pi$
$152$		0			0
$153$	1.00000	+	1.73205i	0.0808452	+	0.140028i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	17.0000			1.35675			0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000			1.35675			0.678374	+	0.734717i	$0.262685\pi$
$158$		0			0
$159$	4.62250	+	8.00640i	0.366588	+	0.634949i
$160$		0			0
$161$		−	18.9111i		−	1.49040i
$162$		0			0
$163$	9.49000	+	5.47905i	0.743314	+	0.429152i	0.823273	−	0.567646i	$-0.192146\pi$
$163$	9.49000	+	5.47905i	0.743314	+	0.429152i	−0.0799591	+	0.996798i	$0.525479\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.37750	+	4.25940i	−0.570888	+	0.329602i	−0.757504	−	0.652831i	$-0.773581\pi$
$167$	−7.37750	+	4.25940i	−0.570888	+	0.329602i	0.186616	+	0.982433i	$0.440248\pi$
$168$		0			0
$169$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$170$		0			0
$171$	−6.12250	+	3.53483i	−0.468199	+	0.270315i
$172$		0			0
$173$	12.6225	−	21.8628i	0.959671	−	1.66220i	0.236373	−	0.971662i	$-0.424041\pi$
$173$	12.6225	−	21.8628i	0.959671	−	1.66220i	0.723298	−	0.690536i	$-0.242625\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			5.33760i			0.401199i
$178$		0			0
$179$	−3.24500	−	5.62050i	−0.242543	−	0.420096i	0.718895	−	0.695118i	$-0.244648\pi$
$179$	−3.24500	−	5.62050i	−0.242543	−	0.420096i	−0.961438	+	0.275022i	$0.911315\pi$
$180$		0			0
$181$	13.4900			1.00270			0.501352	−	0.865244i	$-0.332836\pi$
$181$	13.4900			1.00270			0.501352	+	0.865244i	$0.332836\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	6.92820i		−	0.506640i
$188$		0			0
$189$	−3.12250	−	1.80278i	−0.227128	−	0.131133i
$190$		0			0
$191$	−6.24500	+	10.8167i	−0.451872	+	0.782666i	−0.998502	−	0.0547085i	$-0.982577\pi$
$191$	−6.24500	+	10.8167i	−0.451872	+	0.782666i	0.546630	+	0.837374i	$0.315910\pi$
$192$		0			0
$193$	7.50000	−	4.33013i	0.539862	−	0.311689i	−0.205161	−	0.978728i	$-0.565772\pi$
$193$	7.50000	−	4.33013i	0.539862	−	0.311689i	0.745023	+	0.667039i	$0.232439\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	−12.1225	+	20.9968i	−0.859341	+	1.48842i	0.0132172	+	0.999913i	$0.495793\pi$
$199$	−12.1225	+	20.9968i	−0.859341	+	1.48842i	−0.872558	+	0.488510i	$0.837541\pi$
$200$		0			0
$201$	0.122499	+	0.0707248i	0.00864041	+	0.00498854i
$202$		0			0
$203$		−	26.1222i		−	1.83342i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−5.24500			−0.364553
$208$		0			0
$209$	24.4900			1.69401
$210$		0			0
$211$	−9.24500	−	16.0128i	−0.636452	−	1.10237i	−0.986206	−	0.165525i	$-0.947068\pi$
$211$	−9.24500	−	16.0128i	−0.636452	−	1.10237i	0.349754	−	0.936842i	$-0.386265\pi$
$212$		0			0
$213$			7.21110i			0.494097i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−6.24500	+	10.8167i	−0.423938	+	0.734282i
$218$		0			0
$219$	1.74500	−	1.00748i	0.117916	−	0.0680789i
$220$		0			0
$221$	−7.00000	−	1.73205i	−0.470871	−	0.116510i
$222$		0			0
$223$	−21.3675	+	12.3365i	−1.43087	+	0.826115i	−0.997187	−	0.0749489i	$-0.976121\pi$
$223$	−21.3675	+	12.3365i	−1.43087	+	0.826115i	−0.433686	+	0.901064i	$0.642787\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−22.8675	−	13.2026i	−1.51777	−	0.876284i	−0.999782	−	0.0208962i	$-0.993348\pi$
$227$	−22.8675	−	13.2026i	−1.51777	−	0.876284i	−0.517987	−	0.855388i	$-0.673319\pi$
$228$		0			0
$229$			19.9013i			1.31511i	0.753406	+	0.657556i	$0.228410\pi$
$229$			19.9013i			1.31511i	−0.753406	+	0.657556i	$0.771590\pi$
$230$		0			0
$231$	6.24500	+	10.8167i	0.410891	+	0.711684i
$232$		0			0
$233$	2.75500			0.180486			0.0902431	−	0.995920i	$-0.471236\pi$
$233$	2.75500			0.180486			0.0902431	+	0.995920i	$0.471236\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	2.12250	+	3.67628i	0.137871	+	0.238800i
$238$		0			0
$239$			16.0128i			1.03578i	0.855447	+	0.517891i	$0.173283\pi$
$239$			16.0128i			1.03578i	−0.855447	+	0.517891i	$0.826717\pi$
$240$		0			0
$241$	−22.7450	−	13.1318i	−1.46513	−	0.845896i	−0.465893	−	0.884841i	$-0.654267\pi$
$241$	−22.7450	−	13.1318i	−1.46513	−	0.845896i	−0.999241	+	0.0389456i	$0.987600\pi$
$242$		0			0
$243$	−0.500000	+	0.866025i	−0.0320750	+	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	6.12250	−	24.7438i	0.389565	−	1.57441i
$248$		0			0
$249$	−10.6225	+	6.13290i	−0.673174	+	0.388657i
$250$		0			0
$251$	0.377501	−	0.653851i	0.0238277	−	0.0412707i	−0.853866	−	0.520493i	$-0.825748\pi$
$251$	0.377501	−	0.653851i	0.0238277	−	0.0412707i	0.877693	+	0.479223i	$0.159081\pi$
$252$		0			0
$253$	15.7350	+	9.08460i	0.989251	+	0.571144i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.4900	−	18.1692i	−0.654348	−	1.13336i	−0.982057	−	0.188585i	$-0.939610\pi$
$257$	−10.4900	−	18.1692i	−0.654348	−	1.13336i	0.327709	−	0.944779i	$-0.393723\pi$
$258$		0			0
$259$	−26.0000			−1.61556
$260$		0			0
$261$	−7.24500			−0.448454
$262$		0			0
$263$	9.49000	+	16.4372i	0.585178	+	1.01356i	0.994853	+	0.101327i	$0.0323088\pi$
$263$	9.49000	+	16.4372i	0.585178	+	1.01356i	−0.409675	+	0.912232i	$0.634358\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−4.37750	−	2.52735i	−0.267899	−	0.154671i
$268$		0			0
$269$	12.0000	−	20.7846i	0.731653	−	1.26726i	−0.224523	−	0.974469i	$-0.572083\pi$
$269$	12.0000	−	20.7846i	0.731653	−	1.26726i	0.956176	−	0.292791i	$-0.0945841\pi$
$270$		0			0
$271$	−12.1225	+	6.99893i	−0.736389	+	0.425155i	−0.820755	−	0.571280i	$-0.806447\pi$
$271$	−12.1225	+	6.99893i	−0.736389	+	0.425155i	0.0843657	+	0.996435i	$0.473114\pi$
$272$		0			0
$273$	12.4900	−	3.60555i	0.755929	−	0.218218i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.25500	+	2.17373i	−0.0754058	+	0.130607i	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−1.25500	+	2.17373i	−0.0754058	+	0.130607i	0.825857	+	0.563880i	$0.190692\pi$
$278$		0			0
$279$	3.00000	+	1.73205i	0.179605	+	0.103695i
$280$		0			0
$281$		−	22.6581i		−	1.35167i	−0.737053	−	0.675835i	$-0.763783\pi$
$281$		−	22.6581i		−	1.35167i	0.737053	−	0.675835i	$-0.236217\pi$
$282$		0			0
$283$	−12.4900	−	21.6333i	−0.742453	−	1.28597i	−0.951375	−	0.308034i	$-0.900329\pi$
$283$	−12.4900	−	21.6333i	−0.742453	−	1.28597i	0.208922	−	0.977932i	$-0.433005\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−26.0000			−1.53473
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$		−	0.282899i		−	0.0165839i
$292$		0			0
$293$	−1.86750	−	1.07820i	−0.109100	−	0.0629891i	0.444457	−	0.895800i	$-0.353397\pi$
$293$	−1.86750	−	1.07820i	−0.109100	−	0.0629891i	−0.553557	+	0.832811i	$0.686730\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	3.00000	−	1.73205i	0.174078	−	0.100504i
$298$		0			0
$299$	13.1125	−	13.6269i	0.758315	−	0.788064i
$300$		0			0
$301$	−25.7450	+	14.8639i	−1.48392	+	0.856740i
$302$		0			0
$303$	−5.62250	+	9.73845i	−0.323004	+	0.559460i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			7.49400i			0.427705i	0.976866	+	0.213853i	$0.0686012\pi$
$307$			7.49400i			0.427705i	−0.976866	+	0.213853i	$0.931399\pi$
$308$		0			0
$309$	7.12250	+	12.3365i	0.405185	+	0.701801i
$310$		0			0
$311$	9.73499			0.552021			0.276010	−	0.961155i	$-0.410988\pi$
$311$	9.73499			0.552021			0.276010	+	0.961155i	$0.410988\pi$
$312$		0			0
$313$	7.00000			0.395663			0.197832	−	0.980236i	$-0.436610\pi$
$313$	7.00000			0.395663			0.197832	+	0.980236i	$0.436610\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	9.08460i		−	0.510242i	−0.966909	−	0.255121i	$-0.917885\pi$
$317$		−	9.08460i		−	0.510242i	0.966909	−	0.255121i	$-0.0821153\pi$
$318$		0			0
$319$	21.7350	+	12.5487i	1.21693	+	0.702593i
$320$		0			0
$321$	1.62250	−	2.81025i	0.0905591	−	0.156853i
$322$		0			0
$323$	12.2450	−	7.06965i	0.681330	−	0.393366i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−9.49000	+	5.47905i	−0.524798	+	0.302992i
$328$		0			0
$329$	19.2450	−	33.3333i	1.06101	−	1.83773i
$330$		0			0
$331$	12.1225	+	6.99893i	0.666313	+	0.384696i	0.794678	−	0.607031i	$-0.207640\pi$
$331$	12.1225	+	6.99893i	0.666313	+	0.384696i	−0.128365	+	0.991727i	$0.540973\pi$
$332$		0			0
$333$			7.21110i			0.395166i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−19.9800			−1.08838			−0.544190	−	0.838962i	$-0.683163\pi$
$337$	−19.9800			−1.08838			−0.544190	+	0.838962i	$0.683163\pi$
$338$		0			0
$339$	−12.0000			−0.651751
$340$		0			0
$341$	−6.00000	−	10.3923i	−0.324918	−	0.562775i
$342$		0			0
$343$			3.60555i			0.194681i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	−0.980921	−	0.194409i	$-0.937721\pi$
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	0.658824	+	0.752297i	$0.271054\pi$
$348$		0			0
$349$	−1.50000	+	0.866025i	−0.0802932	+	0.0463573i	−0.539609	−	0.841916i	$-0.681428\pi$
$349$	−1.50000	+	0.866025i	−0.0802932	+	0.0463573i	0.459316	+	0.888273i	$0.348095\pi$
$350$		0			0
$351$	−1.00000	−	3.46410i	−0.0533761	−	0.184900i
$352$		0			0
$353$	7.62250	−	4.40085i	0.405705	−	0.234234i	−0.283238	−	0.959050i	$-0.591409\pi$
$353$	7.62250	−	4.40085i	0.405705	−	0.234234i	0.688943	+	0.724816i	$0.258075\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	6.24500	+	3.60555i	0.330520	+	0.190826i
$358$		0			0
$359$		−	8.23591i		−	0.434674i	−0.976097	−	0.217337i	$-0.930263\pi$
$359$		−	8.23591i		−	0.434674i	0.976097	−	0.217337i	$-0.0697371\pi$
$360$		0			0
$361$	15.4900	+	26.8295i	0.815263	+	1.41208i
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.36750	+	2.36857i	0.0713828	+	0.123639i	0.899508	−	0.436905i	$-0.143925\pi$
$367$	1.36750	+	2.36857i	0.0713828	+	0.123639i	−0.828125	+	0.560544i	$0.810592\pi$
$368$		0			0
$369$			7.21110i			0.375395i
$370$		0			0
$371$	28.8675	+	16.6667i	1.49873	+	0.865290i
$372$		0			0
$373$	−3.50000	+	6.06218i	−0.181223	+	0.313888i	−0.942297	−	0.334777i	$-0.891339\pi$
$373$	−3.50000	+	6.06218i	−0.181223	+	0.313888i	0.761074	+	0.648665i	$0.224672\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	18.1125	−	18.8231i	0.932841	−	0.969437i
$378$		0			0
$379$	−9.12250	+	5.26688i	−0.468591	+	0.270541i	−0.715650	−	0.698459i	$-0.753869\pi$
$379$	−9.12250	+	5.26688i	−0.468591	+	0.270541i	0.247059	+	0.969001i	$0.420536\pi$
$380$		0			0
$381$	−8.12250	+	14.0686i	−0.416128	+	0.720755i
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.12250	+	7.14038i	0.209558	+	0.362966i
$388$		0			0
$389$	−16.9800			−0.860920			−0.430460	−	0.902610i	$-0.641649\pi$
$389$	−16.9800			−0.860920			−0.430460	+	0.902610i	$0.641649\pi$
$390$		0			0
$391$	10.4900			0.530502
$392$		0			0
$393$	−5.62250	−	9.73845i	−0.283618	−	0.491240i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	16.5000	+	9.52628i	0.828111	+	0.478110i	0.853206	−	0.521575i	$-0.174655\pi$
$397$	16.5000	+	9.52628i	0.828111	+	0.478110i	−0.0250943	+	0.999685i	$0.507989\pi$
$398$		0			0
$399$	−12.7450	+	22.0750i	−0.638048	+	1.10513i
$400$		0			0
$401$	−24.9800	+	14.4222i	−1.24744	+	0.720211i	−0.970598	−	0.240705i	$-0.922621\pi$
$401$	−24.9800	+	14.4222i	−1.24744	+	0.720211i	−0.276843	+	0.960915i	$0.589288\pi$
$402$		0			0
$403$	−12.0000	+	3.46410i	−0.597763	+	0.172559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.4900	−	21.6333i	0.619106	−	1.07232i
$408$		0			0
$409$	11.7550	+	6.78675i	0.581247	+	0.335583i	0.761629	−	0.648013i	$-0.224400\pi$
$409$	11.7550	+	6.78675i	0.581247	+	0.335583i	−0.180382	+	0.983597i	$0.557733\pi$
$410$		0			0
$411$		−	16.2957i		−	0.803808i
$412$		0			0
$413$	9.62250	+	16.6667i	0.473492	+	0.820113i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.244998			−0.0119976
$418$		0			0
$419$	−16.8675	−	29.2154i	−0.824031	−	1.42726i	−0.902657	−	0.430360i	$-0.858387\pi$
$419$	−16.8675	−	29.2154i	−0.824031	−	1.42726i	0.0786263	−	0.996904i	$-0.474947\pi$
$420$		0			0
$421$		−	5.19615i		−	0.253245i	−0.991951	−	0.126622i	$-0.959586\pi$
$421$		−	5.19615i		−	0.253245i	0.991951	−	0.126622i	$-0.0404137\pi$
$422$		0			0
$423$	−9.24500	−	5.33760i	−0.449507	−	0.259523i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−3.00000	+	12.1244i	−0.144841	+	0.585369i
$430$		0			0
$431$	22.8675	−	13.2026i	1.10149	−	0.635945i	0.164877	−	0.986314i	$-0.447277\pi$
$431$	22.8675	−	13.2026i	1.10149	−	0.635945i	0.936612	+	0.350369i	$0.113944\pi$
$432$		0			0
$433$	−5.24500	+	9.08460i	−0.252059	+	0.436578i	−0.964092	−	0.265567i	$-0.914441\pi$
$433$	−5.24500	+	9.08460i	−0.252059	+	0.436578i	0.712034	+	0.702145i	$0.247774\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			37.0803i			1.77379i
$438$		0			0
$439$	−13.1225	−	22.7288i	−0.626303	−	1.08479i	−0.988287	−	0.152604i	$-0.951234\pi$
$439$	−13.1225	−	22.7288i	−0.626303	−	1.08479i	0.361985	−	0.932184i	$-0.382099\pi$
$440$		0			0
$441$	−6.00000			−0.285714
$442$		0			0
$443$	18.0000			0.855206			0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000			0.855206			0.427603	+	0.903967i	$0.359358\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	22.6581i		−	1.07169i
$448$		0			0
$449$	7.86750	+	4.54230i	0.371290	+	0.214364i	0.674022	−	0.738711i	$-0.264565\pi$
$449$	7.86750	+	4.54230i	0.371290	+	0.214364i	−0.302732	+	0.953076i	$0.597899\pi$
$450$		0			0
$451$	12.4900	−	21.6333i	0.588131	−	1.01867i
$452$		0			0
$453$	−18.3675	+	10.6045i	−0.862980	+	0.498242i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.01000	−	2.31518i	0.187580	−	0.108299i	−0.403269	−	0.915081i	$-0.632126\pi$
$457$	4.01000	−	2.31518i	0.187580	−	0.108299i	0.590849	+	0.806782i	$0.298793\pi$
$458$		0			0
$459$	1.00000	−	1.73205i	0.0466760	−	0.0808452i
$460$		0			0
$461$	−24.7350	−	14.2808i	−1.15202	−	0.665121i	−0.202644	−	0.979252i	$-0.564953\pi$
$461$	−24.7350	−	14.2808i	−1.15202	−	0.665121i	−0.949380	+	0.314131i	$0.898287\pi$
$462$		0			0
$463$		−	10.5338i		−	0.489545i	−0.969581	−	0.244773i	$-0.921287\pi$
$463$		−	10.5338i		−	0.489545i	0.969581	−	0.244773i	$-0.0787133\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.4900			0.948164			0.474082	−	0.880481i	$-0.342780\pi$
$467$	20.4900			0.948164			0.474082	+	0.880481i	$0.342780\pi$
$468$		0			0
$469$	0.510004			0.0235498
$470$		0			0
$471$	−8.50000	−	14.7224i	−0.391659	−	0.678374i
$472$		0			0
$473$		−	28.5615i		−	1.31326i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4.62250	−	8.00640i	0.211650	−	0.366588i
$478$		0			0
$479$	32.1125	−	18.5402i	1.46726	−	0.847121i	0.467929	−	0.883766i	$-0.345000\pi$
$479$	32.1125	−	18.5402i	1.46726	−	0.847121i	0.999328	+	0.0366446i	$0.0116670\pi$
$480$		0			0
$481$	−18.7350	−	18.0278i	−0.854242	−	0.821995i
$482$		0			0
$483$	−16.3775	+	9.45555i	−0.745202	+	0.430243i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	6.61249	+	3.81773i	0.299641	+	0.172998i	0.642281	−	0.766469i	$-0.277988\pi$
$487$	6.61249	+	3.81773i	0.299641	+	0.172998i	−0.342641	+	0.939467i	$0.611321\pi$
$488$		0			0
$489$		−	10.9581i		−	0.495543i
$490$		0			0
$491$	5.75500	+	9.96796i	0.259720	+	0.449848i	0.966167	−	0.257918i	$-0.0830365\pi$
$491$	5.75500	+	9.96796i	0.259720	+	0.449848i	−0.706447	+	0.707766i	$0.749703\pi$
$492$		0			0
$493$	14.4900			0.652597
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13.0000	+	22.5167i	0.583130	+	1.01001i
$498$		0			0
$499$			20.6432i			0.924115i	0.886850	+	0.462057i	$0.152889\pi$
$499$			20.6432i			0.924115i	−0.886850	+	0.462057i	$0.847111\pi$
$500$		0			0
$501$	7.37750	+	4.25940i	0.329602	+	0.190296i
$502$		0			0
$503$	1.86750	−	3.23460i	0.0832676	−	0.144224i	−0.821384	−	0.570375i	$-0.806798\pi$
$503$	1.86750	−	3.23460i	0.0832676	−	0.144224i	0.904652	+	0.426152i	$0.140131\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	11.5000	+	6.06218i	0.510733	+	0.269231i
$508$		0			0
$509$	9.00000	−	5.19615i	0.398918	−	0.230315i	−0.287099	−	0.957901i	$-0.592691\pi$
$509$	9.00000	−	5.19615i	0.398918	−	0.230315i	0.686017	+	0.727586i	$0.259358\pi$
$510$		0			0
$511$	3.63250	−	6.29168i	0.160692	−	0.278328i
$512$		0			0
$513$	6.12250	+	3.53483i	0.270315	+	0.156066i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	18.4900	+	32.0256i	0.813189	+	1.40848i
$518$		0			0
$519$	−25.2450			−1.10813
$520$		0			0
$521$	8.75500			0.383564			0.191782	−	0.981438i	$-0.438573\pi$
$521$	8.75500			0.383564			0.191782	+	0.981438i	$0.438573\pi$
$522$		0			0
$523$	9.36750	+	16.2250i	0.409612	+	0.709469i	0.994846	−	0.101395i	$-0.0323307\pi$
$523$	9.36750	+	16.2250i	0.409612	+	0.709469i	−0.585234	+	0.810864i	$0.698997\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.00000	−	3.46410i	−0.261364	−	0.150899i
$528$		0			0
$529$	−2.25500	+	3.90578i	−0.0980436	+	0.169816i
$530$		0			0
$531$	4.62250	−	2.66880i	0.200599	−	0.115816i
$532$		0			0
$533$	−18.7350	−	18.0278i	−0.811503	−	0.780869i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−3.24500	+	5.62050i	−0.140032	+	0.242543i
$538$		0			0
$539$	18.0000	+	10.3923i	0.775315	+	0.447628i
$540$		0			0
$541$			6.64530i			0.285704i	0.989744	+	0.142852i	$0.0456273\pi$
$541$			6.64530i			0.285704i	−0.989744	+	0.142852i	$0.954373\pi$
$542$		0			0
$543$	−6.74500	−	11.6827i	−0.289456	−	0.501352i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−22.2450			−0.951127			−0.475564	−	0.879681i	$-0.657756\pi$
$547$	−22.2450			−0.951127			−0.475564	+	0.879681i	$0.657756\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			51.2196i			2.18203i
$552$		0			0
$553$	13.2550	+	7.65278i	0.563660	+	0.325429i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.9800	+	12.6902i	−0.931322	+	0.537699i	−0.887229	−	0.461328i	$-0.847373\pi$
$557$	−21.9800	+	12.6902i	−0.931322	+	0.537699i	−0.0440927	+	0.999027i	$0.514040\pi$
$558$		0			0
$559$	−28.8575	−	7.14038i	−1.22054	−	0.302006i
$560$		0			0
$561$	−6.00000	+	3.46410i	−0.253320	+	0.146254i
$562$		0			0
$563$	6.75500	−	11.7000i	0.284689	−	0.493096i	−0.687844	−	0.725858i	$-0.741443\pi$
$563$	6.75500	−	11.7000i	0.284689	−	0.493096i	0.972534	+	0.232762i	$0.0747762\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			3.60555i			0.151419i
$568$		0			0
$569$	−19.6225	−	33.9872i	−0.822618	−	1.42482i	−0.903726	−	0.428111i	$-0.859179\pi$
$569$	−19.6225	−	33.9872i	−0.822618	−	1.42482i	0.0811083	−	0.996705i	$-0.474154\pi$
$570$		0			0
$571$	−10.7350			−0.449246			−0.224623	−	0.974446i	$-0.572115\pi$
$571$	−10.7350			−0.449246			−0.224623	+	0.974446i	$0.572115\pi$
$572$		0			0
$573$	12.4900			0.521777
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	19.6184i		−	0.816723i	−0.912820	−	0.408361i	$-0.866100\pi$
$577$		−	19.6184i		−	0.816723i	0.912820	−	0.408361i	$-0.133900\pi$
$578$		0			0
$579$	−7.50000	−	4.33013i	−0.311689	−	0.179954i
$580$		0			0
$581$	−22.1125	+	38.3000i	−0.917381	+	1.58895i
$582$		0			0
$583$	−27.7350	+	16.0128i	−1.14867	+	0.663183i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−33.2450	+	19.1940i	−1.37217	+	0.792222i	−0.991201	−	0.132367i	$-0.957742\pi$
$587$	−33.2450	+	19.1940i	−1.37217	+	0.792222i	−0.380967	+	0.924589i	$0.624409\pi$
$588$		0			0
$589$	12.2450	−	21.2090i	0.504546	−	0.873900i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			5.33760i			0.219189i	0.993976	+	0.109595i	$0.0349552\pi$
$593$			5.33760i			0.219189i	−0.993976	+	0.109595i	$0.965045\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	24.2450			0.992282
$598$		0			0
$599$	−21.7350			−0.888068			−0.444034	−	0.896010i	$-0.646453\pi$
$599$	−21.7350			−0.888068			−0.444034	+	0.896010i	$0.646453\pi$
$600$		0			0
$601$	5.50000	+	9.52628i	0.224350	+	0.388585i	0.956124	−	0.292962i	$-0.0946409\pi$
$601$	5.50000	+	9.52628i	0.224350	+	0.388585i	−0.731774	+	0.681547i	$0.761308\pi$
$602$		0			0
$603$		−	0.141450i		−	0.00576028i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	21.7350	−	37.6461i	0.882196	−	1.52801i	0.0333023	−	0.999445i	$-0.489398\pi$
$607$	21.7350	−	37.6461i	0.882196	−	1.52801i	0.848894	−	0.528563i	$-0.177269\pi$
$608$		0			0
$609$	−22.6225	+	13.0611i	−0.916710	+	0.529263i
$610$		0			0
$611$	36.9800	−	10.6752i	1.49605	−	0.431873i
$612$		0			0
$613$	−22.2550	+	12.8489i	−0.898871	+	0.518963i	−0.876834	−	0.480794i	$-0.840349\pi$
$613$	−22.2550	+	12.8489i	−0.898871	+	0.518963i	−0.0220373	+	0.999757i	$0.507015\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	11.3575	+	6.55725i	0.457235	+	0.263985i	0.710881	−	0.703312i	$-0.248296\pi$
$617$	11.3575	+	6.55725i	0.457235	+	0.263985i	−0.253646	+	0.967297i	$0.581630\pi$
$618$		0			0
$619$		−	38.5295i		−	1.54863i	−0.632800	−	0.774315i	$-0.718095\pi$
$619$		−	38.5295i		−	1.54863i	0.632800	−	0.774315i	$-0.281905\pi$
$620$		0			0
$621$	2.62250	+	4.54230i	0.105237	+	0.182276i
$622$		0			0
$623$	−18.2250			−0.730169
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−12.2450	−	21.2090i	−0.489018	−	0.847004i
$628$		0			0
$629$		−	14.4222i		−	0.575051i
$630$		0			0
$631$	24.1225	+	13.9271i	0.960302	+	0.554430i	0.896266	−	0.443517i	$-0.146270\pi$
$631$	24.1225	+	13.9271i	0.960302	+	0.554430i	0.0640357	+	0.997948i	$0.479603\pi$
$632$		0			0
$633$	−9.24500	+	16.0128i	−0.367456	+	0.636452i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	15.0000	−	15.5885i	0.594322	−	0.617637i
$638$		0			0
$639$	6.24500	−	3.60555i	0.247048	−	0.142633i
$640$		0			0
$641$	22.1125	−	38.3000i	0.873391	−	1.51276i	0.0149242	−	0.999889i	$-0.495249\pi$
$641$	22.1125	−	38.3000i	0.873391	−	1.51276i	0.858467	−	0.512869i	$-0.171417\pi$
$642$		0			0
$643$	−22.1025	−	12.7609i	−0.871637	−	0.503240i	−0.00374522	−	0.999993i	$-0.501192\pi$
$643$	−22.1025	−	12.7609i	−0.871637	−	0.503240i	−0.867892	+	0.496753i	$0.834525\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12.3775	+	21.4385i	0.486610	+	0.842833i	0.999882	−	0.0153933i	$-0.00490003\pi$
$647$	12.3775	+	21.4385i	0.486610	+	0.842833i	−0.513272	+	0.858226i	$0.671567\pi$
$648$		0			0
$649$	−18.4900			−0.725796
$650$		0			0
$651$	12.4900			0.489522
$652$		0			0
$653$	17.2450	+	29.8692i	0.674849	+	1.16887i	0.976513	+	0.215458i	$0.0691246\pi$
$653$	17.2450	+	29.8692i	0.674849	+	1.16887i	−0.301664	+	0.953414i	$0.597542\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.74500	−	1.00748i	−0.0680789	−	0.0393053i
$658$		0			0
$659$	−5.49000	+	9.50895i	−0.213860	+	0.370416i	−0.952919	−	0.303224i	$-0.901937\pi$
$659$	−5.49000	+	9.50895i	−0.213860	+	0.370416i	0.739059	+	0.673640i	$0.235270\pi$
$660$		0			0
$661$	−4.01000	+	2.31518i	−0.155971	+	0.0900499i	−0.575954	−	0.817482i	$-0.695369\pi$
$661$	−4.01000	+	2.31518i	−0.155971	+	0.0900499i	0.419983	+	0.907532i	$0.362036\pi$
$662$		0			0
$663$	2.00000	+	6.92820i	0.0776736	+	0.269069i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−19.0000	+	32.9090i	−0.735683	+	1.27424i
$668$		0			0
$669$	21.3675	+	12.3365i	0.826115	+	0.476958i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	8.50000	+	14.7224i	0.327651	+	0.567508i	0.982045	−	0.188645i	$-0.0604097\pi$
$673$	8.50000	+	14.7224i	0.327651	+	0.567508i	−0.654394	+	0.756153i	$0.727076\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−31.7350			−1.21967			−0.609837	−	0.792527i	$-0.708765\pi$
$677$	−31.7350			−1.21967			−0.609837	+	0.792527i	$0.708765\pi$
$678$		0			0
$679$	−0.510004	−	0.883353i	−0.0195722	−	0.0339000i
$680$		0			0
$681$			26.4051i			1.01185i
$682$		0			0
$683$	20.1125	+	11.6120i	0.769583	+	0.444319i	0.832726	−	0.553685i	$-0.186779\pi$
$683$	20.1125	+	11.6120i	0.769583	+	0.444319i	−0.0631427	+	0.998005i	$0.520112\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	17.2350	−	9.95063i	0.657556	−	0.379640i
$688$		0			0
$689$	9.24500	+	32.0256i	0.352206	+	1.22008i
$690$		0			0
$691$	−5.87750	+	3.39338i	−0.223591	+	0.129090i	−0.607612	−	0.794234i	$-0.707872\pi$
$691$	−5.87750	+	3.39338i	−0.223591	+	0.129090i	0.384021	+	0.923324i	$0.374539\pi$
$692$		0			0
$693$	6.24500	−	10.8167i	0.237228	−	0.410891i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	14.4222i		−	0.546280i
$698$		0			0
$699$	−1.37750	−	2.38590i	−0.0521019	−	0.0902431i
$700$		0			0
$701$	22.9800			0.867942			0.433971	−	0.900927i	$-0.357112\pi$
$701$	22.9800			0.867942			0.433971	+	0.900927i	$0.357112\pi$
$702$		0			0
$703$	50.9800			1.92275
$704$		0			0
$705$		0			0
$706$		0			0
$707$			40.5444i			1.52483i
$708$		0			0
$709$	13.7450	+	7.93568i	0.516204	+	0.298031i	0.735380	−	0.677655i	$-0.237004\pi$
$709$	13.7450	+	7.93568i	0.516204	+	0.298031i	−0.219176	+	0.975685i	$0.570337\pi$
$710$		0			0
$711$	2.12250	−	3.67628i	0.0795999	−	0.137871i
$712$		0			0
$713$	15.7350	−	9.08460i	0.589280	−	0.340221i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	13.8675	−	8.00640i	0.517891	−	0.299005i
$718$		0			0
$719$	19.3775	−	33.5628i	0.722659	−	1.25168i	−0.237272	−	0.971443i	$-0.576253\pi$
$719$	19.3775	−	33.5628i	0.722659	−	1.25168i	0.959930	−	0.280238i	$-0.0904135\pi$
$720$		0			0
$721$	44.4800	+	25.6805i	1.65652	+	0.956393i
$722$		0			0
$723$			26.2637i			0.976756i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−51.2250			−1.89983			−0.949915	−	0.312509i	$-0.898831\pi$
$727$	−51.2250			−1.89983			−0.949915	+	0.312509i	$0.898831\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−8.24500	−	14.2808i	−0.304952	−	0.528193i
$732$		0			0
$733$			22.2338i			0.821223i	0.911810	+	0.410611i	$0.134685\pi$
$733$			22.2338i			0.821223i	−0.911810	+	0.410611i	$0.865315\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.244998	+	0.424349i	−0.00902462	+	0.0156311i
$738$		0			0
$739$	−2.51000	+	1.44915i	−0.0923320	+	0.0533079i	−0.545455	−	0.838140i	$-0.683643\pi$
$739$	−2.51000	+	1.44915i	−0.0923320	+	0.0533079i	0.453123	+	0.891448i	$0.350310\pi$
$740$		0			0
$741$	−24.4900	+	7.06965i	−0.899662	+	0.259710i
$742$		0			0
$743$	−30.2450	+	17.4620i	−1.10958	+	0.640617i	−0.938721	−	0.344678i	$-0.887988\pi$
$743$	−30.2450	+	17.4620i	−1.10958	+	0.640617i	−0.170860	+	0.985295i	$0.554655\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	10.6225	+	6.13290i	0.388657	+	0.224391i
$748$		0			0
$749$		−	11.7000i		−	0.427509i
$750$		0			0
$751$	5.75500	+	9.96796i	0.210003	+	0.363736i	0.951715	−	0.306982i	$-0.0993193\pi$
$751$	5.75500	+	9.96796i	0.210003	+	0.363736i	−0.741712	+	0.670718i	$0.765986\pi$
$752$		0			0
$753$	−0.755002			−0.0275138
$754$		0			0
$755$		0			0
$756$		0			0
$757$	13.5000	+	23.3827i	0.490666	+	0.849858i	0.999942	−	0.0107448i	$-0.00342025\pi$
$757$	13.5000	+	23.3827i	0.490666	+	0.849858i	−0.509276	+	0.860603i	$0.670087\pi$
$758$		0			0
$759$		−	18.1692i		−	0.659500i
$760$		0			0
$761$	3.00000	+	1.73205i	0.108750	+	0.0627868i	0.553388	−	0.832923i	$-0.313335\pi$
$761$	3.00000	+	1.73205i	0.108750	+	0.0627868i	−0.444639	+	0.895710i	$0.646668\pi$
$762$		0			0
$763$	−19.7550	+	34.2167i	−0.715179	+	1.23873i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.62250	+	18.6816i	−0.166909	+	0.674554i
$768$		0			0
$769$	−1.74500	+	1.00748i	−0.0629262	+	0.0363305i	−0.531133	−	0.847288i	$-0.678234\pi$
$769$	−1.74500	+	1.00748i	−0.0629262	+	0.0363305i	0.468207	+	0.883619i	$0.344900\pi$
$770$		0			0
$771$	−10.4900	+	18.1692i	−0.377788	+	0.654348i
$772$		0			0
$773$	39.7350	+	22.9410i	1.42917	+	0.825131i	0.997055	−	0.0766886i	$-0.0244347\pi$
$773$	39.7350	+	22.9410i	1.42917	+	0.825131i	0.432113	+	0.901819i	$0.357768\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	13.0000	+	22.5167i	0.466372	+	0.807781i
$778$		0			0
$779$	50.9800			1.82655
$780$		0			0
$781$	−24.9800			−0.893854
$782$		0			0
$783$	3.62250	+	6.27435i	0.129458	+	0.224227i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	21.0000	+	12.1244i	0.748569	+	0.432187i	0.825177	−	0.564875i	$-0.191076\pi$
$787$	21.0000	+	12.1244i	0.748569	+	0.432187i	−0.0766075	+	0.997061i	$0.524409\pi$
$788$		0			0
$789$	9.49000	−	16.4372i	0.337853	−	0.585178i
$790$		0			0
$791$	−37.4700	+	21.6333i	−1.33228	+	0.769192i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7.62250	+	13.2026i	−0.270003	+	0.467659i	−0.968862	−	0.247601i	$-0.920358\pi$
$797$	−7.62250	+	13.2026i	−0.270003	+	0.467659i	0.698859	+	0.715259i	$0.253691\pi$
$798$		0			0
$799$	18.4900	+	10.6752i	0.654129	+	0.377662i
$800$		0			0
$801$			5.05470i			0.178599i
$802$		0			0
$803$	3.49000	+	6.04485i	0.123159	+	0.213318i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−24.0000			−0.844840
$808$		0			0
$809$	−2.13250	−	3.69360i	−0.0749748	−	0.129860i	0.826101	−	0.563523i	$-0.190554\pi$
$809$	−2.13250	−	3.69360i	−0.0749748	−	0.129860i	−0.901075	+	0.433663i	$0.857221\pi$
$810$		0			0
$811$			34.7825i			1.22138i	0.791871	+	0.610689i	$0.209107\pi$
$811$			34.7825i			1.22138i	−0.791871	+	0.610689i	$0.790893\pi$
$812$		0			0
$813$	12.1225	+	6.99893i	0.425155	+	0.245463i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	50.4800	−	29.1446i	1.76607	−	1.01964i
$818$		0			0
$819$	−9.36750	−	9.01388i	−0.327327	−	0.314970i
$820$		0			0
$821$	36.0925	−	20.8380i	1.25964	−	0.727251i	0.286633	−	0.958041i	$-0.407464\pi$
$821$	36.0925	−	20.8380i	1.25964	−	0.727251i	0.973004	+	0.230789i	$0.0741307\pi$
$822$		0			0
$823$	4.36750	−	7.56473i	0.152241	−	0.263690i	−0.779810	−	0.626017i	$-0.784684\pi$
$823$	4.36750	−	7.56473i	0.152241	−	0.263690i	0.932051	+	0.362327i	$0.118018\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			6.92820i			0.240917i	0.992718	+	0.120459i	$0.0384365\pi$
$827$			6.92820i			0.240917i	−0.992718	+	0.120459i	$0.961563\pi$
$828$		0			0
$829$	−14.2550	−	24.6904i	−0.495097	−	0.857533i	0.504887	−	0.863185i	$-0.331534\pi$
$829$	−14.2550	−	24.6904i	−0.495097	−	0.857533i	−0.999984	+	0.00565264i	$0.998201\pi$
$830$		0			0
$831$	2.51000			0.0870711
$832$		0			0
$833$	12.0000			0.415775
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	3.46410i		−	0.119737i
$838$		0			0
$839$	7.62250	+	4.40085i	0.263158	+	0.151934i	0.625774	−	0.780004i	$-0.284783\pi$
$839$	7.62250	+	4.40085i	0.263158	+	0.151934i	−0.362616	+	0.931939i	$0.618116\pi$
$840$		0			0
$841$	−11.7450	+	20.3429i	−0.405000	+	0.701480i
$842$		0			0
$843$	−19.6225	+	11.3291i	−0.675835	+	0.390193i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−3.12250	+	1.80278i	−0.107290	+	0.0619441i
$848$		0			0
$849$	−12.4900	+	21.6333i	−0.428656	+	0.742453i
$850$		0			0
$851$	32.7550	+	18.9111i	1.12283	+	0.648264i
$852$		0			0
$853$			38.6709i			1.32407i	0.749474	+	0.662033i	$0.230306\pi$
$853$			38.6709i			1.32407i	−0.749474	+	0.662033i	$0.769694\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	40.4900			1.38311			0.691556	−	0.722323i	$-0.256926\pi$
$857$	40.4900			1.38311			0.691556	+	0.722323i	$0.256926\pi$
$858$		0			0
$859$	14.7350			0.502752			0.251376	−	0.967890i	$-0.419117\pi$
$859$	14.7350			0.502752			0.251376	+	0.967890i	$0.419117\pi$
$860$		0			0
$861$	13.0000	+	22.5167i	0.443039	+	0.767366i
$862$		0			0
$863$			38.9538i			1.32600i	0.748618	+	0.663002i	$0.230718\pi$
$863$			38.9538i			1.32600i	−0.748618	+	0.663002i	$0.769282\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	6.50000	−	11.2583i	0.220752	−	0.382353i
$868$		0			0
$869$	−12.7350	+	7.35255i	−0.432005	+	0.249418i
$870$		0			0
$871$	0.367497	+	0.353624i	0.0124522	+	0.0119821i
$872$		0			0
$873$	−0.244998	+	0.141450i	−0.00829193	+	0.00478735i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−26.4800	−	15.2882i	−0.894166	−	0.516247i	−0.0188630	−	0.999822i	$-0.506005\pi$
$877$	−26.4800	−	15.2882i	−0.894166	−	0.516247i	−0.875303	+	0.483575i	$0.839338\pi$
$878$		0			0
$879$			2.15640i			0.0727336i
$880$		0			0
$881$	−23.7350	−	41.1102i	−0.799652	−	1.38504i	−0.919843	−	0.392287i	$-0.871684\pi$
$881$	−23.7350	−	41.1102i	−0.799652	−	1.38504i	0.120190	−	0.992751i	$-0.461649\pi$
$882$		0			0
$883$	4.49000			0.151100			0.0755502	−	0.997142i	$-0.475929\pi$
$883$	4.49000			0.151100			0.0755502	+	0.997142i	$0.475929\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	11.6225	+	20.1308i	0.390245	+	0.675925i	0.992482	−	0.122393i	$-0.0390569\pi$
$887$	11.6225	+	20.1308i	0.390245	+	0.675925i	−0.602236	+	0.798318i	$0.705724\pi$
$888$		0			0
$889$			58.5722i			1.96445i
$890$		0			0
$891$	−3.00000	−	1.73205i	−0.100504	−	0.0580259i
$892$		0			0
$893$	−37.7350	+	65.3589i	−1.26275	+	2.18715i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−18.3575	−	4.54230i	−0.612939	−	0.151663i
$898$		0			0
$899$	21.7350	−	12.5487i	0.724903	−	0.418523i
$900$		0			0
$901$	−9.24500	+	16.0128i	−0.307996	+	0.533464i
$902$		0			0
$903$	25.7450	+	14.8639i	0.856740	+	0.494639i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.24500	−	3.88845i	−0.0745439	−	0.129114i	0.826344	−	0.563166i	$-0.190417\pi$
$907$	−2.24500	−	3.88845i	−0.0745439	−	0.129114i	−0.900888	+	0.434052i	$0.857083\pi$
$908$		0			0
$909$	11.2450			0.372973
$910$		0			0
$911$	14.7550			0.488855			0.244428	−	0.969668i	$-0.421400\pi$
$911$	14.7550			0.488855			0.244428	+	0.969668i	$0.421400\pi$
$912$		0			0
$913$	−21.2450	−	36.7974i	−0.703107	−	1.21782i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−35.1125	−	20.2722i	−1.15952	−	0.669447i
$918$		0			0
$919$	−7.87750	+	13.6442i	−0.259855	+	0.450082i	−0.966203	−	0.257783i	$-0.917008\pi$
$919$	−7.87750	+	13.6442i	−0.259855	+	0.450082i	0.706348	+	0.707865i	$0.250341\pi$
$920$		0			0
$921$	6.49000	−	3.74700i	0.213853	−	0.123468i
$922$		0			0
$923$	−6.24500	+	25.2389i	−0.205557	+	0.830747i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	7.12250	−	12.3365i	0.233934	−	0.405185i
$928$		0			0
$929$	−3.97999	−	2.29785i	−0.130579	−	0.0753900i	0.433287	−	0.901256i	$-0.357354\pi$
$929$	−3.97999	−	2.29785i	−0.130579	−	0.0753900i	−0.563867	+	0.825866i	$0.690687\pi$
$930$		0			0
$931$			42.4179i			1.39019i
$932$		0			0
$933$	−4.86750	−	8.43075i	−0.159355	−	0.276010i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	59.4900			1.94345			0.971727	−	0.236108i	$-0.0758720\pi$
$937$	59.4900			1.94345			0.971727	+	0.236108i	$0.0758720\pi$
$938$		0			0
$939$	−3.50000	−	6.06218i	−0.114218	−	0.197832i
$940$		0			0
$941$		−	36.7974i		−	1.19956i	−0.800164	−	0.599781i	$-0.795254\pi$
$941$		−	36.7974i		−	1.19956i	0.800164	−	0.599781i	$-0.204746\pi$
$942$		0			0
$943$	32.7550	+	18.9111i	1.06665	+	0.615830i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−12.7350	+	7.35255i	−0.413832	+	0.238926i	−0.692435	−	0.721480i	$-0.743462\pi$
$947$	−12.7350	+	7.35255i	−0.413832	+	0.238926i	0.278603	+	0.960406i	$0.410129\pi$
$948$		0			0
$949$	6.97999	−	2.01495i	0.226580	−	0.0654080i
$950$		0			0
$951$	−7.86750	+	4.54230i	−0.255121	+	0.147294i
$952$		0			0
$953$	13.7550	−	23.8244i	0.445568	−	0.771747i	−0.552523	−	0.833497i	$-0.686335\pi$
$953$	13.7550	−	23.8244i	0.445568	−	0.771747i	0.998092	+	0.0617506i	$0.0196683\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	25.0974i		−	0.811284i
$958$		0			0
$959$	−29.3775	−	50.8833i	−0.948649	−	1.64311i
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$	−3.24500			−0.104569
$964$		0			0
$965$		0			0
$966$		0			0
$967$			0.565799i			0.0181949i	0.999959	+	0.00909743i	$0.00289584\pi$
$967$			0.565799i			0.0181949i	−0.999959	+	0.00909743i	$0.997104\pi$
$968$		0			0
$969$	−12.2450	−	7.06965i	−0.393366	−	0.227110i
$970$		0			0
$971$	4.11249	−	7.12305i	0.131976	−	0.228590i	−0.792462	−	0.609921i	$-0.791201\pi$
$971$	4.11249	−	7.12305i	0.131976	−	0.228590i	0.924438	+	0.381332i	$0.124534\pi$
$972$		0			0
$973$	−0.765006	+	0.441676i	−0.0245250	+	0.0141595i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.2250	−	12.8316i	0.711040	−	0.410519i	−0.100406	−	0.994947i	$-0.532014\pi$
$977$	22.2250	−	12.8316i	0.711040	−	0.410519i	0.811446	+	0.584427i	$0.198681\pi$
$978$		0			0
$979$	8.75500	−	15.1641i	0.279811	−	0.484647i
$980$		0			0
$981$	9.49000	+	5.47905i	0.302992	+	0.174933i
$982$		0			0
$983$		−	44.8572i		−	1.43072i	−0.698755	−	0.715362i	$-0.746262\pi$
$983$		−	44.8572i		−	1.43072i	0.698755	−	0.715362i	$-0.253738\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−38.4900			−1.22515
$988$		0			0
$989$	43.2450			1.37511
$990$		0			0
$991$	15.6125	+	27.0416i	0.495947	+	0.859006i	0.999989	−	0.00467341i	$-0.00148760\pi$
$991$	15.6125	+	27.0416i	0.495947	+	0.859006i	−0.504042	+	0.863679i	$0.668154\pi$
$992$		0			0
$993$		−	13.9979i		−	0.444209i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.50000	−	2.59808i	0.0475055	−	0.0822819i	−0.841295	−	0.540576i	$-0.818206\pi$
$997$	1.50000	−	2.59808i	0.0475055	−	0.0822819i	0.888800	+	0.458295i	$0.151540\pi$
$998$		0			0
$999$	6.24500	−	3.60555i	0.197583	−	0.114075i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3900.2.cd.f.2701.1	yes	4
5.2	odd	4		3900.2.bw.i.49.2		8
5.3	odd	4		3900.2.bw.i.49.3		8
5.4	even	2		3900.2.cd.h.2701.2	yes	4
13.4	even	6	inner	3900.2.cd.f.901.1	✓	4
65.4	even	6		3900.2.cd.h.901.2	yes	4
65.17	odd	12		3900.2.bw.i.2149.3		8
65.43	odd	12		3900.2.bw.i.2149.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3900.2.bw.i.49.2		8	5.2	odd	4
3900.2.bw.i.49.3		8	5.3	odd	4
3900.2.bw.i.2149.2		8	65.43	odd	12
3900.2.bw.i.2149.3		8	65.17	odd	12
3900.2.cd.f.901.1	✓	4	13.4	even	6	inner
3900.2.cd.f.2701.1	yes	4	1.1	even	1	trivial
3900.2.cd.h.901.2	yes	4	65.4	even	6
3900.2.cd.h.2701.2	yes	4	5.4	even	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:	\( N \)	\(=\)	\( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3900.cd (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +(-3.12250 - 1.80278i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(3.00000 - 1.73205i) q^{11} +(-1.00000 - 3.46410i) q^{13} +(1.00000 - 1.73205i) q^{17} +(6.12250 + 3.53483i) q^{19} +3.60555i q^{21} +(2.62250 + 4.54230i) q^{23} +1.00000 q^{27} +(3.62250 + 6.27435i) q^{29} -3.46410i q^{31} +(-3.00000 - 1.73205i) q^{33} +(6.24500 - 3.60555i) q^{37} +(-2.50000 + 2.59808i) q^{39} +(6.24500 - 3.60555i) q^{41} +(4.12250 - 7.14038i) q^{43} +10.6752i q^{47} +(3.00000 + 5.19615i) q^{49} -2.00000 q^{51} -9.24500 q^{53} -7.06965i q^{57} +(-4.62250 - 2.66880i) q^{59} +(3.12250 - 1.80278i) q^{63} +(-0.122499 + 0.0707248i) q^{67} +(2.62250 - 4.54230i) q^{69} +(-6.24500 - 3.60555i) q^{71} +2.01495i q^{73} -12.4900 q^{77} -4.24500 q^{79} +(-0.500000 - 0.866025i) q^{81} -12.2658i q^{83} +(3.62250 - 6.27435i) q^{87} +(4.37750 - 2.52735i) q^{89} +(-3.12250 + 12.6194i) q^{91} +(-3.00000 + 1.73205i) q^{93} +(0.244998 + 0.141450i) q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{9} + 12 q^{11} - 4 q^{13} + 4 q^{17} + 12 q^{19} - 2 q^{23} + 4 q^{27} + 2 q^{29} - 12 q^{33} - 10 q^{39} + 4 q^{43} + 12 q^{49} - 8 q^{51} - 12 q^{53} - 6 q^{59} + 12 q^{67} - 2 q^{69}+ \cdots - 24 q^{97}+O(q^{100}) \) 4 * q - 2 * q^3 - 2 * q^9 + 12 * q^11 - 4 * q^13 + 4 * q^17 + 12 * q^19 - 2 * q^23 + 4 * q^27 + 2 * q^29 - 12 * q^33 - 10 * q^39 + 4 * q^43 + 12 * q^49 - 8 * q^51 - 12 * q^53 - 6 * q^59 + 12 * q^67 - 2 * q^69 + 8 * q^79 - 2 * q^81 + 2 * q^87 + 30 * q^89 - 12 * q^93 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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