LMFDB - Embedded newform 208.2.i.b.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,2,Mod(81,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("208.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	208.i (of order $3$, degree $2$, 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:	$1.66088836204$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			113.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	208.113
Dual form			208.2.i.b.81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +(2.00000 + 3.46410i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})$	$q-1.00000 q^{5} +(2.00000 + 3.46410i) q^{7} +(1.50000 + 2.59808i) q^{9} +(2.00000 - 3.46410i) q^{11} +(3.50000 + 0.866025i) q^{13} +(-1.50000 - 2.59808i) q^{17} +(-2.00000 + 3.46410i) q^{23} -4.00000 q^{25} +(0.500000 - 0.866025i) q^{29} -4.00000 q^{31} +(-2.00000 - 3.46410i) q^{35} +(-1.50000 + 2.59808i) q^{37} +(4.50000 - 7.79423i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(-1.50000 - 2.59808i) q^{45} +8.00000 q^{47} +(-4.50000 + 7.79423i) q^{49} -9.00000 q^{53} +(-2.00000 + 3.46410i) q^{55} +(-2.00000 - 3.46410i) q^{59} +(-3.50000 - 6.06218i) q^{61} +(-6.00000 + 10.3923i) q^{63} +(-3.50000 - 0.866025i) q^{65} +(2.00000 - 3.46410i) q^{67} +(-4.00000 - 6.92820i) q^{71} +11.0000 q^{73} +16.0000 q^{77} +4.00000 q^{79} +(-4.50000 + 7.79423i) q^{81} +(1.50000 + 2.59808i) q^{85} +(3.00000 - 5.19615i) q^{89} +(4.00000 + 13.8564i) q^{91} +(-1.00000 - 1.73205i) q^{97} +12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 + 4 * q^7 + 3 * q^9	$ 2 q - 2 q^{5} + 4 q^{7} + 3 q^{9} + 4 q^{11} + 7 q^{13} - 3 q^{17} - 4 q^{23} - 8 q^{25} + q^{29} - 8 q^{31} - 4 q^{35} - 3 q^{37} + 9 q^{41} - 8 q^{43} - 3 q^{45} + 16 q^{47} - 9 q^{49} - 18 q^{53} - 4 q^{55} - 4 q^{59} - 7 q^{61} - 12 q^{63} - 7 q^{65} + 4 q^{67} - 8 q^{71} + 22 q^{73} + 32 q^{77} + 8 q^{79} - 9 q^{81} + 3 q^{85} + 6 q^{89} + 8 q^{91} - 2 q^{97} + 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^7 + 3 * q^9 + 4 * q^11 + 7 * q^13 - 3 * q^17 - 4 * q^23 - 8 * q^25 + q^29 - 8 * q^31 - 4 * q^35 - 3 * q^37 + 9 * q^41 - 8 * q^43 - 3 * q^45 + 16 * q^47 - 9 * q^49 - 18 * q^53 - 4 * q^55 - 4 * q^59 - 7 * q^61 - 12 * q^63 - 7 * q^65 + 4 * q^67 - 8 * q^71 + 22 * q^73 + 32 * q^77 + 8 * q^79 - 9 * q^81 + 3 * q^85 + 6 * q^89 + 8 * q^91 - 2 * q^97 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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			0		−0.866025	−	0.500000i	$-0.833333\pi$
$3$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$4$		0			0
$5$	−1.00000			−0.447214			−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000			−0.447214			−0.223607	+	0.974679i	$0.571783\pi$
$6$		0			0
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	0.944911	+	0.327327i	$0.106148\pi$
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$	2.00000	−	3.46410i	0.603023	−	1.04447i	−0.389338	−	0.921095i	$-0.627296\pi$
$11$	2.00000	−	3.46410i	0.603023	−	1.04447i	0.992361	−	0.123371i	$-0.0393705\pi$
$12$		0			0
$13$	3.50000	+	0.866025i	0.970725	+	0.240192i
$13$	3.50000	+	0.866025i	0.970725	+	0.240192i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	0.624780	−	0.780801i	$-0.285189\pi$
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	−0.988583	+	0.150675i	$0.951855\pi$
$18$		0			0
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	−0.995639	−	0.0932891i	$-0.970262\pi$
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	0.578610	+	0.815604i	$0.303595\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.500000	−	0.866025i	0.0928477	−	0.160817i	−0.815861	−	0.578249i	$-0.803736\pi$
$29$	0.500000	−	0.866025i	0.0928477	−	0.160817i	0.908708	+	0.417432i	$0.137070\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.00000	−	3.46410i	−0.338062	−	0.585540i
$36$		0			0
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	−0.962580	−	0.270998i	$-0.912646\pi$
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	0.715981	+	0.698119i	$0.245980\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.50000	−	7.79423i	0.702782	−	1.21725i	−0.264704	−	0.964330i	$-0.585274\pi$
$41$	4.50000	−	7.79423i	0.702782	−	1.21725i	0.967486	−	0.252924i	$-0.0813924\pi$
$42$		0			0
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	0.381246	−	0.924473i	$-0.375495\pi$
$44$		0			0
$45$	−1.50000	−	2.59808i	−0.223607	−	0.387298i
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	−4.50000	+	7.79423i	−0.642857	+	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.00000			−1.23625			−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000			−1.23625			−0.618123	+	0.786082i	$0.712106\pi$
$54$		0			0
$55$	−2.00000	+	3.46410i	−0.269680	+	0.467099i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	0.705965	−	0.708247i	$-0.250514\pi$
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	−0.966342	+	0.257260i	$0.917180\pi$
$60$		0			0
$61$	−3.50000	−	6.06218i	−0.448129	−	0.776182i	0.550135	−	0.835076i	$-0.314576\pi$
$61$	−3.50000	−	6.06218i	−0.448129	−	0.776182i	−0.998264	+	0.0588933i	$0.981243\pi$
$62$		0			0
$63$	−6.00000	+	10.3923i	−0.755929	+	1.30931i
$64$		0			0
$65$	−3.50000	−	0.866025i	−0.434122	−	0.107417i
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.00000	−	6.92820i	−0.474713	−	0.822226i	0.524868	−	0.851184i	$-0.324115\pi$
$71$	−4.00000	−	6.92820i	−0.474713	−	0.822226i	−0.999581	+	0.0289572i	$0.990781\pi$
$72$		0			0
$73$	11.0000			1.28745			0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000			1.28745			0.643726	+	0.765256i	$0.277388\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	16.0000			1.82337
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	1.50000	+	2.59808i	0.162698	+	0.281801i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.00000	−	5.19615i	0.317999	−	0.550791i	−0.662071	−	0.749441i	$-0.730322\pi$
$89$	3.00000	−	5.19615i	0.317999	−	0.550791i	0.980071	+	0.198650i	$0.0636557\pi$
$90$		0			0
$91$	4.00000	+	13.8564i	0.419314	+	1.45255i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$	12.0000			1.20605
$100$		0			0
$101$	−3.50000	+	6.06218i	−0.348263	+	0.603209i	−0.985941	−	0.167094i	$-0.946562\pi$
$101$	−3.50000	+	6.06218i	−0.348263	+	0.603209i	0.637678	+	0.770303i	$0.279895\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.00000	+	3.46410i	−0.193347	+	0.334887i	−0.946357	−	0.323122i	$-0.895268\pi$
$107$	−2.00000	+	3.46410i	−0.193347	+	0.334887i	0.753010	+	0.658009i	$0.228601\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.500000	+	0.866025i	0.0470360	+	0.0814688i	0.888585	−	0.458712i	$-0.151689\pi$
$113$	0.500000	+	0.866025i	0.0470360	+	0.0814688i	−0.841549	+	0.540181i	$0.818356\pi$
$114$		0			0
$115$	2.00000	−	3.46410i	0.186501	−	0.323029i
$116$		0			0
$117$	3.00000	+	10.3923i	0.277350	+	0.960769i
$118$		0			0
$119$	6.00000	−	10.3923i	0.550019	−	0.952661i
$120$		0			0
$121$	−2.50000	−	4.33013i	−0.227273	−	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$	−4.00000	+	6.92820i	−0.354943	+	0.614779i	−0.987108	−	0.160055i	$-0.948833\pi$
$127$	−4.00000	+	6.92820i	−0.354943	+	0.614779i	0.632166	+	0.774833i	$0.282166\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−20.0000			−1.74741			−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000			−1.74741			−0.873704	+	0.486458i	$0.838289\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.50000	+	7.79423i	0.384461	+	0.665906i	0.991694	−	0.128618i	$-0.0410540\pi$
$137$	4.50000	+	7.79423i	0.384461	+	0.665906i	−0.607233	+	0.794524i	$0.707721\pi$
$138$		0			0
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	0.975417	+	0.220366i	$0.0707252\pi$
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	−0.296866	+	0.954919i	$0.595942\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	10.0000	−	10.3923i	0.836242	−	0.869048i
$144$		0			0
$145$	−0.500000	+	0.866025i	−0.0415227	+	0.0719195i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.50000	−	12.9904i	−0.614424	−	1.06421i	−0.990485	−	0.137619i	$-0.956055\pi$
$149$	−7.50000	−	12.9904i	−0.614424	−	1.06421i	0.376061	−	0.926595i	$-0.377278\pi$
$150$		0			0
$151$	−12.0000			−0.976546			−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000			−0.976546			−0.488273	+	0.872691i	$0.662373\pi$
$152$		0			0
$153$	4.50000	−	7.79423i	0.363803	−	0.630126i
$154$		0			0
$155$	4.00000			0.321288
$156$		0			0
$157$	11.0000			0.877896			0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000			0.877896			0.438948	+	0.898513i	$0.355351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−16.0000			−1.26098
$162$		0			0
$163$	4.00000	+	6.92820i	0.313304	+	0.542659i	0.979076	−	0.203497i	$-0.0652307\pi$
$163$	4.00000	+	6.92820i	0.313304	+	0.542659i	−0.665771	+	0.746156i	$0.731897\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	−0.534875	−	0.844931i	$-0.679641\pi$
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	0.999169	+	0.0407502i	$0.0129748\pi$
$168$		0			0
$169$	11.5000	+	6.06218i	0.884615	+	0.466321i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.00000	−	12.1244i	−0.532200	−	0.921798i	−0.999293	−	0.0375896i	$-0.988032\pi$
$173$	−7.00000	−	12.1244i	−0.532200	−	0.921798i	0.467093	−	0.884208i	$-0.345301\pi$
$174$		0			0
$175$	−8.00000	−	13.8564i	−0.604743	−	1.04745i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000	+	20.7846i	−0.896922	+	1.55351i	−0.0655145	+	0.997852i	$0.520869\pi$
$179$	−12.0000	+	20.7846i	−0.896922	+	1.55351i	−0.831408	+	0.555663i	$0.812464\pi$
$180$		0			0
$181$	−21.0000			−1.56092			−0.780459	−	0.625207i	$-0.785014\pi$
$181$	−21.0000			−1.56092			−0.780459	+	0.625207i	$0.785014\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.50000	−	2.59808i	0.110282	−	0.191014i
$186$		0			0
$187$	−12.0000			−0.877527
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.0000	−	17.3205i	−0.723575	−	1.25327i	−0.959558	−	0.281511i	$-0.909164\pi$
$191$	−10.0000	−	17.3205i	−0.723575	−	1.25327i	0.235983	−	0.971757i	$-0.424169\pi$
$192$		0			0
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	−0.993215	−	0.116289i	$-0.962900\pi$
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	0.597317	+	0.802005i	$0.296234\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.00000	+	5.19615i	−0.213741	+	0.370211i	−0.952882	−	0.303340i	$-0.901898\pi$
$197$	−3.00000	+	5.19615i	−0.213741	+	0.370211i	0.739141	+	0.673550i	$0.235232\pi$
$198$		0			0
$199$	−2.00000	−	3.46410i	−0.141776	−	0.245564i	0.786389	−	0.617731i	$-0.211948\pi$
$199$	−2.00000	−	3.46410i	−0.141776	−	0.245564i	−0.928166	+	0.372168i	$0.878615\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.00000			0.280745
$204$		0			0
$205$	−4.50000	+	7.79423i	−0.314294	+	0.544373i
$206$		0			0
$207$	−12.0000			−0.834058
$208$		0			0
$209$		0			0
$210$		0			0
$211$	10.0000	−	17.3205i	0.688428	−	1.19239i	−0.283918	−	0.958849i	$-0.591634\pi$
$211$	10.0000	−	17.3205i	0.688428	−	1.19239i	0.972346	−	0.233544i	$-0.0750324\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000	+	6.92820i	0.272798	+	0.472500i
$216$		0			0
$217$	−8.00000	−	13.8564i	−0.543075	−	0.940634i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.00000	−	10.3923i	−0.201802	−	0.699062i
$222$		0			0
$223$	6.00000	−	10.3923i	0.401790	−	0.695920i	−0.592152	−	0.805826i	$-0.701722\pi$
$223$	6.00000	−	10.3923i	0.401790	−	0.695920i	0.993942	+	0.109906i	$0.0350549\pi$
$224$		0			0
$225$	−6.00000	−	10.3923i	−0.400000	−	0.692820i
$226$		0			0
$227$	12.0000	+	20.7846i	0.796468	+	1.37952i	0.921903	+	0.387421i	$0.126634\pi$
$227$	12.0000	+	20.7846i	0.796468	+	1.37952i	−0.125435	+	0.992102i	$0.540033\pi$
$228$		0			0
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.0000			0.655122			0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000			0.655122			0.327561	+	0.944830i	$0.393773\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	−1.50000	−	2.59808i	−0.0966235	−	0.167357i	0.813662	−	0.581339i	$-0.197471\pi$
$241$	−1.50000	−	2.59808i	−0.0966235	−	0.167357i	−0.910285	+	0.413982i	$0.864138\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.50000	−	7.79423i	0.287494	−	0.497955i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.00000	−	10.3923i	−0.378717	−	0.655956i	0.612159	−	0.790735i	$-0.290301\pi$
$251$	−6.00000	−	10.3923i	−0.378717	−	0.655956i	−0.990876	+	0.134778i	$0.956968\pi$
$252$		0			0
$253$	8.00000	+	13.8564i	0.502956	+	0.871145i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.50000	+	12.9904i	−0.467837	+	0.810318i	−0.999325	−	0.0367485i	$-0.988300\pi$
$257$	−7.50000	+	12.9904i	−0.467837	+	0.810318i	0.531487	+	0.847066i	$0.321633\pi$
$258$		0			0
$259$	−12.0000			−0.745644
$260$		0			0
$261$	3.00000			0.185695
$262$		0			0
$263$	−12.0000	+	20.7846i	−0.739952	+	1.28163i	0.212565	+	0.977147i	$0.431818\pi$
$263$	−12.0000	+	20.7846i	−0.739952	+	1.28163i	−0.952517	+	0.304487i	$0.901515\pi$
$264$		0			0
$265$	9.00000			0.552866
$266$		0			0
$267$		0			0
$268$		0			0
$269$	9.00000	+	15.5885i	0.548740	+	0.950445i	0.998361	+	0.0572259i	$0.0182255\pi$
$269$	9.00000	+	15.5885i	0.548740	+	0.950445i	−0.449622	+	0.893219i	$0.648441\pi$
$270$		0			0
$271$	−10.0000	+	17.3205i	−0.607457	+	1.05215i	0.384201	+	0.923249i	$0.374477\pi$
$271$	−10.0000	+	17.3205i	−0.607457	+	1.05215i	−0.991658	+	0.128897i	$0.958856\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.00000	+	13.8564i	−0.482418	+	0.835573i
$276$		0			0
$277$	4.50000	+	7.79423i	0.270379	+	0.468310i	0.968959	−	0.247222i	$-0.0795177\pi$
$277$	4.50000	+	7.79423i	0.270379	+	0.468310i	−0.698580	+	0.715532i	$0.746184\pi$
$278$		0			0
$279$	−6.00000	−	10.3923i	−0.359211	−	0.622171i
$280$		0			0
$281$	−5.00000			−0.298275			−0.149137	−	0.988816i	$-0.547650\pi$
$281$	−5.00000			−0.298275			−0.149137	+	0.988816i	$0.547650\pi$
$282$		0			0
$283$	14.0000	−	24.2487i	0.832214	−	1.44144i	−0.0640654	−	0.997946i	$-0.520407\pi$
$283$	14.0000	−	24.2487i	0.832214	−	1.44144i	0.896279	−	0.443491i	$-0.146260\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	36.0000			2.12501
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.50000	+	4.33013i	0.146052	+	0.252969i	0.929765	−	0.368154i	$-0.120010\pi$
$293$	2.50000	+	4.33013i	0.146052	+	0.252969i	−0.783713	+	0.621123i	$0.786677\pi$
$294$		0			0
$295$	2.00000	+	3.46410i	0.116445	+	0.201688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.0000	+	10.3923i	−0.578315	+	0.601003i
$300$		0			0
$301$	16.0000	−	27.7128i	0.922225	−	1.59734i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.50000	+	6.06218i	0.200409	+	0.347119i
$306$		0			0
$307$	28.0000			1.59804			0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000			1.59804			0.799022	+	0.601302i	$0.205351\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$	−22.0000			−1.24351			−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000			−1.24351			−0.621757	+	0.783210i	$0.713581\pi$
$314$		0			0
$315$	6.00000	−	10.3923i	0.338062	−	0.585540i
$316$		0			0
$317$	3.00000			0.168497			0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000			0.168497			0.0842484	+	0.996445i	$0.473151\pi$
$318$		0			0
$319$	−2.00000	−	3.46410i	−0.111979	−	0.193952i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−14.0000	−	3.46410i	−0.776580	−	0.192154i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	16.0000	+	27.7128i	0.882109	+	1.52786i
$330$		0			0
$331$	4.00000	+	6.92820i	0.219860	+	0.380808i	0.954765	−	0.297361i	$-0.0961066\pi$
$331$	4.00000	+	6.92820i	0.219860	+	0.380808i	−0.734905	+	0.678170i	$0.762773\pi$
$332$		0			0
$333$	−9.00000			−0.493197
$334$		0			0
$335$	−2.00000	+	3.46410i	−0.109272	+	0.189264i
$336$		0			0
$337$	23.0000			1.25289			0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000			1.25289			0.626445	+	0.779466i	$0.284509\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.00000	+	13.8564i	−0.433224	+	0.750366i
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	0.980921	−	0.194409i	$-0.0622790\pi$
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	−0.658824	+	0.752297i	$0.728946\pi$
$348$		0			0
$349$	1.00000	−	1.73205i	0.0535288	−	0.0927146i	−0.838019	−	0.545640i	$-0.816286\pi$
$349$	1.00000	−	1.73205i	0.0535288	−	0.0927146i	0.891548	+	0.452926i	$0.149620\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.50000	+	6.06218i	−0.186286	+	0.322657i	−0.944009	−	0.329919i	$-0.892979\pi$
$353$	−3.50000	+	6.06218i	−0.186286	+	0.322657i	0.757723	+	0.652576i	$0.226312\pi$
$354$		0			0
$355$	4.00000	+	6.92820i	0.212298	+	0.367711i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	9.50000	−	16.4545i	0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−11.0000			−0.575766
$366$		0			0
$367$	−14.0000	+	24.2487i	−0.730794	+	1.26577i	0.225750	+	0.974185i	$0.427517\pi$
$367$	−14.0000	+	24.2487i	−0.730794	+	1.26577i	−0.956544	+	0.291587i	$0.905817\pi$
$368$		0			0
$369$	27.0000			1.40556
$370$		0			0
$371$	−18.0000	−	31.1769i	−0.934513	−	1.61862i
$372$		0			0
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.50000	−	2.59808i	0.128757	−	0.133808i
$378$		0			0
$379$	−4.00000	+	6.92820i	−0.205466	+	0.355878i	−0.950281	−	0.311393i	$-0.899204\pi$
$379$	−4.00000	+	6.92820i	−0.205466	+	0.355878i	0.744815	+	0.667271i	$0.232538\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$	−16.0000			−0.815436
$386$		0			0
$387$	12.0000	−	20.7846i	0.609994	−	1.05654i
$388$		0			0
$389$	−9.00000			−0.456318			−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000			−0.456318			−0.228159	+	0.973624i	$0.573271\pi$
$390$		0			0
$391$	12.0000			0.606866
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.00000			−0.201262
$396$		0			0
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	0.635161	−	0.772380i	$-0.280934\pi$
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	−0.986481	+	0.163876i	$0.947600\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.50000	+	2.59808i	−0.0749064	+	0.129742i	−0.901046	−	0.433724i	$-0.857199\pi$
$401$	−1.50000	+	2.59808i	−0.0749064	+	0.129742i	0.826139	+	0.563466i	$0.190532\pi$
$402$		0			0
$403$	−14.0000	−	3.46410i	−0.697390	−	0.172559i
$404$		0			0
$405$	4.50000	−	7.79423i	0.223607	−	0.387298i
$406$		0			0
$407$	6.00000	+	10.3923i	0.297409	+	0.515127i
$408$		0			0
$409$	−15.5000	−	26.8468i	−0.766426	−	1.32749i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−15.5000	−	26.8468i	−0.766426	−	1.32749i	0.173064	−	0.984911i	$-0.444633\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.00000	−	13.8564i	0.393654	−	0.681829i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	−0.681426	−	0.731887i	$-0.738640\pi$
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	0.974546	+	0.224189i	$0.0719734\pi$
$420$		0			0
$421$	−5.00000			−0.243685			−0.121843	−	0.992549i	$-0.538880\pi$
$421$	−5.00000			−0.243685			−0.121843	+	0.992549i	$0.538880\pi$
$422$		0			0
$423$	12.0000	+	20.7846i	0.583460	+	1.01058i
$424$		0			0
$425$	6.00000	+	10.3923i	0.291043	+	0.504101i
$426$		0			0
$427$	14.0000	−	24.2487i	0.677507	−	1.17348i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.0000	+	20.7846i	−0.578020	+	1.00116i	0.417687	+	0.908591i	$0.362841\pi$
$431$	−12.0000	+	20.7846i	−0.578020	+	1.00116i	−0.995706	+	0.0925683i	$0.970492\pi$
$432$		0			0
$433$	2.50000	+	4.33013i	0.120142	+	0.208093i	0.919824	−	0.392332i	$-0.128332\pi$
$433$	2.50000	+	4.33013i	0.120142	+	0.208093i	−0.799681	+	0.600425i	$0.794998\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	4.00000	−	6.92820i	0.190910	−	0.330665i	−0.754642	−	0.656136i	$-0.772190\pi$
$439$	4.00000	−	6.92820i	0.190910	−	0.330665i	0.945552	+	0.325471i	$0.105523\pi$
$440$		0			0
$441$	−27.0000			−1.28571
$442$		0			0
$443$	−24.0000			−1.14027			−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000			−1.14027			−0.570137	+	0.821549i	$0.693110\pi$
$444$		0			0
$445$	−3.00000	+	5.19615i	−0.142214	+	0.246321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−17.0000	−	29.4449i	−0.802280	−	1.38959i	−0.918112	−	0.396320i	$-0.870287\pi$
$449$	−17.0000	−	29.4449i	−0.802280	−	1.38959i	0.115833	−	0.993269i	$-0.463046\pi$
$450$		0			0
$451$	−18.0000	−	31.1769i	−0.847587	−	1.46806i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.00000	−	13.8564i	−0.187523	−	0.649598i
$456$		0			0
$457$	−15.5000	+	26.8468i	−0.725059	+	1.25584i	0.233890	+	0.972263i	$0.424854\pi$
$457$	−15.5000	+	26.8468i	−0.725059	+	1.25584i	−0.958950	+	0.283577i	$0.908479\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.5000	+	28.5788i	0.768482	+	1.33105i	0.938386	+	0.345589i	$0.112321\pi$
$461$	16.5000	+	28.5788i	0.768482	+	1.33105i	−0.169904	+	0.985461i	$0.554346\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.0000			−0.925490			−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000			−0.925490			−0.462745	+	0.886492i	$0.653135\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−32.0000			−1.47136
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−13.5000	−	23.3827i	−0.618123	−	1.07062i
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	−7.50000	+	7.79423i	−0.341971	+	0.355386i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.00000	+	1.73205i	0.0454077	+	0.0786484i
$486$		0			0
$487$	−8.00000	−	13.8564i	−0.362515	−	0.627894i	0.625859	−	0.779936i	$-0.284748\pi$
$487$	−8.00000	−	13.8564i	−0.362515	−	0.627894i	−0.988374	+	0.152042i	$0.951415\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.00000	+	6.92820i	−0.180517	+	0.312665i	−0.942057	−	0.335453i	$-0.891111\pi$
$491$	−4.00000	+	6.92820i	−0.180517	+	0.312665i	0.761539	+	0.648119i	$0.224444\pi$
$492$		0			0
$493$	−3.00000			−0.135113
$494$		0			0
$495$	−12.0000			−0.539360
$496$		0			0
$497$	16.0000	−	27.7128i	0.717698	−	1.24309i
$498$		0			0
$499$	32.0000			1.43252			0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000			1.43252			0.716258	+	0.697835i	$0.245853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	8.00000	+	13.8564i	0.356702	+	0.617827i	0.987408	−	0.158196i	$-0.0505677\pi$
$503$	8.00000	+	13.8564i	0.356702	+	0.617827i	−0.630705	+	0.776022i	$0.717234\pi$
$504$		0			0
$505$	3.50000	−	6.06218i	0.155748	−	0.269763i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−21.5000	+	37.2391i	−0.952971	+	1.65059i	−0.214026	+	0.976828i	$0.568658\pi$
$509$	−21.5000	+	37.2391i	−0.952971	+	1.65059i	−0.738945	+	0.673766i	$0.764676\pi$
$510$		0			0
$511$	22.0000	+	38.1051i	0.973223	+	1.68567i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.00000			−0.352522
$516$		0			0
$517$	16.0000	−	27.7128i	0.703679	−	1.21881i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	39.0000			1.70862			0.854311	−	0.519763i	$-0.173980\pi$
$521$	39.0000			1.70862			0.854311	+	0.519763i	$0.173980\pi$
$522$		0			0
$523$	18.0000	−	31.1769i	0.787085	−	1.36327i	−0.140660	−	0.990058i	$-0.544923\pi$
$523$	18.0000	−	31.1769i	0.787085	−	1.36327i	0.927746	−	0.373213i	$-0.121744\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.00000	+	10.3923i	0.261364	+	0.452696i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$	6.00000	−	10.3923i	0.260378	−	0.450988i
$532$		0			0
$533$	22.5000	−	23.3827i	0.974583	−	1.01282i
$534$		0			0
$535$	2.00000	−	3.46410i	0.0864675	−	0.149766i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18.0000	+	31.1769i	0.775315	+	1.34288i
$540$		0			0
$541$	−25.0000			−1.07483			−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000			−1.07483			−0.537417	+	0.843317i	$0.680600\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.00000			0.0856706
$546$		0			0
$547$	−16.0000			−0.684111			−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000			−0.684111			−0.342055	+	0.939680i	$0.611123\pi$
$548$		0			0
$549$	10.5000	−	18.1865i	0.448129	−	0.776182i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	8.00000	+	13.8564i	0.340195	+	0.589234i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.5000	+	33.7750i	−0.826242	+	1.43109i	0.0747252	+	0.997204i	$0.476192\pi$
$557$	−19.5000	+	33.7750i	−0.826242	+	1.43109i	−0.900967	+	0.433888i	$0.857141\pi$
$558$		0			0
$559$	−8.00000	−	27.7128i	−0.338364	−	1.17213i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.00000	−	3.46410i	−0.0842900	−	0.145994i	0.820798	−	0.571218i	$-0.193529\pi$
$563$	−2.00000	−	3.46410i	−0.0842900	−	0.145994i	−0.905088	+	0.425223i	$0.860196\pi$
$564$		0			0
$565$	−0.500000	−	0.866025i	−0.0210352	−	0.0364340i
$566$		0			0
$567$	−36.0000			−1.51186
$568$		0			0
$569$	−5.00000	+	8.66025i	−0.209611	+	0.363057i	−0.951592	−	0.307364i	$-0.900553\pi$
$569$	−5.00000	+	8.66025i	−0.209611	+	0.363057i	0.741981	+	0.670421i	$0.233886\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.00000	−	13.8564i	0.333623	−	0.577852i
$576$		0			0
$577$	39.0000			1.62359			0.811796	−	0.583942i	$-0.198490\pi$
$577$	39.0000			1.62359			0.811796	+	0.583942i	$0.198490\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−18.0000	+	31.1769i	−0.745484	+	1.29122i
$584$		0			0
$585$	−3.00000	−	10.3923i	−0.124035	−	0.429669i
$586$		0			0
$587$	−8.00000	+	13.8564i	−0.330195	+	0.571915i	−0.982550	−	0.185999i	$-0.940448\pi$
$587$	−8.00000	+	13.8564i	−0.330195	+	0.571915i	0.652355	+	0.757914i	$0.273781\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.00000			−0.0410651			−0.0205325	−	0.999789i	$-0.506536\pi$
$593$	−1.00000			−0.0410651			−0.0205325	+	0.999789i	$0.506536\pi$
$594$		0			0
$595$	−6.00000	+	10.3923i	−0.245976	+	0.426043i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	44.0000			1.79779			0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000			1.79779			0.898896	+	0.438163i	$0.144371\pi$
$600$		0			0
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	−0.992114	−	0.125336i	$-0.959999\pi$
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	0.604601	+	0.796528i	$0.293332\pi$
$602$		0			0
$603$	12.0000			0.488678
$604$		0			0
$605$	2.50000	+	4.33013i	0.101639	+	0.176045i
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	28.0000	+	6.92820i	1.13276	+	0.280285i
$612$		0			0
$613$	−5.50000	+	9.52628i	−0.222143	+	0.384763i	−0.955458	−	0.295126i	$-0.904638\pi$
$613$	−5.50000	+	9.52628i	−0.222143	+	0.384763i	0.733316	+	0.679888i	$0.237972\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14.5000	+	25.1147i	0.583748	+	1.01108i	0.995030	+	0.0995732i	$0.0317477\pi$
$617$	14.5000	+	25.1147i	0.583748	+	1.01108i	−0.411282	+	0.911508i	$0.634919\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	24.0000			0.961540
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9.00000			0.358854
$630$		0			0
$631$	−4.00000	−	6.92820i	−0.159237	−	0.275807i	0.775356	−	0.631524i	$-0.217570\pi$
$631$	−4.00000	−	6.92820i	−0.159237	−	0.275807i	−0.934594	+	0.355716i	$0.884237\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.00000	−	6.92820i	0.158735	−	0.274937i
$636$		0			0
$637$	−22.5000	+	23.3827i	−0.891482	+	0.926456i
$638$		0			0
$639$	12.0000	−	20.7846i	0.474713	−	0.822226i
$640$		0			0
$641$	−9.50000	−	16.4545i	−0.375227	−	0.649913i	0.615134	−	0.788423i	$-0.289102\pi$
$641$	−9.50000	−	16.4545i	−0.375227	−	0.649913i	−0.990361	+	0.138510i	$0.955769\pi$
$642$		0			0
$643$	22.0000	+	38.1051i	0.867595	+	1.50272i	0.864447	+	0.502724i	$0.167669\pi$
$643$	22.0000	+	38.1051i	0.867595	+	1.50272i	0.00314839	+	0.999995i	$0.498998\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.0000	−	24.2487i	0.550397	−	0.953315i	−0.447849	−	0.894109i	$-0.647810\pi$
$647$	14.0000	−	24.2487i	0.550397	−	0.953315i	0.998246	−	0.0592060i	$-0.0188569\pi$
$648$		0			0
$649$	−16.0000			−0.628055
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.00000	−	15.5885i	0.352197	−	0.610023i	−0.634437	−	0.772975i	$-0.718768\pi$
$653$	9.00000	−	15.5885i	0.352197	−	0.610023i	0.986634	+	0.162951i	$0.0521013\pi$
$654$		0			0
$655$	20.0000			0.781465
$656$		0			0
$657$	16.5000	+	28.5788i	0.643726	+	1.11497i
$658$		0			0
$659$	24.0000	+	41.5692i	0.934907	+	1.61931i	0.774799	+	0.632207i	$0.217851\pi$
$659$	24.0000	+	41.5692i	0.934907	+	1.61931i	0.160108	+	0.987099i	$0.448816\pi$
$660$		0			0
$661$	24.5000	−	42.4352i	0.952940	−	1.65054i	0.213925	−	0.976850i	$-0.431375\pi$
$661$	24.5000	−	42.4352i	0.952940	−	1.65054i	0.739014	−	0.673690i	$-0.235292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2.00000	+	3.46410i	0.0774403	+	0.134131i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−28.0000			−1.08093
$672$		0			0
$673$	14.5000	−	25.1147i	0.558934	−	0.968102i	−0.438652	−	0.898657i	$-0.644544\pi$
$673$	14.5000	−	25.1147i	0.558934	−	0.968102i	0.997586	−	0.0694449i	$-0.0221228\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$	4.00000	−	6.92820i	0.153506	−	0.265880i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−22.0000	−	38.1051i	−0.841807	−	1.45805i	−0.888366	−	0.459136i	$-0.848159\pi$
$683$	−22.0000	−	38.1051i	−0.841807	−	1.45805i	0.0465592	−	0.998916i	$-0.485174\pi$
$684$		0			0
$685$	−4.50000	−	7.79423i	−0.171936	−	0.297802i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−31.5000	−	7.79423i	−1.20005	−	0.296936i
$690$		0			0
$691$	−10.0000	+	17.3205i	−0.380418	+	0.658903i	−0.991122	−	0.132956i	$-0.957553\pi$
$691$	−10.0000	+	17.3205i	−0.380418	+	0.658903i	0.610704	+	0.791859i	$0.290887\pi$
$692$		0			0
$693$	24.0000	+	41.5692i	0.911685	+	1.57908i
$694$		0			0
$695$	−8.00000	−	13.8564i	−0.303457	−	0.525603i
$696$		0			0
$697$	−27.0000			−1.02270
$698$		0			0
$699$		0			0
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−28.0000			−1.05305
$708$		0			0
$709$	6.50000	+	11.2583i	0.244113	+	0.422815i	0.961882	−	0.273466i	$-0.0881700\pi$
$709$	6.50000	+	11.2583i	0.244113	+	0.422815i	−0.717769	+	0.696281i	$0.754837\pi$
$710$		0			0
$711$	6.00000	+	10.3923i	0.225018	+	0.389742i
$712$		0			0
$713$	8.00000	−	13.8564i	0.299602	−	0.518927i
$714$		0			0
$715$	−10.0000	+	10.3923i	−0.373979	+	0.388650i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−14.0000	−	24.2487i	−0.522112	−	0.904324i	−0.999669	−	0.0257237i	$-0.991811\pi$
$719$	−14.0000	−	24.2487i	−0.522112	−	0.904324i	0.477557	−	0.878601i	$-0.341522\pi$
$720$		0			0
$721$	16.0000	+	27.7128i	0.595871	+	1.03208i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.00000	+	3.46410i	−0.0742781	+	0.128654i
$726$		0			0
$727$	−28.0000			−1.03846			−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000			−1.03846			−0.519231	+	0.854634i	$0.673782\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	−12.0000	+	20.7846i	−0.443836	+	0.768747i
$732$		0			0
$733$	−1.00000			−0.0369358			−0.0184679	−	0.999829i	$-0.505879\pi$
$733$	−1.00000			−0.0369358			−0.0184679	+	0.999829i	$0.505879\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.00000	−	13.8564i	−0.294684	−	0.510407i
$738$		0			0
$739$	−12.0000	+	20.7846i	−0.441427	+	0.764574i	−0.997796	−	0.0663614i	$-0.978861\pi$
$739$	−12.0000	+	20.7846i	−0.441427	+	0.764574i	0.556369	+	0.830936i	$0.312194\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−4.00000	+	6.92820i	−0.146746	+	0.254171i	−0.930023	−	0.367502i	$-0.880213\pi$
$743$	−4.00000	+	6.92820i	−0.146746	+	0.254171i	0.783277	+	0.621673i	$0.213547\pi$
$744$		0			0
$745$	7.50000	+	12.9904i	0.274779	+	0.475931i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−16.0000			−0.584627
$750$		0			0
$751$	12.0000	−	20.7846i	0.437886	−	0.758441i	−0.559640	−	0.828736i	$-0.689061\pi$
$751$	12.0000	−	20.7846i	0.437886	−	0.758441i	0.997526	+	0.0702946i	$0.0223939\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0000			0.436725
$756$		0			0
$757$	−3.00000	+	5.19615i	−0.109037	+	0.188857i	−0.915380	−	0.402590i	$-0.868110\pi$
$757$	−3.00000	+	5.19615i	−0.109037	+	0.188857i	0.806343	+	0.591448i	$0.201443\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−21.0000	−	36.3731i	−0.761249	−	1.31852i	−0.942207	−	0.335032i	$-0.891253\pi$
$761$	−21.0000	−	36.3731i	−0.761249	−	1.31852i	0.180957	−	0.983491i	$-0.442080\pi$
$762$		0			0
$763$	−4.00000	−	6.92820i	−0.144810	−	0.250818i
$764$		0			0
$765$	−4.50000	+	7.79423i	−0.162698	+	0.281801i
$766$		0			0
$767$	−4.00000	−	13.8564i	−0.144432	−	0.500326i
$768$		0			0
$769$	15.0000	−	25.9808i	0.540914	−	0.936890i	−0.457938	−	0.888984i	$-0.651412\pi$
$769$	15.0000	−	25.9808i	0.540914	−	0.936890i	0.998852	−	0.0479061i	$-0.0152548\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.0000	−	32.9090i	−0.683383	−	1.18365i	−0.973942	−	0.226796i	$-0.927175\pi$
$773$	−19.0000	−	32.9090i	−0.683383	−	1.18365i	0.290560	−	0.956857i	$-0.406159\pi$
$774$		0			0
$775$	16.0000			0.574737
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−32.0000			−1.14505
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11.0000			−0.392607
$786$		0			0
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	0.828176	−	0.560469i	$-0.189379\pi$
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	−0.899468	+	0.436987i	$0.856046\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.00000	+	3.46410i	−0.0711118	+	0.123169i
$792$		0			0
$793$	−7.00000	−	24.2487i	−0.248577	−	0.861097i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15.0000	−	25.9808i	−0.531327	−	0.920286i	−0.999331	−	0.0365596i	$-0.988360\pi$
$797$	−15.0000	−	25.9808i	−0.531327	−	0.920286i	0.468004	−	0.883726i	$-0.344973\pi$
$798$		0			0
$799$	−12.0000	−	20.7846i	−0.424529	−	0.735307i
$800$		0			0
$801$	18.0000			0.635999
$802$		0			0
$803$	22.0000	−	38.1051i	0.776363	−	1.34470i
$804$		0			0
$805$	16.0000			0.563926
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−19.5000	+	33.7750i	−0.685583	+	1.18747i	0.287670	+	0.957730i	$0.407120\pi$
$809$	−19.5000	+	33.7750i	−0.685583	+	1.18747i	−0.973253	+	0.229736i	$0.926214\pi$
$810$		0			0
$811$	28.0000			0.983213			0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000			0.983213			0.491606	+	0.870817i	$0.336410\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.00000	−	6.92820i	−0.140114	−	0.242684i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−30.0000	+	31.1769i	−1.04828	+	1.08941i
$820$		0			0
$821$	−19.0000	+	32.9090i	−0.663105	+	1.14853i	0.316691	+	0.948529i	$0.397428\pi$
$821$	−19.0000	+	32.9090i	−0.663105	+	1.14853i	−0.979795	+	0.200002i	$0.935905\pi$
$822$		0			0
$823$	6.00000	+	10.3923i	0.209147	+	0.362253i	0.951446	−	0.307816i	$-0.0995980\pi$
$823$	6.00000	+	10.3923i	0.209147	+	0.362253i	−0.742299	+	0.670069i	$0.766265\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.0000			0.417281			0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000			0.417281			0.208640	+	0.977992i	$0.433096\pi$
$828$		0			0
$829$	−3.50000	+	6.06218i	−0.121560	+	0.210548i	−0.920383	−	0.391018i	$-0.872123\pi$
$829$	−3.50000	+	6.06218i	−0.121560	+	0.210548i	0.798823	+	0.601566i	$0.205456\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	27.0000			0.935495
$834$		0			0
$835$	−6.00000	+	10.3923i	−0.207639	+	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−20.0000	−	34.6410i	−0.690477	−	1.19594i	−0.971682	−	0.236293i	$-0.924067\pi$
$839$	−20.0000	−	34.6410i	−0.690477	−	1.19594i	0.281205	−	0.959648i	$-0.409266\pi$
$840$		0			0
$841$	14.0000	+	24.2487i	0.482759	+	0.836162i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−11.5000	−	6.06218i	−0.395612	−	0.208545i
$846$		0			0
$847$	10.0000	−	17.3205i	0.343604	−	0.595140i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.00000	−	10.3923i	−0.205677	−	0.356244i
$852$		0			0
$853$	7.00000			0.239675			0.119838	−	0.992793i	$-0.461763\pi$
$853$	7.00000			0.239675			0.119838	+	0.992793i	$0.461763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−17.0000			−0.580709			−0.290354	−	0.956919i	$-0.593773\pi$
$857$	−17.0000			−0.580709			−0.290354	+	0.956919i	$0.593773\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−52.0000			−1.77010			−0.885050	−	0.465495i	$-0.845876\pi$
$863$	−52.0000			−1.77010			−0.885050	+	0.465495i	$0.845876\pi$
$864$		0			0
$865$	7.00000	+	12.1244i	0.238007	+	0.412240i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	8.00000	−	13.8564i	0.271381	−	0.470046i
$870$		0			0
$871$	10.0000	−	10.3923i	0.338837	−	0.352130i
$872$		0			0
$873$	3.00000	−	5.19615i	0.101535	−	0.175863i
$874$		0			0
$875$	18.0000	+	31.1769i	0.608511	+	1.05397i
$876$		0			0
$877$	−7.50000	−	12.9904i	−0.253257	−	0.438654i	0.711164	−	0.703027i	$-0.248168\pi$
$877$	−7.50000	−	12.9904i	−0.253257	−	0.438654i	−0.964421	+	0.264373i	$0.914835\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	4.50000	−	7.79423i	0.151609	−	0.262594i	−0.780210	−	0.625517i	$-0.784888\pi$
$881$	4.50000	−	7.79423i	0.151609	−	0.262594i	0.931819	+	0.362923i	$0.118221\pi$
$882$		0			0
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.0000	+	17.3205i	−0.335767	+	0.581566i	−0.983632	−	0.180190i	$-0.942329\pi$
$887$	−10.0000	+	17.3205i	−0.335767	+	0.581566i	0.647865	+	0.761755i	$0.275662\pi$
$888$		0			0
$889$	−32.0000			−1.07325
$890$		0			0
$891$	18.0000	+	31.1769i	0.603023	+	1.04447i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	12.0000	−	20.7846i	0.401116	−	0.694753i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.00000	+	3.46410i	−0.0667037	+	0.115534i
$900$		0			0
$901$	13.5000	+	23.3827i	0.449750	+	0.778990i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	21.0000			0.698064
$906$		0			0
$907$	−12.0000	+	20.7846i	−0.398453	+	0.690142i	−0.993535	−	0.113523i	$-0.963786\pi$
$907$	−12.0000	+	20.7846i	−0.398453	+	0.690142i	0.595082	+	0.803665i	$0.297120\pi$
$908$		0			0
$909$	−21.0000			−0.696526
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−40.0000	−	69.2820i	−1.32092	−	2.28789i
$918$		0			0
$919$	−24.0000	−	41.5692i	−0.791687	−	1.37124i	−0.924922	−	0.380158i	$-0.875870\pi$
$919$	−24.0000	−	41.5692i	−0.791687	−	1.37124i	0.133235	−	0.991084i	$-0.457464\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.00000	−	27.7128i	−0.263323	−	0.912178i
$924$		0			0
$925$	6.00000	−	10.3923i	0.197279	−	0.341697i
$926$		0			0
$927$	12.0000	+	20.7846i	0.394132	+	0.682656i
$928$		0			0
$929$	10.5000	+	18.1865i	0.344494	+	0.596681i	0.985262	−	0.171054i	$-0.0547172\pi$
$929$	10.5000	+	18.1865i	0.344494	+	0.596681i	−0.640768	+	0.767735i	$0.721384\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.0000			0.392442
$936$		0			0
$937$	−21.0000			−0.686040			−0.343020	−	0.939328i	$-0.611450\pi$
$937$	−21.0000			−0.686040			−0.343020	+	0.939328i	$0.611450\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	46.0000			1.49956			0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000			1.49956			0.749779	+	0.661689i	$0.230160\pi$
$942$		0			0
$943$	18.0000	+	31.1769i	0.586161	+	1.01526i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.0000	−	20.7846i	0.389948	−	0.675409i	−0.602494	−	0.798123i	$-0.705826\pi$
$947$	12.0000	−	20.7846i	0.389948	−	0.675409i	0.992442	+	0.122714i	$0.0391598\pi$
$948$		0			0
$949$	38.5000	+	9.52628i	1.24976	+	0.309236i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	11.0000	+	19.0526i	0.356325	+	0.617173i	0.987344	−	0.158595i	$-0.0506963\pi$
$953$	11.0000	+	19.0526i	0.356325	+	0.617173i	−0.631019	+	0.775768i	$0.717363\pi$
$954$		0			0
$955$	10.0000	+	17.3205i	0.323592	+	0.560478i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.0000	+	31.1769i	−0.581250	+	1.00676i
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$	−12.0000			−0.386695
$964$		0			0
$965$	5.50000	−	9.52628i	0.177051	−	0.306662i
$966$		0			0
$967$	40.0000			1.28631			0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000			1.28631			0.643157	+	0.765735i	$0.277624\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$	−32.0000	+	55.4256i	−1.02587	+	1.77686i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.5000	−	38.9711i	0.719839	−	1.24680i	−0.241225	−	0.970469i	$-0.577549\pi$
$977$	22.5000	−	38.9711i	0.719839	−	1.24680i	0.961063	−	0.276328i	$-0.0891176\pi$
$978$		0			0
$979$	−12.0000	−	20.7846i	−0.383522	−	0.664279i
$980$		0			0
$981$	−3.00000	−	5.19615i	−0.0957826	−	0.165900i
$982$		0			0
$983$	52.0000			1.65854			0.829271	−	0.558846i	$-0.188756\pi$
$983$	52.0000			1.65854			0.829271	+	0.558846i	$0.188756\pi$
$984$		0			0
$985$	3.00000	−	5.19615i	0.0955879	−	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	32.0000			1.01754
$990$		0			0
$991$	−12.0000	+	20.7846i	−0.381193	+	0.660245i	−0.991233	−	0.132125i	$-0.957820\pi$
$991$	−12.0000	+	20.7846i	−0.381193	+	0.660245i	0.610040	+	0.792370i	$0.291153\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.00000	+	3.46410i	0.0634043	+	0.109819i
$996$		0			0
$997$	12.5000	+	21.6506i	0.395879	+	0.685682i	0.993213	−	0.116310i	$-0.0371066\pi$
$997$	12.5000	+	21.6506i	0.395879	+	0.685682i	−0.597334	+	0.801993i	$0.703773\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.2.i.b.113.1		2
3.2	odd	2		1872.2.t.k.1153.1		2
4.3	odd	2		26.2.c.a.9.1	yes	2
8.3	odd	2		832.2.i.e.321.1		2
8.5	even	2		832.2.i.f.321.1		2
12.11	even	2		234.2.h.c.217.1		2
13.3	even	3	inner	208.2.i.b.81.1		2
13.4	even	6		2704.2.a.i.1.1		1
13.6	odd	12		2704.2.f.g.337.2		2
13.7	odd	12		2704.2.f.g.337.1		2
13.9	even	3		2704.2.a.h.1.1		1
20.3	even	4		650.2.o.c.399.2		4
20.7	even	4		650.2.o.c.399.1		4
20.19	odd	2		650.2.e.c.451.1		2
28.3	even	6		1274.2.h.a.373.1		2
28.11	odd	6		1274.2.h.b.373.1		2
28.19	even	6		1274.2.e.m.165.1		2
28.23	odd	6		1274.2.e.n.165.1		2
28.27	even	2		1274.2.g.a.295.1		2
39.29	odd	6		1872.2.t.k.289.1		2
52.3	odd	6		26.2.c.a.3.1	✓	2
52.7	even	12		338.2.b.b.337.2		2
52.11	even	12		338.2.e.b.23.2		4
52.15	even	12		338.2.e.b.23.1		4
52.19	even	12		338.2.b.b.337.1		2
52.23	odd	6		338.2.c.e.315.1		2
52.31	even	4		338.2.e.b.147.2		4
52.35	odd	6		338.2.a.e.1.1		1
52.43	odd	6		338.2.a.c.1.1		1
52.47	even	4		338.2.e.b.147.1		4
52.51	odd	2		338.2.c.e.191.1		2
104.3	odd	6		832.2.i.e.705.1		2
104.29	even	6		832.2.i.f.705.1		2
156.35	even	6		3042.2.a.e.1.1		1
156.59	odd	12		3042.2.b.e.1351.1		2
156.71	odd	12		3042.2.b.e.1351.2		2
156.95	even	6		3042.2.a.k.1.1		1
156.107	even	6		234.2.h.c.55.1		2
260.3	even	12		650.2.o.c.549.1		4
260.107	even	12		650.2.o.c.549.2		4
260.139	odd	6		8450.2.a.f.1.1		1
260.159	odd	6		650.2.e.c.601.1		2
260.199	odd	6		8450.2.a.s.1.1		1
364.3	even	6		1274.2.e.m.471.1		2
364.55	even	6		1274.2.g.a.393.1		2
364.107	odd	6		1274.2.h.b.263.1		2
364.159	even	6		1274.2.h.a.263.1		2
364.263	odd	6		1274.2.e.n.471.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.c.a.3.1	✓	2	52.3	odd	6
26.2.c.a.9.1	yes	2	4.3	odd	2
208.2.i.b.81.1		2	13.3	even	3	inner
208.2.i.b.113.1		2	1.1	even	1	trivial
234.2.h.c.55.1		2	156.107	even	6
234.2.h.c.217.1		2	12.11	even	2
338.2.a.c.1.1		1	52.43	odd	6
338.2.a.e.1.1		1	52.35	odd	6
338.2.b.b.337.1		2	52.19	even	12
338.2.b.b.337.2		2	52.7	even	12
338.2.c.e.191.1		2	52.51	odd	2
338.2.c.e.315.1		2	52.23	odd	6
338.2.e.b.23.1		4	52.15	even	12
338.2.e.b.23.2		4	52.11	even	12
338.2.e.b.147.1		4	52.47	even	4
338.2.e.b.147.2		4	52.31	even	4
650.2.e.c.451.1		2	20.19	odd	2
650.2.e.c.601.1		2	260.159	odd	6
650.2.o.c.399.1		4	20.7	even	4
650.2.o.c.399.2		4	20.3	even	4
650.2.o.c.549.1		4	260.3	even	12
650.2.o.c.549.2		4	260.107	even	12
832.2.i.e.321.1		2	8.3	odd	2
832.2.i.e.705.1		2	104.3	odd	6
832.2.i.f.321.1		2	8.5	even	2
832.2.i.f.705.1		2	104.29	even	6
1274.2.e.m.165.1		2	28.19	even	6
1274.2.e.m.471.1		2	364.3	even	6
1274.2.e.n.165.1		2	28.23	odd	6
1274.2.e.n.471.1		2	364.263	odd	6
1274.2.g.a.295.1		2	28.27	even	2
1274.2.g.a.393.1		2	364.55	even	6
1274.2.h.a.263.1		2	364.159	even	6
1274.2.h.a.373.1		2	28.3	even	6
1274.2.h.b.263.1		2	364.107	odd	6
1274.2.h.b.373.1		2	28.11	odd	6
1872.2.t.k.289.1		2	39.29	odd	6
1872.2.t.k.1153.1		2	3.2	odd	2
2704.2.a.h.1.1		1	13.9	even	3
2704.2.a.i.1.1		1	13.4	even	6
2704.2.f.g.337.1		2	13.7	odd	12
2704.2.f.g.337.2		2	13.6	odd	12
3042.2.a.e.1.1		1	156.35	even	6
3042.2.a.k.1.1		1	156.95	even	6
3042.2.b.e.1351.1		2	156.59	odd	12
3042.2.b.e.1351.2		2	156.71	odd	12
8450.2.a.f.1.1		1	260.139	odd	6
8450.2.a.s.1.1		1	260.199	odd	6

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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	208.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +(2.00000 + 3.46410i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q-1.00000 q^{5} +(2.00000 + 3.46410i) q^{7} +(1.50000 + 2.59808i) q^{9} +(2.00000 - 3.46410i) q^{11} +(3.50000 + 0.866025i) q^{13} +(-1.50000 - 2.59808i) q^{17} +(-2.00000 + 3.46410i) q^{23} -4.00000 q^{25} +(0.500000 - 0.866025i) q^{29} -4.00000 q^{31} +(-2.00000 - 3.46410i) q^{35} +(-1.50000 + 2.59808i) q^{37} +(4.50000 - 7.79423i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(-1.50000 - 2.59808i) q^{45} +8.00000 q^{47} +(-4.50000 + 7.79423i) q^{49} -9.00000 q^{53} +(-2.00000 + 3.46410i) q^{55} +(-2.00000 - 3.46410i) q^{59} +(-3.50000 - 6.06218i) q^{61} +(-6.00000 + 10.3923i) q^{63} +(-3.50000 - 0.866025i) q^{65} +(2.00000 - 3.46410i) q^{67} +(-4.00000 - 6.92820i) q^{71} +11.0000 q^{73} +16.0000 q^{77} +4.00000 q^{79} +(-4.50000 + 7.79423i) q^{81} +(1.50000 + 2.59808i) q^{85} +(3.00000 - 5.19615i) q^{89} +(4.00000 + 13.8564i) q^{91} +(-1.00000 - 1.73205i) q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 + 4 * q^7 + 3 * q^9	\( 2 q - 2 q^{5} + 4 q^{7} + 3 q^{9} + 4 q^{11} + 7 q^{13} - 3 q^{17} - 4 q^{23} - 8 q^{25} + q^{29} - 8 q^{31} - 4 q^{35} - 3 q^{37} + 9 q^{41} - 8 q^{43} - 3 q^{45} + 16 q^{47} - 9 q^{49} - 18 q^{53} - 4 q^{55} - 4 q^{59} - 7 q^{61} - 12 q^{63} - 7 q^{65} + 4 q^{67} - 8 q^{71} + 22 q^{73} + 32 q^{77} + 8 q^{79} - 9 q^{81} + 3 q^{85} + 6 q^{89} + 8 q^{91} - 2 q^{97} + 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^7 + 3 * q^9 + 4 * q^11 + 7 * q^13 - 3 * q^17 - 4 * q^23 - 8 * q^25 + q^29 - 8 * q^31 - 4 * q^35 - 3 * q^37 + 9 * q^41 - 8 * q^43 - 3 * q^45 + 16 * q^47 - 9 * q^49 - 18 * q^53 - 4 * q^55 - 4 * q^59 - 7 * q^61 - 12 * q^63 - 7 * q^65 + 4 * q^67 - 8 * q^71 + 22 * q^73 + 32 * q^77 + 8 * q^79 - 9 * q^81 + 3 * q^85 + 6 * q^89 + 8 * q^91 - 2 * q^97 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more