LMFDB - Embedded newform 513.2.m.d.350.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [513,2,Mod(107,513)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("513.107"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 513 = 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	513.m (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,4,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$4.09632562369$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-14})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 14x^{2} + 196 $ x^4 - 14*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			350.2
Root			$3.24037 - 1.87083i$ of defining polynomial
Character	$\chi$	$=$	513.350
Dual form			513.2.m.d.107.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.73205i) q^{4} +(3.24037 + 1.87083i) q^{5} +2.00000 q^{7} -3.74166i q^{11} +(4.50000 - 2.59808i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(-6.48074 - 3.74166i) q^{17} +(-3.50000 + 2.59808i) q^{19} +7.48331i q^{20} +(-3.24037 + 1.87083i) q^{23} +(4.50000 + 7.79423i) q^{25} +(2.00000 + 3.46410i) q^{28} +(-3.24037 - 5.61249i) q^{29} -1.73205i q^{31} +(6.48074 + 3.74166i) q^{35} -5.19615i q^{37} +(3.24037 - 5.61249i) q^{41} +(-2.50000 + 4.33013i) q^{43} +(6.48074 - 3.74166i) q^{44} +(-3.24037 + 1.87083i) q^{47} -3.00000 q^{49} +(9.00000 + 5.19615i) q^{52} +(3.24037 + 5.61249i) q^{53} +(7.00000 - 12.1244i) q^{55} +(-6.48074 + 11.2250i) q^{59} +(3.50000 + 6.06218i) q^{61} -8.00000 q^{64} +19.4422 q^{65} -14.9666i q^{68} +(-3.24037 + 5.61249i) q^{71} +(2.00000 - 3.46410i) q^{73} +(-8.00000 - 3.46410i) q^{76} -7.48331i q^{77} +(-4.50000 - 2.59808i) q^{79} +(-12.9615 + 7.48331i) q^{80} +7.48331i q^{83} +(-14.0000 - 24.2487i) q^{85} +(-6.48074 - 11.2250i) q^{89} +(9.00000 - 5.19615i) q^{91} +(-6.48074 - 3.74166i) q^{92} +(-16.2019 + 1.87083i) q^{95} +(4.50000 + 2.59808i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 8 q^{7} + 18 q^{13} - 8 q^{16} - 14 q^{19} + 18 q^{25} + 8 q^{28} - 10 q^{43} - 12 q^{49} + 36 q^{52} + 28 q^{55} + 14 q^{61} - 32 q^{64} + 8 q^{73} - 32 q^{76} - 18 q^{79} - 56 q^{85} + 36 q^{91}+ \cdots + 18 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 + 8 * q^7 + 18 * q^13 - 8 * q^16 - 14 * q^19 + 18 * q^25 + 8 * q^28 - 10 * q^43 - 12 * q^49 + 36 * q^52 + 28 * q^55 + 14 * q^61 - 32 * q^64 + 8 * q^73 - 32 * q^76 - 18 * q^79 - 56 * q^85 + 36 * q^91 + 18 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/513\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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		−0.866025	−	0.500000i	$-0.833333\pi$
$2$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$3$		0			0
$3$		0			0
$4$	1.00000	+	1.73205i	0.500000	+	0.866025i
$5$	3.24037	+	1.87083i	1.44914	+	0.836660i	0.998430	−	0.0560116i	$-0.0178384\pi$
$5$	3.24037	+	1.87083i	1.44914	+	0.836660i	0.450708	+	0.892672i	$0.351172\pi$
$6$		0			0
$7$	2.00000			0.755929			0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000			0.755929			0.377964	+	0.925820i	$0.376624\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.74166i		−	1.12815i	−0.825723	−	0.564076i	$-0.809232\pi$
$11$		−	3.74166i		−	1.12815i	0.825723	−	0.564076i	$-0.190768\pi$
$12$		0			0
$13$	4.50000	−	2.59808i	1.24808	−	0.720577i	0.277350	−	0.960769i	$-0.410544\pi$
$13$	4.50000	−	2.59808i	1.24808	−	0.720577i	0.970725	+	0.240192i	$0.0772105\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$	−6.48074	−	3.74166i	−1.57181	−	0.907485i	−0.995947	−	0.0899392i	$-0.971333\pi$
$17$	−6.48074	−	3.74166i	−1.57181	−	0.907485i	−0.575863	−	0.817546i	$-0.695334\pi$
$18$		0			0
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$20$			7.48331i			1.67332i
$21$		0			0
$22$		0			0
$23$	−3.24037	+	1.87083i	−0.675664	+	0.390095i	−0.798219	−	0.602367i	$-0.794224\pi$
$23$	−3.24037	+	1.87083i	−0.675664	+	0.390095i	0.122555	+	0.992462i	$0.460891\pi$
$24$		0			0
$25$	4.50000	+	7.79423i	0.900000	+	1.55885i
$26$		0			0
$27$		0			0
$28$	2.00000	+	3.46410i	0.377964	+	0.654654i
$29$	−3.24037	−	5.61249i	−0.601722	−	1.04221i	−0.992560	−	0.121753i	$-0.961148\pi$
$29$	−3.24037	−	5.61249i	−0.601722	−	1.04221i	0.390839	−	0.920459i	$-0.372185\pi$
$30$		0			0
$31$		−	1.73205i		−	0.311086i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$		−	1.73205i		−	0.311086i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.48074	+	3.74166i	1.09545	+	0.632456i
$36$		0			0
$37$		−	5.19615i		−	0.854242i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$		−	5.19615i		−	0.854242i	0.904194	−	0.427121i	$-0.140472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.24037	−	5.61249i	0.506061	−	0.876523i	−0.493915	−	0.869510i	$-0.664434\pi$
$41$	3.24037	−	5.61249i	0.506061	−	0.876523i	0.999975	−	0.00701264i	$-0.00223221\pi$
$42$		0			0
$43$	−2.50000	+	4.33013i	−0.381246	+	0.660338i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−2.50000	+	4.33013i	−0.381246	+	0.660338i	0.609994	+	0.792406i	$0.291172\pi$
$44$	6.48074	−	3.74166i	0.977008	−	0.564076i
$45$		0			0
$46$		0			0
$47$	−3.24037	+	1.87083i	−0.472657	+	0.272888i	−0.717351	−	0.696712i	$-0.754646\pi$
$47$	−3.24037	+	1.87083i	−0.472657	+	0.272888i	0.244695	+	0.969600i	$0.421312\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$		0			0
$51$		0			0
$52$	9.00000	+	5.19615i	1.24808	+	0.720577i
$53$	3.24037	+	5.61249i	0.445099	+	0.770934i	0.998059	−	0.0622735i	$-0.0198351\pi$
$53$	3.24037	+	5.61249i	0.445099	+	0.770934i	−0.552960	+	0.833208i	$0.686502\pi$
$54$		0			0
$55$	7.00000	−	12.1244i	0.943880	−	1.63485i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.48074	+	11.2250i	−0.843721	+	1.46137i	0.0430071	+	0.999075i	$0.486306\pi$
$59$	−6.48074	+	11.2250i	−0.843721	+	1.46137i	−0.886728	+	0.462292i	$0.847027\pi$
$60$		0			0
$61$	3.50000	+	6.06218i	0.448129	+	0.776182i	0.998264	−	0.0588933i	$-0.0187572\pi$
$61$	3.50000	+	6.06218i	0.448129	+	0.776182i	−0.550135	+	0.835076i	$0.685424\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	19.4422			2.41151
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		−	14.9666i		−	1.81497i
$69$		0			0
$70$		0			0
$71$	−3.24037	+	5.61249i	−0.384561	+	0.666080i	−0.991708	−	0.128510i	$-0.958981\pi$
$71$	−3.24037	+	5.61249i	−0.384561	+	0.666080i	0.607147	+	0.794590i	$0.292314\pi$
$72$		0			0
$73$	2.00000	−	3.46410i	0.234082	−	0.405442i	−0.724923	−	0.688830i	$-0.758125\pi$
$73$	2.00000	−	3.46410i	0.234082	−	0.405442i	0.959006	+	0.283387i	$0.0914581\pi$
$74$		0			0
$75$		0			0
$76$	−8.00000	−	3.46410i	−0.917663	−	0.397360i
$77$		−	7.48331i		−	0.852803i
$78$		0			0
$79$	−4.50000	−	2.59808i	−0.506290	−	0.292306i	0.225018	−	0.974355i	$-0.427756\pi$
$79$	−4.50000	−	2.59808i	−0.506290	−	0.292306i	−0.731307	+	0.682048i	$0.761089\pi$
$80$	−12.9615	+	7.48331i	−1.44914	+	0.836660i
$81$		0			0
$82$		0			0
$83$			7.48331i			0.821401i	0.911770	+	0.410700i	$0.134716\pi$
$83$			7.48331i			0.821401i	−0.911770	+	0.410700i	$0.865284\pi$
$84$		0			0
$85$	−14.0000	−	24.2487i	−1.51851	−	2.63014i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.48074	−	11.2250i	−0.686957	−	1.18984i	−0.972817	−	0.231573i	$-0.925613\pi$
$89$	−6.48074	−	11.2250i	−0.686957	−	1.18984i	0.285860	−	0.958271i	$-0.407721\pi$
$90$		0			0
$91$	9.00000	−	5.19615i	0.943456	−	0.544705i
$92$	−6.48074	−	3.74166i	−0.675664	−	0.390095i
$93$		0			0
$94$		0			0
$95$	−16.2019	+	1.87083i	−1.66227	+	0.191943i
$96$		0			0
$97$	4.50000	+	2.59808i	0.456906	+	0.263795i	0.710742	−	0.703452i	$-0.248359\pi$
$97$	4.50000	+	2.59808i	0.456906	+	0.263795i	−0.253837	+	0.967247i	$0.581693\pi$
$98$		0			0
$99$		0			0
$100$	−9.00000	+	15.5885i	−0.900000	+	1.55885i
$101$	−3.24037	+	1.87083i	−0.322429	+	0.186154i	−0.652475	−	0.757811i	$-0.726269\pi$
$101$	−3.24037	+	1.87083i	−0.322429	+	0.186154i	0.330046	+	0.943965i	$0.392936\pi$
$102$		0			0
$103$		−	6.92820i		−	0.682656i	−0.939944	−	0.341328i	$-0.889123\pi$
$103$		−	6.92820i		−	0.682656i	0.939944	−	0.341328i	$-0.110877\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	19.4422			1.87955			0.939775	−	0.341793i	$-0.111034\pi$
$107$	19.4422			1.87955			0.939775	+	0.341793i	$0.111034\pi$
$108$		0			0
$109$	−3.00000	−	1.73205i	−0.287348	−	0.165900i	0.349397	−	0.936975i	$-0.386386\pi$
$109$	−3.00000	−	1.73205i	−0.287348	−	0.165900i	−0.636745	+	0.771074i	$0.719720\pi$
$110$		0			0
$111$		0			0
$112$	−4.00000	+	6.92820i	−0.377964	+	0.654654i
$113$	6.48074			0.609657			0.304828	−	0.952407i	$-0.401401\pi$
$113$	6.48074			0.609657			0.304828	+	0.952407i	$0.401401\pi$
$114$		0			0
$115$	−14.0000			−1.30551
$116$	6.48074	−	11.2250i	0.601722	−	1.04221i
$117$		0			0
$118$		0			0
$119$	−12.9615	−	7.48331i	−1.18818	−	0.685994i
$120$		0			0
$121$	−3.00000			−0.272727
$122$		0			0
$123$		0			0
$124$	3.00000	−	1.73205i	0.269408	−	0.155543i
$125$			14.9666i			1.33866i
$126$		0			0
$127$	−12.0000	+	6.92820i	−1.06483	+	0.614779i	−0.926764	−	0.375645i	$-0.877421\pi$
$127$	−12.0000	+	6.92820i	−1.06483	+	0.614779i	−0.138064	+	0.990423i	$0.544088\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.9615	+	7.48331i	1.13245	+	0.653820i	0.944550	−	0.328368i	$-0.106499\pi$
$131$	12.9615	+	7.48331i	1.13245	+	0.653820i	0.187900	+	0.982188i	$0.439832\pi$
$132$		0			0
$133$	−7.00000	+	5.19615i	−0.606977	+	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.24037	+	1.87083i	−0.276844	+	0.159836i	−0.631994	−	0.774974i	$-0.717763\pi$
$137$	−3.24037	+	1.87083i	−0.276844	+	0.159836i	0.355150	+	0.934809i	$0.384430\pi$
$138$		0			0
$139$	−7.00000	−	12.1244i	−0.593732	−	1.02837i	−0.993724	−	0.111856i	$-0.964321\pi$
$139$	−7.00000	−	12.1244i	−0.593732	−	1.02837i	0.399992	−	0.916519i	$-0.369013\pi$
$140$			14.9666i			1.26491i
$141$		0			0
$142$		0			0
$143$	−9.72111	−	16.8375i	−0.812920	−	1.40802i
$144$		0			0
$145$		−	24.2487i		−	2.01375i
$146$		0			0
$147$		0			0
$148$	9.00000	−	5.19615i	0.739795	−	0.427121i
$149$	12.9615	+	7.48331i	1.06185	+	0.613057i	0.925942	−	0.377665i	$-0.123273\pi$
$149$	12.9615	+	7.48331i	1.06185	+	0.613057i	0.135904	+	0.990722i	$0.456606\pi$
$150$		0			0
$151$		−	8.66025i		−	0.704761i	−0.935857	−	0.352381i	$-0.885372\pi$
$151$		−	8.66025i		−	0.704761i	0.935857	−	0.352381i	$-0.114628\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.24037	−	5.61249i	0.260273	−	0.450806i
$156$		0			0
$157$	−5.50000	+	9.52628i	−0.438948	+	0.760280i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−5.50000	+	9.52628i	−0.438948	+	0.760280i	0.558661	+	0.829396i	$0.311315\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.48074	+	3.74166i	−0.510754	+	0.294884i
$162$		0			0
$163$	−1.00000			−0.0783260			−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000			−0.0783260			−0.0391630	+	0.999233i	$0.512469\pi$
$164$	12.9615			1.01212
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	7.00000	−	12.1244i	0.538462	−	0.932643i
$170$		0			0
$171$		0			0
$172$	−10.0000			−0.762493
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	9.00000	+	15.5885i	0.680336	+	1.17838i
$176$	12.9615	+	7.48331i	0.977008	+	0.564076i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	19.5000	−	11.2583i	1.44942	−	0.836825i	0.450977	−	0.892536i	$-0.351076\pi$
$181$	19.5000	−	11.2583i	1.44942	−	0.836825i	0.998447	+	0.0557107i	$0.0177424\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9.72111	−	16.8375i	0.714710	−	1.23791i
$186$		0			0
$187$	−14.0000	+	24.2487i	−1.02378	+	1.77324i
$188$	−6.48074	−	3.74166i	−0.472657	−	0.272888i
$189$		0			0
$190$		0			0
$191$		−	3.74166i		−	0.270737i	−0.990795	−	0.135368i	$-0.956778\pi$
$191$		−	3.74166i		−	0.270737i	0.990795	−	0.135368i	$-0.0432218\pi$
$192$		0			0
$193$	21.0000	+	12.1244i	1.51161	+	0.872730i	0.999908	+	0.0135691i	$0.00431931\pi$
$193$	21.0000	+	12.1244i	1.51161	+	0.872730i	0.511705	+	0.859161i	$0.329014\pi$
$194$		0			0
$195$		0			0
$196$	−3.00000	−	5.19615i	−0.214286	−	0.371154i
$197$		−	3.74166i		−	0.266582i	−0.991077	−	0.133291i	$-0.957446\pi$
$197$		−	3.74166i		−	0.266582i	0.991077	−	0.133291i	$-0.0425545\pi$
$198$		0			0
$199$	2.00000	+	3.46410i	0.141776	+	0.245564i	0.928166	−	0.372168i	$-0.121385\pi$
$199$	2.00000	+	3.46410i	0.141776	+	0.245564i	−0.786389	+	0.617731i	$0.788052\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.48074	−	11.2250i	−0.454859	−	0.787839i
$204$		0			0
$205$	21.0000	−	12.1244i	1.46670	−	0.846802i
$206$		0			0
$207$		0			0
$208$			20.7846i			1.44115i
$209$	9.72111	+	13.0958i	0.672423	+	0.905855i
$210$		0			0
$211$	12.0000	+	6.92820i	0.826114	+	0.476957i	0.852520	−	0.522694i	$-0.175073\pi$
$211$	12.0000	+	6.92820i	0.826114	+	0.476957i	−0.0264062	+	0.999651i	$0.508406\pi$
$212$	−6.48074	+	11.2250i	−0.445099	+	0.770934i
$213$		0			0
$214$		0			0
$215$	−16.2019	+	9.35414i	−1.10496	+	0.637947i
$216$		0			0
$217$		−	3.46410i		−	0.235159i
$218$		0			0
$219$		0			0
$220$	28.0000			1.88776
$221$	−38.8844			−2.61565
$222$		0			0
$223$	−4.50000	−	2.59808i	−0.301342	−	0.173980i	0.341703	−	0.939808i	$-0.388996\pi$
$223$	−4.50000	−	2.59808i	−0.301342	−	0.173980i	−0.643046	+	0.765828i	$0.722329\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.48074			−0.430142			−0.215071	−	0.976598i	$-0.568998\pi$
$227$	−6.48074			−0.430142			−0.215071	+	0.976598i	$0.568998\pi$
$228$		0			0
$229$	−1.00000			−0.0660819			−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000			−0.0660819			−0.0330409	+	0.999454i	$0.510519\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.48074	−	3.74166i	−0.424567	−	0.245124i	0.272462	−	0.962167i	$-0.412162\pi$
$233$	−6.48074	−	3.74166i	−0.424567	−	0.245124i	−0.697030	+	0.717042i	$0.745495\pi$
$234$		0			0
$235$	−14.0000			−0.913259
$236$	−25.9230			−1.68744
$237$		0			0
$238$		0			0
$239$		−	14.9666i		−	0.968111i	−0.875037	−	0.484055i	$-0.839163\pi$
$239$		−	14.9666i		−	0.968111i	0.875037	−	0.484055i	$-0.160837\pi$
$240$		0			0
$241$	12.0000	−	6.92820i	0.772988	−	0.446285i	−0.0609515	−	0.998141i	$-0.519414\pi$
$241$	12.0000	−	6.92820i	0.772988	−	0.446285i	0.833939	+	0.551856i	$0.186080\pi$
$242$		0			0
$243$		0			0
$244$	−7.00000	+	12.1244i	−0.448129	+	0.776182i
$245$	−9.72111	−	5.61249i	−0.621059	−	0.358569i
$246$		0			0
$247$	−9.00000	+	20.7846i	−0.572656	+	1.32249i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	16.2019	−	9.35414i	1.02265	−	0.590428i	0.107781	−	0.994175i	$-0.465626\pi$
$251$	16.2019	−	9.35414i	1.02265	−	0.590428i	0.914871	+	0.403746i	$0.132292\pi$
$252$		0			0
$253$	7.00000	+	12.1244i	0.440086	+	0.762252i
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	9.72111	+	16.8375i	0.606386	+	1.05029i	0.991831	+	0.127561i	$0.0407148\pi$
$257$	9.72111	+	16.8375i	0.606386	+	1.05029i	−0.385445	+	0.922731i	$0.625952\pi$
$258$		0			0
$259$		−	10.3923i		−	0.645746i
$260$	19.4422	+	33.6749i	1.20576	+	2.08843i
$261$		0			0
$262$		0			0
$263$	12.9615	+	7.48331i	0.799239	+	0.461441i	0.843205	−	0.537592i	$-0.180666\pi$
$263$	12.9615	+	7.48331i	0.799239	+	0.461441i	−0.0439659	+	0.999033i	$0.513999\pi$
$264$		0			0
$265$			24.2487i			1.48959i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.48074	−	11.2250i	0.395138	−	0.684399i	−0.597981	−	0.801510i	$-0.704030\pi$
$269$	6.48074	−	11.2250i	0.395138	−	0.684399i	0.993119	+	0.117112i	$0.0373635\pi$
$270$		0			0
$271$	6.50000	−	11.2583i	0.394847	−	0.683895i	−0.598235	−	0.801321i	$-0.704131\pi$
$271$	6.50000	−	11.2583i	0.394847	−	0.683895i	0.993082	+	0.117426i	$0.0374643\pi$
$272$	25.9230	−	14.9666i	1.57181	−	0.907485i
$273$		0			0
$274$		0			0
$275$	29.1633	−	16.8375i	1.75862	−	1.01534i
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.24037	−	5.61249i	−0.193304	−	0.334813i	0.753039	−	0.657976i	$-0.228587\pi$
$281$	−3.24037	−	5.61249i	−0.193304	−	0.334813i	−0.946343	+	0.323163i	$0.895254\pi$
$282$		0			0
$283$	5.00000	−	8.66025i	0.297219	−	0.514799i	−0.678280	−	0.734804i	$-0.737274\pi$
$283$	5.00000	−	8.66025i	0.297219	−	0.514799i	0.975499	+	0.220005i	$0.0706075\pi$
$284$	−12.9615			−0.769122
$285$		0			0
$286$		0			0
$287$	6.48074	−	11.2250i	0.382546	−	0.662589i
$288$		0			0
$289$	19.5000	+	33.7750i	1.14706	+	1.98676i
$290$		0			0
$291$		0			0
$292$	8.00000			0.468165
$293$	−12.9615			−0.757218			−0.378609	−	0.925557i	$-0.623597\pi$
$293$	−12.9615			−0.757218			−0.378609	+	0.925557i	$0.623597\pi$
$294$		0			0
$295$	−42.0000	+	24.2487i	−2.44533	+	1.41181i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.72111	+	16.8375i	−0.562186	+	0.973735i
$300$		0			0
$301$	−5.00000	+	8.66025i	−0.288195	+	0.499169i
$302$		0			0
$303$		0			0
$304$	−2.00000	−	17.3205i	−0.114708	−	0.993399i
$305$			26.1916i			1.49973i
$306$		0			0
$307$	6.00000	+	3.46410i	0.342438	+	0.197707i	0.661350	−	0.750078i	$-0.269984\pi$
$307$	6.00000	+	3.46410i	0.342438	+	0.197707i	−0.318912	+	0.947784i	$0.603317\pi$
$308$	12.9615	−	7.48331i	0.738549	−	0.426401i
$309$		0			0
$310$		0			0
$311$		−	3.74166i		−	0.212170i	−0.994357	−	0.106085i	$-0.966168\pi$
$311$		−	3.74166i		−	0.212170i	0.994357	−	0.106085i	$-0.0338316\pi$
$312$		0			0
$313$	3.50000	+	6.06218i	0.197832	+	0.342655i	0.947825	−	0.318791i	$-0.103277\pi$
$313$	3.50000	+	6.06218i	0.197832	+	0.342655i	−0.749993	+	0.661445i	$0.769943\pi$
$314$		0			0
$315$		0			0
$316$		−	10.3923i		−	0.584613i
$317$	6.48074	+	11.2250i	0.363995	+	0.630457i	0.988614	−	0.150472i	$-0.0480793\pi$
$317$	6.48074	+	11.2250i	0.363995	+	0.630457i	−0.624620	+	0.780929i	$0.714746\pi$
$318$		0			0
$319$	−21.0000	+	12.1244i	−1.17577	+	0.678834i
$320$	−25.9230	−	14.9666i	−1.44914	−	0.836660i
$321$		0			0
$322$		0			0
$323$	32.4037	−	3.74166i	1.80299	−	0.208191i
$324$		0			0
$325$	40.5000	+	23.3827i	2.24654	+	1.29704i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.48074	+	3.74166i	−0.357295	+	0.206284i
$330$		0			0
$331$		−	29.4449i		−	1.61844i	−0.587508	−	0.809218i	$-0.699891\pi$
$331$		−	29.4449i		−	1.61844i	0.587508	−	0.809218i	$-0.300109\pi$
$332$	−12.9615	+	7.48331i	−0.711354	+	0.410700i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	12.0000	+	6.92820i	0.653682	+	0.377403i	0.789865	−	0.613280i	$-0.210150\pi$
$337$	12.0000	+	6.92820i	0.653682	+	0.377403i	−0.136184	+	0.990684i	$0.543484\pi$
$338$		0			0
$339$		0			0
$340$	28.0000	−	48.4974i	1.51851	−	2.63014i
$341$	−6.48074			−0.350952
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.24037	+	1.87083i	0.173952	+	0.100431i	0.584448	−	0.811431i	$-0.301311\pi$
$347$	3.24037	+	1.87083i	0.173952	+	0.100431i	−0.410496	+	0.911862i	$0.634644\pi$
$348$		0			0
$349$	−7.00000			−0.374701			−0.187351	−	0.982293i	$-0.559990\pi$
$349$	−7.00000			−0.374701			−0.187351	+	0.982293i	$0.559990\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			29.9333i			1.59319i	0.604516	+	0.796593i	$0.293367\pi$
$353$			29.9333i			1.59319i	−0.604516	+	0.796593i	$0.706633\pi$
$354$		0			0
$355$	−21.0000	+	12.1244i	−1.11456	+	0.643494i
$356$	12.9615	−	22.4499i	0.686957	−	1.18984i
$357$		0			0
$358$		0			0
$359$	12.9615	+	7.48331i	0.684081	+	0.394954i	0.801391	−	0.598141i	$-0.204094\pi$
$359$	12.9615	+	7.48331i	0.684081	+	0.394954i	−0.117310	+	0.993095i	$0.537427\pi$
$360$		0			0
$361$	5.50000	−	18.1865i	0.289474	−	0.957186i
$362$		0			0
$363$		0			0
$364$	18.0000	+	10.3923i	0.943456	+	0.544705i
$365$	12.9615	−	7.48331i	0.678435	−	0.391695i
$366$		0			0
$367$	12.5000	+	21.6506i	0.652495	+	1.13015i	0.982516	+	0.186180i	$0.0596109\pi$
$367$	12.5000	+	21.6506i	0.652495	+	1.13015i	−0.330021	+	0.943974i	$0.607056\pi$
$368$		−	14.9666i		−	0.780189i
$369$		0			0
$370$		0			0
$371$	6.48074	+	11.2250i	0.336463	+	0.582772i
$372$		0			0
$373$		−	12.1244i		−	0.627775i	−0.949460	−	0.313888i	$-0.898368\pi$
$373$		−	12.1244i		−	0.627775i	0.949460	−	0.313888i	$-0.101632\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−29.1633	−	16.8375i	−1.50199	−	0.867173i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	−19.4422	−	26.1916i	−0.997365	−	1.34360i
$381$		0			0
$382$		0			0
$383$	−16.2019	+	28.0624i	−0.827876	+	1.43392i	0.0718254	+	0.997417i	$0.477118\pi$
$383$	−16.2019	+	28.0624i	−0.827876	+	1.43392i	−0.899701	+	0.436506i	$0.856216\pi$
$384$		0			0
$385$	14.0000	−	24.2487i	0.713506	−	1.23583i
$386$		0			0
$387$		0			0
$388$			10.3923i			0.527589i
$389$	6.48074	−	3.74166i	0.328587	−	0.189710i	−0.326627	−	0.945153i	$-0.605912\pi$
$389$	6.48074	−	3.74166i	0.328587	−	0.189710i	0.655213	+	0.755444i	$0.272579\pi$
$390$		0			0
$391$	28.0000			1.41602
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−9.72111	−	16.8375i	−0.489122	−	0.847184i
$396$		0			0
$397$	0.500000	−	0.866025i	0.0250943	−	0.0434646i	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	0.500000	−	0.866025i	0.0250943	−	0.0434646i	0.878300	+	0.478110i	$0.158678\pi$
$398$		0			0
$399$		0			0
$400$	−36.0000			−1.80000
$401$	−16.2019	+	28.0624i	−0.809082	+	1.40137i	0.104419	+	0.994533i	$0.466702\pi$
$401$	−16.2019	+	28.0624i	−0.809082	+	1.40137i	−0.913501	+	0.406837i	$0.866632\pi$
$402$		0			0
$403$	−4.50000	−	7.79423i	−0.224161	−	0.388258i
$404$	−6.48074	−	3.74166i	−0.322429	−	0.186154i
$405$		0			0
$406$		0			0
$407$	−19.4422			−0.963715
$408$		0			0
$409$	−22.5000	+	12.9904i	−1.11255	+	0.642333i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−22.5000	+	12.9904i	−1.11255	+	0.642333i	−0.173064	+	0.984911i	$0.555367\pi$
$410$		0			0
$411$		0			0
$412$	12.0000	−	6.92820i	0.591198	−	0.341328i
$413$	−12.9615	+	22.4499i	−0.637793	+	1.10469i
$414$		0			0
$415$	−14.0000	+	24.2487i	−0.687233	+	1.19032i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	14.9666i		−	0.731168i	−0.930778	−	0.365584i	$-0.880869\pi$
$419$		−	14.9666i		−	0.731168i	0.930778	−	0.365584i	$-0.119131\pi$
$420$		0			0
$421$	−31.5000	−	18.1865i	−1.53522	−	0.886357i	−0.999109	−	0.0422075i	$-0.986561\pi$
$421$	−31.5000	−	18.1865i	−1.53522	−	0.886357i	−0.536107	−	0.844150i	$-0.680106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	67.3498i		−	3.26695i
$426$		0			0
$427$	7.00000	+	12.1244i	0.338754	+	0.586739i
$428$	19.4422	+	33.6749i	0.939775	+	1.62774i
$429$		0			0
$430$		0			0
$431$	−6.48074	−	11.2250i	−0.312166	−	0.540688i	0.666665	−	0.745358i	$-0.267721\pi$
$431$	−6.48074	−	11.2250i	−0.312166	−	0.540688i	−0.978831	+	0.204670i	$0.934388\pi$
$432$		0			0
$433$	19.5000	−	11.2583i	0.937110	−	0.541041i	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	19.5000	−	11.2583i	0.937110	−	0.541041i	0.889053	+	0.457804i	$0.151364\pi$
$434$		0			0
$435$		0			0
$436$		−	6.92820i		−	0.331801i
$437$	6.48074	−	14.9666i	0.310016	−	0.715951i
$438$		0			0
$439$	13.5000	+	7.79423i	0.644320	+	0.371998i	0.786277	−	0.617875i	$-0.212006\pi$
$439$	13.5000	+	7.79423i	0.644320	+	0.371998i	−0.141957	+	0.989873i	$0.545339\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−22.6826	+	13.0958i	−1.07768	+	0.622200i	−0.930270	−	0.366875i	$-0.880428\pi$
$443$	−22.6826	+	13.0958i	−1.07768	+	0.622200i	−0.147412	+	0.989075i	$0.547094\pi$
$444$		0			0
$445$		−	48.4974i		−	2.29900i
$446$		0			0
$447$		0			0
$448$	−16.0000			−0.755929
$449$	−19.4422			−0.917535			−0.458768	−	0.888556i	$-0.651709\pi$
$449$	−19.4422			−0.917535			−0.458768	+	0.888556i	$0.651709\pi$
$450$		0			0
$451$	−21.0000	−	12.1244i	−0.988851	−	0.570914i
$452$	6.48074	+	11.2250i	0.304828	+	0.527978i
$453$		0			0
$454$		0			0
$455$	38.8844			1.82293
$456$		0			0
$457$	−31.0000			−1.45012			−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000			−1.45012			−0.725059	+	0.688686i	$0.758188\pi$
$458$		0			0
$459$		0			0
$460$	−14.0000	−	24.2487i	−0.652753	−	1.13060i
$461$	22.6826	+	13.0958i	1.05643	+	0.609932i	0.924444	−	0.381319i	$-0.124530\pi$
$461$	22.6826	+	13.0958i	1.05643	+	0.609932i	0.131990	+	0.991251i	$0.457863\pi$
$462$		0			0
$463$	−1.00000			−0.0464739			−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000			−0.0464739			−0.0232370	+	0.999730i	$0.507397\pi$
$464$	25.9230			1.20344
$465$		0			0
$466$		0			0
$467$			18.7083i			0.865716i	0.901462	+	0.432858i	$0.142495\pi$
$467$			18.7083i			0.865716i	−0.901462	+	0.432858i	$0.857505\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	16.2019	+	9.35414i	0.744962	+	0.430104i
$474$		0			0
$475$	−36.0000	−	15.5885i	−1.65179	−	0.715247i
$476$		−	29.9333i		−	1.37199i
$477$		0			0
$478$		0			0
$479$	−3.24037	+	1.87083i	−0.148056	+	0.0854803i	−0.572198	−	0.820115i	$-0.693909\pi$
$479$	−3.24037	+	1.87083i	−0.148056	+	0.0854803i	0.424142	+	0.905596i	$0.360576\pi$
$480$		0			0
$481$	−13.5000	−	23.3827i	−0.615547	−	1.06616i
$482$		0			0
$483$		0			0
$484$	−3.00000	−	5.19615i	−0.136364	−	0.236189i
$485$	9.72111	+	16.8375i	0.441413	+	0.764550i
$486$		0			0
$487$		−	36.3731i		−	1.64822i	−0.566429	−	0.824110i	$-0.691675\pi$
$487$		−	36.3731i		−	1.64822i	0.566429	−	0.824110i	$-0.308325\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.24037	+	1.87083i	0.146236	+	0.0844293i	0.571333	−	0.820719i	$-0.306427\pi$
$491$	3.24037	+	1.87083i	0.146236	+	0.0844293i	−0.425097	+	0.905148i	$0.639760\pi$
$492$		0			0
$493$			48.4974i			2.18421i
$494$		0			0
$495$		0			0
$496$	6.00000	+	3.46410i	0.269408	+	0.155543i
$497$	−6.48074	+	11.2250i	−0.290701	+	0.503509i
$498$		0			0
$499$	−8.50000	+	14.7224i	−0.380512	+	0.659067i	−0.991136	−	0.132855i	$-0.957586\pi$
$499$	−8.50000	+	14.7224i	−0.380512	+	0.659067i	0.610623	+	0.791921i	$0.290919\pi$
$500$	−25.9230	+	14.9666i	−1.15931	+	0.669328i
$501$		0			0
$502$		0			0
$503$	6.48074	−	3.74166i	0.288962	−	0.166832i	−0.348512	−	0.937304i	$-0.613313\pi$
$503$	6.48074	−	3.74166i	0.288962	−	0.166832i	0.637474	+	0.770472i	$0.279979\pi$
$504$		0			0
$505$	−14.0000			−0.622992
$506$		0			0
$507$		0			0
$508$	−24.0000	−	13.8564i	−1.06483	−	0.614779i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	4.00000	−	6.92820i	0.176950	−	0.306486i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.9615	−	22.4499i	0.571151	−	0.989263i
$516$		0			0
$517$	7.00000	+	12.1244i	0.307860	+	0.533229i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	19.4422			0.851779			0.425890	−	0.904775i	$-0.359961\pi$
$521$	19.4422			0.851779			0.425890	+	0.904775i	$0.359961\pi$
$522$		0			0
$523$	−34.5000	+	19.9186i	−1.50858	+	0.870979i	−0.508630	+	0.860985i	$0.669848\pi$
$523$	−34.5000	+	19.9186i	−1.50858	+	0.870979i	−0.999950	+	0.00999344i	$0.996819\pi$
$524$			29.9333i			1.30764i
$525$		0			0
$526$		0			0
$527$	−6.48074	+	11.2250i	−0.282305	+	0.488967i
$528$		0			0
$529$	−4.50000	+	7.79423i	−0.195652	+	0.338880i
$530$		0			0
$531$		0			0
$532$	−16.0000	−	6.92820i	−0.693688	−	0.300376i
$533$		−	33.6749i		−	1.45862i
$534$		0			0
$535$	63.0000	+	36.3731i	2.72373	+	1.57254i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			11.2250i			0.483494i
$540$		0			0
$541$	−8.50000	−	14.7224i	−0.365444	−	0.632967i	0.623404	−	0.781900i	$-0.285749\pi$
$541$	−8.50000	−	14.7224i	−0.365444	−	0.632967i	−0.988847	+	0.148933i	$0.952416\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.48074	−	11.2250i	−0.277604	−	0.480825i
$546$		0			0
$547$	−10.5000	+	6.06218i	−0.448948	+	0.259200i	−0.707386	−	0.706828i	$-0.750126\pi$
$547$	−10.5000	+	6.06218i	−0.448948	+	0.259200i	0.258438	+	0.966028i	$0.416792\pi$
$548$	−6.48074	−	3.74166i	−0.276844	−	0.159836i
$549$		0			0
$550$		0			0
$551$	25.9230	+	11.2250i	1.10436	+	0.478200i
$552$		0			0
$553$	−9.00000	−	5.19615i	−0.382719	−	0.220963i
$554$		0			0
$555$		0			0
$556$	14.0000	−	24.2487i	0.593732	−	1.02837i
$557$	35.6441	−	20.5791i	1.51029	−	0.871965i	0.510360	−	0.859961i	$-0.329512\pi$
$557$	35.6441	−	20.5791i	1.51029	−	0.871965i	0.999928	−	0.0120042i	$-0.00382115\pi$
$558$		0			0
$559$			25.9808i			1.09887i
$560$	−25.9230	+	14.9666i	−1.09545	+	0.632456i
$561$		0			0
$562$		0			0
$563$	−12.9615			−0.546261			−0.273131	−	0.961977i	$-0.588059\pi$
$563$	−12.9615			−0.546261			−0.273131	+	0.961977i	$0.588059\pi$
$564$		0			0
$565$	21.0000	+	12.1244i	0.883477	+	0.510075i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.48074			−0.271687			−0.135843	−	0.990730i	$-0.543374\pi$
$569$	−6.48074			−0.271687			−0.135843	+	0.990730i	$0.543374\pi$
$570$		0			0
$571$	−37.0000			−1.54840			−0.774201	−	0.632940i	$-0.781848\pi$
$571$	−37.0000			−1.54840			−0.774201	+	0.632940i	$0.781848\pi$
$572$	19.4422	−	33.6749i	0.812920	−	1.40802i
$573$		0			0
$574$		0			0
$575$	−29.1633	−	16.8375i	−1.21620	−	0.702171i
$576$		0			0
$577$	17.0000			0.707719			0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000			0.707719			0.353860	+	0.935299i	$0.384869\pi$
$578$		0			0
$579$		0			0
$580$	42.0000	−	24.2487i	1.74396	−	1.00687i
$581$			14.9666i			0.620920i
$582$		0			0
$583$	21.0000	−	12.1244i	0.869731	−	0.502140i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.9615	+	7.48331i	0.534978	+	0.308869i	0.743041	−	0.669246i	$-0.233383\pi$
$587$	12.9615	+	7.48331i	0.534978	+	0.308869i	−0.208063	+	0.978115i	$0.566716\pi$
$588$		0			0
$589$	4.50000	+	6.06218i	0.185419	+	0.249788i
$590$		0			0
$591$		0			0
$592$	18.0000	+	10.3923i	0.739795	+	0.427121i
$593$	35.6441	−	20.5791i	1.46373	−	0.845083i	0.464546	−	0.885549i	$-0.346218\pi$
$593$	35.6441	−	20.5791i	1.46373	−	0.845083i	0.999181	+	0.0404661i	$0.0128843\pi$
$594$		0			0
$595$	−28.0000	−	48.4974i	−1.14789	−	1.98820i
$596$			29.9333i			1.22611i
$597$		0			0
$598$		0			0
$599$	−19.4422	−	33.6749i	−0.794388	−	1.37592i	−0.923227	−	0.384254i	$-0.874459\pi$
$599$	−19.4422	−	33.6749i	−0.794388	−	1.37592i	0.128840	−	0.991665i	$-0.458875\pi$
$600$		0			0
$601$		−	29.4449i		−	1.20108i	−0.799594	−	0.600541i	$-0.794952\pi$
$601$		−	29.4449i		−	1.20108i	0.799594	−	0.600541i	$-0.205048\pi$
$602$		0			0
$603$		0			0
$604$	15.0000	−	8.66025i	0.610341	−	0.352381i
$605$	−9.72111	−	5.61249i	−0.395219	−	0.228180i
$606$		0			0
$607$			3.46410i			0.140604i	0.997526	+	0.0703018i	$0.0223962\pi$
$607$			3.46410i			0.140604i	−0.997526	+	0.0703018i	$0.977604\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.72111	+	16.8375i	−0.393274	+	0.681171i
$612$		0			0
$613$	−11.5000	+	19.9186i	−0.464481	+	0.804504i	−0.999178	−	0.0405396i	$-0.987092\pi$
$613$	−11.5000	+	19.9186i	−0.464481	+	0.804504i	0.534697	+	0.845044i	$0.320426\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−32.4037	+	18.7083i	−1.30452	+	0.753167i	−0.981177	−	0.193113i	$-0.938142\pi$
$617$	−32.4037	+	18.7083i	−1.30452	+	0.753167i	−0.323347	+	0.946280i	$0.604808\pi$
$618$		0			0
$619$	−25.0000			−1.00483			−0.502417	−	0.864625i	$-0.667556\pi$
$619$	−25.0000			−1.00483			−0.502417	+	0.864625i	$0.667556\pi$
$620$	12.9615			0.520546
$621$		0			0
$622$		0			0
$623$	−12.9615	−	22.4499i	−0.519291	−	0.899438i
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$	−22.0000			−0.877896
$629$	−19.4422	+	33.6749i	−0.775212	+	1.34271i
$630$		0			0
$631$	−2.50000	−	4.33013i	−0.0995234	−	0.172380i	0.811964	−	0.583707i	$-0.198398\pi$
$631$	−2.50000	−	4.33013i	−0.0995234	−	0.172380i	−0.911487	+	0.411328i	$0.865065\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−51.8459			−2.05744
$636$		0			0
$637$	−13.5000	+	7.79423i	−0.534889	+	0.308819i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	18.5000	−	32.0429i	0.729569	−	1.26365i	−0.227497	−	0.973779i	$-0.573054\pi$
$643$	18.5000	−	32.0429i	0.729569	−	1.26365i	0.957066	−	0.289871i	$-0.0936125\pi$
$644$	−12.9615	−	7.48331i	−0.510754	−	0.294884i
$645$		0			0
$646$		0			0
$647$			18.7083i			0.735499i	0.929925	+	0.367749i	$0.119872\pi$
$647$			18.7083i			0.735499i	−0.929925	+	0.367749i	$0.880128\pi$
$648$		0			0
$649$	42.0000	+	24.2487i	1.64864	+	0.951845i
$650$		0			0
$651$		0			0
$652$	−1.00000	−	1.73205i	−0.0391630	−	0.0678323i
$653$		−	14.9666i		−	0.585689i	−0.956160	−	0.292845i	$-0.905398\pi$
$653$		−	14.9666i		−	0.585689i	0.956160	−	0.292845i	$-0.0946019\pi$
$654$		0			0
$655$	28.0000	+	48.4974i	1.09405	+	1.89495i
$656$	12.9615	+	22.4499i	0.506061	+	0.876523i
$657$		0			0
$658$		0			0
$659$	−3.24037	−	5.61249i	−0.126227	−	0.218631i	0.795985	−	0.605316i	$-0.206953\pi$
$659$	−3.24037	−	5.61249i	−0.126227	−	0.218631i	−0.922212	+	0.386685i	$0.873620\pi$
$660$		0			0
$661$	13.5000	−	7.79423i	0.525089	−	0.303160i	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	13.5000	−	7.79423i	0.525089	−	0.303160i	0.739014	+	0.673690i	$0.235292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−32.4037	+	3.74166i	−1.25656	+	0.145095i
$666$		0			0
$667$	21.0000	+	12.1244i	0.813123	+	0.469457i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	22.6826	−	13.0958i	0.875652	−	0.505558i
$672$		0			0
$673$		−	13.8564i		−	0.534125i	−0.963679	−	0.267063i	$-0.913947\pi$
$673$		−	13.8564i		−	0.534125i	0.963679	−	0.267063i	$-0.0860531\pi$
$674$		0			0
$675$		0			0
$676$	28.0000			1.07692
$677$	−6.48074			−0.249075			−0.124538	−	0.992215i	$-0.539745\pi$
$677$	−6.48074			−0.249075			−0.124538	+	0.992215i	$0.539745\pi$
$678$		0			0
$679$	9.00000	+	5.19615i	0.345388	+	0.199410i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−19.4422			−0.743936			−0.371968	−	0.928246i	$-0.621317\pi$
$683$	−19.4422			−0.743936			−0.371968	+	0.928246i	$0.621317\pi$
$684$		0			0
$685$	−14.0000			−0.534913
$686$		0			0
$687$		0			0
$688$	−10.0000	−	17.3205i	−0.381246	−	0.660338i
$689$	29.1633	+	16.8375i	1.11103	+	0.641456i
$690$		0			0
$691$	50.0000			1.90209			0.951045	−	0.309053i	$-0.100012\pi$
$691$	50.0000			1.90209			0.951045	+	0.309053i	$0.100012\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	52.3832i		−	1.98701i
$696$		0			0
$697$	−42.0000	+	24.2487i	−1.59086	+	0.918485i
$698$		0			0
$699$		0			0
$700$	−18.0000	+	31.1769i	−0.680336	+	1.17838i
$701$	−16.2019	−	9.35414i	−0.611935	−	0.353301i	0.161787	−	0.986826i	$-0.448274\pi$
$701$	−16.2019	−	9.35414i	−0.611935	−	0.353301i	−0.773723	+	0.633525i	$0.781608\pi$
$702$		0			0
$703$	13.5000	+	18.1865i	0.509162	+	0.685918i
$704$			29.9333i			1.12815i
$705$		0			0
$706$		0			0
$707$	−6.48074	+	3.74166i	−0.243733	+	0.140720i
$708$		0			0
$709$	26.0000	+	45.0333i	0.976450	+	1.69126i	0.675063	+	0.737760i	$0.264116\pi$
$709$	26.0000	+	45.0333i	0.976450	+	1.69126i	0.301388	+	0.953502i	$0.402550\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.24037	+	5.61249i	0.121353	+	0.210189i
$714$		0			0
$715$		−	72.7461i		−	2.72055i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	42.1248	+	24.3208i	1.57099	+	0.907012i	0.996048	+	0.0888181i	$0.0283090\pi$
$719$	42.1248	+	24.3208i	1.57099	+	0.907012i	0.574943	+	0.818194i	$0.305024\pi$
$720$		0			0
$721$		−	13.8564i		−	0.516040i
$722$		0			0
$723$		0			0
$724$	39.0000	+	22.5167i	1.44942	+	0.836825i
$725$	29.1633	−	50.5124i	1.08310	−	1.87598i
$726$		0			0
$727$	−17.5000	+	30.3109i	−0.649039	+	1.12417i	0.334314	+	0.942462i	$0.391496\pi$
$727$	−17.5000	+	30.3109i	−0.649039	+	1.12417i	−0.983353	+	0.181707i	$0.941838\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	32.4037	−	18.7083i	1.19849	−	0.691951i
$732$		0			0
$733$	8.00000			0.295487			0.147743	−	0.989026i	$-0.452799\pi$
$733$	8.00000			0.295487			0.147743	+	0.989026i	$0.452799\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−8.50000	+	14.7224i	−0.312678	+	0.541573i	−0.978941	−	0.204143i	$-0.934559\pi$
$739$	−8.50000	+	14.7224i	−0.312678	+	0.541573i	0.666264	+	0.745716i	$0.267893\pi$
$740$	38.8844			1.42942
$741$		0			0
$742$		0			0
$743$	3.24037	−	5.61249i	0.118878	−	0.205902i	−0.800446	−	0.599406i	$-0.795404\pi$
$743$	3.24037	−	5.61249i	0.118878	−	0.205902i	0.919323	+	0.393503i	$0.128737\pi$
$744$		0			0
$745$	28.0000	+	48.4974i	1.02584	+	1.77681i
$746$		0			0
$747$		0			0
$748$	−56.0000			−2.04756
$749$	38.8844			1.42081
$750$		0			0
$751$	4.50000	−	2.59808i	0.164207	−	0.0948051i	−0.415644	−	0.909527i	$-0.636444\pi$
$751$	4.50000	−	2.59808i	0.164207	−	0.0948051i	0.579852	+	0.814722i	$0.303111\pi$
$752$		−	14.9666i		−	0.545777i
$753$		0			0
$754$		0			0
$755$	16.2019	−	28.0624i	0.589646	−	1.02130i
$756$		0			0
$757$	9.50000	−	16.4545i	0.345283	−	0.598048i	−0.640122	−	0.768273i	$-0.721116\pi$
$757$	9.50000	−	16.4545i	0.345283	−	0.598048i	0.985405	+	0.170225i	$0.0544495\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			29.9333i			1.08508i	0.840030	+	0.542540i	$0.182537\pi$
$761$			29.9333i			1.08508i	−0.840030	+	0.542540i	$0.817463\pi$
$762$		0			0
$763$	−6.00000	−	3.46410i	−0.217215	−	0.125409i
$764$	6.48074	−	3.74166i	0.234465	−	0.135368i
$765$		0			0
$766$		0			0
$767$			67.3498i			2.43186i
$768$		0			0
$769$	−23.5000	−	40.7032i	−0.847432	−	1.46779i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−23.5000	−	40.7032i	−0.847432	−	1.46779i	0.0360609	−	0.999350i	$-0.488519\pi$
$770$		0			0
$771$		0			0
$772$			48.4974i			1.74546i
$773$	12.9615	+	22.4499i	0.466192	+	0.807468i	0.999254	−	0.0386076i	$-0.0122922\pi$
$773$	12.9615	+	22.4499i	0.466192	+	0.807468i	−0.533062	+	0.846076i	$0.678959\pi$
$774$		0			0
$775$	13.5000	−	7.79423i	0.484934	−	0.279977i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.24037	+	28.0624i	0.116098	+	1.00544i
$780$		0			0
$781$	21.0000	+	12.1244i	0.751439	+	0.433844i
$782$		0			0
$783$		0			0
$784$	6.00000	−	10.3923i	0.214286	−	0.371154i
$785$	−35.6441	+	20.5791i	−1.27219	+	0.734500i
$786$		0			0
$787$			39.8372i			1.42004i	0.704181	+	0.710021i	$0.251315\pi$
$787$			39.8372i			1.42004i	−0.704181	+	0.710021i	$0.748685\pi$
$788$	6.48074	−	3.74166i	0.230867	−	0.133291i
$789$		0			0
$790$		0			0
$791$	12.9615			0.460857
$792$		0			0
$793$	31.5000	+	18.1865i	1.11860	+	0.645823i
$794$		0			0
$795$		0			0
$796$	−4.00000	+	6.92820i	−0.141776	+	0.245564i
$797$	−51.8459			−1.83648			−0.918238	−	0.396028i	$-0.870388\pi$
$797$	−51.8459			−1.83648			−0.918238	+	0.396028i	$0.870388\pi$
$798$		0			0
$799$	28.0000			0.990569
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−12.9615	−	7.48331i	−0.457401	−	0.264080i
$804$		0			0
$805$	−28.0000			−0.986870
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	37.4166i		−	1.31550i	−0.753238	−	0.657748i	$-0.771509\pi$
$809$		−	37.4166i		−	1.31550i	0.753238	−	0.657748i	$-0.228491\pi$
$810$		0			0
$811$	−12.0000	+	6.92820i	−0.421377	+	0.243282i	−0.695666	−	0.718365i	$-0.744891\pi$
$811$	−12.0000	+	6.92820i	−0.421377	+	0.243282i	0.274289	+	0.961647i	$0.411557\pi$
$812$	12.9615	−	22.4499i	0.454859	−	0.787839i
$813$		0			0
$814$		0			0
$815$	−3.24037	−	1.87083i	−0.113505	−	0.0655323i
$816$		0			0
$817$	−2.50000	−	21.6506i	−0.0874639	−	0.757460i
$818$		0			0
$819$		0			0
$820$	42.0000	+	24.2487i	1.46670	+	0.846802i
$821$	−32.4037	+	18.7083i	−1.13090	+	0.652924i	−0.944160	−	0.329486i	$-0.893124\pi$
$821$	−32.4037	+	18.7083i	−1.13090	+	0.652924i	−0.186737	+	0.982410i	$0.559791\pi$
$822$		0			0
$823$	−11.5000	−	19.9186i	−0.400865	−	0.694318i	0.592966	−	0.805228i	$-0.297957\pi$
$823$	−11.5000	−	19.9186i	−0.400865	−	0.694318i	−0.993831	+	0.110910i	$0.964624\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−16.2019	−	28.0624i	−0.563394	−	0.975826i	−0.997197	−	0.0748188i	$-0.976162\pi$
$827$	−16.2019	−	28.0624i	−0.563394	−	0.975826i	0.433804	−	0.901007i	$-0.357171\pi$
$828$		0			0
$829$		−	10.3923i		−	0.360940i	−0.983581	−	0.180470i	$-0.942238\pi$
$829$		−	10.3923i		−	0.360940i	0.983581	−	0.180470i	$-0.0577618\pi$
$830$		0			0
$831$		0			0
$832$	−36.0000	+	20.7846i	−1.24808	+	0.720577i
$833$	19.4422	+	11.2250i	0.673633	+	0.388922i
$834$		0			0
$835$		0			0
$836$	−12.9615	+	29.9333i	−0.448282	+	1.03526i
$837$		0			0
$838$		0			0
$839$	9.72111	−	16.8375i	0.335610	−	0.581294i	−0.647992	−	0.761647i	$-0.724391\pi$
$839$	9.72111	−	16.8375i	0.335610	−	0.581294i	0.983602	+	0.180354i	$0.0577242\pi$
$840$		0			0
$841$	−6.50000	+	11.2583i	−0.224138	+	0.388218i
$842$		0			0
$843$		0			0
$844$			27.7128i			0.953914i
$845$	45.3652	−	26.1916i	1.56061	−	0.901018i
$846$		0			0
$847$	−6.00000			−0.206162
$848$	−25.9230			−0.890198
$849$		0			0
$850$		0			0
$851$	9.72111	+	16.8375i	0.333235	+	0.577181i
$852$		0			0
$853$	20.0000	−	34.6410i	0.684787	−	1.18609i	−0.288717	−	0.957415i	$-0.593229\pi$
$853$	20.0000	−	34.6410i	0.684787	−	1.18609i	0.973504	−	0.228671i	$-0.0734381\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	3.50000	+	6.06218i	0.119418	+	0.206839i	0.919537	−	0.393003i	$-0.128564\pi$
$859$	3.50000	+	6.06218i	0.119418	+	0.206839i	−0.800119	+	0.599841i	$0.795230\pi$
$860$	−32.4037	−	18.7083i	−1.10496	−	0.637947i
$861$		0			0
$862$		0			0
$863$	−38.8844			−1.32364			−0.661821	−	0.749662i	$-0.730216\pi$
$863$	−38.8844			−1.32364			−0.661821	+	0.749662i	$0.730216\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	6.00000	−	3.46410i	0.203653	−	0.117579i
$869$	−9.72111	+	16.8375i	−0.329766	+	0.571172i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$			29.9333i			1.01193i
$876$		0			0
$877$	−10.5000	−	6.06218i	−0.354560	−	0.204705i	0.312132	−	0.950039i	$-0.398957\pi$
$877$	−10.5000	−	6.06218i	−0.354560	−	0.204705i	−0.666692	+	0.745334i	$0.732290\pi$
$878$		0			0
$879$		0			0
$880$	28.0000	+	48.4974i	0.943880	+	1.63485i
$881$		−	14.9666i		−	0.504239i	−0.967696	−	0.252119i	$-0.918872\pi$
$881$		−	14.9666i		−	0.504239i	0.967696	−	0.252119i	$-0.0811275\pi$
$882$		0			0
$883$	−20.5000	−	35.5070i	−0.689880	−	1.19491i	−0.971876	−	0.235492i	$-0.924330\pi$
$883$	−20.5000	−	35.5070i	−0.689880	−	1.19491i	0.281996	−	0.959415i	$-0.409003\pi$
$884$	−38.8844	−	67.3498i	−1.30783	−	2.26522i
$885$		0			0
$886$		0			0
$887$	29.1633	+	50.5124i	0.979209	+	1.69604i	0.665282	+	0.746593i	$0.268311\pi$
$887$	29.1633	+	50.5124i	0.979209	+	1.69604i	0.313927	+	0.949447i	$0.398355\pi$
$888$		0			0
$889$	−24.0000	+	13.8564i	−0.804934	+	0.464729i
$890$		0			0
$891$		0			0
$892$		−	10.3923i		−	0.347960i
$893$	6.48074	−	14.9666i	0.216870	−	0.500839i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9.72111	+	5.61249i	−0.324217	+	0.187187i
$900$		0			0
$901$		−	48.4974i		−	1.61568i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	84.2496			2.80055
$906$		0			0
$907$	−13.5000	−	7.79423i	−0.448260	−	0.258803i	0.258835	−	0.965922i	$-0.416661\pi$
$907$	−13.5000	−	7.79423i	−0.448260	−	0.258803i	−0.707095	+	0.707118i	$0.749995\pi$
$908$	−6.48074	−	11.2250i	−0.215071	−	0.372514i
$909$		0			0
$910$		0			0
$911$	−25.9230			−0.858866			−0.429433	−	0.903099i	$-0.641287\pi$
$911$	−25.9230			−0.858866			−0.429433	+	0.903099i	$0.641287\pi$
$912$		0			0
$913$	28.0000			0.926665
$914$		0			0
$915$		0			0
$916$	−1.00000	−	1.73205i	−0.0330409	−	0.0572286i
$917$	25.9230	+	14.9666i	0.856052	+	0.494242i
$918$		0			0
$919$	17.0000			0.560778			0.280389	−	0.959886i	$-0.409536\pi$
$919$	17.0000			0.560778			0.280389	+	0.959886i	$0.409536\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			33.6749i			1.10842i
$924$		0			0
$925$	40.5000	−	23.3827i	1.33163	−	0.768818i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−35.6441	−	20.5791i	−1.16944	−	0.675179i	−0.215895	−	0.976417i	$-0.569267\pi$
$929$	−35.6441	−	20.5791i	−1.16944	−	0.675179i	−0.953549	+	0.301238i	$0.902600\pi$
$930$		0			0
$931$	10.5000	−	7.79423i	0.344124	−	0.255446i
$932$		−	14.9666i		−	0.490248i
$933$		0			0
$934$		0			0
$935$	−90.7304	+	52.3832i	−2.96720	+	1.71311i
$936$		0			0
$937$	18.5000	+	32.0429i	0.604369	+	1.04680i	0.992151	+	0.125046i	$0.0399079\pi$
$937$	18.5000	+	32.0429i	0.604369	+	1.04680i	−0.387782	+	0.921751i	$0.626759\pi$
$938$		0			0
$939$		0			0
$940$	−14.0000	−	24.2487i	−0.456630	−	0.790906i
$941$	−29.1633	−	50.5124i	−0.950697	−	1.64666i	−0.743920	−	0.668268i	$-0.767036\pi$
$941$	−29.1633	−	50.5124i	−0.950697	−	1.64666i	−0.206777	−	0.978388i	$-0.566298\pi$
$942$		0			0
$943$			24.2487i			0.789647i
$944$	−25.9230	−	44.8999i	−0.843721	−	1.46137i
$945$		0			0
$946$		0			0
$947$	−6.48074	−	3.74166i	−0.210596	−	0.121588i	0.390992	−	0.920394i	$-0.372132\pi$
$947$	−6.48074	−	3.74166i	−0.210596	−	0.121588i	−0.601588	+	0.798806i	$0.705465\pi$
$948$		0			0
$949$		−	20.7846i		−	0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−3.24037	+	5.61249i	−0.104966	+	0.181806i	−0.913724	−	0.406335i	$-0.866807\pi$
$953$	−3.24037	+	5.61249i	−0.104966	+	0.181806i	0.808758	+	0.588141i	$0.200140\pi$
$954$		0			0
$955$	7.00000	−	12.1244i	0.226515	−	0.392335i
$956$	25.9230	−	14.9666i	0.838409	−	0.484055i
$957$		0			0
$958$		0			0
$959$	−6.48074	+	3.74166i	−0.209274	+	0.120824i
$960$		0			0
$961$	28.0000			0.903226
$962$		0			0
$963$		0			0
$964$	24.0000	+	13.8564i	0.772988	+	0.446285i
$965$	45.3652	+	78.5748i	1.46036	+	2.52941i
$966$		0			0
$967$	−13.0000	+	22.5167i	−0.418052	+	0.724087i	−0.995743	−	0.0921681i	$-0.970620\pi$
$967$	−13.0000	+	22.5167i	−0.418052	+	0.724087i	0.577692	+	0.816255i	$0.303954\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.24037	+	5.61249i	−0.103988	+	0.180113i	−0.913324	−	0.407233i	$-0.866494\pi$
$971$	−3.24037	+	5.61249i	−0.103988	+	0.180113i	0.809336	+	0.587346i	$0.199827\pi$
$972$		0			0
$973$	−14.0000	−	24.2487i	−0.448819	−	0.777378i
$974$		0			0
$975$		0			0
$976$	−28.0000			−0.896258
$977$	45.3652			1.45136			0.725680	−	0.688032i	$-0.241525\pi$
$977$	45.3652			1.45136			0.725680	+	0.688032i	$0.241525\pi$
$978$		0			0
$979$	−42.0000	+	24.2487i	−1.34233	+	0.774992i
$980$		−	22.4499i		−	0.717137i
$981$		0			0
$982$		0			0
$983$	19.4422	−	33.6749i	0.620111	−	1.07406i	−0.369354	−	0.929289i	$-0.620421\pi$
$983$	19.4422	−	33.6749i	0.620111	−	1.07406i	0.989465	−	0.144774i	$-0.0462456\pi$
$984$		0			0
$985$	7.00000	−	12.1244i	0.223039	−	0.386314i
$986$		0			0
$987$		0			0
$988$	−45.0000	+	5.19615i	−1.43164	+	0.165312i
$989$		−	18.7083i		−	0.594889i
$990$		0			0
$991$	−34.5000	−	19.9186i	−1.09593	−	0.632735i	−0.160780	−	0.986990i	$-0.551401\pi$
$991$	−34.5000	−	19.9186i	−1.09593	−	0.632735i	−0.935149	+	0.354256i	$0.884734\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			14.9666i			0.474474i
$996$		0			0
$997$	24.5000	+	42.4352i	0.775923	+	1.34394i	0.934274	+	0.356555i	$0.116049\pi$
$997$	24.5000	+	42.4352i	0.775923	+	1.34394i	−0.158352	+	0.987383i	$0.550618\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.m.d.350.2	yes	4
3.2	odd	2	inner	513.2.m.d.350.1	yes	4
19.12	odd	6	inner	513.2.m.d.107.1	✓	4
57.50	even	6	inner	513.2.m.d.107.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.m.d.107.1	✓	4	19.12	odd	6	inner
513.2.m.d.107.2	yes	4	57.50	even	6	inner
513.2.m.d.350.1	yes	4	3.2	odd	2	inner
513.2.m.d.350.2	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.m (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{4} +(3.24037 + 1.87083i) q^{5} +2.00000 q^{7} -3.74166i q^{11} +(4.50000 - 2.59808i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(-6.48074 - 3.74166i) q^{17} +(-3.50000 + 2.59808i) q^{19} +7.48331i q^{20} +(-3.24037 + 1.87083i) q^{23} +(4.50000 + 7.79423i) q^{25} +(2.00000 + 3.46410i) q^{28} +(-3.24037 - 5.61249i) q^{29} -1.73205i q^{31} +(6.48074 + 3.74166i) q^{35} -5.19615i q^{37} +(3.24037 - 5.61249i) q^{41} +(-2.50000 + 4.33013i) q^{43} +(6.48074 - 3.74166i) q^{44} +(-3.24037 + 1.87083i) q^{47} -3.00000 q^{49} +(9.00000 + 5.19615i) q^{52} +(3.24037 + 5.61249i) q^{53} +(7.00000 - 12.1244i) q^{55} +(-6.48074 + 11.2250i) q^{59} +(3.50000 + 6.06218i) q^{61} -8.00000 q^{64} +19.4422 q^{65} -14.9666i q^{68} +(-3.24037 + 5.61249i) q^{71} +(2.00000 - 3.46410i) q^{73} +(-8.00000 - 3.46410i) q^{76} -7.48331i q^{77} +(-4.50000 - 2.59808i) q^{79} +(-12.9615 + 7.48331i) q^{80} +7.48331i q^{83} +(-14.0000 - 24.2487i) q^{85} +(-6.48074 - 11.2250i) q^{89} +(9.00000 - 5.19615i) q^{91} +(-6.48074 - 3.74166i) q^{92} +(-16.2019 + 1.87083i) q^{95} +(4.50000 + 2.59808i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 8 q^{7} + 18 q^{13} - 8 q^{16} - 14 q^{19} + 18 q^{25} + 8 q^{28} - 10 q^{43} - 12 q^{49} + 36 q^{52} + 28 q^{55} + 14 q^{61} - 32 q^{64} + 8 q^{73} - 32 q^{76} - 18 q^{79} - 56 q^{85} + 36 q^{91}+ \cdots + 18 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 + 8 * q^7 + 18 * q^13 - 8 * q^16 - 14 * q^19 + 18 * q^25 + 8 * q^28 - 10 * q^43 - 12 * q^49 + 36 * q^52 + 28 * q^55 + 14 * q^61 - 32 * q^64 + 8 * q^73 - 32 * q^76 - 18 * q^79 - 56 * q^85 + 36 * q^91 + 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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