LMFDB - Embedded newform 1520.2.bq.i.31.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1520,2,Mod(31,1520)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1520, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1520.31"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.bq (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,-1,0,0,0,-1,0,0,0,6,0,2,0,6,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$12.1372611072$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1520.31
Dual form			1520.2.bq.i.1471.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.73205i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{9} -1.73205i q^{11} +(3.00000 + 1.73205i) q^{13} +(1.00000 - 1.73205i) q^{15} +(3.00000 + 5.19615i) q^{17} +(0.500000 - 4.33013i) q^{19} +(-0.500000 + 0.866025i) q^{25} +4.00000 q^{27} +(1.50000 + 0.866025i) q^{29} -5.00000 q^{31} +(3.00000 - 1.73205i) q^{33} +6.92820i q^{37} +6.92820i q^{39} +(6.00000 - 3.46410i) q^{41} +(6.00000 - 3.46410i) q^{43} +1.00000 q^{45} +(3.00000 + 1.73205i) q^{47} +7.00000 q^{49} +(-6.00000 + 10.3923i) q^{51} +(3.00000 + 1.73205i) q^{53} +(-1.50000 + 0.866025i) q^{55} +(8.00000 - 3.46410i) q^{57} +(4.50000 + 7.79423i) q^{59} +(-0.500000 + 0.866025i) q^{61} -3.46410i q^{65} +(-5.00000 + 8.66025i) q^{67} +(-7.50000 - 12.9904i) q^{71} +(-2.00000 - 3.46410i) q^{73} -2.00000 q^{75} +(0.500000 + 0.866025i) q^{79} +(5.50000 + 9.52628i) q^{81} +6.92820i q^{83} +(3.00000 - 5.19615i) q^{85} +3.46410i q^{87} +(10.5000 + 6.06218i) q^{89} +(-5.00000 - 8.66025i) q^{93} +(-4.00000 + 1.73205i) q^{95} +(-12.0000 + 6.92820i) q^{97} +(1.50000 + 0.866025i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - q^{5} - q^{9} + 6 q^{13} + 2 q^{15} + 6 q^{17} + q^{19} - q^{25} + 8 q^{27} + 3 q^{29} - 10 q^{31} + 6 q^{33} + 12 q^{41} + 12 q^{43} + 2 q^{45} + 6 q^{47} + 14 q^{49} - 12 q^{51} + 6 q^{53}+ \cdots + 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - q^5 - q^9 + 6 * q^13 + 2 * q^15 + 6 * q^17 + q^19 - q^25 + 8 * q^27 + 3 * q^29 - 10 * q^31 + 6 * q^33 + 12 * q^41 + 12 * q^43 + 2 * q^45 + 6 * q^47 + 14 * q^49 - 12 * q^51 + 6 * q^53 - 3 * q^55 + 16 * q^57 + 9 * q^59 - q^61 - 10 * q^67 - 15 * q^71 - 4 * q^73 - 4 * q^75 + q^79 + 11 * q^81 + 6 * q^85 + 21 * q^89 - 10 * q^93 - 8 * q^95 - 24 * q^97 + 3 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.00000	+	1.73205i	0.577350	+	1.00000i	0.995782	+	0.0917517i	$0.0292466\pi$
$3$	1.00000	+	1.73205i	0.577350	+	1.00000i	−0.418432	+	0.908248i	$0.637420\pi$
$4$		0			0
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$		−	1.73205i		−	0.522233i	−0.965307	−	0.261116i	$-0.915909\pi$
$11$		−	1.73205i		−	0.522233i	0.965307	−	0.261116i	$-0.0840907\pi$
$12$		0			0
$13$	3.00000	+	1.73205i	0.832050	+	0.480384i	0.854554	−	0.519362i	$-0.173830\pi$
$13$	3.00000	+	1.73205i	0.832050	+	0.480384i	−0.0225039	+	0.999747i	$0.507164\pi$
$14$		0			0
$15$	1.00000	−	1.73205i	0.258199	−	0.447214i
$16$		0			0
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	0.957892	+	0.287129i	$0.0927008\pi$
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	−0.230285	+	0.973123i	$0.573966\pi$
$18$		0			0
$19$	0.500000	−	4.33013i	0.114708	−	0.993399i
$19$	0.500000	−	4.33013i	0.114708	−	0.993399i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$	4.00000			0.769800
$28$		0			0
$29$	1.50000	+	0.866025i	0.278543	+	0.160817i	0.632764	−	0.774345i	$-0.281920\pi$
$29$	1.50000	+	0.866025i	0.278543	+	0.160817i	−0.354221	+	0.935162i	$0.615254\pi$
$30$		0			0
$31$	−5.00000			−0.898027			−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000			−0.898027			−0.449013	+	0.893525i	$0.648224\pi$
$32$		0			0
$33$	3.00000	−	1.73205i	0.522233	−	0.301511i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			6.92820i			1.13899i	0.821995	+	0.569495i	$0.192861\pi$
$37$			6.92820i			1.13899i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$			6.92820i			1.10940i
$40$		0			0
$41$	6.00000	−	3.46410i	0.937043	−	0.541002i	0.0480106	−	0.998847i	$-0.484712\pi$
$41$	6.00000	−	3.46410i	0.937043	−	0.541002i	0.889032	+	0.457845i	$0.151379\pi$
$42$		0			0
$43$	6.00000	−	3.46410i	0.914991	−	0.528271i	0.0329577	−	0.999457i	$-0.489507\pi$
$43$	6.00000	−	3.46410i	0.914991	−	0.528271i	0.882034	+	0.471186i	$0.156174\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$	3.00000	+	1.73205i	0.437595	+	0.252646i	0.702577	−	0.711608i	$-0.252033\pi$
$47$	3.00000	+	1.73205i	0.437595	+	0.252646i	−0.264982	+	0.964253i	$0.585366\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−6.00000	+	10.3923i	−0.840168	+	1.45521i
$52$		0			0
$53$	3.00000	+	1.73205i	0.412082	+	0.237915i	0.691684	−	0.722200i	$-0.256869\pi$
$53$	3.00000	+	1.73205i	0.412082	+	0.237915i	−0.279602	+	0.960116i	$0.590203\pi$
$54$		0			0
$55$	−1.50000	+	0.866025i	−0.202260	+	0.116775i
$56$		0			0
$57$	8.00000	−	3.46410i	1.05963	−	0.458831i
$58$		0			0
$59$	4.50000	+	7.79423i	0.585850	+	1.01472i	0.994769	+	0.102151i	$0.0325726\pi$
$59$	4.50000	+	7.79423i	0.585850	+	1.01472i	−0.408919	+	0.912571i	$0.634094\pi$
$60$		0			0
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	0.832240	+	0.554416i	$0.187058\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	3.46410i		−	0.429669i
$66$		0			0
$67$	−5.00000	+	8.66025i	−0.610847	+	1.05802i	0.380251	+	0.924883i	$0.375838\pi$
$67$	−5.00000	+	8.66025i	−0.610847	+	1.05802i	−0.991098	+	0.133135i	$0.957496\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.50000	−	12.9904i	−0.890086	−	1.54167i	−0.839771	−	0.542941i	$-0.817311\pi$
$71$	−7.50000	−	12.9904i	−0.890086	−	1.54167i	−0.0503155	−	0.998733i	$-0.516023\pi$
$72$		0			0
$73$	−2.00000	−	3.46410i	−0.234082	−	0.405442i	0.724923	−	0.688830i	$-0.241875\pi$
$73$	−2.00000	−	3.46410i	−0.234082	−	0.405442i	−0.959006	+	0.283387i	$0.908542\pi$
$74$		0			0
$75$	−2.00000			−0.230940
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	0.892781	−	0.450490i	$-0.148751\pi$
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	−0.836527	+	0.547926i	$0.815418\pi$
$80$		0			0
$81$	5.50000	+	9.52628i	0.611111	+	1.05848i
$82$		0			0
$83$			6.92820i			0.760469i	0.924890	+	0.380235i	$0.124157\pi$
$83$			6.92820i			0.760469i	−0.924890	+	0.380235i	$0.875843\pi$
$84$		0			0
$85$	3.00000	−	5.19615i	0.325396	−	0.563602i
$86$		0			0
$87$			3.46410i			0.371391i
$88$		0			0
$89$	10.5000	+	6.06218i	1.11300	+	0.642590i	0.939604	−	0.342263i	$-0.111193\pi$
$89$	10.5000	+	6.06218i	1.11300	+	0.642590i	0.173394	+	0.984853i	$0.444527\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−5.00000	−	8.66025i	−0.518476	−	0.898027i
$94$		0			0
$95$	−4.00000	+	1.73205i	−0.410391	+	0.177705i
$96$		0			0
$97$	−12.0000	+	6.92820i	−1.21842	+	0.703452i	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	−12.0000	+	6.92820i	−1.21842	+	0.703452i	−0.253837	+	0.967247i	$0.581693\pi$
$98$		0			0
$99$	1.50000	+	0.866025i	0.150756	+	0.0870388i
$100$		0			0
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	−0.781697	−	0.623658i	$-0.785646\pi$
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	0.930953	+	0.365140i	$0.118979\pi$
$102$		0			0
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	−13.5000	+	7.79423i	−1.29307	+	0.746552i	−0.979196	−	0.202915i	$-0.934959\pi$
$109$	−13.5000	+	7.79423i	−1.29307	+	0.746552i	−0.313869	+	0.949466i	$0.601625\pi$
$110$		0			0
$111$	−12.0000	+	6.92820i	−1.13899	+	0.657596i
$112$		0			0
$113$		−	17.3205i		−	1.62938i	−0.579899	−	0.814688i	$-0.696908\pi$
$113$		−	17.3205i		−	1.62938i	0.579899	−	0.814688i	$-0.303092\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.00000	+	1.73205i	−0.277350	+	0.160128i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.00000			0.727273
$122$		0			0
$123$	12.0000	+	6.92820i	1.08200	+	0.624695i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	1.00000	−	1.73205i	0.0887357	−	0.153695i	−0.818241	−	0.574875i	$-0.805051\pi$
$127$	1.00000	−	1.73205i	0.0887357	−	0.153695i	0.906977	+	0.421180i	$0.138384\pi$
$128$		0			0
$129$	12.0000	+	6.92820i	1.05654	+	0.609994i
$130$		0			0
$131$	−9.00000	+	5.19615i	−0.786334	+	0.453990i	−0.838670	−	0.544640i	$-0.816666\pi$
$131$	−9.00000	+	5.19615i	−0.786334	+	0.453990i	0.0523366	+	0.998630i	$0.483333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−2.00000	−	3.46410i	−0.172133	−	0.298142i
$136$		0			0
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	−0.487278	−	0.873247i	$-0.662010\pi$
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	0.999893	−	0.0146279i	$-0.00465636\pi$
$138$		0			0
$139$	−9.00000	−	5.19615i	−0.763370	−	0.440732i	0.0671344	−	0.997744i	$-0.478614\pi$
$139$	−9.00000	−	5.19615i	−0.763370	−	0.440732i	−0.830504	+	0.557012i	$0.811948\pi$
$140$		0			0
$141$			6.92820i			0.583460i
$142$		0			0
$143$	3.00000	−	5.19615i	0.250873	−	0.434524i
$144$		0			0
$145$		−	1.73205i		−	0.143839i
$146$		0			0
$147$	7.00000	+	12.1244i	0.577350	+	1.00000i
$148$		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
$151$	−19.0000			−1.54620			−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000			−1.54620			−0.773099	+	0.634285i	$0.781294\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$	2.50000	+	4.33013i	0.200805	+	0.347804i
$156$		0			0
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	0.997609	+	0.0691164i	$0.0220180\pi$
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$			6.92820i			0.549442i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	3.46410i		−	0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$		−	3.46410i		−	0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$		0			0
$165$	−3.00000	−	1.73205i	−0.233550	−	0.134840i
$166$		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
$169$	−0.500000	−	0.866025i	−0.0384615	−	0.0666173i
$170$		0			0
$171$	3.50000	+	2.59808i	0.267652	+	0.198680i
$172$		0			0
$173$	−12.0000	+	6.92820i	−0.912343	+	0.526742i	−0.881184	−	0.472773i	$-0.843253\pi$
$173$	−12.0000	+	6.92820i	−0.912343	+	0.526742i	−0.0311588	+	0.999514i	$0.509920\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−9.00000	+	15.5885i	−0.676481	+	1.17170i
$178$		0			0
$179$	9.00000			0.672692			0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000			0.672692			0.336346	+	0.941739i	$0.390809\pi$
$180$		0			0
$181$	12.0000	+	6.92820i	0.891953	+	0.514969i	0.874581	−	0.484880i	$-0.161137\pi$
$181$	12.0000	+	6.92820i	0.891953	+	0.514969i	0.0173722	+	0.999849i	$0.494470\pi$
$182$		0			0
$183$	−2.00000			−0.147844
$184$		0			0
$185$	6.00000	−	3.46410i	0.441129	−	0.254686i
$186$		0			0
$187$	9.00000	−	5.19615i	0.658145	−	0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.1244i		−	0.877288i	−0.898661	−	0.438644i	$-0.855459\pi$
$191$		−	12.1244i		−	0.877288i	0.898661	−	0.438644i	$-0.144541\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$	6.00000	−	3.46410i	0.429669	−	0.248069i
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	−4.50000	−	2.59808i	−0.318997	−	0.184173i	0.331949	−	0.943297i	$-0.392294\pi$
$199$	−4.50000	−	2.59808i	−0.318997	−	0.184173i	−0.650945	+	0.759125i	$0.725627\pi$
$200$		0			0
$201$	−20.0000			−1.41069
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−6.00000	−	3.46410i	−0.419058	−	0.241943i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.50000	−	0.866025i	−0.518786	−	0.0599042i
$210$		0			0
$211$	−0.500000	−	0.866025i	−0.0344214	−	0.0596196i	0.848301	−	0.529514i	$-0.177626\pi$
$211$	−0.500000	−	0.866025i	−0.0344214	−	0.0596196i	−0.882723	+	0.469894i	$0.844292\pi$
$212$		0			0
$213$	15.0000	−	25.9808i	1.02778	−	1.78017i
$214$		0			0
$215$	−6.00000	−	3.46410i	−0.409197	−	0.236250i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	4.00000	−	6.92820i	0.270295	−	0.468165i
$220$		0			0
$221$			20.7846i			1.39812i
$222$		0			0
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	0.700449	−	0.713702i	$-0.252983\pi$
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	−0.968309	+	0.249756i	$0.919650\pi$
$224$		0			0
$225$	−0.500000	−	0.866025i	−0.0333333	−	0.0577350i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	−13.0000			−0.859064			−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000			−0.859064			−0.429532	+	0.903052i	$0.641321\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.00000	−	15.5885i	−0.589610	−	1.02123i	−0.994283	−	0.106773i	$-0.965948\pi$
$233$	−9.00000	−	15.5885i	−0.589610	−	1.02123i	0.404674	−	0.914461i	$-0.367385\pi$
$234$		0			0
$235$		−	3.46410i		−	0.225973i
$236$		0			0
$237$	−1.00000	+	1.73205i	−0.0649570	+	0.112509i
$238$		0			0
$239$		−	8.66025i		−	0.560185i	−0.959973	−	0.280093i	$-0.909635\pi$
$239$		−	8.66025i		−	0.560185i	0.959973	−	0.280093i	$-0.0903652\pi$
$240$		0			0
$241$	4.50000	+	2.59808i	0.289870	+	0.167357i	0.637883	−	0.770133i	$-0.279810\pi$
$241$	4.50000	+	2.59808i	0.289870	+	0.167357i	−0.348013	+	0.937490i	$0.613143\pi$
$242$		0			0
$243$	−5.00000	+	8.66025i	−0.320750	+	0.555556i
$244$		0			0
$245$	−3.50000	−	6.06218i	−0.223607	−	0.387298i
$246$		0			0
$247$	9.00000	−	12.1244i	0.572656	−	0.771454i
$248$		0			0
$249$	−12.0000	+	6.92820i	−0.760469	+	0.439057i
$250$		0			0
$251$	4.50000	+	2.59808i	0.284037	+	0.163989i	0.635250	−	0.772307i	$-0.280897\pi$
$251$	4.50000	+	2.59808i	0.284037	+	0.163989i	−0.351212	+	0.936296i	$0.614230\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	12.0000			0.751469
$256$		0			0
$257$	9.00000	+	5.19615i	0.561405	+	0.324127i	0.753709	−	0.657208i	$-0.228263\pi$
$257$	9.00000	+	5.19615i	0.561405	+	0.324127i	−0.192304	+	0.981335i	$0.561596\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−1.50000	+	0.866025i	−0.0928477	+	0.0536056i
$262$		0			0
$263$	21.0000	−	12.1244i	1.29492	−	0.747620i	0.315394	−	0.948961i	$-0.397863\pi$
$263$	21.0000	−	12.1244i	1.29492	−	0.747620i	0.979521	+	0.201341i	$0.0645299\pi$
$264$		0			0
$265$		−	3.46410i		−	0.212798i
$266$		0			0
$267$			24.2487i			1.48400i
$268$		0			0
$269$	13.5000	−	7.79423i	0.823110	−	0.475223i	−0.0283781	−	0.999597i	$-0.509034\pi$
$269$	13.5000	−	7.79423i	0.823110	−	0.475223i	0.851488	+	0.524375i	$0.175701\pi$
$270$		0			0
$271$	16.5000	−	9.52628i	1.00230	−	0.578680i	0.0933746	−	0.995631i	$-0.470235\pi$
$271$	16.5000	−	9.52628i	1.00230	−	0.578680i	0.908929	+	0.416951i	$0.136901\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.50000	+	0.866025i	0.0904534	+	0.0522233i
$276$		0			0
$277$	−16.0000			−0.961347			−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000			−0.961347			−0.480673	+	0.876900i	$0.659608\pi$
$278$		0			0
$279$	2.50000	−	4.33013i	0.149671	−	0.259238i
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	27.0000	−	15.5885i	1.60498	−	0.926638i	0.614514	−	0.788906i	$-0.289352\pi$
$283$	27.0000	−	15.5885i	1.60498	−	0.926638i	0.990470	−	0.137732i	$-0.0439811\pi$
$284$		0			0
$285$	−7.00000	−	5.19615i	−0.414644	−	0.307794i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$		0			0
$291$	−24.0000	−	13.8564i	−1.40690	−	0.812277i
$292$		0			0
$293$		−	6.92820i		−	0.404750i	−0.979308	−	0.202375i	$-0.935134\pi$
$293$		−	6.92820i		−	0.404750i	0.979308	−	0.202375i	$-0.0648660\pi$
$294$		0			0
$295$	4.50000	−	7.79423i	0.262000	−	0.453798i
$296$		0			0
$297$		−	6.92820i		−	0.402015i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	6.00000			0.344691
$304$		0			0
$305$	1.00000			0.0572598
$306$		0			0
$307$	−7.00000	−	12.1244i	−0.399511	−	0.691974i	0.594154	−	0.804351i	$-0.297487\pi$
$307$	−7.00000	−	12.1244i	−0.399511	−	0.691974i	−0.993666	+	0.112377i	$0.964153\pi$
$308$		0			0
$309$	−4.00000	−	6.92820i	−0.227552	−	0.394132i
$310$		0			0
$311$			3.46410i			0.196431i	0.995165	+	0.0982156i	$0.0313135\pi$
$311$			3.46410i			0.196431i	−0.995165	+	0.0982156i	$0.968687\pi$
$312$		0			0
$313$	13.0000	−	22.5167i	0.734803	−	1.27272i	−0.220006	−	0.975499i	$-0.570608\pi$
$313$	13.0000	−	22.5167i	0.734803	−	1.27272i	0.954810	−	0.297218i	$-0.0960589\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.0000	+	10.3923i	1.01098	+	0.583690i	0.911479	−	0.411347i	$-0.134942\pi$
$317$	18.0000	+	10.3923i	1.01098	+	0.583690i	0.0995021	+	0.995037i	$0.468275\pi$
$318$		0			0
$319$	1.50000	−	2.59808i	0.0839839	−	0.145464i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	24.0000	−	10.3923i	1.33540	−	0.578243i
$324$		0			0
$325$	−3.00000	+	1.73205i	−0.166410	+	0.0960769i
$326$		0			0
$327$	−27.0000	−	15.5885i	−1.49310	−	0.862044i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$		0			0
$333$	−6.00000	−	3.46410i	−0.328798	−	0.189832i
$334$		0			0
$335$	10.0000			0.546358
$336$		0			0
$337$	−18.0000	+	10.3923i	−0.980522	+	0.566105i	−0.902428	−	0.430841i	$-0.858217\pi$
$337$	−18.0000	+	10.3923i	−0.980522	+	0.566105i	−0.0780946	+	0.996946i	$0.524884\pi$
$338$		0			0
$339$	30.0000	−	17.3205i	1.62938	−	0.940721i
$340$		0			0
$341$			8.66025i			0.468979i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.0000	+	10.3923i	−0.966291	+	0.557888i	−0.898103	−	0.439784i	$-0.855055\pi$
$347$	−18.0000	+	10.3923i	−0.966291	+	0.557888i	−0.0681872	+	0.997673i	$0.521722\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$	12.0000	+	6.92820i	0.640513	+	0.369800i
$352$		0			0
$353$	−24.0000			−1.27739			−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000			−1.27739			−0.638696	+	0.769460i	$0.720526\pi$
$354$		0			0
$355$	−7.50000	+	12.9904i	−0.398059	+	0.689458i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−27.0000	+	15.5885i	−1.42501	+	0.822727i	−0.996721	−	0.0809166i	$-0.974215\pi$
$359$	−27.0000	+	15.5885i	−1.42501	+	0.822727i	−0.428285	+	0.903644i	$0.640882\pi$
$360$		0			0
$361$	−18.5000	−	4.33013i	−0.973684	−	0.227901i
$362$		0			0
$363$	8.00000	+	13.8564i	0.419891	+	0.727273i
$364$		0			0
$365$	−2.00000	+	3.46410i	−0.104685	+	0.181319i
$366$		0			0
$367$	−30.0000	−	17.3205i	−1.56599	−	0.904123i	−0.996630	−	0.0820332i	$-0.973859\pi$
$367$	−30.0000	−	17.3205i	−1.56599	−	0.904123i	−0.569358	−	0.822090i	$-0.692808\pi$
$368$		0			0
$369$			6.92820i			0.360668i
$370$		0			0
$371$		0			0
$372$		0			0
$373$			13.8564i			0.717458i	0.933442	+	0.358729i	$0.116790\pi$
$373$			13.8564i			0.717458i	−0.933442	+	0.358729i	$0.883210\pi$
$374$		0			0
$375$	1.00000	+	1.73205i	0.0516398	+	0.0894427i
$376$		0			0
$377$	3.00000	+	5.19615i	0.154508	+	0.267615i
$378$		0			0
$379$	−13.0000			−0.667765			−0.333883	−	0.942615i	$-0.608359\pi$
$379$	−13.0000			−0.667765			−0.333883	+	0.942615i	$0.608359\pi$
$380$		0			0
$381$	4.00000			0.204926
$382$		0			0
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	0.779143	−	0.626846i	$-0.215654\pi$
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	−0.932436	+	0.361335i	$0.882321\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			6.92820i			0.352180i
$388$		0			0
$389$	13.5000	−	23.3827i	0.684477	−	1.18555i	−0.289124	−	0.957292i	$-0.593364\pi$
$389$	13.5000	−	23.3827i	0.684477	−	1.18555i	0.973601	−	0.228257i	$-0.0733028\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−18.0000	−	10.3923i	−0.907980	−	0.524222i
$394$		0			0
$395$	0.500000	−	0.866025i	0.0251577	−	0.0435745i
$396$		0			0
$397$	−10.0000	−	17.3205i	−0.501886	−	0.869291i	−0.999998	−	0.00217869i	$-0.999307\pi$
$397$	−10.0000	−	17.3205i	−0.501886	−	0.869291i	0.498112	−	0.867113i	$-0.334027\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−22.5000	+	12.9904i	−1.12360	+	0.648709i	−0.942317	−	0.334723i	$-0.891357\pi$
$401$	−22.5000	+	12.9904i	−1.12360	+	0.648709i	−0.181280	+	0.983432i	$0.558024\pi$
$402$		0			0
$403$	−15.0000	−	8.66025i	−0.747203	−	0.431398i
$404$		0			0
$405$	5.50000	−	9.52628i	0.273297	−	0.473365i
$406$		0			0
$407$	12.0000			0.594818
$408$		0			0
$409$	25.5000	+	14.7224i	1.26089	+	0.727977i	0.973247	−	0.229759i	$-0.0737939\pi$
$409$	25.5000	+	14.7224i	1.26089	+	0.727977i	0.287646	+	0.957737i	$0.407127\pi$
$410$		0			0
$411$	24.0000			1.18383
$412$		0			0
$413$		0			0
$414$		0			0
$415$	6.00000	−	3.46410i	0.294528	−	0.170046i
$416$		0			0
$417$		−	20.7846i		−	1.01783i
$418$		0			0
$419$			12.1244i			0.592314i	0.955139	+	0.296157i	$0.0957051\pi$
$419$			12.1244i			0.592314i	−0.955139	+	0.296157i	$0.904295\pi$
$420$		0			0
$421$	−10.5000	+	6.06218i	−0.511739	+	0.295452i	−0.733548	−	0.679638i	$-0.762137\pi$
$421$	−10.5000	+	6.06218i	−0.511739	+	0.295452i	0.221809	+	0.975090i	$0.428804\pi$
$422$		0			0
$423$	−3.00000	+	1.73205i	−0.145865	+	0.0842152i
$424$		0			0
$425$	−6.00000			−0.291043
$426$		0			0
$427$		0			0
$428$		0			0
$429$	12.0000			0.579365
$430$		0			0
$431$	−10.5000	+	18.1865i	−0.505767	+	0.876014i	0.494211	+	0.869342i	$0.335457\pi$
$431$	−10.5000	+	18.1865i	−0.505767	+	0.876014i	−0.999978	+	0.00667224i	$0.997876\pi$
$432$		0			0
$433$	−30.0000	−	17.3205i	−1.44171	−	0.832370i	−0.443744	−	0.896153i	$-0.646350\pi$
$433$	−30.0000	−	17.3205i	−1.44171	−	0.832370i	−0.997964	+	0.0637830i	$0.979683\pi$
$434$		0			0
$435$	3.00000	−	1.73205i	0.143839	−	0.0830455i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−8.50000	−	14.7224i	−0.405683	−	0.702663i	0.588718	−	0.808339i	$-0.299633\pi$
$439$	−8.50000	−	14.7224i	−0.405683	−	0.702663i	−0.994401	+	0.105675i	$0.966300\pi$
$440$		0			0
$441$	−3.50000	+	6.06218i	−0.166667	+	0.288675i
$442$		0			0
$443$	−6.00000	−	3.46410i	−0.285069	−	0.164584i	0.350647	−	0.936508i	$-0.385962\pi$
$443$	−6.00000	−	3.46410i	−0.285069	−	0.164584i	−0.635716	+	0.771923i	$0.719295\pi$
$444$		0			0
$445$		−	12.1244i		−	0.574750i
$446$		0			0
$447$	15.0000	−	25.9808i	0.709476	−	1.22885i
$448$		0			0
$449$			1.73205i			0.0817405i	0.999164	+	0.0408703i	$0.0130130\pi$
$449$			1.73205i			0.0817405i	−0.999164	+	0.0408703i	$0.986987\pi$
$450$		0			0
$451$	−6.00000	−	10.3923i	−0.282529	−	0.489355i
$452$		0			0
$453$	−19.0000	−	32.9090i	−0.892698	−	1.54620i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−26.0000			−1.21623			−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000			−1.21623			−0.608114	+	0.793849i	$0.708074\pi$
$458$		0			0
$459$	12.0000	+	20.7846i	0.560112	+	0.970143i
$460$		0			0
$461$	10.5000	+	18.1865i	0.489034	+	0.847031i	0.999920	−	0.0126168i	$-0.00401615\pi$
$461$	10.5000	+	18.1865i	0.489034	+	0.847031i	−0.510887	+	0.859648i	$0.670683\pi$
$462$		0			0
$463$			13.8564i			0.643962i	0.946746	+	0.321981i	$0.104349\pi$
$463$			13.8564i			0.643962i	−0.946746	+	0.321981i	$0.895651\pi$
$464$		0			0
$465$	−5.00000	+	8.66025i	−0.231869	+	0.401610i
$466$		0			0
$467$			27.7128i			1.28240i	0.767375	+	0.641198i	$0.221562\pi$
$467$			27.7128i			1.28240i	−0.767375	+	0.641198i	$0.778438\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−14.0000	+	24.2487i	−0.645086	+	1.11732i
$472$		0			0
$473$	−6.00000	−	10.3923i	−0.275880	−	0.477839i
$474$		0			0
$475$	3.50000	+	2.59808i	0.160591	+	0.119208i
$476$		0			0
$477$	−3.00000	+	1.73205i	−0.137361	+	0.0793052i
$478$		0			0
$479$	−13.5000	−	7.79423i	−0.616831	−	0.356127i	0.158803	−	0.987310i	$-0.449236\pi$
$479$	−13.5000	−	7.79423i	−0.616831	−	0.356127i	−0.775634	+	0.631183i	$0.782570\pi$
$480$		0			0
$481$	−12.0000	+	20.7846i	−0.547153	+	0.947697i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.0000	+	6.92820i	0.544892	+	0.314594i
$486$		0			0
$487$	−38.0000			−1.72194			−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000			−1.72194			−0.860972	+	0.508652i	$0.830144\pi$
$488$		0			0
$489$	6.00000	−	3.46410i	0.271329	−	0.156652i
$490$		0			0
$491$	19.5000	−	11.2583i	0.880023	−	0.508081i	0.00935679	−	0.999956i	$-0.497022\pi$
$491$	19.5000	−	11.2583i	0.880023	−	0.508081i	0.870666	+	0.491875i	$0.163688\pi$
$492$		0			0
$493$			10.3923i			0.468046i
$494$		0			0
$495$		−	1.73205i		−	0.0778499i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−21.0000	+	12.1244i	−0.940089	+	0.542761i	−0.889988	−	0.455983i	$-0.849288\pi$
$499$	−21.0000	+	12.1244i	−0.940089	+	0.542761i	−0.0501009	+	0.998744i	$0.515954\pi$
$500$		0			0
$501$	24.0000			1.07224
$502$		0			0
$503$	−3.00000	−	1.73205i	−0.133763	−	0.0772283i	0.431625	−	0.902053i	$-0.357940\pi$
$503$	−3.00000	−	1.73205i	−0.133763	−	0.0772283i	−0.565388	+	0.824825i	$0.691274\pi$
$504$		0			0
$505$	−3.00000			−0.133498
$506$		0			0
$507$	1.00000	−	1.73205i	0.0444116	−	0.0769231i
$508$		0			0
$509$	−24.0000	−	13.8564i	−1.06378	−	0.614174i	−0.137305	−	0.990529i	$-0.543844\pi$
$509$	−24.0000	−	13.8564i	−1.06378	−	0.614174i	−0.926476	+	0.376354i	$0.877178\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	2.00000	−	17.3205i	0.0883022	−	0.764719i
$514$		0			0
$515$	2.00000	+	3.46410i	0.0881305	+	0.152647i
$516$		0			0
$517$	3.00000	−	5.19615i	0.131940	−	0.228527i
$518$		0			0
$519$	−24.0000	−	13.8564i	−1.05348	−	0.608229i
$520$		0			0
$521$			5.19615i			0.227648i	0.993501	+	0.113824i	$0.0363099\pi$
$521$			5.19615i			0.227648i	−0.993501	+	0.113824i	$0.963690\pi$
$522$		0			0
$523$	19.0000	−	32.9090i	0.830812	−	1.43901i	−0.0665832	−	0.997781i	$-0.521210\pi$
$523$	19.0000	−	32.9090i	0.830812	−	1.43901i	0.897395	−	0.441228i	$-0.145457\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−15.0000	−	25.9808i	−0.653410	−	1.13174i
$528$		0			0
$529$	−11.5000	−	19.9186i	−0.500000	−	0.866025i
$530$		0			0
$531$	−9.00000			−0.390567
$532$		0			0
$533$	24.0000			1.03956
$534$		0			0
$535$		0			0
$536$		0			0
$537$	9.00000	+	15.5885i	0.388379	+	0.672692i
$538$		0			0
$539$		−	12.1244i		−	0.522233i
$540$		0			0
$541$	−12.5000	+	21.6506i	−0.537417	+	0.930834i	0.461625	+	0.887075i	$0.347267\pi$
$541$	−12.5000	+	21.6506i	−0.537417	+	0.930834i	−0.999042	+	0.0437584i	$0.986067\pi$
$542$		0			0
$543$			27.7128i			1.18927i
$544$		0			0
$545$	13.5000	+	7.79423i	0.578276	+	0.333868i
$546$		0			0
$547$	−16.0000	+	27.7128i	−0.684111	+	1.18491i	0.289605	+	0.957146i	$0.406476\pi$
$547$	−16.0000	+	27.7128i	−0.684111	+	1.18491i	−0.973715	+	0.227768i	$0.926857\pi$
$548$		0			0
$549$	−0.500000	−	0.866025i	−0.0213395	−	0.0369611i
$550$		0			0
$551$	4.50000	−	6.06218i	0.191706	−	0.258257i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	12.0000	+	6.92820i	0.509372	+	0.294086i
$556$		0			0
$557$	6.00000	−	10.3923i	0.254228	−	0.440336i	−0.710457	−	0.703740i	$-0.751512\pi$
$557$	6.00000	−	10.3923i	0.254228	−	0.440336i	0.964686	+	0.263404i	$0.0848453\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$	18.0000	+	10.3923i	0.759961	+	0.438763i
$562$		0			0
$563$	18.0000			0.758610			0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000			0.758610			0.379305	+	0.925272i	$0.376163\pi$
$564$		0			0
$565$	−15.0000	+	8.66025i	−0.631055	+	0.364340i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	15.5885i		−	0.653502i	−0.945110	−	0.326751i	$-0.894046\pi$
$569$		−	15.5885i		−	0.653502i	0.945110	−	0.326751i	$-0.105954\pi$
$570$		0			0
$571$			32.9090i			1.37720i	0.725143	+	0.688599i	$0.241774\pi$
$571$			32.9090i			1.37720i	−0.725143	+	0.688599i	$0.758226\pi$
$572$		0			0
$573$	21.0000	−	12.1244i	0.877288	−	0.506502i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−8.00000			−0.333044			−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000			−0.333044			−0.166522	+	0.986038i	$0.553254\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	3.00000	−	5.19615i	0.124247	−	0.215203i
$584$		0			0
$585$	3.00000	+	1.73205i	0.124035	+	0.0716115i
$586$		0			0
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$	−2.50000	+	21.6506i	−0.103011	+	0.892099i
$590$		0			0
$591$	6.00000	+	10.3923i	0.246807	+	0.427482i
$592$		0			0
$593$	18.0000	−	31.1769i	0.739171	−	1.28028i	−0.213697	−	0.976900i	$-0.568551\pi$
$593$	18.0000	−	31.1769i	0.739171	−	1.28028i	0.952869	−	0.303383i	$-0.0981160\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	10.3923i		−	0.425329i
$598$		0			0
$599$	−24.0000	+	41.5692i	−0.980613	+	1.69847i	−0.320607	+	0.947212i	$0.603887\pi$
$599$	−24.0000	+	41.5692i	−0.980613	+	1.69847i	−0.660006	+	0.751260i	$0.729446\pi$
$600$		0			0
$601$			39.8372i			1.62499i	0.582967	+	0.812496i	$0.301892\pi$
$601$			39.8372i			1.62499i	−0.582967	+	0.812496i	$0.698108\pi$
$602$		0			0
$603$	−5.00000	−	8.66025i	−0.203616	−	0.352673i
$604$		0			0
$605$	−4.00000	−	6.92820i	−0.162623	−	0.281672i
$606$		0			0
$607$	10.0000			0.405887			0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000			0.405887			0.202944	+	0.979190i	$0.434949\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.00000	+	10.3923i	0.242734	+	0.420428i
$612$		0			0
$613$	13.0000	+	22.5167i	0.525065	+	0.909439i	0.999574	+	0.0291886i	$0.00929235\pi$
$613$	13.0000	+	22.5167i	0.525065	+	0.909439i	−0.474509	+	0.880251i	$0.657374\pi$
$614$		0			0
$615$		−	13.8564i		−	0.558744i
$616$		0			0
$617$	12.0000	−	20.7846i	0.483102	−	0.836757i	−0.516710	−	0.856161i	$-0.672843\pi$
$617$	12.0000	−	20.7846i	0.483102	−	0.836757i	0.999812	+	0.0194037i	$0.00617676\pi$
$618$		0			0
$619$			3.46410i			0.139234i	0.997574	+	0.0696170i	$0.0221777\pi$
$619$			3.46410i			0.139234i	−0.997574	+	0.0696170i	$0.977822\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$	−6.00000	−	13.8564i	−0.239617	−	0.553372i
$628$		0			0
$629$	−36.0000	+	20.7846i	−1.43541	+	0.828737i
$630$		0			0
$631$	19.5000	+	11.2583i	0.776283	+	0.448187i	0.835111	−	0.550081i	$-0.185403\pi$
$631$	19.5000	+	11.2583i	0.776283	+	0.448187i	−0.0588285	+	0.998268i	$0.518737\pi$
$632$		0			0
$633$	1.00000	−	1.73205i	0.0397464	−	0.0688428i
$634$		0			0
$635$	−2.00000			−0.0793676
$636$		0			0
$637$	21.0000	+	12.1244i	0.832050	+	0.480384i
$638$		0			0
$639$	15.0000			0.593391
$640$		0			0
$641$	34.5000	−	19.9186i	1.36267	−	0.786737i	0.372690	−	0.927956i	$-0.378436\pi$
$641$	34.5000	−	19.9186i	1.36267	−	0.786737i	0.989978	+	0.141219i	$0.0451022\pi$
$642$		0			0
$643$	−15.0000	+	8.66025i	−0.591542	+	0.341527i	−0.765707	−	0.643189i	$-0.777611\pi$
$643$	−15.0000	+	8.66025i	−0.591542	+	0.341527i	0.174165	+	0.984717i	$0.444277\pi$
$644$		0			0
$645$		−	13.8564i		−	0.545595i
$646$		0			0
$647$			17.3205i			0.680939i	0.940255	+	0.340470i	$0.110586\pi$
$647$			17.3205i			0.680939i	−0.940255	+	0.340470i	$0.889414\pi$
$648$		0			0
$649$	13.5000	−	7.79423i	0.529921	−	0.305950i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	36.0000			1.40879			0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000			1.40879			0.704394	+	0.709809i	$0.251219\pi$
$654$		0			0
$655$	9.00000	+	5.19615i	0.351659	+	0.203030i
$656$		0			0
$657$	4.00000			0.156055
$658$		0			0
$659$	18.0000	−	31.1769i	0.701180	−	1.21448i	−0.266872	−	0.963732i	$-0.585990\pi$
$659$	18.0000	−	31.1769i	0.701180	−	1.21448i	0.968052	−	0.250748i	$-0.0806766\pi$
$660$		0			0
$661$	25.5000	+	14.7224i	0.991835	+	0.572636i	0.905822	−	0.423658i	$-0.139254\pi$
$661$	25.5000	+	14.7224i	0.991835	+	0.572636i	0.0860127	+	0.996294i	$0.472587\pi$
$662$		0			0
$663$	−36.0000	+	20.7846i	−1.39812	+	0.807207i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	8.00000	−	13.8564i	0.309298	−	0.535720i
$670$		0			0
$671$	1.50000	+	0.866025i	0.0579069	+	0.0334325i
$672$		0			0
$673$			20.7846i			0.801188i	0.916256	+	0.400594i	$0.131196\pi$
$673$			20.7846i			0.801188i	−0.916256	+	0.400594i	$0.868804\pi$
$674$		0			0
$675$	−2.00000	+	3.46410i	−0.0769800	+	0.133333i
$676$		0			0
$677$			20.7846i			0.798817i	0.916773	+	0.399409i	$0.130785\pi$
$677$			20.7846i			0.798817i	−0.916773	+	0.399409i	$0.869215\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−30.0000			−1.14792			−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000			−1.14792			−0.573959	+	0.818884i	$0.694593\pi$
$684$		0			0
$685$	−12.0000			−0.458496
$686$		0			0
$687$	−13.0000	−	22.5167i	−0.495981	−	0.859064i
$688$		0			0
$689$	6.00000	+	10.3923i	0.228582	+	0.395915i
$690$		0			0
$691$		−	15.5885i		−	0.593013i	−0.955031	−	0.296506i	$-0.904178\pi$
$691$		−	15.5885i		−	0.593013i	0.955031	−	0.296506i	$-0.0958216\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			10.3923i			0.394203i
$696$		0			0
$697$	36.0000	+	20.7846i	1.36360	+	0.787273i
$698$		0			0
$699$	18.0000	−	31.1769i	0.680823	−	1.17922i
$700$		0			0
$701$	−3.00000	−	5.19615i	−0.113308	−	0.196256i	0.803794	−	0.594908i	$-0.202811\pi$
$701$	−3.00000	−	5.19615i	−0.113308	−	0.196256i	−0.917102	+	0.398652i	$0.869478\pi$
$702$		0			0
$703$	30.0000	+	3.46410i	1.13147	+	0.130651i
$704$		0			0
$705$	6.00000	−	3.46410i	0.225973	−	0.130466i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	17.5000	−	30.3109i	0.657226	−	1.13835i	−0.324104	−	0.946021i	$-0.605063\pi$
$709$	17.5000	−	30.3109i	0.657226	−	1.13835i	0.981331	−	0.192328i	$-0.0616038\pi$
$710$		0			0
$711$	−1.00000			−0.0375029
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−6.00000			−0.224387
$716$		0			0
$717$	15.0000	−	8.66025i	0.560185	−	0.323423i
$718$		0			0
$719$	1.50000	−	0.866025i	0.0559406	−	0.0322973i	−0.471769	−	0.881722i	$-0.656384\pi$
$719$	1.50000	−	0.866025i	0.0559406	−	0.0322973i	0.527709	+	0.849425i	$0.323051\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			10.3923i			0.386494i
$724$		0			0
$725$	−1.50000	+	0.866025i	−0.0557086	+	0.0321634i
$726$		0			0
$727$	39.0000	−	22.5167i	1.44643	−	0.835097i	0.448163	−	0.893952i	$-0.352078\pi$
$727$	39.0000	−	22.5167i	1.44643	−	0.835097i	0.998267	+	0.0588549i	$0.0187449\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	36.0000	+	20.7846i	1.33151	+	0.768747i
$732$		0			0
$733$	−44.0000			−1.62518			−0.812589	−	0.582838i	$-0.801942\pi$
$733$	−44.0000			−1.62518			−0.812589	+	0.582838i	$0.801942\pi$
$734$		0			0
$735$	7.00000	−	12.1244i	0.258199	−	0.447214i
$736$		0			0
$737$	15.0000	+	8.66025i	0.552532	+	0.319005i
$738$		0			0
$739$	−37.5000	+	21.6506i	−1.37946	+	0.796431i	−0.992094	−	0.125495i	$-0.959948\pi$
$739$	−37.5000	+	21.6506i	−1.37946	+	0.796431i	−0.387366	+	0.921926i	$0.626615\pi$
$740$		0			0
$741$	30.0000	+	3.46410i	1.10208	+	0.127257i
$742$		0			0
$743$	6.00000	+	10.3923i	0.220119	+	0.381257i	0.954844	−	0.297108i	$-0.0960222\pi$
$743$	6.00000	+	10.3923i	0.220119	+	0.381257i	−0.734725	+	0.678365i	$0.762689\pi$
$744$		0			0
$745$	−7.50000	+	12.9904i	−0.274779	+	0.475931i
$746$		0			0
$747$	−6.00000	−	3.46410i	−0.219529	−	0.126745i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	21.5000	−	37.2391i	0.784546	−	1.35887i	−0.144724	−	0.989472i	$-0.546229\pi$
$751$	21.5000	−	37.2391i	0.784546	−	1.35887i	0.929270	−	0.369402i	$-0.120437\pi$
$752$		0			0
$753$			10.3923i			0.378717i
$754$		0			0
$755$	9.50000	+	16.4545i	0.345740	+	0.598840i
$756$		0			0
$757$	−1.00000	−	1.73205i	−0.0363456	−	0.0629525i	0.847280	−	0.531146i	$-0.178238\pi$
$757$	−1.00000	−	1.73205i	−0.0363456	−	0.0629525i	−0.883626	+	0.468193i	$0.844905\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	3.00000	+	5.19615i	0.108465	+	0.187867i
$766$		0			0
$767$			31.1769i			1.12573i
$768$		0			0
$769$	−21.5000	+	37.2391i	−0.775310	+	1.34288i	0.159310	+	0.987229i	$0.449073\pi$
$769$	−21.5000	+	37.2391i	−0.775310	+	1.34288i	−0.934620	+	0.355647i	$0.884260\pi$
$770$		0			0
$771$			20.7846i			0.748539i
$772$		0			0
$773$	3.00000	+	1.73205i	0.107903	+	0.0622975i	0.552980	−	0.833194i	$-0.313491\pi$
$773$	3.00000	+	1.73205i	0.107903	+	0.0622975i	−0.445078	+	0.895492i	$0.646824\pi$
$774$		0			0
$775$	2.50000	−	4.33013i	0.0898027	−	0.155543i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.0000	−	27.7128i	−0.429945	−	0.992915i
$780$		0			0
$781$	−22.5000	+	12.9904i	−0.805113	+	0.464832i
$782$		0			0
$783$	6.00000	+	3.46410i	0.214423	+	0.123797i
$784$		0			0
$785$	7.00000	−	12.1244i	0.249841	−	0.432737i
$786$		0			0
$787$	−14.0000			−0.499046			−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000			−0.499046			−0.249523	+	0.968369i	$0.580274\pi$
$788$		0			0
$789$	42.0000	+	24.2487i	1.49524	+	0.863277i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.00000	+	1.73205i	−0.106533	+	0.0615069i
$794$		0			0
$795$	6.00000	−	3.46410i	0.212798	−	0.122859i
$796$		0			0
$797$			17.3205i			0.613524i	0.951786	+	0.306762i	$0.0992455\pi$
$797$			17.3205i			0.613524i	−0.951786	+	0.306762i	$0.900754\pi$
$798$		0			0
$799$			20.7846i			0.735307i
$800$		0			0
$801$	−10.5000	+	6.06218i	−0.370999	+	0.214197i
$802$		0			0
$803$	−6.00000	+	3.46410i	−0.211735	+	0.122245i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	27.0000	+	15.5885i	0.950445	+	0.548740i
$808$		0			0
$809$	−3.00000			−0.105474			−0.0527372	−	0.998608i	$-0.516795\pi$
$809$	−3.00000			−0.105474			−0.0527372	+	0.998608i	$0.516795\pi$
$810$		0			0
$811$	6.50000	−	11.2583i	0.228246	−	0.395333i	−0.729042	−	0.684468i	$-0.760034\pi$
$811$	6.50000	−	11.2583i	0.228246	−	0.395333i	0.957288	+	0.289135i	$0.0933677\pi$
$812$		0			0
$813$	33.0000	+	19.0526i	1.15736	+	0.668202i
$814$		0			0
$815$	−3.00000	+	1.73205i	−0.105085	+	0.0606711i
$816$		0			0
$817$	−12.0000	−	27.7128i	−0.419827	−	0.969549i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	4.50000	−	7.79423i	0.157051	−	0.272020i	−0.776753	−	0.629805i	$-0.783135\pi$
$821$	4.50000	−	7.79423i	0.157051	−	0.272020i	0.933804	+	0.357785i	$0.116468\pi$
$822$		0			0
$823$	−15.0000	−	8.66025i	−0.522867	−	0.301877i	0.215240	−	0.976561i	$-0.430947\pi$
$823$	−15.0000	−	8.66025i	−0.522867	−	0.301877i	−0.738107	+	0.674684i	$0.764280\pi$
$824$		0			0
$825$			3.46410i			0.120605i
$826$		0			0
$827$	3.00000	−	5.19615i	0.104320	−	0.180688i	−0.809140	−	0.587616i	$-0.800067\pi$
$827$	3.00000	−	5.19615i	0.104320	−	0.180688i	0.913460	+	0.406928i	$0.133400\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$	−16.0000	−	27.7128i	−0.555034	−	0.961347i
$832$		0			0
$833$	21.0000	+	36.3731i	0.727607	+	1.26025i
$834$		0			0
$835$	−12.0000			−0.415277
$836$		0			0
$837$	−20.0000			−0.691301
$838$		0			0
$839$	12.0000	+	20.7846i	0.414286	+	0.717564i	0.995353	−	0.0962912i	$-0.0306980\pi$
$839$	12.0000	+	20.7846i	0.414286	+	0.717564i	−0.581067	+	0.813856i	$0.697365\pi$
$840$		0			0
$841$	−13.0000	−	22.5167i	−0.448276	−	0.776437i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.500000	+	0.866025i	−0.0172005	+	0.0297922i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	54.0000	+	31.1769i	1.85328	+	1.06999i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.00000	−	1.73205i	−0.0342393	−	0.0593043i	0.848398	−	0.529359i	$-0.177568\pi$
$853$	−1.00000	−	1.73205i	−0.0342393	−	0.0593043i	−0.882637	+	0.470055i	$0.844234\pi$
$854$		0			0
$855$	0.500000	−	4.33013i	0.0170996	−	0.148087i
$856$		0			0
$857$	30.0000	−	17.3205i	1.02478	−	0.591657i	0.109295	−	0.994009i	$-0.465141\pi$
$857$	30.0000	−	17.3205i	1.02478	−	0.591657i	0.915485	+	0.402352i	$0.131807\pi$
$858$		0			0
$859$	−28.5000	−	16.4545i	−0.972407	−	0.561420i	−0.0724381	−	0.997373i	$-0.523078\pi$
$859$	−28.5000	−	16.4545i	−0.972407	−	0.561420i	−0.899969	+	0.435953i	$0.856411\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	54.0000			1.83818			0.919091	−	0.394046i	$-0.128925\pi$
$863$	54.0000			1.83818			0.919091	+	0.394046i	$0.128925\pi$
$864$		0			0
$865$	12.0000	+	6.92820i	0.408012	+	0.235566i
$866$		0			0
$867$	−38.0000			−1.29055
$868$		0			0
$869$	1.50000	−	0.866025i	0.0508840	−	0.0293779i
$870$		0			0
$871$	−30.0000	+	17.3205i	−1.01651	+	0.586883i
$872$		0			0
$873$		−	13.8564i		−	0.468968i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	12.0000	−	6.92820i	0.405211	−	0.233949i	−0.283519	−	0.958967i	$-0.591502\pi$
$877$	12.0000	−	6.92820i	0.405211	−	0.233949i	0.688730	+	0.725018i	$0.258169\pi$
$878$		0			0
$879$	12.0000	−	6.92820i	0.404750	−	0.233682i
$880$		0			0
$881$	33.0000			1.11180			0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000			1.11180			0.555899	+	0.831250i	$0.312374\pi$
$882$		0			0
$883$	12.0000	+	6.92820i	0.403832	+	0.233153i	0.688136	−	0.725582i	$-0.258429\pi$
$883$	12.0000	+	6.92820i	0.403832	+	0.233153i	−0.284304	+	0.958734i	$0.591763\pi$
$884$		0			0
$885$	18.0000			0.605063
$886$		0			0
$887$	−6.00000	+	10.3923i	−0.201460	+	0.348939i	−0.948999	−	0.315279i	$-0.897902\pi$
$887$	−6.00000	+	10.3923i	−0.201460	+	0.348939i	0.747539	+	0.664218i	$0.231235\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	16.5000	−	9.52628i	0.552771	−	0.319142i
$892$		0			0
$893$	9.00000	−	12.1244i	0.301174	−	0.405726i
$894$		0			0
$895$	−4.50000	−	7.79423i	−0.150418	−	0.260532i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−7.50000	−	4.33013i	−0.250139	−	0.144418i
$900$		0			0
$901$			20.7846i			0.692436i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	13.8564i		−	0.460603i
$906$		0			0
$907$	16.0000	+	27.7128i	0.531271	+	0.920189i	0.999334	+	0.0364935i	$0.0116188\pi$
$907$	16.0000	+	27.7128i	0.531271	+	0.920189i	−0.468063	+	0.883695i	$0.655048\pi$
$908$		0			0
$909$	1.50000	+	2.59808i	0.0497519	+	0.0861727i
$910$		0			0
$911$	−15.0000			−0.496972			−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000			−0.496972			−0.248486	+	0.968635i	$0.579933\pi$
$912$		0			0
$913$	12.0000			0.397142
$914$		0			0
$915$	1.00000	+	1.73205i	0.0330590	+	0.0572598i
$916$		0			0
$917$		0			0
$918$		0			0
$919$			45.0333i			1.48551i	0.669562	+	0.742756i	$0.266482\pi$
$919$			45.0333i			1.48551i	−0.669562	+	0.742756i	$0.733518\pi$
$920$		0			0
$921$	14.0000	−	24.2487i	0.461316	−	0.799022i
$922$		0			0
$923$		−	51.9615i		−	1.71033i
$924$		0			0
$925$	−6.00000	−	3.46410i	−0.197279	−	0.113899i
$926$		0			0
$927$	2.00000	−	3.46410i	0.0656886	−	0.113776i
$928$		0			0
$929$	−22.5000	−	38.9711i	−0.738201	−	1.27860i	−0.953305	−	0.302010i	$-0.902342\pi$
$929$	−22.5000	−	38.9711i	−0.738201	−	1.27860i	0.215104	−	0.976591i	$-0.430991\pi$
$930$		0			0
$931$	3.50000	−	30.3109i	0.114708	−	0.993399i
$932$		0			0
$933$	−6.00000	+	3.46410i	−0.196431	+	0.113410i
$934$		0			0
$935$	−9.00000	−	5.19615i	−0.294331	−	0.169932i
$936$		0			0
$937$	−1.00000	+	1.73205i	−0.0326686	+	0.0565836i	−0.881897	−	0.471441i	$-0.843734\pi$
$937$	−1.00000	+	1.73205i	−0.0326686	+	0.0565836i	0.849229	+	0.528025i	$0.177067\pi$
$938$		0			0
$939$	52.0000			1.69696
$940$		0			0
$941$	−40.5000	−	23.3827i	−1.32026	−	0.762254i	−0.336492	−	0.941686i	$-0.609241\pi$
$941$	−40.5000	−	23.3827i	−1.32026	−	0.762254i	−0.983770	+	0.179433i	$0.942574\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−51.0000	+	29.4449i	−1.65728	+	0.956830i	−0.683314	+	0.730125i	$0.739462\pi$
$947$	−51.0000	+	29.4449i	−1.65728	+	0.956830i	−0.973964	+	0.226705i	$0.927205\pi$
$948$		0			0
$949$		−	13.8564i		−	0.449798i
$950$		0			0
$951$			41.5692i			1.34797i
$952$		0			0
$953$	−30.0000	+	17.3205i	−0.971795	+	0.561066i	−0.899783	−	0.436337i	$-0.856275\pi$
$953$	−30.0000	+	17.3205i	−0.971795	+	0.561066i	−0.0720122	+	0.997404i	$0.522942\pi$
$954$		0			0
$955$	−10.5000	+	6.06218i	−0.339772	+	0.196167i
$956$		0			0
$957$	6.00000			0.193952
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−6.00000			−0.193548
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$	42.0000	+	31.1769i	1.34923	+	1.00155i
$970$		0			0
$971$	18.0000	+	31.1769i	0.577647	+	1.00051i	0.995748	+	0.0921142i	$0.0293625\pi$
$971$	18.0000	+	31.1769i	0.577647	+	1.00051i	−0.418101	+	0.908401i	$0.637304\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−6.00000	−	3.46410i	−0.192154	−	0.110940i
$976$		0			0
$977$			27.7128i			0.886611i	0.896370	+	0.443306i	$0.146194\pi$
$977$			27.7128i			0.886611i	−0.896370	+	0.443306i	$0.853806\pi$
$978$		0			0
$979$	10.5000	−	18.1865i	0.335581	−	0.581244i
$980$		0			0
$981$		−	15.5885i		−	0.497701i
$982$		0			0
$983$	−18.0000	−	31.1769i	−0.574111	−	0.994389i	−0.996138	−	0.0878058i	$-0.972015\pi$
$983$	−18.0000	−	31.1769i	−0.574111	−	0.994389i	0.422027	−	0.906583i	$-0.361319\pi$
$984$		0			0
$985$	−3.00000	−	5.19615i	−0.0955879	−	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	4.00000	+	6.92820i	0.127064	+	0.220082i	0.922538	−	0.385906i	$-0.126111\pi$
$991$	4.00000	+	6.92820i	0.127064	+	0.220082i	−0.795474	+	0.605988i	$0.792778\pi$
$992$		0			0
$993$	−4.00000	−	6.92820i	−0.126936	−	0.219860i
$994$		0			0
$995$			5.19615i			0.164729i
$996$		0			0
$997$	−11.0000	+	19.0526i	−0.348373	+	0.603401i	−0.985961	−	0.166978i	$-0.946599\pi$
$997$	−11.0000	+	19.0526i	−0.348373	+	0.603401i	0.637587	+	0.770378i	$0.279933\pi$
$998$		0			0
$999$			27.7128i			0.876795i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.bq.i.31.1	yes	2
4.3	odd	2		1520.2.bq.c.31.1	✓	2
19.8	odd	6		1520.2.bq.c.1471.1	yes	2
76.27	even	6	inner	1520.2.bq.i.1471.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1520.2.bq.c.31.1	✓	2	4.3	odd	2
1520.2.bq.c.1471.1	yes	2	19.8	odd	6
1520.2.bq.i.31.1	yes	2	1.1	even	1	trivial
1520.2.bq.i.1471.1	yes	2	76.27	even	6	inner

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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.bq (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{9} -1.73205i q^{11} +(3.00000 + 1.73205i) q^{13} +(1.00000 - 1.73205i) q^{15} +(3.00000 + 5.19615i) q^{17} +(0.500000 - 4.33013i) q^{19} +(-0.500000 + 0.866025i) q^{25} +4.00000 q^{27} +(1.50000 + 0.866025i) q^{29} -5.00000 q^{31} +(3.00000 - 1.73205i) q^{33} +6.92820i q^{37} +6.92820i q^{39} +(6.00000 - 3.46410i) q^{41} +(6.00000 - 3.46410i) q^{43} +1.00000 q^{45} +(3.00000 + 1.73205i) q^{47} +7.00000 q^{49} +(-6.00000 + 10.3923i) q^{51} +(3.00000 + 1.73205i) q^{53} +(-1.50000 + 0.866025i) q^{55} +(8.00000 - 3.46410i) q^{57} +(4.50000 + 7.79423i) q^{59} +(-0.500000 + 0.866025i) q^{61} -3.46410i q^{65} +(-5.00000 + 8.66025i) q^{67} +(-7.50000 - 12.9904i) q^{71} +(-2.00000 - 3.46410i) q^{73} -2.00000 q^{75} +(0.500000 + 0.866025i) q^{79} +(5.50000 + 9.52628i) q^{81} +6.92820i q^{83} +(3.00000 - 5.19615i) q^{85} +3.46410i q^{87} +(10.5000 + 6.06218i) q^{89} +(-5.00000 - 8.66025i) q^{93} +(-4.00000 + 1.73205i) q^{95} +(-12.0000 + 6.92820i) q^{97} +(1.50000 + 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - q^{5} - q^{9} + 6 q^{13} + 2 q^{15} + 6 q^{17} + q^{19} - q^{25} + 8 q^{27} + 3 q^{29} - 10 q^{31} + 6 q^{33} + 12 q^{41} + 12 q^{43} + 2 q^{45} + 6 q^{47} + 14 q^{49} - 12 q^{51} + 6 q^{53}+ \cdots + 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - q^5 - q^9 + 6 * q^13 + 2 * q^15 + 6 * q^17 + q^19 - q^25 + 8 * q^27 + 3 * q^29 - 10 * q^31 + 6 * q^33 + 12 * q^41 + 12 * q^43 + 2 * q^45 + 6 * q^47 + 14 * q^49 - 12 * q^51 + 6 * q^53 - 3 * q^55 + 16 * q^57 + 9 * q^59 - q^61 - 10 * q^67 - 15 * q^71 - 4 * q^73 - 4 * q^75 + q^79 + 11 * q^81 + 6 * q^85 + 21 * q^89 - 10 * q^93 - 8 * q^95 - 24 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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