LMFDB - Embedded newform 2028.2.q.b.1837.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2028,2,Mod(361,2028)] 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("2028.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2028, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,0,0,0,0,-1,0,-6,0,0,0,6,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$16.1936615299$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1837.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2028.1837
Dual form			2028.2.q.b.361.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} +3.46410i q^{5} +(-0.500000 - 0.866025i) q^{9} +(-3.00000 - 1.73205i) q^{11} +(3.00000 + 1.73205i) q^{15} +(-3.00000 - 5.19615i) q^{17} +(-6.00000 + 3.46410i) q^{19} -7.00000 q^{25} -1.00000 q^{27} +(3.00000 - 5.19615i) q^{29} -6.92820i q^{31} +(-3.00000 + 1.73205i) q^{33} +(3.00000 + 1.73205i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(3.00000 - 1.73205i) q^{45} +3.46410i q^{47} +(-3.50000 + 6.06218i) q^{49} -6.00000 q^{51} +6.00000 q^{53} +(6.00000 - 10.3923i) q^{55} +6.92820i q^{57} +(-3.00000 + 1.73205i) q^{59} +(-5.00000 - 8.66025i) q^{61} +(-12.0000 - 6.92820i) q^{67} +(9.00000 - 5.19615i) q^{71} -6.92820i q^{73} +(-3.50000 + 6.06218i) q^{75} -8.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} -3.46410i q^{83} +(18.0000 - 10.3923i) q^{85} +(-3.00000 - 5.19615i) q^{87} +(15.0000 + 8.66025i) q^{89} +(-6.00000 - 3.46410i) q^{93} +(-12.0000 - 20.7846i) q^{95} +(6.00000 - 3.46410i) q^{97} +3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - q^{9} - 6 q^{11} + 6 q^{15} - 6 q^{17} - 12 q^{19} - 14 q^{25} - 2 q^{27} + 6 q^{29} - 6 q^{33} + 6 q^{41} - 8 q^{43} + 6 q^{45} - 7 q^{49} - 12 q^{51} + 12 q^{53} + 12 q^{55} - 6 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) $ 2 * q + q^3 - q^9 - 6 * q^11 + 6 * q^15 - 6 * q^17 - 12 * q^19 - 14 * q^25 - 2 * q^27 + 6 * q^29 - 6 * q^33 + 6 * q^41 - 8 * q^43 + 6 * q^45 - 7 * q^49 - 12 * q^51 + 12 * q^53 + 12 * q^55 - 6 * q^59 - 10 * q^61 - 24 * q^67 + 18 * q^71 - 7 * q^75 - 16 * q^79 - q^81 + 36 * q^85 - 6 * q^87 + 30 * q^89 - 12 * q^93 - 24 * q^95 + 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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
$2$		0			0
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$4$		0			0
$5$			3.46410i			1.54919i	0.632456	+	0.774597i	$0.282047\pi$
$5$			3.46410i			1.54919i	−0.632456	+	0.774597i	$0.717953\pi$
$6$		0			0
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.0258656	−	0.999665i	$-0.508234\pi$
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.878668	+	0.477432i	$0.841568\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$		0			0
$15$	3.00000	+	1.73205i	0.774597	+	0.447214i
$16$		0			0
$17$	−3.00000	−	5.19615i	−0.727607	−	1.26025i	−0.957892	−	0.287129i	$-0.907299\pi$
$17$	−3.00000	−	5.19615i	−0.727607	−	1.26025i	0.230285	−	0.973123i	$-0.426034\pi$
$18$		0			0
$19$	−6.00000	+	3.46410i	−1.37649	+	0.794719i	−0.991736	−	0.128298i	$-0.959049\pi$
$19$	−6.00000	+	3.46410i	−1.37649	+	0.794719i	−0.384759	+	0.923017i	$0.625715\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−7.00000			−1.40000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	−0.440652	−	0.897678i	$-0.645253\pi$
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	0.997738	−	0.0672232i	$-0.0214140\pi$
$30$		0			0
$31$		−	6.92820i		−	1.24434i	−0.782881	−	0.622171i	$-0.786251\pi$
$31$		−	6.92820i		−	1.24434i	0.782881	−	0.622171i	$-0.213749\pi$
$32$		0			0
$33$	−3.00000	+	1.73205i	−0.522233	+	0.301511i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.00000	+	1.73205i	0.468521	+	0.270501i	0.715621	−	0.698489i	$-0.246144\pi$
$41$	3.00000	+	1.73205i	0.468521	+	0.270501i	−0.247099	+	0.968990i	$0.579477\pi$
$42$		0			0
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	0.381246	−	0.924473i	$-0.375495\pi$
$44$		0			0
$45$	3.00000	−	1.73205i	0.447214	−	0.258199i
$46$		0			0
$47$			3.46410i			0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$			3.46410i			0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$		0			0
$49$	−3.50000	+	6.06218i	−0.500000	+	0.866025i
$50$		0			0
$51$	−6.00000			−0.840168
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	6.00000	−	10.3923i	0.809040	−	1.40130i
$56$		0			0
$57$			6.92820i			0.917663i
$58$		0			0
$59$	−3.00000	+	1.73205i	−0.390567	+	0.225494i	−0.682406	−	0.730974i	$-0.739066\pi$
$59$	−3.00000	+	1.73205i	−0.390567	+	0.225494i	0.291839	+	0.956467i	$0.405733\pi$
$60$		0			0
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	−0.985391	−	0.170305i	$-0.945525\pi$
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	0.345207	−	0.938527i	$-0.387809\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−12.0000	−	6.92820i	−1.46603	−	0.846415i	−0.466755	−	0.884387i	$-0.654577\pi$
$67$	−12.0000	−	6.92820i	−1.46603	−	0.846415i	−0.999279	+	0.0379722i	$0.987910\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	9.00000	−	5.19615i	1.06810	−	0.616670i	0.140441	−	0.990089i	$-0.455148\pi$
$71$	9.00000	−	5.19615i	1.06810	−	0.616670i	0.927663	+	0.373419i	$0.121815\pi$
$72$		0			0
$73$		−	6.92820i		−	0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$		−	6.92820i		−	0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$		0			0
$75$	−3.50000	+	6.06218i	−0.404145	+	0.700000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$		−	3.46410i		−	0.380235i	−0.981761	−	0.190117i	$-0.939113\pi$
$83$		−	3.46410i		−	0.380235i	0.981761	−	0.190117i	$-0.0608868\pi$
$84$		0			0
$85$	18.0000	−	10.3923i	1.95237	−	1.12720i
$86$		0			0
$87$	−3.00000	−	5.19615i	−0.321634	−	0.557086i
$88$		0			0
$89$	15.0000	+	8.66025i	1.59000	+	0.917985i	0.993306	+	0.115514i	$0.0368515\pi$
$89$	15.0000	+	8.66025i	1.59000	+	0.917985i	0.596691	+	0.802471i	$0.296482\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−6.00000	−	3.46410i	−0.622171	−	0.359211i
$94$		0			0
$95$	−12.0000	−	20.7846i	−1.23117	−	2.13246i
$96$		0			0
$97$	6.00000	−	3.46410i	0.609208	−	0.351726i	−0.163448	−	0.986552i	$-0.552261\pi$
$97$	6.00000	−	3.46410i	0.609208	−	0.351726i	0.772655	+	0.634826i	$0.218928\pi$
$98$		0			0
$99$			3.46410i			0.348155i
$100$		0			0
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	−0.975796	−	0.218685i	$-0.929823\pi$
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	0.677284	+	0.735721i	$0.263157\pi$
$102$		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
$107$	6.00000	−	10.3923i	0.580042	−	1.00466i	−0.415432	−	0.909624i	$-0.636370\pi$
$107$	6.00000	−	10.3923i	0.580042	−	1.00466i	0.995474	−	0.0950377i	$-0.0302972\pi$
$108$		0			0
$109$		−	6.92820i		−	0.663602i	−0.943349	−	0.331801i	$-0.892344\pi$
$109$		−	6.92820i		−	0.663602i	0.943349	−	0.331801i	$-0.107656\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.00000	+	15.5885i	0.846649	+	1.46644i	0.884182	+	0.467143i	$0.154717\pi$
$113$	9.00000	+	15.5885i	0.846649	+	1.46644i	−0.0375328	+	0.999295i	$0.511950\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.500000	+	0.866025i	0.0454545	+	0.0787296i
$122$		0			0
$123$	3.00000	−	1.73205i	0.270501	−	0.156174i
$124$		0			0
$125$		−	6.92820i		−	0.619677i
$126$		0			0
$127$	−8.00000	+	13.8564i	−0.709885	+	1.22956i	0.255014	+	0.966937i	$0.417920\pi$
$127$	−8.00000	+	13.8564i	−0.709885	+	1.22956i	−0.964899	+	0.262620i	$0.915413\pi$
$128$		0			0
$129$	−8.00000			−0.704361
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		−	3.46410i		−	0.298142i
$136$		0			0
$137$	−9.00000	+	5.19615i	−0.768922	+	0.443937i	−0.832490	−	0.554040i	$-0.813085\pi$
$137$	−9.00000	+	5.19615i	−0.768922	+	0.443937i	0.0635680	+	0.997978i	$0.479752\pi$
$138$		0			0
$139$	−4.00000	−	6.92820i	−0.339276	−	0.587643i	0.645021	−	0.764165i	$-0.276849\pi$
$139$	−4.00000	−	6.92820i	−0.339276	−	0.587643i	−0.984297	+	0.176522i	$0.943515\pi$
$140$		0			0
$141$	3.00000	+	1.73205i	0.252646	+	0.145865i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	18.0000	+	10.3923i	1.49482	+	0.863034i
$146$		0			0
$147$	3.50000	+	6.06218i	0.288675	+	0.500000i
$148$		0			0
$149$	−15.0000	+	8.66025i	−1.22885	+	0.709476i	−0.966789	−	0.255576i	$-0.917735\pi$
$149$	−15.0000	+	8.66025i	−1.22885	+	0.709476i	−0.262059	+	0.965052i	$0.584401\pi$
$150$		0			0
$151$			6.92820i			0.563809i	0.959442	+	0.281905i	$0.0909662\pi$
$151$			6.92820i			0.563809i	−0.959442	+	0.281905i	$0.909034\pi$
$152$		0			0
$153$	−3.00000	+	5.19615i	−0.242536	+	0.420084i
$154$		0			0
$155$	24.0000			1.92773
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$	3.00000	−	5.19615i	0.237915	−	0.412082i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−18.0000	+	10.3923i	−1.40987	+	0.813988i	−0.995375	−	0.0960641i	$-0.969375\pi$
$163$	−18.0000	+	10.3923i	−1.40987	+	0.813988i	−0.414494	+	0.910052i	$0.636041\pi$
$164$		0			0
$165$	−6.00000	−	10.3923i	−0.467099	−	0.809040i
$166$		0			0
$167$	−15.0000	−	8.66025i	−1.16073	−	0.670151i	−0.209255	−	0.977861i	$-0.567104\pi$
$167$	−15.0000	−	8.66025i	−1.16073	−	0.670151i	−0.951480	+	0.307711i	$0.900437\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	6.00000	+	3.46410i	0.458831	+	0.264906i
$172$		0			0
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	0.729155	−	0.684349i	$-0.239913\pi$
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	−0.957241	+	0.289292i	$0.906580\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			3.46410i			0.260378i
$178$		0			0
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	−0.549825	−	0.835280i	$-0.685306\pi$
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	0.998286	+	0.0585225i	$0.0186389\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$	−10.0000			−0.739221
$184$		0			0
$185$		0			0
$186$		0			0
$187$			20.7846i			1.51992i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	−12.0000	−	6.92820i	−0.863779	−	0.498703i	0.00149702	−	0.999999i	$-0.499523\pi$
$193$	−12.0000	−	6.92820i	−0.863779	−	0.498703i	−0.865276	+	0.501296i	$0.832857\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.0000	−	8.66025i	−1.06871	−	0.617018i	−0.140878	−	0.990027i	$-0.544993\pi$
$197$	−15.0000	−	8.66025i	−1.06871	−	0.617018i	−0.927828	+	0.373009i	$0.878326\pi$
$198$		0			0
$199$	4.00000	+	6.92820i	0.283552	+	0.491127i	0.972257	−	0.233915i	$-0.0751537\pi$
$199$	4.00000	+	6.92820i	0.283552	+	0.491127i	−0.688705	+	0.725042i	$0.741820\pi$
$200$		0			0
$201$	−12.0000	+	6.92820i	−0.846415	+	0.488678i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−6.00000	+	10.3923i	−0.419058	+	0.725830i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$	−2.00000	+	3.46410i	−0.137686	+	0.238479i	−0.926620	−	0.375999i	$-0.877300\pi$
$211$	−2.00000	+	3.46410i	−0.137686	+	0.238479i	0.788935	+	0.614477i	$0.210633\pi$
$212$		0			0
$213$		−	10.3923i		−	0.712069i
$214$		0			0
$215$	24.0000	−	13.8564i	1.63679	−	0.944999i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−6.00000	−	3.46410i	−0.405442	−	0.234082i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	6.00000	+	3.46410i	0.401790	+	0.231973i	0.687256	−	0.726415i	$-0.258815\pi$
$223$	6.00000	+	3.46410i	0.401790	+	0.231973i	−0.285466	+	0.958389i	$0.592148\pi$
$224$		0			0
$225$	3.50000	+	6.06218i	0.233333	+	0.404145i
$226$		0			0
$227$	−9.00000	+	5.19615i	−0.597351	+	0.344881i	−0.767999	−	0.640451i	$-0.778747\pi$
$227$	−9.00000	+	5.19615i	−0.597351	+	0.344881i	0.170648	+	0.985332i	$0.445414\pi$
$228$		0			0
$229$			6.92820i			0.457829i	0.973447	+	0.228914i	$0.0735176\pi$
$229$			6.92820i			0.457829i	−0.973447	+	0.228914i	$0.926482\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	−12.0000			−0.782794
$236$		0			0
$237$	−4.00000	+	6.92820i	−0.259828	+	0.450035i
$238$		0			0
$239$			10.3923i			0.672222i	0.941822	+	0.336111i	$0.109112\pi$
$239$			10.3923i			0.672222i	−0.941822	+	0.336111i	$0.890888\pi$
$240$		0			0
$241$	−18.0000	+	10.3923i	−1.15948	+	0.669427i	−0.951180	−	0.308637i	$-0.900127\pi$
$241$	−18.0000	+	10.3923i	−1.15948	+	0.669427i	−0.208302	+	0.978065i	$0.566794\pi$
$242$		0			0
$243$	0.500000	+	0.866025i	0.0320750	+	0.0555556i
$244$		0			0
$245$	−21.0000	−	12.1244i	−1.34164	−	0.774597i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−3.00000	−	1.73205i	−0.190117	−	0.109764i
$250$		0			0
$251$	−6.00000	−	10.3923i	−0.378717	−	0.655956i	0.612159	−	0.790735i	$-0.290301\pi$
$251$	−6.00000	−	10.3923i	−0.378717	−	0.655956i	−0.990876	+	0.134778i	$0.956968\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		−	20.7846i		−	1.30158i
$256$		0			0
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	−0.757159	−	0.653231i	$-0.773413\pi$
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	0.944294	+	0.329104i	$0.106747\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$			20.7846i			1.27679i
$266$		0			0
$267$	15.0000	−	8.66025i	0.917985	−	0.529999i
$268$		0			0
$269$	15.0000	+	25.9808i	0.914566	+	1.58408i	0.807535	+	0.589819i	$0.200801\pi$
$269$	15.0000	+	25.9808i	0.914566	+	1.58408i	0.107031	+	0.994256i	$0.465866\pi$
$270$		0			0
$271$	24.0000	+	13.8564i	1.45790	+	0.841717i	0.998908	−	0.0467255i	$-0.0148786\pi$
$271$	24.0000	+	13.8564i	1.45790	+	0.841717i	0.458988	+	0.888442i	$0.348212\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	21.0000	+	12.1244i	1.26635	+	0.731126i
$276$		0			0
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	−0.675810	+	0.737075i	$0.736206\pi$
$278$		0			0
$279$	−6.00000	+	3.46410i	−0.359211	+	0.207390i
$280$		0			0
$281$			17.3205i			1.03325i	0.856210	+	0.516627i	$0.172813\pi$
$281$			17.3205i			1.03325i	−0.856210	+	0.516627i	$0.827187\pi$
$282$		0			0
$283$	10.0000	−	17.3205i	0.594438	−	1.02960i	−0.399188	−	0.916869i	$-0.630708\pi$
$283$	10.0000	−	17.3205i	0.594438	−	1.02960i	0.993626	−	0.112728i	$-0.0359589\pi$
$284$		0			0
$285$	−24.0000			−1.42164
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$		0			0
$291$		−	6.92820i		−	0.406138i
$292$		0			0
$293$	3.00000	−	1.73205i	0.175262	−	0.101187i	−0.409803	−	0.912174i	$-0.634402\pi$
$293$	3.00000	−	1.73205i	0.175262	−	0.101187i	0.585065	+	0.810987i	$0.301069\pi$
$294$		0			0
$295$	−6.00000	−	10.3923i	−0.349334	−	0.605063i
$296$		0			0
$297$	3.00000	+	1.73205i	0.174078	+	0.100504i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	3.00000	+	5.19615i	0.172345	+	0.298511i
$304$		0			0
$305$	30.0000	−	17.3205i	1.71780	−	0.991769i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	−2.00000	+	3.46410i	−0.113776	+	0.197066i
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	24.2487i		−	1.36194i	−0.732310	−	0.680972i	$-0.761558\pi$
$317$		−	24.2487i		−	1.36194i	0.732310	−	0.680972i	$-0.238442\pi$
$318$		0			0
$319$	−18.0000	+	10.3923i	−1.00781	+	0.581857i
$320$		0			0
$321$	−6.00000	−	10.3923i	−0.334887	−	0.580042i
$322$		0			0
$323$	36.0000	+	20.7846i	2.00309	+	1.15649i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−6.00000	−	3.46410i	−0.331801	−	0.191565i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	12.0000	−	6.92820i	0.659580	−	0.380808i	−0.132537	−	0.991178i	$-0.542312\pi$
$331$	12.0000	−	6.92820i	0.659580	−	0.380808i	0.792117	+	0.610370i	$0.208979\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	24.0000	−	41.5692i	1.31126	−	2.27117i
$336$		0			0
$337$	10.0000			0.544735			0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000			0.544735			0.272367	+	0.962193i	$0.412193\pi$
$338$		0			0
$339$	18.0000			0.977626
$340$		0			0
$341$	−12.0000	+	20.7846i	−0.649836	+	1.12555i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	0.980921	−	0.194409i	$-0.0622790\pi$
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	−0.658824	+	0.752297i	$0.728946\pi$
$348$		0			0
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.00000	−	5.19615i	−0.479022	−	0.276563i	0.240987	−	0.970528i	$-0.422529\pi$
$353$	−9.00000	−	5.19615i	−0.479022	−	0.276563i	−0.720009	+	0.693965i	$0.755862\pi$
$354$		0			0
$355$	18.0000	+	31.1769i	0.955341	+	1.65470i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	24.2487i		−	1.27980i	−0.768459	−	0.639899i	$-0.778976\pi$
$359$		−	24.2487i		−	1.27980i	0.768459	−	0.639899i	$-0.221024\pi$
$360$		0			0
$361$	14.5000	−	25.1147i	0.763158	−	1.32183i
$362$		0			0
$363$	1.00000			0.0524864
$364$		0			0
$365$	24.0000			1.25622
$366$		0			0
$367$	14.0000	−	24.2487i	0.730794	−	1.26577i	−0.225750	−	0.974185i	$-0.572483\pi$
$367$	14.0000	−	24.2487i	0.730794	−	1.26577i	0.956544	−	0.291587i	$-0.0941834\pi$
$368$		0			0
$369$		−	3.46410i		−	0.180334i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.00000	+	1.73205i	0.0517780	+	0.0896822i	0.890753	−	0.454488i	$-0.150178\pi$
$373$	1.00000	+	1.73205i	0.0517780	+	0.0896822i	−0.838975	+	0.544170i	$0.816844\pi$
$374$		0			0
$375$	−6.00000	−	3.46410i	−0.309839	−	0.178885i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−18.0000	−	10.3923i	−0.924598	−	0.533817i	−0.0394989	−	0.999220i	$-0.512576\pi$
$379$	−18.0000	−	10.3923i	−0.924598	−	0.533817i	−0.885099	+	0.465403i	$0.845909\pi$
$380$		0			0
$381$	8.00000	+	13.8564i	0.409852	+	0.709885i
$382$		0			0
$383$	−21.0000	+	12.1244i	−1.07305	+	0.619526i	−0.929013	−	0.370047i	$-0.879342\pi$
$383$	−21.0000	+	12.1244i	−1.07305	+	0.619526i	−0.144037	+	0.989572i	$0.546008\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.00000	+	6.92820i	−0.203331	+	0.352180i
$388$		0			0
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−6.00000	+	10.3923i	−0.302660	+	0.524222i
$394$		0			0
$395$		−	27.7128i		−	1.39438i
$396$		0			0
$397$	12.0000	−	6.92820i	0.602263	−	0.347717i	−0.167668	−	0.985843i	$-0.553624\pi$
$397$	12.0000	−	6.92820i	0.602263	−	0.347717i	0.769931	+	0.638127i	$0.220290\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	−	5.19615i	−0.449439	−	0.259483i	0.258154	−	0.966104i	$-0.416886\pi$
$401$	−9.00000	−	5.19615i	−0.449439	−	0.259483i	−0.707593	+	0.706620i	$0.750219\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−3.00000	−	1.73205i	−0.149071	−	0.0860663i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	18.0000	−	10.3923i	0.890043	−	0.513866i	0.0160862	−	0.999871i	$-0.494879\pi$
$409$	18.0000	−	10.3923i	0.890043	−	0.513866i	0.873956	+	0.486004i	$0.161546\pi$
$410$		0			0
$411$			10.3923i			0.512615i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	12.0000			0.589057
$416$		0			0
$417$	−8.00000			−0.391762
$418$		0			0
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	−0.974546	−	0.224189i	$-0.928027\pi$
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	0.681426	+	0.731887i	$0.261360\pi$
$420$		0			0
$421$		−	20.7846i		−	1.01298i	−0.862246	−	0.506490i	$-0.830943\pi$
$421$		−	20.7846i		−	1.01298i	0.862246	−	0.506490i	$-0.169057\pi$
$422$		0			0
$423$	3.00000	−	1.73205i	0.145865	−	0.0842152i
$424$		0			0
$425$	21.0000	+	36.3731i	1.01865	+	1.76435i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−27.0000	−	15.5885i	−1.30054	−	0.750870i	−0.320047	−	0.947402i	$-0.603699\pi$
$431$	−27.0000	−	15.5885i	−1.30054	−	0.750870i	−0.980497	+	0.196532i	$0.937032\pi$
$432$		0			0
$433$	1.00000	+	1.73205i	0.0480569	+	0.0832370i	0.889053	−	0.457804i	$-0.151364\pi$
$433$	1.00000	+	1.73205i	0.0480569	+	0.0832370i	−0.840996	+	0.541041i	$0.818030\pi$
$434$		0			0
$435$	18.0000	−	10.3923i	0.863034	−	0.498273i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−10.0000	+	17.3205i	−0.477274	+	0.826663i	−0.999661	−	0.0260459i	$-0.991708\pi$
$439$	−10.0000	+	17.3205i	−0.477274	+	0.826663i	0.522387	+	0.852709i	$0.325042\pi$
$440$		0			0
$441$	7.00000			0.333333
$442$		0			0
$443$	−12.0000			−0.570137			−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000			−0.570137			−0.285069	+	0.958507i	$0.592016\pi$
$444$		0			0
$445$	−30.0000	+	51.9615i	−1.42214	+	2.46321i
$446$		0			0
$447$			17.3205i			0.819232i
$448$		0			0
$449$	−3.00000	+	1.73205i	−0.141579	+	0.0817405i	−0.569116	−	0.822257i	$-0.692715\pi$
$449$	−3.00000	+	1.73205i	−0.141579	+	0.0817405i	0.427537	+	0.903998i	$0.359381\pi$
$450$		0			0
$451$	−6.00000	−	10.3923i	−0.282529	−	0.489355i
$452$		0			0
$453$	6.00000	+	3.46410i	0.281905	+	0.162758i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6.00000	−	3.46410i	−0.280668	−	0.162044i	0.353058	−	0.935602i	$-0.385142\pi$
$457$	−6.00000	−	3.46410i	−0.280668	−	0.162044i	−0.633726	+	0.773558i	$0.718475\pi$
$458$		0			0
$459$	3.00000	+	5.19615i	0.140028	+	0.242536i
$460$		0			0
$461$	−9.00000	+	5.19615i	−0.419172	+	0.242009i	−0.694723	−	0.719277i	$-0.744473\pi$
$461$	−9.00000	+	5.19615i	−0.419172	+	0.242009i	0.275551	+	0.961286i	$0.411140\pi$
$462$		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
$465$	12.0000	−	20.7846i	0.556487	−	0.963863i
$466$		0			0
$467$	36.0000			1.66588			0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000			1.66588			0.832941	+	0.553362i	$0.186655\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.00000	−	1.73205i	0.0460776	−	0.0798087i
$472$		0			0
$473$			27.7128i			1.27424i
$474$		0			0
$475$	42.0000	−	24.2487i	1.92709	−	1.11261i
$476$		0			0
$477$	−3.00000	−	5.19615i	−0.137361	−	0.237915i
$478$		0			0
$479$	3.00000	+	1.73205i	0.137073	+	0.0791394i	0.566969	−	0.823739i	$-0.308116\pi$
$479$	3.00000	+	1.73205i	0.137073	+	0.0791394i	−0.429895	+	0.902879i	$0.641449\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.0000	+	20.7846i	0.544892	+	0.943781i
$486$		0			0
$487$	6.00000	−	3.46410i	0.271886	−	0.156973i	−0.357858	−	0.933776i	$-0.616493\pi$
$487$	6.00000	−	3.46410i	0.271886	−	0.156973i	0.629744	+	0.776802i	$0.283160\pi$
$488$		0			0
$489$			20.7846i			0.939913i
$490$		0			0
$491$	18.0000	−	31.1769i	0.812329	−	1.40699i	−0.0989017	−	0.995097i	$-0.531533\pi$
$491$	18.0000	−	31.1769i	0.812329	−	1.40699i	0.911230	−	0.411897i	$-0.135134\pi$
$492$		0			0
$493$	−36.0000			−1.62136
$494$		0			0
$495$	−12.0000			−0.539360
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$	−15.0000	+	8.66025i	−0.670151	+	0.386912i
$502$		0			0
$503$	−12.0000	−	20.7846i	−0.535054	−	0.926740i	−0.999161	−	0.0409609i	$-0.986958\pi$
$503$	−12.0000	−	20.7846i	−0.535054	−	0.926740i	0.464107	−	0.885779i	$-0.346375\pi$
$504$		0			0
$505$	−18.0000	−	10.3923i	−0.800989	−	0.462451i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	21.0000	+	12.1244i	0.930809	+	0.537403i	0.887067	−	0.461640i	$-0.152739\pi$
$509$	21.0000	+	12.1244i	0.930809	+	0.537403i	0.0437414	+	0.999043i	$0.486072\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	6.00000	−	3.46410i	0.264906	−	0.152944i
$514$		0			0
$515$		−	13.8564i		−	0.610586i
$516$		0			0
$517$	6.00000	−	10.3923i	0.263880	−	0.457053i
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$	−4.00000	+	6.92820i	−0.174908	+	0.302949i	−0.940129	−	0.340818i	$-0.889296\pi$
$523$	−4.00000	+	6.92820i	−0.174908	+	0.302949i	0.765222	+	0.643767i	$0.222629\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−36.0000	+	20.7846i	−1.56818	+	0.905392i
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$	3.00000	+	1.73205i	0.130189	+	0.0751646i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	36.0000	+	20.7846i	1.55642	+	0.898597i
$536$		0			0
$537$	−6.00000	−	10.3923i	−0.258919	−	0.448461i
$538$		0			0
$539$	21.0000	−	12.1244i	0.904534	−	0.522233i
$540$		0			0
$541$		−	20.7846i		−	0.893600i	−0.894634	−	0.446800i	$-0.852564\pi$
$541$		−	20.7846i		−	0.893600i	0.894634	−	0.446800i	$-0.147436\pi$
$542$		0			0
$543$	−11.0000	+	19.0526i	−0.472055	+	0.817624i
$544$		0			0
$545$	24.0000			1.02805
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$		0			0
$549$	−5.00000	+	8.66025i	−0.213395	+	0.369611i
$550$		0			0
$551$			41.5692i			1.77091i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	21.0000	+	12.1244i	0.889799	+	0.513725i	0.873877	−	0.486148i	$-0.161598\pi$
$557$	21.0000	+	12.1244i	0.889799	+	0.513725i	0.0159220	+	0.999873i	$0.494932\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	18.0000	+	10.3923i	0.759961	+	0.438763i
$562$		0			0
$563$	18.0000	+	31.1769i	0.758610	+	1.31395i	0.943560	+	0.331202i	$0.107454\pi$
$563$	18.0000	+	31.1769i	0.758610	+	1.31395i	−0.184950	+	0.982748i	$0.559212\pi$
$564$		0			0
$565$	−54.0000	+	31.1769i	−2.27180	+	1.31162i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.00000	+	15.5885i	−0.377300	+	0.653502i	−0.990668	−	0.136295i	$-0.956481\pi$
$569$	−9.00000	+	15.5885i	−0.377300	+	0.653502i	0.613369	+	0.789797i	$0.289814\pi$
$570$		0			0
$571$	28.0000			1.17176			0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000			1.17176			0.585882	+	0.810397i	$0.300748\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	13.8564i		−	0.576850i	−0.957503	−	0.288425i	$-0.906868\pi$
$577$		−	13.8564i		−	0.576850i	0.957503	−	0.288425i	$-0.0931316\pi$
$578$		0			0
$579$	−12.0000	+	6.92820i	−0.498703	+	0.287926i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−18.0000	−	10.3923i	−0.745484	−	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	27.0000	+	15.5885i	1.11441	+	0.643404i	0.939968	−	0.341263i	$-0.110855\pi$
$587$	27.0000	+	15.5885i	1.11441	+	0.643404i	0.174441	+	0.984668i	$0.444188\pi$
$588$		0			0
$589$	24.0000	+	41.5692i	0.988903	+	1.71283i
$590$		0			0
$591$	−15.0000	+	8.66025i	−0.617018	+	0.356235i
$592$		0			0
$593$			24.2487i			0.995775i	0.867242	+	0.497888i	$0.165891\pi$
$593$			24.2487i			0.995775i	−0.867242	+	0.497888i	$0.834109\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	8.00000			0.327418
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−5.00000	+	8.66025i	−0.203954	+	0.353259i	−0.949799	−	0.312861i	$-0.898713\pi$
$601$	−5.00000	+	8.66025i	−0.203954	+	0.353259i	0.745845	+	0.666120i	$0.232046\pi$
$602$		0			0
$603$			13.8564i			0.564276i
$604$		0			0
$605$	−3.00000	+	1.73205i	−0.121967	+	0.0704179i
$606$		0			0
$607$	14.0000	+	24.2487i	0.568242	+	0.984225i	0.996740	+	0.0806818i	$0.0257098\pi$
$607$	14.0000	+	24.2487i	0.568242	+	0.984225i	−0.428497	+	0.903543i	$0.640957\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−12.0000	−	6.92820i	−0.484675	−	0.279827i	0.237687	−	0.971342i	$-0.423611\pi$
$613$	−12.0000	−	6.92820i	−0.484675	−	0.279827i	−0.722363	+	0.691514i	$0.756944\pi$
$614$		0			0
$615$	6.00000	+	10.3923i	0.241943	+	0.419058i
$616$		0			0
$617$	−27.0000	+	15.5885i	−1.08698	+	0.627568i	−0.932771	−	0.360471i	$-0.882616\pi$
$617$	−27.0000	+	15.5885i	−1.08698	+	0.627568i	−0.154209	+	0.988038i	$0.549283\pi$
$618$		0			0
$619$		−	27.7128i		−	1.11387i	−0.830555	−	0.556936i	$-0.811977\pi$
$619$		−	27.7128i		−	1.11387i	0.830555	−	0.556936i	$-0.188023\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$	12.0000	−	20.7846i	0.479234	−	0.830057i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	18.0000	−	10.3923i	0.716569	−	0.413711i	−0.0969198	−	0.995292i	$-0.530899\pi$
$631$	18.0000	−	10.3923i	0.716569	−	0.413711i	0.813488	+	0.581581i	$0.197566\pi$
$632$		0			0
$633$	2.00000	+	3.46410i	0.0794929	+	0.137686i
$634$		0			0
$635$	−48.0000	−	27.7128i	−1.90482	−	1.09975i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−9.00000	−	5.19615i	−0.356034	−	0.205557i
$640$		0			0
$641$	−15.0000	−	25.9808i	−0.592464	−	1.02618i	−0.993899	−	0.110291i	$-0.964822\pi$
$641$	−15.0000	−	25.9808i	−0.592464	−	1.02618i	0.401435	−	0.915888i	$-0.368512\pi$
$642$		0			0
$643$	−36.0000	+	20.7846i	−1.41970	+	0.819665i	−0.996272	−	0.0862642i	$-0.972507\pi$
$643$	−36.0000	+	20.7846i	−1.41970	+	0.819665i	−0.423429	+	0.905929i	$0.639174\pi$
$644$		0			0
$645$		−	27.7128i		−	1.09119i
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.00000	−	5.19615i	0.117399	−	0.203341i	−0.801337	−	0.598213i	$-0.795878\pi$
$653$	3.00000	−	5.19615i	0.117399	−	0.203341i	0.918736	+	0.394872i	$0.129211\pi$
$654$		0			0
$655$		−	41.5692i		−	1.62424i
$656$		0			0
$657$	−6.00000	+	3.46410i	−0.234082	+	0.135147i
$658$		0			0
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	0.725175	−	0.688565i	$-0.241759\pi$
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	−0.958902	+	0.283738i	$0.908425\pi$
$660$		0			0
$661$	12.0000	+	6.92820i	0.466746	+	0.269476i	0.714877	−	0.699251i	$-0.246483\pi$
$661$	12.0000	+	6.92820i	0.466746	+	0.269476i	−0.248131	+	0.968727i	$0.579816\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	6.00000	−	3.46410i	0.231973	−	0.133930i
$670$		0			0
$671$			34.6410i			1.33730i
$672$		0			0
$673$	−7.00000	+	12.1244i	−0.269830	+	0.467360i	−0.968818	−	0.247774i	$-0.920301\pi$
$673$	−7.00000	+	12.1244i	−0.269830	+	0.467360i	0.698988	+	0.715134i	$0.253634\pi$
$674$		0			0
$675$	7.00000			0.269430
$676$		0			0
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			10.3923i			0.398234i
$682$		0			0
$683$	3.00000	−	1.73205i	0.114792	−	0.0662751i	−0.441505	−	0.897259i	$-0.645555\pi$
$683$	3.00000	−	1.73205i	0.114792	−	0.0662751i	0.556297	+	0.830984i	$0.312222\pi$
$684$		0			0
$685$	−18.0000	−	31.1769i	−0.687745	−	1.19121i
$686$		0			0
$687$	6.00000	+	3.46410i	0.228914	+	0.132164i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−24.0000	−	13.8564i	−0.913003	−	0.527123i	−0.0316069	−	0.999500i	$-0.510062\pi$
$691$	−24.0000	−	13.8564i	−0.913003	−	0.527123i	−0.881396	+	0.472378i	$0.843396\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	24.0000	−	13.8564i	0.910372	−	0.525603i
$696$		0			0
$697$		−	20.7846i		−	0.787273i
$698$		0			0
$699$	9.00000	−	15.5885i	0.340411	−	0.589610i
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−6.00000	+	10.3923i	−0.225973	+	0.391397i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−6.00000	+	3.46410i	−0.225335	+	0.130097i	−0.608418	−	0.793617i	$-0.708196\pi$
$709$	−6.00000	+	3.46410i	−0.225335	+	0.130097i	0.383083	+	0.923714i	$0.374862\pi$
$710$		0			0
$711$	4.00000	+	6.92820i	0.150012	+	0.259828i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	9.00000	+	5.19615i	0.336111	+	0.194054i
$718$		0			0
$719$	−12.0000	−	20.7846i	−0.447524	−	0.775135i	0.550700	−	0.834703i	$-0.314361\pi$
$719$	−12.0000	−	20.7846i	−0.447524	−	0.775135i	−0.998224	+	0.0595683i	$0.981028\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			20.7846i			0.772988i
$724$		0			0
$725$	−21.0000	+	36.3731i	−0.779920	+	1.35086i
$726$		0			0
$727$	16.0000			0.593407			0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000			0.593407			0.296704	+	0.954970i	$0.404113\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−24.0000	+	41.5692i	−0.887672	+	1.53749i
$732$		0			0
$733$			6.92820i			0.255899i	0.991781	+	0.127950i	$0.0408395\pi$
$733$			6.92820i			0.255899i	−0.991781	+	0.127950i	$0.959160\pi$
$734$		0			0
$735$	−21.0000	+	12.1244i	−0.774597	+	0.447214i
$736$		0			0
$737$	24.0000	+	41.5692i	0.884051	+	1.53122i
$738$		0			0
$739$	−24.0000	−	13.8564i	−0.882854	−	0.509716i	−0.0112558	−	0.999937i	$-0.503583\pi$
$739$	−24.0000	−	13.8564i	−0.882854	−	0.509716i	−0.871598	+	0.490221i	$0.836916\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9.00000	−	5.19615i	−0.330178	−	0.190628i	0.325742	−	0.945459i	$-0.394386\pi$
$743$	−9.00000	−	5.19615i	−0.330178	−	0.190628i	−0.655920	+	0.754830i	$0.727719\pi$
$744$		0			0
$745$	−30.0000	−	51.9615i	−1.09911	−	1.90372i
$746$		0			0
$747$	−3.00000	+	1.73205i	−0.109764	+	0.0633724i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	14.0000	−	24.2487i	0.510867	−	0.884848i	−0.489053	−	0.872254i	$-0.662658\pi$
$751$	14.0000	−	24.2487i	0.510867	−	0.884848i	0.999921	−	0.0125942i	$-0.00400897\pi$
$752$		0			0
$753$	−12.0000			−0.437304
$754$		0			0
$755$	−24.0000			−0.873449
$756$		0			0
$757$	19.0000	−	32.9090i	0.690567	−	1.19610i	−0.281086	−	0.959683i	$-0.590695\pi$
$757$	19.0000	−	32.9090i	0.690567	−	1.19610i	0.971652	−	0.236414i	$-0.0759722\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	27.0000	−	15.5885i	0.978749	−	0.565081i	0.0768569	−	0.997042i	$-0.475512\pi$
$761$	27.0000	−	15.5885i	0.978749	−	0.565081i	0.901892	+	0.431961i	$0.142178\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−18.0000	−	10.3923i	−0.650791	−	0.375735i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−24.0000	−	13.8564i	−0.865462	−	0.499675i	0.000375472	−	1.00000i	$-0.499880\pi$
$769$	−24.0000	−	13.8564i	−0.865462	−	0.499675i	−0.865838	+	0.500325i	$0.833214\pi$
$770$		0			0
$771$	−3.00000	−	5.19615i	−0.108042	−	0.187135i
$772$		0			0
$773$	45.0000	−	25.9808i	1.61854	−	0.934463i	0.631239	−	0.775589i	$-0.282547\pi$
$773$	45.0000	−	25.9808i	1.61854	−	0.934463i	0.987299	−	0.158874i	$-0.0507865\pi$
$774$		0			0
$775$			48.4974i			1.74208i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.0000			−0.859889
$780$		0			0
$781$	−36.0000			−1.28818
$782$		0			0
$783$	−3.00000	+	5.19615i	−0.107211	+	0.185695i
$784$		0			0
$785$			6.92820i			0.247278i
$786$		0			0
$787$	−24.0000	+	13.8564i	−0.855508	+	0.493928i	−0.862505	−	0.506048i	$-0.831106\pi$
$787$	−24.0000	+	13.8564i	−0.855508	+	0.493928i	0.00699773	+	0.999976i	$0.497773\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	18.0000	+	10.3923i	0.638394	+	0.368577i
$796$		0			0
$797$	15.0000	+	25.9808i	0.531327	+	0.920286i	0.999331	+	0.0365596i	$0.0116399\pi$
$797$	15.0000	+	25.9808i	0.531327	+	0.920286i	−0.468004	+	0.883726i	$0.655027\pi$
$798$		0			0
$799$	18.0000	−	10.3923i	0.636794	−	0.367653i
$800$		0			0
$801$		−	17.3205i		−	0.611990i
$802$		0			0
$803$	−12.0000	+	20.7846i	−0.423471	+	0.733473i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	30.0000			1.05605
$808$		0			0
$809$	9.00000	−	15.5885i	0.316423	−	0.548061i	−0.663316	−	0.748340i	$-0.730851\pi$
$809$	9.00000	−	15.5885i	0.316423	−	0.548061i	0.979739	+	0.200279i	$0.0641847\pi$
$810$		0			0
$811$			6.92820i			0.243282i	0.992574	+	0.121641i	$0.0388157\pi$
$811$			6.92820i			0.243282i	−0.992574	+	0.121641i	$0.961184\pi$
$812$		0			0
$813$	24.0000	−	13.8564i	0.841717	−	0.485965i
$814$		0			0
$815$	−36.0000	−	62.3538i	−1.26102	−	2.18416i
$816$		0			0
$817$	48.0000	+	27.7128i	1.67931	+	0.969549i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.00000	+	1.73205i	0.104701	+	0.0604490i	0.551436	−	0.834217i	$-0.314080\pi$
$821$	3.00000	+	1.73205i	0.104701	+	0.0604490i	−0.446735	+	0.894666i	$0.647413\pi$
$822$		0			0
$823$	2.00000	+	3.46410i	0.0697156	+	0.120751i	0.898776	−	0.438408i	$-0.144457\pi$
$823$	2.00000	+	3.46410i	0.0697156	+	0.120751i	−0.829060	+	0.559159i	$0.811124\pi$
$824$		0			0
$825$	21.0000	−	12.1244i	0.731126	−	0.422116i
$826$		0			0
$827$			17.3205i			0.602293i	0.953578	+	0.301147i	$0.0973693\pi$
$827$			17.3205i			0.602293i	−0.953578	+	0.301147i	$0.902631\pi$
$828$		0			0
$829$	−17.0000	+	29.4449i	−0.590434	+	1.02266i	0.403739	+	0.914874i	$0.367710\pi$
$829$	−17.0000	+	29.4449i	−0.590434	+	1.02266i	−0.994174	+	0.107788i	$0.965623\pi$
$830$		0			0
$831$	10.0000			0.346896
$832$		0			0
$833$	42.0000			1.45521
$834$		0			0
$835$	30.0000	−	51.9615i	1.03819	−	1.79820i
$836$		0			0
$837$			6.92820i			0.239474i
$838$		0			0
$839$	9.00000	−	5.19615i	0.310715	−	0.179391i	−0.336532	−	0.941672i	$-0.609254\pi$
$839$	9.00000	−	5.19615i	0.310715	−	0.179391i	0.647246	+	0.762281i	$0.275921\pi$
$840$		0			0
$841$	−3.50000	−	6.06218i	−0.120690	−	0.209041i
$842$		0			0
$843$	15.0000	+	8.66025i	0.516627	+	0.298275i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−10.0000	−	17.3205i	−0.343199	−	0.594438i
$850$		0			0
$851$		0			0
$852$		0			0
$853$			13.8564i			0.474434i	0.971457	+	0.237217i	$0.0762353\pi$
$853$			13.8564i			0.474434i	−0.971457	+	0.237217i	$0.923765\pi$
$854$		0			0
$855$	−12.0000	+	20.7846i	−0.410391	+	0.710819i
$856$		0			0
$857$	−6.00000			−0.204956			−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000			−0.204956			−0.102478	+	0.994735i	$0.532677\pi$
$858$		0			0
$859$	28.0000			0.955348			0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000			0.955348			0.477674	+	0.878537i	$0.341480\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	38.1051i		−	1.29711i	−0.761166	−	0.648557i	$-0.775373\pi$
$863$		−	38.1051i		−	1.29711i	0.761166	−	0.648557i	$-0.224627\pi$
$864$		0			0
$865$	18.0000	−	10.3923i	0.612018	−	0.353349i
$866$		0			0
$867$	9.50000	+	16.4545i	0.322637	+	0.558824i
$868$		0			0
$869$	24.0000	+	13.8564i	0.814144	+	0.470046i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−6.00000	−	3.46410i	−0.203069	−	0.117242i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	36.0000	−	20.7846i	1.21563	−	0.701846i	0.251653	−	0.967818i	$-0.419026\pi$
$877$	36.0000	−	20.7846i	1.21563	−	0.701846i	0.963981	+	0.265971i	$0.0856926\pi$
$878$		0			0
$879$		−	3.46410i		−	0.116841i
$880$		0			0
$881$	3.00000	−	5.19615i	0.101073	−	0.175063i	−0.811054	−	0.584971i	$-0.801106\pi$
$881$	3.00000	−	5.19615i	0.101073	−	0.175063i	0.912127	+	0.409908i	$0.134439\pi$
$882$		0			0
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$		0			0
$885$	−12.0000			−0.403376
$886$		0			0
$887$	24.0000	−	41.5692i	0.805841	−	1.39576i	−0.109881	−	0.993945i	$-0.535047\pi$
$887$	24.0000	−	41.5692i	0.805841	−	1.39576i	0.915722	−	0.401813i	$-0.131620\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	3.00000	−	1.73205i	0.100504	−	0.0580259i
$892$		0			0
$893$	−12.0000	−	20.7846i	−0.401565	−	0.695530i
$894$		0			0
$895$	36.0000	+	20.7846i	1.20335	+	0.694753i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−36.0000	−	20.7846i	−1.20067	−	0.693206i
$900$		0			0
$901$	−18.0000	−	31.1769i	−0.599667	−	1.03865i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	76.2102i		−	2.53331i
$906$		0			0
$907$	−14.0000	+	24.2487i	−0.464862	+	0.805165i	−0.999195	−	0.0401089i	$-0.987230\pi$
$907$	−14.0000	+	24.2487i	−0.464862	+	0.805165i	0.534333	+	0.845274i	$0.320563\pi$
$908$		0			0
$909$	6.00000			0.199007
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$	−6.00000	+	10.3923i	−0.198571	+	0.343935i
$914$		0			0
$915$		−	34.6410i		−	1.14520i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	16.0000	+	27.7128i	0.527791	+	0.914161i	0.999475	+	0.0323936i	$0.0103130\pi$
$919$	16.0000	+	27.7128i	0.527791	+	0.914161i	−0.471684	+	0.881768i	$0.656354\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	2.00000	+	3.46410i	0.0656886	+	0.113776i
$928$		0			0
$929$	−27.0000	+	15.5885i	−0.885841	+	0.511441i	−0.872580	−	0.488471i	$-0.837555\pi$
$929$	−27.0000	+	15.5885i	−0.885841	+	0.511441i	−0.0132613	+	0.999912i	$0.504221\pi$
$930$		0			0
$931$		−	48.4974i		−	1.58944i
$932$		0			0
$933$	12.0000	−	20.7846i	0.392862	−	0.680458i
$934$		0			0
$935$	−72.0000			−2.35465
$936$		0			0
$937$	22.0000			0.718709			0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000			0.718709			0.359354	+	0.933201i	$0.382997\pi$
$938$		0			0
$939$	5.00000	−	8.66025i	0.163169	−	0.282617i
$940$		0			0
$941$			10.3923i			0.338779i	0.985549	+	0.169390i	$0.0541797\pi$
$941$			10.3923i			0.338779i	−0.985549	+	0.169390i	$0.945820\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	39.0000	+	22.5167i	1.26733	+	0.731693i	0.974482	−	0.224466i	$-0.0720637\pi$
$947$	39.0000	+	22.5167i	1.26733	+	0.731693i	0.292848	+	0.956159i	$0.405397\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−21.0000	−	12.1244i	−0.680972	−	0.393159i
$952$		0			0
$953$	3.00000	+	5.19615i	0.0971795	+	0.168320i	0.910516	−	0.413473i	$-0.135685\pi$
$953$	3.00000	+	5.19615i	0.0971795	+	0.168320i	−0.813337	+	0.581793i	$0.802351\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			20.7846i			0.671871i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−17.0000			−0.548387
$962$		0			0
$963$	−12.0000			−0.386695
$964$		0			0
$965$	24.0000	−	41.5692i	0.772587	−	1.33816i
$966$		0			0
$967$		−	34.6410i		−	1.11398i	−0.830519	−	0.556990i	$-0.811956\pi$
$967$		−	34.6410i		−	1.11398i	0.830519	−	0.556990i	$-0.188044\pi$
$968$		0			0
$969$	36.0000	−	20.7846i	1.15649	−	0.667698i
$970$		0			0
$971$	−30.0000	−	51.9615i	−0.962746	−	1.66752i	−0.715553	−	0.698558i	$-0.753825\pi$
$971$	−30.0000	−	51.9615i	−0.962746	−	1.66752i	−0.247193	−	0.968966i	$-0.579508\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	21.0000	+	12.1244i	0.671850	+	0.387893i	0.796777	−	0.604273i	$-0.206537\pi$
$977$	21.0000	+	12.1244i	0.671850	+	0.387893i	−0.124928	+	0.992166i	$0.539870\pi$
$978$		0			0
$979$	−30.0000	−	51.9615i	−0.958804	−	1.66070i
$980$		0			0
$981$	−6.00000	+	3.46410i	−0.191565	+	0.110600i
$982$		0			0
$983$		−	31.1769i		−	0.994389i	−0.867639	−	0.497195i	$-0.834364\pi$
$983$		−	31.1769i		−	0.994389i	0.867639	−	0.497195i	$-0.165636\pi$
$984$		0			0
$985$	30.0000	−	51.9615i	0.955879	−	1.65563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−10.0000	+	17.3205i	−0.317660	+	0.550204i	−0.979999	−	0.199000i	$-0.936231\pi$
$991$	−10.0000	+	17.3205i	−0.317660	+	0.550204i	0.662339	+	0.749204i	$0.269564\pi$
$992$		0			0
$993$		−	13.8564i		−	0.439720i
$994$		0			0
$995$	−24.0000	+	13.8564i	−0.760851	+	0.439278i
$996$		0			0
$997$	−19.0000	−	32.9090i	−0.601736	−	1.04224i	−0.992558	−	0.121771i	$-0.961143\pi$
$997$	−19.0000	−	32.9090i	−0.601736	−	1.04224i	0.390822	−	0.920466i	$-0.372191\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.q.b.1837.1		2
13.2	odd	12		2028.2.i.i.2005.2		4
13.3	even	3		2028.2.q.c.361.1		2
13.4	even	6		156.2.b.a.25.1	✓	2
13.5	odd	4		2028.2.i.i.529.2		4
13.6	odd	12		2028.2.a.g.1.2		2
13.7	odd	12		2028.2.a.g.1.1		2
13.8	odd	4		2028.2.i.i.529.1		4
13.9	even	3		156.2.b.a.25.2	yes	2
13.10	even	6	inner	2028.2.q.b.361.1		2
13.11	odd	12		2028.2.i.i.2005.1		4
13.12	even	2		2028.2.q.c.1837.1		2
39.17	odd	6		468.2.b.a.181.2		2
39.20	even	12		6084.2.a.v.1.2		2
39.32	even	12		6084.2.a.v.1.1		2
39.35	odd	6		468.2.b.a.181.1		2
52.7	even	12		8112.2.a.bs.1.1		2
52.19	even	12		8112.2.a.bs.1.2		2
52.35	odd	6		624.2.c.f.337.2		2
52.43	odd	6		624.2.c.f.337.1		2
65.4	even	6		3900.2.c.c.3301.1		2
65.9	even	6		3900.2.c.c.3301.2		2
65.17	odd	12		3900.2.j.h.649.3		4
65.22	odd	12		3900.2.j.h.649.4		4
65.43	odd	12		3900.2.j.h.649.1		4
65.48	odd	12		3900.2.j.h.649.2		4
91.48	odd	6		7644.2.e.g.4705.1		2
91.69	odd	6		7644.2.e.g.4705.2		2
104.35	odd	6		2496.2.c.e.961.1		2
104.43	odd	6		2496.2.c.e.961.2		2
104.61	even	6		2496.2.c.l.961.1		2
104.69	even	6		2496.2.c.l.961.2		2
156.35	even	6		1872.2.c.c.1585.1		2
156.95	even	6		1872.2.c.c.1585.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.b.a.25.1	✓	2	13.4	even	6
156.2.b.a.25.2	yes	2	13.9	even	3
468.2.b.a.181.1		2	39.35	odd	6
468.2.b.a.181.2		2	39.17	odd	6
624.2.c.f.337.1		2	52.43	odd	6
624.2.c.f.337.2		2	52.35	odd	6
1872.2.c.c.1585.1		2	156.35	even	6
1872.2.c.c.1585.2		2	156.95	even	6
2028.2.a.g.1.1		2	13.7	odd	12
2028.2.a.g.1.2		2	13.6	odd	12
2028.2.i.i.529.1		4	13.8	odd	4
2028.2.i.i.529.2		4	13.5	odd	4
2028.2.i.i.2005.1		4	13.11	odd	12
2028.2.i.i.2005.2		4	13.2	odd	12
2028.2.q.b.361.1		2	13.10	even	6	inner
2028.2.q.b.1837.1		2	1.1	even	1	trivial
2028.2.q.c.361.1		2	13.3	even	3
2028.2.q.c.1837.1		2	13.12	even	2
2496.2.c.e.961.1		2	104.35	odd	6
2496.2.c.e.961.2		2	104.43	odd	6
2496.2.c.l.961.1		2	104.61	even	6
2496.2.c.l.961.2		2	104.69	even	6
3900.2.c.c.3301.1		2	65.4	even	6
3900.2.c.c.3301.2		2	65.9	even	6
3900.2.j.h.649.1		4	65.43	odd	12
3900.2.j.h.649.2		4	65.48	odd	12
3900.2.j.h.649.3		4	65.17	odd	12
3900.2.j.h.649.4		4	65.22	odd	12
6084.2.a.v.1.1		2	39.32	even	12
6084.2.a.v.1.2		2	39.20	even	12
7644.2.e.g.4705.1		2	91.48	odd	6
7644.2.e.g.4705.2		2	91.69	odd	6
8112.2.a.bs.1.1		2	52.7	even	12
8112.2.a.bs.1.2		2	52.19	even	12

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:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +3.46410i q^{5} +(-0.500000 - 0.866025i) q^{9} +(-3.00000 - 1.73205i) q^{11} +(3.00000 + 1.73205i) q^{15} +(-3.00000 - 5.19615i) q^{17} +(-6.00000 + 3.46410i) q^{19} -7.00000 q^{25} -1.00000 q^{27} +(3.00000 - 5.19615i) q^{29} -6.92820i q^{31} +(-3.00000 + 1.73205i) q^{33} +(3.00000 + 1.73205i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(3.00000 - 1.73205i) q^{45} +3.46410i q^{47} +(-3.50000 + 6.06218i) q^{49} -6.00000 q^{51} +6.00000 q^{53} +(6.00000 - 10.3923i) q^{55} +6.92820i q^{57} +(-3.00000 + 1.73205i) q^{59} +(-5.00000 - 8.66025i) q^{61} +(-12.0000 - 6.92820i) q^{67} +(9.00000 - 5.19615i) q^{71} -6.92820i q^{73} +(-3.50000 + 6.06218i) q^{75} -8.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} -3.46410i q^{83} +(18.0000 - 10.3923i) q^{85} +(-3.00000 - 5.19615i) q^{87} +(15.0000 + 8.66025i) q^{89} +(-6.00000 - 3.46410i) q^{93} +(-12.0000 - 20.7846i) q^{95} +(6.00000 - 3.46410i) q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - q^{9} - 6 q^{11} + 6 q^{15} - 6 q^{17} - 12 q^{19} - 14 q^{25} - 2 q^{27} + 6 q^{29} - 6 q^{33} + 6 q^{41} - 8 q^{43} + 6 q^{45} - 7 q^{49} - 12 q^{51} + 12 q^{53} + 12 q^{55} - 6 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) \) 2 * q + q^3 - q^9 - 6 * q^11 + 6 * q^15 - 6 * q^17 - 12 * q^19 - 14 * q^25 - 2 * q^27 + 6 * q^29 - 6 * q^33 + 6 * q^41 - 8 * q^43 + 6 * q^45 - 7 * q^49 - 12 * q^51 + 12 * q^53 + 12 * q^55 - 6 * q^59 - 10 * q^61 - 24 * q^67 + 18 * q^71 - 7 * q^75 - 16 * q^79 - q^81 + 36 * q^85 - 6 * q^87 + 30 * q^89 - 12 * q^93 - 24 * q^95 + 12 * q^97

Introduction

L-functions

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