LMFDB - Embedded newform 507.2.k.b.188.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(80,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 1]))

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

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

chi := DirichletCharacter("507.80");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.k (of order $12$, degree $4$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			188.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	507.188
Dual form			507.2.k.b.89.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(1.13397 - 4.23205i) q^{7} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})$	$q+(-0.866025 + 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(1.13397 - 4.23205i) q^{7} +(-1.50000 - 2.59808i) q^{9} -3.46410i q^{12} +(2.00000 - 3.46410i) q^{16} +(3.09808 + 0.830127i) q^{19} +(5.36603 + 5.36603i) q^{21} -5.00000i q^{25} +5.19615 q^{27} +(2.26795 + 8.46410i) q^{28} +(0.830127 - 0.830127i) q^{31} +(5.19615 + 3.00000i) q^{36} +(11.5622 - 3.09808i) q^{37} +(1.50000 - 0.866025i) q^{43} +(3.46410 + 6.00000i) q^{48} +(-10.5622 - 6.09808i) q^{49} +(-3.92820 + 3.92820i) q^{57} +(-4.33013 - 7.50000i) q^{61} +(-12.6962 + 3.40192i) q^{63} +8.00000i q^{64} +(-4.23205 - 15.7942i) q^{67} +(-7.63397 - 7.63397i) q^{73} +(7.50000 + 4.33013i) q^{75} +(-6.19615 + 1.66025i) q^{76} -12.1244 q^{79} +(-4.50000 + 7.79423i) q^{81} +(-14.6603 - 3.92820i) q^{84} +(0.526279 + 1.96410i) q^{93} +(9.59808 + 2.57180i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7} - 6 q^{9}+O(q^{10}) $ 4 * q + 8 * q^7 - 6 * q^9	$ 4 q + 8 q^{7} - 6 q^{9} + 8 q^{16} + 2 q^{19} + 18 q^{21} + 16 q^{28} - 14 q^{31} + 22 q^{37} + 6 q^{43} - 18 q^{49} + 12 q^{57} - 30 q^{63} - 10 q^{67} - 34 q^{73} + 30 q^{75} - 4 q^{76} - 18 q^{81} - 24 q^{84} - 36 q^{93} + 28 q^{97}+O(q^{100}) $ 4 * q + 8 * q^7 - 6 * q^9 + 8 * q^16 + 2 * q^19 + 18 * q^21 + 16 * q^28 - 14 * q^31 + 22 * q^37 + 6 * q^43 - 18 * q^49 + 12 * q^57 - 30 * q^63 - 10 * q^67 - 34 * q^73 + 30 * q^75 - 4 * q^76 - 18 * q^81 - 24 * q^84 - 36 * q^93 + 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{5}{12}\right)$

Coefficient data

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

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

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

$2$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$2$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$3$	−0.866025	+	1.50000i	−0.500000	+	0.866025i
$3$	−0.866025	+	1.50000i	−0.500000	+	0.866025i
$4$	−1.73205	+	1.00000i	−0.866025	+	0.500000i
$5$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$5$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$6$		0			0
$7$	1.13397	−	4.23205i	0.428602	−	1.59956i	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	1.13397	−	4.23205i	0.428602	−	1.59956i	0.755929	−	0.654654i	$-0.227186\pi$
$8$		0			0
$9$	−1.50000	−	2.59808i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$11$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$12$		−	3.46410i		−	1.00000i
$13$		0			0
$13$		0			0
$14$		0			0
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	3.09808	+	0.830127i	0.710747	+	0.190444i	0.596040	−	0.802955i	$-0.296740\pi$
$19$	3.09808	+	0.830127i	0.710747	+	0.190444i	0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$	5.36603	+	5.36603i	1.17096	+	1.17096i
$22$		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
$25$		−	5.00000i		−	1.00000i
$26$		0			0
$27$	5.19615			1.00000
$28$	2.26795	+	8.46410i	0.428602	+	1.59956i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	0.830127	−	0.830127i	0.149095	−	0.149095i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	0.830127	−	0.830127i	0.149095	−	0.149095i	0.777714	+	0.628619i	$0.216379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	5.19615	+	3.00000i	0.866025	+	0.500000i
$37$	11.5622	−	3.09808i	1.90081	−	0.509321i	0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.5622	−	3.09808i	1.90081	−	0.509321i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$41$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$42$		0			0
$43$	1.50000	−	0.866025i	0.228748	−	0.132068i	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	1.50000	−	0.866025i	0.228748	−	0.132068i	0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	3.46410	+	6.00000i	0.500000	+	0.866025i
$49$	−10.5622	−	6.09808i	−1.50888	−	0.871154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−3.92820	+	3.92820i	−0.520303	+	0.520303i
$58$		0			0
$59$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$59$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$60$		0			0
$61$	−4.33013	−	7.50000i	−0.554416	−	0.960277i	−0.997949	−	0.0640184i	$-0.979608\pi$
$61$	−4.33013	−	7.50000i	−0.554416	−	0.960277i	0.443533	−	0.896258i	$-0.353725\pi$
$62$		0			0
$63$	−12.6962	+	3.40192i	−1.59956	+	0.428602i
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	−4.23205	−	15.7942i	−0.517027	−	1.92957i	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−4.23205	−	15.7942i	−0.517027	−	1.92957i	−0.211604	−	0.977356i	$-0.567869\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$71$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$72$		0			0
$73$	−7.63397	−	7.63397i	−0.893489	−	0.893489i	0.101361	−	0.994850i	$-0.467680\pi$
$73$	−7.63397	−	7.63397i	−0.893489	−	0.893489i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$	7.50000	+	4.33013i	0.866025	+	0.500000i
$76$	−6.19615	+	1.66025i	−0.710747	+	0.190444i
$77$		0			0
$78$		0			0
$79$	−12.1244			−1.36410			−0.682048	−	0.731307i	$-0.738911\pi$
$79$	−12.1244			−1.36410			−0.682048	+	0.731307i	$0.738911\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$	−14.6603	−	3.92820i	−1.59956	−	0.428602i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$89$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	0.526279	+	1.96410i	0.0545726	+	0.203668i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	9.59808	+	2.57180i	0.974537	+	0.261126i	0.710742	−	0.703452i	$-0.248359\pi$
$97$	9.59808	+	2.57180i	0.974537	+	0.261126i	0.263795	+	0.964579i	$0.415026\pi$
$98$		0			0
$99$		0			0
$100$	5.00000	+	8.66025i	0.500000	+	0.866025i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$			15.5885i			1.53598i	0.640464	+	0.767988i	$0.278742\pi$
$103$			15.5885i			1.53598i	−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	−9.00000	+	5.19615i	−0.866025	+	0.500000i
$109$	13.8301	−	13.8301i	1.32469	−	1.32469i	0.414751	−	0.909935i	$-0.363869\pi$
$109$	13.8301	−	13.8301i	1.32469	−	1.32469i	0.909935	−	0.414751i	$-0.136131\pi$
$110$		0			0
$111$	−5.36603	+	20.0263i	−0.509321	+	1.90081i
$112$	−12.3923	−	12.3923i	−1.17096	−	1.17096i
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	9.52628	−	5.50000i	0.866025	−	0.500000i
$122$		0			0
$123$		0			0
$124$	−0.607695	+	2.26795i	−0.0545726	+	0.203668i
$125$		0			0
$126$		0			0
$127$	−0.866025	−	0.500000i	−0.0768473	−	0.0443678i	0.461084	−	0.887357i	$-0.347461\pi$
$127$	−0.866025	−	0.500000i	−0.0768473	−	0.0443678i	−0.537931	+	0.842989i	$0.680794\pi$
$128$		0			0
$129$			3.00000i			0.264135i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	7.02628	−	12.1699i	0.609256	−	1.05526i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$137$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$138$		0			0
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	0.975417	−	0.220366i	$-0.0707252\pi$
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$	18.2942	−	10.5622i	1.50888	−	0.871154i
$148$	−16.9282	+	16.9282i	−1.39149	+	1.39149i
$149$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$149$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$150$		0			0
$151$	10.1244	+	10.1244i	0.823908	+	0.823908i	0.986666	−	0.162758i	$-0.0520389\pi$
$151$	10.1244	+	10.1244i	0.823908	+	0.823908i	−0.162758	+	0.986666i	$0.552039\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−11.0000			−0.877896			−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000			−0.877896			−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3.62436	+	13.5263i	−0.283881	+	1.05946i	0.665771	+	0.746156i	$0.268103\pi$
$163$	−3.62436	+	13.5263i	−0.283881	+	1.05946i	−0.949653	+	0.313304i	$0.898564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$167$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	−2.49038	−	9.29423i	−0.190444	−	0.710747i
$172$	−1.73205	+	3.00000i	−0.132068	+	0.228748i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−21.1603	−	5.66987i	−1.59956	−	0.428602i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$		−	6.92820i		−	0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$		−	6.92820i		−	0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$	15.0000			1.10883
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	5.89230	−	21.9904i	0.428602	−	1.59956i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−12.0000	−	6.92820i	−0.866025	−	0.500000i
$193$	5.06218	−	1.35641i	0.364384	−	0.0976363i	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	5.06218	−	1.35641i	0.364384	−	0.0976363i	0.436365	+	0.899770i	$0.356266\pi$
$194$		0			0
$195$		0			0
$196$	24.3923			1.74231
$197$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$197$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$198$		0			0
$199$	−14.7224	+	8.50000i	−1.04365	+	0.602549i	−0.920864	−	0.389885i	$-0.872515\pi$
$199$	−14.7224	+	8.50000i	−1.04365	+	0.602549i	−0.122782	+	0.992434i	$0.539182\pi$
$200$		0			0
$201$	27.3564	+	7.33013i	1.92957	+	0.517027i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−12.9904	+	22.5000i	−0.894295	+	1.54896i	−0.0596196	+	0.998221i	$0.518989\pi$
$211$	−12.9904	+	22.5000i	−0.894295	+	1.54896i	−0.834675	+	0.550743i	$0.814345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.57180	−	4.45448i	−0.174585	−	0.302390i
$218$		0			0
$219$	18.0622	−	4.83975i	1.22053	−	0.327040i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	7.02628	+	26.2224i	0.470514	+	1.75598i	0.637927	+	0.770097i	$0.279792\pi$
$223$	7.02628	+	26.2224i	0.470514	+	1.75598i	−0.167412	+	0.985887i	$0.553541\pi$
$224$		0			0
$225$	−12.9904	+	7.50000i	−0.866025	+	0.500000i
$226$		0			0
$227$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$227$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$228$	2.87564	−	10.7321i	0.190444	−	0.710747i
$229$	0.607695	+	0.607695i	0.0401576	+	0.0401576i	0.726900	−	0.686743i	$-0.240960\pi$
$229$	0.607695	+	0.607695i	0.0401576	+	0.0401576i	−0.686743	+	0.726900i	$0.740960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	10.5000	−	18.1865i	0.682048	−	1.18134i
$238$		0			0
$239$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$239$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	7.63397	−	28.4904i	0.491748	−	1.83523i	−0.0557856	−	0.998443i	$-0.517766\pi$
$241$	7.63397	−	28.4904i	0.491748	−	1.83523i	0.547533	−	0.836784i	$-0.315567\pi$
$242$		0			0
$243$	−7.79423	−	13.5000i	−0.500000	−	0.866025i
$244$	15.0000	+	8.66025i	0.960277	+	0.554416i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$	18.5885	−	18.5885i	1.17096	−	1.17096i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		−	52.4449i		−	3.25877i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	23.1244	+	23.1244i	1.41254	+	1.41254i
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	−30.4545	+	8.16025i	−1.84998	+	0.495700i	−0.999539	−	0.0303728i	$-0.990331\pi$
$271$	−30.4545	+	8.16025i	−1.84998	+	0.495700i	−0.850439	+	0.526073i	$0.823664\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−18.0000	+	10.3923i	−1.08152	+	0.624413i	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−18.0000	+	10.3923i	−1.08152	+	0.624413i	−0.150210	+	0.988654i	$0.547995\pi$
$278$		0			0
$279$	−3.40192	−	0.911543i	−0.203668	−	0.0545726i
$280$		0			0
$281$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$281$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$282$		0			0
$283$	21.6506	+	12.5000i	1.28700	+	0.743048i	0.978117	−	0.208053i	$-0.0667128\pi$
$283$	21.6506	+	12.5000i	1.28700	+	0.743048i	0.308879	+	0.951101i	$0.400046\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$	−12.1699	+	12.1699i	−0.713411	+	0.713411i
$292$	20.8564	+	5.58846i	1.22053	+	0.327040i
$293$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$293$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−17.3205			−1.00000
$301$	−1.96410	−	7.33013i	−0.113209	−	0.422501i
$302$		0			0
$303$		0			0
$304$	9.07180	−	9.07180i	0.520303	−	0.520303i
$305$		0			0
$306$		0			0
$307$	18.3660	+	18.3660i	1.04820	+	1.04820i	0.998778	+	0.0494267i	$0.0157394\pi$
$307$	18.3660	+	18.3660i	1.04820	+	1.04820i	0.0494267	+	0.998778i	$0.484261\pi$
$308$		0			0
$309$	−23.3827	−	13.5000i	−1.33019	−	0.767988i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	32.9090			1.86012			0.930062	−	0.367402i	$-0.119753\pi$
$313$	32.9090			1.86012			0.930062	+	0.367402i	$0.119753\pi$
$314$		0			0
$315$		0			0
$316$	21.0000	−	12.1244i	1.18134	−	0.682048i
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	18.0000i		−	1.00000i
$325$		0			0
$326$		0			0
$327$	8.76795	+	32.7224i	0.484869	+	1.80955i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−8.16025	−	2.18653i	−0.448528	−	0.120183i	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	−8.16025	−	2.18653i	−0.448528	−	0.120183i	−0.476011	+	0.879440i	$0.657918\pi$
$332$		0			0
$333$	−25.3923	−	25.3923i	−1.39149	−	1.39149i
$334$		0			0
$335$		0			0
$336$	29.3205	−	7.85641i	1.59956	−	0.428602i
$337$			29.0000i			1.57973i	0.613280	+	0.789865i	$0.289850\pi$
$337$			29.0000i			1.57973i	−0.613280	+	0.789865i	$0.710150\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−16.0981	+	16.0981i	−0.869214	+	0.869214i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	35.8205	−	9.59808i	1.91743	−	0.513773i	0.927146	−	0.374701i	$-0.122255\pi$
$349$	35.8205	−	9.59808i	1.91743	−	0.513773i	0.990282	−	0.139072i	$-0.0444119\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$353$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$		0			0
$361$	−7.54552	−	4.35641i	−0.397132	−	0.229285i
$362$		0			0
$363$			19.0526i			1.00000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−15.5000	+	26.8468i	−0.809093	+	1.40139i	0.104399	+	0.994535i	$0.466708\pi$
$367$	−15.5000	+	26.8468i	−0.809093	+	1.40139i	−0.913493	+	0.406855i	$0.866625\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−2.87564	−	2.87564i	−0.149095	−	0.149095i
$373$	18.1865	+	31.5000i	0.941663	+	1.63101i	0.762299	+	0.647225i	$0.224071\pi$
$373$	18.1865	+	31.5000i	0.941663	+	1.63101i	0.179364	+	0.983783i	$0.442596\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−8.99038	−	33.5526i	−0.461805	−	1.72348i	−0.667271	−	0.744815i	$-0.732538\pi$
$379$	−8.99038	−	33.5526i	−0.461805	−	1.72348i	0.205466	−	0.978664i	$-0.434129\pi$
$380$		0			0
$381$	1.50000	−	0.866025i	0.0768473	−	0.0443678i
$382$		0			0
$383$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$383$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.50000	−	2.59808i	−0.228748	−	0.132068i
$388$	−19.1962	+	5.14359i	−0.974537	+	0.261126i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−7.10770	+	26.5263i	−0.356725	+	1.33132i	0.521575	+	0.853206i	$0.325345\pi$
$397$	−7.10770	+	26.5263i	−0.356725	+	1.33132i	−0.878300	+	0.478110i	$0.841322\pi$
$398$		0			0
$399$	12.1699	+	21.0788i	0.609256	+	1.05526i
$400$	−17.3205	−	10.0000i	−0.866025	−	0.500000i
$401$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$401$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−38.9186	−	10.4282i	−1.92440	−	0.515641i	−0.984911	−	0.173064i	$-0.944633\pi$
$409$	−38.9186	−	10.4282i	−1.92440	−	0.515641i	−0.939490	−	0.342578i	$-0.888700\pi$
$410$		0			0
$411$		0			0
$412$	−15.5885	−	27.0000i	−0.767988	−	1.33019i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−12.1244			−0.593732
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	−8.68653	+	8.68653i	−0.423356	+	0.423356i	−0.886357	−	0.463002i	$-0.846772\pi$
$421$	−8.68653	+	8.68653i	−0.423356	+	0.423356i	0.463002	+	0.886357i	$0.346772\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−36.6506	+	9.82051i	−1.77365	+	0.475248i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$431$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$432$	10.3923	−	18.0000i	0.500000	−	0.866025i
$433$	30.3109	−	17.5000i	1.45665	−	0.840996i	0.457804	−	0.889053i	$-0.348636\pi$
$433$	30.3109	−	17.5000i	1.45665	−	0.840996i	0.998845	+	0.0480569i	$0.0153029\pi$
$434$		0			0
$435$		0			0
$436$	−10.1244	+	37.7846i	−0.484869	+	1.80955i
$437$		0			0
$438$		0			0
$439$	34.5000	+	19.9186i	1.64660	+	0.950662i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	34.5000	+	19.9186i	1.64660	+	0.950662i	0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$			36.5885i			1.74231i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	−10.7321	−	40.0526i	−0.509321	−	1.90081i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	33.8564	+	9.07180i	1.59956	+	0.428602i
$449$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$449$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−23.9545	+	6.41858i	−1.12548	+	0.301571i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5.28461	+	19.7224i	0.247204	+	0.922576i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	5.28461	+	19.7224i	0.247204	+	0.922576i	−0.725059	+	0.688686i	$0.758188\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$461$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$462$		0			0
$463$	−20.6340	−	20.6340i	−0.958942	−	0.958942i	0.0402476	−	0.999190i	$-0.487185\pi$
$463$	−20.6340	−	20.6340i	−0.958942	−	0.958942i	−0.999190	+	0.0402476i	$0.987185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−71.6410			−3.30807
$470$		0			0
$471$	9.52628	−	16.5000i	0.438948	−	0.760280i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	4.15064	−	15.4904i	0.190444	−	0.710747i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$479$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−32.4186	−	8.68653i	−1.46903	−	0.393624i	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	−32.4186	−	8.68653i	−1.46903	−	0.393624i	−0.902597	+	0.430486i	$0.858342\pi$
$488$		0			0
$489$	−17.1506	−	17.1506i	−0.775579	−	0.775579i
$490$		0			0
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.21539	−	4.53590i	−0.0545726	−	0.203668i
$497$		0			0
$498$		0			0
$499$	31.5885	−	31.5885i	1.41409	−	1.41409i	0.697835	−	0.716258i	$-0.254147\pi$
$499$	31.5885	−	31.5885i	1.41409	−	1.41409i	0.716258	−	0.697835i	$-0.245853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	2.00000			0.0887357
$509$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$509$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$510$		0			0
$511$	−40.9641	+	23.6506i	−1.81215	+	1.04624i
$512$		0			0
$513$	16.0981	+	4.31347i	0.710747	+	0.190444i
$514$		0			0
$515$		0			0
$516$	−3.00000	−	5.19615i	−0.132068	−	0.228748i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	4.00000	−	6.92820i	0.174908	−	0.302949i	−0.765222	−	0.643767i	$-0.777371\pi$
$523$	4.00000	−	6.92820i	0.174908	−	0.302949i	0.940129	+	0.340818i	$0.110704\pi$
$524$		0			0
$525$	26.8301	−	26.8301i	1.17096	−	1.17096i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$			28.1051i			1.21851i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−30.1506	−	30.1506i	−1.29628	−	1.29628i	−0.930834	−	0.365444i	$-0.880917\pi$
$541$	−30.1506	−	30.1506i	−1.29628	−	1.29628i	−0.365444	−	0.930834i	$-0.619083\pi$
$542$		0			0
$543$	10.3923	+	6.00000i	0.445976	+	0.257485i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	41.0000			1.75303			0.876517	−	0.481371i	$-0.159861\pi$
$547$	41.0000			1.75303			0.876517	+	0.481371i	$0.159861\pi$
$548$		0			0
$549$	−12.9904	+	22.5000i	−0.554416	+	0.960277i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−13.7487	+	51.3109i	−0.584655	+	2.18196i
$554$		0			0
$555$		0			0
$556$	−12.1244	−	7.00000i	−0.514187	−	0.296866i
$557$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$557$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	27.8827	+	27.8827i	1.17096	+	1.17096i
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$			16.0000i			0.669579i	0.942293	+	0.334790i	$0.108665\pi$
$571$			16.0000i			0.669579i	−0.942293	+	0.334790i	$0.891335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	20.7846	−	12.0000i	0.866025	−	0.500000i
$577$	−29.9282	+	29.9282i	−1.24593	+	1.24593i	−0.288425	+	0.957503i	$0.593132\pi$
$577$	−29.9282	+	29.9282i	−1.24593	+	1.24593i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		0			0
$579$	−2.34936	+	8.76795i	−0.0976363	+	0.364384i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$587$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$588$	−21.1244	+	36.5885i	−0.871154	+	1.50888i
$589$	3.26091	−	1.88269i	0.134363	−	0.0775747i
$590$		0			0
$591$		0			0
$592$	12.3923	−	46.2487i	0.509321	−	1.90081i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	29.4449i		−	1.20510i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	20.7846	−	36.0000i	0.847822	−	1.46847i	−0.0353259	−	0.999376i	$-0.511247\pi$
$601$	20.7846	−	36.0000i	0.847822	−	1.46847i	0.883148	−	0.469095i	$-0.155420\pi$
$602$		0			0
$603$	−34.6865	+	34.6865i	−1.41254	+	1.41254i
$604$	−27.6603	−	7.41154i	−1.12548	−	0.301571i
$605$		0			0
$606$		0			0
$607$	10.0000	+	17.3205i	0.405887	+	0.703018i	0.994424	−	0.105453i	$-0.0336291\pi$
$607$	10.0000	+	17.3205i	0.405887	+	0.703018i	−0.588537	+	0.808470i	$0.700296\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.748711	−	2.79423i	−0.0302402	−	0.112858i	0.949156	−	0.314806i	$-0.101939\pi$
$613$	−0.748711	−	2.79423i	−0.0302402	−	0.112858i	−0.979396	+	0.201948i	$0.935273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$617$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$618$		0			0
$619$	14.8827	+	14.8827i	0.598186	+	0.598186i	0.939829	−	0.341644i	$-0.110984\pi$
$619$	14.8827	+	14.8827i	0.598186	+	0.598186i	−0.341644	+	0.939829i	$0.610984\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$	19.0526	−	11.0000i	0.760280	−	0.438948i
$629$		0			0
$630$		0			0
$631$	9.37564	−	34.9904i	0.373239	−	1.39295i	−0.482663	−	0.875806i	$-0.660330\pi$
$631$	9.37564	−	34.9904i	0.373239	−	1.39295i	0.855901	−	0.517139i	$-0.173003\pi$
$632$		0			0
$633$	−22.5000	−	38.9711i	−0.894295	−	1.54896i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$	45.1147	+	12.0885i	1.77915	+	0.476722i	0.990429	−	0.138027i	$-0.0440759\pi$
$643$	45.1147	+	12.0885i	1.77915	+	0.476722i	0.788723	+	0.614749i	$0.210743\pi$
$644$		0			0
$645$		0			0
$646$		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
$649$		0			0
$650$		0			0
$651$	8.90897			0.349170
$652$	−7.24871	−	27.0526i	−0.283881	−	1.05946i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−8.38269	+	31.2846i	−0.327040	+	1.22053i
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	22.8205	−	6.11474i	0.887615	−	0.237836i	0.213925	−	0.976850i	$-0.431375\pi$
$661$	22.8205	−	6.11474i	0.887615	−	0.237836i	0.673690	+	0.739014i	$0.264708\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−45.4186	−	12.1699i	−1.75598	−	0.470514i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−43.5000	−	25.1147i	−1.67680	−	0.968102i	−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−43.5000	−	25.1147i	−1.67680	−	0.968102i	−0.713123	−	0.701039i	$-0.752720\pi$
$674$		0			0
$675$		−	25.9808i		−	1.00000i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	21.7679	−	37.7032i	0.835377	−	1.44692i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$683$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$684$	13.6077	+	13.6077i	0.520303	+	0.520303i
$685$		0			0
$686$		0			0
$687$	−1.43782	+	0.385263i	−0.0548563	+	0.0146987i
$688$		−	6.92820i		−	0.264135i
$689$		0			0
$690$		0			0
$691$	13.5263	+	50.4808i	0.514564	+	1.92038i	0.362397	+	0.932024i	$0.381959\pi$
$691$	13.5263	+	50.4808i	0.514564	+	1.92038i	0.152167	+	0.988355i	$0.451375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	42.3205	−	11.3397i	1.59956	−	0.428602i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	38.3923			1.44799
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	10.6506	−	39.7487i	0.399993	−	1.49279i	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	10.6506	−	39.7487i	0.399993	−	1.49279i	0.813107	−	0.582115i	$-0.197775\pi$
$710$		0			0
$711$	18.1865	+	31.5000i	0.682048	+	1.18134i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	65.9711	+	17.6769i	2.45689	+	0.658323i
$722$		0			0
$723$	36.1244	+	36.1244i	1.34348	+	1.34348i
$724$	6.92820	+	12.0000i	0.257485	+	0.445976i
$725$		0			0
$726$		0			0
$727$		−	49.0000i		−	1.81731i	−0.417548	−	0.908655i	$-0.637111\pi$
$727$		−	49.0000i		−	1.81731i	0.417548	−	0.908655i	$-0.362889\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$	−25.9808	+	15.0000i	−0.960277	+	0.554416i
$733$	23.3468	−	23.3468i	0.862333	−	0.862333i	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	23.3468	−	23.3468i	0.862333	−	0.862333i	0.991609	+	0.129275i	$0.0412651\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	24.5622	−	6.58142i	0.903534	−	0.242101i	0.223001	−	0.974818i	$-0.428415\pi$
$739$	24.5622	−	6.58142i	0.903534	−	0.242101i	0.680534	+	0.732717i	$0.261748\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$743$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	15.0000	+	8.66025i	0.547358	+	0.316017i	0.748056	−	0.663636i	$-0.230988\pi$
$751$	15.0000	+	8.66025i	0.547358	+	0.316017i	−0.200698	+	0.979653i	$0.564321\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	11.7846	+	43.9808i	0.428602	+	1.59956i
$757$	−24.2487	+	42.0000i	−0.881334	+	1.52652i	−0.0314762	+	0.999505i	$0.510021\pi$
$757$	−24.2487	+	42.0000i	−0.881334	+	1.52652i	−0.849858	+	0.527011i	$0.823312\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$761$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$762$		0			0
$763$	−42.8468	−	74.2128i	−1.55116	−	2.68668i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	27.7128			1.00000
$769$	10.5096	+	39.2224i	0.378987	+	1.41440i	0.847432	+	0.530904i	$0.178148\pi$
$769$	10.5096	+	39.2224i	0.378987	+	1.41440i	−0.468445	+	0.883493i	$0.655186\pi$
$770$		0			0
$771$		0			0
$772$	−7.41154	+	7.41154i	−0.266747	+	0.266747i
$773$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$773$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$774$		0			0
$775$	−4.15064	−	4.15064i	−0.149095	−	0.149095i
$776$		0			0
$777$	78.6673	+	45.4186i	2.82217	+	1.62938i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−42.2487	+	24.3923i	−1.50888	+	0.871154i
$785$		0			0
$786$		0			0
$787$	4.61731	−	17.2321i	0.164589	−	0.614256i	−0.833503	−	0.552515i	$-0.813668\pi$
$787$	4.61731	−	17.2321i	0.164589	−	0.614256i	0.998092	−	0.0617409i	$-0.0196653\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	17.0000	−	29.4449i	0.602549	−	1.04365i
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−54.7128	+	14.6603i	−1.92957	+	0.517027i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	36.3468	−	36.3468i	1.27631	−	1.27631i	0.333590	−	0.942718i	$-0.391740\pi$
$811$	36.3468	−	36.3468i	1.27631	−	1.27631i	0.942718	−	0.333590i	$-0.108260\pi$
$812$		0			0
$813$	14.1340	−	52.7487i	0.495700	−	1.84998i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	5.36603	−	1.43782i	0.187733	−	0.0503030i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$821$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$822$		0			0
$823$	21.0000	−	12.1244i	0.732014	−	0.422628i	−0.0871445	−	0.996196i	$-0.527774\pi$
$823$	21.0000	−	12.1244i	0.732014	−	0.422628i	0.819159	+	0.573567i	$0.194441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$	−45.8993	−	26.5000i	−1.59415	−	0.920383i	−0.992584	−	0.121560i	$-0.961210\pi$
$829$	−45.8993	−	26.5000i	−1.59415	−	0.920383i	−0.601566	−	0.798823i	$-0.705456\pi$
$830$		0			0
$831$		−	36.0000i		−	1.24883i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	4.31347	−	4.31347i	0.149095	−	0.149095i
$838$		0			0
$839$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$839$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		−	51.9615i		−	1.78859i
$845$		0			0
$846$		0			0
$847$	−12.4737	−	46.5526i	−0.428602	−	1.59956i
$848$		0			0
$849$	−37.5000	+	21.6506i	−1.28700	+	0.743048i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	40.8827	+	40.8827i	1.39980	+	1.39980i	0.800608	+	0.599189i	$0.204510\pi$
$853$	40.8827	+	40.8827i	1.39980	+	1.39980i	0.599189	+	0.800608i	$0.295490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−57.1577			−1.95019			−0.975097	−	0.221777i	$-0.928814\pi$
$859$	−57.1577			−1.95019			−0.975097	+	0.221777i	$0.928814\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	14.7224	+	25.5000i	0.500000	+	0.866025i
$868$	8.90897	+	5.14359i	0.302390	+	0.174585i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−7.71539	−	28.7942i	−0.261126	−	0.974537i
$874$		0			0
$875$		0			0
$876$	−26.4449	+	26.4449i	−0.893489	+	0.893489i
$877$	−9.90192	−	2.65321i	−0.334364	−	0.0895926i	0.0877308	−	0.996144i	$-0.472038\pi$
$877$	−9.90192	−	2.65321i	−0.334364	−	0.0895926i	−0.422095	+	0.906552i	$0.638705\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$881$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$882$		0			0
$883$			55.0000i			1.85090i	0.378873	+	0.925449i	$0.376312\pi$
$883$			55.0000i			1.85090i	−0.378873	+	0.925449i	$0.623688\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−3.09808	+	3.09808i	−0.103906	+	0.103906i
$890$		0			0
$891$		0			0
$892$	−38.3923	−	38.3923i	−1.28547	−	1.28547i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	15.0000	−	25.9808i	0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	12.6962	+	3.40192i	0.422501	+	0.113209i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−34.6410	−	20.0000i	−1.15024	−	0.664089i	−0.201291	−	0.979531i	$-0.564514\pi$
$907$	−34.6410	−	20.0000i	−1.15024	−	0.664089i	−0.948945	+	0.315442i	$0.897847\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	5.75129	+	21.4641i	0.190444	+	0.710747i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.66025	−	0.444864i	−0.0548563	−	0.0146987i
$917$		0			0
$918$		0			0
$919$	−15.5885	−	27.0000i	−0.514216	−	0.890648i	−0.999864	−	0.0164935i	$-0.994750\pi$
$919$	−15.5885	−	27.0000i	−0.514216	−	0.890648i	0.485648	−	0.874154i	$-0.338584\pi$
$920$		0			0
$921$	−43.4545	+	11.6436i	−1.43187	+	0.383669i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−15.4904	−	57.8109i	−0.509321	−	1.90081i
$926$		0			0
$927$	40.5000	−	23.3827i	1.33019	−	0.767988i
$928$		0			0
$929$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$929$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$930$		0			0
$931$	−27.6603	−	27.6603i	−0.906528	−	0.906528i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	55.4256			1.81068			0.905338	−	0.424691i	$-0.139617\pi$
$937$	55.4256			1.81068			0.905338	+	0.424691i	$0.139617\pi$
$938$		0			0
$939$	−28.5000	+	49.3634i	−0.930062	+	1.61092i
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$947$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$948$			42.0000i			1.36410i
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$953$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$			29.6218i			0.955541i
$962$		0			0
$963$		0			0
$964$	15.2679	+	56.9808i	0.491748	+	1.83523i
$965$		0			0
$966$		0			0
$967$	−39.4449	+	39.4449i	−1.26846	+	1.26846i	−0.321578	+	0.946883i	$0.604213\pi$
$967$	−39.4449	+	39.4449i	−1.26846	+	1.26846i	−0.946883	+	0.321578i	$0.895787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	27.0000	+	15.5885i	0.866025	+	0.500000i
$973$	29.6244	−	7.93782i	0.949713	−	0.254475i
$974$		0			0
$975$		0			0
$976$	−34.6410			−1.10883
$977$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$977$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−56.6769	−	15.1865i	−1.80955	−	0.484869i
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−22.0000	+	38.1051i	−0.698853	+	1.21045i	0.270011	+	0.962857i	$0.412973\pi$
$991$	−22.0000	+	38.1051i	−0.698853	+	1.21045i	−0.968864	+	0.247592i	$0.920361\pi$
$992$		0			0
$993$	10.3468	−	10.3468i	0.328345	−	0.328345i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	29.5000	+	51.0955i	0.934274	+	1.61821i	0.775923	+	0.630828i	$0.217285\pi$
$997$	29.5000	+	51.0955i	0.934274	+	1.61821i	0.158352	+	0.987383i	$0.449382\pi$
$998$		0			0
$999$	60.0788	−	16.0981i	1.90081	−	0.509321i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.k.b.188.1		4
3.2	odd	2	CM	507.2.k.b.188.1		4
13.2	odd	12		507.2.k.a.89.1		4
13.3	even	3		39.2.k.a.2.1	✓	4
13.4	even	6		507.2.f.b.239.2		4
13.5	odd	4		507.2.k.c.488.1		4
13.6	odd	12		507.2.f.b.437.2		4
13.7	odd	12		507.2.f.c.437.2		4
13.8	odd	4		39.2.k.a.20.1	yes	4
13.9	even	3		507.2.f.c.239.2		4
13.10	even	6		507.2.k.c.80.1		4
13.11	odd	12	inner	507.2.k.b.89.1		4
13.12	even	2		507.2.k.a.188.1		4
39.2	even	12		507.2.k.a.89.1		4
39.5	even	4		507.2.k.c.488.1		4
39.8	even	4		39.2.k.a.20.1	yes	4
39.11	even	12	inner	507.2.k.b.89.1		4
39.17	odd	6		507.2.f.b.239.2		4
39.20	even	12		507.2.f.c.437.2		4
39.23	odd	6		507.2.k.c.80.1		4
39.29	odd	6		39.2.k.a.2.1	✓	4
39.32	even	12		507.2.f.b.437.2		4
39.35	odd	6		507.2.f.c.239.2		4
39.38	odd	2		507.2.k.a.188.1		4
52.3	odd	6		624.2.cn.b.353.1		4
52.47	even	4		624.2.cn.b.449.1		4
65.3	odd	12		975.2.bp.d.899.1		4
65.8	even	4		975.2.bp.a.449.1		4
65.29	even	6		975.2.bo.c.626.1		4
65.34	odd	4		975.2.bo.c.176.1		4
65.42	odd	12		975.2.bp.a.899.1		4
65.47	even	4		975.2.bp.d.449.1		4
156.47	odd	4		624.2.cn.b.449.1		4
156.107	even	6		624.2.cn.b.353.1		4
195.8	odd	4		975.2.bp.a.449.1		4
195.29	odd	6		975.2.bo.c.626.1		4
195.47	odd	4		975.2.bp.d.449.1		4
195.68	even	12		975.2.bp.d.899.1		4
195.107	even	12		975.2.bp.a.899.1		4
195.164	even	4		975.2.bo.c.176.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.k.a.2.1	✓	4	13.3	even	3
39.2.k.a.2.1	✓	4	39.29	odd	6
39.2.k.a.20.1	yes	4	13.8	odd	4
39.2.k.a.20.1	yes	4	39.8	even	4
507.2.f.b.239.2		4	13.4	even	6
507.2.f.b.239.2		4	39.17	odd	6
507.2.f.b.437.2		4	13.6	odd	12
507.2.f.b.437.2		4	39.32	even	12
507.2.f.c.239.2		4	13.9	even	3
507.2.f.c.239.2		4	39.35	odd	6
507.2.f.c.437.2		4	13.7	odd	12
507.2.f.c.437.2		4	39.20	even	12
507.2.k.a.89.1		4	13.2	odd	12
507.2.k.a.89.1		4	39.2	even	12
507.2.k.a.188.1		4	13.12	even	2
507.2.k.a.188.1		4	39.38	odd	2
507.2.k.b.89.1		4	13.11	odd	12	inner
507.2.k.b.89.1		4	39.11	even	12	inner
507.2.k.b.188.1		4	1.1	even	1	trivial
507.2.k.b.188.1		4	3.2	odd	2	CM
507.2.k.c.80.1		4	13.10	even	6
507.2.k.c.80.1		4	39.23	odd	6
507.2.k.c.488.1		4	13.5	odd	4
507.2.k.c.488.1		4	39.5	even	4
624.2.cn.b.353.1		4	52.3	odd	6
624.2.cn.b.353.1		4	156.107	even	6
624.2.cn.b.449.1		4	52.47	even	4
624.2.cn.b.449.1		4	156.47	odd	4
975.2.bo.c.176.1		4	65.34	odd	4
975.2.bo.c.176.1		4	195.164	even	4
975.2.bo.c.626.1		4	65.29	even	6
975.2.bo.c.626.1		4	195.29	odd	6
975.2.bp.a.449.1		4	65.8	even	4
975.2.bp.a.449.1		4	195.8	odd	4
975.2.bp.a.899.1		4	65.42	odd	12
975.2.bp.a.899.1		4	195.107	even	12
975.2.bp.d.449.1		4	65.47	even	4
975.2.bp.d.449.1		4	195.47	odd	4
975.2.bp.d.899.1		4	65.3	odd	12
975.2.bp.d.899.1		4	195.68	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.k (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(1.13397 - 4.23205i) q^{7} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(-0.866025 + 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(1.13397 - 4.23205i) q^{7} +(-1.50000 - 2.59808i) q^{9} -3.46410i q^{12} +(2.00000 - 3.46410i) q^{16} +(3.09808 + 0.830127i) q^{19} +(5.36603 + 5.36603i) q^{21} -5.00000i q^{25} +5.19615 q^{27} +(2.26795 + 8.46410i) q^{28} +(0.830127 - 0.830127i) q^{31} +(5.19615 + 3.00000i) q^{36} +(11.5622 - 3.09808i) q^{37} +(1.50000 - 0.866025i) q^{43} +(3.46410 + 6.00000i) q^{48} +(-10.5622 - 6.09808i) q^{49} +(-3.92820 + 3.92820i) q^{57} +(-4.33013 - 7.50000i) q^{61} +(-12.6962 + 3.40192i) q^{63} +8.00000i q^{64} +(-4.23205 - 15.7942i) q^{67} +(-7.63397 - 7.63397i) q^{73} +(7.50000 + 4.33013i) q^{75} +(-6.19615 + 1.66025i) q^{76} -12.1244 q^{79} +(-4.50000 + 7.79423i) q^{81} +(-14.6603 - 3.92820i) q^{84} +(0.526279 + 1.96410i) q^{93} +(9.59808 + 2.57180i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7} - 6 q^{9}+O(q^{10}) \) 4 * q + 8 * q^7 - 6 * q^9	\( 4 q + 8 q^{7} - 6 q^{9} + 8 q^{16} + 2 q^{19} + 18 q^{21} + 16 q^{28} - 14 q^{31} + 22 q^{37} + 6 q^{43} - 18 q^{49} + 12 q^{57} - 30 q^{63} - 10 q^{67} - 34 q^{73} + 30 q^{75} - 4 q^{76} - 18 q^{81} - 24 q^{84} - 36 q^{93} + 28 q^{97}+O(q^{100}) \) 4 * q + 8 * q^7 - 6 * q^9 + 8 * q^16 + 2 * q^19 + 18 * q^21 + 16 * q^28 - 14 * q^31 + 22 * q^37 + 6 * q^43 - 18 * q^49 + 12 * q^57 - 30 * q^63 - 10 * q^67 - 34 * q^73 + 30 * q^75 - 4 * q^76 - 18 * q^81 - 24 * q^84 - 36 * q^93 + 28 * q^97

Introduction

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