LMFDB - Embedded newform 507.2.k.b.80.1

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			80.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	507.80
Dual form			507.2.k.b.488.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 1.50000i) q^{3} +(1.73205 + 1.00000i) q^{4} +(2.86603 - 0.767949i) q^{7} +(-1.50000 + 2.59808i) q^{9} +3.46410i q^{12} +(2.00000 + 3.46410i) q^{16} +(-2.09808 - 7.83013i) q^{19} +(3.63397 + 3.63397i) q^{21} -5.00000i q^{25} -5.19615 q^{27} +(5.73205 + 1.53590i) q^{28} +(-7.83013 + 7.83013i) q^{31} +(-5.19615 + 3.00000i) q^{36} +(-0.562178 + 2.09808i) q^{37} +(1.50000 + 0.866025i) q^{43} +(-3.46410 + 6.00000i) q^{48} +(1.56218 - 0.901924i) q^{49} +(9.92820 - 9.92820i) q^{57} +(4.33013 - 7.50000i) q^{61} +(-2.30385 + 8.59808i) q^{63} +8.00000i q^{64} +(-0.767949 - 0.205771i) q^{67} +(-9.36603 - 9.36603i) q^{73} +(7.50000 - 4.33013i) q^{75} +(4.19615 - 15.6603i) q^{76} +12.1244 q^{79} +(-4.50000 - 7.79423i) q^{81} +(2.66025 + 9.92820i) q^{84} +(-18.5263 - 4.96410i) q^{93} +(4.40192 + 16.4282i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

Character values

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

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

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		−0.965926	−	0.258819i	$-0.916667\pi$
$2$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$3$	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
$7$	2.86603	−	0.767949i	1.08326	−	0.290258i	0.327327	−	0.944911i	$-0.393852\pi$
$7$	2.86603	−	0.767949i	1.08326	−	0.290258i	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		0.258819	−	0.965926i	$-0.416667\pi$
$11$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$12$			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	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−2.09808	−	7.83013i	−0.481332	−	1.79635i	−0.596040	−	0.802955i	$-0.703260\pi$
$19$	−2.09808	−	7.83013i	−0.481332	−	1.79635i	0.114708	−	0.993399i	$-0.463407\pi$
$20$		0			0
$21$	3.63397	+	3.63397i	0.792998	+	0.792998i
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$		−	5.00000i		−	1.00000i
$26$		0			0
$27$	−5.19615			−1.00000
$28$	5.73205	+	1.53590i	1.08326	+	0.290258i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	−7.83013	+	7.83013i	−1.40633	+	1.40633i	−0.628619	+	0.777714i	$0.716379\pi$
$31$	−7.83013	+	7.83013i	−1.40633	+	1.40633i	−0.777714	+	0.628619i	$0.783621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.19615	+	3.00000i	−0.866025	+	0.500000i
$37$	−0.562178	+	2.09808i	−0.0924215	+	0.344922i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	−0.562178	+	2.09808i	−0.0924215	+	0.344922i	0.904194	+	0.427121i	$0.140472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$41$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$42$		0			0
$43$	1.50000	+	0.866025i	0.228748	+	0.132068i	0.609994	−	0.792406i	$-0.291172\pi$
$43$	1.50000	+	0.866025i	0.228748	+	0.132068i	−0.381246	+	0.924473i	$0.624505\pi$
$44$		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
$49$	1.56218	−	0.901924i	0.223168	−	0.128846i
$50$		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
$57$	9.92820	−	9.92820i	1.31502	−	1.31502i
$58$		0			0
$59$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$59$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$60$		0			0
$61$	4.33013	−	7.50000i	0.554416	−	0.960277i	−0.443533	−	0.896258i	$-0.646275\pi$
$61$	4.33013	−	7.50000i	0.554416	−	0.960277i	0.997949	−	0.0640184i	$-0.0203916\pi$
$62$		0			0
$63$	−2.30385	+	8.59808i	−0.290258	+	1.08326i
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	−0.767949	−	0.205771i	−0.0938199	−	0.0251390i	0.211604	−	0.977356i	$-0.432131\pi$
$67$	−0.767949	−	0.205771i	−0.0938199	−	0.0251390i	−0.305424	+	0.952217i	$0.598798\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$71$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$72$		0			0
$73$	−9.36603	−	9.36603i	−1.09621	−	1.09621i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−9.36603	−	9.36603i	−1.09621	−	1.09621i	−0.101361	−	0.994850i	$-0.532320\pi$
$74$		0			0
$75$	7.50000	−	4.33013i	0.866025	−	0.500000i
$76$	4.19615	−	15.6603i	0.481332	−	1.79635i
$77$		0			0
$78$		0			0
$79$	12.1244			1.36410			0.682048	−	0.731307i	$-0.261089\pi$
$79$	12.1244			1.36410			0.682048	+	0.731307i	$0.261089\pi$
$80$		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	$0.750000\pi$
$84$	2.66025	+	9.92820i	0.290258	+	1.08326i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$89$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−18.5263	−	4.96410i	−1.92109	−	0.514753i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.40192	+	16.4282i	0.446948	+	1.66803i	0.710742	+	0.703452i	$0.248359\pi$
$97$	4.40192	+	16.4282i	0.446948	+	1.66803i	−0.263795	+	0.964579i	$0.584974\pi$
$98$		0			0
$99$		0			0
$100$	5.00000	−	8.66025i	0.500000	−	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		−	15.5885i		−	1.53598i	−0.640464	−	0.767988i	$-0.721258\pi$
$103$		−	15.5885i		−	1.53598i	0.640464	−	0.767988i	$-0.278742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	−9.00000	−	5.19615i	−0.866025	−	0.500000i
$109$	5.16987	−	5.16987i	0.495184	−	0.495184i	−0.414751	−	0.909935i	$-0.636131\pi$
$109$	5.16987	−	5.16987i	0.495184	−	0.495184i	0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$	−3.63397	+	0.973721i	−0.344922	+	0.0924215i
$112$	8.39230	+	8.39230i	0.792998	+	0.792998i
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		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
$124$	−21.3923	+	5.73205i	−1.92109	+	0.514753i
$125$		0			0
$126$		0			0
$127$	0.866025	−	0.500000i	0.0768473	−	0.0443678i	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	0.866025	−	0.500000i	0.0768473	−	0.0443678i	0.537931	+	0.842989i	$0.319206\pi$
$128$		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
$133$	−12.0263	−	20.8301i	−1.04281	−	1.80620i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$137$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$138$		0			0
$139$	3.50000	−	6.06218i	0.296866	−	0.514187i	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	3.50000	−	6.06218i	0.296866	−	0.514187i	0.975417	+	0.220366i	$0.0707252\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$	2.70577	+	1.56218i	0.223168	+	0.128846i
$148$	−3.07180	+	3.07180i	−0.252500	+	0.252500i
$149$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$149$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$150$		0			0
$151$	−14.1244	−	14.1244i	−1.14942	−	1.14942i	−0.986666	−	0.162758i	$-0.947961\pi$
$151$	−14.1244	−	14.1244i	−1.14942	−	1.14942i	−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
$163$	20.6244	−	5.52628i	1.61542	−	0.432852i	0.665771	−	0.746156i	$-0.268103\pi$
$163$	20.6244	−	5.52628i	1.61542	−	0.432852i	0.949653	+	0.313304i	$0.101436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$167$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	23.4904	+	6.29423i	1.79635	+	0.481332i
$172$	1.73205	+	3.00000i	0.132068	+	0.228748i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	−3.83975	−	14.3301i	−0.290258	−	1.08326i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$		0			0
$181$			6.92820i			0.514969i	0.966282	+	0.257485i	$0.0828937\pi$
$181$			6.92820i			0.514969i	−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$	15.0000			1.10883
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−14.8923	+	3.99038i	−1.08326	+	0.290258i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−12.0000	+	6.92820i	−0.866025	+	0.500000i
$193$	−7.06218	+	26.3564i	−0.508347	+	1.89718i	−0.0719816	+	0.997406i	$0.522932\pi$
$193$	−7.06218	+	26.3564i	−0.508347	+	1.89718i	−0.436365	+	0.899770i	$0.643734\pi$
$194$		0			0
$195$		0			0
$196$	3.60770			0.257693
$197$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$197$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$198$		0			0
$199$	14.7224	+	8.50000i	1.04365	+	0.602549i	0.920864	−	0.389885i	$-0.127485\pi$
$199$	14.7224	+	8.50000i	1.04365	+	0.602549i	0.122782	+	0.992434i	$0.460818\pi$
$200$		0			0
$201$	−0.356406	−	1.33013i	−0.0251390	−	0.0938199i
$202$		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	0.834675	+	0.550743i	$0.185655\pi$
$211$	12.9904	+	22.5000i	0.894295	+	1.54896i	0.0596196	+	0.998221i	$0.481011\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−16.4282	+	28.4545i	−1.11522	+	1.93162i
$218$		0			0
$219$	5.93782	−	22.1603i	0.401241	−	1.49745i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−12.0263	−	3.22243i	−0.805339	−	0.215790i	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	−12.0263	−	3.22243i	−0.805339	−	0.215790i	−0.637927	+	0.770097i	$0.720208\pi$
$224$		0			0
$225$	12.9904	+	7.50000i	0.866025	+	0.500000i
$226$		0			0
$227$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$227$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$228$	27.1244	−	7.26795i	1.79635	−	0.481332i
$229$	21.3923	+	21.3923i	1.41364	+	1.41364i	0.726900	+	0.686743i	$0.240960\pi$
$229$	21.3923	+	21.3923i	1.41364	+	1.41364i	0.686743	+	0.726900i	$0.259040\pi$
$230$		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
$241$	9.36603	−	2.50962i	0.603319	−	0.161659i	0.0557856	−	0.998443i	$-0.482234\pi$
$241$	9.36603	−	2.50962i	0.603319	−	0.161659i	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	$-0.833333\pi$
$251$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$252$	−12.5885	+	12.5885i	−0.792998	+	0.792998i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$			6.44486i			0.400464i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.12436	−	1.12436i	−0.0686810	−	0.0686810i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	2.45448	−	9.16025i	0.149099	−	0.556446i	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	2.45448	−	9.16025i	0.149099	−	0.556446i	0.999539	−	0.0303728i	$-0.00966946\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−18.0000	−	10.3923i	−1.08152	−	0.624413i	−0.150210	−	0.988654i	$-0.547995\pi$
$277$	−18.0000	−	10.3923i	−1.08152	−	0.624413i	−0.931305	+	0.364241i	$0.881328\pi$
$278$		0			0
$279$	−8.59808	−	32.0885i	−0.514753	−	1.92109i
$280$		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	$-0.933287\pi$
$283$	−21.6506	+	12.5000i	−1.28700	+	0.743048i	−0.308879	+	0.951101i	$0.599954\pi$
$284$		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
$291$	−20.8301	+	20.8301i	−1.22108	+	1.22108i
$292$	−6.85641	−	25.5885i	−0.401241	−	1.49745i
$293$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$293$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	17.3205			1.00000
$301$	4.96410	+	1.33013i	0.286126	+	0.0766672i
$302$		0			0
$303$		0			0
$304$	22.9282	−	22.9282i	1.31502	−	1.31502i
$305$		0			0
$306$		0			0
$307$	16.6340	+	16.6340i	0.949351	+	0.949351i	0.998778	−	0.0494267i	$-0.0157394\pi$
$307$	16.6340	+	16.6340i	0.949351	+	0.949351i	−0.0494267	+	0.998778i	$0.515739\pi$
$308$		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	$-0.880247\pi$
$313$	−32.9090			−1.86012			−0.930062	+	0.367402i	$0.880247\pi$
$314$		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
$327$	12.2321	+	3.27757i	0.676434	+	0.181250i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	9.16025	+	34.1865i	0.503493	+	1.87906i	0.476011	+	0.879440i	$0.342082\pi$
$331$	9.16025	+	34.1865i	0.503493	+	1.87906i	0.0274825	+	0.999622i	$0.491251\pi$
$332$		0			0
$333$	−4.60770	−	4.60770i	−0.252500	−	0.252500i
$334$		0			0
$335$		0			0
$336$	−5.32051	+	19.8564i	−0.290258	+	1.08326i
$337$			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
$343$	−10.9019	+	10.9019i	−0.588649	+	0.588649i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	1.17949	−	4.40192i	0.0631368	−	0.235630i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$	1.17949	−	4.40192i	0.0631368	−	0.235630i	0.990282	+	0.139072i	$0.0444119\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$353$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$354$		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
$361$	−40.4545	+	23.3564i	−2.12918	+	1.22928i
$362$		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	−0.913493	−	0.406855i	$-0.866625\pi$
$367$	−15.5000	−	26.8468i	−0.809093	−	1.40139i	0.104399	−	0.994535i	$-0.466708\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−27.1244	−	27.1244i	−1.40633	−	1.40633i
$373$	−18.1865	+	31.5000i	−0.941663	+	1.63101i	−0.179364	+	0.983783i	$0.557404\pi$
$373$	−18.1865	+	31.5000i	−0.941663	+	1.63101i	−0.762299	+	0.647225i	$0.775929\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	16.9904	+	4.55256i	0.872737	+	0.233849i	0.667271	−	0.744815i	$-0.267462\pi$
$379$	16.9904	+	4.55256i	0.872737	+	0.233849i	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		−0.258819	−	0.965926i	$-0.583333\pi$
$383$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.50000	+	2.59808i	−0.228748	+	0.132068i
$388$	−8.80385	+	32.8564i	−0.446948	+	1.66803i
$389$		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
$397$	−27.8923	+	7.47372i	−1.39987	+	0.375095i	−0.878300	−	0.478110i	$-0.841322\pi$
$397$	−27.8923	+	7.47372i	−1.39987	+	0.375095i	−0.521575	+	0.853206i	$0.674655\pi$
$398$		0			0
$399$	20.8301	−	36.0788i	1.04281	−	1.80620i
$400$	17.3205	−	10.0000i	0.866025	−	0.500000i
$401$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$401$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.918584	+	3.42820i	0.0454211	+	0.169514i	0.984911	−	0.173064i	$-0.0553667\pi$
$409$	0.918584	+	3.42820i	0.0454211	+	0.169514i	−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	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$	27.6865	−	27.6865i	1.34936	−	1.34936i	0.463002	−	0.886357i	$-0.346772\pi$
$421$	27.6865	−	27.6865i	1.34936	−	1.34936i	0.886357	−	0.463002i	$-0.153228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	6.65064	−	24.8205i	0.321847	−	1.20115i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$431$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$432$	−10.3923	−	18.0000i	−0.500000	−	0.866025i
$433$	−30.3109	−	17.5000i	−1.45665	−	0.840996i	−0.457804	−	0.889053i	$-0.651364\pi$
$433$	−30.3109	−	17.5000i	−1.45665	−	0.840996i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$		0			0
$435$		0			0
$436$	14.1244	−	3.78461i	0.676434	−	0.181250i
$437$		0			0
$438$		0			0
$439$	34.5000	−	19.9186i	1.64660	−	0.950662i	0.668184	−	0.743996i	$-0.267072\pi$
$439$	34.5000	−	19.9186i	1.64660	−	0.950662i	0.978412	−	0.206666i	$-0.0662612\pi$
$440$		0			0
$441$			5.41154i			0.257693i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	−7.26795	−	1.94744i	−0.344922	−	0.0924215i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	6.14359	+	22.9282i	0.290258	+	1.08326i
$449$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$449$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	8.95448	−	33.4186i	0.420718	−	1.57014i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−36.2846	−	9.72243i	−1.69732	−	0.454796i	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−36.2846	−	9.72243i	−1.69732	−	0.454796i	−0.972263	+	0.233890i	$0.924854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$461$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$462$		0			0
$463$	−22.3660	−	22.3660i	−1.03944	−	1.03944i	−0.999190	−	0.0402476i	$-0.987185\pi$
$463$	−22.3660	−	22.3660i	−1.03944	−	1.03944i	−0.0402476	−	0.999190i	$-0.512815\pi$
$464$		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
$469$	−2.35898			−0.108928
$470$		0			0
$471$	−9.52628	−	16.5000i	−0.438948	−	0.760280i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−39.1506	+	10.4904i	−1.79635	+	0.481332i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$479$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$480$		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
$487$	7.41858	+	27.6865i	0.336168	+	1.25460i	0.902597	+	0.430486i	$0.141658\pi$
$487$	7.41858	+	27.6865i	0.336168	+	1.25460i	−0.566429	+	0.824110i	$0.691675\pi$
$488$		0			0
$489$	26.1506	+	26.1506i	1.18257	+	1.18257i
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−42.7846	−	11.4641i	−1.92109	−	0.514753i
$497$		0			0
$498$		0			0
$499$	0.411543	−	0.411543i	0.0184232	−	0.0184232i	−0.697835	−	0.716258i	$-0.745853\pi$
$499$	0.411543	−	0.411543i	0.0184232	−	0.0184232i	0.716258	+	0.697835i	$0.245853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	2.00000			0.0887357
$509$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$509$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$510$		0			0
$511$	−34.0359	−	19.6506i	−1.50566	−	0.869293i
$512$		0			0
$513$	10.9019	+	40.6865i	0.481332	+	1.79635i
$514$		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	0.940129	−	0.340818i	$-0.110704\pi$
$523$	4.00000	+	6.92820i	0.174908	+	0.302949i	−0.765222	+	0.643767i	$0.777371\pi$
$524$		0			0
$525$	18.1699	−	18.1699i	0.792998	−	0.792998i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		−	48.1051i		−	2.08562i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	13.1506	+	13.1506i	0.565390	+	0.565390i	0.930834	−	0.365444i	$-0.119083\pi$
$541$	13.1506	+	13.1506i	0.565390	+	0.565390i	−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
$553$	34.7487	−	9.31089i	1.47767	−	0.395939i
$554$		0			0
$555$		0			0
$556$	12.1244	−	7.00000i	0.514187	−	0.296866i
$557$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$557$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−18.8827	−	18.8827i	−0.792998	−	0.792998i
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		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
$577$	−16.0718	+	16.0718i	−0.669078	+	0.669078i	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−16.0718	+	16.0718i	−0.669078	+	0.669078i	0.288425	+	0.957503i	$0.406868\pi$
$578$		0			0
$579$	−45.6506	+	12.2321i	−1.89718	+	0.508347i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$587$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$588$	3.12436	+	5.41154i	0.128846	+	0.223168i
$589$	77.7391	+	44.8827i	3.20318	+	1.84936i
$590$		0			0
$591$		0			0
$592$	−8.39230	+	2.24871i	−0.344922	+	0.0924215i
$593$		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	−0.883148	−	0.469095i	$-0.844580\pi$
$601$	−20.7846	−	36.0000i	−0.847822	−	1.46847i	0.0353259	−	0.999376i	$-0.488753\pi$
$602$		0			0
$603$	1.68653	−	1.68653i	0.0686810	−	0.0686810i
$604$	−10.3397	−	38.5885i	−0.420718	−	1.57014i
$605$		0			0
$606$		0			0
$607$	10.0000	−	17.3205i	0.405887	−	0.703018i	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	10.0000	−	17.3205i	0.405887	−	0.703018i	0.994424	+	0.105453i	$0.0336291\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	47.7487	+	12.7942i	1.92855	+	0.516754i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	47.7487	+	12.7942i	1.92855	+	0.516754i	0.949156	+	0.314806i	$0.101939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$617$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$618$		0			0
$619$	−31.8827	−	31.8827i	−1.28147	−	1.28147i	−0.939829	−	0.341644i	$-0.889016\pi$
$619$	−31.8827	−	31.8827i	−1.28147	−	1.28147i	−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
$631$	33.6244	−	9.00962i	1.33856	−	0.358667i	0.482663	−	0.875806i	$-0.339670\pi$
$631$	33.6244	−	9.00962i	1.33856	−	0.358667i	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	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	−5.11474	−	19.0885i	−0.201706	−	0.752775i	−0.990429	−	0.138027i	$-0.955924\pi$
$643$	−5.11474	−	19.0885i	−0.201706	−	0.752775i	0.788723	−	0.614749i	$-0.210743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−56.9090			−2.23044
$652$	41.2487	+	11.0526i	1.61542	+	0.432852i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	38.3827	−	10.2846i	1.49745	−	0.401241i
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	−11.8205	+	44.1147i	−0.459764	+	1.71586i	0.213925	+	0.976850i	$0.431375\pi$
$661$	−11.8205	+	44.1147i	−0.459764	+	1.71586i	−0.673690	+	0.739014i	$0.735292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−5.58142	−	20.8301i	−0.215790	−	0.805339i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−43.5000	+	25.1147i	−1.67680	+	0.968102i	−0.713123	+	0.701039i	$0.752720\pi$
$673$	−43.5000	+	25.1147i	−1.67680	+	0.968102i	−0.963679	+	0.267063i	$0.913947\pi$
$674$		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
$679$	25.2321	+	43.7032i	0.968317	+	1.67717i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$683$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$684$	34.3923	+	34.3923i	1.31502	+	1.31502i
$685$		0			0
$686$		0			0
$687$	−13.5622	+	50.6147i	−0.517429	+	1.93107i
$688$			6.92820i			0.264135i
$689$		0			0
$690$		0			0
$691$	−5.52628	−	1.48076i	−0.210230	−	0.0563308i	0.152167	−	0.988355i	$-0.451375\pi$
$691$	−5.52628	−	1.48076i	−0.210230	−	0.0563308i	−0.362397	+	0.932024i	$0.618041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	7.67949	−	28.6603i	0.290258	−	1.08326i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	17.6077			0.664087
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−32.6506	+	8.74871i	−1.22622	+	0.328565i	−0.813107	−	0.582115i	$-0.802225\pi$
$709$	−32.6506	+	8.74871i	−1.22622	+	0.328565i	−0.413114	+	0.910679i	$0.635559\pi$
$710$		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	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	−11.9711	−	44.6769i	−0.445829	−	1.66386i
$722$		0			0
$723$	11.8756	+	11.8756i	0.441660	+	0.441660i
$724$	−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
$733$	−30.3468	+	30.3468i	−1.12088	+	1.12088i	−0.129275	+	0.991609i	$0.541265\pi$
$733$	−30.3468	+	30.3468i	−1.12088	+	1.12088i	−0.991609	+	0.129275i	$0.958735\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	12.4378	−	46.4186i	0.457533	−	1.70754i	−0.223001	−	0.974818i	$-0.571585\pi$
$739$	12.4378	−	46.4186i	0.457533	−	1.70754i	0.680534	−	0.732717i	$-0.261748\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$743$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$744$		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	−0.200698	−	0.979653i	$-0.564321\pi$
$751$	15.0000	−	8.66025i	0.547358	−	0.316017i	0.748056	+	0.663636i	$0.230988\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−29.7846	−	7.98076i	−1.08326	−	0.290258i
$757$	24.2487	+	42.0000i	0.881334	+	1.52652i	0.849858	+	0.527011i	$0.176688\pi$
$757$	24.2487	+	42.0000i	0.881334	+	1.52652i	0.0314762	+	0.999505i	$0.489979\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$761$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$762$		0			0
$763$	10.8468	−	18.7872i	0.392680	−	0.680142i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−27.7128			−1.00000
$769$	36.4904	+	9.77757i	1.31588	+	0.352588i	0.847432	−	0.530904i	$-0.178148\pi$
$769$	36.4904	+	9.77757i	1.31588	+	0.352588i	0.468445	+	0.883493i	$0.344814\pi$
$770$		0			0
$771$		0			0
$772$	−38.5885	+	38.5885i	−1.38883	+	1.38883i
$773$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$773$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$774$		0			0
$775$	39.1506	+	39.1506i	1.40633	+	1.40633i
$776$		0			0
$777$	−9.66730	+	5.58142i	−0.346812	+	0.200232i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	6.24871	+	3.60770i	0.223168	+	0.128846i
$785$		0			0
$786$		0			0
$787$	51.3827	−	13.7679i	1.83159	−	0.490774i	0.833503	−	0.552515i	$-0.186332\pi$
$787$	51.3827	−	13.7679i	1.83159	−	0.490774i	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	$-0.833333\pi$
$797$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.712813	−	2.66025i	0.0251390	−	0.0938199i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$	−17.3468	+	17.3468i	−0.609128	+	0.609128i	−0.942718	−	0.333590i	$-0.891740\pi$
$811$	−17.3468	+	17.3468i	−0.609128	+	0.609128i	0.333590	+	0.942718i	$0.391740\pi$
$812$		0			0
$813$	15.8660	−	4.25129i	0.556446	−	0.149099i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	3.63397	−	13.5622i	0.127137	−	0.474481i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$821$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$822$		0			0
$823$	21.0000	+	12.1244i	0.732014	+	0.422628i	0.819159	−	0.573567i	$-0.194441\pi$
$823$	21.0000	+	12.1244i	0.732014	+	0.422628i	−0.0871445	+	0.996196i	$0.527774\pi$
$824$		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	0.601566	−	0.798823i	$-0.294544\pi$
$829$	45.8993	−	26.5000i	1.59415	−	0.920383i	0.992584	−	0.121560i	$-0.0387897\pi$
$830$		0			0
$831$		−	36.0000i		−	1.24883i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	40.6865	−	40.6865i	1.40633	−	1.40633i
$838$		0			0
$839$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$839$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$840$		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
$847$	−31.5263	−	8.44744i	−1.08326	−	0.290258i
$848$		0			0
$849$	−37.5000	−	21.6506i	−1.28700	−	0.743048i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−5.88269	−	5.88269i	−0.201419	−	0.201419i	0.599189	−	0.800608i	$-0.295490\pi$
$853$	−5.88269	−	5.88269i	−0.201419	−	0.201419i	−0.800608	+	0.599189i	$0.795490\pi$
$854$		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	$-0.0711857\pi$
$859$	57.1577			1.95019			0.975097	+	0.221777i	$0.0711857\pi$
$860$		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
$868$	−56.9090	+	32.8564i	−1.93162	+	1.11522i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−49.2846	−	13.2058i	−1.66803	−	0.446948i
$874$		0			0
$875$		0			0
$876$	32.4449	−	32.4449i	1.09621	−	1.09621i
$877$	−15.0981	−	56.3468i	−0.509826	−	1.90270i	−0.422095	−	0.906552i	$-0.638705\pi$
$877$	−15.0981	−	56.3468i	−0.509826	−	1.90270i	−0.0877308	−	0.996144i	$-0.527962\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$881$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$882$		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	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	2.09808	−	2.09808i	0.0703672	−	0.0703672i
$890$		0			0
$891$		0			0
$892$	−17.6077	−	17.6077i	−0.589549	−	0.589549i
$893$		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
$903$	2.30385	+	8.59808i	0.0766672	+	0.286126i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	34.6410	−	20.0000i	1.15024	−	0.664089i	0.201291	−	0.979531i	$-0.435486\pi$
$907$	34.6410	−	20.0000i	1.15024	−	0.664089i	0.948945	+	0.315442i	$0.102153\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	54.2487	+	14.5359i	1.79635	+	0.481332i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	15.6603	+	58.4449i	0.517429	+	1.93107i
$917$		0			0
$918$		0			0
$919$	15.5885	−	27.0000i	0.514216	−	0.890648i	−0.485648	−	0.874154i	$-0.661416\pi$
$919$	15.5885	−	27.0000i	0.514216	−	0.890648i	0.999864	−	0.0164935i	$-0.00525028\pi$
$920$		0			0
$921$	−10.5455	+	39.3564i	−0.347487	+	1.29684i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	10.4904	+	2.81089i	0.344922	+	0.0924215i
$926$		0			0
$927$	40.5000	+	23.3827i	1.33019	+	0.767988i
$928$		0			0
$929$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$929$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$930$		0			0
$931$	−10.3397	−	10.3397i	−0.338871	−	0.338871i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−55.4256			−1.81068			−0.905338	−	0.424691i	$-0.860383\pi$
$937$	−55.4256			−1.81068			−0.905338	+	0.424691i	$0.860383\pi$
$938$		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		0.258819	−	0.965926i	$-0.416667\pi$
$947$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$948$			42.0000i			1.36410i
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$953$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	91.6218i		−	2.95554i
$962$		0			0
$963$		0			0
$964$	18.7321	+	5.01924i	0.603319	+	0.161659i
$965$		0			0
$966$		0			0
$967$	19.4449	−	19.4449i	0.625305	−	0.625305i	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	19.4449	−	19.4449i	0.625305	−	0.625305i	0.946883	+	0.321578i	$0.104213\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	27.0000	−	15.5885i	0.866025	−	0.500000i
$973$	5.37564	−	20.0622i	0.172335	−	0.643164i
$974$		0			0
$975$		0			0
$976$	34.6410			1.10883
$977$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$977$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	5.67691	+	21.1865i	0.181250	+	0.676434i
$982$		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	−0.968864	−	0.247592i	$-0.920361\pi$
$991$	−22.0000	−	38.1051i	−0.698853	−	1.21045i	0.270011	−	0.962857i	$-0.412973\pi$
$992$		0			0
$993$	−43.3468	+	43.3468i	−1.37557	+	1.37557i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	29.5000	−	51.0955i	0.934274	−	1.61821i	0.158352	−	0.987383i	$-0.449382\pi$
$997$	29.5000	−	51.0955i	0.934274	−	1.61821i	0.775923	−	0.630828i	$-0.217285\pi$
$998$		0			0
$999$	2.92116	−	10.9019i	0.0924215	−	0.344922i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.k.b.80.1		4
3.2	odd	2	CM	507.2.k.b.80.1		4
13.2	odd	12		507.2.f.b.437.1		4
13.3	even	3		507.2.f.c.239.1		4
13.4	even	6		507.2.k.c.188.1		4
13.5	odd	4		507.2.k.c.89.1		4
13.6	odd	12		507.2.k.a.488.1		4
13.7	odd	12	inner	507.2.k.b.488.1		4
13.8	odd	4		39.2.k.a.11.1	✓	4
13.9	even	3		39.2.k.a.32.1	yes	4
13.10	even	6		507.2.f.b.239.1		4
13.11	odd	12		507.2.f.c.437.1		4
13.12	even	2		507.2.k.a.80.1		4
39.2	even	12		507.2.f.b.437.1		4
39.5	even	4		507.2.k.c.89.1		4
39.8	even	4		39.2.k.a.11.1	✓	4
39.11	even	12		507.2.f.c.437.1		4
39.17	odd	6		507.2.k.c.188.1		4
39.20	even	12	inner	507.2.k.b.488.1		4
39.23	odd	6		507.2.f.b.239.1		4
39.29	odd	6		507.2.f.c.239.1		4
39.32	even	12		507.2.k.a.488.1		4
39.35	odd	6		39.2.k.a.32.1	yes	4
39.38	odd	2		507.2.k.a.80.1		4
52.35	odd	6		624.2.cn.b.305.1		4
52.47	even	4		624.2.cn.b.401.1		4
65.8	even	4		975.2.bp.a.674.1		4
65.9	even	6		975.2.bo.c.851.1		4
65.22	odd	12		975.2.bp.a.149.1		4
65.34	odd	4		975.2.bo.c.401.1		4
65.47	even	4		975.2.bp.d.674.1		4
65.48	odd	12		975.2.bp.d.149.1		4
156.35	even	6		624.2.cn.b.305.1		4
156.47	odd	4		624.2.cn.b.401.1		4
195.8	odd	4		975.2.bp.a.674.1		4
195.47	odd	4		975.2.bp.d.674.1		4
195.74	odd	6		975.2.bo.c.851.1		4
195.113	even	12		975.2.bp.d.149.1		4
195.152	even	12		975.2.bp.a.149.1		4
195.164	even	4		975.2.bo.c.401.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.k.a.11.1	✓	4	13.8	odd	4
39.2.k.a.11.1	✓	4	39.8	even	4
39.2.k.a.32.1	yes	4	13.9	even	3
39.2.k.a.32.1	yes	4	39.35	odd	6
507.2.f.b.239.1		4	13.10	even	6
507.2.f.b.239.1		4	39.23	odd	6
507.2.f.b.437.1		4	13.2	odd	12
507.2.f.b.437.1		4	39.2	even	12
507.2.f.c.239.1		4	13.3	even	3
507.2.f.c.239.1		4	39.29	odd	6
507.2.f.c.437.1		4	13.11	odd	12
507.2.f.c.437.1		4	39.11	even	12
507.2.k.a.80.1		4	13.12	even	2
507.2.k.a.80.1		4	39.38	odd	2
507.2.k.a.488.1		4	13.6	odd	12
507.2.k.a.488.1		4	39.32	even	12
507.2.k.b.80.1		4	1.1	even	1	trivial
507.2.k.b.80.1		4	3.2	odd	2	CM
507.2.k.b.488.1		4	13.7	odd	12	inner
507.2.k.b.488.1		4	39.20	even	12	inner
507.2.k.c.89.1		4	13.5	odd	4
507.2.k.c.89.1		4	39.5	even	4
507.2.k.c.188.1		4	13.4	even	6
507.2.k.c.188.1		4	39.17	odd	6
624.2.cn.b.305.1		4	52.35	odd	6
624.2.cn.b.305.1		4	156.35	even	6
624.2.cn.b.401.1		4	52.47	even	4
624.2.cn.b.401.1		4	156.47	odd	4
975.2.bo.c.401.1		4	65.34	odd	4
975.2.bo.c.401.1		4	195.164	even	4
975.2.bo.c.851.1		4	65.9	even	6
975.2.bo.c.851.1		4	195.74	odd	6
975.2.bp.a.149.1		4	65.22	odd	12
975.2.bp.a.149.1		4	195.152	even	12
975.2.bp.a.674.1		4	65.8	even	4
975.2.bp.a.674.1		4	195.8	odd	4
975.2.bp.d.149.1		4	65.48	odd	12
975.2.bp.d.149.1		4	195.113	even	12
975.2.bp.d.674.1		4	65.47	even	4
975.2.bp.d.674.1		4	195.47	odd	4

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)

\(f(q)\)	\(=\)	\(q+(0.866025 + 1.50000i) q^{3} +(1.73205 + 1.00000i) q^{4} +(2.86603 - 0.767949i) q^{7} +(-1.50000 + 2.59808i) q^{9} +3.46410i q^{12} +(2.00000 + 3.46410i) q^{16} +(-2.09808 - 7.83013i) q^{19} +(3.63397 + 3.63397i) q^{21} -5.00000i q^{25} -5.19615 q^{27} +(5.73205 + 1.53590i) q^{28} +(-7.83013 + 7.83013i) q^{31} +(-5.19615 + 3.00000i) q^{36} +(-0.562178 + 2.09808i) q^{37} +(1.50000 + 0.866025i) q^{43} +(-3.46410 + 6.00000i) q^{48} +(1.56218 - 0.901924i) q^{49} +(9.92820 - 9.92820i) q^{57} +(4.33013 - 7.50000i) q^{61} +(-2.30385 + 8.59808i) q^{63} +8.00000i q^{64} +(-0.767949 - 0.205771i) q^{67} +(-9.36603 - 9.36603i) q^{73} +(7.50000 - 4.33013i) q^{75} +(4.19615 - 15.6603i) q^{76} +12.1244 q^{79} +(-4.50000 - 7.79423i) q^{81} +(2.66025 + 9.92820i) q^{84} +(-18.5263 - 4.96410i) q^{93} +(4.40192 + 16.4282i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7} - 6 q^{9} + 8 q^{16} + 2 q^{19} + 18 q^{21} + 16 q^{28} - 14 q^{31} + 22 q^{37} + 6 q^{43} - 18 q^{49} + 12 q^{57} - 30 q^{63} - 10 q^{67} - 34 q^{73} + 30 q^{75} - 4 q^{76} - 18 q^{81} - 24 q^{84}+ \cdots + 28 q^{97}+O(q^{100}) \) 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

L-functions

Modular forms

Varieties

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Representations

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Database

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