LMFDB - Embedded newform 525.2.bf.c.107.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(32,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 3, 8]))

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

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

chi := DirichletCharacter("525.32");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.bf (of order $12$, degree $4$, 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:	$4.19214610612$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			107.1
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	525.107
Dual form			525.2.bf.c.368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.67303 - 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-2.63896 + 0.189469i) q^{7} +(2.59808 + 1.50000i) q^{9} +O(q^{10})$	$q+(-1.67303 - 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-2.63896 + 0.189469i) q^{7} +(2.59808 + 1.50000i) q^{9} +(-3.34607 + 0.896575i) q^{12} +(4.89898 - 4.89898i) q^{13} +(2.00000 - 3.46410i) q^{16} +(-6.92820 - 4.00000i) q^{19} +(4.50000 + 0.866025i) q^{21} +(-3.67423 - 3.67423i) q^{27} +(-4.38134 + 2.96713i) q^{28} +(-3.50000 - 6.06218i) q^{31} +6.00000 q^{36} +(-5.01910 + 1.34486i) q^{37} +(-10.3923 + 6.00000i) q^{39} +(8.57321 - 8.57321i) q^{43} +(-4.89898 + 4.89898i) q^{48} +(6.92820 - 1.00000i) q^{49} +(3.58630 - 13.3843i) q^{52} +(9.79796 + 9.79796i) q^{57} +(0.500000 - 0.866025i) q^{61} +(-7.14042 - 3.46618i) q^{63} -8.00000i q^{64} +(-0.896575 + 3.34607i) q^{67} +(1.67303 + 0.448288i) q^{73} -16.0000 q^{76} +(14.7224 + 8.50000i) q^{79} +(4.50000 + 7.79423i) q^{81} +(8.66025 - 3.00000i) q^{84} +(-12.0000 + 13.8564i) q^{91} +(3.13801 + 11.7112i) q^{93} +(3.67423 + 3.67423i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{16} + 36 q^{21} - 28 q^{31} + 48 q^{36} + 4 q^{61} - 128 q^{76} + 36 q^{81} - 96 q^{91}+O(q^{100}) $ 8 * q + 16 * q^16 + 36 * q^21 - 28 * q^31 + 48 * q^36 + 4 * q^61 - 128 * q^76 + 36 * q^81 - 96 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$e\left(\frac{1}{3}\right)$

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	$-0.0833333\pi$
$2$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$3$	−1.67303	−	0.448288i	−0.965926	−	0.258819i
$3$	−1.67303	−	0.448288i	−0.965926	−	0.258819i
$4$	1.73205	−	1.00000i	0.866025	−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.63896	+	0.189469i	−0.997433	+	0.0716124i
$7$	−2.63896	+	0.189469i	−0.997433	+	0.0716124i
$8$		0			0
$9$	2.59808	+	1.50000i	0.866025	+	0.500000i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−3.34607	+	0.896575i	−0.965926	+	0.258819i
$13$	4.89898	−	4.89898i	1.35873	−	1.35873i	0.483250	−	0.875482i	$-0.339456\pi$
$13$	4.89898	−	4.89898i	1.35873	−	1.35873i	0.875482	−	0.483250i	$-0.160544\pi$
$14$		0			0
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$17$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$18$		0			0
$19$	−6.92820	−	4.00000i	−1.58944	−	0.917663i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$	−6.92820	−	4.00000i	−1.58944	−	0.917663i	−0.596040	−	0.802955i	$-0.703260\pi$
$20$		0			0
$21$	4.50000	+	0.866025i	0.981981	+	0.188982i
$22$		0			0
$23$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$23$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	−	3.67423i	−0.707107	−	0.707107i
$28$	−4.38134	+	2.96713i	−0.827996	+	0.560734i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−3.50000	−	6.06218i	−0.628619	−	1.08880i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−3.50000	−	6.06218i	−0.628619	−	1.08880i	0.359211	−	0.933257i	$-0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	6.00000			1.00000
$37$	−5.01910	+	1.34486i	−0.825135	+	0.221094i	−0.646590	−	0.762838i	$-0.723806\pi$
$37$	−5.01910	+	1.34486i	−0.825135	+	0.221094i	−0.178545	+	0.983932i	$0.557139\pi$
$38$		0			0
$39$	−10.3923	+	6.00000i	−1.66410	+	0.960769i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	8.57321	−	8.57321i	1.30740	−	1.30740i	0.384120	−	0.923283i	$-0.374505\pi$
$43$	8.57321	−	8.57321i	1.30740	−	1.30740i	0.923283	−	0.384120i	$-0.125495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$47$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$48$	−4.89898	+	4.89898i	−0.707107	+	0.707107i
$49$	6.92820	−	1.00000i	0.989743	−	0.142857i
$50$		0			0
$51$		0			0
$52$	3.58630	−	13.3843i	0.497331	−	1.85606i
$53$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$53$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	9.79796	+	9.79796i	1.29777	+	1.29777i
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$	−7.14042	−	3.46618i	−0.899608	−	0.436698i
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	−0.896575	+	3.34607i	−0.109534	+	0.408787i	−0.998820	−	0.0485648i	$-0.984535\pi$
$67$	−0.896575	+	3.34607i	−0.109534	+	0.408787i	0.889286	+	0.457352i	$0.151202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	1.67303	+	0.448288i	0.195814	+	0.0524681i	0.355393	−	0.934717i	$-0.384347\pi$
$73$	1.67303	+	0.448288i	0.195814	+	0.0524681i	−0.159579	+	0.987185i	$0.551014\pi$
$74$		0			0
$75$		0			0
$76$	−16.0000			−1.83533
$77$		0			0
$78$		0			0
$79$	14.7224	+	8.50000i	1.65640	+	0.956325i	0.974355	+	0.225018i	$0.0722440\pi$
$79$	14.7224	+	8.50000i	1.65640	+	0.956325i	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	$0.750000\pi$
$84$	8.66025	−	3.00000i	0.944911	−	0.327327i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	−12.0000	+	13.8564i	−1.25794	+	1.45255i
$92$		0			0
$93$	3.13801	+	11.7112i	0.325397	+	1.21440i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.67423	+	3.67423i	0.373062	+	0.373062i	0.868591	−	0.495529i	$-0.165026\pi$
$97$	3.67423	+	3.67423i	0.373062	+	0.373062i	−0.495529	+	0.868591i	$0.665026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	4.03459	+	15.0573i	0.397540	+	1.48364i	0.817411	+	0.576055i	$0.195409\pi$
$103$	4.03459	+	15.0573i	0.397540	+	1.48364i	−0.419871	+	0.907584i	$0.637925\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$107$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$108$	−10.0382	−	2.68973i	−0.965926	−	0.258819i
$109$	−1.73205	+	1.00000i	−0.165900	+	0.0957826i	−0.580651	−	0.814152i	$-0.697202\pi$
$109$	−1.73205	+	1.00000i	−0.165900	+	0.0957826i	0.414751	+	0.909935i	$0.363869\pi$
$110$		0			0
$111$	9.00000			0.854242
$112$	−4.62158	+	9.52056i	−0.436698	+	0.899608i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	20.0764	−	5.37945i	1.85606	−	0.497331i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	+	9.52628i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	−12.1244	−	7.00000i	−1.08880	−	0.628619i
$125$		0			0
$126$		0			0
$127$	15.9217	+	15.9217i	1.41282	+	1.41282i	0.737783	+	0.675038i	$0.235873\pi$
$127$	15.9217	+	15.9217i	1.41282	+	1.41282i	0.675038	+	0.737783i	$0.264127\pi$
$128$		0			0
$129$	−18.1865	+	10.5000i	−1.60123	+	0.924473i
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	19.0411	+	9.24316i	1.65107	+	0.801483i
$134$		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
$139$			23.0000i			1.95083i	0.220366	+	0.975417i	$0.429275\pi$
$139$			23.0000i			1.95083i	−0.220366	+	0.975417i	$0.570725\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	10.3923	−	6.00000i	0.866025	−	0.500000i
$145$		0			0
$146$		0			0
$147$	−12.0394	−	1.43280i	−0.992993	−	0.118175i
$148$	−7.34847	+	7.34847i	−0.604040	+	0.604040i
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	−9.50000	−	16.4545i	−0.773099	−	1.33905i	−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−9.50000	−	16.4545i	−0.773099	−	1.33905i	0.162758	−	0.986666i	$-0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−12.0000	+	20.7846i	−0.960769	+	1.66410i
$157$	5.82774	−	21.7494i	0.465104	−	1.73579i	−0.191439	−	0.981505i	$-0.561315\pi$
$157$	5.82774	−	21.7494i	0.465104	−	1.73579i	0.656543	−	0.754288i	$-0.272018\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.93117	−	18.4034i	−0.386239	−	1.44146i	−0.836205	−	0.548417i	$-0.815231\pi$
$163$	−4.93117	−	18.4034i	−0.386239	−	1.44146i	0.449966	−	0.893045i	$-0.351436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	35.0000i		−	2.69231i
$170$		0			0
$171$	−12.0000	−	20.7846i	−0.917663	−	1.58944i
$172$	6.27603	−	23.4225i	0.478543	−	1.78595i
$173$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$173$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$174$		0			0
$175$		0			0
$176$		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
$181$	7.00000			0.520306			0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000			0.520306			0.260153	+	0.965567i	$0.416227\pi$
$182$		0			0
$183$	−1.22474	+	1.22474i	−0.0905357	+	0.0905357i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	10.3923	+	9.00000i	0.755929	+	0.654654i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−3.58630	+	13.3843i	−0.258819	+	0.965926i
$193$	−11.7112	−	3.13801i	−0.842993	−	0.225879i	−0.188619	−	0.982050i	$-0.560401\pi$
$193$	−11.7112	−	3.13801i	−0.842993	−	0.225879i	−0.654374	+	0.756171i	$0.727068\pi$
$194$		0			0
$195$		0			0
$196$	11.0000	−	8.66025i	0.785714	−	0.618590i
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		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
$201$	3.00000	−	5.19615i	0.211604	−	0.366508i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−7.17260	−	26.7685i	−0.497331	−	1.85606i
$209$		0			0
$210$		0			0
$211$	29.0000			1.99644			0.998221	−	0.0596196i	$-0.0189888\pi$
$211$	29.0000			1.99644			0.998221	+	0.0596196i	$0.0189888\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	10.3849	+	15.3347i	0.704976	+	1.04099i
$218$		0			0
$219$	−2.59808	−	1.50000i	−0.175562	−	0.101361i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−13.4722	+	13.4722i	−0.902165	+	0.902165i	−0.995623	−	0.0934584i	$-0.970208\pi$
$223$	−13.4722	+	13.4722i	−0.902165	+	0.902165i	0.0934584	+	0.995623i	$0.470208\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$227$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$228$	26.7685	+	7.17260i	1.77279	+	0.475017i
$229$	19.0526	+	11.0000i	1.25903	+	0.726900i	0.972886	−	0.231287i	$-0.0742935\pi$
$229$	19.0526	+	11.0000i	1.25903	+	0.726900i	0.286143	+	0.958187i	$0.407627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$233$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−20.8207	−	20.8207i	−1.35245	−	1.35245i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	15.5000	+	26.8468i	0.998443	+	1.72935i	0.547533	+	0.836784i	$0.315567\pi$
$241$	15.5000	+	26.8468i	0.998443	+	1.72935i	0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$	−4.03459	−	15.0573i	−0.258819	−	0.965926i
$244$		−	2.00000i		−	0.128037i
$245$		0			0
$246$		0			0
$247$	−53.5370	+	14.3452i	−3.40648	+	0.912764i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−15.8338	+	1.13681i	−0.997433	+	0.0716124i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$257$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$258$		0			0
$259$	12.9904	−	4.50000i	0.807183	−	0.279616i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$263$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.79315	+	6.69213i	0.109534	+	0.408787i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	14.0000	−	24.2487i	0.850439	−	1.47300i	−0.0303728	−	0.999539i	$-0.509669\pi$
$271$	14.0000	−	24.2487i	0.850439	−	1.47300i	0.880812	−	0.473466i	$-0.156997\pi$
$272$		0			0
$273$	26.2880	−	17.8028i	1.59103	−	1.07747i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.37945	+	20.0764i	−0.323220	+	1.20627i	0.592869	+	0.805299i	$0.297995\pi$
$277$	−5.37945	+	20.0764i	−0.323220	+	1.20627i	−0.916089	+	0.400975i	$0.868672\pi$
$278$		0			0
$279$		−	21.0000i		−	1.25724i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	10.0382	+	2.68973i	0.596709	+	0.159888i	0.544518	−	0.838749i	$-0.316713\pi$
$283$	10.0382	+	2.68973i	0.596709	+	0.159888i	0.0521913	+	0.998637i	$0.483379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−14.7224	−	8.50000i	−0.866025	−	0.500000i
$290$		0			0
$291$	−4.50000	−	7.79423i	−0.263795	−	0.456906i
$292$	3.34607	−	0.896575i	0.195814	−	0.0524681i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−21.0000	+	24.2487i	−1.21042	+	1.39767i
$302$		0			0
$303$		0			0
$304$	−27.7128	+	16.0000i	−1.58944	+	0.917663i
$305$		0			0
$306$		0			0
$307$	−1.22474	−	1.22474i	−0.0698999	−	0.0698999i	0.671293	−	0.741192i	$-0.265739\pi$
$307$	−1.22474	−	1.22474i	−0.0698999	−	0.0698999i	−0.741192	+	0.671293i	$0.765739\pi$
$308$		0			0
$309$		−	27.0000i		−	1.53598i
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−1.34486	−	5.01910i	−0.0760162	−	0.283696i	0.917446	−	0.397861i	$-0.130247\pi$
$313$	−1.34486	−	5.01910i	−0.0760162	−	0.283696i	−0.993462	+	0.114165i	$0.963581\pi$
$314$		0			0
$315$		0			0
$316$	34.0000			1.91265
$317$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$317$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	15.5885	+	9.00000i	0.866025	+	0.500000i
$325$		0			0
$326$		0			0
$327$	3.34607	−	0.896575i	0.185038	−	0.0495807i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	15.5000	−	26.8468i	0.851957	−	1.47563i	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	15.5000	−	26.8468i	0.851957	−	1.47563i	0.879440	−	0.476011i	$-0.157918\pi$
$332$		0			0
$333$	−15.0573	−	4.03459i	−0.825135	−	0.221094i
$334$		0			0
$335$		0			0
$336$	12.0000	−	13.8564i	0.654654	−	0.755929i
$337$	−25.7196	−	25.7196i	−1.40104	−	1.40104i	−0.796815	−	0.604223i	$-0.793484\pi$
$337$	−25.7196	−	25.7196i	−1.40104	−	1.40104i	−0.604223	−	0.796815i	$-0.706516\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18.0938	+	3.95164i	−0.976972	+	0.213368i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$347$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$348$		0			0
$349$			14.0000i			0.749403i	0.927146	+	0.374701i	$0.122255\pi$
$349$			14.0000i			0.749403i	−0.927146	+	0.374701i	$0.877745\pi$
$350$		0			0
$351$	−36.0000			−1.92154
$352$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	22.5000	+	38.9711i	1.18421	+	2.05111i
$362$		0			0
$363$	13.4722	−	13.4722i	0.707107	−	0.707107i
$364$	−6.92820	+	36.0000i	−0.363137	+	1.88691i
$365$		0			0
$366$		0			0
$367$	4.03459	−	15.0573i	0.210604	−	0.785984i	−0.777064	−	0.629421i	$-0.783292\pi$
$367$	4.03459	−	15.0573i	0.210604	−	0.785984i	0.987668	−	0.156563i	$-0.0500414\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	17.1464	+	17.1464i	0.889001	+	0.889001i
$373$	1.79315	+	6.69213i	0.0928458	+	0.346505i	0.996684	−	0.0813690i	$-0.0259292\pi$
$373$	1.79315	+	6.69213i	0.0928458	+	0.346505i	−0.903838	+	0.427874i	$0.859263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			8.00000i			0.410932i	0.978664	+	0.205466i	$0.0658711\pi$
$379$			8.00000i			0.410932i	−0.978664	+	0.205466i	$0.934129\pi$
$380$		0			0
$381$	−19.5000	−	33.7750i	−0.999015	−	1.73035i
$382$		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
$387$	35.1337	−	9.41404i	1.78595	−	0.478543i
$388$	10.0382	+	2.68973i	0.509612	+	0.136550i
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	18.4034	−	4.93117i	0.923638	−	0.247488i	0.234498	−	0.972117i	$-0.424655\pi$
$397$	18.4034	−	4.93117i	0.923638	−	0.247488i	0.689140	+	0.724628i	$0.257989\pi$
$398$		0			0
$399$	−27.7128	−	24.0000i	−1.38738	−	1.20150i
$400$		0			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−46.8449	−	12.5521i	−2.33351	−	0.625262i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	32.9090	−	19.0000i	1.62724	−	0.939490i	0.642333	−	0.766426i	$-0.277967\pi$
$409$	32.9090	−	19.0000i	1.62724	−	0.939490i	0.984911	−	0.173064i	$-0.0553667\pi$
$410$		0			0
$411$		0			0
$412$	22.0454	+	22.0454i	1.08610	+	1.08610i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	10.3106	−	38.4797i	0.504913	−	1.88436i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−41.0000			−1.99822			−0.999109	−	0.0422075i	$-0.986561\pi$
$421$	−41.0000			−1.99822			−0.999109	+	0.0422075i	$0.986561\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.15539	+	2.38014i	−0.0559135	+	0.115183i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	−20.0764	+	5.37945i	−0.965926	+	0.258819i
$433$	−15.9217	+	15.9217i	−0.765147	+	0.765147i	−0.977248	−	0.212101i	$-0.931970\pi$
$433$	−15.9217	+	15.9217i	−0.765147	+	0.765147i	0.212101	+	0.977248i	$0.431970\pi$
$434$		0			0
$435$		0			0
$436$	−2.00000	+	3.46410i	−0.0957826	+	0.165900i
$437$		0			0
$438$		0			0
$439$	−11.2583	−	6.50000i	−0.537331	−	0.310228i	0.206666	−	0.978412i	$-0.433739\pi$
$439$	−11.2583	−	6.50000i	−0.537331	−	0.310228i	−0.743996	+	0.668184i	$0.767072\pi$
$440$		0			0
$441$	19.5000	+	7.79423i	0.928571	+	0.371154i
$442$		0			0
$443$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$443$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$444$	15.5885	−	9.00000i	0.739795	−	0.427121i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.51575	+	21.1117i	0.0716124	+	0.997433i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	8.51747	+	31.7876i	0.400186	+	1.49351i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	28.4416	−	7.62089i	1.33044	−	0.356490i	0.477561	−	0.878599i	$-0.341521\pi$
$457$	28.4416	−	7.62089i	1.33044	−	0.356490i	0.852879	+	0.522108i	$0.174854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−1.22474	+	1.22474i	−0.0569187	+	0.0569187i	−0.734993	−	0.678074i	$-0.762815\pi$
$463$	−1.22474	+	1.22474i	−0.0569187	+	0.0569187i	0.678074	+	0.734993i	$0.262815\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$467$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$468$	29.3939	−	29.3939i	1.35873	−	1.35873i
$469$	1.73205	−	9.00000i	0.0799787	−	0.415581i
$470$		0			0
$471$	−19.5000	+	33.7750i	−0.898513	+	1.55627i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−18.0000	+	31.1769i	−0.820729	+	1.42154i
$482$		0			0
$483$		0			0
$484$			22.0000i			1.00000i
$485$		0			0
$486$		0			0
$487$	−9.41404	+	35.1337i	−0.426591	+	1.59206i	0.333833	+	0.942632i	$0.391658\pi$
$487$	−9.41404	+	35.1337i	−0.426591	+	1.59206i	−0.760424	+	0.649427i	$0.775009\pi$
$488$		0			0
$489$			33.0000i			1.49231i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−28.0000			−1.25724
$497$		0			0
$498$		0			0
$499$	27.7128	+	16.0000i	1.24060	+	0.716258i	0.969216	−	0.246214i	$-0.0791865\pi$
$499$	27.7128	+	16.0000i	1.24060	+	0.716258i	0.271380	+	0.962472i	$0.412520\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−15.6901	+	58.5561i	−0.696821	+	2.60057i
$508$	43.4988	+	11.6555i	1.92995	+	0.517128i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−4.50000	−	0.866025i	−0.199068	−	0.0383107i
$512$		0			0
$513$	10.7589	+	40.1528i	0.475017	+	1.77279i
$514$		0			0
$515$		0			0
$516$	−21.0000	+	36.3731i	−0.924473	+	1.60123i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	7.62089	+	28.4416i	0.333238	+	1.24366i	0.905766	+	0.423777i	$0.139296\pi$
$523$	7.62089	+	28.4416i	0.333238	+	1.24366i	−0.572528	+	0.819885i	$0.694037\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−19.9186	+	11.5000i	−0.866025	+	0.500000i
$530$		0			0
$531$		0			0
$532$	42.2233	−	3.03150i	1.83061	−	0.131432i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	23.0000	−	39.8372i	0.988847	−	1.71273i	0.365444	−	0.930834i	$-0.380917\pi$
$541$	23.0000	−	39.8372i	0.988847	−	1.71273i	0.623404	−	0.781900i	$-0.285749\pi$
$542$		0			0
$543$	−11.7112	−	3.13801i	−0.502577	−	0.134665i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	17.1464	+	17.1464i	0.733128	+	0.733128i	0.971238	−	0.238110i	$-0.0765278\pi$
$547$	17.1464	+	17.1464i	0.733128	+	0.733128i	−0.238110	+	0.971238i	$0.576528\pi$
$548$		0			0
$549$	2.59808	−	1.50000i	0.110883	−	0.0640184i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−40.4624	−	19.6417i	−1.72063	−	0.835250i
$554$		0			0
$555$		0			0
$556$	23.0000	+	39.8372i	0.975417	+	1.68947i
$557$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$557$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$558$		0			0
$559$		−	84.0000i		−	3.55282i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$563$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−13.3521	−	19.7160i	−0.560734	−	0.827996i
$568$		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
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	0.983444	−	0.181210i	$-0.0580014\pi$
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	−0.648655	+	0.761083i	$0.724668\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	12.0000	−	20.7846i	0.500000	−	0.866025i
$577$	8.51747	−	31.7876i	0.354587	−	1.32334i	−0.526417	−	0.850227i	$-0.676465\pi$
$577$	8.51747	−	31.7876i	0.354587	−	1.32334i	0.881004	−	0.473109i	$-0.156868\pi$
$578$		0			0
$579$	18.1865	+	10.5000i	0.755807	+	0.436365i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$	−22.2856	+	9.55772i	−0.919044	+	0.394154i
$589$			56.0000i			2.30744i
$590$		0			0
$591$		0			0
$592$	−5.37945	+	20.0764i	−0.221094	+	0.825135i
$593$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$593$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	28.4416	−	7.62089i	1.16404	−	0.311902i
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$	−7.34847	+	7.34847i	−0.299253	+	0.299253i
$604$	−32.9090	−	19.0000i	−1.33905	−	0.773099i
$605$		0			0
$606$		0			0
$607$	−38.4797	+	10.3106i	−1.56184	+	0.418495i	−0.933247	−	0.359235i	$-0.883038\pi$
$607$	−38.4797	+	10.3106i	−1.56184	+	0.418495i	−0.628598	+	0.777730i	$0.716371\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	46.8449	+	12.5521i	1.89205	+	0.506973i	0.998292	+	0.0584195i	$0.0186061\pi$
$613$	46.8449	+	12.5521i	1.89205	+	0.506973i	0.893756	+	0.448553i	$0.148061\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$	42.4352	−	24.5000i	1.70562	−	0.984738i	0.765787	−	0.643094i	$-0.222350\pi$
$619$	42.4352	−	24.5000i	1.70562	−	0.984738i	0.939829	−	0.341644i	$-0.110984\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$			48.0000i			1.92154i
$625$		0			0
$626$		0			0
$627$		0			0
$628$	−11.6555	−	43.4988i	−0.465104	−	1.73579i
$629$		0			0
$630$		0			0
$631$	−1.00000			−0.0398094			−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000			−0.0398094			−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$	−48.5179	−	13.0003i	−1.92842	−	0.516717i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	29.0421	−	38.8401i	1.15069	−	1.53890i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	−13.4722	+	13.4722i	−0.531291	+	0.531291i	−0.920957	−	0.389665i	$-0.872591\pi$
$643$	−13.4722	+	13.4722i	−0.531291	+	0.531291i	0.389665	+	0.920957i	$0.372591\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$647$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−10.5000	−	30.3109i	−0.411527	−	1.18798i
$652$	−26.9444	−	26.9444i	−1.05522	−	1.05522i
$653$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$653$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	3.67423	+	3.67423i	0.143346	+	0.143346i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	5.50000	+	9.52628i	0.213925	+	0.370529i	0.952940	−	0.303160i	$-0.0980418\pi$
$661$	5.50000	+	9.52628i	0.213925	+	0.370529i	−0.739014	+	0.673690i	$0.764708\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	28.5788	−	16.5000i	1.10492	−	0.637927i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	35.5176	−	35.5176i	1.36910	−	1.36910i	0.507381	−	0.861722i	$-0.330614\pi$
$673$	35.5176	−	35.5176i	1.36910	−	1.36910i	0.861722	−	0.507381i	$-0.169386\pi$
$674$		0			0
$675$		0			0
$676$	−35.0000	−	60.6218i	−1.34615	−	2.33161i
$677$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$677$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$678$		0			0
$679$	−10.3923	−	9.00000i	−0.398820	−	0.345388i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$683$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$684$	−41.5692	−	24.0000i	−1.58944	−	0.917663i
$685$		0			0
$686$		0			0
$687$	−26.9444	−	26.9444i	−1.02799	−	1.02799i
$688$	−12.5521	−	46.8449i	−0.478543	−	1.78595i
$689$		0			0
$690$		0			0
$691$	20.5000	−	35.5070i	0.779857	−	1.35075i	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	20.5000	−	35.5070i	0.779857	−	1.35075i	0.932024	−	0.362397i	$-0.118041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	40.1528	+	10.7589i	1.51439	+	0.405780i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−45.8993	−	26.5000i	−1.72379	−	0.995228i	−0.910679	−	0.413114i	$-0.864441\pi$
$709$	−45.8993	−	26.5000i	−1.72379	−	0.995228i	−0.813107	−	0.582115i	$-0.802225\pi$
$710$		0			0
$711$	25.5000	+	44.1673i	0.956325	+	1.65640i
$712$		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
$721$	−13.5000	−	38.9711i	−0.502766	−	1.45136i
$722$		0			0
$723$	−13.8969	−	51.8640i	−0.516832	−	1.92884i
$724$	12.1244	−	7.00000i	0.450598	−	0.260153i
$725$		0			0
$726$		0			0
$727$	22.0454	+	22.0454i	0.817619	+	0.817619i	0.985763	−	0.168144i	$-0.0537772\pi$
$727$	22.0454	+	22.0454i	0.817619	+	0.817619i	−0.168144	+	0.985763i	$0.553777\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$		0			0
$732$	−0.896575	+	3.34607i	−0.0331384	+	0.123674i
$733$	8.51747	+	31.7876i	0.314600	+	1.17410i	0.924362	+	0.381518i	$0.124598\pi$
$733$	8.51747	+	31.7876i	0.314600	+	1.17410i	−0.609762	+	0.792585i	$0.708735\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−32.0429	+	18.5000i	−1.17872	+	0.680534i	−0.955718	−	0.294285i	$-0.904919\pi$
$739$	−32.0429	+	18.5000i	−1.17872	+	0.680534i	−0.223001	+	0.974818i	$0.571585\pi$
$740$		0			0
$741$	96.0000			3.52665
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	5.50000	−	9.52628i	0.200698	−	0.347619i	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	5.50000	−	9.52628i	0.200698	−	0.347619i	0.948753	+	0.316017i	$0.102346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	27.0000	+	5.19615i	0.981981	+	0.188982i
$757$	−33.0681	−	33.0681i	−1.20188	−	1.20188i	−0.973594	−	0.228287i	$-0.926688\pi$
$757$	−33.0681	−	33.0681i	−1.20188	−	1.20188i	−0.228287	−	0.973594i	$-0.573312\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	4.38134	−	2.96713i	0.158615	−	0.107417i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	7.17260	+	26.7685i	0.258819	+	0.965926i
$769$			49.0000i			1.76699i	0.468445	+	0.883493i	$0.344814\pi$
$769$			49.0000i			1.76699i	−0.468445	+	0.883493i	$0.655186\pi$
$770$		0			0
$771$		0			0
$772$	−23.4225	+	6.27603i	−0.842993	+	0.225879i
$773$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$773$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−23.7506	+	1.70522i	−0.852049	+	0.0611744i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	10.3923	−	26.0000i	0.371154	−	0.928571i
$785$		0			0
$786$		0			0
$787$	−12.1038	+	45.1719i	−0.431453	+	1.61020i	0.317962	+	0.948103i	$0.397001\pi$
$787$	−12.1038	+	45.1719i	−0.431453	+	1.61020i	−0.749415	+	0.662100i	$0.769665\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.79315	−	6.69213i	−0.0636767	−	0.237645i
$794$		0			0
$795$		0			0
$796$	−17.0000	+	29.4449i	−0.602549	+	1.04365i
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		−	12.0000i		−	0.423207i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$809$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$810$		0			0
$811$	19.0000			0.667180			0.333590	−	0.942718i	$-0.391740\pi$
$811$	19.0000			0.667180			0.333590	+	0.942718i	$0.391740\pi$
$812$		0			0
$813$	−34.2929	+	34.2929i	−1.20270	+	1.20270i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−93.6898	+	25.1041i	−3.27779	+	0.878282i
$818$		0			0
$819$	−51.9615	+	18.0000i	−1.81568	+	0.628971i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	−31.7876	−	8.51747i	−1.10805	−	0.296900i	−0.342010	−	0.939696i	$-0.611108\pi$
$823$	−31.7876	−	8.51747i	−1.10805	−	0.296900i	−0.766037	+	0.642796i	$0.777774\pi$
$824$		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
$831$	18.0000	−	31.1769i	0.624413	−	1.08152i
$832$	−39.1918	−	39.1918i	−1.35873	−	1.35873i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−9.41404	+	35.1337i	−0.325397	+	1.21440i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$	50.2295	−	29.0000i	1.72897	−	0.998221i
$845$		0			0
$846$		0			0
$847$	12.7093	−	26.1815i	0.436698	−	0.899608i
$848$		0			0
$849$	−15.5885	−	9.00000i	−0.534994	−	0.308879i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	33.0681	−	33.0681i	1.13223	−	1.13223i	0.142425	−	0.989806i	$-0.454510\pi$
$853$	33.0681	−	33.0681i	1.13223	−	1.13223i	0.989806	−	0.142425i	$-0.0454900\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$857$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$858$		0			0
$859$	48.4974	+	28.0000i	1.65471	+	0.955348i	0.975097	+	0.221777i	$0.0711857\pi$
$859$	48.4974	+	28.0000i	1.65471	+	0.955348i	0.679613	+	0.733571i	$0.262148\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$863$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	20.8207	+	20.8207i	0.707107	+	0.707107i
$868$	33.3220	+	16.1755i	1.13102	+	0.549033i
$869$		0			0
$870$		0			0
$871$	12.0000	+	20.7846i	0.406604	+	0.704260i
$872$		0			0
$873$	4.03459	+	15.0573i	0.136550	+	0.509612i
$874$		0			0
$875$		0			0
$876$	−6.00000			−0.202721
$877$	−5.01910	+	1.34486i	−0.169483	+	0.0454128i	−0.342563	−	0.939495i	$-0.611295\pi$
$877$	−5.01910	+	1.34486i	−0.169483	+	0.0454128i	0.173080	+	0.984908i	$0.444628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	41.6413	−	41.6413i	1.40134	−	1.40134i	0.605487	−	0.795855i	$-0.292978\pi$
$883$	41.6413	−	41.6413i	1.40134	−	1.40134i	0.795855	−	0.605487i	$-0.207022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$887$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$888$		0			0
$889$	−45.0333	−	39.0000i	−1.51037	−	1.30802i
$890$		0			0
$891$		0			0
$892$	−9.86233	+	36.8067i	−0.330215	+	1.23238i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	46.0041	−	31.1548i	1.53092	−	1.03677i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	14.7935	−	55.2101i	0.491210	−	1.83322i	−0.0590889	−	0.998253i	$-0.518820\pi$
$907$	14.7935	−	55.2101i	0.491210	−	1.83322i	0.550299	−	0.834968i	$-0.314514\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	53.5370	−	14.3452i	1.77279	−	0.475017i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	44.0000			1.45380
$917$		0			0
$918$		0			0
$919$	0.866025	+	0.500000i	0.0285675	+	0.0164935i	0.514216	−	0.857661i	$-0.328083\pi$
$919$	0.866025	+	0.500000i	0.0285675	+	0.0164935i	−0.485648	+	0.874154i	$0.661416\pi$
$920$		0			0
$921$	1.50000	+	2.59808i	0.0494267	+	0.0856095i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−12.1038	+	45.1719i	−0.397540	+	1.48364i
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$	−52.0000	−	20.7846i	−1.70423	−	0.681188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	35.5176	+	35.5176i	1.16031	+	1.16031i	0.984408	+	0.175902i	$0.0562841\pi$
$937$	35.5176	+	35.5176i	1.16031	+	1.16031i	0.175902	+	0.984408i	$0.443716\pi$
$938$		0			0
$939$			9.00000i			0.293704i
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		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	$0.916667\pi$
$948$	−56.8831	−	15.2418i	−1.84748	−	0.495030i
$949$	10.3923	−	6.00000i	0.337348	−	0.194768i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−9.00000	+	15.5885i	−0.290323	+	0.502853i
$962$		0			0
$963$		0			0
$964$	53.6936	+	31.0000i	1.72935	+	0.998443i
$965$		0			0
$966$		0			0
$967$	−33.0681	−	33.0681i	−1.06340	−	1.06340i	−0.997849	−	0.0655495i	$-0.979120\pi$
$967$	−33.0681	−	33.0681i	−1.06340	−	1.06340i	−0.0655495	−	0.997849i	$-0.520880\pi$
$968$		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	$0.666667\pi$
$972$	−22.0454	−	22.0454i	−0.707107	−	0.707107i
$973$	−4.35778	−	60.6960i	−0.139704	−	1.94583i
$974$		0			0
$975$		0			0
$976$	−2.00000	−	3.46410i	−0.0640184	−	0.110883i
$977$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$977$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−6.00000			−0.191565
$982$		0			0
$983$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$983$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−78.3837	+	78.3837i	−2.49372	+	2.49372i
$989$		0			0
$990$		0			0
$991$	−8.50000	−	14.7224i	−0.270011	−	0.467673i	0.698853	−	0.715265i	$-0.253694\pi$
$991$	−8.50000	−	14.7224i	−0.270011	−	0.467673i	−0.968864	+	0.247592i	$0.920361\pi$
$992$		0			0
$993$	−37.9671	+	37.9671i	−1.20485	+	1.20485i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.82774	−	21.7494i	0.184566	−	0.688811i	−0.810157	−	0.586214i	$-0.800618\pi$
$997$	5.82774	−	21.7494i	0.184566	−	0.688811i	0.994723	−	0.102598i	$-0.0327154\pi$
$998$		0			0
$999$	23.3827	+	13.5000i	0.739795	+	0.427121i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bf.c.107.1	yes	8
3.2	odd	2	CM	525.2.bf.c.107.1	yes	8
5.2	odd	4	inner	525.2.bf.c.443.1	yes	8
5.3	odd	4	inner	525.2.bf.c.443.2	yes	8
5.4	even	2	inner	525.2.bf.c.107.2	yes	8
7.4	even	3	inner	525.2.bf.c.32.2	yes	8
15.2	even	4	inner	525.2.bf.c.443.1	yes	8
15.8	even	4	inner	525.2.bf.c.443.2	yes	8
15.14	odd	2	inner	525.2.bf.c.107.2	yes	8
21.11	odd	6	inner	525.2.bf.c.32.2	yes	8
35.4	even	6	inner	525.2.bf.c.32.1	✓	8
35.18	odd	12	inner	525.2.bf.c.368.1	yes	8
35.32	odd	12	inner	525.2.bf.c.368.2	yes	8
105.32	even	12	inner	525.2.bf.c.368.2	yes	8
105.53	even	12	inner	525.2.bf.c.368.1	yes	8
105.74	odd	6	inner	525.2.bf.c.32.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bf.c.32.1	✓	8	35.4	even	6	inner
525.2.bf.c.32.1	✓	8	105.74	odd	6	inner
525.2.bf.c.32.2	yes	8	7.4	even	3	inner
525.2.bf.c.32.2	yes	8	21.11	odd	6	inner
525.2.bf.c.107.1	yes	8	1.1	even	1	trivial
525.2.bf.c.107.1	yes	8	3.2	odd	2	CM
525.2.bf.c.107.2	yes	8	5.4	even	2	inner
525.2.bf.c.107.2	yes	8	15.14	odd	2	inner
525.2.bf.c.368.1	yes	8	35.18	odd	12	inner
525.2.bf.c.368.1	yes	8	105.53	even	12	inner
525.2.bf.c.368.2	yes	8	35.32	odd	12	inner
525.2.bf.c.368.2	yes	8	105.32	even	12	inner
525.2.bf.c.443.1	yes	8	5.2	odd	4	inner
525.2.bf.c.443.1	yes	8	15.2	even	4	inner
525.2.bf.c.443.2	yes	8	5.3	odd	4	inner
525.2.bf.c.443.2	yes	8	15.8	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bf (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.67303 - 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-2.63896 + 0.189469i) q^{7} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)	\(q+(-1.67303 - 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-2.63896 + 0.189469i) q^{7} +(2.59808 + 1.50000i) q^{9} +(-3.34607 + 0.896575i) q^{12} +(4.89898 - 4.89898i) q^{13} +(2.00000 - 3.46410i) q^{16} +(-6.92820 - 4.00000i) q^{19} +(4.50000 + 0.866025i) q^{21} +(-3.67423 - 3.67423i) q^{27} +(-4.38134 + 2.96713i) q^{28} +(-3.50000 - 6.06218i) q^{31} +6.00000 q^{36} +(-5.01910 + 1.34486i) q^{37} +(-10.3923 + 6.00000i) q^{39} +(8.57321 - 8.57321i) q^{43} +(-4.89898 + 4.89898i) q^{48} +(6.92820 - 1.00000i) q^{49} +(3.58630 - 13.3843i) q^{52} +(9.79796 + 9.79796i) q^{57} +(0.500000 - 0.866025i) q^{61} +(-7.14042 - 3.46618i) q^{63} -8.00000i q^{64} +(-0.896575 + 3.34607i) q^{67} +(1.67303 + 0.448288i) q^{73} -16.0000 q^{76} +(14.7224 + 8.50000i) q^{79} +(4.50000 + 7.79423i) q^{81} +(8.66025 - 3.00000i) q^{84} +(-12.0000 + 13.8564i) q^{91} +(3.13801 + 11.7112i) q^{93} +(3.67423 + 3.67423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{16} + 36 q^{21} - 28 q^{31} + 48 q^{36} + 4 q^{61} - 128 q^{76} + 36 q^{81} - 96 q^{91}+O(q^{100}) \) 8 * q + 16 * q^16 + 36 * q^21 - 28 * q^31 + 48 * q^36 + 4 * q^61 - 128 * q^76 + 36 * q^81 - 96 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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