LMFDB - Embedded newform 525.2.bf.c.32.2

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			32.2
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	525.32
Dual form			525.2.bf.c.443.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.448288 + 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(0.189469 - 2.63896i) q^{7} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})$	$q+(0.448288 + 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(0.189469 - 2.63896i) q^{7} +(-2.59808 + 1.50000i) q^{9} +(0.896575 - 3.34607i) q^{12} +(4.89898 - 4.89898i) q^{13} +(2.00000 + 3.46410i) q^{16} +(6.92820 - 4.00000i) q^{19} +(4.50000 - 0.866025i) q^{21} +(-3.67423 - 3.67423i) q^{27} +(-2.96713 + 4.38134i) q^{28} +(-3.50000 + 6.06218i) q^{31} +6.00000 q^{36} +(1.34486 - 5.01910i) q^{37} +(10.3923 + 6.00000i) q^{39} +(8.57321 - 8.57321i) q^{43} +(-4.89898 + 4.89898i) q^{48} +(-6.92820 - 1.00000i) q^{49} +(-13.3843 + 3.58630i) q^{52} +(9.79796 + 9.79796i) q^{57} +(0.500000 + 0.866025i) q^{61} +(3.46618 + 7.14042i) q^{63} -8.00000i q^{64} +(3.34607 - 0.896575i) q^{67} +(-0.448288 - 1.67303i) q^{73} -16.0000 q^{76} +(-14.7224 + 8.50000i) q^{79} +(4.50000 - 7.79423i) q^{81} +(-8.66025 - 3.00000i) q^{84} +(-12.0000 - 13.8564i) q^{91} +(-11.7112 - 3.13801i) q^{93} +(3.67423 + 3.67423i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$e\left(\frac{2}{3}\right)$

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		0.258819	−	0.965926i	$-0.416667\pi$
$2$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$3$	0.448288	+	1.67303i	0.258819	+	0.965926i
$3$	0.448288	+	1.67303i	0.258819	+	0.965926i
$4$	−1.73205	−	1.00000i	−0.866025	−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.189469	−	2.63896i	0.0716124	−	0.997433i
$7$	0.189469	−	2.63896i	0.0716124	−	0.997433i
$8$		0			0
$9$	−2.59808	+	1.50000i	−0.866025	+	0.500000i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	0.896575	−	3.34607i	0.258819	−	0.965926i
$13$	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		0.965926	−	0.258819i	$-0.0833333\pi$
$17$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$18$		0			0
$19$	6.92820	−	4.00000i	1.58944	−	0.917663i	0.596040	−	0.802955i	$-0.296740\pi$
$19$	6.92820	−	4.00000i	1.58944	−	0.917663i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$		0			0
$21$	4.50000	−	0.866025i	0.981981	−	0.188982i
$22$		0			0
$23$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$23$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	−	3.67423i	−0.707107	−	0.707107i
$28$	−2.96713	+	4.38134i	−0.560734	+	0.827996i
$29$		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	0.359211	+	0.933257i	$0.383046\pi$
$31$	−3.50000	+	6.06218i	−0.628619	+	1.08880i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	6.00000			1.00000
$37$	1.34486	−	5.01910i	0.221094	−	0.825135i	−0.762838	−	0.646590i	$-0.776194\pi$
$37$	1.34486	−	5.01910i	0.221094	−	0.825135i	0.983932	−	0.178545i	$-0.0571389\pi$
$38$		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		0.258819	−	0.965926i	$-0.416667\pi$
$47$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$48$	−4.89898	+	4.89898i	−0.707107	+	0.707107i
$49$	−6.92820	−	1.00000i	−0.989743	−	0.142857i
$50$		0			0
$51$		0			0
$52$	−13.3843	+	3.58630i	−1.85606	+	0.497331i
$53$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$53$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$54$		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	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$	3.46618	+	7.14042i	0.436698	+	0.899608i
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	3.34607	−	0.896575i	0.408787	−	0.109534i	−0.0485648	−	0.998820i	$-0.515465\pi$
$67$	3.34607	−	0.896575i	0.408787	−	0.109534i	0.457352	+	0.889286i	$0.348798\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−0.448288	−	1.67303i	−0.0524681	−	0.195814i	0.934717	−	0.355393i	$-0.115653\pi$
$73$	−0.448288	−	1.67303i	−0.0524681	−	0.195814i	−0.987185	+	0.159579i	$0.948986\pi$
$74$		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	−0.682048	+	0.731307i	$0.738911\pi$
$79$	−14.7224	+	8.50000i	−1.65640	+	0.956325i	−0.974355	+	0.225018i	$0.927756\pi$
$80$		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	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	−12.0000	−	13.8564i	−1.25794	−	1.45255i
$92$		0			0
$93$	−11.7112	−	3.13801i	−1.21440	−	0.325397i
$94$		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	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−15.0573	−	4.03459i	−1.48364	−	0.397540i	−0.576055	−	0.817411i	$-0.695409\pi$
$103$	−15.0573	−	4.03459i	−1.48364	−	0.397540i	−0.907584	+	0.419871i	$0.862075\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$107$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$108$	2.68973	+	10.0382i	0.258819	+	0.965926i
$109$	1.73205	+	1.00000i	0.165900	+	0.0957826i	0.580651	−	0.814152i	$-0.302798\pi$
$109$	1.73205	+	1.00000i	0.165900	+	0.0957826i	−0.414751	+	0.909935i	$0.636131\pi$
$110$		0			0
$111$	9.00000			0.854242
$112$	9.52056	−	4.62158i	0.899608	−	0.436698i
$113$		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
$117$	−5.37945	+	20.0764i	−0.497331	+	1.85606i
$118$		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	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	−9.24316	−	19.0411i	−0.801483	−	1.65107i
$134$		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
$139$			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
$147$	−1.43280	−	12.0394i	−0.118175	−	0.992993i
$148$	−7.34847	+	7.34847i	−0.604040	+	0.604040i
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	−9.50000	+	16.4545i	−0.773099	+	1.33905i	0.162758	+	0.986666i	$0.447961\pi$
$151$	−9.50000	+	16.4545i	−0.773099	+	1.33905i	−0.935857	+	0.352381i	$0.885372\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−12.0000	−	20.7846i	−0.960769	−	1.66410i
$157$	−21.7494	+	5.82774i	−1.73579	+	0.465104i	−0.981505	−	0.191439i	$-0.938685\pi$
$157$	−21.7494	+	5.82774i	−1.73579	+	0.465104i	−0.754288	+	0.656543i	$0.772018\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	18.4034	+	4.93117i	1.44146	+	0.386239i	0.893045	−	0.449966i	$-0.148564\pi$
$163$	18.4034	+	4.93117i	1.44146	+	0.386239i	0.548417	+	0.836205i	$0.315231\pi$
$164$		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
$172$	−23.4225	+	6.27603i	−1.78595	+	0.478543i
$173$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$173$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$174$		0			0
$175$		0			0
$176$		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
$181$	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	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	13.3843	−	3.58630i	0.965926	−	0.258819i
$193$	3.13801	+	11.7112i	0.225879	+	0.842993i	0.982050	+	0.188619i	$0.0604011\pi$
$193$	3.13801	+	11.7112i	0.225879	+	0.842993i	−0.756171	+	0.654374i	$0.772932\pi$
$194$		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	$-0.127485\pi$
$199$	14.7224	+	8.50000i	1.04365	+	0.602549i	0.122782	+	0.992434i	$0.460818\pi$
$200$		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
$208$	26.7685	+	7.17260i	1.85606	+	0.497331i
$209$		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
$217$	15.3347	+	10.3849i	1.04099	+	0.704976i
$218$		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		0.965926	−	0.258819i	$-0.0833333\pi$
$227$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$228$	−7.17260	−	26.7685i	−0.475017	−	1.77279i
$229$	−19.0526	+	11.0000i	−1.25903	+	0.726900i	−0.972886	−	0.231287i	$-0.925707\pi$
$229$	−19.0526	+	11.0000i	−1.25903	+	0.726900i	−0.286143	+	0.958187i	$0.592373\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$233$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$234$		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	0.450910	−	0.892570i	$-0.351100\pi$
$241$	15.5000	−	26.8468i	0.998443	−	1.72935i	0.547533	−	0.836784i	$-0.315567\pi$
$242$		0			0
$243$	15.0573	+	4.03459i	0.965926	+	0.258819i
$244$		−	2.00000i		−	0.128037i
$245$		0			0
$246$		0			0
$247$	14.3452	−	53.5370i	0.912764	−	3.40648i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	1.13681	−	15.8338i	0.0716124	−	0.997433i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$257$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$258$		0			0
$259$	−12.9904	−	4.50000i	−0.807183	−	0.279616i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$263$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−6.69213	−	1.79315i	−0.408787	−	0.109534i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	14.0000	+	24.2487i	0.850439	+	1.47300i	0.880812	+	0.473466i	$0.156997\pi$
$271$	14.0000	+	24.2487i	0.850439	+	1.47300i	−0.0303728	+	0.999539i	$0.509669\pi$
$272$		0			0
$273$	17.8028	−	26.2880i	1.07747	−	1.59103i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	20.0764	−	5.37945i	1.20627	−	0.323220i	0.400975	−	0.916089i	$-0.368672\pi$
$277$	20.0764	−	5.37945i	1.20627	−	0.323220i	0.805299	+	0.592869i	$0.202005\pi$
$278$		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
$283$	−2.68973	−	10.0382i	−0.159888	−	0.596709i	−0.998637	−	0.0521913i	$-0.983379\pi$
$283$	−2.68973	−	10.0382i	−0.159888	−	0.596709i	0.838749	−	0.544518i	$-0.183287\pi$
$284$		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
$292$	−0.896575	+	3.34607i	−0.0524681	+	0.195814i
$293$		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	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	5.01910	+	1.34486i	0.283696	+	0.0760162i	0.397861	−	0.917446i	$-0.369753\pi$
$313$	5.01910	+	1.34486i	0.283696	+	0.0760162i	−0.114165	+	0.993462i	$0.536419\pi$
$314$		0			0
$315$		0			0
$316$	34.0000			1.91265
$317$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$317$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$318$		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
$327$	−0.896575	+	3.34607i	−0.0495807	+	0.185038i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	15.5000	+	26.8468i	0.851957	+	1.47563i	0.879440	+	0.476011i	$0.157918\pi$
$331$	15.5000	+	26.8468i	0.851957	+	1.47563i	−0.0274825	+	0.999622i	$0.508749\pi$
$332$		0			0
$333$	4.03459	+	15.0573i	0.221094	+	0.825135i
$334$		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
$343$	−3.95164	+	18.0938i	−0.213368	+	0.976972i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$347$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$348$		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		−0.258819	−	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		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		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
$367$	−15.0573	+	4.03459i	−0.785984	+	0.210604i	−0.629421	−	0.777064i	$-0.716708\pi$
$367$	−15.0573	+	4.03459i	−0.785984	+	0.210604i	−0.156563	+	0.987668i	$0.550041\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	17.1464	+	17.1464i	0.889001	+	0.889001i
$373$	−6.69213	−	1.79315i	−0.346505	−	0.0928458i	0.0813690	−	0.996684i	$-0.474071\pi$
$373$	−6.69213	−	1.79315i	−0.346505	−	0.0928458i	−0.427874	+	0.903838i	$0.640737\pi$
$374$		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		−0.965926	−	0.258819i	$-0.916667\pi$
$383$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−9.41404	+	35.1337i	−0.478543	+	1.78595i
$388$	−2.68973	−	10.0382i	−0.136550	−	0.509612i
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−4.93117	+	18.4034i	−0.247488	+	0.923638i	0.724628	+	0.689140i	$0.242011\pi$
$397$	−4.93117	+	18.4034i	−0.247488	+	0.923638i	−0.972117	+	0.234498i	$0.924655\pi$
$398$		0			0
$399$	27.7128	−	24.0000i	1.38738	−	1.20150i
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	12.5521	+	46.8449i	0.625262	+	2.33351i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−32.9090	−	19.0000i	−1.62724	−	0.939490i	−0.984911	−	0.173064i	$-0.944633\pi$
$409$	−32.9090	−	19.0000i	−1.62724	−	0.939490i	−0.642333	−	0.766426i	$-0.722033\pi$
$410$		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
$417$	−38.4797	+	10.3106i	−1.88436	+	0.504913i
$418$		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
$427$	2.38014	−	1.15539i	0.115183	−	0.0559135i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	5.37945	−	20.0764i	0.258819	−	0.965926i
$433$	−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	$-0.566261\pi$
$439$	11.2583	−	6.50000i	0.537331	−	0.310228i	0.743996	+	0.668184i	$0.232928\pi$
$440$		0			0
$441$	19.5000	−	7.79423i	0.928571	−	0.371154i
$442$		0			0
$443$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$443$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$444$	−15.5885	−	9.00000i	−0.739795	−	0.427121i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−21.1117	−	1.51575i	−0.997433	−	0.0716124i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−31.7876	−	8.51747i	−1.49351	−	0.400186i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7.62089	+	28.4416i	−0.356490	+	1.33044i	0.522108	+	0.852879i	$0.325146\pi$
$457$	−7.62089	+	28.4416i	−0.356490	+	1.33044i	−0.878599	+	0.477561i	$0.841521\pi$
$458$		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		0.258819	−	0.965926i	$-0.416667\pi$
$467$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$468$	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	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		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
$487$	35.1337	−	9.41404i	1.59206	−	0.426591i	0.649427	−	0.760424i	$-0.275009\pi$
$487$	35.1337	−	9.41404i	1.59206	−	0.426591i	0.942632	+	0.333833i	$0.108342\pi$
$488$		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	$-0.920813\pi$
$499$	−27.7128	+	16.0000i	−1.24060	+	0.716258i	−0.271380	+	0.962472i	$0.587480\pi$
$500$		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
$507$	58.5561	−	15.6901i	2.60057	−	0.696821i
$508$	−11.6555	−	43.4988i	−0.517128	−	1.92995i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−4.50000	+	0.866025i	−0.199068	+	0.0383107i
$512$		0			0
$513$	−40.1528	−	10.7589i	−1.77279	−	0.475017i
$514$		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	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−28.4416	−	7.62089i	−1.24366	−	0.333238i	−0.423777	−	0.905766i	$-0.639296\pi$
$523$	−28.4416	−	7.62089i	−1.24366	−	0.333238i	−0.819885	+	0.572528i	$0.805963\pi$
$524$		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
$532$	−3.03150	+	42.2233i	−0.131432	+	1.83061i
$533$		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	0.623404	+	0.781900i	$0.285749\pi$
$541$	23.0000	+	39.8372i	0.988847	+	1.71273i	0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$	3.13801	+	11.7112i	0.134665	+	0.502577i
$544$		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
$553$	19.6417	+	40.4624i	0.835250	+	1.72063i
$554$		0			0
$555$		0			0
$556$	23.0000	−	39.8372i	0.975417	−	1.68947i
$557$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$557$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$558$		0			0
$559$		−	84.0000i		−	3.55282i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$563$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−19.7160	−	13.3521i	−0.827996	−	0.560734i
$568$		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
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	0.983444	+	0.181210i	$0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	12.0000	+	20.7846i	0.500000	+	0.866025i
$577$	−31.7876	+	8.51747i	−1.32334	+	0.354587i	−0.850227	−	0.526417i	$-0.823535\pi$
$577$	−31.7876	+	8.51747i	−1.32334	+	0.354587i	−0.473109	+	0.881004i	$0.656868\pi$
$578$		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	$0.250000\pi$
$588$	−9.55772	+	22.2856i	−0.394154	+	0.919044i
$589$			56.0000i			2.30744i
$590$		0			0
$591$		0			0
$592$	20.0764	−	5.37945i	0.825135	−	0.221094i
$593$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$593$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.62089	+	28.4416i	−0.311902	+	1.16404i
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		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
$607$	10.3106	−	38.4797i	0.418495	−	1.56184i	−0.359235	−	0.933247i	$-0.616962\pi$
$607$	10.3106	−	38.4797i	0.418495	−	1.56184i	0.777730	−	0.628598i	$-0.216371\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−12.5521	−	46.8449i	−0.506973	−	1.89205i	−0.448553	−	0.893756i	$-0.648061\pi$
$613$	−12.5521	−	46.8449i	−0.506973	−	1.89205i	−0.0584195	−	0.998292i	$-0.518606\pi$
$614$		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	−0.939829	−	0.341644i	$-0.889016\pi$
$619$	−42.4352	−	24.5000i	−1.70562	−	0.984738i	−0.765787	−	0.643094i	$-0.777650\pi$
$620$		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
$628$	43.4988	+	11.6555i	1.73579	+	0.465104i
$629$		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
$633$	13.0003	+	48.5179i	0.516717	+	1.92842i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−38.8401	+	29.0421i	−1.53890	+	1.15069i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		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		0.965926	−	0.258819i	$-0.0833333\pi$
$647$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$648$		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		−0.965926	−	0.258819i	$-0.916667\pi$
$653$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$654$		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	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	5.50000	−	9.52628i	0.213925	−	0.370529i	0.952940	+	0.303160i	$0.0980418\pi$
$662$		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		0.258819	−	0.965926i	$-0.416667\pi$
$677$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$678$		0			0
$679$	10.3923	−	9.00000i	0.398820	−	0.345388i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$683$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$684$	41.5692	−	24.0000i	1.58944	−	0.917663i
$685$		0			0
$686$		0			0
$687$	−26.9444	−	26.9444i	−1.02799	−	1.02799i
$688$	46.8449	+	12.5521i	1.78595	+	0.478543i
$689$		0			0
$690$		0			0
$691$	20.5000	+	35.5070i	0.779857	+	1.35075i	0.932024	+	0.362397i	$0.118041\pi$
$691$	20.5000	+	35.5070i	0.779857	+	1.35075i	−0.152167	+	0.988355i	$0.548625\pi$
$692$		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
$703$	−10.7589	−	40.1528i	−0.405780	−	1.51439i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	45.8993	−	26.5000i	1.72379	−	0.995228i	0.813107	−	0.582115i	$-0.197775\pi$
$709$	45.8993	−	26.5000i	1.72379	−	0.995228i	0.910679	−	0.413114i	$-0.135559\pi$
$710$		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	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	−13.5000	+	38.9711i	−0.502766	+	1.45136i
$722$		0			0
$723$	51.8640	+	13.8969i	1.92884	+	0.516832i
$724$	−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
$732$	3.34607	−	0.896575i	0.123674	−	0.0331384i
$733$	−31.7876	−	8.51747i	−1.17410	−	0.314600i	−0.381518	−	0.924362i	$-0.624598\pi$
$733$	−31.7876	−	8.51747i	−1.17410	−	0.314600i	−0.792585	+	0.609762i	$0.791265\pi$
$734$		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	$-0.0950814\pi$
$739$	32.0429	+	18.5000i	1.17872	+	0.680534i	0.223001	+	0.974818i	$0.428415\pi$
$740$		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	0.948753	−	0.316017i	$-0.102346\pi$
$751$	5.50000	+	9.52628i	0.200698	+	0.347619i	−0.748056	+	0.663636i	$0.769012\pi$
$752$		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	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	2.96713	−	4.38134i	0.107417	−	0.158615i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−26.7685	−	7.17260i	−0.965926	−	0.258819i
$769$			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
$772$	6.27603	−	23.4225i	0.225879	−	0.842993i
$773$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$773$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	1.70522	−	23.7506i	0.0611744	−	0.852049i
$778$		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
$787$	45.1719	−	12.1038i	1.61020	−	0.431453i	0.662100	−	0.749415i	$-0.269665\pi$
$787$	45.1719	−	12.1038i	1.61020	−	0.431453i	0.948103	+	0.317962i	$0.102999\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.69213	+	1.79315i	0.237645	+	0.0636767i
$794$		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	$-0.833333\pi$
$809$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$810$		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
$817$	25.1041	−	93.6898i	0.878282	−	3.27779i
$818$		0			0
$819$	51.9615	+	18.0000i	1.81568	+	0.628971i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	8.51747	+	31.7876i	0.296900	+	1.10805i	0.939696	+	0.342010i	$0.111108\pi$
$823$	8.51747	+	31.7876i	0.296900	+	1.10805i	−0.642796	+	0.766037i	$0.722226\pi$
$824$		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
$831$	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
$837$	35.1337	−	9.41404i	1.21440	−	0.325397i
$838$		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
$847$	−26.1815	+	12.7093i	−0.899608	+	0.436698i
$848$		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		0.965926	−	0.258819i	$-0.0833333\pi$
$857$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$858$		0			0
$859$	−48.4974	+	28.0000i	−1.65471	+	0.955348i	−0.679613	+	0.733571i	$0.737852\pi$
$859$	−48.4974	+	28.0000i	−1.65471	+	0.955348i	−0.975097	+	0.221777i	$0.928814\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$863$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	20.8207	+	20.8207i	0.707107	+	0.707107i
$868$	−16.1755	−	33.3220i	−0.549033	−	1.13102i
$869$		0			0
$870$		0			0
$871$	12.0000	−	20.7846i	0.406604	−	0.704260i
$872$		0			0
$873$	−15.0573	−	4.03459i	−0.509612	−	0.136550i
$874$		0			0
$875$		0			0
$876$	−6.00000			−0.202721
$877$	1.34486	−	5.01910i	0.0454128	−	0.169483i	−0.939495	−	0.342563i	$-0.888705\pi$
$877$	1.34486	−	5.01910i	0.0454128	−	0.169483i	0.984908	+	0.173080i	$0.0553718\pi$
$878$		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		0.258819	−	0.965926i	$-0.416667\pi$
$887$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$888$		0			0
$889$	45.0333	−	39.0000i	1.51037	−	1.30802i
$890$		0			0
$891$		0			0
$892$	36.8067	−	9.86233i	1.23238	−	0.330215i
$893$		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
$903$	31.1548	−	46.0041i	1.03677	−	1.53092i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−55.2101	+	14.7935i	−1.83322	+	0.491210i	−0.998253	−	0.0590889i	$-0.981180\pi$
$907$	−55.2101	+	14.7935i	−1.83322	+	0.491210i	−0.834968	+	0.550299i	$0.814514\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−14.3452	+	53.5370i	−0.475017	+	1.77279i
$913$		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	$-0.671917\pi$
$919$	−0.866025	+	0.500000i	−0.0285675	+	0.0164935i	0.485648	+	0.874154i	$0.338584\pi$
$920$		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
$927$	45.1719	−	12.1038i	1.48364	−	0.397540i
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		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	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		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	$0.583333\pi$
$948$	15.2418	+	56.8831i	0.495030	+	1.84748i
$949$	−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	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	−22.0454	−	22.0454i	−0.707107	−	0.707107i
$973$	60.6960	+	4.35778i	1.94583	+	0.139704i
$974$		0			0
$975$		0			0
$976$	−2.00000	+	3.46410i	−0.0640184	+	0.110883i
$977$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$977$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−6.00000			−0.191565
$982$		0			0
$983$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$983$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$984$		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	−0.968864	−	0.247592i	$-0.920361\pi$
$991$	−8.50000	+	14.7224i	−0.270011	+	0.467673i	0.698853	+	0.715265i	$0.253694\pi$
$992$		0			0
$993$	−37.9671	+	37.9671i	−1.20485	+	1.20485i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−21.7494	+	5.82774i	−0.688811	+	0.184566i	−0.586214	−	0.810157i	$-0.699382\pi$
$997$	−21.7494	+	5.82774i	−0.688811	+	0.184566i	−0.102598	+	0.994723i	$0.532715\pi$
$998$		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.32.2	yes	8
3.2	odd	2	CM	525.2.bf.c.32.2	yes	8
5.2	odd	4	inner	525.2.bf.c.368.2	yes	8
5.3	odd	4	inner	525.2.bf.c.368.1	yes	8
5.4	even	2	inner	525.2.bf.c.32.1	✓	8
7.2	even	3	inner	525.2.bf.c.107.1	yes	8
15.2	even	4	inner	525.2.bf.c.368.2	yes	8
15.8	even	4	inner	525.2.bf.c.368.1	yes	8
15.14	odd	2	inner	525.2.bf.c.32.1	✓	8
21.2	odd	6	inner	525.2.bf.c.107.1	yes	8
35.2	odd	12	inner	525.2.bf.c.443.1	yes	8
35.9	even	6	inner	525.2.bf.c.107.2	yes	8
35.23	odd	12	inner	525.2.bf.c.443.2	yes	8
105.2	even	12	inner	525.2.bf.c.443.1	yes	8
105.23	even	12	inner	525.2.bf.c.443.2	yes	8
105.44	odd	6	inner	525.2.bf.c.107.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bf.c.32.1	✓	8	5.4	even	2	inner
525.2.bf.c.32.1	✓	8	15.14	odd	2	inner
525.2.bf.c.32.2	yes	8	1.1	even	1	trivial
525.2.bf.c.32.2	yes	8	3.2	odd	2	CM
525.2.bf.c.107.1	yes	8	7.2	even	3	inner
525.2.bf.c.107.1	yes	8	21.2	odd	6	inner
525.2.bf.c.107.2	yes	8	35.9	even	6	inner
525.2.bf.c.107.2	yes	8	105.44	odd	6	inner
525.2.bf.c.368.1	yes	8	5.3	odd	4	inner
525.2.bf.c.368.1	yes	8	15.8	even	4	inner
525.2.bf.c.368.2	yes	8	5.2	odd	4	inner
525.2.bf.c.368.2	yes	8	15.2	even	4	inner
525.2.bf.c.443.1	yes	8	35.2	odd	12	inner
525.2.bf.c.443.1	yes	8	105.2	even	12	inner
525.2.bf.c.443.2	yes	8	35.23	odd	12	inner
525.2.bf.c.443.2	yes	8	105.23	even	12	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)

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