LMFDB - Embedded newform 525.2.bf.b.32.1

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			32.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	525.32
Dual form			525.2.bf.b.443.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.448288 - 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(-1.48356 - 2.19067i) q^{7} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})$	$q+(-0.448288 - 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(-1.48356 - 2.19067i) q^{7} +(-2.59808 + 1.50000i) q^{9} +(-0.896575 + 3.34607i) q^{12} +(1.22474 - 1.22474i) q^{13} +(2.00000 + 3.46410i) q^{16} +(-6.06218 + 3.50000i) q^{19} +(-3.00000 + 3.46410i) q^{21} +(3.67423 + 3.67423i) q^{27} +(0.378937 + 5.27792i) q^{28} +(-3.50000 + 6.06218i) q^{31} +6.00000 q^{36} +(3.13801 - 11.7112i) q^{37} +(-2.59808 - 1.50000i) q^{39} +(-8.57321 + 8.57321i) q^{43} +(4.89898 - 4.89898i) q^{48} +(-2.59808 + 6.50000i) q^{49} +(-3.34607 + 0.896575i) q^{52} +(8.57321 + 8.57321i) q^{57} +(-7.00000 - 12.1244i) q^{61} +(7.14042 + 3.46618i) q^{63} -8.00000i q^{64} +(-11.7112 + 3.13801i) q^{67} +(-4.03459 - 15.0573i) q^{73} +14.0000 q^{76} +(11.2583 - 6.50000i) q^{79} +(4.50000 - 7.79423i) q^{81} +(8.66025 - 3.00000i) q^{84} +(-4.50000 - 0.866025i) q^{91} +(11.7112 + 3.13801i) q^{93} +(-9.79796 - 9.79796i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{16} - 24 q^{21} - 28 q^{31} + 48 q^{36} - 56 q^{61} + 112 q^{76} + 36 q^{81} - 36 q^{91}+O(q^{100}) $ 8 * q + 16 * q^16 - 24 * q^21 - 28 * q^31 + 48 * q^36 - 56 * q^61 + 112 * q^76 + 36 * q^81 - 36 * 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
$7$	−1.48356	−	2.19067i	−0.560734	−	0.827996i
$7$	−1.48356	−	2.19067i	−0.560734	−	0.827996i
$8$		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
$13$	1.22474	−	1.22474i	0.339683	−	0.339683i	−0.516565	−	0.856248i	$-0.672790\pi$
$13$	1.22474	−	1.22474i	0.339683	−	0.339683i	0.856248	+	0.516565i	$0.172790\pi$
$14$		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
$19$	−6.06218	+	3.50000i	−1.39076	+	0.802955i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$	−6.06218	+	3.50000i	−1.39076	+	0.802955i	−0.397360	+	0.917663i	$0.630073\pi$
$20$		0			0
$21$	−3.00000	+	3.46410i	−0.654654	+	0.755929i
$22$		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
$28$	0.378937	+	5.27792i	0.0716124	+	0.997433i
$29$		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
$37$	3.13801	−	11.7112i	0.515886	−	1.92531i	0.178545	−	0.983932i	$-0.442861\pi$
$37$	3.13801	−	11.7112i	0.515886	−	1.92531i	0.337342	−	0.941382i	$-0.390472\pi$
$38$		0			0
$39$	−2.59808	−	1.50000i	−0.416025	−	0.240192i
$40$		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	$0.625495\pi$
$43$	−8.57321	+	8.57321i	−1.30740	+	1.30740i	−0.923283	+	0.384120i	$0.874505\pi$
$44$		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
$49$	−2.59808	+	6.50000i	−0.371154	+	0.928571i
$50$		0			0
$51$		0			0
$52$	−3.34607	+	0.896575i	−0.464016	+	0.124333i
$53$		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
$57$	8.57321	+	8.57321i	1.13555	+	1.13555i
$58$		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
$61$	−7.00000	−	12.1244i	−0.896258	−	1.55236i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−7.00000	−	12.1244i	−0.896258	−	1.55236i	−0.0640184	−	0.997949i	$-0.520392\pi$
$62$		0			0
$63$	7.14042	+	3.46618i	0.899608	+	0.436698i
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	−11.7112	+	3.13801i	−1.43075	+	0.383369i	−0.889286	−	0.457352i	$-0.848798\pi$
$67$	−11.7112	+	3.13801i	−1.43075	+	0.383369i	−0.541468	+	0.840721i	$0.682131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−4.03459	−	15.0573i	−0.472213	−	1.76232i	−0.631792	−	0.775138i	$-0.717680\pi$
$73$	−4.03459	−	15.0573i	−0.472213	−	1.76232i	0.159579	−	0.987185i	$-0.448986\pi$
$74$		0			0
$75$		0			0
$76$	14.0000			1.60591
$77$		0			0
$78$		0			0
$79$	11.2583	−	6.50000i	1.26666	−	0.731307i	0.292306	−	0.956325i	$-0.405577\pi$
$79$	11.2583	−	6.50000i	1.26666	−	0.731307i	0.974355	+	0.225018i	$0.0722440\pi$
$80$		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
$91$	−4.50000	−	0.866025i	−0.471728	−	0.0907841i
$92$		0			0
$93$	11.7112	+	3.13801i	1.21440	+	0.325397i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−9.79796	−	9.79796i	−0.994832	−	0.994832i	0.00515471	−	0.999987i	$-0.498359\pi$
$97$	−9.79796	−	9.79796i	−0.994832	−	0.994832i	−0.999987	+	0.00515471i	$0.998359\pi$
$98$		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
$103$	−18.4034	−	4.93117i	−1.81334	−	0.485882i	−0.817411	−	0.576055i	$-0.804591\pi$
$103$	−18.4034	−	4.93117i	−1.81334	−	0.485882i	−0.995926	+	0.0901732i	$0.971258\pi$
$104$		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
$109$	14.7224	+	8.50000i	1.41015	+	0.814152i	0.995402	−	0.0957826i	$-0.0305354\pi$
$109$	14.7224	+	8.50000i	1.41015	+	0.814152i	0.414751	+	0.909935i	$0.363869\pi$
$110$		0			0
$111$	−21.0000			−1.99323
$112$	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
$117$	−1.34486	+	5.01910i	−0.124333	+	0.464016i
$118$		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
$127$	8.57321	+	8.57321i	0.760750	+	0.760750i	0.976458	−	0.215708i	$-0.0692060\pi$
$127$	8.57321	+	8.57321i	0.760750	+	0.760750i	−0.215708	+	0.976458i	$0.569206\pi$
$128$		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
$133$	16.6610	+	8.08776i	1.44469	+	0.701298i
$134$		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
$139$		−	7.00000i		−	0.593732i	−0.954919	−	0.296866i	$-0.904058\pi$
$139$		−	7.00000i		−	0.593732i	0.954919	−	0.296866i	$-0.0959415\pi$
$140$		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
$148$	−17.1464	+	17.1464i	−1.40943	+	1.40943i
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	−2.00000	+	3.46410i	−0.162758	+	0.281905i	−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−2.00000	+	3.46410i	−0.162758	+	0.281905i	0.773099	+	0.634285i	$0.218706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	3.00000	+	5.19615i	0.240192	+	0.416025i
$157$	−20.0764	+	5.37945i	−1.60227	+	0.429327i	−0.945727	−	0.324961i	$-0.894649\pi$
$157$	−20.0764	+	5.37945i	−1.60227	+	0.429327i	−0.656543	+	0.754288i	$0.727982\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	23.4225	+	6.27603i	1.83459	+	0.491576i	0.998383	−	0.0568404i	$-0.0181026\pi$
$163$	23.4225	+	6.27603i	1.83459	+	0.491576i	0.836205	+	0.548417i	$0.184769\pi$
$164$		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
$169$			10.0000i			0.769231i
$170$		0			0
$171$	10.5000	−	18.1865i	0.802955	−	1.39076i
$172$	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
$183$	−17.1464	+	17.1464i	−1.26750	+	1.26750i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	2.59808	−	13.5000i	0.188982	−	0.981981i
$190$		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	$-0.939599\pi$
$193$	−3.13801	−	11.7112i	−0.225879	−	0.842993i	0.756171	−	0.654374i	$-0.227068\pi$
$194$		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
$199$	−24.2487	−	14.0000i	−1.71895	−	0.992434i	−0.920864	−	0.389885i	$-0.872515\pi$
$199$	−24.2487	−	14.0000i	−1.71895	−	0.992434i	−0.798082	−	0.602549i	$-0.794152\pi$
$200$		0			0
$201$	10.5000	+	18.1865i	0.740613	+	1.28278i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	6.69213	+	1.79315i	0.464016	+	0.124333i
$209$		0			0
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	18.4727	−	1.32628i	1.25401	−	0.0900338i
$218$		0			0
$219$	−23.3827	+	13.5000i	−1.58006	+	0.912245i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	7.34847	−	7.34847i	0.492090	−	0.492090i	−0.416874	−	0.908964i	$-0.636874\pi$
$223$	7.34847	−	7.34847i	0.492090	−	0.492090i	0.908964	+	0.416874i	$0.136874\pi$
$224$		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	$0.916667\pi$
$228$	−6.27603	−	23.4225i	−0.415640	−	1.55119i
$229$	−6.06218	+	3.50000i	−0.400600	+	0.231287i	−0.686743	−	0.726900i	$-0.740960\pi$
$229$	−6.06218	+	3.50000i	−0.400600	+	0.231287i	0.286143	+	0.958187i	$0.407627\pi$
$230$		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
$237$	−15.9217	−	15.9217i	−1.03422	−	1.03422i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−7.00000	+	12.1244i	−0.450910	+	0.780998i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−7.00000	+	12.1244i	−0.450910	+	0.780998i	0.547533	+	0.836784i	$0.315567\pi$
$242$		0			0
$243$	−15.0573	−	4.03459i	−0.965926	−	0.258819i
$244$			28.0000i			1.79252i
$245$		0			0
$246$		0			0
$247$	−3.13801	+	11.7112i	−0.199667	+	0.745168i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−8.90138	−	13.1440i	−0.560734	−	0.827996i
$253$		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
$259$	−30.3109	+	10.5000i	−1.88343	+	0.652438i
$260$		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
$268$	23.4225	+	6.27603i	1.43075	+	0.383369i
$269$		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
$273$	0.568406	+	7.91688i	0.0344015	+	0.479151i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−11.7112	+	3.13801i	−0.703660	+	0.188545i	−0.592869	−	0.805299i	$-0.702005\pi$
$277$	−11.7112	+	3.13801i	−0.703660	+	0.188545i	−0.110790	+	0.993844i	$0.535338\pi$
$278$		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
$283$	−8.51747	−	31.7876i	−0.506311	−	1.88958i	−0.454120	−	0.890941i	$-0.650046\pi$
$283$	−8.51747	−	31.7876i	−0.506311	−	1.88958i	−0.0521913	−	0.998637i	$-0.516621\pi$
$284$		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
$291$	−12.0000	+	20.7846i	−0.703452	+	1.21842i
$292$	−8.06918	+	30.1146i	−0.472213	+	1.76232i
$293$		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
$301$	31.5000	+	6.06218i	1.81563	+	0.349418i
$302$		0			0
$303$		0			0
$304$	−24.2487	−	14.0000i	−1.39076	−	0.802955i
$305$		0			0
$306$		0			0
$307$	1.22474	+	1.22474i	0.0698999	+	0.0698999i	0.741192	−	0.671293i	$-0.234261\pi$
$307$	1.22474	+	1.22474i	0.0698999	+	0.0698999i	−0.671293	+	0.741192i	$0.734261\pi$
$308$		0			0
$309$			33.0000i			1.87730i
$310$		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	0.114165	−	0.993462i	$-0.463581\pi$
$313$	−5.01910	−	1.34486i	−0.283696	−	0.0760162i	−0.397861	+	0.917446i	$0.630247\pi$
$314$		0			0
$315$		0			0
$316$	−26.0000			−1.46261
$317$		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
$327$	7.62089	−	28.4416i	0.421436	−	1.57282i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.500000	+	0.866025i	0.0274825	+	0.0476011i	0.879440	−	0.476011i	$-0.157918\pi$
$331$	0.500000	+	0.866025i	0.0274825	+	0.0476011i	−0.851957	+	0.523612i	$0.824584\pi$
$332$		0			0
$333$	9.41404	+	35.1337i	0.515886	+	1.92531i
$334$		0			0
$335$		0			0
$336$	−18.0000	−	3.46410i	−0.981981	−	0.188982i
$337$	25.7196	+	25.7196i	1.40104	+	1.40104i	0.796815	+	0.604223i	$0.206516\pi$
$337$	25.7196	+	25.7196i	1.40104	+	1.40104i	0.604223	+	0.796815i	$0.293484\pi$
$338$		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		0.965926	−	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
$351$	9.00000			0.480384
$352$		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
$361$	15.0000	−	25.9808i	0.789474	−	1.36741i
$362$		0			0
$363$	−13.4722	+	13.4722i	−0.707107	+	0.707107i
$364$	6.92820	+	6.00000i	0.363137	+	0.314485i
$365$		0			0
$366$		0			0
$367$	15.0573	−	4.03459i	0.785984	−	0.210604i	0.156563	−	0.987668i	$-0.449959\pi$
$367$	15.0573	−	4.03459i	0.785984	−	0.210604i	0.629421	+	0.777064i	$0.283292\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−17.1464	−	17.1464i	−0.889001	−	0.889001i
$373$	−35.1337	−	9.41404i	−1.81915	−	0.487441i	−0.822469	−	0.568810i	$-0.807404\pi$
$373$	−35.1337	−	9.41404i	−1.81915	−	0.487441i	−0.996684	+	0.0813690i	$0.974071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	37.0000i		−	1.90056i	−0.311393	−	0.950281i	$-0.600796\pi$
$379$		−	37.0000i		−	1.90056i	0.311393	−	0.950281i	$-0.399204\pi$
$380$		0			0
$381$	10.5000	−	18.1865i	0.537931	−	0.931724i
$382$		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
$388$	7.17260	+	26.7685i	0.364134	+	1.35897i
$389$		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	$-0.757989\pi$
$397$	4.93117	−	18.4034i	0.247488	−	0.923638i	0.972117	−	0.234498i	$-0.0753447\pi$
$398$		0			0
$399$	6.06218	−	31.5000i	0.303488	−	1.57697i
$400$		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
$403$	3.13801	+	11.7112i	0.156316	+	0.583378i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	6.06218	+	3.50000i	0.299755	+	0.173064i	0.642333	−	0.766426i	$-0.277967\pi$
$409$	6.06218	+	3.50000i	0.299755	+	0.173064i	−0.342578	+	0.939490i	$0.611300\pi$
$410$		0			0
$411$		0			0
$412$	26.9444	+	26.9444i	1.32745	+	1.32745i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−11.7112	+	3.13801i	−0.573501	+	0.153669i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	19.0000			0.926003			0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000			0.926003			0.463002	+	0.886357i	$0.346772\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−16.1755	+	33.3220i	−0.782788	+	1.61256i
$428$		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	−0.212101	−	0.977248i	$-0.568030\pi$
$433$	15.9217	−	15.9217i	0.765147	−	0.765147i	0.977248	+	0.212101i	$0.0680304\pi$
$434$		0			0
$435$		0			0
$436$	−17.0000	−	29.4449i	−0.814152	−	1.41015i
$437$		0			0
$438$		0			0
$439$	24.2487	−	14.0000i	1.15733	−	0.668184i	0.206666	−	0.978412i	$-0.433739\pi$
$439$	24.2487	−	14.0000i	1.15733	−	0.668184i	0.950662	+	0.310228i	$0.100405\pi$
$440$		0			0
$441$	−3.00000	−	20.7846i	−0.142857	−	0.989743i
$442$		0			0
$443$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$443$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$444$	36.3731	+	21.0000i	1.72619	+	0.996616i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−17.5254	+	11.8685i	−0.827996	+	0.560734i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	6.69213	+	1.79315i	0.314424	+	0.0842496i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.13801	−	11.7112i	0.146790	−	0.547828i	−0.852879	−	0.522108i	$-0.825146\pi$
$457$	3.13801	−	11.7112i	0.146790	−	0.547828i	0.999669	−	0.0257197i	$-0.00818773\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	25.7196	−	25.7196i	1.19529	−	1.19529i	0.219733	−	0.975560i	$-0.429481\pi$
$463$	25.7196	−	25.7196i	1.19529	−	1.19529i	0.975560	−	0.219733i	$-0.0705187\pi$
$464$		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	$0.583333\pi$
$468$	7.34847	−	7.34847i	0.339683	−	0.339683i
$469$	24.2487	+	21.0000i	1.11970	+	0.969690i
$470$		0			0
$471$	18.0000	+	31.1769i	0.829396	+	1.43656i
$472$		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
$481$	−10.5000	−	18.1865i	−0.478759	−	0.829235i
$482$		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	−0.942632	−	0.333833i	$-0.891658\pi$
$487$	−35.1337	+	9.41404i	−1.59206	+	0.426591i	−0.649427	+	0.760424i	$0.724991\pi$
$488$		0			0
$489$		−	42.0000i		−	1.89931i
$490$		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
$499$	37.2391	−	21.5000i	1.66705	−	0.962472i	0.697835	−	0.716258i	$-0.254147\pi$
$499$	37.2391	−	21.5000i	1.66705	−	0.962472i	0.969216	−	0.246214i	$-0.0791865\pi$
$500$		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
$507$	16.7303	−	4.48288i	0.743020	−	0.199092i
$508$	−6.27603	−	23.4225i	−0.278454	−	1.03920i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−27.0000	+	31.1769i	−1.19441	+	1.37919i
$512$		0			0
$513$	−35.1337	−	9.41404i	−1.55119	−	0.415640i
$514$		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	0.819885	−	0.572528i	$-0.194037\pi$
$523$	28.4416	+	7.62089i	1.24366	+	0.333238i	0.423777	+	0.905766i	$0.360704\pi$
$524$		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
$532$	−20.7699	−	30.6694i	−0.900489	−	1.32969i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−14.5000	−	25.1147i	−0.623404	−	1.07977i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−14.5000	−	25.1147i	−0.623404	−	1.07977i	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	0.238110	−	0.971238i	$-0.423472\pi$
$547$	−17.1464	−	17.1464i	−0.733128	−	0.733128i	−0.971238	+	0.238110i	$0.923472\pi$
$548$		0			0
$549$	36.3731	+	21.0000i	1.55236	+	0.896258i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−30.9418	−	15.0201i	−1.31578	−	0.638721i
$554$		0			0
$555$		0			0
$556$	−7.00000	+	12.1244i	−0.296866	+	0.514187i
$557$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$557$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$558$		0			0
$559$			21.0000i			0.888205i
$560$		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
$567$	−23.7506	+	1.70522i	−0.997433	+	0.0716124i
$568$		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
$571$	15.5000	−	26.8468i	0.648655	−	1.12350i	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	15.5000	−	26.8468i	0.648655	−	1.12350i	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	0.473109	−	0.881004i	$-0.343132\pi$
$577$	31.7876	−	8.51747i	1.32334	−	0.354587i	0.850227	+	0.526417i	$0.176465\pi$
$578$		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	$0.250000\pi$
$588$	−19.4201	−	14.5211i	−0.800869	−	0.598839i
$589$		−	49.0000i		−	2.01901i
$590$		0			0
$591$		0			0
$592$	46.8449	−	12.5521i	1.92531	−	0.515886i
$593$		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
$597$	−12.5521	+	46.8449i	−0.513721	+	1.91723i
$598$		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
$601$	49.0000			1.99875			0.999376	−	0.0353259i	$-0.0112469\pi$
$601$	49.0000			1.99875			0.999376	+	0.0353259i	$0.0112469\pi$
$602$		0			0
$603$	25.7196	−	25.7196i	1.04738	−	1.04738i
$604$	6.92820	−	4.00000i	0.281905	−	0.162758i
$605$		0			0
$606$		0			0
$607$	−1.34486	+	5.01910i	−0.0545863	+	0.203719i	−0.987833	−	0.155517i	$-0.950296\pi$
$607$	−1.34486	+	5.01910i	−0.0545863	+	0.203719i	0.933247	+	0.359235i	$0.116962\pi$
$608$		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	$0.351939\pi$
$613$	12.5521	+	46.8449i	0.506973	+	1.89205i	0.0584195	+	0.998292i	$0.481394\pi$
$614$		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
$624$		−	12.0000i		−	0.480384i
$625$		0			0
$626$		0			0
$627$		0			0
$628$	40.1528	+	10.7589i	1.60227	+	0.429327i
$629$		0			0
$630$		0			0
$631$	44.0000			1.75161			0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000			1.75161			0.875806	+	0.482663i	$0.160330\pi$
$632$		0			0
$633$	7.17260	+	26.7685i	0.285085	+	1.06395i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	4.77886	+	11.1428i	0.189345	+	0.441495i
$638$		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
$643$	−35.5176	+	35.5176i	−1.40068	+	1.40068i	−0.602739	+	0.797938i	$0.705924\pi$
$643$	−35.5176	+	35.5176i	−1.40068	+	1.40068i	−0.797938	+	0.602739i	$0.794076\pi$
$644$		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
$652$	−34.2929	−	34.2929i	−1.34301	−	1.34301i
$653$		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
$657$	33.0681	+	33.0681i	1.29011	+	1.29011i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−24.5000	+	42.4352i	−0.952940	+	1.65054i	−0.213925	+	0.976850i	$0.568625\pi$
$661$	−24.5000	+	42.4352i	−0.952940	+	1.65054i	−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
$669$	−15.5885	−	9.00000i	−0.602685	−	0.347960i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	25.7196	−	25.7196i	0.991419	−	0.991419i	−0.00854415	−	0.999963i	$-0.502720\pi$
$673$	25.7196	−	25.7196i	0.991419	−	0.991419i	0.999963	+	0.00854415i	$0.00271972\pi$
$674$		0			0
$675$		0			0
$676$	10.0000	−	17.3205i	0.384615	−	0.666173i
$677$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$677$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$678$		0			0
$679$	−6.92820	+	36.0000i	−0.265880	+	1.38155i
$680$		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	$0.416667\pi$
$684$	−36.3731	+	21.0000i	−1.39076	+	0.802955i
$685$		0			0
$686$		0			0
$687$	8.57321	+	8.57321i	0.327089	+	0.327089i
$688$	−46.8449	−	12.5521i	−1.78595	−	0.478543i
$689$		0			0
$690$		0			0
$691$	−24.5000	−	42.4352i	−0.932024	−	1.61431i	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	−24.5000	−	42.4352i	−0.932024	−	1.61431i	−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
$703$	21.9661	+	81.9786i	0.828467	+	3.09188i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−19.0526	+	11.0000i	−0.715534	+	0.413114i	−0.813107	−	0.582115i	$-0.802225\pi$
$709$	−19.0526	+	11.0000i	−0.715534	+	0.413114i	0.0975728	+	0.995228i	$0.468892\pi$
$710$		0			0
$711$	−19.5000	+	33.7750i	−0.731307	+	1.26666i
$712$		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
$721$	16.5000	+	47.6314i	0.614492	+	1.77389i
$722$		0			0
$723$	23.4225	+	6.27603i	0.871091	+	0.233408i
$724$	−12.1244	−	7.00000i	−0.450598	−	0.260153i
$725$		0			0
$726$		0			0
$727$	−15.9217	−	15.9217i	−0.590503	−	0.590503i	0.347265	−	0.937767i	$-0.387111\pi$
$727$	−15.9217	−	15.9217i	−0.590503	−	0.590503i	−0.937767	+	0.347265i	$0.887111\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$		0			0
$732$	46.8449	−	12.5521i	1.73144	−	0.463937i
$733$	−51.8640	−	13.8969i	−1.91564	−	0.513294i	−0.991279	−	0.131777i	$-0.957932\pi$
$733$	−51.8640	−	13.8969i	−1.91564	−	0.513294i	−0.924362	−	0.381518i	$-0.875402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−45.8993	−	26.5000i	−1.68843	−	0.974818i	−0.955718	−	0.294285i	$-0.904919\pi$
$739$	−45.8993	−	26.5000i	−1.68843	−	0.974818i	−0.732717	−	0.680534i	$-0.761748\pi$
$740$		0			0
$741$	21.0000			0.771454
$742$		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
$751$	20.5000	+	35.5070i	0.748056	+	1.29567i	0.948753	+	0.316017i	$0.102346\pi$
$751$	20.5000	+	35.5070i	0.748056	+	1.29567i	−0.200698	+	0.979653i	$0.564321\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−18.0000	+	20.7846i	−0.654654	+	0.755929i
$757$	−34.2929	−	34.2929i	−1.24640	−	1.24640i	−0.957301	−	0.289095i	$-0.906646\pi$
$757$	−34.2929	−	34.2929i	−1.24640	−	1.24640i	−0.289095	−	0.957301i	$-0.593354\pi$
$758$		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
$763$	−3.22097	−	44.8623i	−0.116607	−	1.62412i
$764$		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
$777$	31.1548	+	46.0041i	1.11767	+	1.65039i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−27.7128	+	4.00000i	−0.989743	+	0.142857i
$785$		0			0
$786$		0			0
$787$	−3.34607	+	0.896575i	−0.119274	+	0.0319595i	−0.317962	−	0.948103i	$-0.602999\pi$
$787$	−3.34607	+	0.896575i	−0.119274	+	0.0319595i	0.198688	+	0.980063i	$0.436332\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−23.4225	−	6.27603i	−0.831756	−	0.222868i
$794$		0			0
$795$		0			0
$796$	28.0000	+	48.4974i	0.992434	+	1.71895i
$797$		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
$804$		−	42.0000i		−	1.48123i
$805$		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
$811$	−56.0000			−1.96643			−0.983213	−	0.182462i	$-0.941593\pi$
$811$	−56.0000			−1.96643			−0.983213	+	0.182462i	$0.941593\pi$
$812$		0			0
$813$	34.2929	−	34.2929i	1.20270	−	1.20270i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	21.9661	−	81.9786i	0.768497	−	2.86807i
$818$		0			0
$819$	12.9904	−	4.50000i	0.453921	−	0.157243i
$820$		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
$823$	−6.27603	−	23.4225i	−0.218769	−	0.816456i	−0.984806	−	0.173659i	$-0.944441\pi$
$823$	−6.27603	−	23.4225i	−0.218769	−	0.816456i	0.766037	−	0.642796i	$-0.222226\pi$
$824$		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
$829$	6.06218	+	3.50000i	0.210548	+	0.121560i	0.601566	−	0.798823i	$-0.294544\pi$
$829$	6.06218	+	3.50000i	0.210548	+	0.121560i	−0.391018	+	0.920383i	$0.627877\pi$
$830$		0			0
$831$	10.5000	+	18.1865i	0.364241	+	0.630884i
$832$	−9.79796	−	9.79796i	−0.339683	−	0.339683i
$833$		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
$844$	27.7128	+	16.0000i	0.953914	+	0.550743i
$845$		0			0
$846$		0			0
$847$	−12.7093	+	26.1815i	−0.436698	+	0.899608i
$848$		0			0
$849$	−49.3634	+	28.5000i	−1.69415	+	0.978117i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−33.0681	+	33.0681i	−1.13223	+	1.13223i	−0.142425	+	0.989806i	$0.545490\pi$
$853$	−33.0681	+	33.0681i	−1.13223	+	1.13223i	−0.989806	+	0.142425i	$0.954510\pi$
$854$		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
$868$	−33.3220	−	16.1755i	−1.13102	−	0.549033i
$869$		0			0
$870$		0			0
$871$	−10.5000	+	18.1865i	−0.355779	+	0.616227i
$872$		0			0
$873$	40.1528	+	10.7589i	1.35897	+	0.364134i
$874$		0			0
$875$		0			0
$876$	54.0000			1.82449
$877$	−12.5521	+	46.8449i	−0.423853	+	1.58184i	0.342563	+	0.939495i	$0.388705\pi$
$877$	−12.5521	+	46.8449i	−0.423853	+	1.58184i	−0.766415	+	0.642345i	$0.777962\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	25.7196	−	25.7196i	0.865535	−	0.865535i	−0.126439	−	0.991974i	$-0.540355\pi$
$883$	25.7196	−	25.7196i	0.865535	−	0.865535i	0.991974	+	0.126439i	$0.0403549\pi$
$884$		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
$889$	6.06218	−	31.5000i	0.203319	−	1.05648i
$890$		0			0
$891$		0			0
$892$	−20.0764	+	5.37945i	−0.672207	+	0.180117i
$893$		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
$903$	−3.97884	−	55.4181i	−0.132408	−	1.84420i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−11.7112	+	3.13801i	−0.388865	+	0.104196i	−0.447954	−	0.894057i	$-0.647847\pi$
$907$	−11.7112	+	3.13801i	−0.388865	+	0.104196i	0.0590889	+	0.998253i	$0.481180\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−12.5521	+	46.8449i	−0.415640	+	1.55119i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	14.0000			0.462573
$917$		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
$927$	55.2101	−	14.7935i	1.81334	−	0.485882i
$928$		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
$931$	−7.00000	−	48.4974i	−0.229416	−	1.58944i
$932$		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	$-0.943716\pi$
$937$	−35.5176	−	35.5176i	−1.16031	−	1.16031i	−0.175902	−	0.984408i	$-0.556284\pi$
$938$		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	$0.583333\pi$
$948$	11.6555	+	43.4988i	0.378552	+	1.41278i
$949$	−23.3827	−	13.5000i	−0.759034	−	0.438229i
$950$		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
$964$	24.2487	−	14.0000i	0.780998	−	0.450910i
$965$		0			0
$966$		0			0
$967$	8.57321	+	8.57321i	0.275696	+	0.275696i	0.831388	−	0.555692i	$-0.187547\pi$
$967$	8.57321	+	8.57321i	0.275696	+	0.275696i	−0.555692	+	0.831388i	$0.687547\pi$
$968$		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
$973$	−15.3347	+	10.3849i	−0.491608	+	0.332926i
$974$		0			0
$975$		0			0
$976$	28.0000	−	48.4974i	0.896258	−	1.55236i
$977$		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
$981$	−51.0000			−1.62830
$982$		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
$988$	17.1464	−	17.1464i	0.545501	−	0.545501i
$989$		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
$993$	1.22474	−	1.22474i	0.0388661	−	0.0388661i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	38.4797	−	10.3106i	1.21867	−	0.326541i	0.408509	−	0.912754i	$-0.366049\pi$
$997$	38.4797	−	10.3106i	1.21867	−	0.326541i	0.810157	+	0.586214i	$0.199382\pi$
$998$		0			0
$999$	54.5596	−	31.5000i	1.72619	−	0.996616i

Twists

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

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

\(f(q)\)	\(=\)	\(q+(-0.448288 - 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(-1.48356 - 2.19067i) q^{7} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})\)	\(q+(-0.448288 - 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(-1.48356 - 2.19067i) q^{7} +(-2.59808 + 1.50000i) q^{9} +(-0.896575 + 3.34607i) q^{12} +(1.22474 - 1.22474i) q^{13} +(2.00000 + 3.46410i) q^{16} +(-6.06218 + 3.50000i) q^{19} +(-3.00000 + 3.46410i) q^{21} +(3.67423 + 3.67423i) q^{27} +(0.378937 + 5.27792i) q^{28} +(-3.50000 + 6.06218i) q^{31} +6.00000 q^{36} +(3.13801 - 11.7112i) q^{37} +(-2.59808 - 1.50000i) q^{39} +(-8.57321 + 8.57321i) q^{43} +(4.89898 - 4.89898i) q^{48} +(-2.59808 + 6.50000i) q^{49} +(-3.34607 + 0.896575i) q^{52} +(8.57321 + 8.57321i) q^{57} +(-7.00000 - 12.1244i) q^{61} +(7.14042 + 3.46618i) q^{63} -8.00000i q^{64} +(-11.7112 + 3.13801i) q^{67} +(-4.03459 - 15.0573i) q^{73} +14.0000 q^{76} +(11.2583 - 6.50000i) q^{79} +(4.50000 - 7.79423i) q^{81} +(8.66025 - 3.00000i) q^{84} +(-4.50000 - 0.866025i) q^{91} +(11.7112 + 3.13801i) q^{93} +(-9.79796 - 9.79796i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{16} - 24 q^{21} - 28 q^{31} + 48 q^{36} - 56 q^{61} + 112 q^{76} + 36 q^{81} - 36 q^{91}+O(q^{100}) \) 8 * q + 16 * q^16 - 24 * q^21 - 28 * q^31 + 48 * q^36 - 56 * q^61 + 112 * q^76 + 36 * q^81 - 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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