LMFDB - Embedded newform 525.2.q.d.374.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(299,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 5]))

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

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

chi := DirichletCharacter("525.299");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.19214610612$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			374.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	525.374
Dual form			525.2.q.d.299.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 1.50000i) q^{3} +(1.00000 - 1.73205i) q^{4} +(2.59808 + 0.500000i) q^{7} +(-1.50000 - 2.59808i) q^{9} +(1.73205 + 3.00000i) q^{12} +1.73205 q^{13} +(-2.00000 - 3.46410i) q^{16} +(4.50000 - 2.59808i) q^{19} +(-3.00000 + 3.46410i) q^{21} +5.19615 q^{27} +(3.46410 - 4.00000i) q^{28} +(7.50000 + 4.33013i) q^{31} -6.00000 q^{36} +(0.866025 - 0.500000i) q^{37} +(-1.50000 + 2.59808i) q^{39} +5.00000i q^{43} +6.92820 q^{48} +(6.50000 + 2.59808i) q^{49} +(1.73205 - 3.00000i) q^{52} +9.00000i q^{57} +(6.00000 - 3.46410i) q^{61} +(-2.59808 - 7.50000i) q^{63} -8.00000 q^{64} +(-9.52628 - 5.50000i) q^{67} +(-7.79423 + 13.5000i) q^{73} -10.3923i q^{76} +(-6.50000 - 11.2583i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(3.00000 + 8.66025i) q^{84} +(4.50000 + 0.866025i) q^{91} +(-12.9904 + 7.50000i) q^{93} -13.8564 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 6 q^{9} - 8 q^{16} + 18 q^{19} - 12 q^{21} + 30 q^{31} - 24 q^{36} - 6 q^{39} + 26 q^{49} + 24 q^{61} - 32 q^{64} - 26 q^{79} - 18 q^{81} + 12 q^{84} + 18 q^{91}+O(q^{100}) $ 4 * q + 4 * q^4 - 6 * q^9 - 8 * q^16 + 18 * q^19 - 12 * q^21 + 30 * q^31 - 24 * q^36 - 6 * q^39 + 26 * q^49 + 24 * q^61 - 32 * q^64 - 26 * q^79 - 18 * q^81 + 12 * q^84 + 18 * q^91

Character values

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

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

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		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$	−0.866025	+	1.50000i	−0.500000	+	0.866025i
$3$	−0.866025	+	1.50000i	−0.500000	+	0.866025i
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.59808	+	0.500000i	0.981981	+	0.188982i
$7$	2.59808	+	0.500000i	0.981981	+	0.188982i
$8$		0			0
$9$	−1.50000	−	2.59808i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	1.73205	+	3.00000i	0.500000	+	0.866025i
$13$	1.73205			0.480384			0.240192	−	0.970725i	$-0.422790\pi$
$13$	1.73205			0.480384			0.240192	+	0.970725i	$0.422790\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	4.50000	−	2.59808i	1.03237	−	0.596040i	0.114708	−	0.993399i	$-0.463407\pi$
$19$	4.50000	−	2.59808i	1.03237	−	0.596040i	0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$	−3.00000	+	3.46410i	−0.654654	+	0.755929i
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	5.19615			1.00000
$28$	3.46410	−	4.00000i	0.654654	−	0.755929i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	7.50000	+	4.33013i	1.34704	+	0.777714i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	7.50000	+	4.33013i	1.34704	+	0.777714i	0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−6.00000			−1.00000
$37$	0.866025	−	0.500000i	0.142374	−	0.0821995i	−0.427121	−	0.904194i	$-0.640472\pi$
$37$	0.866025	−	0.500000i	0.142374	−	0.0821995i	0.569495	+	0.821995i	$0.307139\pi$
$38$		0			0
$39$	−1.50000	+	2.59808i	−0.240192	+	0.416025i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$			5.00000i			0.762493i	0.924473	+	0.381246i	$0.124505\pi$
$43$			5.00000i			0.762493i	−0.924473	+	0.381246i	$0.875495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	6.92820			1.00000
$49$	6.50000	+	2.59808i	0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$	1.73205	−	3.00000i	0.240192	−	0.416025i
$53$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$53$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			9.00000i			1.19208i
$58$		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
$61$	6.00000	−	3.46410i	0.768221	−	0.443533i	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	6.00000	−	3.46410i	0.768221	−	0.443533i	0.832240	+	0.554416i	$0.187058\pi$
$62$		0			0
$63$	−2.59808	−	7.50000i	−0.327327	−	0.944911i
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	−9.52628	−	5.50000i	−1.16382	−	0.671932i	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−9.52628	−	5.50000i	−1.16382	−	0.671932i	−0.952217	+	0.305424i	$0.901202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−7.79423	+	13.5000i	−0.912245	+	1.58006i	−0.101361	+	0.994850i	$0.532320\pi$
$73$	−7.79423	+	13.5000i	−0.912245	+	1.58006i	−0.810885	+	0.585206i	$0.801014\pi$
$74$		0			0
$75$		0			0
$76$		−	10.3923i		−	1.19208i
$77$		0			0
$78$		0			0
$79$	−6.50000	−	11.2583i	−0.731307	−	1.26666i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−6.50000	−	11.2583i	−0.731307	−	1.26666i	0.225018	−	0.974355i	$-0.427756\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	3.00000	+	8.66025i	0.327327	+	0.944911i
$85$		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
$93$	−12.9904	+	7.50000i	−1.34704	+	0.777714i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−13.8564			−1.40690			−0.703452	−	0.710742i	$-0.748359\pi$
$97$	−13.8564			−1.40690			−0.703452	+	0.710742i	$0.748359\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−9.52628	−	16.5000i	−0.938652	−	1.62579i	−0.767988	−	0.640464i	$-0.778742\pi$
$103$	−9.52628	−	16.5000i	−0.938652	−	1.62579i	−0.170664	−	0.985329i	$-0.554591\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$	5.19615	−	9.00000i	0.500000	−	0.866025i
$109$	−8.50000	+	14.7224i	−0.814152	+	1.41015i	0.0957826	+	0.995402i	$0.469465\pi$
$109$	−8.50000	+	14.7224i	−0.814152	+	1.41015i	−0.909935	+	0.414751i	$0.863869\pi$
$110$		0			0
$111$			1.73205i			0.164399i
$112$	−3.46410	−	10.0000i	−0.327327	−	0.944911i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.59808	−	4.50000i	−0.240192	−	0.416025i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	15.0000	−	8.66025i	1.34704	−	0.777714i
$125$		0			0
$126$		0			0
$127$		−	19.0000i		−	1.68598i	−0.537931	−	0.842989i	$-0.680794\pi$
$127$		−	19.0000i		−	1.68598i	0.537931	−	0.842989i	$-0.319206\pi$
$128$		0			0
$129$	−7.50000	−	4.33013i	−0.660338	−	0.381246i
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	12.9904	−	4.50000i	1.12641	−	0.390199i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$			22.5167i			1.90984i	0.296866	+	0.954919i	$0.404058\pi$
$139$			22.5167i			1.90984i	−0.296866	+	0.954919i	$0.595942\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−6.00000	+	10.3923i	−0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	−9.52628	+	7.50000i	−0.785714	+	0.618590i
$148$		−	2.00000i		−	0.164399i
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		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
$157$	−10.3923	+	18.0000i	−0.829396	+	1.43656i	0.0691164	+	0.997609i	$0.477982\pi$
$157$	−10.3923	+	18.0000i	−0.829396	+	1.43656i	−0.898513	+	0.438948i	$0.855351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.92820	+	4.00000i	−0.542659	+	0.313304i	−0.746156	−	0.665771i	$-0.768103\pi$
$163$	−6.92820	+	4.00000i	−0.542659	+	0.313304i	0.203497	+	0.979076i	$0.434769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−10.0000			−0.769231
$170$		0			0
$171$	−13.5000	−	7.79423i	−1.03237	−	0.596040i
$172$	8.66025	+	5.00000i	0.660338	+	0.381246i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$			25.9808i			1.93113i	0.260153	+	0.965567i	$0.416227\pi$
$181$			25.9808i			1.93113i	−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$			12.0000i			0.887066i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	13.5000	+	2.59808i	0.981981	+	0.188982i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	6.92820	−	12.0000i	0.500000	−	0.866025i
$193$	21.6506	+	12.5000i	1.55845	+	0.899770i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	21.6506	+	12.5000i	1.55845	+	0.899770i	0.561041	+	0.827788i	$0.310401\pi$
$194$		0			0
$195$		0			0
$196$	11.0000	−	8.66025i	0.785714	−	0.618590i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	3.00000	+	1.73205i	0.212664	+	0.122782i	0.602549	−	0.798082i	$-0.294152\pi$
$199$	3.00000	+	1.73205i	0.212664	+	0.122782i	−0.389885	+	0.920864i	$0.627485\pi$
$200$		0			0
$201$	16.5000	−	9.52628i	1.16382	−	0.671932i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−3.46410	−	6.00000i	−0.240192	−	0.416025i
$209$		0			0
$210$		0			0
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	17.3205	+	15.0000i	1.17579	+	1.01827i
$218$		0			0
$219$	−13.5000	−	23.3827i	−0.912245	−	1.58006i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−10.3923			−0.695920			−0.347960	−	0.937509i	$-0.613126\pi$
$223$	−10.3923			−0.695920			−0.347960	+	0.937509i	$0.613126\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	15.5885	+	9.00000i	1.03237	+	0.596040i
$229$	25.5000	−	14.7224i	1.68509	−	0.972886i	0.726900	−	0.686743i	$-0.240960\pi$
$229$	25.5000	−	14.7224i	1.68509	−	0.972886i	0.958187	−	0.286143i	$-0.0923732\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	22.5167			1.46261
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−24.0000	−	13.8564i	−1.54598	−	0.892570i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−24.0000	−	13.8564i	−1.54598	−	0.892570i	−0.547533	−	0.836784i	$-0.684433\pi$
$242$		0			0
$243$	−7.79423	−	13.5000i	−0.500000	−	0.866025i
$244$		−	13.8564i		−	0.887066i
$245$		0			0
$246$		0			0
$247$	7.79423	−	4.50000i	0.495935	−	0.286328i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−15.5885	−	3.00000i	−0.981981	−	0.188982i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	2.50000	−	0.866025i	0.155342	−	0.0538122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−19.0526	+	11.0000i	−1.16382	+	0.671932i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	−15.0000	+	8.66025i	−0.911185	+	0.526073i	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−15.0000	+	8.66025i	−0.911185	+	0.526073i	−0.0303728	+	0.999539i	$0.509669\pi$
$272$		0			0
$273$	−5.19615	+	6.00000i	−0.314485	+	0.363137i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	26.8468	+	15.5000i	1.61307	+	0.931305i	0.988654	+	0.150210i	$0.0479951\pi$
$277$	26.8468	+	15.5000i	1.61307	+	0.931305i	0.624413	+	0.781094i	$0.285338\pi$
$278$		0			0
$279$		−	25.9808i		−	1.55543i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	16.4545	−	28.5000i	0.978117	−	1.69415i	0.308879	−	0.951101i	$-0.400046\pi$
$283$	16.4545	−	28.5000i	0.978117	−	1.69415i	0.669238	−	0.743048i	$-0.266621\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.50000	−	14.7224i	−0.500000	−	0.866025i
$290$		0			0
$291$	12.0000	−	20.7846i	0.703452	−	1.21842i
$292$	15.5885	+	27.0000i	0.912245	+	1.58006i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−2.50000	+	12.9904i	−0.144098	+	0.748753i
$302$		0			0
$303$		0			0
$304$	−18.0000	−	10.3923i	−1.03237	−	0.596040i
$305$		0			0
$306$		0			0
$307$	−1.73205			−0.0988534			−0.0494267	−	0.998778i	$-0.515739\pi$
$307$	−1.73205			−0.0988534			−0.0494267	+	0.998778i	$0.515739\pi$
$308$		0			0
$309$	33.0000			1.87730
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	2.59808	+	4.50000i	0.146852	+	0.254355i	0.930062	−	0.367402i	$-0.119753\pi$
$313$	2.59808	+	4.50000i	0.146852	+	0.254355i	−0.783210	+	0.621757i	$0.786419\pi$
$314$		0			0
$315$		0			0
$316$	−26.0000			−1.46261
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	9.00000	+	15.5885i	0.500000	+	0.866025i
$325$		0			0
$326$		0			0
$327$	−14.7224	−	25.5000i	−0.814152	−	1.41015i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.500000	−	0.866025i	−0.0274825	−	0.0476011i	0.851957	−	0.523612i	$-0.175416\pi$
$331$	−0.500000	−	0.866025i	−0.0274825	−	0.0476011i	−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$	−2.59808	−	1.50000i	−0.142374	−	0.0821995i
$334$		0			0
$335$		0			0
$336$	18.0000	+	3.46410i	0.981981	+	0.188982i
$337$		−	5.00000i		−	0.272367i	−0.990684	−	0.136184i	$-0.956516\pi$
$337$		−	5.00000i		−	0.272367i	0.990684	−	0.136184i	$-0.0434837\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	15.5885	+	10.0000i	0.841698	+	0.539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$			34.6410i			1.85429i	0.374701	+	0.927146i	$0.377745\pi$
$349$			34.6410i			1.85429i	−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	9.00000			0.480384
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	4.00000	−	6.92820i	0.210526	−	0.364642i
$362$		0			0
$363$	19.0526			1.00000
$364$	6.00000	−	6.92820i	0.314485	−	0.363137i
$365$		0			0
$366$		0			0
$367$	7.79423	−	13.5000i	0.406855	−	0.704694i	−0.587680	−	0.809093i	$-0.699959\pi$
$367$	7.79423	−	13.5000i	0.406855	−	0.704694i	0.994535	+	0.104399i	$0.0332919\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$			30.0000i			1.55543i
$373$	11.2583	−	6.50000i	0.582934	−	0.336557i	−0.179364	−	0.983783i	$-0.557404\pi$
$373$	11.2583	−	6.50000i	0.582934	−	0.336557i	0.762299	+	0.647225i	$0.224071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−37.0000			−1.90056			−0.950281	−	0.311393i	$-0.899204\pi$
$379$	−37.0000			−1.90056			−0.950281	+	0.311393i	$0.899204\pi$
$380$		0			0
$381$	28.5000	+	16.4545i	1.46010	+	0.842989i
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	12.9904	−	7.50000i	0.660338	−	0.381246i
$388$	−13.8564	+	24.0000i	−0.703452	+	1.21842i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	9.52628	+	16.5000i	0.478110	+	0.828111i	0.999685	−	0.0250943i	$-0.00798860\pi$
$397$	9.52628	+	16.5000i	0.478110	+	0.828111i	−0.521575	+	0.853206i	$0.674655\pi$
$398$		0			0
$399$	−4.50000	+	23.3827i	−0.225282	+	1.17060i
$400$		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
$403$	12.9904	+	7.50000i	0.647097	+	0.373602i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	34.5000	+	19.9186i	1.70592	+	0.984911i	0.939490	+	0.342578i	$0.111300\pi$
$409$	34.5000	+	19.9186i	1.70592	+	0.984911i	0.766426	+	0.642333i	$0.222033\pi$
$410$		0			0
$411$		0			0
$412$	−38.1051			−1.87730
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−33.7750	−	19.5000i	−1.65397	−	0.954919i
$418$		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	$-0.653228\pi$
$421$	−19.0000			−0.926003			−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	17.3205	−	6.00000i	0.838198	−	0.290360i
$428$		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	$0.666667\pi$
$432$	−10.3923	−	18.0000i	−0.500000	−	0.866025i
$433$	−22.5167			−1.08208			−0.541041	−	0.840996i	$-0.681970\pi$
$433$	−22.5167			−1.08208			−0.541041	+	0.840996i	$0.681970\pi$
$434$		0			0
$435$		0			0
$436$	17.0000	+	29.4449i	0.814152	+	1.41015i
$437$		0			0
$438$		0			0
$439$	−27.0000	+	15.5885i	−1.28864	+	0.743996i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−27.0000	+	15.5885i	−1.28864	+	0.743996i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$	−3.00000	−	20.7846i	−0.142857	−	0.989743i
$442$		0			0
$443$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$443$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$444$	3.00000	+	1.73205i	0.142374	+	0.0821995i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−20.7846	−	4.00000i	−0.981981	−	0.188982i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−3.46410	−	6.00000i	−0.162758	−	0.281905i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−35.5070	+	20.5000i	−1.66095	+	0.958950i	−0.688686	+	0.725059i	$0.741812\pi$
$457$	−35.5070	+	20.5000i	−1.66095	+	0.958950i	−0.972263	+	0.233890i	$0.924854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		−	23.0000i		−	1.06890i	−0.845200	−	0.534450i	$-0.820519\pi$
$463$		−	23.0000i		−	1.06890i	0.845200	−	0.534450i	$-0.179481\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$	−10.3923			−0.480384
$469$	−22.0000	−	19.0526i	−1.01587	−	0.879765i
$470$		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
$481$	1.50000	−	0.866025i	0.0683941	−	0.0394874i
$482$		0			0
$483$		0			0
$484$	−22.0000			−1.00000
$485$		0			0
$486$		0			0
$487$	−21.6506	−	12.5000i	−0.981084	−	0.566429i	−0.0784867	−	0.996915i	$-0.525009\pi$
$487$	−21.6506	−	12.5000i	−0.981084	−	0.566429i	−0.902597	+	0.430486i	$0.858342\pi$
$488$		0			0
$489$		−	13.8564i		−	0.626608i
$490$		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
$496$		−	34.6410i		−	1.55543i
$497$		0			0
$498$		0			0
$499$	21.5000	+	37.2391i	0.962472	+	1.66705i	0.716258	+	0.697835i	$0.245853\pi$
$499$	21.5000	+	37.2391i	0.962472	+	1.66705i	0.246214	+	0.969216i	$0.420813\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	8.66025	−	15.0000i	0.384615	−	0.666173i
$508$	−32.9090	−	19.0000i	−1.46010	−	0.842989i
$509$		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
$513$	23.3827	−	13.5000i	1.03237	−	0.596040i
$514$		0			0
$515$		0			0
$516$	−15.0000	+	8.66025i	−0.660338	+	0.381246i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	14.7224	+	25.5000i	0.643767	+	1.11504i	0.984585	+	0.174908i	$0.0559627\pi$
$523$	14.7224	+	25.5000i	0.643767	+	1.11504i	−0.340818	+	0.940129i	$0.610704\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	5.19615	−	27.0000i	0.225282	−	1.17060i
$533$		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
$543$	−38.9711	−	22.5000i	−1.67241	−	0.965567i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	40.0000i		−	1.71028i	−0.518400	−	0.855138i	$-0.673472\pi$
$547$		−	40.0000i		−	1.71028i	0.518400	−	0.855138i	$-0.326528\pi$
$548$		0			0
$549$	−18.0000	−	10.3923i	−0.768221	−	0.443533i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−11.2583	−	32.5000i	−0.478753	−	1.38204i
$554$		0			0
$555$		0			0
$556$	39.0000	+	22.5167i	1.65397	+	0.954919i
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$			8.66025i			0.366290i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−15.5885	+	18.0000i	−0.654654	+	0.755929i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		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
$577$	−16.4545	+	28.5000i	−0.685009	+	1.18647i	0.288425	+	0.957503i	$0.406868\pi$
$577$	−16.4545	+	28.5000i	−0.685009	+	1.18647i	−0.973434	+	0.228968i	$0.926465\pi$
$578$		0			0
$579$	−37.5000	+	21.6506i	−1.55845	+	0.899770i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	3.46410	+	24.0000i	0.142857	+	0.989743i
$589$	45.0000			1.85419
$590$		0			0
$591$		0			0
$592$	−3.46410	−	2.00000i	−0.142374	−	0.0821995i
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−5.19615	+	3.00000i	−0.212664	+	0.122782i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$			1.73205i			0.0706518i	0.999376	+	0.0353259i	$0.0112469\pi$
$601$			1.73205i			0.0706518i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$		0			0
$603$			33.0000i			1.34386i
$604$	4.00000	+	6.92820i	0.162758	+	0.281905i
$605$		0			0
$606$		0			0
$607$	−2.59808	−	4.50000i	−0.105453	−	0.182649i	0.808470	−	0.588537i	$-0.200296\pi$
$607$	−2.59808	−	4.50000i	−0.105453	−	0.182649i	−0.913923	+	0.405887i	$0.866962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−8.66025	−	5.00000i	−0.349784	−	0.201948i	0.314806	−	0.949156i	$-0.398061\pi$
$613$	−8.66025	−	5.00000i	−0.349784	−	0.201948i	−0.664590	+	0.747208i	$0.731394\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	−7.50000	−	4.33013i	−0.301450	−	0.174042i	0.341644	−	0.939829i	$-0.389016\pi$
$619$	−7.50000	−	4.33013i	−0.301450	−	0.174042i	−0.643094	+	0.765787i	$0.722350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	12.0000			0.480384
$625$		0			0
$626$		0			0
$627$		0			0
$628$	20.7846	+	36.0000i	0.829396	+	1.43656i
$629$		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
$633$	−13.8564	+	24.0000i	−0.550743	+	0.953914i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	11.2583	+	4.50000i	0.446071	+	0.178296i
$638$		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
$643$	50.2295			1.98086			0.990429	−	0.138027i	$-0.0440759\pi$
$643$	50.2295			1.98086			0.990429	+	0.138027i	$0.0440759\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−37.5000	+	12.9904i	−1.46974	+	0.509133i
$652$			16.0000i			0.626608i
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	46.7654			1.82449
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−13.5000	−	7.79423i	−0.525089	−	0.303160i	0.213925	−	0.976850i	$-0.431375\pi$
$661$	−13.5000	−	7.79423i	−0.525089	−	0.303160i	−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
$669$	9.00000	−	15.5885i	0.347960	−	0.602685i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	37.0000i		−	1.42625i	−0.701039	−	0.713123i	$-0.747280\pi$
$673$		−	37.0000i		−	1.42625i	0.701039	−	0.713123i	$-0.252720\pi$
$674$		0			0
$675$		0			0
$676$	−10.0000	+	17.3205i	−0.384615	+	0.666173i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	−36.0000	−	6.92820i	−1.38155	−	0.265880i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$683$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$684$	−27.0000	+	15.5885i	−1.03237	+	0.596040i
$685$		0			0
$686$		0			0
$687$			51.0000i			1.94577i
$688$	17.3205	−	10.0000i	0.660338	−	0.381246i
$689$		0			0
$690$		0			0
$691$	16.5000	−	9.52628i	0.627690	−	0.362397i	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	16.5000	−	9.52628i	0.627690	−	0.362397i	0.779857	+	0.625958i	$0.215292\pi$
$692$		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
$703$	2.59808	−	4.50000i	0.0979883	−	0.169721i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	0.995228	−	0.0975728i	$-0.0311079\pi$
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	−0.582115	+	0.813107i	$0.697775\pi$
$710$		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
$723$	41.5692	−	24.0000i	1.54598	−	0.892570i
$724$	45.0000	+	25.9808i	1.67241	+	0.965567i
$725$		0			0
$726$		0			0
$727$	22.5167			0.835097			0.417548	−	0.908655i	$-0.362889\pi$
$727$	22.5167			0.835097			0.417548	+	0.908655i	$0.362889\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$	20.7846	+	12.0000i	0.768221	+	0.443533i
$733$	26.8468	+	46.5000i	0.991609	+	1.71752i	0.607760	+	0.794121i	$0.292068\pi$
$733$	26.8468	+	46.5000i	0.991609	+	1.71752i	0.383849	+	0.923396i	$0.374598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	26.5000	−	45.8993i	0.974818	−	1.68843i	0.294285	−	0.955718i	$-0.404919\pi$
$739$	26.5000	−	45.8993i	0.974818	−	1.68843i	0.680534	−	0.732717i	$-0.261748\pi$
$740$		0			0
$741$			15.5885i			0.572656i
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		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
$757$		−	26.0000i		−	0.944986i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$		−	26.0000i		−	0.944986i	0.881334	−	0.472493i	$-0.156646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	−29.4449	+	34.0000i	−1.06598	+	1.23088i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−13.8564	−	24.0000i	−0.500000	−	0.866025i
$769$		−	25.9808i		−	0.936890i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$		−	25.9808i		−	0.936890i	0.883493	−	0.468445i	$-0.155186\pi$
$770$		0			0
$771$		0			0
$772$	43.3013	−	25.0000i	1.55845	−	0.899770i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−0.866025	+	4.50000i	−0.0310685	+	0.161437i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−4.00000	−	27.7128i	−0.142857	−	0.989743i
$785$		0			0
$786$		0			0
$787$	1.73205	−	3.00000i	0.0617409	−	0.106938i	−0.833503	−	0.552515i	$-0.813668\pi$
$787$	1.73205	−	3.00000i	0.0617409	−	0.106938i	0.895244	+	0.445577i	$0.147001\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	10.3923	−	6.00000i	0.369042	−	0.213066i
$794$		0			0
$795$		0			0
$796$	6.00000	−	3.46410i	0.212664	−	0.122782i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		−	38.1051i		−	1.34386i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$		−	10.3923i		−	0.364923i	−0.983213	−	0.182462i	$-0.941593\pi$
$811$		−	10.3923i		−	0.364923i	0.983213	−	0.182462i	$-0.0584065\pi$
$812$		0			0
$813$		−	30.0000i		−	1.05215i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	12.9904	+	22.5000i	0.454476	+	0.787175i
$818$		0			0
$819$	−4.50000	−	12.9904i	−0.157243	−	0.453921i
$820$		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
$823$	−45.0333	−	26.0000i	−1.56976	−	0.906303i	−0.996196	−	0.0871445i	$-0.972226\pi$
$823$	−45.0333	−	26.0000i	−1.56976	−	0.906303i	−0.573567	−	0.819159i	$-0.694441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−49.5000	−	28.5788i	−1.71921	−	0.992584i	−0.920383	−	0.391018i	$-0.872123\pi$
$829$	−49.5000	−	28.5788i	−1.71921	−	0.992584i	−0.798823	−	0.601566i	$-0.794544\pi$
$830$		0			0
$831$	−46.5000	+	26.8468i	−1.61307	+	0.931305i
$832$	−13.8564			−0.480384
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	38.9711	+	22.5000i	1.34704	+	0.777714i
$838$		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
$844$	16.0000	−	27.7128i	0.550743	−	0.953914i
$845$		0			0
$846$		0			0
$847$	−9.52628	−	27.5000i	−0.327327	−	0.944911i
$848$		0			0
$849$	28.5000	+	49.3634i	0.978117	+	1.69415i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−46.7654			−1.60122			−0.800608	−	0.599189i	$-0.795490\pi$
$853$	−46.7654			−1.60122			−0.800608	+	0.599189i	$0.795490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	15.0000	−	8.66025i	0.511793	−	0.295484i	−0.221777	−	0.975097i	$-0.571186\pi$
$859$	15.0000	−	8.66025i	0.511793	−	0.295484i	0.733571	+	0.679613i	$0.237852\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	29.4449			1.00000
$868$	43.3013	−	15.0000i	1.46974	−	0.509133i
$869$		0			0
$870$		0			0
$871$	−16.5000	−	9.52628i	−0.559081	−	0.322786i
$872$		0			0
$873$	20.7846	+	36.0000i	0.703452	+	1.21842i
$874$		0			0
$875$		0			0
$876$	−54.0000			−1.82449
$877$	−29.4449	+	17.0000i	−0.994282	+	0.574049i	−0.906552	−	0.422095i	$-0.861295\pi$
$877$	−29.4449	+	17.0000i	−0.994282	+	0.574049i	−0.0877308	+	0.996144i	$0.527962\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$			47.0000i			1.58168i	0.612026	+	0.790838i	$0.290355\pi$
$883$			47.0000i			1.58168i	−0.612026	+	0.790838i	$0.709645\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	9.50000	−	49.3634i	0.318620	−	1.65560i
$890$		0			0
$891$		0			0
$892$	−10.3923	+	18.0000i	−0.347960	+	0.602685i
$893$		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
$903$	−17.3205	−	15.0000i	−0.576390	−	0.499169i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	51.0955	+	29.5000i	1.69660	+	0.979531i	0.948945	+	0.315442i	$0.102153\pi$
$907$	51.0955	+	29.5000i	1.69660	+	0.979531i	0.747653	+	0.664089i	$0.231180\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	31.1769	−	18.0000i	1.03237	−	0.596040i
$913$		0			0
$914$		0			0
$915$		0			0
$916$		−	58.8897i		−	1.94577i
$917$		0			0
$918$		0			0
$919$	0.500000	+	0.866025i	0.0164935	+	0.0285675i	0.874154	−	0.485648i	$-0.161416\pi$
$919$	0.500000	+	0.866025i	0.0164935	+	0.0285675i	−0.857661	+	0.514216i	$0.828083\pi$
$920$		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
$927$	−28.5788	+	49.5000i	−0.938652	+	1.62579i
$928$		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
$931$	36.0000	−	5.19615i	1.17985	−	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−50.2295			−1.64093			−0.820463	−	0.571700i	$-0.806284\pi$
$937$	−50.2295			−1.64093			−0.820463	+	0.571700i	$0.806284\pi$
$938$		0			0
$939$	−9.00000			−0.293704
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$	22.5167	−	39.0000i	0.731307	−	1.26666i
$949$	−13.5000	+	23.3827i	−0.438229	+	0.759034i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	22.0000	+	38.1051i	0.709677	+	1.22920i
$962$		0			0
$963$		0			0
$964$	−48.0000	+	27.7128i	−1.54598	+	0.892570i
$965$		0			0
$966$		0			0
$967$		−	61.0000i		−	1.96163i	−0.194946	−	0.980814i	$-0.562453\pi$
$967$		−	61.0000i		−	1.96163i	0.194946	−	0.980814i	$-0.437547\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$	−31.1769			−1.00000
$973$	−11.2583	+	58.5000i	−0.360925	+	1.87542i
$974$		0			0
$975$		0			0
$976$	−24.0000	−	13.8564i	−0.768221	−	0.443533i
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	51.0000			1.62830
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		−	18.0000i		−	0.572656i
$989$		0			0
$990$		0			0
$991$	8.50000	−	14.7224i	0.270011	−	0.467673i	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	8.50000	−	14.7224i	0.270011	−	0.467673i	0.968864	+	0.247592i	$0.0796392\pi$
$992$		0			0
$993$	1.73205			0.0549650
$994$		0			0
$995$		0			0
$996$		0			0
$997$	19.9186	−	34.5000i	0.630828	−	1.09263i	−0.356555	−	0.934274i	$-0.616049\pi$
$997$	19.9186	−	34.5000i	0.630828	−	1.09263i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$		0			0
$999$	4.50000	−	2.59808i	0.142374	−	0.0821995i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.q.d.374.1		4
3.2	odd	2	CM	525.2.q.d.374.1		4
5.2	odd	4		525.2.t.c.101.1		2
5.3	odd	4		21.2.g.a.17.1	yes	2
5.4	even	2	inner	525.2.q.d.374.2		4
7.5	odd	6	inner	525.2.q.d.299.2		4
15.2	even	4		525.2.t.c.101.1		2
15.8	even	4		21.2.g.a.17.1	yes	2
15.14	odd	2	inner	525.2.q.d.374.2		4
20.3	even	4		336.2.bc.c.17.1		2
21.5	even	6	inner	525.2.q.d.299.2		4
35.3	even	12		147.2.c.a.146.1		2
35.12	even	12		525.2.t.c.26.1		2
35.13	even	4		147.2.g.a.80.1		2
35.18	odd	12		147.2.c.a.146.2		2
35.19	odd	6	inner	525.2.q.d.299.1		4
35.23	odd	12		147.2.g.a.68.1		2
35.33	even	12		21.2.g.a.5.1	✓	2
45.13	odd	12		567.2.i.b.269.1		2
45.23	even	12		567.2.i.b.269.1		2
45.38	even	12		567.2.s.a.458.1		2
45.43	odd	12		567.2.s.a.458.1		2
60.23	odd	4		336.2.bc.c.17.1		2
105.23	even	12		147.2.g.a.68.1		2
105.38	odd	12		147.2.c.a.146.1		2
105.47	odd	12		525.2.t.c.26.1		2
105.53	even	12		147.2.c.a.146.2		2
105.68	odd	12		21.2.g.a.5.1	✓	2
105.83	odd	4		147.2.g.a.80.1		2
105.89	even	6	inner	525.2.q.d.299.1		4
140.3	odd	12		2352.2.k.c.881.2		2
140.103	odd	12		336.2.bc.c.257.1		2
140.123	even	12		2352.2.k.c.881.1		2
315.68	odd	12		567.2.s.a.26.1		2
315.103	even	12		567.2.s.a.26.1		2
315.173	odd	12		567.2.i.b.215.1		2
315.313	even	12		567.2.i.b.215.1		2
420.143	even	12		2352.2.k.c.881.2		2
420.263	odd	12		2352.2.k.c.881.1		2
420.383	even	12		336.2.bc.c.257.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.2.g.a.5.1	✓	2	35.33	even	12
21.2.g.a.5.1	✓	2	105.68	odd	12
21.2.g.a.17.1	yes	2	5.3	odd	4
21.2.g.a.17.1	yes	2	15.8	even	4
147.2.c.a.146.1		2	35.3	even	12
147.2.c.a.146.1		2	105.38	odd	12
147.2.c.a.146.2		2	35.18	odd	12
147.2.c.a.146.2		2	105.53	even	12
147.2.g.a.68.1		2	35.23	odd	12
147.2.g.a.68.1		2	105.23	even	12
147.2.g.a.80.1		2	35.13	even	4
147.2.g.a.80.1		2	105.83	odd	4
336.2.bc.c.17.1		2	20.3	even	4
336.2.bc.c.17.1		2	60.23	odd	4
336.2.bc.c.257.1		2	140.103	odd	12
336.2.bc.c.257.1		2	420.383	even	12
525.2.q.d.299.1		4	35.19	odd	6	inner
525.2.q.d.299.1		4	105.89	even	6	inner
525.2.q.d.299.2		4	7.5	odd	6	inner
525.2.q.d.299.2		4	21.5	even	6	inner
525.2.q.d.374.1		4	1.1	even	1	trivial
525.2.q.d.374.1		4	3.2	odd	2	CM
525.2.q.d.374.2		4	5.4	even	2	inner
525.2.q.d.374.2		4	15.14	odd	2	inner
525.2.t.c.26.1		2	35.12	even	12
525.2.t.c.26.1		2	105.47	odd	12
525.2.t.c.101.1		2	5.2	odd	4
525.2.t.c.101.1		2	15.2	even	4
567.2.i.b.215.1		2	315.173	odd	12
567.2.i.b.215.1		2	315.313	even	12
567.2.i.b.269.1		2	45.13	odd	12
567.2.i.b.269.1		2	45.23	even	12
567.2.s.a.26.1		2	315.68	odd	12
567.2.s.a.26.1		2	315.103	even	12
567.2.s.a.458.1		2	45.38	even	12
567.2.s.a.458.1		2	45.43	odd	12
2352.2.k.c.881.1		2	140.123	even	12
2352.2.k.c.881.1		2	420.263	odd	12
2352.2.k.c.881.2		2	140.3	odd	12
2352.2.k.c.881.2		2	420.143	even	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 1.50000i) q^{3} +(1.00000 - 1.73205i) q^{4} +(2.59808 + 0.500000i) q^{7} +(-1.50000 - 2.59808i) q^{9} +(1.73205 + 3.00000i) q^{12} +1.73205 q^{13} +(-2.00000 - 3.46410i) q^{16} +(4.50000 - 2.59808i) q^{19} +(-3.00000 + 3.46410i) q^{21} +5.19615 q^{27} +(3.46410 - 4.00000i) q^{28} +(7.50000 + 4.33013i) q^{31} -6.00000 q^{36} +(0.866025 - 0.500000i) q^{37} +(-1.50000 + 2.59808i) q^{39} +5.00000i q^{43} +6.92820 q^{48} +(6.50000 + 2.59808i) q^{49} +(1.73205 - 3.00000i) q^{52} +9.00000i q^{57} +(6.00000 - 3.46410i) q^{61} +(-2.59808 - 7.50000i) q^{63} -8.00000 q^{64} +(-9.52628 - 5.50000i) q^{67} +(-7.79423 + 13.5000i) q^{73} -10.3923i q^{76} +(-6.50000 - 11.2583i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(3.00000 + 8.66025i) q^{84} +(4.50000 + 0.866025i) q^{91} +(-12.9904 + 7.50000i) q^{93} -13.8564 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 6 q^{9} - 8 q^{16} + 18 q^{19} - 12 q^{21} + 30 q^{31} - 24 q^{36} - 6 q^{39} + 26 q^{49} + 24 q^{61} - 32 q^{64} - 26 q^{79} - 18 q^{81} + 12 q^{84} + 18 q^{91}+O(q^{100}) \) 4 * q + 4 * q^4 - 6 * q^9 - 8 * q^16 + 18 * q^19 - 12 * q^21 + 30 * q^31 - 24 * q^36 - 6 * q^39 + 26 * q^49 + 24 * q^61 - 32 * q^64 - 26 * q^79 - 18 * q^81 + 12 * q^84 + 18 * q^91

Introduction

L-functions

Modular forms

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Representations

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Database

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