LMFDB - Embedded newform 507.2.f.b.239.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(239,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("507.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			239.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	507.239
Dual form			507.2.f.b.437.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} -2.00000i q^{4} +(-3.09808 - 3.09808i) q^{7} +3.00000 q^{9} -3.46410i q^{12} -4.00000 q^{16} +(2.26795 - 2.26795i) q^{19} +(-5.36603 - 5.36603i) q^{21} -5.00000i q^{25} +5.19615 q^{27} +(-6.19615 + 6.19615i) q^{28} +(-0.830127 + 0.830127i) q^{31} -6.00000i q^{36} +(8.46410 + 8.46410i) q^{37} +1.73205i q^{43} -6.92820 q^{48} +12.1962i q^{49} +(3.92820 - 3.92820i) q^{57} +8.66025 q^{61} +(-9.29423 - 9.29423i) q^{63} +8.00000i q^{64} +(11.5622 - 11.5622i) q^{67} +(7.63397 + 7.63397i) q^{73} -8.66025i q^{75} +(-4.53590 - 4.53590i) q^{76} -12.1244 q^{79} +9.00000 q^{81} +(-10.7321 + 10.7321i) q^{84} +(-1.43782 + 1.43782i) q^{93} +(7.02628 - 7.02628i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{7} + 12 q^{9} - 16 q^{16} + 16 q^{19} - 18 q^{21} - 4 q^{28} + 14 q^{31} + 20 q^{37} - 12 q^{57} - 6 q^{63} + 22 q^{67} + 34 q^{73} - 32 q^{76} + 36 q^{81} - 36 q^{84} - 30 q^{93} - 10 q^{97}+O(q^{100}) $ 4 * q - 2 * q^7 + 12 * q^9 - 16 * q^16 + 16 * q^19 - 18 * q^21 - 4 * q^28 + 14 * q^31 + 20 * q^37 - 12 * q^57 - 6 * q^63 + 22 * q^67 + 34 * q^73 - 32 * q^76 + 36 * q^81 - 36 * q^84 - 30 * q^93 - 10 * q^97

Character values

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

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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		0.707107	−	0.707107i	$-0.250000\pi$
$2$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.73205			1.00000
$3$	1.73205			1.00000
$4$		−	2.00000i		−	1.00000i
$5$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$5$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$6$		0			0
$7$	−3.09808	−	3.09808i	−1.17096	−	1.17096i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	−3.09808	−	3.09808i	−1.17096	−	1.17096i	−0.188982	−	0.981981i	$-0.560519\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$11$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$12$		−	3.46410i		−	1.00000i
$13$		0			0
$13$		0			0
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	2.26795	−	2.26795i	0.520303	−	0.520303i	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	2.26795	−	2.26795i	0.520303	−	0.520303i	0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$	−5.36603	−	5.36603i	−1.17096	−	1.17096i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$		−	5.00000i		−	1.00000i
$26$		0			0
$27$	5.19615			1.00000
$28$	−6.19615	+	6.19615i	−1.17096	+	1.17096i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−0.830127	+	0.830127i	−0.149095	+	0.149095i	−0.777714	−	0.628619i	$-0.783621\pi$
$31$	−0.830127	+	0.830127i	−0.149095	+	0.149095i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		−	6.00000i		−	1.00000i
$37$	8.46410	+	8.46410i	1.39149	+	1.39149i	0.821995	+	0.569495i	$0.192861\pi$
$37$	8.46410	+	8.46410i	1.39149	+	1.39149i	0.569495	+	0.821995i	$0.307139\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$41$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$42$		0			0
$43$			1.73205i			0.264135i	0.991241	+	0.132068i	$0.0421616\pi$
$43$			1.73205i			0.264135i	−0.991241	+	0.132068i	$0.957838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	−6.92820			−1.00000
$49$			12.1962i			1.74231i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.92820	−	3.92820i	0.520303	−	0.520303i
$58$		0			0
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$	8.66025			1.10883			0.554416	−	0.832240i	$-0.312942\pi$
$61$	8.66025			1.10883			0.554416	+	0.832240i	$0.312942\pi$
$62$		0			0
$63$	−9.29423	−	9.29423i	−1.17096	−	1.17096i
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	11.5622	−	11.5622i	1.41254	−	1.41254i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.5622	−	11.5622i	1.41254	−	1.41254i	0.740613	−	0.671932i	$-0.234535\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$71$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$72$		0			0
$73$	7.63397	+	7.63397i	0.893489	+	0.893489i	0.994850	−	0.101361i	$-0.0323196\pi$
$73$	7.63397	+	7.63397i	0.893489	+	0.893489i	−0.101361	+	0.994850i	$0.532320\pi$
$74$		0			0
$75$		−	8.66025i		−	1.00000i
$76$	−4.53590	−	4.53590i	−0.520303	−	0.520303i
$77$		0			0
$78$		0			0
$79$	−12.1244			−1.36410			−0.682048	−	0.731307i	$-0.738911\pi$
$79$	−12.1244			−1.36410			−0.682048	+	0.731307i	$0.738911\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$	−10.7321	+	10.7321i	−1.17096	+	1.17096i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$89$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−1.43782	+	1.43782i	−0.149095	+	0.149095i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	7.02628	−	7.02628i	0.713411	−	0.713411i	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	7.02628	−	7.02628i	0.713411	−	0.713411i	0.967247	+	0.253837i	$0.0816925\pi$
$98$		0			0
$99$		0			0
$100$	−10.0000			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$			15.5885i			1.53598i	0.640464	+	0.767988i	$0.278742\pi$
$103$			15.5885i			1.53598i	−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		−	10.3923i		−	1.00000i
$109$	−13.8301	+	13.8301i	−1.32469	+	1.32469i	−0.414751	+	0.909935i	$0.636131\pi$
$109$	−13.8301	+	13.8301i	−1.32469	+	1.32469i	−0.909935	+	0.414751i	$0.863869\pi$
$110$		0			0
$111$	14.6603	+	14.6603i	1.39149	+	1.39149i
$112$	12.3923	+	12.3923i	1.17096	+	1.17096i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$			11.0000i			1.00000i
$122$		0			0
$123$		0			0
$124$	1.66025	+	1.66025i	0.149095	+	0.149095i
$125$		0			0
$126$		0			0
$127$			1.00000i			0.0887357i	0.999015	+	0.0443678i	$0.0141274\pi$
$127$			1.00000i			0.0887357i	−0.999015	+	0.0443678i	$0.985873\pi$
$128$		0			0
$129$			3.00000i			0.264135i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	−14.0526			−1.21851
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$	−7.00000			−0.593732			−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000			−0.593732			−0.296866	+	0.954919i	$0.595942\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$			21.1244i			1.74231i
$148$	16.9282	−	16.9282i	1.39149	−	1.39149i
$149$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$149$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$150$		0			0
$151$	−10.1244	−	10.1244i	−0.823908	−	0.823908i	0.162758	−	0.986666i	$-0.447961\pi$
$151$	−10.1244	−	10.1244i	−0.823908	−	0.823908i	−0.986666	+	0.162758i	$0.947961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−11.0000			−0.877896			−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000			−0.877896			−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	9.90192	+	9.90192i	0.775579	+	0.775579i	0.979076	−	0.203497i	$-0.0652307\pi$
$163$	9.90192	+	9.90192i	0.775579	+	0.775579i	−0.203497	+	0.979076i	$0.565231\pi$
$164$		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
$169$		0			0
$170$		0			0
$171$	6.80385	−	6.80385i	0.520303	−	0.520303i
$172$	3.46410			0.264135
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−15.4904	+	15.4904i	−1.17096	+	1.17096i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		−	6.92820i		−	0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$		−	6.92820i		−	0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$	15.0000			1.10883
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−16.0981	−	16.0981i	−1.17096	−	1.17096i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$			13.8564i			1.00000i
$193$	3.70577	+	3.70577i	0.266747	+	0.266747i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	3.70577	+	3.70577i	0.266747	+	0.266747i	−0.561041	+	0.827788i	$0.689599\pi$
$194$		0			0
$195$		0			0
$196$	24.3923			1.74231
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		−	17.0000i		−	1.20510i	−0.798082	−	0.602549i	$-0.794152\pi$
$199$		−	17.0000i		−	1.20510i	0.798082	−	0.602549i	$-0.205848\pi$
$200$		0			0
$201$	20.0263	−	20.0263i	1.41254	−	1.41254i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	25.9808			1.78859			0.894295	−	0.447478i	$-0.147678\pi$
$211$	25.9808			1.78859			0.894295	+	0.447478i	$0.147678\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	5.14359			0.349170
$218$		0			0
$219$	13.2224	+	13.2224i	0.893489	+	0.893489i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−19.1962	+	19.1962i	−1.28547	+	1.28547i	−0.347960	+	0.937509i	$0.613126\pi$
$223$	−19.1962	+	19.1962i	−1.28547	+	1.28547i	−0.937509	+	0.347960i	$0.886874\pi$
$224$		0			0
$225$		−	15.0000i		−	1.00000i
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$	−7.85641	−	7.85641i	−0.520303	−	0.520303i
$229$	−0.607695	−	0.607695i	−0.0401576	−	0.0401576i	0.686743	−	0.726900i	$-0.259040\pi$
$229$	−0.607695	−	0.607695i	−0.0401576	−	0.0401576i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−21.0000			−1.36410
$238$		0			0
$239$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$239$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	−20.8564	−	20.8564i	−1.34348	−	1.34348i	−0.892570	−	0.450910i	$-0.851100\pi$
$241$	−20.8564	−	20.8564i	−1.34348	−	1.34348i	−0.450910	−	0.892570i	$-0.648900\pi$
$242$		0			0
$243$	15.5885			1.00000
$244$		−	17.3205i		−	1.10883i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−18.5885	+	18.5885i	−1.17096	+	1.17096i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		−	52.4449i		−	3.25877i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−23.1244	−	23.1244i	−1.41254	−	1.41254i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−22.2942	−	22.2942i	−1.35428	−	1.35428i	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−22.2942	−	22.2942i	−1.35428	−	1.35428i	−0.473466	−	0.880812i	$-0.656997\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	20.7846i		−	1.24883i	−0.781094	−	0.624413i	$-0.785338\pi$
$277$		−	20.7846i		−	1.24883i	0.781094	−	0.624413i	$-0.214662\pi$
$278$		0			0
$279$	−2.49038	+	2.49038i	−0.149095	+	0.149095i
$280$		0			0
$281$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$281$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$282$		0			0
$283$		−	25.0000i		−	1.48610i	−0.669238	−	0.743048i	$-0.733379\pi$
$283$		−	25.0000i		−	1.48610i	0.669238	−	0.743048i	$-0.266621\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	12.1699	−	12.1699i	0.713411	−	0.713411i
$292$	15.2679	−	15.2679i	0.893489	−	0.893489i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−17.3205			−1.00000
$301$	5.36603	−	5.36603i	0.309293	−	0.309293i
$302$		0			0
$303$		0			0
$304$	−9.07180	+	9.07180i	−0.520303	+	0.520303i
$305$		0			0
$306$		0			0
$307$	−18.3660	−	18.3660i	−1.04820	−	1.04820i	−0.998778	−	0.0494267i	$-0.984261\pi$
$307$	−18.3660	−	18.3660i	−1.04820	−	1.04820i	−0.0494267	−	0.998778i	$-0.515739\pi$
$308$		0			0
$309$			27.0000i			1.53598i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	32.9090			1.86012			0.930062	−	0.367402i	$-0.119753\pi$
$313$	32.9090			1.86012			0.930062	+	0.367402i	$0.119753\pi$
$314$		0			0
$315$		0			0
$316$			24.2487i			1.36410i
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	18.0000i		−	1.00000i
$325$		0			0
$326$		0			0
$327$	−23.9545	+	23.9545i	−1.32469	+	1.32469i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−5.97372	+	5.97372i	−0.328345	+	0.328345i	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−5.97372	+	5.97372i	−0.328345	+	0.328345i	0.523612	+	0.851957i	$0.324584\pi$
$332$		0			0
$333$	25.3923	+	25.3923i	1.39149	+	1.39149i
$334$		0			0
$335$		0			0
$336$	21.4641	+	21.4641i	1.17096	+	1.17096i
$337$			29.0000i			1.57973i	0.613280	+	0.789865i	$0.289850\pi$
$337$			29.0000i			1.57973i	−0.613280	+	0.789865i	$0.710150\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	16.0981	−	16.0981i	0.869214	−	0.869214i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	26.2224	+	26.2224i	1.40365	+	1.40365i	0.788074	+	0.615581i	$0.211079\pi$
$349$	26.2224	+	26.2224i	1.40365	+	1.40365i	0.615581	+	0.788074i	$0.288921\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$		0			0
$361$			8.71281i			0.458569i
$362$		0			0
$363$			19.0526i			1.00000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	31.0000			1.61819			0.809093	−	0.587680i	$-0.199959\pi$
$367$	31.0000			1.61819			0.809093	+	0.587680i	$0.199959\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	2.87564	+	2.87564i	0.149095	+	0.149095i
$373$	−36.3731			−1.88333			−0.941663	−	0.336557i	$-0.890737\pi$
$373$	−36.3731			−1.88333			−0.941663	+	0.336557i	$0.890737\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	24.5622	−	24.5622i	1.26167	−	1.26167i	0.311393	−	0.950281i	$-0.399204\pi$
$379$	24.5622	−	24.5622i	1.26167	−	1.26167i	0.950281	−	0.311393i	$-0.100796\pi$
$380$		0			0
$381$			1.73205i			0.0887357i
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			5.19615i			0.264135i
$388$	−14.0526	−	14.0526i	−0.713411	−	0.713411i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	19.4186	+	19.4186i	0.974591	+	0.974591i	0.999685	−	0.0250943i	$-0.00798860\pi$
$397$	19.4186	+	19.4186i	0.974591	+	0.974591i	−0.0250943	+	0.999685i	$0.507989\pi$
$398$		0			0
$399$	−24.3397			−1.21851
$400$			20.0000i			1.00000i
$401$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$401$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−28.4904	+	28.4904i	−1.40876	+	1.40876i	−0.642333	+	0.766426i	$0.722033\pi$
$409$	−28.4904	+	28.4904i	−1.40876	+	1.40876i	−0.766426	+	0.642333i	$0.777967\pi$
$410$		0			0
$411$		0			0
$412$	31.1769			1.53598
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−12.1244			−0.593732
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	8.68653	−	8.68653i	0.423356	−	0.423356i	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	8.68653	−	8.68653i	0.423356	−	0.423356i	0.886357	+	0.463002i	$0.153228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−26.8301	−	26.8301i	−1.29840	−	1.29840i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$431$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$432$	−20.7846			−1.00000
$433$			35.0000i			1.68199i	0.541041	+	0.840996i	$0.318030\pi$
$433$			35.0000i			1.68199i	−0.541041	+	0.840996i	$0.681970\pi$
$434$		0			0
$435$		0			0
$436$	27.6603	+	27.6603i	1.32469	+	1.32469i
$437$		0			0
$438$		0			0
$439$		−	39.8372i		−	1.90132i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$		−	39.8372i		−	1.90132i	0.310228	−	0.950662i	$-0.399595\pi$
$440$		0			0
$441$			36.5885i			1.74231i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	29.3205	−	29.3205i	1.39149	−	1.39149i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	24.7846	−	24.7846i	1.17096	−	1.17096i
$449$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$449$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−17.5359	−	17.5359i	−0.823908	−	0.823908i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−14.4378	+	14.4378i	−0.675373	+	0.675373i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−14.4378	+	14.4378i	−0.675373	+	0.675373i	0.283577	+	0.958950i	$0.408479\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$461$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$	20.6340	+	20.6340i	0.958942	+	0.958942i	0.999190	−	0.0402476i	$-0.0128147\pi$
$463$	20.6340	+	20.6340i	0.958942	+	0.958942i	−0.0402476	+	0.999190i	$0.512815\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−71.6410			−3.30807
$470$		0			0
$471$	−19.0526			−0.877896
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−11.3397	−	11.3397i	−0.520303	−	0.520303i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$479$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	−23.7321	+	23.7321i	−1.07540	+	1.07540i	−0.0784867	+	0.996915i	$0.525009\pi$
$487$	−23.7321	+	23.7321i	−1.07540	+	1.07540i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$	17.1506	+	17.1506i	0.775579	+	0.775579i
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	3.32051	−	3.32051i	0.149095	−	0.149095i
$497$		0			0
$498$		0			0
$499$	−31.5885	+	31.5885i	−1.41409	+	1.41409i	−0.697835	+	0.716258i	$0.745853\pi$
$499$	−31.5885	+	31.5885i	−1.41409	+	1.41409i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		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
$507$		0			0
$508$	2.00000			0.0887357
$509$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$509$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$510$		0			0
$511$		−	47.3013i		−	2.09248i
$512$		0			0
$513$	11.7846	−	11.7846i	0.520303	−	0.520303i
$514$		0			0
$515$		0			0
$516$	6.00000			0.264135
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−8.00000			−0.349816			−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000			−0.349816			−0.174908	+	0.984585i	$0.555963\pi$
$524$		0			0
$525$	−26.8301	+	26.8301i	−1.17096	+	1.17096i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$			28.1051i			1.21851i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	30.1506	+	30.1506i	1.29628	+	1.29628i	0.930834	+	0.365444i	$0.119083\pi$
$541$	30.1506	+	30.1506i	1.29628	+	1.29628i	0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$		−	12.0000i		−	0.514969i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	41.0000			1.75303			0.876517	−	0.481371i	$-0.159861\pi$
$547$	41.0000			1.75303			0.876517	+	0.481371i	$0.159861\pi$
$548$		0			0
$549$	25.9808			1.10883
$550$		0			0
$551$		0			0
$552$		0			0
$553$	37.5622	+	37.5622i	1.59731	+	1.59731i
$554$		0			0
$555$		0			0
$556$			14.0000i			0.593732i
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−27.8827	−	27.8827i	−1.17096	−	1.17096i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$			16.0000i			0.669579i	0.942293	+	0.334790i	$0.108665\pi$
$571$			16.0000i			0.669579i	−0.942293	+	0.334790i	$0.891335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$			24.0000i			1.00000i
$577$	29.9282	−	29.9282i	1.24593	−	1.24593i	0.288425	−	0.957503i	$-0.406868\pi$
$577$	29.9282	−	29.9282i	1.24593	−	1.24593i	0.957503	−	0.288425i	$-0.0931316\pi$
$578$		0			0
$579$	6.41858	+	6.41858i	0.266747	+	0.266747i
$580$		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	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$	42.2487			1.74231
$589$			3.76537i			0.155149i
$590$		0			0
$591$		0			0
$592$	−33.8564	−	33.8564i	−1.39149	−	1.39149i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	29.4449i		−	1.20510i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−41.5692			−1.69564			−0.847822	−	0.530281i	$-0.822086\pi$
$601$	−41.5692			−1.69564			−0.847822	+	0.530281i	$0.822086\pi$
$602$		0			0
$603$	34.6865	−	34.6865i	1.41254	−	1.41254i
$604$	−20.2487	+	20.2487i	−0.823908	+	0.823908i
$605$		0			0
$606$		0			0
$607$	−20.0000			−0.811775			−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000			−0.811775			−0.405887	+	0.913923i	$0.633038\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	2.04552	−	2.04552i	0.0826177	−	0.0826177i	−0.664590	−	0.747208i	$-0.731394\pi$
$613$	2.04552	−	2.04552i	0.0826177	−	0.0826177i	0.747208	+	0.664590i	$0.231394\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$	−14.8827	−	14.8827i	−0.598186	−	0.598186i	0.341644	−	0.939829i	$-0.389016\pi$
$619$	−14.8827	−	14.8827i	−0.598186	−	0.598186i	−0.939829	+	0.341644i	$0.889016\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$			22.0000i			0.877896i
$629$		0			0
$630$		0			0
$631$	−25.6147	−	25.6147i	−1.01971	−	1.01971i	−0.999802	−	0.0199047i	$-0.993664\pi$
$631$	−25.6147	−	25.6147i	−1.01971	−	1.01971i	−0.0199047	−	0.999802i	$-0.506336\pi$
$632$		0			0
$633$	45.0000			1.78859
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	33.0263	−	33.0263i	1.30243	−	1.30243i	0.375680	−	0.926750i	$-0.377409\pi$
$643$	33.0263	−	33.0263i	1.30243	−	1.30243i	0.926750	−	0.375680i	$-0.122591\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	8.90897			0.349170
$652$	19.8038	−	19.8038i	0.775579	−	0.775579i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	22.9019	+	22.9019i	0.893489	+	0.893489i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	16.7058	+	16.7058i	0.649779	+	0.649779i	0.952940	−	0.303160i	$-0.0980418\pi$
$661$	16.7058	+	16.7058i	0.649779	+	0.649779i	−0.303160	+	0.952940i	$0.598042\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−33.2487	+	33.2487i	−1.28547	+	1.28547i
$670$		0			0
$671$		0			0
$672$		0			0
$673$			50.2295i			1.93620i	0.250557	+	0.968102i	$0.419386\pi$
$673$			50.2295i			1.93620i	−0.250557	+	0.968102i	$0.580614\pi$
$674$		0			0
$675$		−	25.9808i		−	1.00000i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	−43.5359			−1.67075
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$	−13.6077	−	13.6077i	−0.520303	−	0.520303i
$685$		0			0
$686$		0			0
$687$	−1.05256	−	1.05256i	−0.0401576	−	0.0401576i
$688$		−	6.92820i		−	0.264135i
$689$		0			0
$690$		0			0
$691$	−36.9545	+	36.9545i	−1.40581	+	1.40581i	−0.625958	+	0.779857i	$0.715292\pi$
$691$	−36.9545	+	36.9545i	−1.40581	+	1.40581i	−0.779857	+	0.625958i	$0.784708\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	30.9808	+	30.9808i	1.17096	+	1.17096i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	38.3923			1.44799
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−29.0981	−	29.0981i	−1.09280	−	1.09280i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	−29.0981	−	29.0981i	−1.09280	−	1.09280i	−0.0975728	−	0.995228i	$-0.531108\pi$
$710$		0			0
$711$	−36.3731			−1.36410
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	48.2942	−	48.2942i	1.79857	−	1.79857i
$722$		0			0
$723$	−36.1244	−	36.1244i	−1.34348	−	1.34348i
$724$	−13.8564			−0.514969
$725$		0			0
$726$		0			0
$727$		−	49.0000i		−	1.81731i	−0.417548	−	0.908655i	$-0.637111\pi$
$727$		−	49.0000i		−	1.81731i	0.417548	−	0.908655i	$-0.362889\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		−	30.0000i		−	1.10883i
$733$	−23.3468	+	23.3468i	−0.862333	+	0.862333i	−0.991609	−	0.129275i	$-0.958735\pi$
$733$	−23.3468	+	23.3468i	−0.862333	+	0.862333i	0.129275	+	0.991609i	$0.458735\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	17.9808	+	17.9808i	0.661433	+	0.661433i	0.955718	−	0.294285i	$-0.0950814\pi$
$739$	17.9808	+	17.9808i	0.661433	+	0.661433i	−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		0			0
$742$		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
$751$		−	17.3205i		−	0.632034i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$		−	17.3205i		−	0.632034i	0.948753	−	0.316017i	$-0.102346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−32.1962	+	32.1962i	−1.17096	+	1.17096i
$757$	48.4974			1.76267			0.881334	−	0.472493i	$-0.156646\pi$
$757$	48.4974			1.76267			0.881334	+	0.472493i	$0.156646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$761$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$762$		0			0
$763$	85.6936			3.10232
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	27.7128			1.00000
$769$	−28.7128	+	28.7128i	−1.03541	+	1.03541i	−0.0360609	+	0.999350i	$0.511481\pi$
$769$	−28.7128	+	28.7128i	−1.03541	+	1.03541i	−0.999350	+	0.0360609i	$0.988519\pi$
$770$		0			0
$771$		0			0
$772$	7.41154	−	7.41154i	0.266747	−	0.266747i
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$	4.15064	+	4.15064i	0.149095	+	0.149095i
$776$		0			0
$777$		−	90.8372i		−	3.25877i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	48.7846i		−	1.74231i
$785$		0			0
$786$		0			0
$787$	−12.6147	−	12.6147i	−0.449667	−	0.449667i	0.445577	−	0.895244i	$-0.352999\pi$
$787$	−12.6147	−	12.6147i	−0.449667	−	0.449667i	−0.895244	+	0.445577i	$0.852999\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−34.0000			−1.20510
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−40.0526	−	40.0526i	−1.41254	−	1.41254i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−36.3468	+	36.3468i	−1.27631	+	1.27631i	−0.333590	+	0.942718i	$0.608260\pi$
$811$	−36.3468	+	36.3468i	−1.27631	+	1.27631i	−0.942718	+	0.333590i	$0.891740\pi$
$812$		0			0
$813$	−38.6147	−	38.6147i	−1.35428	−	1.35428i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	3.92820	+	3.92820i	0.137430	+	0.137430i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$821$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$822$		0			0
$823$			24.2487i			0.845257i	0.906303	+	0.422628i	$0.138892\pi$
$823$			24.2487i			0.845257i	−0.906303	+	0.422628i	$0.861108\pi$
$824$		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
$829$			53.0000i			1.84077i	0.391018	+	0.920383i	$0.372123\pi$
$829$			53.0000i			1.84077i	−0.391018	+	0.920383i	$0.627877\pi$
$830$		0			0
$831$		−	36.0000i		−	1.24883i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−4.31347	+	4.31347i	−0.149095	+	0.149095i
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		−	51.9615i		−	1.78859i
$845$		0			0
$846$		0			0
$847$	34.0788	−	34.0788i	1.17096	−	1.17096i
$848$		0			0
$849$		−	43.3013i		−	1.48610i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−40.8827	−	40.8827i	−1.39980	−	1.39980i	−0.800608	−	0.599189i	$-0.795490\pi$
$853$	−40.8827	−	40.8827i	−1.39980	−	1.39980i	−0.599189	−	0.800608i	$-0.704510\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−57.1577			−1.95019			−0.975097	−	0.221777i	$-0.928814\pi$
$859$	−57.1577			−1.95019			−0.975097	+	0.221777i	$0.928814\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−29.4449			−1.00000
$868$		−	10.2872i		−	0.349170i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	21.0788	−	21.0788i	0.713411	−	0.713411i
$874$		0			0
$875$		0			0
$876$	26.4449	−	26.4449i	0.893489	−	0.893489i
$877$	−7.24871	+	7.24871i	−0.244772	+	0.244772i	−0.818821	−	0.574049i	$-0.805372\pi$
$877$	−7.24871	+	7.24871i	−0.244772	+	0.244772i	0.574049	+	0.818821i	$0.305372\pi$
$878$		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
$883$			55.0000i			1.85090i	0.378873	+	0.925449i	$0.376312\pi$
$883$			55.0000i			1.85090i	−0.378873	+	0.925449i	$0.623688\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	3.09808	−	3.09808i	0.103906	−	0.103906i
$890$		0			0
$891$		0			0
$892$	38.3923	+	38.3923i	1.28547	+	1.28547i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−30.0000			−1.00000
$901$		0			0
$902$		0			0
$903$	9.29423	−	9.29423i	0.309293	−	0.309293i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			40.0000i			1.32818i	0.747653	+	0.664089i	$0.231180\pi$
$907$			40.0000i			1.32818i	−0.747653	+	0.664089i	$0.768820\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−15.7128	+	15.7128i	−0.520303	+	0.520303i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.21539	+	1.21539i	−0.0401576	+	0.0401576i
$917$		0			0
$918$		0			0
$919$	31.1769			1.02843			0.514216	−	0.857661i	$-0.328083\pi$
$919$	31.1769			1.02843			0.514216	+	0.857661i	$0.328083\pi$
$920$		0			0
$921$	−31.8109	−	31.8109i	−1.04820	−	1.04820i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	42.3205	−	42.3205i	1.39149	−	1.39149i
$926$		0			0
$927$			46.7654i			1.53598i
$928$		0			0
$929$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$929$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$930$		0			0
$931$	27.6603	+	27.6603i	0.906528	+	0.906528i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	55.4256			1.81068			0.905338	−	0.424691i	$-0.139617\pi$
$937$	55.4256			1.81068			0.905338	+	0.424691i	$0.139617\pi$
$938$		0			0
$939$	57.0000			1.86012
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$			42.0000i			1.36410i
$949$		0			0
$950$		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
$961$			29.6218i			0.955541i
$962$		0			0
$963$		0			0
$964$	−41.7128	+	41.7128i	−1.34348	+	1.34348i
$965$		0			0
$966$		0			0
$967$	39.4449	−	39.4449i	1.26846	−	1.26846i	0.321578	−	0.946883i	$-0.395787\pi$
$967$	39.4449	−	39.4449i	1.26846	−	1.26846i	0.946883	−	0.321578i	$-0.104213\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		−	31.1769i		−	1.00000i
$973$	21.6865	+	21.6865i	0.695238	+	0.695238i
$974$		0			0
$975$		0			0
$976$	−34.6410			−1.10883
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−41.4904	+	41.4904i	−1.32469	+	1.32469i
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	44.0000			1.39771			0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000			1.39771			0.698853	+	0.715265i	$0.253694\pi$
$992$		0			0
$993$	−10.3468	+	10.3468i	−0.328345	+	0.328345i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−59.0000			−1.86855			−0.934274	−	0.356555i	$-0.883951\pi$
$997$	−59.0000			−1.86855			−0.934274	+	0.356555i	$0.883951\pi$
$998$		0			0
$999$	43.9808	+	43.9808i	1.39149	+	1.39149i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.f.b.239.2		4
3.2	odd	2	CM	507.2.f.b.239.2		4
13.2	odd	12		39.2.k.a.20.1	yes	4
13.3	even	3		507.2.k.a.188.1		4
13.4	even	6		39.2.k.a.2.1	✓	4
13.5	odd	4		507.2.f.c.437.2		4
13.6	odd	12		507.2.k.b.89.1		4
13.7	odd	12		507.2.k.a.89.1		4
13.8	odd	4	inner	507.2.f.b.437.2		4
13.9	even	3		507.2.k.c.80.1		4
13.10	even	6		507.2.k.b.188.1		4
13.11	odd	12		507.2.k.c.488.1		4
13.12	even	2		507.2.f.c.239.2		4
39.2	even	12		39.2.k.a.20.1	yes	4
39.5	even	4		507.2.f.c.437.2		4
39.8	even	4	inner	507.2.f.b.437.2		4
39.11	even	12		507.2.k.c.488.1		4
39.17	odd	6		39.2.k.a.2.1	✓	4
39.20	even	12		507.2.k.a.89.1		4
39.23	odd	6		507.2.k.b.188.1		4
39.29	odd	6		507.2.k.a.188.1		4
39.32	even	12		507.2.k.b.89.1		4
39.35	odd	6		507.2.k.c.80.1		4
39.38	odd	2		507.2.f.c.239.2		4
52.15	even	12		624.2.cn.b.449.1		4
52.43	odd	6		624.2.cn.b.353.1		4
65.2	even	12		975.2.bp.d.449.1		4
65.4	even	6		975.2.bo.c.626.1		4
65.17	odd	12		975.2.bp.a.899.1		4
65.28	even	12		975.2.bp.a.449.1		4
65.43	odd	12		975.2.bp.d.899.1		4
65.54	odd	12		975.2.bo.c.176.1		4
156.95	even	6		624.2.cn.b.353.1		4
156.119	odd	12		624.2.cn.b.449.1		4
195.2	odd	12		975.2.bp.d.449.1		4
195.17	even	12		975.2.bp.a.899.1		4
195.119	even	12		975.2.bo.c.176.1		4
195.134	odd	6		975.2.bo.c.626.1		4
195.158	odd	12		975.2.bp.a.449.1		4
195.173	even	12		975.2.bp.d.899.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.k.a.2.1	✓	4	13.4	even	6
39.2.k.a.2.1	✓	4	39.17	odd	6
39.2.k.a.20.1	yes	4	13.2	odd	12
39.2.k.a.20.1	yes	4	39.2	even	12
507.2.f.b.239.2		4	1.1	even	1	trivial
507.2.f.b.239.2		4	3.2	odd	2	CM
507.2.f.b.437.2		4	13.8	odd	4	inner
507.2.f.b.437.2		4	39.8	even	4	inner
507.2.f.c.239.2		4	13.12	even	2
507.2.f.c.239.2		4	39.38	odd	2
507.2.f.c.437.2		4	13.5	odd	4
507.2.f.c.437.2		4	39.5	even	4
507.2.k.a.89.1		4	13.7	odd	12
507.2.k.a.89.1		4	39.20	even	12
507.2.k.a.188.1		4	13.3	even	3
507.2.k.a.188.1		4	39.29	odd	6
507.2.k.b.89.1		4	13.6	odd	12
507.2.k.b.89.1		4	39.32	even	12
507.2.k.b.188.1		4	13.10	even	6
507.2.k.b.188.1		4	39.23	odd	6
507.2.k.c.80.1		4	13.9	even	3
507.2.k.c.80.1		4	39.35	odd	6
507.2.k.c.488.1		4	13.11	odd	12
507.2.k.c.488.1		4	39.11	even	12
624.2.cn.b.353.1		4	52.43	odd	6
624.2.cn.b.353.1		4	156.95	even	6
624.2.cn.b.449.1		4	52.15	even	12
624.2.cn.b.449.1		4	156.119	odd	12
975.2.bo.c.176.1		4	65.54	odd	12
975.2.bo.c.176.1		4	195.119	even	12
975.2.bo.c.626.1		4	65.4	even	6
975.2.bo.c.626.1		4	195.134	odd	6
975.2.bp.a.449.1		4	65.28	even	12
975.2.bp.a.449.1		4	195.158	odd	12
975.2.bp.a.899.1		4	65.17	odd	12
975.2.bp.a.899.1		4	195.17	even	12
975.2.bp.d.449.1		4	65.2	even	12
975.2.bp.d.449.1		4	195.2	odd	12
975.2.bp.d.899.1		4	65.43	odd	12
975.2.bp.d.899.1		4	195.173	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} -2.00000i q^{4} +(-3.09808 - 3.09808i) q^{7} +3.00000 q^{9} -3.46410i q^{12} -4.00000 q^{16} +(2.26795 - 2.26795i) q^{19} +(-5.36603 - 5.36603i) q^{21} -5.00000i q^{25} +5.19615 q^{27} +(-6.19615 + 6.19615i) q^{28} +(-0.830127 + 0.830127i) q^{31} -6.00000i q^{36} +(8.46410 + 8.46410i) q^{37} +1.73205i q^{43} -6.92820 q^{48} +12.1962i q^{49} +(3.92820 - 3.92820i) q^{57} +8.66025 q^{61} +(-9.29423 - 9.29423i) q^{63} +8.00000i q^{64} +(11.5622 - 11.5622i) q^{67} +(7.63397 + 7.63397i) q^{73} -8.66025i q^{75} +(-4.53590 - 4.53590i) q^{76} -12.1244 q^{79} +9.00000 q^{81} +(-10.7321 + 10.7321i) q^{84} +(-1.43782 + 1.43782i) q^{93} +(7.02628 - 7.02628i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{7} + 12 q^{9} - 16 q^{16} + 16 q^{19} - 18 q^{21} - 4 q^{28} + 14 q^{31} + 20 q^{37} - 12 q^{57} - 6 q^{63} + 22 q^{67} + 34 q^{73} - 32 q^{76} + 36 q^{81} - 36 q^{84} - 30 q^{93} - 10 q^{97}+O(q^{100}) \) 4 * q - 2 * q^7 + 12 * q^9 - 16 * q^16 + 16 * q^19 - 18 * q^21 - 4 * q^28 + 14 * q^31 + 20 * q^37 - 12 * q^57 - 6 * q^63 + 22 * q^67 + 34 * q^73 - 32 * q^76 + 36 * q^81 - 36 * q^84 - 30 * q^93 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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