LMFDB - Embedded newform 507.2.f.b.437.1

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			437.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	507.437
Dual form			507.2.f.b.239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +2.00000i q^{4} +(2.09808 - 2.09808i) q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} +2.00000i q^{4} +(2.09808 - 2.09808i) q^{7} +3.00000 q^{9} -3.46410i q^{12} -4.00000 q^{16} +(5.73205 + 5.73205i) q^{19} +(-3.63397 + 3.63397i) q^{21} +5.00000i q^{25} -5.19615 q^{27} +(4.19615 + 4.19615i) q^{28} +(7.83013 + 7.83013i) q^{31} +6.00000i q^{36} +(1.53590 - 1.53590i) q^{37} +1.73205i q^{43} +6.92820 q^{48} -1.80385i q^{49} +(-9.92820 - 9.92820i) q^{57} -8.66025 q^{61} +(6.29423 - 6.29423i) q^{63} -8.00000i q^{64} +(-0.562178 - 0.562178i) q^{67} +(9.36603 - 9.36603i) q^{73} -8.66025i q^{75} +(-11.4641 + 11.4641i) q^{76} +12.1244 q^{79} +9.00000 q^{81} +(-7.26795 - 7.26795i) q^{84} +(-13.5622 - 13.5622i) q^{93} +(-12.0263 - 12.0263i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{7} + 12 q^{9}+O(q^{10}) $ 4 * q - 2 * q^7 + 12 * q^9	$ 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

Display 100 coefficients 10 coefficients

Character values

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

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

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	$-0.750000\pi$
$2$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$3$	−1.73205			−1.00000
$3$	−1.73205			−1.00000
$4$			2.00000i			1.00000i
$5$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$5$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$6$		0			0
$7$	2.09808	−	2.09808i	0.792998	−	0.792998i	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	2.09808	−	2.09808i	0.792998	−	0.792998i	0.981981	+	0.188982i	$0.0605189\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$		−	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
$19$	5.73205	+	5.73205i	1.31502	+	1.31502i	0.917663	+	0.397360i	$0.130073\pi$
$19$	5.73205	+	5.73205i	1.31502	+	1.31502i	0.397360	+	0.917663i	$0.369927\pi$
$20$		0			0
$21$	−3.63397	+	3.63397i	−0.792998	+	0.792998i
$22$		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
$28$	4.19615	+	4.19615i	0.792998	+	0.792998i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	7.83013	+	7.83013i	1.40633	+	1.40633i	0.777714	+	0.628619i	$0.216379\pi$
$31$	7.83013	+	7.83013i	1.40633	+	1.40633i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$			6.00000i			1.00000i
$37$	1.53590	−	1.53590i	0.252500	−	0.252500i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	1.53590	−	1.53590i	0.252500	−	0.252500i	0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$41$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$42$		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	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	6.92820			1.00000
$49$		−	1.80385i		−	0.257693i
$50$		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
$57$	−9.92820	−	9.92820i	−1.31502	−	1.31502i
$58$		0			0
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	−8.66025			−1.10883			−0.554416	−	0.832240i	$-0.687058\pi$
$61$	−8.66025			−1.10883			−0.554416	+	0.832240i	$0.687058\pi$
$62$		0			0
$63$	6.29423	−	6.29423i	0.792998	−	0.792998i
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	−0.562178	−	0.562178i	−0.0686810	−	0.0686810i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	−0.562178	−	0.562178i	−0.0686810	−	0.0686810i	−0.740613	+	0.671932i	$0.765465\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$71$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$72$		0			0
$73$	9.36603	−	9.36603i	1.09621	−	1.09621i	0.101361	−	0.994850i	$-0.467680\pi$
$73$	9.36603	−	9.36603i	1.09621	−	1.09621i	0.994850	−	0.101361i	$-0.0323196\pi$
$74$		0			0
$75$		−	8.66025i		−	1.00000i
$76$	−11.4641	+	11.4641i	−1.31502	+	1.31502i
$77$		0			0
$78$		0			0
$79$	12.1244			1.36410			0.682048	−	0.731307i	$-0.261089\pi$
$79$	12.1244			1.36410			0.682048	+	0.731307i	$0.261089\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$	−7.26795	−	7.26795i	−0.792998	−	0.792998i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−13.5622	−	13.5622i	−1.40633	−	1.40633i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−12.0263	−	12.0263i	−1.22108	−	1.22108i	−0.967247	−	0.253837i	$-0.918307\pi$
$97$	−12.0263	−	12.0263i	−1.22108	−	1.22108i	−0.253837	−	0.967247i	$-0.581693\pi$
$98$		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
$109$	−5.16987	−	5.16987i	−0.495184	−	0.495184i	0.414751	−	0.909935i	$-0.363869\pi$
$109$	−5.16987	−	5.16987i	−0.495184	−	0.495184i	−0.909935	+	0.414751i	$0.863869\pi$
$110$		0			0
$111$	−2.66025	+	2.66025i	−0.252500	+	0.252500i
$112$	−8.39230	+	8.39230i	−0.792998	+	0.792998i
$113$		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
$124$	−15.6603	+	15.6603i	−1.40633	+	1.40633i
$125$		0			0
$126$		0			0
$127$		−	1.00000i		−	0.0887357i	−0.999015	−	0.0443678i	$-0.985873\pi$
$127$		−	1.00000i		−	0.0887357i	0.999015	−	0.0443678i	$-0.0141274\pi$
$128$		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
$133$	24.0526			2.08562
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		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
$147$			3.12436i			0.257693i
$148$	3.07180	+	3.07180i	0.252500	+	0.252500i
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$	14.1244	−	14.1244i	1.14942	−	1.14942i	0.162758	−	0.986666i	$-0.447961\pi$
$151$	14.1244	−	14.1244i	1.14942	−	1.14942i	0.986666	−	0.162758i	$-0.0520389\pi$
$152$		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
$163$	15.0981	−	15.0981i	1.18257	−	1.18257i	0.203497	−	0.979076i	$-0.434769\pi$
$163$	15.0981	−	15.0981i	1.18257	−	1.18257i	0.979076	−	0.203497i	$-0.0652307\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	17.1962	+	17.1962i	1.31502	+	1.31502i
$172$	−3.46410			−0.264135
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	10.4904	+	10.4904i	0.792998	+	0.792998i
$176$		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
$189$	−10.9019	+	10.9019i	−0.792998	+	0.792998i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$			13.8564i			1.00000i
$193$	19.2942	−	19.2942i	1.38883	−	1.38883i	0.561041	−	0.827788i	$-0.310401\pi$
$193$	19.2942	−	19.2942i	1.38883	−	1.38883i	0.827788	−	0.561041i	$-0.189599\pi$
$194$		0			0
$195$		0			0
$196$	3.60770			0.257693
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$			17.0000i			1.20510i	0.798082	+	0.602549i	$0.205848\pi$
$199$			17.0000i			1.20510i	−0.798082	+	0.602549i	$0.794152\pi$
$200$		0			0
$201$	0.973721	+	0.973721i	0.0686810	+	0.0686810i
$202$		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	$-0.852322\pi$
$211$	−25.9808			−1.78859			−0.894295	+	0.447478i	$0.852322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	32.8564			2.23044
$218$		0			0
$219$	−16.2224	+	16.2224i	−1.09621	+	1.09621i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−8.80385	−	8.80385i	−0.589549	−	0.589549i	0.347960	−	0.937509i	$-0.386874\pi$
$223$	−8.80385	−	8.80385i	−0.589549	−	0.589549i	−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	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$	19.8564	−	19.8564i	1.31502	−	1.31502i
$229$	−21.3923	+	21.3923i	−1.41364	+	1.41364i	−0.686743	+	0.726900i	$0.740960\pi$
$229$	−21.3923	+	21.3923i	−1.41364	+	1.41364i	−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	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	6.85641	−	6.85641i	0.441660	−	0.441660i	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	6.85641	−	6.85641i	0.441660	−	0.441660i	0.892570	+	0.450910i	$0.148900\pi$
$242$		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	$0.5\pi$
$252$	12.5885	+	12.5885i	0.792998	+	0.792998i
$253$		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
$259$		−	6.44486i		−	0.400464i
$260$		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
$268$	1.12436	−	1.12436i	0.0686810	−	0.0686810i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−6.70577	+	6.70577i	−0.407347	+	0.407347i	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−6.70577	+	6.70577i	−0.407347	+	0.407347i	0.473466	+	0.880812i	$0.343003\pi$
$272$		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
$279$	23.4904	+	23.4904i	1.40633	+	1.40633i
$280$		0			0
$281$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$281$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$282$		0			0
$283$			25.0000i			1.48610i	0.669238	+	0.743048i	$0.266621\pi$
$283$			25.0000i			1.48610i	−0.669238	+	0.743048i	$0.733379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	20.8301	+	20.8301i	1.22108	+	1.22108i
$292$	18.7321	+	18.7321i	1.09621	+	1.09621i
$293$		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
$300$	17.3205			1.00000
$301$	3.63397	+	3.63397i	0.209459	+	0.209459i
$302$		0			0
$303$		0			0
$304$	−22.9282	−	22.9282i	−1.31502	−	1.31502i
$305$		0			0
$306$		0			0
$307$	−16.6340	+	16.6340i	−0.949351	+	0.949351i	−0.998778	−	0.0494267i	$-0.984261\pi$
$307$	−16.6340	+	16.6340i	−0.949351	+	0.949351i	0.0494267	+	0.998778i	$0.484261\pi$
$308$		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	$-0.880247\pi$
$313$	−32.9090			−1.86012			−0.930062	+	0.367402i	$0.880247\pi$
$314$		0			0
$315$		0			0
$316$			24.2487i			1.36410i
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		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
$327$	8.95448	+	8.95448i	0.495184	+	0.495184i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−25.0263	−	25.0263i	−1.37557	−	1.37557i	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−25.0263	−	25.0263i	−1.37557	−	1.37557i	−0.523612	−	0.851957i	$-0.675416\pi$
$332$		0			0
$333$	4.60770	−	4.60770i	0.252500	−	0.252500i
$334$		0			0
$335$		0			0
$336$	14.5359	−	14.5359i	0.792998	−	0.792998i
$337$		−	29.0000i		−	1.57973i	−0.613280	−	0.789865i	$-0.710150\pi$
$337$		−	29.0000i		−	1.57973i	0.613280	−	0.789865i	$-0.289850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	10.9019	+	10.9019i	0.588649	+	0.588649i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	−3.22243	+	3.22243i	−0.172493	+	0.172493i	−0.788074	−	0.615581i	$-0.788921\pi$
$349$	−3.22243	+	3.22243i	−0.172493	+	0.172493i	0.615581	+	0.788074i	$0.288921\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$		0			0
$361$			46.7128i			2.45857i
$362$		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
$372$	27.1244	−	27.1244i	1.40633	−	1.40633i
$373$	36.3731			1.88333			0.941663	−	0.336557i	$-0.109263\pi$
$373$	36.3731			1.88333			0.941663	+	0.336557i	$0.109263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	12.4378	+	12.4378i	0.638888	+	0.638888i	0.950281	−	0.311393i	$-0.100796\pi$
$379$	12.4378	+	12.4378i	0.638888	+	0.638888i	−0.311393	+	0.950281i	$0.600796\pi$
$380$		0			0
$381$			1.73205i			0.0887357i
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			5.19615i			0.264135i
$388$	24.0526	−	24.0526i	1.22108	−	1.22108i
$389$		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
$397$	−20.4186	+	20.4186i	−1.02478	+	1.02478i	−0.0250943	+	0.999685i	$0.507989\pi$
$397$	−20.4186	+	20.4186i	−1.02478	+	1.02478i	−0.999685	+	0.0250943i	$0.992011\pi$
$398$		0			0
$399$	−41.6603			−2.08562
$400$		−	20.0000i		−	1.00000i
$401$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$401$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−2.50962	−	2.50962i	−0.124093	−	0.124093i	0.642333	−	0.766426i	$-0.277967\pi$
$409$	−2.50962	−	2.50962i	−0.124093	−	0.124093i	−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
$421$	−27.6865	−	27.6865i	−1.34936	−	1.34936i	−0.886357	−	0.463002i	$-0.846772\pi$
$421$	−27.6865	−	27.6865i	−1.34936	−	1.34936i	−0.463002	−	0.886357i	$-0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−18.1699	+	18.1699i	−0.879302	+	0.879302i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$	20.7846			1.00000
$433$		−	35.0000i		−	1.68199i	−0.541041	−	0.840996i	$-0.681970\pi$
$433$		−	35.0000i		−	1.68199i	0.541041	−	0.840996i	$-0.318030\pi$
$434$		0			0
$435$		0			0
$436$	10.3397	−	10.3397i	0.495184	−	0.495184i
$437$		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
$441$		−	5.41154i		−	0.257693i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	−5.32051	−	5.32051i	−0.252500	−	0.252500i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−16.7846	−	16.7846i	−0.792998	−	0.792998i
$449$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$449$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−24.4641	+	24.4641i	−1.14942	+	1.14942i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−26.5622	−	26.5622i	−1.24253	−	1.24253i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−26.5622	−	26.5622i	−1.24253	−	1.24253i	−0.283577	−	0.958950i	$-0.591521\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$	22.3660	−	22.3660i	1.03944	−	1.03944i	0.0402476	−	0.999190i	$-0.487185\pi$
$463$	22.3660	−	22.3660i	1.03944	−	1.03944i	0.999190	−	0.0402476i	$-0.0128147\pi$
$464$		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
$469$	−2.35898			−0.108928
$470$		0			0
$471$	19.0526			0.877896
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−28.6603	+	28.6603i	−1.31502	+	1.31502i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	−20.2679	−	20.2679i	−0.918428	−	0.918428i	0.0784867	−	0.996915i	$-0.474991\pi$
$487$	−20.2679	−	20.2679i	−0.918428	−	0.918428i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$	−26.1506	+	26.1506i	−1.18257	+	1.18257i
$490$		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
$496$	−31.3205	−	31.3205i	−1.40633	−	1.40633i
$497$		0			0
$498$		0			0
$499$	−0.411543	−	0.411543i	−0.0184232	−	0.0184232i	0.697835	−	0.716258i	$-0.254147\pi$
$499$	−0.411543	−	0.411543i	−0.0184232	−	0.0184232i	−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	$-0.750000\pi$
$509$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$510$		0			0
$511$		−	39.3013i		−	1.73859i
$512$		0			0
$513$	−29.7846	−	29.7846i	−1.31502	−	1.31502i
$514$		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
$525$	−18.1699	−	18.1699i	−0.792998	−	0.792998i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$			48.1051i			2.08562i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−13.1506	+	13.1506i	−0.565390	+	0.565390i	−0.930834	−	0.365444i	$-0.880917\pi$
$541$	−13.1506	+	13.1506i	−0.565390	+	0.565390i	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
$553$	25.4378	−	25.4378i	1.08173	−	1.08173i
$554$		0			0
$555$		0			0
$556$		−	14.0000i		−	0.593732i
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		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
$567$	18.8827	−	18.8827i	0.792998	−	0.792998i
$568$		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	$-0.891335\pi$
$571$		−	16.0000i		−	0.669579i	0.942293	−	0.334790i	$-0.108665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		−	24.0000i		−	1.00000i
$577$	16.0718	+	16.0718i	0.669078	+	0.669078i	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	16.0718	+	16.0718i	0.669078	+	0.669078i	−0.288425	+	0.957503i	$0.593132\pi$
$578$		0			0
$579$	−33.4186	+	33.4186i	−1.38883	+	1.38883i
$580$		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	$0.250000\pi$
$588$	−6.24871			−0.257693
$589$			89.7654i			3.69872i
$590$		0			0
$591$		0			0
$592$	−6.14359	+	6.14359i	−0.252500	+	0.252500i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		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	$-0.177914\pi$
$601$	41.5692			1.69564			0.847822	+	0.530281i	$0.177914\pi$
$602$		0			0
$603$	−1.68653	−	1.68653i	−0.0686810	−	0.0686810i
$604$	28.2487	+	28.2487i	1.14942	+	1.14942i
$605$		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
$613$	34.9545	+	34.9545i	1.41180	+	1.41180i	0.747208	+	0.664590i	$0.231394\pi$
$613$	34.9545	+	34.9545i	1.41180	+	1.41180i	0.664590	+	0.747208i	$0.268606\pi$
$614$		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
$619$	31.8827	−	31.8827i	1.28147	−	1.28147i	0.341644	−	0.939829i	$-0.389016\pi$
$619$	31.8827	−	31.8827i	1.28147	−	1.28147i	0.939829	−	0.341644i	$-0.110984\pi$
$620$		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
$631$	24.6147	−	24.6147i	0.979897	−	0.979897i	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	24.6147	−	24.6147i	0.979897	−	0.979897i	0.999802	+	0.0199047i	$0.00633628\pi$
$632$		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
$643$	13.9737	+	13.9737i	0.551070	+	0.551070i	0.926750	−	0.375680i	$-0.122591\pi$
$643$	13.9737	+	13.9737i	0.551070	+	0.551070i	−0.375680	+	0.926750i	$0.622591\pi$
$644$		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
$651$	−56.9090			−2.23044
$652$	30.1962	+	30.1962i	1.18257	+	1.18257i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	28.0981	−	28.0981i	1.09621	−	1.09621i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	32.2942	−	32.2942i	1.25610	−	1.25610i	0.303160	−	0.952940i	$-0.401958\pi$
$661$	32.2942	−	32.2942i	1.25610	−	1.25610i	0.952940	−	0.303160i	$-0.0980418\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	15.2487	+	15.2487i	0.589549	+	0.589549i
$670$		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
$679$	−50.4641			−1.93663
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$	−34.3923	+	34.3923i	−1.31502	+	1.31502i
$685$		0			0
$686$		0			0
$687$	37.0526	−	37.0526i	1.41364	−	1.41364i
$688$		−	6.92820i		−	0.264135i
$689$		0			0
$690$		0			0
$691$	−4.04552	−	4.04552i	−0.153899	−	0.153899i	0.625958	−	0.779857i	$-0.284708\pi$
$691$	−4.04552	−	4.04552i	−0.153899	−	0.153899i	−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
$700$	−20.9808	+	20.9808i	−0.792998	+	0.792998i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	17.6077			0.664087
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−23.9019	+	23.9019i	−0.897656	+	0.897656i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	−23.9019	+	23.9019i	−0.897656	+	0.897656i	0.0975728	+	0.995228i	$0.468892\pi$
$710$		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
$721$	32.7058	+	32.7058i	1.21803	+	1.21803i
$722$		0			0
$723$	−11.8756	+	11.8756i	−0.441660	+	0.441660i
$724$	13.8564			0.514969
$725$		0			0
$726$		0			0
$727$			49.0000i			1.81731i	0.417548	+	0.908655i	$0.362889\pi$
$727$			49.0000i			1.81731i	−0.417548	+	0.908655i	$0.637111\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$			30.0000i			1.10883i
$733$	30.3468	+	30.3468i	1.12088	+	1.12088i	0.991609	+	0.129275i	$0.0412651\pi$
$733$	30.3468	+	30.3468i	1.12088	+	1.12088i	0.129275	+	0.991609i	$0.458735\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−33.9808	+	33.9808i	−1.25000	+	1.25000i	−0.294285	+	0.955718i	$0.595081\pi$
$739$	−33.9808	+	33.9808i	−1.25000	+	1.25000i	−0.955718	+	0.294285i	$0.904919\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		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
$756$	−21.8038	−	21.8038i	−0.792998	−	0.792998i
$757$	−48.4974			−1.76267			−0.881334	−	0.472493i	$-0.843354\pi$
$757$	−48.4974			−1.76267			−0.881334	+	0.472493i	$0.843354\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$761$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$762$		0			0
$763$	−21.6936			−0.785360
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−27.7128			−1.00000
$769$	26.7128	+	26.7128i	0.963289	+	0.963289i	0.999350	−	0.0360609i	$-0.0114810\pi$
$769$	26.7128	+	26.7128i	0.963289	+	0.963289i	−0.0360609	+	0.999350i	$0.511481\pi$
$770$		0			0
$771$		0			0
$772$	38.5885	+	38.5885i	1.38883	+	1.38883i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$	−39.1506	+	39.1506i	−1.40633	+	1.40633i
$776$		0			0
$777$			11.1628i			0.400464i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			7.21539i			0.257693i
$785$		0			0
$786$		0			0
$787$	37.6147	−	37.6147i	1.34082	−	1.34082i	0.445577	−	0.895244i	$-0.352999\pi$
$787$	37.6147	−	37.6147i	1.34082	−	1.34082i	0.895244	−	0.445577i	$-0.147001\pi$
$788$		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
$804$	−1.94744	+	1.94744i	−0.0686810	+	0.0686810i
$805$		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
$811$	17.3468	+	17.3468i	0.609128	+	0.609128i	0.942718	−	0.333590i	$-0.108260\pi$
$811$	17.3468	+	17.3468i	0.609128	+	0.609128i	−0.333590	+	0.942718i	$0.608260\pi$
$812$		0			0
$813$	11.6147	−	11.6147i	0.407347	−	0.407347i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−9.92820	+	9.92820i	−0.347344	+	0.347344i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$		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	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$		−	53.0000i		−	1.84077i	−0.391018	−	0.920383i	$-0.627877\pi$
$829$		−	53.0000i		−	1.84077i	0.391018	−	0.920383i	$-0.372123\pi$
$830$		0			0
$831$			36.0000i			1.24883i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−40.6865	−	40.6865i	−1.40633	−	1.40633i
$838$		0			0
$839$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$839$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		−	51.9615i		−	1.78859i
$845$		0			0
$846$		0			0
$847$	−23.0788	−	23.0788i	−0.792998	−	0.792998i
$848$		0			0
$849$		−	43.3013i		−	1.48610i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	5.88269	−	5.88269i	0.201419	−	0.201419i	−0.599189	−	0.800608i	$-0.704510\pi$
$853$	5.88269	−	5.88269i	0.201419	−	0.201419i	0.800608	+	0.599189i	$0.204510\pi$
$854$		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	$-0.0711857\pi$
$859$	57.1577			1.95019			0.975097	+	0.221777i	$0.0711857\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	29.4449			1.00000
$868$			65.7128i			2.23044i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−36.0788	−	36.0788i	−1.22108	−	1.22108i
$874$		0			0
$875$		0			0
$876$	−32.4449	−	32.4449i	−1.09621	−	1.09621i
$877$	41.2487	+	41.2487i	1.39287	+	1.39287i	0.818821	+	0.574049i	$0.194628\pi$
$877$	41.2487	+	41.2487i	1.39287	+	1.39287i	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	$-0.623688\pi$
$883$		−	55.0000i		−	1.85090i	0.378873	−	0.925449i	$-0.376312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	−2.09808	−	2.09808i	−0.0703672	−	0.0703672i
$890$		0			0
$891$		0			0
$892$	17.6077	−	17.6077i	0.589549	−	0.589549i
$893$		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
$903$	−6.29423	−	6.29423i	−0.209459	−	0.209459i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	40.0000i		−	1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$		−	40.0000i		−	1.32818i	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			$\pi$
$912$	39.7128	+	39.7128i	1.31502	+	1.31502i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−42.7846	−	42.7846i	−1.41364	−	1.41364i
$917$		0			0
$918$		0			0
$919$	−31.1769			−1.02843			−0.514216	−	0.857661i	$-0.671917\pi$
$919$	−31.1769			−1.02843			−0.514216	+	0.857661i	$0.671917\pi$
$920$		0			0
$921$	28.8109	−	28.8109i	0.949351	−	0.949351i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	7.67949	+	7.67949i	0.252500	+	0.252500i
$926$		0			0
$927$			46.7654i			1.53598i
$928$		0			0
$929$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$929$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$930$		0			0
$931$	10.3397	−	10.3397i	0.338871	−	0.338871i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−55.4256			−1.81068			−0.905338	−	0.424691i	$-0.860383\pi$
$937$	−55.4256			−1.81068			−0.905338	+	0.424691i	$0.860383\pi$
$938$		0			0
$939$	57.0000			1.86012
$940$		0			0
$941$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$941$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		−	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
$961$			91.6218i			2.95554i
$962$		0			0
$963$		0			0
$964$	13.7128	+	13.7128i	0.441660	+	0.441660i
$965$		0			0
$966$		0			0
$967$	−19.4449	−	19.4449i	−0.625305	−	0.625305i	0.321578	−	0.946883i	$-0.395787\pi$
$967$	−19.4449	−	19.4449i	−0.625305	−	0.625305i	−0.946883	+	0.321578i	$0.895787\pi$
$968$		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
$973$	−14.6865	+	14.6865i	−0.470829	+	0.470829i
$974$		0			0
$975$		0			0
$976$	34.6410			1.10883
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−15.5096	−	15.5096i	−0.495184	−	0.495184i
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		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
$993$	43.3468	+	43.3468i	1.37557	+	1.37557i
$994$		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
$999$	−7.98076	+	7.98076i	−0.252500	+	0.252500i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.f.b.437.1		4
3.2	odd	2	CM	507.2.f.b.437.1		4
13.2	odd	12		507.2.k.c.188.1		4
13.3	even	3		507.2.k.a.488.1		4
13.4	even	6		39.2.k.a.11.1	✓	4
13.5	odd	4	inner	507.2.f.b.239.1		4
13.6	odd	12		507.2.k.a.80.1		4
13.7	odd	12		507.2.k.b.80.1		4
13.8	odd	4		507.2.f.c.239.1		4
13.9	even	3		507.2.k.c.89.1		4
13.10	even	6		507.2.k.b.488.1		4
13.11	odd	12		39.2.k.a.32.1	yes	4
13.12	even	2		507.2.f.c.437.1		4
39.2	even	12		507.2.k.c.188.1		4
39.5	even	4	inner	507.2.f.b.239.1		4
39.8	even	4		507.2.f.c.239.1		4
39.11	even	12		39.2.k.a.32.1	yes	4
39.17	odd	6		39.2.k.a.11.1	✓	4
39.20	even	12		507.2.k.b.80.1		4
39.23	odd	6		507.2.k.b.488.1		4
39.29	odd	6		507.2.k.a.488.1		4
39.32	even	12		507.2.k.a.80.1		4
39.35	odd	6		507.2.k.c.89.1		4
39.38	odd	2		507.2.f.c.437.1		4
52.11	even	12		624.2.cn.b.305.1		4
52.43	odd	6		624.2.cn.b.401.1		4
65.4	even	6		975.2.bo.c.401.1		4
65.17	odd	12		975.2.bp.d.674.1		4
65.24	odd	12		975.2.bo.c.851.1		4
65.37	even	12		975.2.bp.a.149.1		4
65.43	odd	12		975.2.bp.a.674.1		4
65.63	even	12		975.2.bp.d.149.1		4
156.11	odd	12		624.2.cn.b.305.1		4
156.95	even	6		624.2.cn.b.401.1		4
195.17	even	12		975.2.bp.d.674.1		4
195.89	even	12		975.2.bo.c.851.1		4
195.128	odd	12		975.2.bp.d.149.1		4
195.134	odd	6		975.2.bo.c.401.1		4
195.167	odd	12		975.2.bp.a.149.1		4
195.173	even	12		975.2.bp.a.674.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.k.a.11.1	✓	4	13.4	even	6
39.2.k.a.11.1	✓	4	39.17	odd	6
39.2.k.a.32.1	yes	4	13.11	odd	12
39.2.k.a.32.1	yes	4	39.11	even	12
507.2.f.b.239.1		4	13.5	odd	4	inner
507.2.f.b.239.1		4	39.5	even	4	inner
507.2.f.b.437.1		4	1.1	even	1	trivial
507.2.f.b.437.1		4	3.2	odd	2	CM
507.2.f.c.239.1		4	13.8	odd	4
507.2.f.c.239.1		4	39.8	even	4
507.2.f.c.437.1		4	13.12	even	2
507.2.f.c.437.1		4	39.38	odd	2
507.2.k.a.80.1		4	13.6	odd	12
507.2.k.a.80.1		4	39.32	even	12
507.2.k.a.488.1		4	13.3	even	3
507.2.k.a.488.1		4	39.29	odd	6
507.2.k.b.80.1		4	13.7	odd	12
507.2.k.b.80.1		4	39.20	even	12
507.2.k.b.488.1		4	13.10	even	6
507.2.k.b.488.1		4	39.23	odd	6
507.2.k.c.89.1		4	13.9	even	3
507.2.k.c.89.1		4	39.35	odd	6
507.2.k.c.188.1		4	13.2	odd	12
507.2.k.c.188.1		4	39.2	even	12
624.2.cn.b.305.1		4	52.11	even	12
624.2.cn.b.305.1		4	156.11	odd	12
624.2.cn.b.401.1		4	52.43	odd	6
624.2.cn.b.401.1		4	156.95	even	6
975.2.bo.c.401.1		4	65.4	even	6
975.2.bo.c.401.1		4	195.134	odd	6
975.2.bo.c.851.1		4	65.24	odd	12
975.2.bo.c.851.1		4	195.89	even	12
975.2.bp.a.149.1		4	65.37	even	12
975.2.bp.a.149.1		4	195.167	odd	12
975.2.bp.a.674.1		4	65.43	odd	12
975.2.bp.a.674.1		4	195.173	even	12
975.2.bp.d.149.1		4	65.63	even	12
975.2.bp.d.149.1		4	195.128	odd	12
975.2.bp.d.674.1		4	65.17	odd	12
975.2.bp.d.674.1		4	195.17	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)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +2.00000i q^{4} +(2.09808 - 2.09808i) q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} +2.00000i q^{4} +(2.09808 - 2.09808i) q^{7} +3.00000 q^{9} -3.46410i q^{12} -4.00000 q^{16} +(5.73205 + 5.73205i) q^{19} +(-3.63397 + 3.63397i) q^{21} +5.00000i q^{25} -5.19615 q^{27} +(4.19615 + 4.19615i) q^{28} +(7.83013 + 7.83013i) q^{31} +6.00000i q^{36} +(1.53590 - 1.53590i) q^{37} +1.73205i q^{43} +6.92820 q^{48} -1.80385i q^{49} +(-9.92820 - 9.92820i) q^{57} -8.66025 q^{61} +(6.29423 - 6.29423i) q^{63} -8.00000i q^{64} +(-0.562178 - 0.562178i) q^{67} +(9.36603 - 9.36603i) q^{73} -8.66025i q^{75} +(-11.4641 + 11.4641i) q^{76} +12.1244 q^{79} +9.00000 q^{81} +(-7.26795 - 7.26795i) q^{84} +(-13.5622 - 13.5622i) q^{93} +(-12.0263 - 12.0263i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{7} + 12 q^{9}+O(q^{10}) \) 4 * q - 2 * q^7 + 12 * q^9	\( 4 q - 2 q^{7} + 12 q^{9} - 16 q^{16} + 16 q^{19} - 18 q^{21} - 4 q^{28} + 14 q^{31} + 20 q^{37} - 12 q^{57} - 6 q^{63} + 22 q^{67} + 34 q^{73} - 32 q^{76} + 36 q^{81} - 36 q^{84} - 30 q^{93} - 10 q^{97}+O(q^{100}) \) 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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