LMFDB - Embedded newform 507.2.f.d.437.3

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:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			437.3
Root			$0.500000 - 0.564882i$ of defining polynomial
Character	$\chi$	$=$	507.437
Dual form			507.2.f.d.239.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.06488 - 1.06488i) q^{2} +1.73205 q^{3} -0.267949i q^{4} +(-2.90931 + 2.90931i) q^{5} +(1.84443 - 1.84443i) q^{6} +(1.84443 + 1.84443i) q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+(1.06488 - 1.06488i) q^{2} +1.73205 q^{3} -0.267949i q^{4} +(-2.90931 + 2.90931i) q^{5} +(1.84443 - 1.84443i) q^{6} +(1.84443 + 1.84443i) q^{8} +3.00000 q^{9} +6.19615i q^{10} +(0.779548 + 0.779548i) q^{11} -0.464102i q^{12} +(-5.03908 + 5.03908i) q^{15} +4.46410 q^{16} +(3.19465 - 3.19465i) q^{18} +(0.779548 + 0.779548i) q^{20} +1.66025 q^{22} +(3.19465 + 3.19465i) q^{24} -11.9282i q^{25} +5.19615 q^{27} +10.7321i q^{30} +(1.06488 - 1.06488i) q^{32} +(1.35022 + 1.35022i) q^{33} -0.803848i q^{36} -10.7321 q^{40} +(7.16884 - 7.16884i) q^{41} -4.00000i q^{43} +(0.208879 - 0.208879i) q^{44} +(-8.72794 + 8.72794i) q^{45} +(-6.59817 - 6.59817i) q^{47} +7.73205 q^{48} +7.00000i q^{49} +(-12.7021 - 12.7021i) q^{50} +(5.53329 - 5.53329i) q^{54} -4.53590 q^{55} +(-10.8577 - 10.8577i) q^{59} +(1.35022 + 1.35022i) q^{60} -13.8564 q^{61} +6.66025i q^{64} +2.87564 q^{66} +(3.47998 - 3.47998i) q^{71} +(5.53329 + 5.53329i) q^{72} -20.6603i q^{75} +10.3923 q^{79} +(-12.9875 + 12.9875i) q^{80} +9.00000 q^{81} -15.2679i q^{82} +(9.29861 - 9.29861i) q^{83} +(-4.25953 - 4.25953i) q^{86} +2.87564i q^{88} +(12.9875 + 12.9875i) q^{89} +18.5885i q^{90} -14.0526 q^{94} +(1.84443 - 1.84443i) q^{96} +(7.45418 + 7.45418i) q^{98} +(2.33864 + 2.33864i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} + 8 q^{16} - 56 q^{22} - 72 q^{40} + 48 q^{48} - 64 q^{55} + 120 q^{66} + 72 q^{81} + 40 q^{94}+O(q^{100}) $ 8 * q + 24 * q^9 + 8 * q^16 - 56 * q^22 - 72 * q^40 + 48 * q^48 - 64 * q^55 + 120 * q^66 + 72 * q^81 + 40 * q^94

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 $

$2$	1.06488	−	1.06488i	0.752986	−	0.752986i	−0.222050	−	0.975035i	$-0.571275\pi$
$2$	1.06488	−	1.06488i	0.752986	−	0.752986i	0.975035	+	0.222050i	$0.0712747\pi$
$3$	1.73205			1.00000
$3$	1.73205			1.00000
$4$		−	0.267949i		−	0.133975i
$5$	−2.90931	+	2.90931i	−1.30108	+	1.30108i	−0.373423	+	0.927661i	$0.621816\pi$
$5$	−2.90931	+	2.90931i	−1.30108	+	1.30108i	−0.927661	+	0.373423i	$0.878184\pi$
$6$	1.84443	−	1.84443i	0.752986	−	0.752986i
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	1.84443	+	1.84443i	0.652105	+	0.652105i
$9$	3.00000			1.00000
$10$			6.19615i			1.95940i
$11$	0.779548	+	0.779548i	0.235043	+	0.235043i	0.814794	−	0.579751i	$-0.196850\pi$
$11$	0.779548	+	0.779548i	0.235043	+	0.235043i	−0.579751	+	0.814794i	$0.696850\pi$
$12$		−	0.464102i		−	0.133975i
$13$		0			0
$13$		0			0
$14$		0			0
$15$	−5.03908	+	5.03908i	−1.30108	+	1.30108i
$16$	4.46410			1.11603
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	3.19465	−	3.19465i	0.752986	−	0.752986i
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$	0.779548	+	0.779548i	0.174312	+	0.174312i
$21$		0			0
$22$	1.66025			0.353967
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	3.19465	+	3.19465i	0.652105	+	0.652105i
$25$		−	11.9282i		−	2.38564i
$26$		0			0
$27$	5.19615			1.00000
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$			10.7321i			1.95940i
$31$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$31$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$32$	1.06488	−	1.06488i	0.188246	−	0.188246i
$33$	1.35022	+	1.35022i	0.235043	+	0.235043i
$34$		0			0
$35$		0			0
$36$		−	0.803848i		−	0.133975i
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$	−10.7321			−1.69689
$41$	7.16884	−	7.16884i	1.11959	−	1.11959i	0.127783	−	0.991802i	$-0.459214\pi$
$41$	7.16884	−	7.16884i	1.11959	−	1.11959i	0.991802	−	0.127783i	$-0.0407861\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0.208879	−	0.208879i	0.0314897	−	0.0314897i
$45$	−8.72794	+	8.72794i	−1.30108	+	1.30108i
$46$		0			0
$47$	−6.59817	−	6.59817i	−0.962443	−	0.962443i	0.0368772	−	0.999320i	$-0.488259\pi$
$47$	−6.59817	−	6.59817i	−0.962443	−	0.962443i	−0.999320	+	0.0368772i	$0.988259\pi$
$48$	7.73205			1.11603
$49$			7.00000i			1.00000i
$50$	−12.7021	−	12.7021i	−1.79635	−	1.79635i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	5.53329	−	5.53329i	0.752986	−	0.752986i
$55$	−4.53590			−0.611620
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.8577	−	10.8577i	−1.41355	−	1.41355i	−0.728392	−	0.685160i	$-0.759732\pi$
$59$	−10.8577	−	10.8577i	−1.41355	−	1.41355i	−0.685160	−	0.728392i	$-0.740268\pi$
$60$	1.35022	+	1.35022i	0.174312	+	0.174312i
$61$	−13.8564			−1.77413			−0.887066	−	0.461644i	$-0.847260\pi$
$61$	−13.8564			−1.77413			−0.887066	+	0.461644i	$0.847260\pi$
$62$		0			0
$63$		0			0
$64$			6.66025i			0.832532i
$65$		0			0
$66$	2.87564			0.353967
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.47998	−	3.47998i	0.412998	−	0.412998i	−0.469784	−	0.882782i	$-0.655668\pi$
$71$	3.47998	−	3.47998i	0.412998	−	0.412998i	0.882782	+	0.469784i	$0.155668\pi$
$72$	5.53329	+	5.53329i	0.652105	+	0.652105i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$		−	20.6603i		−	2.38564i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	10.3923			1.16923			0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923			1.16923			0.584613	+	0.811312i	$0.301246\pi$
$80$	−12.9875	+	12.9875i	−1.45204	+	1.45204i
$81$	9.00000			1.00000
$82$		−	15.2679i		−	1.68606i
$83$	9.29861	−	9.29861i	1.02065	−	1.02065i	0.0208726	−	0.999782i	$-0.493356\pi$
$83$	9.29861	−	9.29861i	1.02065	−	1.02065i	0.999782	−	0.0208726i	$-0.00664445\pi$
$84$		0			0
$85$		0			0
$86$	−4.25953	−	4.25953i	−0.459317	−	0.459317i
$87$		0			0
$88$			2.87564i			0.306545i
$89$	12.9875	+	12.9875i	1.37667	+	1.37667i	0.850185	+	0.526484i	$0.176490\pi$
$89$	12.9875	+	12.9875i	1.37667	+	1.37667i	0.526484	+	0.850185i	$0.323510\pi$
$90$			18.5885i			1.95940i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−14.0526			−1.44941
$95$		0			0
$96$	1.84443	−	1.84443i	0.188246	−	0.188246i
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	7.45418	+	7.45418i	0.752986	+	0.752986i
$99$	2.33864	+	2.33864i	0.235043	+	0.235043i
$100$	−3.19615			−0.319615
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$			16.0000i			1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$			16.0000i			1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		−	1.39230i		−	0.133975i
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$	−4.83020	+	4.83020i	−0.460541	+	0.460541i
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−23.1244			−2.12877
$119$		0			0
$120$	−18.5885			−1.69689
$121$		−	9.78461i		−	0.889510i
$122$	−14.7554	+	14.7554i	−1.33590	+	1.33590i
$123$	12.4168	−	12.4168i	1.11959	−	1.11959i
$124$		0			0
$125$	20.1563	+	20.1563i	1.80284	+	1.80284i
$126$		0			0
$127$		−	17.3205i		−	1.53695i	−0.639882	−	0.768473i	$-0.721017\pi$
$127$		−	17.3205i		−	1.53695i	0.639882	−	0.768473i	$-0.278983\pi$
$128$	9.22215	+	9.22215i	0.815131	+	0.815131i
$129$		−	6.92820i		−	0.609994i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	0.361790	−	0.361790i	0.0314897	−	0.0314897i
$133$		0			0
$134$		0			0
$135$	−15.1172	+	15.1172i	−1.30108	+	1.30108i
$136$		0			0
$137$	10.2870	+	10.2870i	0.878881	+	0.878881i	0.993419	−	0.114538i	$-0.0365388\pi$
$137$	10.2870	+	10.2870i	0.878881	+	0.878881i	−0.114538	+	0.993419i	$0.536539\pi$
$138$		0			0
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$		0			0
$141$	−11.4284	−	11.4284i	−0.962443	−	0.962443i
$142$		−	7.41154i		−	0.621963i
$143$		0			0
$144$	13.3923			1.11603
$145$		0			0
$146$		0			0
$147$			12.1244i			1.00000i
$148$		0			0
$149$	0.208879	−	0.208879i	0.0171121	−	0.0171121i	−0.698499	−	0.715611i	$-0.746148\pi$
$149$	0.208879	−	0.208879i	0.0171121	−	0.0171121i	0.715611	+	0.698499i	$0.246148\pi$
$150$	−22.0007	−	22.0007i	−1.79635	−	1.79635i
$151$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$151$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$	11.0666	−	11.0666i	0.880410	−	0.880410i
$159$		0			0
$160$			6.19615i			0.489849i
$161$		0			0
$162$	9.58394	−	9.58394i	0.752986	−	0.752986i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$	−1.92089	−	1.92089i	−0.149996	−	0.149996i
$165$	−7.85641			−0.611620
$166$		−	19.8038i		−	1.53708i
$167$	−8.15727	−	8.15727i	−0.631229	−	0.631229i	0.317148	−	0.948376i	$-0.397275\pi$
$167$	−8.15727	−	8.15727i	−0.631229	−	0.631229i	−0.948376	+	0.317148i	$0.897275\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$	−1.07180			−0.0817237
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	3.47998	+	3.47998i	0.262313	+	0.262313i
$177$	−18.8061	−	18.8061i	−1.41355	−	1.41355i
$178$	27.6603			2.07322
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	2.33864	+	2.33864i	0.174312	+	0.174312i
$181$		−	10.0000i		−	0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$		−	10.0000i		−	0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$		0			0
$183$	−24.0000			−1.77413
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−1.76798	+	1.76798i	−0.128943	+	0.128943i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$			11.5359i			0.832532i
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$	1.87564			0.133975
$197$	−17.2470	+	17.2470i	−1.22880	+	1.22880i	−0.264379	+	0.964419i	$0.585167\pi$
$197$	−17.2470	+	17.2470i	−1.22880	+	1.22880i	−0.964419	+	0.264379i	$0.914833\pi$
$198$	4.98076			0.353967
$199$			24.2487i			1.71895i	0.511182	+	0.859473i	$0.329208\pi$
$199$			24.2487i			1.71895i	−0.511182	+	0.859473i	$0.670792\pi$
$200$	22.0007	−	22.0007i	1.55569	−	1.55569i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$			41.7128i			2.91335i
$206$	17.0381	+	17.0381i	1.18710	+	1.18710i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	3.46410			0.238479			0.119239	−	0.992866i	$-0.461954\pi$
$211$	3.46410			0.238479			0.119239	+	0.992866i	$0.461954\pi$
$212$		0			0
$213$	6.02751	−	6.02751i	0.412998	−	0.412998i
$214$		0			0
$215$	11.6373	+	11.6373i	0.793654	+	0.793654i
$216$	9.58394	+	9.58394i	0.652105	+	0.652105i
$217$		0			0
$218$		0			0
$219$		0			0
$220$			1.21539i			0.0819416i
$221$		0			0
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$		−	35.7846i		−	2.38564i
$226$		0			0
$227$	−20.9359	+	20.9359i	−1.38956	+	1.38956i	−0.563329	+	0.826232i	$0.690480\pi$
$227$	−20.9359	+	20.9359i	−1.38956	+	1.38956i	−0.826232	+	0.563329i	$0.809520\pi$
$228$		0			0
$229$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$229$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$230$		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
$235$	38.3923			2.50444
$236$	−2.90931	+	2.90931i	−0.189380	+	0.189380i
$237$	18.0000			1.16923
$238$		0			0
$239$	18.2354	−	18.2354i	1.17955	−	1.17955i	0.199693	−	0.979858i	$-0.436005\pi$
$239$	18.2354	−	18.2354i	1.17955	−	1.17955i	0.979858	−	0.199693i	$-0.0639945\pi$
$240$	−22.4950	+	22.4950i	−1.45204	+	1.45204i
$241$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$241$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$242$	−10.4195	−	10.4195i	−0.669788	−	0.669788i
$243$	15.5885			1.00000
$244$			3.71281i			0.237688i
$245$	−20.3652	−	20.3652i	−1.30108	−	1.30108i
$246$		−	26.4449i		−	1.68606i
$247$		0			0
$248$		0			0
$249$	16.1057	−	16.1057i	1.02065	−	1.02065i
$250$	42.9282			2.71502
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	−18.4443	−	18.4443i	−1.15730	−	1.15730i
$255$		0			0
$256$	6.32051			0.395032
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$	−7.37772	−	7.37772i	−0.459317	−	0.459317i
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$			4.98076i			0.306545i
$265$		0			0
$266$		0			0
$267$	22.4950	+	22.4950i	1.37667	+	1.37667i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$			32.1962i			1.95940i
$271$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$271$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$272$		0			0
$273$		0			0
$274$	21.9090			1.32357
$275$	9.29861	−	9.29861i	0.560727	−	0.560727i
$276$		0			0
$277$			22.0000i			1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$			22.0000i			1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	−21.2976	+	21.2976i	−1.27735	+	1.27735i
$279$		0			0
$280$		0			0
$281$	−9.86928	−	9.86928i	−0.588752	−	0.588752i	0.348542	−	0.937293i	$-0.386677\pi$
$281$	−9.86928	−	9.86928i	−0.588752	−	0.588752i	−0.937293	+	0.348542i	$0.886677\pi$
$282$	−24.3397			−1.44941
$283$		−	17.3205i		−	1.02960i	−0.857311	−	0.514799i	$-0.827867\pi$
$283$		−	17.3205i		−	1.02960i	0.857311	−	0.514799i	$-0.172133\pi$
$284$	−0.932458	−	0.932458i	−0.0553312	−	0.0553312i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	3.19465	−	3.19465i	0.188246	−	0.188246i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−23.0656	−	23.0656i	−1.34751	−	1.34751i	−0.888359	−	0.459149i	$-0.848154\pi$
$293$	−23.0656	−	23.0656i	−1.34751	−	1.34751i	−0.459149	−	0.888359i	$-0.651846\pi$
$294$	12.9110	+	12.9110i	0.752986	+	0.752986i
$295$	63.1769			3.67830
$296$		0			0
$297$	4.05065	+	4.05065i	0.235043	+	0.235043i
$298$		−	0.444864i		−	0.0257703i
$299$		0			0
$300$	−5.53590			−0.319615
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	40.3126	−	40.3126i	2.30829	−	2.30829i
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$			27.7128i			1.57653i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−34.6410			−1.95803			−0.979013	−	0.203798i	$-0.934671\pi$
$313$	−34.6410			−1.95803			−0.979013	+	0.203798i	$0.934671\pi$
$314$	2.12976	−	2.12976i	0.120190	−	0.120190i
$315$		0			0
$316$		−	2.78461i		−	0.156647i
$317$	−11.8461	+	11.8461i	−0.665345	+	0.665345i	−0.956635	−	0.291290i	$-0.905916\pi$
$317$	−11.8461	+	11.8461i	−0.665345	+	0.665345i	0.291290	+	0.956635i	$0.405916\pi$
$318$		0			0
$319$		0			0
$320$	−19.3768	−	19.3768i	−1.08319	−	1.08319i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	2.41154i		−	0.133975i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	26.4449			1.46017
$329$		0			0
$330$	−8.36615	+	8.36615i	−0.460541	+	0.460541i
$331$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$331$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$332$	−2.49155	−	2.49155i	−0.136742	−	0.136742i
$333$		0			0
$334$	−17.3731			−0.950612
$335$		0			0
$336$		0			0
$337$		−	6.92820i		−	0.377403i	−0.982034	−	0.188702i	$-0.939572\pi$
$337$		−	6.92820i		−	0.377403i	0.982034	−	0.188702i	$-0.0604279\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	7.37772	−	7.37772i	0.397780	−	0.397780i
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$		0			0
$351$		0			0
$352$	1.66025			0.0884918
$353$	−1.76798	+	1.76798i	−0.0940998	+	0.0940998i	−0.752590	−	0.658490i	$-0.771196\pi$
$353$	−1.76798	+	1.76798i	−0.0940998	+	0.0940998i	0.658490	+	0.752590i	$0.271196\pi$
$354$	−40.0526			−2.12877
$355$			20.2487i			1.07469i
$356$	3.47998	−	3.47998i	0.184439	−	0.184439i
$357$		0			0
$358$		0			0
$359$	26.7545	+	26.7545i	1.41205	+	1.41205i	0.745143	+	0.666905i	$0.232381\pi$
$359$	26.7545	+	26.7545i	1.41205	+	1.41205i	0.666905	+	0.745143i	$0.267619\pi$
$360$	−32.1962			−1.69689
$361$		−	19.0000i		−	1.00000i
$362$	−10.6488	−	10.6488i	−0.559690	−	0.559690i
$363$		−	16.9474i		−	0.889510i
$364$		0			0
$365$		0			0
$366$	−25.5572	+	25.5572i	−1.33590	+	1.33590i
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		0			0
$369$	21.5065	−	21.5065i	1.11959	−	1.11959i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−13.8564			−0.717458			−0.358729	−	0.933442i	$-0.616790\pi$
$373$	−13.8564			−0.717458			−0.358729	+	0.933442i	$0.616790\pi$
$374$		0			0
$375$	34.9118	+	34.9118i	1.80284	+	1.80284i
$376$		−	24.3397i		−	1.25523i
$377$		0			0
$378$		0			0
$379$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$379$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$380$		0			0
$381$		−	30.0000i		−	1.53695i
$382$		0			0
$383$	−11.9990	+	11.9990i	−0.613122	+	0.613122i	−0.943758	−	0.330636i	$-0.892737\pi$
$383$	−11.9990	+	11.9990i	−0.613122	+	0.613122i	0.330636	+	0.943758i	$0.392737\pi$
$384$	15.9732	+	15.9732i	0.815131	+	0.815131i
$385$		0			0
$386$		0			0
$387$		−	12.0000i		−	0.609994i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	−12.9110	+	12.9110i	−0.652105	+	0.652105i
$393$		0			0
$394$			36.7321i			1.85053i
$395$	−30.2345	+	30.2345i	−1.52126	+	1.52126i
$396$	0.626638	−	0.626638i	0.0314897	−	0.0314897i
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	25.8220	+	25.8220i	1.29434	+	1.29434i
$399$		0			0
$400$		−	53.2487i		−	2.66244i
$401$	21.9243	+	21.9243i	1.09485	+	1.09485i	0.995003	+	0.0998435i	$0.0318342\pi$
$401$	21.9243	+	21.9243i	1.09485	+	1.09485i	0.0998435	+	0.995003i	$0.468166\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−26.1838	+	26.1838i	−1.30108	+	1.30108i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$409$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$410$	44.4192	+	44.4192i	2.19371	+	2.19371i
$411$	17.8177	+	17.8177i	0.878881	+	0.878881i
$412$	4.28719			0.211215
$413$		0			0
$414$		0			0
$415$			54.1051i			2.65592i
$416$		0			0
$417$	−34.6410			−1.69638
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$	3.68886	−	3.68886i	0.179571	−	0.179571i
$423$	−19.7945	−	19.7945i	−0.962443	−	0.962443i
$424$		0			0
$425$		0			0
$426$		−	12.8372i		−	0.621963i
$427$		0			0
$428$		0			0
$429$		0			0
$430$	24.7846			1.19522
$431$	28.3136	−	28.3136i	1.36382	−	1.36382i	0.494824	−	0.868993i	$-0.335233\pi$
$431$	28.3136	−	28.3136i	1.36382	−	1.36382i	0.868993	−	0.494824i	$-0.164767\pi$
$432$	23.1962			1.11603
$433$		−	20.7846i		−	0.998845i	−0.866359	−	0.499422i	$-0.833546\pi$
$433$		−	20.7846i		−	0.998845i	0.866359	−	0.499422i	$-0.166454\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			40.0000i			1.90910i	0.298057	+	0.954548i	$0.403661\pi$
$439$			40.0000i			1.90910i	−0.298057	+	0.954548i	$0.596339\pi$
$440$	−8.36615	−	8.36615i	−0.398841	−	0.398841i
$441$			21.0000i			1.00000i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−75.5692			−3.58232
$446$		0			0
$447$	0.361790	−	0.361790i	0.0171121	−	0.0171121i
$448$		0			0
$449$	−7.58660	−	7.58660i	−0.358034	−	0.358034i	0.505054	−	0.863088i	$-0.331473\pi$
$449$	−7.58660	−	7.58660i	−0.358034	−	0.358034i	−0.863088	+	0.505054i	$0.831473\pi$
$450$	−38.1064	−	38.1064i	−1.79635	−	1.79635i
$451$	11.1769			0.526300
$452$		0			0
$453$		0			0
$454$			44.5885i			2.09264i
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.6648	+	17.6648i	−0.822730	+	0.822730i	−0.986499	−	0.163769i	$-0.947635\pi$
$461$	−17.6648	+	17.6648i	−0.822730	+	0.822730i	0.163769	+	0.986499i	$0.447635\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		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
$469$		0			0
$470$	40.8833	−	40.8833i	1.88581	−	1.88581i
$471$	3.46410			0.159617
$472$		−	40.0526i		−	1.84357i
$473$	3.11819	−	3.11819i	0.143375	−	0.143375i
$474$	19.1679	−	19.1679i	0.880410	−	0.880410i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		−	38.8372i		−	1.77637i
$479$	−23.6363	−	23.6363i	−1.07997	−	1.07997i	−0.996511	−	0.0834585i	$-0.973403\pi$
$479$	−23.6363	−	23.6363i	−1.07997	−	1.07997i	−0.0834585	−	0.996511i	$-0.526597\pi$
$480$			10.7321i			0.489849i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−2.62178			−0.119172
$485$		0			0
$486$	16.5999	−	16.5999i	0.752986	−	0.752986i
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	−25.5572	−	25.5572i	−1.15692	−	1.15692i
$489$		0			0
$490$	−43.3731			−1.95940
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$	−3.32707	−	3.32707i	−0.149996	−	0.149996i
$493$		0			0
$494$		0			0
$495$	−13.6077			−0.611620
$496$		0			0
$497$		0			0
$498$		−	34.3013i		−	1.53708i
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$	5.40087	−	5.40087i	0.241534	−	0.241534i
$501$	−14.1288	−	14.1288i	−0.631229	−	0.631229i
$502$		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
$508$	−4.64102			−0.205912
$509$	−8.31018	+	8.31018i	−0.368342	+	0.368342i	−0.866872	−	0.498530i	$-0.833873\pi$
$509$	−8.31018	+	8.31018i	−0.368342	+	0.368342i	0.498530	+	0.866872i	$0.333873\pi$
$510$		0			0
$511$		0			0
$512$	−11.7137	+	11.7137i	−0.517678	+	0.517678i
$513$		0			0
$514$		0			0
$515$	−46.5490	−	46.5490i	−2.05119	−	2.05119i
$516$	−1.85641			−0.0817237
$517$		−	10.2872i		−	0.452430i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	44.0000			1.92399			0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000			1.92399			0.961993	+	0.273075i	$0.0880406\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	6.02751	+	6.02751i	0.262313	+	0.262313i
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−32.5731	−	32.5731i	−1.41355	−	1.41355i
$532$		0			0
$533$		0			0
$534$	47.9090			2.07322
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.45684	+	5.45684i	−0.235043	+	0.235043i
$540$	4.05065	+	4.05065i	0.174312	+	0.174312i
$541$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$541$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$542$		0			0
$543$		−	17.3205i		−	0.743294i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$	2.75640	−	2.75640i	0.117748	−	0.117748i
$549$	−41.5692			−1.77413
$550$		−	19.8038i		−	0.844439i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	23.4274	+	23.4274i	0.995335	+	0.995335i
$555$		0			0
$556$			5.35898i			0.227272i
$557$	−2.49155	−	2.49155i	−0.105571	−	0.105571i	0.652349	−	0.757919i	$-0.273784\pi$
$557$	−2.49155	−	2.49155i	−0.105571	−	0.105571i	−0.757919	+	0.652349i	$0.773784\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−21.0192			−0.886643
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	−3.06222	+	3.06222i	−0.128943	+	0.128943i
$565$		0			0
$566$	−18.4443	−	18.4443i	−0.775272	−	0.775272i
$567$		0			0
$568$	12.8372			0.538636
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$			38.1051i			1.59465i	0.603550	+	0.797325i	$0.293752\pi$
$571$			38.1051i			1.59465i	−0.603550	+	0.797325i	$0.706248\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$			19.9808i			0.832532i
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	−18.1030	+	18.1030i	−0.752986	+	0.752986i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−49.1244			−2.02931
$587$	−29.4549	+	29.4549i	−1.21573	+	1.21573i	−0.246623	+	0.969111i	$0.579321\pi$
$587$	−29.4549	+	29.4549i	−1.21573	+	1.21573i	−0.969111	+	0.246623i	$0.920679\pi$
$588$	3.24871			0.133975
$589$		0			0
$590$	67.2760	−	67.2760i	2.76971	−	2.76971i
$591$	−29.8727	+	29.8727i	−1.22880	+	1.22880i
$592$		0			0
$593$	−27.7429	−	27.7429i	−1.13926	−	1.13926i	−0.988582	−	0.150683i	$-0.951853\pi$
$593$	−27.7429	−	27.7429i	−1.13926	−	1.13926i	−0.150683	−	0.988582i	$-0.548147\pi$
$594$	8.62693			0.353967
$595$		0			0
$596$	−0.0559690	−	0.0559690i	−0.00229258	−	0.00229258i
$597$			42.0000i			1.71895i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	38.1064	−	38.1064i	1.55569	−	1.55569i
$601$	48.4974			1.97825			0.989126	−	0.147074i	$-0.0469854\pi$
$601$	48.4974			1.97825			0.989126	+	0.147074i	$0.0469854\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	28.4665	+	28.4665i	1.15733	+	1.15733i
$606$		0			0
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$		0			0
$609$		0			0
$610$		−	85.8564i		−	3.47622i
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$			72.2487i			2.91335i
$616$		0			0
$617$	−33.1438	+	33.1438i	−1.33432	+	1.33432i	−0.432855	+	0.901464i	$0.642494\pi$
$617$	−33.1438	+	33.1438i	−1.33432	+	1.33432i	−0.901464	+	0.432855i	$0.857506\pi$
$618$	29.5109	+	29.5109i	1.18710	+	1.18710i
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−57.6410			−2.30564
$626$	−36.8886	+	36.8886i	−1.47437	+	1.47437i
$627$		0			0
$628$		−	0.535898i		−	0.0213847i
$629$		0			0
$630$		0			0
$631$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$631$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$632$	19.1679	+	19.1679i	0.762457	+	0.762457i
$633$	6.00000			0.238479
$634$			25.2295i			1.00199i
$635$	50.3908	+	50.3908i	1.99970	+	1.99970i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	10.4399	−	10.4399i	0.412998	−	0.412998i
$640$	−53.6603			−2.12111
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$	20.1563	+	20.1563i	0.793654	+	0.793654i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	16.5999	+	16.5999i	0.652105	+	0.652105i
$649$		−	16.9282i		−	0.664490i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$	32.0024	−	32.0024i	1.24949	−	1.24949i
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$			2.10512i			0.0819416i
$661$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$661$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$662$		0			0
$663$		0			0
$664$	34.3013			1.33115
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−2.18573	+	2.18573i	−0.0845686	+	0.0845686i
$669$		0			0
$670$		0			0
$671$	−10.8017	−	10.8017i	−0.416996	−	0.416996i
$672$		0			0
$673$			14.0000i			0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$			14.0000i			0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	−7.37772	−	7.37772i	−0.284179	−	0.284179i
$675$		−	61.9808i		−	2.38564i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−36.2620	+	36.2620i	−1.38956	+	1.38956i
$682$		0			0
$683$	4.62132	+	4.62132i	0.176830	+	0.176830i	0.789972	−	0.613142i	$-0.210095\pi$
$683$	4.62132	+	4.62132i	0.176830	+	0.176830i	−0.613142	+	0.789972i	$0.710095\pi$
$684$		0			0
$685$	−59.8564			−2.28700
$686$		0			0
$687$		0			0
$688$		−	17.8564i		−	0.680769i
$689$		0			0
$690$		0			0
$691$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$691$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	58.1863	−	58.1863i	2.20713	−	2.20713i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−5.19199	+	5.19199i	−0.195680	+	0.195680i
$705$	66.4974			2.50444
$706$			3.76537i			0.141712i
$707$		0			0
$708$	−5.03908	+	5.03908i	−0.189380	+	0.189380i
$709$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$709$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$710$	21.5625	+	21.5625i	0.809226	+	0.809226i
$711$	31.1769			1.16923
$712$			47.9090i			1.79546i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	31.5847	−	31.5847i	1.17955	−	1.17955i
$718$	56.9808			2.12650
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−38.9624	+	38.9624i	−1.45204	+	1.45204i
$721$		0			0
$722$	−20.2328	−	20.2328i	−0.752986	−	0.752986i
$723$		0			0
$724$	−2.67949			−0.0995825
$725$		0			0
$726$	−18.0470	−	18.0470i	−0.669788	−	0.669788i
$727$		−	51.9615i		−	1.92715i	−0.267445	−	0.963573i	$-0.586179\pi$
$727$		−	51.9615i		−	1.92715i	0.267445	−	0.963573i	$-0.413821\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$			6.43078i			0.237688i
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$	−8.51906	+	8.51906i	−0.314444	+	0.314444i
$735$	−35.2735	−	35.2735i	−1.30108	−	1.30108i
$736$		0			0
$737$		0			0
$738$		−	45.8038i		−	1.68606i
$739$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$739$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−38.3917	+	38.3917i	−1.40846	+	1.40846i	−0.640488	+	0.767968i	$0.721268\pi$
$743$	−38.3917	+	38.3917i	−1.40846	+	1.40846i	−0.767968	+	0.640488i	$0.778732\pi$
$744$		0			0
$745$			1.21539i			0.0445285i
$746$	−14.7554	+	14.7554i	−0.540235	+	0.540235i
$747$	27.8958	−	27.8958i	1.02065	−	1.02065i
$748$		0			0
$749$		0			0
$750$	74.3538			2.71502
$751$			40.0000i			1.45962i	0.683650	+	0.729810i	$0.260392\pi$
$751$			40.0000i			1.45962i	−0.683650	+	0.729810i	$0.739608\pi$
$752$	−29.4549	−	29.4549i	−1.07411	−	1.07411i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−41.5692			−1.51086			−0.755429	−	0.655230i	$-0.772572\pi$
$757$	−41.5692			−1.51086			−0.755429	+	0.655230i	$0.772572\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	19.2239	+	19.2239i	0.696864	+	0.696864i	0.963733	−	0.266869i	$-0.0859890\pi$
$761$	19.2239	+	19.2239i	0.696864	+	0.696864i	−0.266869	+	0.963733i	$0.585989\pi$
$762$	−31.9465	−	31.9465i	−1.15730	−	1.15730i
$763$		0			0
$764$		0			0
$765$		0			0
$766$			25.5551i			0.923345i
$767$		0			0
$768$	10.9474			0.395032
$769$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$769$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9.14570	−	9.14570i	0.328948	−	0.328948i	−0.523238	−	0.852186i	$-0.675276\pi$
$773$	9.14570	−	9.14570i	0.328948	−	0.328948i	0.852186	+	0.523238i	$0.175276\pi$
$774$	−12.7786	−	12.7786i	−0.459317	−	0.459317i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	5.42563			0.194144
$782$		0			0
$783$		0			0
$784$			31.2487i			1.11603i
$785$	−5.81863	+	5.81863i	−0.207676	+	0.207676i
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	4.62132	+	4.62132i	0.164628	+	0.164628i
$789$		0			0
$790$			64.3923i			2.29098i
$791$		0			0
$792$			8.62693i			0.306545i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	6.49742			0.230295
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−12.7021	−	12.7021i	−0.449088	−	0.449088i
$801$	38.9624	+	38.9624i	1.37667	+	1.37667i
$802$	46.6936			1.64881
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$			55.7654i			1.95940i
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	11.1769			0.390315
$821$	40.5215	−	40.5215i	1.41421	−	1.41421i	0.705552	−	0.708658i	$-0.250699\pi$
$821$	40.5215	−	40.5215i	1.41421	−	1.41421i	0.708658	−	0.705552i	$-0.249301\pi$
$822$	37.9474			1.32357
$823$		−	56.0000i		−	1.95204i	−0.217687	−	0.976019i	$-0.569851\pi$
$823$		−	56.0000i		−	1.95204i	0.217687	−	0.976019i	$-0.430149\pi$
$824$	−29.5109	+	29.5109i	−1.02806	+	1.02806i
$825$	16.1057	−	16.1057i	0.560727	−	0.560727i
$826$		0			0
$827$	35.6913	+	35.6913i	1.24111	+	1.24111i	0.959542	+	0.281566i	$0.0908540\pi$
$827$	35.6913	+	35.6913i	1.24111	+	1.24111i	0.281566	+	0.959542i	$0.409146\pi$
$828$		0			0
$829$			27.7128i			0.962506i	0.876582	+	0.481253i	$0.159818\pi$
$829$			27.7128i			0.962506i	−0.876582	+	0.481253i	$0.840182\pi$
$830$	57.6156	+	57.6156i	1.99987	+	1.99987i
$831$			38.1051i			1.32185i
$832$		0			0
$833$		0			0
$834$	−36.8886	+	36.8886i	−1.27735	+	1.27735i
$835$	47.4641			1.64256
$836$		0			0
$837$		0			0
$838$		0			0
$839$	31.4318	+	31.4318i	1.08515	+	1.08515i	0.996020	+	0.0891249i	$0.0284070\pi$
$839$	31.4318	+	31.4318i	1.08515	+	1.08515i	0.0891249	+	0.996020i	$0.471593\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	−17.0941	−	17.0941i	−0.588752	−	0.588752i
$844$		−	0.928203i		−	0.0319501i
$845$		0			0
$846$	−42.1577			−1.44941
$847$		0			0
$848$		0			0
$849$		−	30.0000i		−	1.02960i
$850$		0			0
$851$		0			0
$852$	−1.61507	−	1.61507i	−0.0553312	−	0.0553312i
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		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
$859$	10.3923			0.354581			0.177290	−	0.984159i	$-0.443267\pi$
$859$	10.3923			0.354581			0.177290	+	0.984159i	$0.443267\pi$
$860$	3.11819	−	3.11819i	0.106329	−	0.106329i
$861$		0			0
$862$		−	60.3013i		−	2.05387i
$863$	33.7144	−	33.7144i	1.14765	−	1.14765i	0.160640	−	0.987013i	$-0.448644\pi$
$863$	33.7144	−	33.7144i	1.14765	−	1.14765i	0.987013	−	0.160640i	$-0.0513559\pi$
$864$	5.53329	−	5.53329i	0.188246	−	0.188246i
$865$		0			0
$866$	−22.1332	−	22.1332i	−0.752116	−	0.752116i
$867$	−29.4449			−1.00000
$868$		0			0
$869$	8.10130	+	8.10130i	0.274818	+	0.274818i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$	42.5953	+	42.5953i	1.43752	+	1.43752i
$879$	−39.9508	−	39.9508i	−1.34751	−	1.34751i
$880$	−20.2487			−0.682584
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	22.3625	+	22.3625i	0.752986	+	0.752986i
$883$		−	51.9615i		−	1.74864i	−0.485346	−	0.874322i	$-0.661306\pi$
$883$		−	51.9615i		−	1.74864i	0.485346	−	0.874322i	$-0.338694\pi$
$884$		0			0
$885$	109.426			3.67830
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$	−80.4723	+	80.4723i	−2.69744	+	2.69744i
$891$	7.01593	+	7.01593i	0.235043	+	0.235043i
$892$		0			0
$893$		0			0
$894$		−	0.770527i		−	0.0257703i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−16.1577			−0.539189
$899$		0			0
$900$	−9.58846			−0.319615
$901$		0			0
$902$	11.9021	−	11.9021i	0.396297	−	0.396297i
$903$		0			0
$904$		0			0
$905$	29.0931	+	29.0931i	0.967088	+	0.967088i
$906$		0			0
$907$		−	17.3205i		−	0.575118i	−0.957763	−	0.287559i	$-0.907156\pi$
$907$		−	17.3205i		−	0.575118i	0.957763	−	0.287559i	$-0.0928437\pi$
$908$	5.60975	+	5.60975i	0.186166	+	0.186166i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	14.4974			0.479795
$914$		0			0
$915$	69.8235	−	69.8235i	2.30829	−	2.30829i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−58.8897			−1.94259			−0.971296	−	0.237872i	$-0.923550\pi$
$919$	−58.8897			−1.94259			−0.971296	+	0.237872i	$0.923550\pi$
$920$		0			0
$921$		0			0
$922$			37.6218i			1.23901i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$			48.0000i			1.57653i
$928$		0			0
$929$	−42.0806	+	42.0806i	−1.38062	+	1.38062i	−0.537103	+	0.843517i	$0.680481\pi$
$929$	−42.0806	+	42.0806i	−1.38062	+	1.38062i	−0.843517	+	0.537103i	$0.819519\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−34.6410			−1.13167			−0.565836	−	0.824518i	$-0.691447\pi$
$937$	−34.6410			−1.13167			−0.565836	+	0.824518i	$0.691447\pi$
$938$		0			0
$939$	−60.0000			−1.95803
$940$		−	10.2872i		−	0.335531i
$941$	14.9643	−	14.9643i	0.487823	−	0.487823i	−0.419796	−	0.907619i	$-0.637898\pi$
$941$	14.9643	−	14.9643i	0.487823	−	0.487823i	0.907619	+	0.419796i	$0.137898\pi$
$942$	3.68886	−	3.68886i	0.120190	−	0.120190i
$943$		0			0
$944$	−48.4699	−	48.4699i	−1.57756	−	1.57756i
$945$		0			0
$946$		−	6.64102i		−	0.215918i
$947$	−14.6995	−	14.6995i	−0.477669	−	0.477669i	0.426717	−	0.904385i	$-0.359670\pi$
$947$	−14.6995	−	14.6995i	−0.477669	−	0.477669i	−0.904385	+	0.426717i	$0.859670\pi$
$948$		−	4.82309i		−	0.156647i
$949$		0			0
$950$		0			0
$951$	−20.5181	+	20.5181i	−0.665345	+	0.665345i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$	−4.88617	−	4.88617i	−0.158030	−	0.158030i
$957$		0			0
$958$	−50.3397			−1.62640
$959$		0			0
$960$	−33.5615	−	33.5615i	−1.08319	−	1.08319i
$961$		−	31.0000i		−	1.00000i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	18.0470	−	18.0470i	0.580054	−	0.580054i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		−	4.17691i		−	0.133975i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−61.8564			−1.97998
$977$	25.0425	−	25.0425i	0.801180	−	0.801180i	−0.182100	−	0.983280i	$-0.558289\pi$
$977$	25.0425	−	25.0425i	0.801180	−	0.801180i	0.983280	+	0.182100i	$0.0582895\pi$
$978$		0			0
$979$			20.2487i			0.647152i
$980$	−5.45684	+	5.45684i	−0.174312	+	0.174312i
$981$		0			0
$982$		0			0
$983$	8.88085	+	8.88085i	0.283255	+	0.283255i	0.834406	−	0.551151i	$-0.185811\pi$
$983$	8.88085	+	8.88085i	0.283255	+	0.283255i	−0.551151	+	0.834406i	$0.685811\pi$
$984$	45.8038			1.46017
$985$		−	100.354i		−	3.19754i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−14.4906	+	14.4906i	−0.460541	+	0.460541i
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−70.5471	−	70.5471i	−2.23649	−	2.23649i
$996$	−4.31550	−	4.31550i	−0.136742	−	0.136742i
$997$	58.0000			1.83688			0.918439	−	0.395562i	$-0.129450\pi$
$997$	58.0000			1.83688			0.918439	+	0.395562i	$0.129450\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.f.d.437.3	yes	8
3.2	odd	2	inner	507.2.f.d.437.2	yes	8
13.2	odd	12		507.2.k.g.188.2		8
13.3	even	3		507.2.k.h.488.2		8
13.4	even	6		507.2.k.g.89.2		8
13.5	odd	4	inner	507.2.f.d.239.2	✓	8
13.6	odd	12		507.2.k.h.80.1		8
13.7	odd	12		507.2.k.h.80.2		8
13.8	odd	4	inner	507.2.f.d.239.3	yes	8
13.9	even	3		507.2.k.g.89.1		8
13.10	even	6		507.2.k.h.488.1		8
13.11	odd	12		507.2.k.g.188.1		8
13.12	even	2	inner	507.2.f.d.437.2	yes	8
39.2	even	12		507.2.k.g.188.1		8
39.5	even	4	inner	507.2.f.d.239.3	yes	8
39.8	even	4	inner	507.2.f.d.239.2	✓	8
39.11	even	12		507.2.k.g.188.2		8
39.17	odd	6		507.2.k.g.89.1		8
39.20	even	12		507.2.k.h.80.1		8
39.23	odd	6		507.2.k.h.488.2		8
39.29	odd	6		507.2.k.h.488.1		8
39.32	even	12		507.2.k.h.80.2		8
39.35	odd	6		507.2.k.g.89.2		8
39.38	odd	2	CM	507.2.f.d.437.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.f.d.239.2	✓	8	13.5	odd	4	inner
507.2.f.d.239.2	✓	8	39.8	even	4	inner
507.2.f.d.239.3	yes	8	13.8	odd	4	inner
507.2.f.d.239.3	yes	8	39.5	even	4	inner
507.2.f.d.437.2	yes	8	3.2	odd	2	inner
507.2.f.d.437.2	yes	8	13.12	even	2	inner
507.2.f.d.437.3	yes	8	1.1	even	1	trivial
507.2.f.d.437.3	yes	8	39.38	odd	2	CM
507.2.k.g.89.1		8	13.9	even	3
507.2.k.g.89.1		8	39.17	odd	6
507.2.k.g.89.2		8	13.4	even	6
507.2.k.g.89.2		8	39.35	odd	6
507.2.k.g.188.1		8	13.11	odd	12
507.2.k.g.188.1		8	39.2	even	12
507.2.k.g.188.2		8	13.2	odd	12
507.2.k.g.188.2		8	39.11	even	12
507.2.k.h.80.1		8	13.6	odd	12
507.2.k.h.80.1		8	39.20	even	12
507.2.k.h.80.2		8	13.7	odd	12
507.2.k.h.80.2		8	39.32	even	12
507.2.k.h.488.1		8	13.10	even	6
507.2.k.h.488.1		8	39.29	odd	6
507.2.k.h.488.2		8	13.3	even	3
507.2.k.h.488.2		8	39.23	odd	6

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)

\(f(q)\)	\(=\)	\(q+(1.06488 - 1.06488i) q^{2} +1.73205 q^{3} -0.267949i q^{4} +(-2.90931 + 2.90931i) q^{5} +(1.84443 - 1.84443i) q^{6} +(1.84443 + 1.84443i) q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+(1.06488 - 1.06488i) q^{2} +1.73205 q^{3} -0.267949i q^{4} +(-2.90931 + 2.90931i) q^{5} +(1.84443 - 1.84443i) q^{6} +(1.84443 + 1.84443i) q^{8} +3.00000 q^{9} +6.19615i q^{10} +(0.779548 + 0.779548i) q^{11} -0.464102i q^{12} +(-5.03908 + 5.03908i) q^{15} +4.46410 q^{16} +(3.19465 - 3.19465i) q^{18} +(0.779548 + 0.779548i) q^{20} +1.66025 q^{22} +(3.19465 + 3.19465i) q^{24} -11.9282i q^{25} +5.19615 q^{27} +10.7321i q^{30} +(1.06488 - 1.06488i) q^{32} +(1.35022 + 1.35022i) q^{33} -0.803848i q^{36} -10.7321 q^{40} +(7.16884 - 7.16884i) q^{41} -4.00000i q^{43} +(0.208879 - 0.208879i) q^{44} +(-8.72794 + 8.72794i) q^{45} +(-6.59817 - 6.59817i) q^{47} +7.73205 q^{48} +7.00000i q^{49} +(-12.7021 - 12.7021i) q^{50} +(5.53329 - 5.53329i) q^{54} -4.53590 q^{55} +(-10.8577 - 10.8577i) q^{59} +(1.35022 + 1.35022i) q^{60} -13.8564 q^{61} +6.66025i q^{64} +2.87564 q^{66} +(3.47998 - 3.47998i) q^{71} +(5.53329 + 5.53329i) q^{72} -20.6603i q^{75} +10.3923 q^{79} +(-12.9875 + 12.9875i) q^{80} +9.00000 q^{81} -15.2679i q^{82} +(9.29861 - 9.29861i) q^{83} +(-4.25953 - 4.25953i) q^{86} +2.87564i q^{88} +(12.9875 + 12.9875i) q^{89} +18.5885i q^{90} -14.0526 q^{94} +(1.84443 - 1.84443i) q^{96} +(7.45418 + 7.45418i) q^{98} +(2.33864 + 2.33864i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} + 8 q^{16} - 56 q^{22} - 72 q^{40} + 48 q^{48} - 64 q^{55} + 120 q^{66} + 72 q^{81} + 40 q^{94}+O(q^{100}) \) 8 * q + 24 * q^9 + 8 * q^16 - 56 * q^22 - 72 * q^40 + 48 * q^48 - 64 * q^55 + 120 * q^66 + 72 * q^81 + 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more