LMFDB - Embedded newform 507.2.f.d.437.2

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:	$8$
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			437.2
Root			$0.500000 + 1.56488i$ of defining polynomial
Character	$\chi$	$=$	507.437
Dual form			507.2.f.d.239.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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	−0.975035	−	0.222050i	$-0.928725\pi$
$2$	−1.06488	+	1.06488i	−0.752986	+	0.752986i	0.222050	+	0.975035i	$0.428725\pi$
$3$	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	$-0.378184\pi$
$5$	2.90931	−	2.90931i	1.30108	−	1.30108i	0.927661	−	0.373423i	$-0.121816\pi$
$6$	−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	0.579751	−	0.814794i	$-0.303150\pi$
$11$	−0.779548	−	0.779548i	−0.235043	−	0.235043i	−0.814794	+	0.579751i	$0.803150\pi$
$12$		−	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	$0.540786\pi$
$41$	−7.16884	+	7.16884i	−1.11959	+	1.11959i	−0.991802	+	0.127783i	$0.959214\pi$
$42$		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	0.999320	−	0.0368772i	$-0.0117410\pi$
$47$	6.59817	+	6.59817i	0.962443	+	0.962443i	−0.0368772	+	0.999320i	$0.511741\pi$
$48$	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	$0.240268\pi$
$59$	10.8577	+	10.8577i	1.41355	+	1.41355i	0.685160	+	0.728392i	$0.259732\pi$
$60$	−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	−0.882782	−	0.469784i	$-0.844332\pi$
$71$	−3.47998	+	3.47998i	−0.412998	+	0.412998i	0.469784	+	0.882782i	$0.344332\pi$
$72$	−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	$0.506644\pi$
$83$	−9.29861	+	9.29861i	−1.02065	+	1.02065i	−0.999782	+	0.0208726i	$0.993356\pi$
$84$		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	$-0.823510\pi$
$89$	−12.9875	−	12.9875i	−1.37667	−	1.37667i	−0.526484	−	0.850185i	$-0.676490\pi$
$90$			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	0.114538	−	0.993419i	$-0.463461\pi$
$137$	−10.2870	−	10.2870i	−0.878881	−	0.878881i	−0.993419	+	0.114538i	$0.963461\pi$
$138$		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	−0.715611	−	0.698499i	$-0.753852\pi$
$149$	−0.208879	+	0.208879i	−0.0171121	+	0.0171121i	0.698499	+	0.715611i	$0.253852\pi$
$150$	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	0.948376	−	0.317148i	$-0.102725\pi$
$167$	8.15727	+	8.15727i	0.631229	+	0.631229i	−0.317148	+	0.948376i	$0.602725\pi$
$168$		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	$-0.414833\pi$
$197$	17.2470	−	17.2470i	1.22880	−	1.22880i	0.964419	−	0.264379i	$-0.0851669\pi$
$198$	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	$-0.309520\pi$
$227$	20.9359	−	20.9359i	1.38956	−	1.38956i	0.826232	−	0.563329i	$-0.190480\pi$
$228$		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	$0.563995\pi$
$239$	−18.2354	+	18.2354i	−1.17955	+	1.17955i	−0.979858	+	0.199693i	$0.936005\pi$
$240$	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	0.937293	−	0.348542i	$-0.113323\pi$
$281$	9.86928	+	9.86928i	0.588752	+	0.588752i	−0.348542	+	0.937293i	$0.613323\pi$
$282$	−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	$0.151846\pi$
$293$	23.0656	+	23.0656i	1.34751	+	1.34751i	0.459149	+	0.888359i	$0.348154\pi$
$294$	−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	−0.291290	−	0.956635i	$-0.594084\pi$
$317$	11.8461	−	11.8461i	0.665345	−	0.665345i	0.956635	+	0.291290i	$0.0940844\pi$
$318$		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	−0.658490	−	0.752590i	$-0.728804\pi$
$353$	1.76798	−	1.76798i	0.0940998	−	0.0940998i	0.752590	+	0.658490i	$0.228804\pi$
$354$	−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	$-0.767619\pi$
$359$	−26.7545	−	26.7545i	−1.41205	−	1.41205i	−0.666905	−	0.745143i	$-0.732381\pi$
$360$	−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	−0.330636	−	0.943758i	$-0.607263\pi$
$383$	11.9990	−	11.9990i	0.613122	−	0.613122i	0.943758	+	0.330636i	$0.107263\pi$
$384$	−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	$-0.968166\pi$
$401$	−21.9243	−	21.9243i	−1.09485	−	1.09485i	−0.0998435	−	0.995003i	$-0.531834\pi$
$402$		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	$0.664767\pi$
$431$	−28.3136	+	28.3136i	−1.36382	+	1.36382i	−0.868993	+	0.494824i	$0.835233\pi$
$432$	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	0.863088	−	0.505054i	$-0.168527\pi$
$449$	7.58660	+	7.58660i	0.358034	+	0.358034i	−0.505054	+	0.863088i	$0.668527\pi$
$450$	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	−0.163769	−	0.986499i	$-0.552365\pi$
$461$	17.6648	−	17.6648i	0.822730	−	0.822730i	0.986499	+	0.163769i	$0.0523652\pi$
$462$		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	$0.0265966\pi$
$479$	23.6363	+	23.6363i	1.07997	+	1.07997i	0.0834585	+	0.996511i	$0.473403\pi$
$480$			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	−0.498530	−	0.866872i	$-0.666127\pi$
$509$	8.31018	−	8.31018i	0.368342	−	0.368342i	0.866872	+	0.498530i	$0.166127\pi$
$510$		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	0.757919	−	0.652349i	$-0.226216\pi$
$557$	2.49155	+	2.49155i	0.105571	+	0.105571i	−0.652349	+	0.757919i	$0.726216\pi$
$558$		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	$-0.420679\pi$
$587$	29.4549	−	29.4549i	1.21573	−	1.21573i	0.969111	−	0.246623i	$-0.0793210\pi$
$588$	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	$0.0481472\pi$
$593$	27.7429	+	27.7429i	1.13926	+	1.13926i	0.150683	+	0.988582i	$0.451853\pi$
$594$	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	$-0.357506\pi$
$617$	33.1438	−	33.1438i	1.33432	−	1.33432i	0.901464	−	0.432855i	$-0.142494\pi$
$618$	−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	0.613142	−	0.789972i	$-0.289905\pi$
$683$	−4.62132	−	4.62132i	−0.176830	−	0.176830i	−0.789972	+	0.613142i	$0.789905\pi$
$684$		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	$-0.278732\pi$
$743$	38.3917	−	38.3917i	1.40846	−	1.40846i	0.767968	−	0.640488i	$-0.221268\pi$
$744$		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	0.266869	−	0.963733i	$-0.414011\pi$
$761$	−19.2239	−	19.2239i	−0.696864	−	0.696864i	−0.963733	+	0.266869i	$0.914011\pi$
$762$	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	−0.852186	−	0.523238i	$-0.824724\pi$
$773$	−9.14570	+	9.14570i	−0.328948	+	0.328948i	0.523238	+	0.852186i	$0.324724\pi$
$774$	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			−

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)

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

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