LMFDB - Embedded newform 507.2.f.d.437.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [507,2,Mod(239,507)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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			437.1
Root			$0.500000 + 2.19293i$ of defining polynomial
Character	$\chi$	$=$	507.437
Dual form			507.2.f.d.239.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.69293 + 1.69293i) q^{2} -1.73205 q^{3} -3.73205i q^{4} +(-1.23931 + 1.23931i) q^{5} +(2.93225 - 2.93225i) q^{6} +(2.93225 + 2.93225i) q^{8} +3.00000 q^{9} -4.19615i q^{10} +(4.62518 + 4.62518i) q^{11} +6.46410i q^{12} +(2.14655 - 2.14655i) q^{15} -2.46410 q^{16} +(-5.07880 + 5.07880i) q^{18} +(4.62518 + 4.62518i) q^{20} -15.6603 q^{22} +(-5.07880 - 5.07880i) q^{24} +1.92820i q^{25} -5.19615 q^{27} +7.26795i q^{30} +(-1.69293 + 1.69293i) q^{32} +(-8.01105 - 8.01105i) q^{33} -11.1962i q^{36} -7.26795 q^{40} +(-5.53242 + 5.53242i) q^{41} -4.00000i q^{43} +(17.2614 - 17.2614i) q^{44} +(-3.71794 + 3.71794i) q^{45} +(-7.10381 - 7.10381i) q^{47} +4.26795 q^{48} +7.00000i q^{49} +(-3.26432 - 3.26432i) q^{50} +(8.79674 - 8.79674i) q^{54} -11.4641 q^{55} +(-0.332073 - 0.332073i) q^{59} +(-8.01105 - 8.01105i) q^{60} +13.8564 q^{61} -10.6603i q^{64} +27.1244 q^{66} +(-11.3969 + 11.3969i) q^{71} +(8.79674 + 8.79674i) q^{72} -3.33975i q^{75} -10.3923 q^{79} +(3.05379 - 3.05379i) q^{80} +9.00000 q^{81} -18.7321i q^{82} +(-8.91829 + 8.91829i) q^{83} +(6.77174 + 6.77174i) q^{86} +27.1244i q^{88} +(-3.05379 - 3.05379i) q^{89} -12.5885i q^{90} +24.0526 q^{94} +(2.93225 - 2.93225i) q^{96} +(-11.8505 - 11.8505i) q^{98} +(13.8755 + 13.8755i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.69293	+	1.69293i	−1.19709	+	1.19709i	−0.222050	+	0.975035i	$0.571275\pi$
$2$	−1.69293	+	1.69293i	−1.19709	+	1.19709i	−0.975035	+	0.222050i	$0.928725\pi$
$3$	−1.73205			−1.00000
$3$	−1.73205			−1.00000
$4$		−	3.73205i		−	1.86603i
$5$	−1.23931	+	1.23931i	−0.554238	+	0.554238i	−0.927661	−	0.373423i	$-0.878184\pi$
$5$	−1.23931	+	1.23931i	−0.554238	+	0.554238i	0.373423	+	0.927661i	$0.378184\pi$
$6$	2.93225	−	2.93225i	1.19709	−	1.19709i
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	2.93225	+	2.93225i	1.03671	+	1.03671i
$9$	3.00000			1.00000
$10$		−	4.19615i		−	1.32694i
$11$	4.62518	+	4.62518i	1.39454	+	1.39454i	0.814794	+	0.579751i	$0.196850\pi$
$11$	4.62518	+	4.62518i	1.39454	+	1.39454i	0.579751	+	0.814794i	$0.303150\pi$
$12$			6.46410i			1.86603i
$13$		0			0
$13$		0			0
$14$		0			0
$15$	2.14655	−	2.14655i	0.554238	−	0.554238i
$16$	−2.46410			−0.616025
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−5.07880	+	5.07880i	−1.19709	+	1.19709i
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$	4.62518	+	4.62518i	1.03422	+	1.03422i
$21$		0			0
$22$	−15.6603			−3.33878
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−5.07880	−	5.07880i	−1.03671	−	1.03671i
$25$			1.92820i			0.385641i
$26$		0			0
$27$	−5.19615			−1.00000
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$			7.26795i			1.32694i
$31$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$31$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$32$	−1.69293	+	1.69293i	−0.299271	+	0.299271i
$33$	−8.01105	−	8.01105i	−1.39454	−	1.39454i
$34$		0			0
$35$		0			0
$36$		−	11.1962i		−	1.86603i
$37$		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
$40$	−7.26795			−1.14916
$41$	−5.53242	+	5.53242i	−0.864019	+	0.864019i	−0.991802	−	0.127783i	$-0.959214\pi$
$41$	−5.53242	+	5.53242i	−0.864019	+	0.864019i	0.127783	+	0.991802i	$0.459214\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	17.2614	−	17.2614i	2.60226	−	2.60226i
$45$	−3.71794	+	3.71794i	−0.554238	+	0.554238i
$46$		0			0
$47$	−7.10381	−	7.10381i	−1.03620	−	1.03620i	−0.999320	−	0.0368772i	$-0.988259\pi$
$47$	−7.10381	−	7.10381i	−1.03620	−	1.03620i	−0.0368772	−	0.999320i	$-0.511741\pi$
$48$	4.26795			0.616025
$49$			7.00000i			1.00000i
$50$	−3.26432	−	3.26432i	−0.461645	−	0.461645i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	8.79674	−	8.79674i	1.19709	−	1.19709i
$55$	−11.4641			−1.54582
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.332073	−	0.332073i	−0.0432322	−	0.0432322i	0.685160	−	0.728392i	$-0.259732\pi$
$59$	−0.332073	−	0.332073i	−0.0432322	−	0.0432322i	−0.728392	+	0.685160i	$0.759732\pi$
$60$	−8.01105	−	8.01105i	−1.03422	−	1.03422i
$61$	13.8564			1.77413			0.887066	−	0.461644i	$-0.152740\pi$
$61$	13.8564			1.77413			0.887066	+	0.461644i	$0.152740\pi$
$62$		0			0
$63$		0			0
$64$		−	10.6603i		−	1.33253i
$65$		0			0
$66$	27.1244			3.33878
$67$		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
$71$	−11.3969	+	11.3969i	−1.35257	+	1.35257i	−0.469784	+	0.882782i	$0.655668\pi$
$71$	−11.3969	+	11.3969i	−1.35257	+	1.35257i	−0.882782	+	0.469784i	$0.844332\pi$
$72$	8.79674	+	8.79674i	1.03671	+	1.03671i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$		−	3.33975i		−	0.385641i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−10.3923			−1.16923			−0.584613	−	0.811312i	$-0.698754\pi$
$79$	−10.3923			−1.16923			−0.584613	+	0.811312i	$0.698754\pi$
$80$	3.05379	−	3.05379i	0.341425	−	0.341425i
$81$	9.00000			1.00000
$82$		−	18.7321i		−	2.06861i
$83$	−8.91829	+	8.91829i	−0.978910	+	0.978910i	−0.999782	−	0.0208726i	$-0.993356\pi$
$83$	−8.91829	+	8.91829i	−0.978910	+	0.978910i	0.0208726	+	0.999782i	$0.493356\pi$
$84$		0			0
$85$		0			0
$86$	6.77174	+	6.77174i	0.730215	+	0.730215i
$87$		0			0
$88$			27.1244i			2.89147i
$89$	−3.05379	−	3.05379i	−0.323702	−	0.323702i	0.526484	−	0.850185i	$-0.323510\pi$
$89$	−3.05379	−	3.05379i	−0.323702	−	0.323702i	−0.850185	+	0.526484i	$0.823510\pi$
$90$		−	12.5885i		−	1.32694i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	24.0526			2.48083
$95$		0			0
$96$	2.93225	−	2.93225i	0.299271	−	0.299271i
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	−11.8505	−	11.8505i	−1.19709	−	1.19709i
$99$	13.8755	+	13.8755i	1.39454	+	1.39454i
$100$	7.19615			0.719615
$101$		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			$\pi$
$108$			19.3923i			1.86603i
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$	19.4080	−	19.4080i	1.85048	−	1.85048i
$111$		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
$118$	1.12436			0.103505
$119$		0			0
$120$	12.5885			1.14916
$121$			31.7846i			2.88951i
$122$	−23.4580	+	23.4580i	−2.12379	+	2.12379i
$123$	9.58244	−	9.58244i	0.864019	−	0.864019i
$124$		0			0
$125$	−8.58622	−	8.58622i	−0.767975	−	0.767975i
$126$		0			0
$127$			17.3205i			1.53695i	0.639882	+	0.768473i	$0.278983\pi$
$127$			17.3205i			1.53695i	−0.639882	+	0.768473i	$0.721017\pi$
$128$	14.6612	+	14.6612i	1.29588	+	1.29588i
$129$			6.92820i			0.609994i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	−29.8976	+	29.8976i	−2.60226	+	2.60226i
$133$		0			0
$134$		0			0
$135$	6.43966	−	6.43966i	0.554238	−	0.554238i
$136$		0			0
$137$	12.9683	+	12.9683i	1.10796	+	1.10796i	0.993419	+	0.114538i	$0.0365388\pi$
$137$	12.9683	+	12.9683i	1.10796	+	1.10796i	0.114538	+	0.993419i	$0.463461\pi$
$138$		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
$141$	12.3042	+	12.3042i	1.03620	+	1.03620i
$142$		−	38.5885i		−	3.23827i
$143$		0			0
$144$	−7.39230			−0.616025
$145$		0			0
$146$		0			0
$147$		−	12.1244i		−	1.00000i
$148$		0			0
$149$	17.2614	−	17.2614i	1.41411	−	1.41411i	0.698499	−	0.715611i	$-0.253852\pi$
$149$	17.2614	−	17.2614i	1.41411	−	1.41411i	0.715611	−	0.698499i	$-0.246148\pi$
$150$	5.65397	+	5.65397i	0.461645	+	0.461645i
$151$		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	$0.474569\pi$
$158$	17.5935	−	17.5935i	1.39966	−	1.39966i
$159$		0			0
$160$		−	4.19615i		−	0.331735i
$161$		0			0
$162$	−15.2364	+	15.2364i	−1.19709	+	1.19709i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$	20.6473	+	20.6473i	1.61228	+	1.61228i
$165$	19.8564			1.54582
$166$		−	30.1962i		−	2.34368i
$167$	−16.3542	−	16.3542i	−1.26552	−	1.26552i	−0.948376	−	0.317148i	$-0.897275\pi$
$167$	−16.3542	−	16.3542i	−1.26552	−	1.26552i	−0.317148	−	0.948376i	$-0.602725\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$	−14.9282			−1.13826
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	−11.3969	−	11.3969i	−0.859075	−	0.859075i
$177$	0.575167	+	0.575167i	0.0432322	+	0.0432322i
$178$	10.3397			0.774997
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	13.8755	+	13.8755i	1.03422	+	1.03422i
$181$		−	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
$188$	−26.5118	+	26.5118i	−1.93357	+	1.93357i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$			18.4641i			1.33253i
$193$		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
$196$	26.1244			1.86603
$197$	9.82553	−	9.82553i	0.700040	−	0.700040i	−0.264379	−	0.964419i	$-0.585167\pi$
$197$	9.82553	−	9.82553i	0.700040	−	0.700040i	0.964419	+	0.264379i	$0.0851669\pi$
$198$	−46.9808			−3.33878
$199$		−	24.2487i		−	1.71895i	−0.511182	−	0.859473i	$-0.670792\pi$
$199$		−	24.2487i		−	1.71895i	0.511182	−	0.859473i	$-0.329208\pi$
$200$	−5.65397	+	5.65397i	−0.399796	+	0.399796i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		−	13.7128i		−	0.957744i
$206$	−27.0869	−	27.0869i	−1.88724	−	1.88724i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−3.46410			−0.238479			−0.119239	−	0.992866i	$-0.538046\pi$
$211$	−3.46410			−0.238479			−0.119239	+	0.992866i	$0.538046\pi$
$212$		0			0
$213$	19.7400	−	19.7400i	1.35257	−	1.35257i
$214$		0			0
$215$	4.95725	+	4.95725i	0.338082	+	0.338082i
$216$	−15.2364	−	15.2364i	−1.03671	−	1.03671i
$217$		0			0
$218$		0			0
$219$		0			0
$220$			42.7846i			2.88454i
$221$		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
$225$			5.78461i			0.385641i
$226$		0			0
$227$	3.96104	−	3.96104i	0.262903	−	0.262903i	−0.563329	−	0.826232i	$-0.690480\pi$
$227$	3.96104	−	3.96104i	0.262903	−	0.262903i	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
$235$	17.6077			1.14860
$236$	−1.23931	+	1.23931i	−0.0806724	+	0.0806724i
$237$	18.0000			1.16923
$238$		0			0
$239$	12.0611	−	12.0611i	0.780165	−	0.780165i	−0.199693	−	0.979858i	$-0.563995\pi$
$239$	12.0611	−	12.0611i	0.780165	−	0.780165i	0.979858	+	0.199693i	$0.0639945\pi$
$240$	−5.28933	+	5.28933i	−0.341425	+	0.341425i
$241$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$241$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$242$	−53.8092	−	53.8092i	−3.45899	−	3.45899i
$243$	−15.5885			−1.00000
$244$		−	51.7128i		−	3.31057i
$245$	−8.67520	−	8.67520i	−0.554238	−	0.554238i
$246$			32.4449i			2.06861i
$247$		0			0
$248$		0			0
$249$	15.4469	−	15.4469i	0.978910	−	0.978910i
$250$	29.0718			1.83866
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	−29.3225	−	29.3225i	−1.83986	−	1.83986i
$255$		0			0
$256$	−28.3205			−1.77003
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$	−11.7290	−	11.7290i	−0.730215	−	0.730215i
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		−	46.9808i		−	2.89147i
$265$		0			0
$266$		0			0
$267$	5.28933	+	5.28933i	0.323702	+	0.323702i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$			21.8038i			1.32694i
$271$		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
$274$	−43.9090			−2.65264
$275$	−8.91829	+	8.91829i	−0.537793	+	0.537793i
$276$		0			0
$277$			22.0000i			1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$			22.0000i			1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	33.8587	−	33.8587i	2.03071	−	2.03071i
$279$		0			0
$280$		0			0
$281$	21.5545	+	21.5545i	1.28583	+	1.28583i	0.937293	+	0.348542i	$0.113323\pi$
$281$	21.5545	+	21.5545i	1.28583	+	1.28583i	0.348542	+	0.937293i	$0.386677\pi$
$282$	−41.6603			−2.48083
$283$			17.3205i			1.02960i	0.857311	+	0.514799i	$0.172133\pi$
$283$			17.3205i			1.02960i	−0.857311	+	0.514799i	$0.827867\pi$
$284$	42.5339	+	42.5339i	2.52392	+	2.52392i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−5.07880	+	5.07880i	−0.299271	+	0.299271i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.34690	+	7.34690i	0.429211	+	0.429211i	0.888359	−	0.459149i	$-0.151846\pi$
$293$	7.34690	+	7.34690i	0.429211	+	0.429211i	−0.459149	+	0.888359i	$0.651846\pi$
$294$	20.5257	+	20.5257i	1.19709	+	1.19709i
$295$	0.823085			0.0479219
$296$		0			0
$297$	−24.0331	−	24.0331i	−1.39454	−	1.39454i
$298$			58.4449i			3.38562i
$299$		0			0
$300$	−12.4641			−0.719615
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−17.1724	+	17.1724i	−0.983291	+	0.983291i
$306$		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	$-0.0653285\pi$
$313$	34.6410			1.95803			0.979013	+	0.203798i	$0.0653285\pi$
$314$	−3.38587	+	3.38587i	−0.191076	+	0.191076i
$315$		0			0
$316$			38.7846i			2.18180i
$317$	−22.2187	+	22.2187i	−1.24792	+	1.24792i	−0.291290	+	0.956635i	$0.594084\pi$
$317$	−22.2187	+	22.2187i	−1.24792	+	1.24792i	−0.956635	+	0.291290i	$0.905916\pi$
$318$		0			0
$319$		0			0
$320$	13.2114	+	13.2114i	0.738540	+	0.738540i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	33.5885i		−	1.86603i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−32.4449			−1.79147
$329$		0			0
$330$	−33.6156	+	33.6156i	−1.85048	+	1.85048i
$331$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$331$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$332$	33.2835	+	33.2835i	1.82667	+	1.82667i
$333$		0			0
$334$	55.3731			3.02988
$335$		0			0
$336$		0			0
$337$			6.92820i			0.377403i	0.982034	+	0.188702i	$0.0604279\pi$
$337$			6.92820i			0.377403i	−0.982034	+	0.188702i	$0.939572\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	11.7290	−	11.7290i	0.632385	−	0.632385i
$345$		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
$352$	−15.6603			−0.834694
$353$	−26.5118	+	26.5118i	−1.41108	+	1.41108i	−0.658490	+	0.752590i	$0.728804\pi$
$353$	−26.5118	+	26.5118i	−1.41108	+	1.41108i	−0.752590	+	0.658490i	$0.771196\pi$
$354$	−1.94744			−0.103505
$355$		−	28.2487i		−	1.49929i
$356$	−11.3969	+	11.3969i	−0.604035	+	0.604035i
$357$		0			0
$358$		0			0
$359$	−1.48241	−	1.48241i	−0.0782385	−	0.0782385i	0.666905	−	0.745143i	$-0.267619\pi$
$359$	−1.48241	−	1.48241i	−0.0782385	−	0.0782385i	−0.745143	+	0.666905i	$0.767619\pi$
$360$	−21.8038			−1.14916
$361$		−	19.0000i		−	1.00000i
$362$	16.9293	+	16.9293i	0.889786	+	0.889786i
$363$		−	55.0526i		−	2.88951i
$364$		0			0
$365$		0			0
$366$	40.6304	−	40.6304i	2.12379	−	2.12379i
$367$	−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
$369$	−16.5973	+	16.5973i	−0.864019	+	0.864019i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	13.8564			0.717458			0.358729	−	0.933442i	$-0.383210\pi$
$373$	13.8564			0.717458			0.358729	+	0.933442i	$0.383210\pi$
$374$		0			0
$375$	14.8718	+	14.8718i	0.767975	+	0.767975i
$376$		−	41.6603i		−	2.14846i
$377$		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
$383$	24.9404	−	24.9404i	1.27439	−	1.27439i	0.330636	−	0.943758i	$-0.392737\pi$
$383$	24.9404	−	24.9404i	1.27439	−	1.27439i	0.943758	−	0.330636i	$-0.107263\pi$
$384$	−25.3940	−	25.3940i	−1.29588	−	1.29588i
$385$		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
$392$	−20.5257	+	20.5257i	−1.03671	+	1.03671i
$393$		0			0
$394$			33.2679i			1.67602i
$395$	12.8793	−	12.8793i	0.648029	−	0.648029i
$396$	51.7842	−	51.7842i	2.60226	−	2.60226i
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	41.0515	+	41.0515i	2.05772	+	2.05772i
$399$		0			0
$400$		−	4.75129i		−	0.237564i
$401$	17.9256	+	17.9256i	0.895160	+	0.895160i	0.995003	−	0.0998435i	$-0.0318342\pi$
$401$	17.9256	+	17.9256i	0.895160	+	0.895160i	−0.0998435	+	0.995003i	$0.531834\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−11.1538	+	11.1538i	−0.554238	+	0.554238i
$406$		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	$0.750000\pi$
$410$	23.2149	+	23.2149i	1.14650	+	1.14650i
$411$	−22.4618	−	22.4618i	−1.10796	−	1.10796i
$412$	59.7128			2.94184
$413$		0			0
$414$		0			0
$415$		−	22.1051i		−	1.08510i
$416$		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	$0.750000\pi$
$422$	5.86450	−	5.86450i	0.285479	−	0.285479i
$423$	−21.3114	−	21.3114i	−1.03620	−	1.03620i
$424$		0			0
$425$		0			0
$426$			66.8372i			3.23827i
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−16.7846			−0.809426
$431$	7.76796	−	7.76796i	0.374169	−	0.374169i	−0.494824	−	0.868993i	$-0.664767\pi$
$431$	7.76796	−	7.76796i	0.374169	−	0.374169i	0.868993	+	0.494824i	$0.164767\pi$
$432$	12.8038			0.616025
$433$			20.7846i			0.998845i	0.866359	+	0.499422i	$0.166454\pi$
$433$			20.7846i			0.998845i	−0.866359	+	0.499422i	$0.833546\pi$
$434$		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	$0.596339\pi$
$440$	−33.6156	−	33.6156i	−1.60256	−	1.60256i
$441$			21.0000i			1.00000i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	7.56922			0.358815
$446$		0			0
$447$	−29.8976	+	29.8976i	−1.41411	+	1.41411i
$448$		0			0
$449$	−28.9904	−	28.9904i	−1.36814	−	1.36814i	−0.863088	−	0.505054i	$-0.831473\pi$
$449$	−28.9904	−	28.9904i	−1.36814	−	1.36814i	−0.505054	−	0.863088i	$-0.668527\pi$
$450$	−9.79296	−	9.79296i	−0.461645	−	0.461645i
$451$	−51.1769			−2.40983
$452$		0			0
$453$		0			0
$454$			13.4115i			0.629435i
$455$		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
$461$	−24.6973	+	24.6973i	−1.15027	+	1.15027i	−0.163769	+	0.986499i	$0.552365\pi$
$461$	−24.6973	+	24.6973i	−1.15027	+	1.15027i	−0.986499	+	0.163769i	$0.947635\pi$
$462$		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
$470$	−29.8087	+	29.8087i	−1.37497	+	1.37497i
$471$	−3.46410			−0.159617
$472$		−	1.94744i		−	0.0896382i
$473$	18.5007	−	18.5007i	0.850664	−	0.850664i
$474$	−30.4728	+	30.4728i	−1.39966	+	1.39966i
$475$		0			0
$476$		0			0
$477$		0			0
$478$			40.8372i			1.86785i
$479$	19.9831	+	19.9831i	0.913053	+	0.913053i	0.996511	−	0.0834585i	$-0.0265966\pi$
$479$	19.9831	+	19.9831i	0.913053	+	0.913053i	−0.0834585	+	0.996511i	$0.526597\pi$
$480$			7.26795i			0.331735i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	118.622			5.39190
$485$		0			0
$486$	26.3902	−	26.3902i	1.19709	−	1.19709i
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	40.6304	+	40.6304i	1.83925	+	1.83925i
$489$		0			0
$490$	29.3731			1.32694
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$	−35.7621	−	35.7621i	−1.61228	−	1.61228i
$493$		0			0
$494$		0			0
$495$	−34.3923			−1.54582
$496$		0			0
$497$		0			0
$498$			52.3013i			2.34368i
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$	−32.0442	+	32.0442i	−1.43306	+	1.43306i
$501$	28.3263	+	28.3263i	1.26552	+	1.26552i
$502$		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
$508$	64.6410			2.86798
$509$	30.8049	−	30.8049i	1.36540	−	1.36540i	0.498530	−	0.866872i	$-0.333873\pi$
$509$	30.8049	−	30.8049i	1.36540	−	1.36540i	0.866872	−	0.498530i	$-0.166127\pi$
$510$		0			0
$511$		0			0
$512$	18.6223	−	18.6223i	0.822996	−	0.822996i
$513$		0			0
$514$		0			0
$515$	−19.8290	−	19.8290i	−0.873771	−	0.873771i
$516$	25.8564			1.13826
$517$		−	65.7128i		−	2.89005i
$518$		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
$528$	19.7400	+	19.7400i	0.859075	+	0.859075i
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−0.996219	−	0.996219i	−0.0432322	−	0.0432322i
$532$		0			0
$533$		0			0
$534$	−17.9090			−0.774997
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−32.3763	+	32.3763i	−1.39454	+	1.39454i
$540$	−24.0331	−	24.0331i	−1.03422	−	1.03422i
$541$		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	$0.295725\pi$
$548$	48.3984	−	48.3984i	2.06748	−	2.06748i
$549$	41.5692			1.77413
$550$		−	30.1962i		−	1.28757i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−37.2445	−	37.2445i	−1.58237	−	1.58237i
$555$		0			0
$556$			74.6410i			3.16548i
$557$	33.2835	+	33.2835i	1.41027	+	1.41027i	0.757919	+	0.652349i	$0.226216\pi$
$557$	33.2835	+	33.2835i	1.41027	+	1.41027i	0.652349	+	0.757919i	$0.273784\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−72.9808			−3.07851
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	45.9197	−	45.9197i	1.93357	−	1.93357i
$565$		0			0
$566$	−29.3225	−	29.3225i	−1.23252	−	1.23252i
$567$		0			0
$568$	−66.8372			−2.80443
$569$		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	$-0.706248\pi$
$571$		−	38.1051i		−	1.59465i	0.603550	−	0.797325i	$-0.293752\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		−	31.9808i		−	1.33253i
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	28.7799	−	28.7799i	1.19709	−	1.19709i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−24.8756			−1.02760
$587$	17.5045	−	17.5045i	0.722488	−	0.722488i	−0.246623	−	0.969111i	$-0.579321\pi$
$587$	17.5045	−	17.5045i	0.722488	−	0.722488i	0.969111	+	0.246623i	$0.0793210\pi$
$588$	−45.2487			−1.86603
$589$		0			0
$590$	−1.39343	+	1.39343i	−0.0573666	+	0.0573666i
$591$	−17.0183	+	17.0183i	−0.700040	+	0.700040i
$592$		0			0
$593$	−20.4042	−	20.4042i	−0.837900	−	0.837900i	0.150683	−	0.988582i	$-0.451853\pi$
$593$	−20.4042	−	20.4042i	−0.837900	−	0.837900i	−0.988582	+	0.150683i	$0.951853\pi$
$594$	81.3731			3.33878
$595$		0			0
$596$	−64.4205	−	64.4205i	−2.63877	−	2.63877i
$597$			42.0000i			1.71895i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	9.79296	−	9.79296i	0.399796	−	0.399796i
$601$	−48.4974			−1.97825			−0.989126	−	0.147074i	$-0.953015\pi$
$601$	−48.4974			−1.97825			−0.989126	+	0.147074i	$0.953015\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−39.3911	−	39.3911i	−1.60148	−	1.60148i
$606$		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
$610$		−	58.1436i		−	2.35417i
$611$		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
$615$			23.7513i			0.957744i
$616$		0			0
$617$	11.6400	−	11.6400i	0.468609	−	0.468609i	−0.432855	−	0.901464i	$-0.642494\pi$
$617$	11.6400	−	11.6400i	0.468609	−	0.468609i	0.901464	+	0.432855i	$0.142494\pi$
$618$	46.9160	+	46.9160i	1.88724	+	1.88724i
$619$		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
$625$	11.6410			0.465641
$626$	−58.6450	+	58.6450i	−2.34392	+	2.34392i
$627$		0			0
$628$		−	7.46410i		−	0.297850i
$629$		0			0
$630$		0			0
$631$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$631$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$632$	−30.4728	−	30.4728i	−1.21214	−	1.21214i
$633$	6.00000			0.238479
$634$		−	75.2295i		−	2.98774i
$635$	−21.4655	−	21.4655i	−0.851834	−	0.851834i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−34.1908	+	34.1908i	−1.35257	+	1.35257i
$640$	−36.3397			−1.43645
$641$		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
$645$	−8.58622	−	8.58622i	−0.338082	−	0.338082i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	26.3902	+	26.3902i	1.03671	+	1.03671i
$649$		−	3.07180i		−	0.120579i
$650$		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
$656$	13.6325	−	13.6325i	0.532258	−	0.532258i
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		−	74.1051i		−	2.88454i
$661$		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
$664$	−52.3013			−2.02968
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−61.0346	+	61.0346i	−2.36150	+	2.36150i
$669$		0			0
$670$		0			0
$671$	64.0884	+	64.0884i	2.47411	+	2.47411i
$672$		0			0
$673$			14.0000i			0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$			14.0000i			0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	−11.7290	−	11.7290i	−0.451784	−	0.451784i
$675$		−	10.0192i		−	0.385641i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−6.86071	+	6.86071i	−0.262903	+	0.262903i
$682$		0			0
$683$	−36.6694	−	36.6694i	−1.40311	−	1.40311i	−0.789972	−	0.613142i	$-0.789905\pi$
$683$	−36.6694	−	36.6694i	−1.40311	−	1.40311i	−0.613142	−	0.789972i	$-0.710095\pi$
$684$		0			0
$685$	−32.1436			−1.22814
$686$		0			0
$687$		0			0
$688$			9.85641i			0.375772i
$689$		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
$695$	24.7863	−	24.7863i	0.940197	−	0.940197i
$696$		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
$704$	49.3056	−	49.3056i	1.85828	−	1.85828i
$705$	−30.4974			−1.14860
$706$		−	89.7654i		−	3.37836i
$707$		0			0
$708$	2.14655	−	2.14655i	0.0806724	−	0.0806724i
$709$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$709$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$710$	47.8232	+	47.8232i	1.79477	+	1.79477i
$711$	−31.1769			−1.16923
$712$		−	17.9090i		−	0.671167i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−20.8904	+	20.8904i	−0.780165	+	0.780165i
$718$	5.01924			0.187316
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	9.16138	−	9.16138i	0.341425	−	0.341425i
$721$		0			0
$722$	32.1657	+	32.1657i	1.19709	+	1.19709i
$723$		0			0
$724$	−37.3205			−1.38701
$725$		0			0
$726$	93.2003	+	93.2003i	3.45899	+	3.45899i
$727$			51.9615i			1.92715i	0.267445	+	0.963573i	$0.413821\pi$
$727$			51.9615i			1.92715i	−0.267445	+	0.963573i	$0.586179\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$			89.5692i			3.31057i
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$	13.5435	−	13.5435i	0.499899	−	0.499899i
$735$	15.0259	+	15.0259i	0.554238	+	0.554238i
$736$		0			0
$737$		0			0
$738$		−	56.1962i		−	2.06861i
$739$		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
$743$	−3.47485	+	3.47485i	−0.127480	+	0.127480i	−0.767968	−	0.640488i	$-0.778732\pi$
$743$	−3.47485	+	3.47485i	−0.127480	+	0.127480i	0.640488	+	0.767968i	$0.278732\pi$
$744$		0			0
$745$			42.7846i			1.56751i
$746$	−23.4580	+	23.4580i	−0.858858	+	0.858858i
$747$	−26.7549	+	26.7549i	−0.978910	+	0.978910i
$748$		0			0
$749$		0			0
$750$	−50.3538			−1.83866
$751$			40.0000i			1.45962i	0.683650	+	0.729810i	$0.260392\pi$
$751$			40.0000i			1.45962i	−0.683650	+	0.729810i	$0.739608\pi$
$752$	17.5045	+	17.5045i	0.638324	+	0.638324i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	41.5692			1.51086			0.755429	−	0.655230i	$-0.227428\pi$
$757$	41.5692			1.51086			0.755429	+	0.655230i	$0.227428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	33.9477	+	33.9477i	1.23060	+	1.23060i	0.963733	+	0.266869i	$0.0859890\pi$
$761$	33.9477	+	33.9477i	1.23060	+	1.23060i	0.266869	+	0.963733i	$0.414011\pi$
$762$	50.7880	+	50.7880i	1.83986	+	1.83986i
$763$		0			0
$764$		0			0
$765$		0			0
$766$			84.4449i			3.05112i
$767$		0			0
$768$	49.0526			1.77003
$769$		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
$773$	38.2408	−	38.2408i	1.37542	−	1.37542i	0.523238	−	0.852186i	$-0.324724\pi$
$773$	38.2408	−	38.2408i	1.37542	−	1.37542i	0.852186	−	0.523238i	$-0.175276\pi$
$774$	20.3152	+	20.3152i	0.730215	+	0.730215i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−105.426			−3.77243
$782$		0			0
$783$		0			0
$784$		−	17.2487i		−	0.616025i
$785$	−2.47863	+	2.47863i	−0.0884660	+	0.0884660i
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	−36.6694	−	36.6694i	−1.30629	−	1.30629i
$789$		0			0
$790$			43.6077i			1.55149i
$791$		0			0
$792$			81.3731i			2.89147i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−90.4974			−3.20760
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−3.26432	−	3.26432i	−0.115411	−	0.115411i
$801$	−9.16138	−	9.16138i	−0.323702	−	0.323702i
$802$	−60.6936			−2.14316
$803$		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			$\pi$
$810$		−	37.7654i		−	1.32694i
$811$		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
$820$	−51.1769			−1.78718
$821$	0.0889787	−	0.0889787i	0.00310538	−	0.00310538i	−0.705552	−	0.708658i	$-0.749301\pi$
$821$	0.0889787	−	0.0889787i	0.00310538	−	0.00310538i	0.708658	+	0.705552i	$0.249301\pi$
$822$	76.0526			2.65264
$823$		−	56.0000i		−	1.95204i	−0.217687	−	0.976019i	$-0.569851\pi$
$823$		−	56.0000i		−	1.95204i	0.217687	−	0.976019i	$-0.430149\pi$
$824$	−46.9160	+	46.9160i	−1.63439	+	1.63439i
$825$	15.4469	−	15.4469i	0.537793	−	0.537793i
$826$		0			0
$827$	19.4969	+	19.4969i	0.677975	+	0.677975i	0.959542	−	0.281566i	$-0.0908540\pi$
$827$	19.4969	+	19.4969i	0.677975	+	0.677975i	−0.281566	+	0.959542i	$0.590854\pi$
$828$		0			0
$829$		−	27.7128i		−	0.962506i	−0.876582	−	0.481253i	$-0.840182\pi$
$829$		−	27.7128i		−	0.962506i	0.876582	−	0.481253i	$-0.159818\pi$
$830$	37.4225	+	37.4225i	1.29895	+	1.29895i
$831$		−	38.1051i		−	1.32185i
$832$		0			0
$833$		0			0
$834$	−58.6450	+	58.6450i	−2.03071	+	2.03071i
$835$	40.5359			1.40280
$836$		0			0
$837$		0			0
$838$		0			0
$839$	26.2687	+	26.2687i	0.906896	+	0.906896i	0.996020	−	0.0891249i	$-0.0284070\pi$
$839$	26.2687	+	26.2687i	0.906896	+	0.906896i	−0.0891249	+	0.996020i	$0.528407\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	−37.3335	−	37.3335i	−1.28583	−	1.28583i
$844$			12.9282i			0.445007i
$845$		0			0
$846$	72.1577			2.48083
$847$		0			0
$848$		0			0
$849$		−	30.0000i		−	1.02960i
$850$		0			0
$851$		0			0
$852$	−73.6708	−	73.6708i	−2.52392	−	2.52392i
$853$		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	$-0.556733\pi$
$859$	−10.3923			−0.354581			−0.177290	+	0.984159i	$0.556733\pi$
$860$	18.5007	−	18.5007i	0.630870	−	0.630870i
$861$		0			0
$862$			26.3013i			0.895825i
$863$	−24.2762	+	24.2762i	−0.826373	+	0.826373i	−0.987013	−	0.160640i	$-0.948644\pi$
$863$	−24.2762	+	24.2762i	−0.826373	+	0.826373i	0.160640	+	0.987013i	$0.448644\pi$
$864$	8.79674	−	8.79674i	0.299271	−	0.299271i
$865$		0			0
$866$	−35.1870	−	35.1870i	−1.19570	−	1.19570i
$867$	29.4449			1.00000
$868$		0			0
$869$	−48.0663	−	48.0663i	−1.63054	−	1.63054i
$870$		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	$0.750000\pi$
$878$	−67.7174	−	67.7174i	−2.28535	−	2.28535i
$879$	−12.7252	−	12.7252i	−0.429211	−	0.429211i
$880$	28.2487			0.952264
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−35.5516	−	35.5516i	−1.19709	−	1.19709i
$883$			51.9615i			1.74864i	0.485346	+	0.874322i	$0.338694\pi$
$883$			51.9615i			1.74864i	−0.485346	+	0.874322i	$0.661306\pi$
$884$		0			0
$885$	−1.42563			−0.0479219
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$	−12.8142	+	12.8142i	−0.429533	+	0.429533i
$891$	41.6266	+	41.6266i	1.39454	+	1.39454i
$892$		0			0
$893$		0			0
$894$		−	101.229i		−	3.38562i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	98.1577			3.27556
$899$		0			0
$900$	21.5885			0.719615
$901$		0			0
$902$	86.6391	−	86.6391i	2.88477	−	2.88477i
$903$		0			0
$904$		0			0
$905$	12.3931	+	12.3931i	0.411962	+	0.411962i
$906$		0			0
$907$			17.3205i			0.575118i	0.957763	+	0.287559i	$0.0928437\pi$
$907$			17.3205i			0.575118i	−0.957763	+	0.287559i	$0.907156\pi$
$908$	−14.7828	−	14.7828i	−0.490584	−	0.490584i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	−82.4974			−2.73027
$914$		0			0
$915$	29.7435	−	29.7435i	0.983291	−	0.983291i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	58.8897			1.94259			0.971296	−	0.237872i	$-0.0764500\pi$
$919$	58.8897			1.94259			0.971296	+	0.237872i	$0.0764500\pi$
$920$		0			0
$921$		0			0
$922$		−	83.6218i		−	2.75394i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$			48.0000i			1.57653i
$928$		0			0
$929$	−9.33934	+	9.33934i	−0.306414	+	0.306414i	−0.843517	−	0.537103i	$-0.819519\pi$
$929$	−9.33934	+	9.33934i	−0.306414	+	0.306414i	0.537103	+	0.843517i	$0.319519\pi$
$930$		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	$-0.308553\pi$
$937$	34.6410			1.13167			0.565836	+	0.824518i	$0.308553\pi$
$938$		0			0
$939$	−60.0000			−1.95803
$940$		−	65.7128i		−	2.14332i
$941$	40.7194	−	40.7194i	1.32741	−	1.32741i	0.419796	−	0.907619i	$-0.362102\pi$
$941$	40.7194	−	40.7194i	1.32741	−	1.32741i	0.907619	−	0.419796i	$-0.137898\pi$
$942$	5.86450	−	5.86450i	0.191076	−	0.191076i
$943$		0			0
$944$	0.818262	+	0.818262i	0.0266322	+	0.0266322i
$945$		0			0
$946$			62.6410i			2.03664i
$947$	40.9625	+	40.9625i	1.33110	+	1.33110i	0.904385	+	0.426717i	$0.140330\pi$
$947$	40.9625	+	40.9625i	1.33110	+	1.33110i	0.426717	+	0.904385i	$0.359670\pi$
$948$		−	67.1769i		−	2.18180i
$949$		0			0
$950$		0			0
$951$	38.4839	−	38.4839i	1.24792	−	1.24792i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$	−45.0125	−	45.0125i	−1.45581	−	1.45581i
$957$		0			0
$958$	−67.6603			−2.18600
$959$		0			0
$960$	−22.8828	−	22.8828i	−0.738540	−	0.738540i
$961$		−	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	$0.750000\pi$
$968$	−93.2003	+	93.2003i	−2.99557	+	2.99557i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$			58.1769i			1.86603i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−34.1436			−1.09291
$977$	36.4263	−	36.4263i	1.16538	−	1.16538i	0.182100	−	0.983280i	$-0.441711\pi$
$977$	36.4263	−	36.4263i	1.16538	−	1.16538i	0.983280	−	0.182100i	$-0.0582895\pi$
$978$		0			0
$979$		−	28.2487i		−	0.902833i
$980$	−32.3763	+	32.3763i	−1.03422	+	1.03422i
$981$		0			0
$982$		0			0
$983$	−43.4411	−	43.4411i	−1.38556	−	1.38556i	−0.834406	−	0.551151i	$-0.814189\pi$
$983$	−43.4411	−	43.4411i	−1.38556	−	1.38556i	−0.551151	−	0.834406i	$-0.685811\pi$
$984$	56.1962			1.79147
$985$			24.3538i			0.775978i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	58.2239	−	58.2239i	1.85048	−	1.85048i
$991$	−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
$995$	30.0518	+	30.0518i	0.952705	+	0.952705i
$996$	−57.6487	−	57.6487i	−1.82667	−	1.82667i
$997$	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.437.1	yes	8
3.2	odd	2	inner	507.2.f.d.437.4	yes	8
13.2	odd	12		507.2.k.h.188.1		8
13.3	even	3		507.2.k.g.488.1		8
13.4	even	6		507.2.k.h.89.1		8
13.5	odd	4	inner	507.2.f.d.239.4	yes	8
13.6	odd	12		507.2.k.g.80.2		8
13.7	odd	12		507.2.k.g.80.1		8
13.8	odd	4	inner	507.2.f.d.239.1	✓	8
13.9	even	3		507.2.k.h.89.2		8
13.10	even	6		507.2.k.g.488.2		8
13.11	odd	12		507.2.k.h.188.2		8
13.12	even	2	inner	507.2.f.d.437.4	yes	8
39.2	even	12		507.2.k.h.188.2		8
39.5	even	4	inner	507.2.f.d.239.1	✓	8
39.8	even	4	inner	507.2.f.d.239.4	yes	8
39.11	even	12		507.2.k.h.188.1		8
39.17	odd	6		507.2.k.h.89.2		8
39.20	even	12		507.2.k.g.80.2		8
39.23	odd	6		507.2.k.g.488.1		8
39.29	odd	6		507.2.k.g.488.2		8
39.32	even	12		507.2.k.g.80.1		8
39.35	odd	6		507.2.k.h.89.1		8
39.38	odd	2	CM	507.2.f.d.437.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.f.d.239.1	✓	8	13.8	odd	4	inner
507.2.f.d.239.1	✓	8	39.5	even	4	inner
507.2.f.d.239.4	yes	8	13.5	odd	4	inner
507.2.f.d.239.4	yes	8	39.8	even	4	inner
507.2.f.d.437.1	yes	8	1.1	even	1	trivial
507.2.f.d.437.1	yes	8	39.38	odd	2	CM
507.2.f.d.437.4	yes	8	3.2	odd	2	inner
507.2.f.d.437.4	yes	8	13.12	even	2	inner
507.2.k.g.80.1		8	13.7	odd	12
507.2.k.g.80.1		8	39.32	even	12
507.2.k.g.80.2		8	13.6	odd	12
507.2.k.g.80.2		8	39.20	even	12
507.2.k.g.488.1		8	13.3	even	3
507.2.k.g.488.1		8	39.23	odd	6
507.2.k.g.488.2		8	13.10	even	6
507.2.k.g.488.2		8	39.29	odd	6
507.2.k.h.89.1		8	13.4	even	6
507.2.k.h.89.1		8	39.35	odd	6
507.2.k.h.89.2		8	13.9	even	3
507.2.k.h.89.2		8	39.17	odd	6
507.2.k.h.188.1		8	13.2	odd	12
507.2.k.h.188.1		8	39.11	even	12
507.2.k.h.188.2		8	13.11	odd	12
507.2.k.h.188.2		8	39.2	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-1.69293 + 1.69293i) q^{2} -1.73205 q^{3} -3.73205i q^{4} +(-1.23931 + 1.23931i) q^{5} +(2.93225 - 2.93225i) q^{6} +(2.93225 + 2.93225i) q^{8} +3.00000 q^{9} -4.19615i q^{10} +(4.62518 + 4.62518i) q^{11} +6.46410i q^{12} +(2.14655 - 2.14655i) q^{15} -2.46410 q^{16} +(-5.07880 + 5.07880i) q^{18} +(4.62518 + 4.62518i) q^{20} -15.6603 q^{22} +(-5.07880 - 5.07880i) q^{24} +1.92820i q^{25} -5.19615 q^{27} +7.26795i q^{30} +(-1.69293 + 1.69293i) q^{32} +(-8.01105 - 8.01105i) q^{33} -11.1962i q^{36} -7.26795 q^{40} +(-5.53242 + 5.53242i) q^{41} -4.00000i q^{43} +(17.2614 - 17.2614i) q^{44} +(-3.71794 + 3.71794i) q^{45} +(-7.10381 - 7.10381i) q^{47} +4.26795 q^{48} +7.00000i q^{49} +(-3.26432 - 3.26432i) q^{50} +(8.79674 - 8.79674i) q^{54} -11.4641 q^{55} +(-0.332073 - 0.332073i) q^{59} +(-8.01105 - 8.01105i) q^{60} +13.8564 q^{61} -10.6603i q^{64} +27.1244 q^{66} +(-11.3969 + 11.3969i) q^{71} +(8.79674 + 8.79674i) q^{72} -3.33975i q^{75} -10.3923 q^{79} +(3.05379 - 3.05379i) q^{80} +9.00000 q^{81} -18.7321i q^{82} +(-8.91829 + 8.91829i) q^{83} +(6.77174 + 6.77174i) q^{86} +27.1244i q^{88} +(-3.05379 - 3.05379i) q^{89} -12.5885i q^{90} +24.0526 q^{94} +(2.93225 - 2.93225i) q^{96} +(-11.8505 - 11.8505i) q^{98} +(13.8755 + 13.8755i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9} + 8 q^{16} - 56 q^{22} - 72 q^{40} + 48 q^{48} - 64 q^{55} + 120 q^{66} + 72 q^{81} + 40 q^{94}+O(q^{100}) \) 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

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