LMFDB - Embedded newform 720.3.bh.b.433.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,3,Mod(433,720)] 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("720.433"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 3])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	720.bh (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-6,0,-16] 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:	$19.6185790339$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			433.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	720.433
Dual form			720.3.bh.b.577.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.00000 - 4.00000i) q^{5} +(-8.00000 + 8.00000i) q^{7} -4.00000 q^{11} +(-3.00000 - 3.00000i) q^{13} +(19.0000 - 19.0000i) q^{17} -8.00000i q^{19} +(20.0000 + 20.0000i) q^{23} +(-7.00000 + 24.0000i) q^{25} +38.0000i q^{29} +44.0000 q^{31} +(56.0000 + 8.00000i) q^{35} +(-3.00000 + 3.00000i) q^{37} +70.0000 q^{41} +(-36.0000 - 36.0000i) q^{43} -79.0000i q^{49} +(17.0000 + 17.0000i) q^{53} +(12.0000 + 16.0000i) q^{55} -92.0000i q^{59} +72.0000 q^{61} +(-3.00000 + 21.0000i) q^{65} +(-44.0000 + 44.0000i) q^{67} +88.0000 q^{71} +(55.0000 + 55.0000i) q^{73} +(32.0000 - 32.0000i) q^{77} +12.0000i q^{79} +(24.0000 + 24.0000i) q^{83} +(-133.000 - 19.0000i) q^{85} -26.0000i q^{89} +48.0000 q^{91} +(-32.0000 + 24.0000i) q^{95} +(-57.0000 + 57.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} - 16 q^{7} - 8 q^{11} - 6 q^{13} + 38 q^{17} + 40 q^{23} - 14 q^{25} + 88 q^{31} + 112 q^{35} - 6 q^{37} + 140 q^{41} - 72 q^{43} + 34 q^{53} + 24 q^{55} + 144 q^{61} - 6 q^{65} - 88 q^{67}+ \cdots - 114 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 - 16 * q^7 - 8 * q^11 - 6 * q^13 + 38 * q^17 + 40 * q^23 - 14 * q^25 + 88 * q^31 + 112 * q^35 - 6 * q^37 + 140 * q^41 - 72 * q^43 + 34 * q^53 + 24 * q^55 + 144 * q^61 - 6 * q^65 - 88 * q^67 + 176 * q^71 + 110 * q^73 + 64 * q^77 + 48 * q^83 - 266 * q^85 + 96 * q^91 - 64 * q^95 - 114 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$5$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$6$		0			0
$7$	−8.00000	+	8.00000i	−1.14286	+	1.14286i	−0.154932	+	0.987925i	$0.549516\pi$
$7$	−8.00000	+	8.00000i	−1.14286	+	1.14286i	−0.987925	+	0.154932i	$0.950484\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.00000			−0.363636			−0.181818	−	0.983332i	$-0.558198\pi$
$11$	−4.00000			−0.363636			−0.181818	+	0.983332i	$0.558198\pi$
$12$		0			0
$13$	−3.00000	−	3.00000i	−0.230769	−	0.230769i	0.582245	−	0.813014i	$-0.302175\pi$
$13$	−3.00000	−	3.00000i	−0.230769	−	0.230769i	−0.813014	+	0.582245i	$0.802175\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	19.0000	−	19.0000i	1.11765	−	1.11765i	0.125561	−	0.992086i	$-0.459927\pi$
$17$	19.0000	−	19.0000i	1.11765	−	1.11765i	0.992086	−	0.125561i	$-0.0400731\pi$
$18$		0			0
$19$		−	8.00000i		−	0.421053i	−0.977588	−	0.210526i	$-0.932482\pi$
$19$		−	8.00000i		−	0.421053i	0.977588	−	0.210526i	$-0.0675178\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	20.0000	+	20.0000i	0.869565	+	0.869565i	0.992424	−	0.122859i	$-0.0392063\pi$
$23$	20.0000	+	20.0000i	0.869565	+	0.869565i	−0.122859	+	0.992424i	$0.539206\pi$
$24$		0			0
$25$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			38.0000i			1.31034i	0.755479	+	0.655172i	$0.227404\pi$
$29$			38.0000i			1.31034i	−0.755479	+	0.655172i	$0.772596\pi$
$30$		0			0
$31$	44.0000			1.41935			0.709677	−	0.704527i	$-0.248841\pi$
$31$	44.0000			1.41935			0.709677	+	0.704527i	$0.248841\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	56.0000	+	8.00000i	1.60000	+	0.228571i
$36$		0			0
$37$	−3.00000	+	3.00000i	−0.0810811	+	0.0810811i	−0.746484	−	0.665403i	$-0.768260\pi$
$37$	−3.00000	+	3.00000i	−0.0810811	+	0.0810811i	0.665403	+	0.746484i	$0.268260\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	70.0000			1.70732			0.853659	−	0.520833i	$-0.174379\pi$
$41$	70.0000			1.70732			0.853659	+	0.520833i	$0.174379\pi$
$42$		0			0
$43$	−36.0000	−	36.0000i	−0.837209	−	0.837209i	0.151281	−	0.988491i	$-0.451660\pi$
$43$	−36.0000	−	36.0000i	−0.837209	−	0.837209i	−0.988491	+	0.151281i	$0.951660\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$		−	79.0000i		−	1.61224i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	17.0000	+	17.0000i	0.320755	+	0.320755i	0.849057	−	0.528302i	$-0.177171\pi$
$53$	17.0000	+	17.0000i	0.320755	+	0.320755i	−0.528302	+	0.849057i	$0.677171\pi$
$54$		0			0
$55$	12.0000	+	16.0000i	0.218182	+	0.290909i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	92.0000i		−	1.55932i	−0.626202	−	0.779661i	$-0.715391\pi$
$59$		−	92.0000i		−	1.55932i	0.626202	−	0.779661i	$-0.284609\pi$
$60$		0			0
$61$	72.0000			1.18033			0.590164	−	0.807283i	$-0.299063\pi$
$61$	72.0000			1.18033			0.590164	+	0.807283i	$0.299063\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.00000	+	21.0000i	−0.0461538	+	0.323077i
$66$		0			0
$67$	−44.0000	+	44.0000i	−0.656716	+	0.656716i	−0.954602	−	0.297885i	$-0.903719\pi$
$67$	−44.0000	+	44.0000i	−0.656716	+	0.656716i	0.297885	+	0.954602i	$0.403719\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	88.0000			1.23944			0.619718	−	0.784824i	$-0.287247\pi$
$71$	88.0000			1.23944			0.619718	+	0.784824i	$0.287247\pi$
$72$		0			0
$73$	55.0000	+	55.0000i	0.753425	+	0.753425i	0.975117	−	0.221692i	$-0.0711580\pi$
$73$	55.0000	+	55.0000i	0.753425	+	0.753425i	−0.221692	+	0.975117i	$0.571158\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	32.0000	−	32.0000i	0.415584	−	0.415584i
$78$		0			0
$79$			12.0000i			0.151899i	0.997112	+	0.0759494i	$0.0241987\pi$
$79$			12.0000i			0.151899i	−0.997112	+	0.0759494i	$0.975801\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	24.0000	+	24.0000i	0.289157	+	0.289157i	0.836747	−	0.547590i	$-0.184455\pi$
$83$	24.0000	+	24.0000i	0.289157	+	0.289157i	−0.547590	+	0.836747i	$0.684455\pi$
$84$		0			0
$85$	−133.000	−	19.0000i	−1.56471	−	0.223529i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	26.0000i		−	0.292135i	−0.989275	−	0.146067i	$-0.953338\pi$
$89$		−	26.0000i		−	0.292135i	0.989275	−	0.146067i	$-0.0466616\pi$
$90$		0			0
$91$	48.0000			0.527473
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−32.0000	+	24.0000i	−0.336842	+	0.252632i
$96$		0			0
$97$	−57.0000	+	57.0000i	−0.587629	+	0.587629i	−0.936989	−	0.349360i	$-0.886399\pi$
$97$	−57.0000	+	57.0000i	−0.587629	+	0.587629i	0.349360	+	0.936989i	$0.386399\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	56.0000			0.554455			0.277228	−	0.960804i	$-0.410584\pi$
$101$	56.0000			0.554455			0.277228	+	0.960804i	$0.410584\pi$
$102$		0			0
$103$	−4.00000	−	4.00000i	−0.0388350	−	0.0388350i	0.687423	−	0.726258i	$-0.258742\pi$
$103$	−4.00000	−	4.00000i	−0.0388350	−	0.0388350i	−0.726258	+	0.687423i	$0.758742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	68.0000	−	68.0000i	0.635514	−	0.635514i	−0.313932	−	0.949446i	$-0.601646\pi$
$107$	68.0000	−	68.0000i	0.635514	−	0.635514i	0.949446	+	0.313932i	$0.101646\pi$
$108$		0			0
$109$			46.0000i			0.422018i	0.977484	+	0.211009i	$0.0676750\pi$
$109$			46.0000i			0.422018i	−0.977484	+	0.211009i	$0.932325\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	53.0000	+	53.0000i	0.469027	+	0.469027i	0.901599	−	0.432573i	$-0.142394\pi$
$113$	53.0000	+	53.0000i	0.469027	+	0.469027i	−0.432573	+	0.901599i	$0.642394\pi$
$114$		0			0
$115$	20.0000	−	140.000i	0.173913	−	1.21739i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			304.000i			2.55462i
$120$		0			0
$121$	−105.000			−0.867769
$122$		0			0
$123$		0			0
$124$		0			0
$125$	117.000	−	44.0000i	0.936000	−	0.352000i
$126$		0			0
$127$	−68.0000	+	68.0000i	−0.535433	+	0.535433i	−0.922184	−	0.386751i	$-0.873597\pi$
$127$	−68.0000	+	68.0000i	−0.535433	+	0.535433i	0.386751	+	0.922184i	$0.373597\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	44.0000			0.335878			0.167939	−	0.985797i	$-0.446289\pi$
$131$	44.0000			0.335878			0.167939	+	0.985797i	$0.446289\pi$
$132$		0			0
$133$	64.0000	+	64.0000i	0.481203	+	0.481203i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−69.0000	+	69.0000i	−0.503650	+	0.503650i	−0.912570	−	0.408920i	$-0.865905\pi$
$137$	−69.0000	+	69.0000i	−0.503650	+	0.503650i	0.408920	+	0.912570i	$0.365905\pi$
$138$		0			0
$139$			80.0000i			0.575540i	0.957700	+	0.287770i	$0.0929138\pi$
$139$			80.0000i			0.575540i	−0.957700	+	0.287770i	$0.907086\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.0000	+	12.0000i	0.0839161	+	0.0839161i
$144$		0			0
$145$	152.000	−	114.000i	1.04828	−	0.786207i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	168.000i		−	1.12752i	−0.825940	−	0.563758i	$-0.809355\pi$
$149$		−	168.000i		−	1.12752i	0.825940	−	0.563758i	$-0.190645\pi$
$150$		0			0
$151$	−4.00000			−0.0264901			−0.0132450	−	0.999912i	$-0.504216\pi$
$151$	−4.00000			−0.0264901			−0.0132450	+	0.999912i	$0.504216\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−132.000	−	176.000i	−0.851613	−	1.13548i
$156$		0			0
$157$	99.0000	−	99.0000i	0.630573	−	0.630573i	−0.317639	−	0.948212i	$-0.602890\pi$
$157$	99.0000	−	99.0000i	0.630573	−	0.630573i	0.948212	+	0.317639i	$0.102890\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−320.000			−1.98758
$162$		0			0
$163$	160.000	+	160.000i	0.981595	+	0.981595i	0.999834	−	0.0182386i	$-0.00580584\pi$
$163$	160.000	+	160.000i	0.981595	+	0.981595i	−0.0182386	+	0.999834i	$0.505806\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−56.0000	+	56.0000i	−0.335329	+	0.335329i	−0.854606	−	0.519277i	$-0.826201\pi$
$167$	−56.0000	+	56.0000i	−0.335329	+	0.335329i	0.519277	+	0.854606i	$0.326201\pi$
$168$		0			0
$169$		−	151.000i		−	0.893491i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−41.0000	−	41.0000i	−0.236994	−	0.236994i	0.578610	−	0.815604i	$-0.303595\pi$
$173$	−41.0000	−	41.0000i	−0.236994	−	0.236994i	−0.815604	+	0.578610i	$0.803595\pi$
$174$		0			0
$175$	−136.000	−	248.000i	−0.777143	−	1.41714i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	172.000i		−	0.960894i	−0.877024	−	0.480447i	$-0.840474\pi$
$179$		−	172.000i		−	0.960894i	0.877024	−	0.480447i	$-0.159526\pi$
$180$		0			0
$181$	62.0000			0.342541			0.171271	−	0.985224i	$-0.445213\pi$
$181$	62.0000			0.342541			0.171271	+	0.985224i	$0.445213\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	21.0000	+	3.00000i	0.113514	+	0.0162162i
$186$		0			0
$187$	−76.0000	+	76.0000i	−0.406417	+	0.406417i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−248.000			−1.29843			−0.649215	−	0.760605i	$-0.724902\pi$
$191$	−248.000			−1.29843			−0.649215	+	0.760605i	$0.724902\pi$
$192$		0			0
$193$	135.000	+	135.000i	0.699482	+	0.699482i	0.964299	−	0.264817i	$-0.0853115\pi$
$193$	135.000	+	135.000i	0.699482	+	0.699482i	−0.264817	+	0.964299i	$0.585312\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−153.000	+	153.000i	−0.776650	+	0.776650i	−0.979260	−	0.202610i	$-0.935058\pi$
$197$	−153.000	+	153.000i	−0.776650	+	0.776650i	0.202610	+	0.979260i	$0.435058\pi$
$198$		0			0
$199$		−	252.000i		−	1.26633i	−0.774016	−	0.633166i	$-0.781755\pi$
$199$		−	252.000i		−	1.26633i	0.774016	−	0.633166i	$-0.218245\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−304.000	−	304.000i	−1.49754	−	1.49754i
$204$		0			0
$205$	−210.000	−	280.000i	−1.02439	−	1.36585i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			32.0000i			0.153110i
$210$		0			0
$211$	64.0000			0.303318			0.151659	−	0.988433i	$-0.451539\pi$
$211$	64.0000			0.303318			0.151659	+	0.988433i	$0.451539\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−36.0000	+	252.000i	−0.167442	+	1.17209i
$216$		0			0
$217$	−352.000	+	352.000i	−1.62212	+	1.62212i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−114.000			−0.515837
$222$		0			0
$223$	228.000	+	228.000i	1.02242	+	1.02242i	0.999743	+	0.0226787i	$0.00721948\pi$
$223$	228.000	+	228.000i	1.02242	+	1.02242i	0.0226787	+	0.999743i	$0.492781\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	100.000	−	100.000i	0.440529	−	0.440529i	−0.451661	−	0.892190i	$-0.649168\pi$
$227$	100.000	−	100.000i	0.440529	−	0.440529i	0.892190	+	0.451661i	$0.149168\pi$
$228$		0			0
$229$			312.000i			1.36245i	0.732076	+	0.681223i	$0.238551\pi$
$229$			312.000i			1.36245i	−0.732076	+	0.681223i	$0.761449\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−93.0000	−	93.0000i	−0.399142	−	0.399142i	0.478789	−	0.877930i	$-0.341076\pi$
$233$	−93.0000	−	93.0000i	−0.399142	−	0.399142i	−0.877930	+	0.478789i	$0.841076\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	96.0000i		−	0.401674i	−0.979625	−	0.200837i	$-0.935634\pi$
$239$		−	96.0000i		−	0.401674i	0.979625	−	0.200837i	$-0.0643661\pi$
$240$		0			0
$241$	160.000			0.663900			0.331950	−	0.943297i	$-0.392293\pi$
$241$	160.000			0.663900			0.331950	+	0.943297i	$0.392293\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−316.000	+	237.000i	−1.28980	+	0.967347i
$246$		0			0
$247$	−24.0000	+	24.0000i	−0.0971660	+	0.0971660i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.0478088			−0.0239044	−	0.999714i	$-0.507610\pi$
$251$	−12.0000			−0.0478088			−0.0239044	+	0.999714i	$0.507610\pi$
$252$		0			0
$253$	−80.0000	−	80.0000i	−0.316206	−	0.316206i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	101.000	−	101.000i	0.392996	−	0.392996i	−0.482758	−	0.875754i	$-0.660365\pi$
$257$	101.000	−	101.000i	0.392996	−	0.392996i	0.875754	+	0.482758i	$0.160365\pi$
$258$		0			0
$259$		−	48.0000i		−	0.185328i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	208.000	+	208.000i	0.790875	+	0.790875i	0.981636	−	0.190762i	$-0.0610958\pi$
$263$	208.000	+	208.000i	0.790875	+	0.790875i	−0.190762	+	0.981636i	$0.561096\pi$
$264$		0			0
$265$	17.0000	−	119.000i	0.0641509	−	0.449057i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			296.000i			1.10037i	0.835042	+	0.550186i	$0.185443\pi$
$269$			296.000i			1.10037i	−0.835042	+	0.550186i	$0.814557\pi$
$270$		0			0
$271$	−108.000			−0.398524			−0.199262	−	0.979946i	$-0.563854\pi$
$271$	−108.000			−0.398524			−0.199262	+	0.979946i	$0.563854\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	28.0000	−	96.0000i	0.101818	−	0.349091i
$276$		0			0
$277$	−243.000	+	243.000i	−0.877256	+	0.877256i	−0.993250	−	0.115994i	$-0.962995\pi$
$277$	−243.000	+	243.000i	−0.877256	+	0.877256i	0.115994	+	0.993250i	$0.462995\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	378.000			1.34520			0.672598	−	0.740008i	$-0.265178\pi$
$281$	378.000			1.34520			0.672598	+	0.740008i	$0.265178\pi$
$282$		0			0
$283$	−92.0000	−	92.0000i	−0.325088	−	0.325088i	0.525627	−	0.850715i	$-0.323831\pi$
$283$	−92.0000	−	92.0000i	−0.325088	−	0.325088i	−0.850715	+	0.525627i	$0.823831\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−560.000	+	560.000i	−1.95122	+	1.95122i
$288$		0			0
$289$		−	433.000i		−	1.49827i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−279.000	−	279.000i	−0.952218	−	0.952218i	0.0466910	−	0.998909i	$-0.485132\pi$
$293$	−279.000	−	279.000i	−0.952218	−	0.952218i	−0.998909	+	0.0466910i	$0.985132\pi$
$294$		0			0
$295$	−368.000	+	276.000i	−1.24746	+	0.935593i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	120.000i		−	0.401338i
$300$		0			0
$301$	576.000			1.91362
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−216.000	−	288.000i	−0.708197	−	0.944262i
$306$		0			0
$307$	−216.000	+	216.000i	−0.703583	+	0.703583i	−0.965178	−	0.261595i	$-0.915752\pi$
$307$	−216.000	+	216.000i	−0.703583	+	0.703583i	0.261595	+	0.965178i	$0.415752\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−272.000			−0.874598			−0.437299	−	0.899316i	$-0.644065\pi$
$311$	−272.000			−0.874598			−0.437299	+	0.899316i	$0.644065\pi$
$312$		0			0
$313$	15.0000	+	15.0000i	0.0479233	+	0.0479233i	0.730662	−	0.682739i	$-0.239211\pi$
$313$	15.0000	+	15.0000i	0.0479233	+	0.0479233i	−0.682739	+	0.730662i	$0.739211\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	87.0000	−	87.0000i	0.274448	−	0.274448i	−0.556440	−	0.830888i	$-0.687833\pi$
$317$	87.0000	−	87.0000i	0.274448	−	0.274448i	0.830888	+	0.556440i	$0.187833\pi$
$318$		0			0
$319$		−	152.000i		−	0.476489i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−152.000	−	152.000i	−0.470588	−	0.470588i
$324$		0			0
$325$	93.0000	−	51.0000i	0.286154	−	0.156923i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	584.000			1.76435			0.882175	−	0.470921i	$-0.156078\pi$
$331$	584.000			1.76435			0.882175	+	0.470921i	$0.156078\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	308.000	+	44.0000i	0.919403	+	0.131343i
$336$		0			0
$337$	−129.000	+	129.000i	−0.382789	+	0.382789i	−0.872106	−	0.489317i	$-0.837246\pi$
$337$	−129.000	+	129.000i	−0.382789	+	0.382789i	0.489317	+	0.872106i	$0.337246\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−176.000			−0.516129
$342$		0			0
$343$	240.000	+	240.000i	0.699708	+	0.699708i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	260.000	−	260.000i	0.749280	−	0.749280i	−0.225064	−	0.974344i	$-0.572259\pi$
$347$	260.000	−	260.000i	0.749280	−	0.749280i	0.974344	+	0.225064i	$0.0722592\pi$
$348$		0			0
$349$		−	136.000i		−	0.389685i	−0.980835	−	0.194842i	$-0.937580\pi$
$349$		−	136.000i		−	0.389685i	0.980835	−	0.194842i	$-0.0624195\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−75.0000	−	75.0000i	−0.212465	−	0.212465i	0.592849	−	0.805314i	$-0.298003\pi$
$353$	−75.0000	−	75.0000i	−0.212465	−	0.212465i	−0.805314	+	0.592849i	$0.798003\pi$
$354$		0			0
$355$	−264.000	−	352.000i	−0.743662	−	0.991549i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			32.0000i			0.0891365i	0.999006	+	0.0445682i	$0.0141912\pi$
$359$			32.0000i			0.0891365i	−0.999006	+	0.0445682i	$0.985809\pi$
$360$		0			0
$361$	297.000			0.822715
$362$		0			0
$363$		0			0
$364$		0			0
$365$	55.0000	−	385.000i	0.150685	−	1.05479i
$366$		0			0
$367$	16.0000	−	16.0000i	0.0435967	−	0.0435967i	−0.684972	−	0.728569i	$-0.740186\pi$
$367$	16.0000	−	16.0000i	0.0435967	−	0.0435967i	0.728569	+	0.684972i	$0.240186\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−272.000			−0.733154
$372$		0			0
$373$	−251.000	−	251.000i	−0.672922	−	0.672922i	0.285466	−	0.958389i	$-0.407851\pi$
$373$	−251.000	−	251.000i	−0.672922	−	0.672922i	−0.958389	+	0.285466i	$0.907851\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	114.000	−	114.000i	0.302387	−	0.302387i
$378$		0			0
$379$		−	560.000i		−	1.47757i	−0.673940	−	0.738786i	$-0.735399\pi$
$379$		−	560.000i		−	1.47757i	0.673940	−	0.738786i	$-0.264601\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	300.000	+	300.000i	0.783290	+	0.783290i	0.980384	−	0.197095i	$-0.0631506\pi$
$383$	300.000	+	300.000i	0.783290	+	0.783290i	−0.197095	+	0.980384i	$0.563151\pi$
$384$		0			0
$385$	−224.000	−	32.0000i	−0.581818	−	0.0831169i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			24.0000i			0.0616967i	0.999524	+	0.0308483i	$0.00982089\pi$
$389$			24.0000i			0.0616967i	−0.999524	+	0.0308483i	$0.990179\pi$
$390$		0			0
$391$	760.000			1.94373
$392$		0			0
$393$		0			0
$394$		0			0
$395$	48.0000	−	36.0000i	0.121519	−	0.0911392i
$396$		0			0
$397$	−299.000	+	299.000i	−0.753149	+	0.753149i	−0.975066	−	0.221917i	$-0.928769\pi$
$397$	−299.000	+	299.000i	−0.753149	+	0.753149i	0.221917	+	0.975066i	$0.428769\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−144.000			−0.359102			−0.179551	−	0.983749i	$-0.557465\pi$
$401$	−144.000			−0.359102			−0.179551	+	0.983749i	$0.557465\pi$
$402$		0			0
$403$	−132.000	−	132.000i	−0.327543	−	0.327543i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.0000	−	12.0000i	0.0294840	−	0.0294840i
$408$		0			0
$409$			354.000i			0.865526i	0.901508	+	0.432763i	$0.142461\pi$
$409$			354.000i			0.865526i	−0.901508	+	0.432763i	$0.857539\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	736.000	+	736.000i	1.78208	+	1.78208i
$414$		0			0
$415$	24.0000	−	168.000i	0.0578313	−	0.404819i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	468.000i		−	1.11695i	−0.829523	−	0.558473i	$-0.811388\pi$
$419$		−	468.000i		−	1.11695i	0.829523	−	0.558473i	$-0.188612\pi$
$420$		0			0
$421$	104.000			0.247031			0.123515	−	0.992343i	$-0.460583\pi$
$421$	104.000			0.247031			0.123515	+	0.992343i	$0.460583\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	323.000	+	589.000i	0.760000	+	1.38588i
$426$		0			0
$427$	−576.000	+	576.000i	−1.34895	+	1.34895i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	680.000			1.57773			0.788863	−	0.614569i	$-0.210670\pi$
$431$	680.000			1.57773			0.788863	+	0.614569i	$0.210670\pi$
$432$		0			0
$433$	41.0000	+	41.0000i	0.0946882	+	0.0946882i	0.752864	−	0.658176i	$-0.228672\pi$
$433$	41.0000	+	41.0000i	0.0946882	+	0.0946882i	−0.658176	+	0.752864i	$0.728672\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	160.000	−	160.000i	0.366133	−	0.366133i
$438$		0			0
$439$		−	364.000i		−	0.829157i	−0.910014	−	0.414579i	$-0.863929\pi$
$439$		−	364.000i		−	0.829157i	0.910014	−	0.414579i	$-0.136071\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	372.000	+	372.000i	0.839729	+	0.839729i	0.988823	−	0.149094i	$-0.0476357\pi$
$443$	372.000	+	372.000i	0.839729	+	0.839729i	−0.149094	+	0.988823i	$0.547636\pi$
$444$		0			0
$445$	−104.000	+	78.0000i	−0.233708	+	0.175281i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			176.000i			0.391982i	0.980606	+	0.195991i	$0.0627924\pi$
$449$			176.000i			0.391982i	−0.980606	+	0.195991i	$0.937208\pi$
$450$		0			0
$451$	−280.000			−0.620843
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−144.000	−	192.000i	−0.316484	−	0.421978i
$456$		0			0
$457$	129.000	−	129.000i	0.282276	−	0.282276i	−0.551740	−	0.834016i	$-0.686036\pi$
$457$	129.000	−	129.000i	0.282276	−	0.282276i	0.834016	+	0.551740i	$0.186036\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	568.000			1.23210			0.616052	−	0.787705i	$-0.288731\pi$
$461$	568.000			1.23210			0.616052	+	0.787705i	$0.288731\pi$
$462$		0			0
$463$	568.000	+	568.000i	1.22678	+	1.22678i	0.965174	+	0.261608i	$0.0842526\pi$
$463$	568.000	+	568.000i	1.22678	+	1.22678i	0.261608	+	0.965174i	$0.415747\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	272.000	−	272.000i	0.582441	−	0.582441i	−0.353132	−	0.935573i	$-0.614883\pi$
$467$	272.000	−	272.000i	0.582441	−	0.582441i	0.935573	+	0.353132i	$0.114883\pi$
$468$		0			0
$469$		−	704.000i		−	1.50107i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	144.000	+	144.000i	0.304440	+	0.304440i
$474$		0			0
$475$	192.000	+	56.0000i	0.404211	+	0.117895i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			928.000i			1.93737i	0.248294	+	0.968685i	$0.420130\pi$
$479$			928.000i			1.93737i	−0.248294	+	0.968685i	$0.579870\pi$
$480$		0			0
$481$	18.0000			0.0374220
$482$		0			0
$483$		0			0
$484$		0			0
$485$	399.000	+	57.0000i	0.822680	+	0.117526i
$486$		0			0
$487$	−252.000	+	252.000i	−0.517454	+	0.517454i	−0.916800	−	0.399346i	$-0.869237\pi$
$487$	−252.000	+	252.000i	−0.517454	+	0.517454i	0.399346	+	0.916800i	$0.369237\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	844.000			1.71894			0.859470	−	0.511185i	$-0.170793\pi$
$491$	844.000			1.71894			0.859470	+	0.511185i	$0.170793\pi$
$492$		0			0
$493$	722.000	+	722.000i	1.46450	+	1.46450i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−704.000	+	704.000i	−1.41650	+	1.41650i
$498$		0			0
$499$			872.000i			1.74749i	0.486380	+	0.873747i	$0.338317\pi$
$499$			872.000i			1.74749i	−0.486380	+	0.873747i	$0.661683\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−480.000	−	480.000i	−0.954274	−	0.954274i	0.0447250	−	0.998999i	$-0.485759\pi$
$503$	−480.000	−	480.000i	−0.954274	−	0.954274i	−0.998999	+	0.0447250i	$0.985759\pi$
$504$		0			0
$505$	−168.000	−	224.000i	−0.332673	−	0.443564i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	694.000i		−	1.36346i	−0.731605	−	0.681729i	$-0.761228\pi$
$509$		−	694.000i		−	1.36346i	0.731605	−	0.681729i	$-0.238772\pi$
$510$		0			0
$511$	−880.000			−1.72211
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.00000	+	28.0000i	−0.00776699	+	0.0543689i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	528.000			1.01344			0.506718	−	0.862112i	$-0.330859\pi$
$521$	528.000			1.01344			0.506718	+	0.862112i	$0.330859\pi$
$522$		0			0
$523$	−552.000	−	552.000i	−1.05545	−	1.05545i	−0.998370	−	0.0570797i	$-0.981821\pi$
$523$	−552.000	−	552.000i	−1.05545	−	1.05545i	−0.0570797	−	0.998370i	$-0.518179\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	836.000	−	836.000i	1.58634	−	1.58634i
$528$		0			0
$529$			271.000i			0.512287i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−210.000	−	210.000i	−0.393996	−	0.393996i
$534$		0			0
$535$	−476.000	−	68.0000i	−0.889720	−	0.127103i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			316.000i			0.586271i
$540$		0			0
$541$	−782.000			−1.44547			−0.722736	−	0.691125i	$-0.757116\pi$
$541$	−782.000			−1.44547			−0.722736	+	0.691125i	$0.757116\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	184.000	−	138.000i	0.337615	−	0.253211i
$546$		0			0
$547$	420.000	−	420.000i	0.767824	−	0.767824i	−0.209899	−	0.977723i	$-0.567313\pi$
$547$	420.000	−	420.000i	0.767824	−	0.767824i	0.977723	+	0.209899i	$0.0673134\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	304.000			0.551724
$552$		0			0
$553$	−96.0000	−	96.0000i	−0.173599	−	0.173599i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	417.000	−	417.000i	0.748654	−	0.748654i	−0.225573	−	0.974226i	$-0.572425\pi$
$557$	417.000	−	417.000i	0.748654	−	0.748654i	0.974226	+	0.225573i	$0.0724254\pi$
$558$		0			0
$559$			216.000i			0.386404i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−228.000	−	228.000i	−0.404973	−	0.404973i	0.475008	−	0.879981i	$-0.342445\pi$
$563$	−228.000	−	228.000i	−0.404973	−	0.404973i	−0.879981	+	0.475008i	$0.842445\pi$
$564$		0			0
$565$	53.0000	−	371.000i	0.0938053	−	0.656637i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	368.000i		−	0.646749i	−0.946271	−	0.323374i	$-0.895183\pi$
$569$		−	368.000i		−	0.646749i	0.946271	−	0.323374i	$-0.104817\pi$
$570$		0			0
$571$	736.000			1.28897			0.644483	−	0.764618i	$-0.277072\pi$
$571$	736.000			1.28897			0.644483	+	0.764618i	$0.277072\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−620.000	+	340.000i	−1.07826	+	0.591304i
$576$		0			0
$577$	−113.000	+	113.000i	−0.195841	+	0.195841i	−0.798214	−	0.602374i	$-0.794222\pi$
$577$	−113.000	+	113.000i	−0.195841	+	0.195841i	0.602374	+	0.798214i	$0.294222\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−384.000			−0.660929
$582$		0			0
$583$	−68.0000	−	68.0000i	−0.116638	−	0.116638i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−684.000	+	684.000i	−1.16525	+	1.16525i	−0.181937	+	0.983310i	$0.558237\pi$
$587$	−684.000	+	684.000i	−1.16525	+	1.16525i	−0.983310	+	0.181937i	$0.941763\pi$
$588$		0			0
$589$		−	352.000i		−	0.597623i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−149.000	−	149.000i	−0.251265	−	0.251265i	0.570224	−	0.821489i	$-0.306856\pi$
$593$	−149.000	−	149.000i	−0.251265	−	0.251265i	−0.821489	+	0.570224i	$0.806856\pi$
$594$		0			0
$595$	1216.00	−	912.000i	2.04370	−	1.53277i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	152.000i		−	0.253756i	−0.991918	−	0.126878i	$-0.959504\pi$
$599$		−	152.000i		−	0.253756i	0.991918	−	0.126878i	$-0.0404957\pi$
$600$		0			0
$601$	320.000			0.532446			0.266223	−	0.963911i	$-0.414224\pi$
$601$	320.000			0.532446			0.266223	+	0.963911i	$0.414224\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	315.000	+	420.000i	0.520661	+	0.694215i
$606$		0			0
$607$	528.000	−	528.000i	0.869852	−	0.869852i	−0.122604	−	0.992456i	$-0.539124\pi$
$607$	528.000	−	528.000i	0.869852	−	0.869852i	0.992456	+	0.122604i	$0.0391245\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	771.000	+	771.000i	1.25775	+	1.25775i	0.952165	+	0.305583i	$0.0988515\pi$
$613$	771.000	+	771.000i	1.25775	+	1.25775i	0.305583	+	0.952165i	$0.401148\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	675.000	−	675.000i	1.09400	−	1.09400i	0.0989065	−	0.995097i	$-0.468466\pi$
$617$	675.000	−	675.000i	1.09400	−	1.09400i	0.995097	−	0.0989065i	$-0.0315345\pi$
$618$		0			0
$619$			600.000i			0.969305i	0.874707	+	0.484653i	$0.161054\pi$
$619$			600.000i			0.969305i	−0.874707	+	0.484653i	$0.838946\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	208.000	+	208.000i	0.333868	+	0.333868i
$624$		0			0
$625$	−527.000	−	336.000i	−0.843200	−	0.537600i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			114.000i			0.181240i
$630$		0			0
$631$	20.0000			0.0316957			0.0158479	−	0.999874i	$-0.494955\pi$
$631$	20.0000			0.0316957			0.0158479	+	0.999874i	$0.494955\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	476.000	+	68.0000i	0.749606	+	0.107087i
$636$		0			0
$637$	−237.000	+	237.000i	−0.372057	+	0.372057i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−694.000			−1.08268			−0.541342	−	0.840803i	$-0.682083\pi$
$641$	−694.000			−1.08268			−0.541342	+	0.840803i	$0.682083\pi$
$642$		0			0
$643$	168.000	+	168.000i	0.261275	+	0.261275i	0.825572	−	0.564297i	$-0.190853\pi$
$643$	168.000	+	168.000i	0.261275	+	0.261275i	−0.564297	+	0.825572i	$0.690853\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	328.000	−	328.000i	0.506955	−	0.506955i	−0.406635	−	0.913591i	$-0.633298\pi$
$647$	328.000	−	328.000i	0.506955	−	0.506955i	0.913591	+	0.406635i	$0.133298\pi$
$648$		0			0
$649$			368.000i			0.567026i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	81.0000	+	81.0000i	0.124043	+	0.124043i	0.766403	−	0.642360i	$-0.222045\pi$
$653$	81.0000	+	81.0000i	0.124043	+	0.124043i	−0.642360	+	0.766403i	$0.722045\pi$
$654$		0			0
$655$	−132.000	−	176.000i	−0.201527	−	0.268702i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			500.000i			0.758725i	0.925248	+	0.379363i	$0.123857\pi$
$659$			500.000i			0.758725i	−0.925248	+	0.379363i	$0.876143\pi$
$660$		0			0
$661$	−568.000			−0.859304			−0.429652	−	0.902995i	$-0.641364\pi$
$661$	−568.000			−0.859304			−0.429652	+	0.902995i	$0.641364\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	64.0000	−	448.000i	0.0962406	−	0.673684i
$666$		0			0
$667$	−760.000	+	760.000i	−1.13943	+	1.13943i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−288.000			−0.429210
$672$		0			0
$673$	73.0000	+	73.0000i	0.108470	+	0.108470i	0.759259	−	0.650789i	$-0.225562\pi$
$673$	73.0000	+	73.0000i	0.108470	+	0.108470i	−0.650789	+	0.759259i	$0.725562\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−839.000	+	839.000i	−1.23929	+	1.23929i	−0.279000	+	0.960291i	$0.590003\pi$
$677$	−839.000	+	839.000i	−1.23929	+	1.23929i	−0.960291	+	0.279000i	$0.909997\pi$
$678$		0			0
$679$		−	912.000i		−	1.34315i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	744.000	+	744.000i	1.08931	+	1.08931i	0.995599	+	0.0937126i	$0.0298735\pi$
$683$	744.000	+	744.000i	1.08931	+	1.08931i	0.0937126	+	0.995599i	$0.470127\pi$
$684$		0			0
$685$	483.000	+	69.0000i	0.705109	+	0.100730i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	102.000i		−	0.148041i
$690$		0			0
$691$	−504.000			−0.729378			−0.364689	−	0.931129i	$-0.618825\pi$
$691$	−504.000			−0.729378			−0.364689	+	0.931129i	$0.618825\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	320.000	−	240.000i	0.460432	−	0.345324i
$696$		0			0
$697$	1330.00	−	1330.00i	1.90818	−	1.90818i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	298.000			0.425107			0.212553	−	0.977149i	$-0.431822\pi$
$701$	298.000			0.425107			0.212553	+	0.977149i	$0.431822\pi$
$702$		0			0
$703$	24.0000	+	24.0000i	0.0341394	+	0.0341394i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−448.000	+	448.000i	−0.633663	+	0.633663i
$708$		0			0
$709$		−	472.000i		−	0.665726i	−0.942975	−	0.332863i	$-0.891985\pi$
$709$		−	472.000i		−	0.665726i	0.942975	−	0.332863i	$-0.108015\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	880.000	+	880.000i	1.23422	+	1.23422i
$714$		0			0
$715$	12.0000	−	84.0000i	0.0167832	−	0.117483i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	872.000i		−	1.21280i	−0.795161	−	0.606398i	$-0.792614\pi$
$719$		−	872.000i		−	1.21280i	0.795161	−	0.606398i	$-0.207386\pi$
$720$		0			0
$721$	64.0000			0.0887656
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−912.000	−	266.000i	−1.25793	−	0.366897i
$726$		0			0
$727$	60.0000	−	60.0000i	0.0825309	−	0.0825309i	−0.664636	−	0.747167i	$-0.731413\pi$
$727$	60.0000	−	60.0000i	0.0825309	−	0.0825309i	0.747167	+	0.664636i	$0.231413\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1368.00			−1.87141
$732$		0			0
$733$	−581.000	−	581.000i	−0.792633	−	0.792633i	0.189289	−	0.981922i	$-0.439382\pi$
$733$	−581.000	−	581.000i	−0.792633	−	0.792633i	−0.981922	+	0.189289i	$0.939382\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	176.000	−	176.000i	0.238806	−	0.238806i
$738$		0			0
$739$			1160.00i			1.56969i	0.619693	+	0.784844i	$0.287257\pi$
$739$			1160.00i			1.56969i	−0.619693	+	0.784844i	$0.712743\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−380.000	−	380.000i	−0.511440	−	0.511440i	0.403527	−	0.914968i	$-0.367784\pi$
$743$	−380.000	−	380.000i	−0.511440	−	0.511440i	−0.914968	+	0.403527i	$0.867784\pi$
$744$		0			0
$745$	−672.000	+	504.000i	−0.902013	+	0.676510i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1088.00i			1.45260i
$750$		0			0
$751$	−780.000			−1.03862			−0.519308	−	0.854587i	$-0.673810\pi$
$751$	−780.000			−1.03862			−0.519308	+	0.854587i	$0.673810\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0000	+	16.0000i	0.0158940	+	0.0211921i
$756$		0			0
$757$	285.000	−	285.000i	0.376486	−	0.376486i	−0.493347	−	0.869833i	$-0.664227\pi$
$757$	285.000	−	285.000i	0.376486	−	0.376486i	0.869833	+	0.493347i	$0.164227\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	304.000			0.399474			0.199737	−	0.979850i	$-0.435991\pi$
$761$	304.000			0.399474			0.199737	+	0.979850i	$0.435991\pi$
$762$		0			0
$763$	−368.000	−	368.000i	−0.482307	−	0.482307i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−276.000	+	276.000i	−0.359844	+	0.359844i
$768$		0			0
$769$			1072.00i			1.39402i	0.717062	+	0.697009i	$0.245486\pi$
$769$			1072.00i			1.39402i	−0.717062	+	0.697009i	$0.754514\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−897.000	−	897.000i	−1.16041	−	1.16041i	−0.984385	−	0.176029i	$-0.943675\pi$
$773$	−897.000	−	897.000i	−1.16041	−	1.16041i	−0.176029	−	0.984385i	$-0.556325\pi$
$774$		0			0
$775$	−308.000	+	1056.00i	−0.397419	+	1.36258i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	560.000i		−	0.718870i
$780$		0			0
$781$	−352.000			−0.450704
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−693.000	−	99.0000i	−0.882803	−	0.126115i
$786$		0			0
$787$	−328.000	+	328.000i	−0.416773	+	0.416773i	−0.884090	−	0.467317i	$-0.845221\pi$
$787$	−328.000	+	328.000i	−0.416773	+	0.416773i	0.467317	+	0.884090i	$0.345221\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−848.000			−1.07206
$792$		0			0
$793$	−216.000	−	216.000i	−0.272383	−	0.272383i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−87.0000	+	87.0000i	−0.109159	+	0.109159i	−0.759577	−	0.650418i	$-0.774594\pi$
$797$	−87.0000	+	87.0000i	−0.109159	+	0.109159i	0.650418	+	0.759577i	$0.274594\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−220.000	−	220.000i	−0.273973	−	0.273973i
$804$		0			0
$805$	960.000	+	1280.00i	1.19255	+	1.59006i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			922.000i			1.13968i	0.821756	+	0.569839i	$0.192995\pi$
$809$			922.000i			1.13968i	−0.821756	+	0.569839i	$0.807005\pi$
$810$		0			0
$811$	1312.00			1.61776			0.808878	−	0.587977i	$-0.200075\pi$
$811$	1312.00			1.61776			0.808878	+	0.587977i	$0.200075\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	160.000	−	1120.00i	0.196319	−	1.37423i
$816$		0			0
$817$	−288.000	+	288.000i	−0.352509	+	0.352509i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−950.000			−1.15713			−0.578563	−	0.815638i	$-0.696386\pi$
$821$	−950.000			−1.15713			−0.578563	+	0.815638i	$0.696386\pi$
$822$		0			0
$823$	−68.0000	−	68.0000i	−0.0826245	−	0.0826245i	0.664587	−	0.747211i	$-0.268608\pi$
$823$	−68.0000	−	68.0000i	−0.0826245	−	0.0826245i	−0.747211	+	0.664587i	$0.768608\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−264.000	+	264.000i	−0.319226	+	0.319226i	−0.848470	−	0.529244i	$-0.822476\pi$
$827$	−264.000	+	264.000i	−0.319226	+	0.319226i	0.529244	+	0.848470i	$0.322476\pi$
$828$		0			0
$829$			178.000i			0.214717i	0.994220	+	0.107358i	$0.0342392\pi$
$829$			178.000i			0.214717i	−0.994220	+	0.107358i	$0.965761\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1501.00	−	1501.00i	−1.80192	−	1.80192i
$834$		0			0
$835$	392.000	+	56.0000i	0.469461	+	0.0670659i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1576.00i		−	1.87843i	−0.343334	−	0.939213i	$-0.611556\pi$
$839$		−	1576.00i		−	1.87843i	0.343334	−	0.939213i	$-0.388444\pi$
$840$		0			0
$841$	−603.000			−0.717004
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−604.000	+	453.000i	−0.714793	+	0.536095i
$846$		0			0
$847$	840.000	−	840.000i	0.991736	−	0.991736i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−120.000			−0.141011
$852$		0			0
$853$	−509.000	−	509.000i	−0.596717	−	0.596717i	0.342720	−	0.939438i	$-0.388652\pi$
$853$	−509.000	−	509.000i	−0.596717	−	0.596717i	−0.939438	+	0.342720i	$0.888652\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	357.000	−	357.000i	0.416569	−	0.416569i	−0.467450	−	0.884019i	$-0.654827\pi$
$857$	357.000	−	357.000i	0.416569	−	0.416569i	0.884019	+	0.467450i	$0.154827\pi$
$858$		0			0
$859$		−	520.000i		−	0.605355i	−0.953093	−	0.302678i	$-0.902119\pi$
$859$		−	520.000i		−	0.605355i	0.953093	−	0.302678i	$-0.0978805\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−952.000	−	952.000i	−1.10313	−	1.10313i	−0.994031	−	0.109098i	$-0.965204\pi$
$863$	−952.000	−	952.000i	−1.10313	−	1.10313i	−0.109098	−	0.994031i	$-0.534796\pi$
$864$		0			0
$865$	−41.0000	+	287.000i	−0.0473988	+	0.331792i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	48.0000i		−	0.0552359i
$870$		0			0
$871$	264.000			0.303100
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−584.000	+	1288.00i	−0.667429	+	1.47200i
$876$		0			0
$877$	−717.000	+	717.000i	−0.817560	+	0.817560i	−0.985754	−	0.168194i	$-0.946206\pi$
$877$	−717.000	+	717.000i	−0.817560	+	0.817560i	0.168194	+	0.985754i	$0.446206\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−554.000			−0.628831			−0.314415	−	0.949285i	$-0.601808\pi$
$881$	−554.000			−0.628831			−0.314415	+	0.949285i	$0.601808\pi$
$882$		0			0
$883$	−952.000	−	952.000i	−1.07814	−	1.07814i	−0.996676	−	0.0814666i	$-0.974040\pi$
$883$	−952.000	−	952.000i	−1.07814	−	1.07814i	−0.0814666	−	0.996676i	$-0.525960\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−648.000	+	648.000i	−0.730552	+	0.730552i	−0.970729	−	0.240177i	$-0.922795\pi$
$887$	−648.000	+	648.000i	−0.730552	+	0.730552i	0.240177	+	0.970729i	$0.422795\pi$
$888$		0			0
$889$		−	1088.00i		−	1.22385i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−688.000	+	516.000i	−0.768715	+	0.576536i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1672.00i			1.85984i
$900$		0			0
$901$	646.000			0.716981
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−186.000	−	248.000i	−0.205525	−	0.274033i
$906$		0			0
$907$	740.000	−	740.000i	0.815877	−	0.815877i	−0.169631	−	0.985508i	$-0.554258\pi$
$907$	740.000	−	740.000i	0.815877	−	0.815877i	0.985508	+	0.169631i	$0.0542576\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	520.000			0.570801			0.285401	−	0.958408i	$-0.407873\pi$
$911$	520.000			0.570801			0.285401	+	0.958408i	$0.407873\pi$
$912$		0			0
$913$	−96.0000	−	96.0000i	−0.105148	−	0.105148i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−352.000	+	352.000i	−0.383860	+	0.383860i
$918$		0			0
$919$		−	844.000i		−	0.918390i	−0.888336	−	0.459195i	$-0.848138\pi$
$919$		−	844.000i		−	0.918390i	0.888336	−	0.459195i	$-0.151862\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−264.000	−	264.000i	−0.286024	−	0.286024i
$924$		0			0
$925$	−51.0000	−	93.0000i	−0.0551351	−	0.100541i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	208.000i		−	0.223897i	−0.993714	−	0.111948i	$-0.964291\pi$
$929$		−	208.000i		−	0.223897i	0.993714	−	0.111948i	$-0.0357091\pi$
$930$		0			0
$931$	−632.000			−0.678840
$932$		0			0
$933$		0			0
$934$		0			0
$935$	532.000	+	76.0000i	0.568984	+	0.0812834i
$936$		0			0
$937$	7.00000	−	7.00000i	0.00747065	−	0.00747065i	−0.703362	−	0.710832i	$-0.748319\pi$
$937$	7.00000	−	7.00000i	0.00747065	−	0.00747065i	0.710832	+	0.703362i	$0.248319\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1752.00			1.86185			0.930925	−	0.365212i	$-0.119003\pi$
$941$	1752.00			1.86185			0.930925	+	0.365212i	$0.119003\pi$
$942$		0			0
$943$	1400.00	+	1400.00i	1.48462	+	1.48462i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−224.000	+	224.000i	−0.236536	+	0.236536i	−0.815414	−	0.578878i	$-0.803491\pi$
$947$	−224.000	+	224.000i	−0.236536	+	0.236536i	0.578878	+	0.815414i	$0.303491\pi$
$948$		0			0
$949$		−	330.000i		−	0.347734i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	645.000	+	645.000i	0.676810	+	0.676810i	0.959277	−	0.282467i	$-0.0911528\pi$
$953$	645.000	+	645.000i	0.676810	+	0.676810i	−0.282467	+	0.959277i	$0.591153\pi$
$954$		0			0
$955$	744.000	+	992.000i	0.779058	+	1.03874i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1104.00i		−	1.15120i
$960$		0			0
$961$	975.000			1.01457
$962$		0			0
$963$		0			0
$964$		0			0
$965$	135.000	−	945.000i	0.139896	−	0.979275i
$966$		0			0
$967$	92.0000	−	92.0000i	0.0951396	−	0.0951396i	−0.657935	−	0.753075i	$-0.728570\pi$
$967$	92.0000	−	92.0000i	0.0951396	−	0.0951396i	0.753075	+	0.657935i	$0.228570\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.00000			0.00411946			0.00205973	−	0.999998i	$-0.499344\pi$
$971$	4.00000			0.00411946			0.00205973	+	0.999998i	$0.499344\pi$
$972$		0			0
$973$	−640.000	−	640.000i	−0.657760	−	0.657760i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1011.00	+	1011.00i	−1.03480	+	1.03480i	−0.0354282	+	0.999372i	$0.511280\pi$
$977$	−1011.00	+	1011.00i	−1.03480	+	1.03480i	−0.999372	+	0.0354282i	$0.988720\pi$
$978$		0			0
$979$			104.000i			0.106231i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	604.000	+	604.000i	0.614446	+	0.614446i	0.944101	−	0.329656i	$-0.106933\pi$
$983$	604.000	+	604.000i	0.614446	+	0.614446i	−0.329656	+	0.944101i	$0.606933\pi$
$984$		0			0
$985$	1071.00	+	153.000i	1.08731	+	0.155330i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1440.00i		−	1.45602i
$990$		0			0
$991$	−652.000			−0.657921			−0.328961	−	0.944344i	$-0.606698\pi$
$991$	−652.000			−0.657921			−0.328961	+	0.944344i	$0.606698\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1008.00	+	756.000i	−1.01307	+	0.759799i
$996$		0			0
$997$	1051.00	−	1051.00i	1.05416	−	1.05416i	0.0557158	−	0.998447i	$-0.482256\pi$
$997$	1051.00	−	1051.00i	1.05416	−	1.05416i	0.998447	−	0.0557158i	$-0.0177441\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.3.bh.b.433.1		2
3.2	odd	2		720.3.bh.d.433.1		2
4.3	odd	2		90.3.g.a.73.1	yes	2
5.2	odd	4	inner	720.3.bh.b.577.1		2
12.11	even	2		90.3.g.c.73.1	yes	2
15.2	even	4		720.3.bh.d.577.1		2
20.3	even	4		450.3.g.d.307.1		2
20.7	even	4		90.3.g.a.37.1	✓	2
20.19	odd	2		450.3.g.d.343.1		2
60.23	odd	4		450.3.g.a.307.1		2
60.47	odd	4		90.3.g.c.37.1	yes	2
60.59	even	2		450.3.g.a.343.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.3.g.a.37.1	✓	2	20.7	even	4
90.3.g.a.73.1	yes	2	4.3	odd	2
90.3.g.c.37.1	yes	2	60.47	odd	4
90.3.g.c.73.1	yes	2	12.11	even	2
450.3.g.a.307.1		2	60.23	odd	4
450.3.g.a.343.1		2	60.59	even	2
450.3.g.d.307.1		2	20.3	even	4
450.3.g.d.343.1		2	20.19	odd	2
720.3.bh.b.433.1		2	1.1	even	1	trivial
720.3.bh.b.577.1		2	5.2	odd	4	inner
720.3.bh.d.433.1		2	3.2	odd	2
720.3.bh.d.577.1		2	15.2	even	4

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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	720.bh (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.00000 - 4.00000i) q^{5} +(-8.00000 + 8.00000i) q^{7} -4.00000 q^{11} +(-3.00000 - 3.00000i) q^{13} +(19.0000 - 19.0000i) q^{17} -8.00000i q^{19} +(20.0000 + 20.0000i) q^{23} +(-7.00000 + 24.0000i) q^{25} +38.0000i q^{29} +44.0000 q^{31} +(56.0000 + 8.00000i) q^{35} +(-3.00000 + 3.00000i) q^{37} +70.0000 q^{41} +(-36.0000 - 36.0000i) q^{43} -79.0000i q^{49} +(17.0000 + 17.0000i) q^{53} +(12.0000 + 16.0000i) q^{55} -92.0000i q^{59} +72.0000 q^{61} +(-3.00000 + 21.0000i) q^{65} +(-44.0000 + 44.0000i) q^{67} +88.0000 q^{71} +(55.0000 + 55.0000i) q^{73} +(32.0000 - 32.0000i) q^{77} +12.0000i q^{79} +(24.0000 + 24.0000i) q^{83} +(-133.000 - 19.0000i) q^{85} -26.0000i q^{89} +48.0000 q^{91} +(-32.0000 + 24.0000i) q^{95} +(-57.0000 + 57.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} - 16 q^{7} - 8 q^{11} - 6 q^{13} + 38 q^{17} + 40 q^{23} - 14 q^{25} + 88 q^{31} + 112 q^{35} - 6 q^{37} + 140 q^{41} - 72 q^{43} + 34 q^{53} + 24 q^{55} + 144 q^{61} - 6 q^{65} - 88 q^{67}+ \cdots - 114 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 - 16 * q^7 - 8 * q^11 - 6 * q^13 + 38 * q^17 + 40 * q^23 - 14 * q^25 + 88 * q^31 + 112 * q^35 - 6 * q^37 + 140 * q^41 - 72 * q^43 + 34 * q^53 + 24 * q^55 + 144 * q^61 - 6 * q^65 - 88 * q^67 + 176 * q^71 + 110 * q^73 + 64 * q^77 + 48 * q^83 - 266 * q^85 + 96 * q^91 - 64 * q^95 - 114 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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