LMFDB - Embedded newform 720.4.w.d.17.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,4,Mod(17,720)] 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("720.17"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 1])) N = Newforms(chi, 4, names="a")

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

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,-24] 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:	$42.4813752041$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944 $ x^12 - 16x^10 - 14x^8 - 512x^6 + 3889x^4 + 126224*x^2 + 506944
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.2
Root			$-0.347140 + 2.27426i$ of defining polynomial
Character	$\chi$	$=$	720.17
Dual form			720.4.w.d.593.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.38994 + 7.38978i) q^{5} +(10.8652 + 10.8652i) q^{7} +37.8905i q^{11} +(-48.1085 + 48.1085i) q^{13} +(-60.3458 + 60.3458i) q^{17} -109.678i q^{19} +(-39.7660 - 39.7660i) q^{23} +(15.7823 - 124.000i) q^{25} +90.3553 q^{29} +233.678 q^{31} +(-171.449 - 10.8670i) q^{35} +(-19.0042 - 19.0042i) q^{37} +260.783i q^{41} +(-176.216 + 176.216i) q^{43} +(145.788 - 145.788i) q^{47} -106.896i q^{49} +(-183.285 - 183.285i) q^{53} +(-280.002 - 317.899i) q^{55} -279.564 q^{59} -390.267 q^{61} +(48.1164 - 759.139i) q^{65} +(-150.444 - 150.444i) q^{67} +470.042i q^{71} +(480.765 - 480.765i) q^{73} +(-411.687 + 411.687i) q^{77} -1322.44i q^{79} +(-456.192 - 456.192i) q^{83} +(60.3558 - 952.240i) q^{85} -1364.85 q^{89} -1045.41 q^{91} +(810.494 + 920.190i) q^{95} +(-785.308 - 785.308i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 24 q^{7} - 108 q^{13} - 144 q^{25} + 1248 q^{31} + 828 q^{37} + 96 q^{43} + 1512 q^{55} + 96 q^{61} - 1632 q^{67} + 3972 q^{73} - 1752 q^{85} - 4752 q^{91} + 2772 q^{97}+O(q^{100}) $ 12 * q - 24 * q^7 - 108 * q^13 - 144 * q^25 + 1248 * q^31 + 828 * q^37 + 96 * q^43 + 1512 * q^55 + 96 * q^61 - 1632 * q^67 + 3972 * q^73 - 1752 * q^85 - 4752 * q^91 + 2772 * q^97

Character values

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

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

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
$5$	−8.38994	+	7.38978i	−0.750419	+	0.660962i
$5$	−8.38994	+	7.38978i	−0.750419	+	0.660962i
$6$		0			0
$7$	10.8652	+	10.8652i	0.586664	+	0.586664i	0.936726	−	0.350062i	$-0.113840\pi$
$7$	10.8652	+	10.8652i	0.586664	+	0.586664i	−0.350062	+	0.936726i	$0.613840\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			37.8905i			1.03858i	0.854597	+	0.519291i	$0.173804\pi$
$11$			37.8905i			1.03858i	−0.854597	+	0.519291i	$0.826196\pi$
$12$		0			0
$13$	−48.1085	+	48.1085i	−1.02638	+	1.02638i	−0.0267344	+	0.999643i	$0.508511\pi$
$13$	−48.1085	+	48.1085i	−1.02638	+	1.02638i	−0.999643	+	0.0267344i	$0.991489\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−60.3458	+	60.3458i	−0.860942	+	0.860942i	−0.991448	−	0.130506i	$-0.958340\pi$
$17$	−60.3458	+	60.3458i	−0.860942	+	0.860942i	0.130506	+	0.991448i	$0.458340\pi$
$18$		0			0
$19$		−	109.678i		−	1.32430i	−0.749369	−	0.662152i	$-0.769643\pi$
$19$		−	109.678i		−	1.32430i	0.749369	−	0.662152i	$-0.230357\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−39.7660	−	39.7660i	−0.360512	−	0.360512i	0.503489	−	0.864002i	$-0.332049\pi$
$23$	−39.7660	−	39.7660i	−0.360512	−	0.360512i	−0.864002	+	0.503489i	$0.832049\pi$
$24$		0			0
$25$	15.7823	−	124.000i	0.126259	−	0.991997i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	90.3553			0.578571			0.289285	−	0.957243i	$-0.406582\pi$
$29$	90.3553			0.578571			0.289285	+	0.957243i	$0.406582\pi$
$30$		0			0
$31$	233.678			1.35386			0.676932	−	0.736046i	$-0.263309\pi$
$31$	233.678			1.35386			0.676932	+	0.736046i	$0.263309\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−171.449	−	10.8670i	−0.828007	−	0.0524815i
$36$		0			0
$37$	−19.0042	−	19.0042i	−0.0844399	−	0.0844399i	0.663625	−	0.748065i	$-0.269017\pi$
$37$	−19.0042	−	19.0042i	−0.0844399	−	0.0844399i	−0.748065	+	0.663625i	$0.769017\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			260.783i			0.993354i	0.867935	+	0.496677i	$0.165447\pi$
$41$			260.783i			0.993354i	−0.867935	+	0.496677i	$0.834553\pi$
$42$		0			0
$43$	−176.216	+	176.216i	−0.624948	+	0.624948i	−0.946792	−	0.321845i	$-0.895697\pi$
$43$	−176.216	+	176.216i	−0.624948	+	0.624948i	0.321845	+	0.946792i	$0.395697\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	145.788	−	145.788i	0.452456	−	0.452456i	−0.443713	−	0.896169i	$-0.646339\pi$
$47$	145.788	−	145.788i	0.452456	−	0.452456i	0.896169	+	0.443713i	$0.146339\pi$
$48$		0			0
$49$		−	106.896i		−	0.311650i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−183.285	−	183.285i	−0.475022	−	0.475022i	0.428513	−	0.903535i	$-0.359038\pi$
$53$	−183.285	−	183.285i	−0.475022	−	0.475022i	−0.903535	+	0.428513i	$0.859038\pi$
$54$		0			0
$55$	−280.002	−	317.899i	−0.686464	−	0.779373i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−279.564			−0.616883			−0.308441	−	0.951243i	$-0.599807\pi$
$59$	−279.564			−0.616883			−0.308441	+	0.951243i	$0.599807\pi$
$60$		0			0
$61$	−390.267			−0.819157			−0.409578	−	0.912275i	$-0.634324\pi$
$61$	−390.267			−0.819157			−0.409578	+	0.912275i	$0.634324\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	48.1164	−	759.139i	0.0918170	−	1.44861i
$66$		0			0
$67$	−150.444	−	150.444i	−0.274324	−	0.274324i	0.556514	−	0.830838i	$-0.312139\pi$
$67$	−150.444	−	150.444i	−0.274324	−	0.274324i	−0.830838	+	0.556514i	$0.812139\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			470.042i			0.785686i	0.919605	+	0.392843i	$0.128508\pi$
$71$			470.042i			0.785686i	−0.919605	+	0.392843i	$0.871492\pi$
$72$		0			0
$73$	480.765	−	480.765i	0.770812	−	0.770812i	−0.207436	−	0.978249i	$-0.566512\pi$
$73$	480.765	−	480.765i	0.770812	−	0.770812i	0.978249	+	0.207436i	$0.0665119\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−411.687	+	411.687i	−0.609299	+	0.609299i
$78$		0			0
$79$		−	1322.44i		−	1.88337i	−0.336497	−	0.941685i	$-0.609242\pi$
$79$		−	1322.44i		−	1.88337i	0.336497	−	0.941685i	$-0.390758\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−456.192	−	456.192i	−0.603296	−	0.603296i	0.337890	−	0.941186i	$-0.390287\pi$
$83$	−456.192	−	456.192i	−0.603296	−	0.603296i	−0.941186	+	0.337890i	$0.890287\pi$
$84$		0			0
$85$	60.3558	−	952.240i	0.0770177	−	1.21512i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1364.85			−1.62555			−0.812774	−	0.582580i	$-0.802043\pi$
$89$	−1364.85			−1.62555			−0.812774	+	0.582580i	$0.802043\pi$
$90$		0			0
$91$	−1045.41			−1.20428
$92$		0			0
$93$		0			0
$94$		0			0
$95$	810.494	+	920.190i	0.875315	+	0.993784i
$96$		0			0
$97$	−785.308	−	785.308i	−0.822020	−	0.822020i	0.164377	−	0.986398i	$-0.447439\pi$
$97$	−785.308	−	785.308i	−0.822020	−	0.822020i	−0.986398	+	0.164377i	$0.947439\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	111.555i		−	0.109902i	−0.998489	−	0.0549510i	$-0.982500\pi$
$101$		−	111.555i		−	0.109902i	0.998489	−	0.0549510i	$-0.0175003\pi$
$102$		0			0
$103$	−13.8820	+	13.8820i	−0.0132800	+	0.0132800i	−0.713716	−	0.700436i	$-0.752989\pi$
$103$	−13.8820	+	13.8820i	−0.0132800	+	0.0132800i	0.700436	+	0.713716i	$0.252989\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−776.165	+	776.165i	−0.701259	+	0.701259i	−0.964681	−	0.263422i	$-0.915149\pi$
$107$	−776.165	+	776.165i	−0.701259	+	0.701259i	0.263422	+	0.964681i	$0.415149\pi$
$108$		0			0
$109$			410.579i			0.360792i	0.983594	+	0.180396i	$0.0577379\pi$
$109$			410.579i			0.360792i	−0.983594	+	0.180396i	$0.942262\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	521.328	+	521.328i	0.434004	+	0.434004i	0.889988	−	0.455984i	$-0.150713\pi$
$113$	521.328	+	521.328i	0.434004	+	0.434004i	−0.455984	+	0.889988i	$0.650713\pi$
$114$		0			0
$115$	627.496	+	39.7725i	0.508820	+	0.0322505i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1311.34			−1.01017
$120$		0			0
$121$	−104.688			−0.0786538
$122$		0			0
$123$		0			0
$124$		0			0
$125$	783.917	+	1156.98i	0.560926	+	0.827866i
$126$		0			0
$127$	−786.555	−	786.555i	−0.549571	−	0.549571i	0.376746	−	0.926317i	$-0.377043\pi$
$127$	−786.555	−	786.555i	−0.549571	−	0.549571i	−0.926317	+	0.376746i	$0.877043\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	448.364i		−	0.299036i	−0.988759	−	0.149518i	$-0.952228\pi$
$131$		−	448.364i		−	0.299036i	0.988759	−	0.149518i	$-0.0477722\pi$
$132$		0			0
$133$	1191.67	−	1191.67i	0.776922	−	0.776922i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−950.120	+	950.120i	−0.592512	+	0.592512i	−0.938309	−	0.345797i	$-0.887609\pi$
$137$	−950.120	+	950.120i	−0.592512	+	0.592512i	0.345797	+	0.938309i	$0.387609\pi$
$138$		0			0
$139$		−	112.735i		−	0.0687918i	−0.999408	−	0.0343959i	$-0.989049\pi$
$139$		−	112.735i		−	0.0687918i	0.999408	−	0.0343959i	$-0.0109507\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1822.85	−	1822.85i	−1.06598	−	1.06598i
$144$		0			0
$145$	−758.076	+	667.706i	−0.434171	+	0.382413i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1492.95			−0.820852			−0.410426	−	0.911894i	$-0.634620\pi$
$149$	−1492.95			−0.820852			−0.410426	+	0.911894i	$0.634620\pi$
$150$		0			0
$151$	2939.54			1.58422			0.792108	−	0.610381i	$-0.208984\pi$
$151$	2939.54			1.58422			0.792108	+	0.610381i	$0.208984\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1960.54	+	1726.83i	−1.01597	+	0.894852i
$156$		0			0
$157$	156.386	+	156.386i	0.0794967	+	0.0794967i	0.745737	−	0.666240i	$-0.232098\pi$
$157$	156.386	+	156.386i	0.0794967	+	0.0794967i	−0.666240	+	0.745737i	$0.732098\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	864.129i		−	0.422999i
$162$		0			0
$163$	239.292	−	239.292i	0.114986	−	0.114986i	−0.647272	−	0.762259i	$-0.724090\pi$
$163$	239.292	−	239.292i	0.114986	−	0.114986i	0.762259	+	0.647272i	$0.224090\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	677.159	−	677.159i	0.313773	−	0.313773i	−0.532596	−	0.846369i	$-0.678784\pi$
$167$	677.159	−	677.159i	0.313773	−	0.313773i	0.846369	+	0.532596i	$0.178784\pi$
$168$		0			0
$169$		−	2431.86i		−	1.10690i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−441.523	−	441.523i	−0.194037	−	0.194037i	0.603401	−	0.797438i	$-0.293812\pi$
$173$	−441.523	−	441.523i	−0.194037	−	0.194037i	−0.797438	+	0.603401i	$0.793812\pi$
$174$		0			0
$175$	1518.76	−	1175.80i	0.656041	−	0.507898i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4724.27			1.97267			0.986337	−	0.164737i	$-0.0526776\pi$
$179$	4724.27			1.97267			0.986337	+	0.164737i	$0.0526776\pi$
$180$		0			0
$181$	−1548.35			−0.635847			−0.317923	−	0.948116i	$-0.602985\pi$
$181$	−1548.35			−0.635847			−0.317923	+	0.948116i	$0.602985\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	299.882	+	19.0074i	0.119177	+	0.00755378i
$186$		0			0
$187$	−2286.53	−	2286.53i	−0.894159	−	0.894159i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	1090.71i		−	0.413200i	−0.978426	−	0.206600i	$-0.933760\pi$
$191$		−	1090.71i		−	0.413200i	0.978426	−	0.206600i	$-0.0662398\pi$
$192$		0			0
$193$	−2825.69	+	2825.69i	−1.05387	+	1.05387i	−0.0554112	+	0.998464i	$0.517647\pi$
$193$	−2825.69	+	2825.69i	−1.05387	+	1.05387i	−0.998464	+	0.0554112i	$0.982353\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1163.99	+	1163.99i	−0.420969	+	0.420969i	−0.885537	−	0.464569i	$-0.846209\pi$
$197$	−1163.99	+	1163.99i	−0.420969	+	0.420969i	0.464569	+	0.885537i	$0.346209\pi$
$198$		0			0
$199$			520.257i			0.185327i	0.995697	+	0.0926633i	$0.0295380\pi$
$199$			520.257i			0.185327i	−0.995697	+	0.0926633i	$0.970462\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	981.726	+	981.726i	0.339427	+	0.339427i
$204$		0			0
$205$	−1927.13	−	2187.96i	−0.656569	−	0.745432i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4155.74			1.37540
$210$		0			0
$211$	−5524.38			−1.80244			−0.901219	−	0.433365i	$-0.857326\pi$
$211$	−5524.38			−1.80244			−0.901219	+	0.433365i	$0.857326\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	176.245	−	2780.65i	0.0559062	−	0.882039i
$216$		0			0
$217$	2538.95	+	2538.95i	0.794263	+	0.794263i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	5806.30i		−	1.76730i
$222$		0			0
$223$	−323.198	+	323.198i	−0.0970537	+	0.0970537i	−0.753967	−	0.656913i	$-0.771862\pi$
$223$	−323.198	+	323.198i	−0.0970537	+	0.0970537i	0.656913	+	0.753967i	$0.271862\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2739.76	+	2739.76i	−0.801077	+	0.801077i	−0.983264	−	0.182187i	$-0.941682\pi$
$227$	−2739.76	+	2739.76i	−0.801077	+	0.801077i	0.182187	+	0.983264i	$0.441682\pi$
$228$		0			0
$229$			230.334i			0.0664667i	0.999448	+	0.0332333i	$0.0105804\pi$
$229$			230.334i			0.0664667i	−0.999448	+	0.0332333i	$0.989420\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	350.156	+	350.156i	0.0984526	+	0.0984526i	0.754618	−	0.656165i	$-0.227822\pi$
$233$	350.156	+	350.156i	0.0984526	+	0.0984526i	−0.656165	+	0.754618i	$0.727822\pi$
$234$		0			0
$235$	−145.812	+	2300.50i	−0.0404755	+	0.638587i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1013.20			0.274218			0.137109	−	0.990556i	$-0.456219\pi$
$239$	1013.20			0.274218			0.137109	+	0.990556i	$0.456219\pi$
$240$		0			0
$241$	2584.30			0.690744			0.345372	−	0.938466i	$-0.387753\pi$
$241$	2584.30			0.690744			0.345372	+	0.938466i	$0.387753\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	789.938	+	896.852i	0.205989	+	0.233868i
$246$		0			0
$247$	5276.43	+	5276.43i	1.35924	+	1.35924i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			1708.34i			0.429599i	0.976658	+	0.214800i	$0.0689099\pi$
$251$			1708.34i			0.429599i	−0.976658	+	0.214800i	$0.931090\pi$
$252$		0			0
$253$	1506.75	−	1506.75i	0.374422	−	0.374422i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2659.48	+	2659.48i	−0.645500	+	0.645500i	−0.951902	−	0.306402i	$-0.900875\pi$
$257$	−2659.48	+	2659.48i	−0.645500	+	0.645500i	0.306402	+	0.951902i	$0.400875\pi$
$258$		0			0
$259$		−	412.969i		−	0.0990757i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2307.03	−	2307.03i	−0.540903	−	0.540903i	0.382891	−	0.923794i	$-0.374929\pi$
$263$	−2307.03	−	2307.03i	−0.540903	−	0.540903i	−0.923794	+	0.382891i	$0.874929\pi$
$264$		0			0
$265$	2892.19	+	183.315i	0.670437	+	0.0424943i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	447.584			0.101449			0.0507243	−	0.998713i	$-0.483847\pi$
$269$	447.584			0.101449			0.0507243	+	0.998713i	$0.483847\pi$
$270$		0			0
$271$	−1573.15			−0.352627			−0.176314	−	0.984334i	$-0.556417\pi$
$271$	−1573.15			−0.352627			−0.176314	+	0.984334i	$0.556417\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4698.41	+	598.000i	1.03027	+	0.131130i
$276$		0			0
$277$	5816.91	+	5816.91i	1.26175	+	1.26175i	0.950245	+	0.311504i	$0.100833\pi$
$277$	5816.91	+	5816.91i	1.26175	+	1.26175i	0.311504	+	0.950245i	$0.399167\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			140.032i			0.0297281i	0.999890	+	0.0148640i	$0.00473155\pi$
$281$			140.032i			0.0297281i	−0.999890	+	0.0148640i	$0.995268\pi$
$282$		0			0
$283$	−4431.85	+	4431.85i	−0.930905	+	0.930905i	−0.997763	−	0.0668572i	$-0.978703\pi$
$283$	−4431.85	+	4431.85i	−0.930905	+	0.930905i	0.0668572	+	0.997763i	$0.478703\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2833.46	+	2833.46i	−0.582765	+	0.582765i
$288$		0			0
$289$		−	2370.24i		−	0.482442i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1244.45	+	1244.45i	0.248127	+	0.248127i	0.820202	−	0.572074i	$-0.193861\pi$
$293$	1244.45	+	1244.45i	0.248127	+	0.248127i	−0.572074	+	0.820202i	$0.693861\pi$
$294$		0			0
$295$	2345.52	−	2065.91i	0.462921	−	0.407736i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3826.17			0.740043
$300$		0			0
$301$	−3829.24			−0.733269
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3274.32	−	2883.99i	0.614711	−	0.541432i
$306$		0			0
$307$	7327.60	+	7327.60i	1.36224	+	1.36224i	0.871053	+	0.491189i	$0.163437\pi$
$307$	7327.60	+	7327.60i	1.36224	+	1.36224i	0.491189	+	0.871053i	$0.336563\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	7489.38i		−	1.36554i	−0.730632	−	0.682771i	$-0.760775\pi$
$311$		−	7489.38i		−	1.36554i	0.730632	−	0.682771i	$-0.239225\pi$
$312$		0			0
$313$	−2456.04	+	2456.04i	−0.443525	+	0.443525i	−0.893195	−	0.449670i	$-0.851542\pi$
$313$	−2456.04	+	2456.04i	−0.443525	+	0.443525i	0.449670	+	0.893195i	$0.351542\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6724.30	−	6724.30i	1.19140	−	1.19140i	0.214728	−	0.976674i	$-0.431113\pi$
$317$	6724.30	−	6724.30i	1.19140	−	1.19140i	0.976674	−	0.214728i	$-0.0688866\pi$
$318$		0			0
$319$			3423.60i			0.600894i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6618.59	+	6618.59i	1.14015	+	1.14015i
$324$		0			0
$325$	5206.18	+	6724.70i	0.888574	+	1.14775i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3168.03			0.530879
$330$		0			0
$331$	199.080			0.0330586			0.0165293	−	0.999863i	$-0.494738\pi$
$331$	199.080			0.0330586			0.0165293	+	0.999863i	$0.494738\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2373.97	+	150.469i	0.387176	+	0.0245403i
$336$		0			0
$337$	3308.77	+	3308.77i	0.534838	+	0.534838i	0.922008	−	0.387170i	$-0.126547\pi$
$337$	3308.77	+	3308.77i	0.534838	+	0.534838i	−0.387170	+	0.922008i	$0.626547\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			8854.16i			1.40610i
$342$		0			0
$343$	4888.20	−	4888.20i	0.769498	−	0.769498i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2371.68	−	2371.68i	0.366912	−	0.366912i	−0.499438	−	0.866350i	$-0.666460\pi$
$347$	2371.68	−	2371.68i	0.366912	−	0.366912i	0.866350	+	0.499438i	$0.166460\pi$
$348$		0			0
$349$			8208.92i			1.25906i	0.776975	+	0.629532i	$0.216753\pi$
$349$			8208.92i			1.25906i	−0.776975	+	0.629532i	$0.783247\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6800.98	−	6800.98i	−1.02544	−	1.02544i	−0.999668	−	0.0257707i	$-0.991796\pi$
$353$	−6800.98	−	6800.98i	−1.02544	−	1.02544i	−0.0257707	−	0.999668i	$-0.508204\pi$
$354$		0			0
$355$	−3473.51	−	3943.63i	−0.519309	−	0.589594i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1778.14			−0.261411			−0.130705	−	0.991421i	$-0.541724\pi$
$359$	−1778.14			−0.261411			−0.130705	+	0.991421i	$0.541724\pi$
$360$		0			0
$361$	−5170.20			−0.753783
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−480.844	+	7586.34i	−0.0689549	+	1.08791i
$366$		0			0
$367$	4673.56	+	4673.56i	0.664735	+	0.664735i	0.956492	−	0.291757i	$-0.0942400\pi$
$367$	4673.56	+	4673.56i	0.664735	+	0.664735i	−0.291757	+	0.956492i	$0.594240\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	3982.85i		−	0.557357i
$372$		0			0
$373$	−5729.37	+	5729.37i	−0.795323	+	0.795323i	−0.982354	−	0.187031i	$-0.940114\pi$
$373$	−5729.37	+	5729.37i	−0.795323	+	0.795323i	0.187031	+	0.982354i	$0.440114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4346.86	+	4346.86i	−0.593832	+	0.593832i
$378$		0			0
$379$		−	153.883i		−	0.0208561i	−0.999946	−	0.0104280i	$-0.996681\pi$
$379$		−	153.883i		−	0.0208561i	0.999946	−	0.0104280i	$-0.00331941\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4297.76	−	4297.76i	−0.573382	−	0.573382i	0.359690	−	0.933072i	$-0.382882\pi$
$383$	−4297.76	−	4297.76i	−0.573382	−	0.573382i	−0.933072	+	0.359690i	$0.882882\pi$
$384$		0			0
$385$	411.754	−	6496.30i	0.0545063	−	0.859954i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7833.00			1.02095			0.510474	−	0.859893i	$-0.329470\pi$
$389$	7833.00			1.02095			0.510474	+	0.859893i	$0.329470\pi$
$390$		0			0
$391$	4799.42			0.620760
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9772.54	+	11095.2i	1.24484	+	1.41332i
$396$		0			0
$397$	−10195.4	−	10195.4i	−1.28889	−	1.28889i	−0.935459	−	0.353435i	$-0.885013\pi$
$397$	−10195.4	−	10195.4i	−1.28889	−	1.28889i	−0.353435	−	0.935459i	$-0.614987\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	10778.3i		−	1.34226i	−0.741341	−	0.671128i	$-0.765810\pi$
$401$		−	10778.3i		−	1.34226i	0.741341	−	0.671128i	$-0.234190\pi$
$402$		0			0
$403$	−11241.9	+	11241.9i	−1.38957	+	1.38957i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	720.079	−	720.079i	0.0876978	−	0.0876978i
$408$		0			0
$409$			4629.92i			0.559743i	0.960037	+	0.279872i	$0.0902919\pi$
$409$			4629.92i			0.559743i	−0.960037	+	0.279872i	$0.909708\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3037.51	−	3037.51i	−0.361903	−	0.361903i
$414$		0			0
$415$	7198.58	+	456.267i	0.851480	+	0.0539693i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11276.5			−1.31478			−0.657389	−	0.753551i	$-0.728339\pi$
$419$	−11276.5			−1.31478			−0.657389	+	0.753551i	$0.728339\pi$
$420$		0			0
$421$	3387.58			0.392163			0.196082	−	0.980588i	$-0.437178\pi$
$421$	3387.58			0.392163			0.196082	+	0.980588i	$0.437178\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6530.47	+	8435.26i	0.745351	+	0.962753i
$426$		0			0
$427$	−4240.32	−	4240.32i	−0.480570	−	0.480570i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	16161.7i		−	1.80623i	−0.429402	−	0.903113i	$-0.641276\pi$
$431$		−	16161.7i		−	1.80623i	0.429402	−	0.903113i	$-0.358724\pi$
$432$		0			0
$433$	−3808.79	+	3808.79i	−0.422722	+	0.422722i	−0.886140	−	0.463418i	$-0.846623\pi$
$433$	−3808.79	+	3808.79i	−0.422722	+	0.422722i	0.463418	+	0.886140i	$0.346623\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4361.44	+	4361.44i	−0.477428	+	0.477428i
$438$		0			0
$439$		−	1607.30i		−	0.174744i	−0.996176	−	0.0873718i	$-0.972153\pi$
$439$		−	1607.30i		−	0.174744i	0.996176	−	0.0873718i	$-0.0278468\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8296.60	−	8296.60i	−0.889805	−	0.889805i	0.104699	−	0.994504i	$-0.466612\pi$
$443$	−8296.60	−	8296.60i	−0.889805	−	0.889805i	−0.994504	+	0.104699i	$0.966612\pi$
$444$		0			0
$445$	11451.0	−	10085.9i	1.21984	−	1.07442i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	15061.8			1.58309			0.791547	−	0.611108i	$-0.209276\pi$
$449$	15061.8			1.58309			0.791547	+	0.611108i	$0.209276\pi$
$450$		0			0
$451$	−9881.20			−1.03168
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8770.97	−	7725.38i	0.903713	−	0.795981i
$456$		0			0
$457$	2671.27	+	2671.27i	0.273428	+	0.273428i	0.830479	−	0.557050i	$-0.188067\pi$
$457$	2671.27	+	2671.27i	0.273428	+	0.273428i	−0.557050	+	0.830479i	$0.688067\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	12809.1i		−	1.29410i	−0.762449	−	0.647049i	$-0.776003\pi$
$461$		−	12809.1i		−	1.29410i	0.762449	−	0.647049i	$-0.223997\pi$
$462$		0			0
$463$	11154.1	−	11154.1i	1.11960	−	1.11960i	0.127804	−	0.991799i	$-0.459207\pi$
$463$	11154.1	−	11154.1i	1.11960	−	1.11960i	0.991799	−	0.127804i	$-0.0407927\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7905.44	+	7905.44i	−0.783341	+	0.783341i	−0.980393	−	0.197052i	$-0.936863\pi$
$467$	−7905.44	+	7905.44i	−0.783341	+	0.783341i	0.197052	+	0.980393i	$0.436863\pi$
$468$		0			0
$469$		−	3269.21i		−	0.321872i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6676.92	−	6676.92i	−0.649060	−	0.649060i
$474$		0			0
$475$	−13600.0	−	1730.97i	−1.31371	−	0.167205i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10715.8			1.02217			0.511085	−	0.859530i	$-0.329244\pi$
$479$	10715.8			1.02217			0.511085	+	0.859530i	$0.329244\pi$
$480$		0			0
$481$	1828.53			0.173334
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12391.9	+	785.437i	1.16018	+	0.0735358i
$486$		0			0
$487$	4489.84	+	4489.84i	0.417770	+	0.417770i	0.884435	−	0.466664i	$-0.154544\pi$
$487$	4489.84	+	4489.84i	0.417770	+	0.417770i	−0.466664	+	0.884435i	$0.654544\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			15174.1i			1.39470i	0.716729	+	0.697352i	$0.245638\pi$
$491$			15174.1i			1.39470i	−0.716729	+	0.697352i	$0.754362\pi$
$492$		0			0
$493$	−5452.56	+	5452.56i	−0.498116	+	0.498116i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5107.09	+	5107.09i	−0.460934	+	0.460934i
$498$		0			0
$499$			8887.20i			0.797286i	0.917106	+	0.398643i	$0.130519\pi$
$499$			8887.20i			0.797286i	−0.917106	+	0.398643i	$0.869481\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12740.5	+	12740.5i	1.12936	+	1.12936i	0.990281	+	0.139084i	$0.0444158\pi$
$503$	12740.5	+	12740.5i	1.12936	+	1.12936i	0.139084	+	0.990281i	$0.455584\pi$
$504$		0			0
$505$	824.364	+	935.937i	0.0726410	+	0.0824726i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9257.00			−0.806108			−0.403054	−	0.915176i	$-0.632051\pi$
$509$	−9257.00			−0.806108			−0.403054	+	0.915176i	$0.632051\pi$
$510$		0			0
$511$	10447.2			0.904416
$512$		0			0
$513$		0			0
$514$		0			0
$515$	13.8843	−	219.055i	0.00118799	−	0.0187431i
$516$		0			0
$517$	5523.99	+	5523.99i	0.469913	+	0.469913i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			13563.3i			1.14054i	0.821458	+	0.570268i	$0.193161\pi$
$521$			13563.3i			1.14054i	−0.821458	+	0.570268i	$0.806839\pi$
$522$		0			0
$523$	−11193.8	+	11193.8i	−0.935894	+	0.935894i	−0.998065	−	0.0621714i	$-0.980197\pi$
$523$	−11193.8	+	11193.8i	−0.935894	+	0.935894i	0.0621714	+	0.998065i	$0.480197\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14101.5	+	14101.5i	−1.16560	+	1.16560i
$528$		0			0
$529$		−	9004.33i		−	0.740062i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12545.9	−	12545.9i	−1.01956	−	1.01956i
$534$		0			0
$535$	776.293	−	12247.7i	0.0627329	−	0.989744i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4050.34			0.323675
$540$		0			0
$541$	1383.33			0.109934			0.0549668	−	0.998488i	$-0.482495\pi$
$541$	1383.33			0.109934			0.0549668	+	0.998488i	$0.482495\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3034.09	−	3444.73i	−0.238470	−	0.270745i
$546$		0			0
$547$	−2181.76	−	2181.76i	−0.170540	−	0.170540i	0.616677	−	0.787217i	$-0.288479\pi$
$547$	−2181.76	−	2181.76i	−0.170540	−	0.170540i	−0.787217	+	0.616677i	$0.788479\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	9909.96i		−	0.766204i
$552$		0			0
$553$	14368.5	−	14368.5i	1.10491	−	1.10491i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15674.9	+	15674.9i	−1.19240	+	1.19240i	−0.216007	+	0.976392i	$0.569303\pi$
$557$	−15674.9	+	15674.9i	−1.19240	+	1.19240i	−0.976392	+	0.216007i	$0.930697\pi$
$558$		0			0
$559$		−	16955.0i		−	1.28286i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−8213.60	−	8213.60i	−0.614852	−	0.614852i	0.329354	−	0.944206i	$-0.393169\pi$
$563$	−8213.60	−	8213.60i	−0.614852	−	0.614852i	−0.944206	+	0.329354i	$0.893169\pi$
$564$		0			0
$565$	−8226.41	−	521.414i	−0.612545	−	0.0388248i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−16902.1			−1.24530			−0.622648	−	0.782502i	$-0.713943\pi$
$569$	−16902.1			−1.24530			−0.622648	+	0.782502i	$0.713943\pi$
$570$		0			0
$571$	−6507.12			−0.476908			−0.238454	−	0.971154i	$-0.576641\pi$
$571$	−6507.12			−0.476908			−0.238454	+	0.971154i	$0.576641\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5558.57	+	4303.37i	−0.403145	+	0.312109i
$576$		0			0
$577$	7884.54	+	7884.54i	0.568870	+	0.568870i	0.931812	−	0.362942i	$-0.118228\pi$
$577$	7884.54	+	7884.54i	0.568870	+	0.568870i	−0.362942	+	0.931812i	$0.618228\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	9913.20i		−	0.707864i
$582$		0			0
$583$	6944.77	−	6944.77i	0.493350	−	0.493350i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9548.44	+	9548.44i	−0.671390	+	0.671390i	−0.958037	−	0.286646i	$-0.907460\pi$
$587$	−9548.44	+	9548.44i	−0.671390	+	0.671390i	0.286646	+	0.958037i	$0.407460\pi$
$588$		0			0
$589$		−	25629.2i		−	1.79293i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3491.97	+	3491.97i	0.241818	+	0.241818i	0.817602	−	0.575784i	$-0.195303\pi$
$593$	3491.97	+	3491.97i	0.241818	+	0.241818i	−0.575784	+	0.817602i	$0.695303\pi$
$594$		0			0
$595$	11002.0	−	9690.48i	0.758049	−	0.667682i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11203.0			−0.764175			−0.382087	−	0.924126i	$-0.624795\pi$
$599$	−11203.0			−0.764175			−0.382087	+	0.924126i	$0.624795\pi$
$600$		0			0
$601$	−13953.5			−0.947048			−0.473524	−	0.880781i	$-0.657018\pi$
$601$	−13953.5			−0.947048			−0.473524	+	0.880781i	$0.657018\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	878.328	−	773.622i	0.0590233	−	0.0519871i
$606$		0			0
$607$	1292.54	+	1292.54i	0.0864293	+	0.0864293i	0.749000	−	0.662570i	$-0.230534\pi$
$607$	1292.54	+	1292.54i	0.0864293	+	0.0864293i	−0.662570	+	0.749000i	$0.730534\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			14027.3i			0.928780i
$612$		0			0
$613$	225.964	−	225.964i	0.0148884	−	0.0148884i	−0.699623	−	0.714512i	$-0.746649\pi$
$613$	225.964	−	225.964i	0.0148884	−	0.0148884i	0.714512	+	0.699623i	$0.246649\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2717.41	−	2717.41i	0.177308	−	0.177308i	−0.612873	−	0.790181i	$-0.709986\pi$
$617$	2717.41	−	2717.41i	0.177308	−	0.177308i	0.790181	+	0.612873i	$0.209986\pi$
$618$		0			0
$619$		−	8870.55i		−	0.575989i	−0.957632	−	0.287995i	$-0.907011\pi$
$619$		−	8870.55i		−	0.575989i	0.957632	−	0.287995i	$-0.0929886\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14829.3	−	14829.3i	−0.953650	−	0.953650i
$624$		0			0
$625$	−15126.8	−	3914.00i	−0.968118	−	0.250496i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2293.65			0.145396
$630$		0			0
$631$	−29762.7			−1.87771			−0.938853	−	0.344318i	$-0.888110\pi$
$631$	−29762.7			−1.87771			−0.938853	+	0.344318i	$0.888110\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	12411.6	+	786.685i	0.775654	+	0.0491632i
$636$		0			0
$637$	5142.61	+	5142.61i	0.319871	+	0.319871i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			5168.75i			0.318492i	0.987239	+	0.159246i	$0.0509063\pi$
$641$			5168.75i			0.318492i	−0.987239	+	0.159246i	$0.949094\pi$
$642$		0			0
$643$	2078.46	−	2078.46i	0.127475	−	0.127475i	−0.640491	−	0.767966i	$-0.721269\pi$
$643$	2078.46	−	2078.46i	0.127475	−	0.127475i	0.767966	+	0.640491i	$0.221269\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10765.7	+	10765.7i	−0.654164	+	0.654164i	−0.953993	−	0.299829i	$-0.903070\pi$
$647$	−10765.7	+	10765.7i	−0.654164	+	0.654164i	0.299829	+	0.953993i	$0.403070\pi$
$648$		0			0
$649$		−	10592.8i		−	0.640684i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12815.1	+	12815.1i	0.767984	+	0.767984i	0.977751	−	0.209767i	$-0.0672707\pi$
$653$	12815.1	+	12815.1i	0.767984	+	0.767984i	−0.209767	+	0.977751i	$0.567271\pi$
$654$		0			0
$655$	3313.31	+	3761.75i	0.197651	+	0.224402i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	16634.8			0.983307			0.491654	−	0.870791i	$-0.336393\pi$
$659$	16634.8			0.983307			0.491654	+	0.870791i	$0.336393\pi$
$660$		0			0
$661$	−10835.6			−0.637601			−0.318800	−	0.947822i	$-0.603280\pi$
$661$	−10835.6			−0.637601			−0.318800	+	0.947822i	$0.603280\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1191.86	+	18804.2i	−0.0695015	+	1.09653i
$666$		0			0
$667$	−3593.07	−	3593.07i	−0.208582	−	0.208582i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	14787.4i		−	0.850762i
$672$		0			0
$673$	−16220.3	+	16220.3i	−0.929045	+	0.929045i	−0.997644	−	0.0685992i	$-0.978147\pi$
$673$	−16220.3	+	16220.3i	−0.929045	+	0.929045i	0.0685992	+	0.997644i	$0.478147\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−11196.6	+	11196.6i	−0.635626	+	0.635626i	−0.949473	−	0.313847i	$-0.898382\pi$
$677$	−11196.6	+	11196.6i	−0.635626	+	0.635626i	0.313847	+	0.949473i	$0.398382\pi$
$678$		0			0
$679$		−	17065.0i		−	0.964500i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−15428.1	−	15428.1i	−0.864331	−	0.864331i	0.127507	−	0.991838i	$-0.459303\pi$
$683$	−15428.1	−	15428.1i	−0.864331	−	0.864331i	−0.991838	+	0.127507i	$0.959303\pi$
$684$		0			0
$685$	950.276	−	14992.6i	0.0530046	−	0.836261i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	17635.2			0.975104
$690$		0			0
$691$	−21858.6			−1.20339			−0.601693	−	0.798728i	$-0.705507\pi$
$691$	−21858.6			−1.20339			−0.601693	+	0.798728i	$0.705507\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	833.087	+	945.841i	0.0454688	+	0.0516227i
$696$		0			0
$697$	−15737.2	−	15737.2i	−0.855220	−	0.855220i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	8846.82i		−	0.476662i	−0.971184	−	0.238331i	$-0.923400\pi$
$701$		−	8846.82i		−	0.476662i	0.971184	−	0.238331i	$-0.0766002\pi$
$702$		0			0
$703$	−2084.34	+	2084.34i	−0.111824	+	0.111824i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1212.06	−	1212.06i	0.0644755	−	0.0644755i
$708$		0			0
$709$		−	23189.8i		−	1.22837i	−0.789163	−	0.614184i	$-0.789485\pi$
$709$		−	23189.8i		−	1.22837i	0.789163	−	0.614184i	$-0.210515\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9292.43	−	9292.43i	−0.488084	−	0.488084i
$714$		0			0
$715$	28764.1	+	1823.15i	1.50450	+	0.0953596i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7005.29			0.363356			0.181678	−	0.983358i	$-0.441847\pi$
$719$	7005.29			0.363356			0.181678	+	0.983358i	$0.441847\pi$
$720$		0			0
$721$	−301.661			−0.0155818
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1426.02	−	11204.0i	0.0730495	−	0.573941i
$726$		0			0
$727$	16025.7	+	16025.7i	0.817553	+	0.817553i	0.985753	−	0.168200i	$-0.0537954\pi$
$727$	16025.7	+	16025.7i	0.817553	+	0.817553i	−0.168200	+	0.985753i	$0.553795\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	21267.8i		−	1.07609i
$732$		0			0
$733$	−3319.43	+	3319.43i	−0.167266	+	0.167266i	−0.785777	−	0.618511i	$-0.787736\pi$
$733$	−3319.43	+	3319.43i	−0.167266	+	0.167266i	0.618511	+	0.785777i	$0.287736\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5700.41	−	5700.41i	0.284908	−	0.284908i
$738$		0			0
$739$		−	19979.4i		−	0.994526i	−0.867600	−	0.497263i	$-0.834338\pi$
$739$		−	19979.4i		−	0.994526i	0.867600	−	0.497263i	$-0.165662\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14574.6	−	14574.6i	−0.719637	−	0.719637i	0.248894	−	0.968531i	$-0.419933\pi$
$743$	−14574.6	−	14574.6i	−0.719637	−	0.719637i	−0.968531	+	0.248894i	$0.919933\pi$
$744$		0			0
$745$	12525.7	−	11032.5i	0.615983	−	0.542552i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−16866.3			−0.822807
$750$		0			0
$751$	11620.1			0.564612			0.282306	−	0.959324i	$-0.408901\pi$
$751$	11620.1			0.564612			0.282306	+	0.959324i	$0.408901\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−24662.6	+	21722.6i	−1.18883	+	1.04711i
$756$		0			0
$757$	10502.0	+	10502.0i	0.504229	+	0.504229i	0.912749	−	0.408520i	$-0.133955\pi$
$757$	10502.0	+	10502.0i	0.504229	+	0.504229i	−0.408520	+	0.912749i	$0.633955\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	36489.6i		−	1.73817i	−0.494664	−	0.869085i	$-0.664709\pi$
$761$		−	36489.6i		−	1.73817i	0.494664	−	0.869085i	$-0.335291\pi$
$762$		0			0
$763$	−4461.01	+	4461.01i	−0.211664	+	0.211664i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13449.4	−	13449.4i	0.633154	−	0.633154i
$768$		0			0
$769$			30323.9i			1.42199i	0.703198	+	0.710994i	$0.251755\pi$
$769$			30323.9i			1.42199i	−0.703198	+	0.710994i	$0.748245\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	10327.2	+	10327.2i	0.480522	+	0.480522i	0.905298	−	0.424777i	$-0.139647\pi$
$773$	10327.2	+	10327.2i	0.480522	+	0.480522i	−0.424777	+	0.905298i	$0.639647\pi$
$774$		0			0
$775$	3687.98	−	28976.0i	0.170937	−	1.34303i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28602.1			1.31550
$780$		0			0
$781$	−17810.1			−0.816000
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2467.73	−	156.412i	−0.112200	−	0.00711157i
$786$		0			0
$787$	−21521.8	−	21521.8i	−0.974802	−	0.974802i	0.0248882	−	0.999690i	$-0.492077\pi$
$787$	−21521.8	−	21521.8i	−0.974802	−	0.974802i	−0.999690	+	0.0248882i	$0.992077\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			11328.6i			0.509229i
$792$		0			0
$793$	18775.2	−	18775.2i	0.840764	−	0.840764i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−17277.8	+	17277.8i	−0.767895	+	0.767895i	−0.977736	−	0.209841i	$-0.932706\pi$
$797$	−17277.8	+	17277.8i	−0.767895	+	0.767895i	0.209841	+	0.977736i	$0.432706\pi$
$798$		0			0
$799$			17595.4i			0.779076i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	18216.4	+	18216.4i	0.800552	+	0.800552i
$804$		0			0
$805$	6385.72	+	7249.99i	0.279586	+	0.317427i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−41260.2			−1.79312			−0.896558	−	0.442926i	$-0.853941\pi$
$809$	−41260.2			−1.79312			−0.896558	+	0.442926i	$0.853941\pi$
$810$		0			0
$811$	23357.1			1.01132			0.505659	−	0.862733i	$-0.331249\pi$
$811$	23357.1			1.01132			0.505659	+	0.862733i	$0.331249\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−239.331	+	3775.96i	−0.0102864	+	0.162290i
$816$		0			0
$817$	19327.0	+	19327.0i	0.827621	+	0.827621i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	8296.46i		−	0.352678i	−0.984330	−	0.176339i	$-0.943575\pi$
$821$		−	8296.46i		−	0.352678i	0.984330	−	0.176339i	$-0.0564254\pi$
$822$		0			0
$823$	−17980.5	+	17980.5i	−0.761555	+	0.761555i	−0.976603	−	0.215048i	$-0.931009\pi$
$823$	−17980.5	+	17980.5i	−0.761555	+	0.761555i	0.215048	+	0.976603i	$0.431009\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−9128.21	+	9128.21i	−0.383820	+	0.383820i	−0.872476	−	0.488656i	$-0.837487\pi$
$827$	−9128.21	+	9128.21i	−0.383820	+	0.383820i	0.488656	+	0.872476i	$0.337487\pi$
$828$		0			0
$829$			24401.3i			1.02231i	0.859489	+	0.511154i	$0.170782\pi$
$829$			24401.3i			1.02231i	−0.859489	+	0.511154i	$0.829218\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6450.73	+	6450.73i	0.268313	+	0.268313i
$834$		0			0
$835$	−677.271	+	10685.4i	−0.0280694	+	0.442854i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−38174.8			−1.57084			−0.785422	−	0.618960i	$-0.787554\pi$
$839$	−38174.8			−1.57084			−0.785422	+	0.618960i	$0.787554\pi$
$840$		0			0
$841$	−16224.9			−0.665256
$842$		0			0
$843$		0			0
$844$		0			0
$845$	17970.9	+	20403.1i	0.731618	+	0.830639i
$846$		0			0
$847$	−1137.45	−	1137.45i	−0.0461433	−	0.0461433i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1511.44i			0.0608833i
$852$		0			0
$853$	1757.73	−	1757.73i	0.0705550	−	0.0705550i	−0.670949	−	0.741504i	$-0.734113\pi$
$853$	1757.73	−	1757.73i	0.0705550	−	0.0705550i	0.741504	+	0.670949i	$0.234113\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7091.77	+	7091.77i	−0.282672	+	0.282672i	−0.834174	−	0.551501i	$-0.814055\pi$
$857$	−7091.77	+	7091.77i	−0.282672	+	0.282672i	0.551501	+	0.834174i	$0.314055\pi$
$858$		0			0
$859$			16842.4i			0.668982i	0.942399	+	0.334491i	$0.108564\pi$
$859$			16842.4i			0.668982i	−0.942399	+	0.334491i	$0.891436\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	33056.1	+	33056.1i	1.30387	+	1.30387i	0.925759	+	0.378114i	$0.123427\pi$
$863$	33056.1	+	33056.1i	1.30387	+	1.30387i	0.378114	+	0.925759i	$0.376573\pi$
$864$		0			0
$865$	6967.12	+	441.596i	0.273860	+	0.0173581i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	50107.9			1.95603
$870$		0			0
$871$	14475.3			0.563120
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4053.37	+	21088.2i	−0.156604	+	0.814754i
$876$		0			0
$877$	1780.97	+	1780.97i	0.0685738	+	0.0685738i	0.740562	−	0.671988i	$-0.234559\pi$
$877$	1780.97	+	1780.97i	0.0685738	+	0.0685738i	−0.671988	+	0.740562i	$0.734559\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			2698.82i			0.103207i	0.998668	+	0.0516036i	$0.0164332\pi$
$881$			2698.82i			0.103207i	−0.998668	+	0.0516036i	$0.983567\pi$
$882$		0			0
$883$	−6744.66	+	6744.66i	−0.257051	+	0.257051i	−0.823854	−	0.566803i	$-0.808180\pi$
$883$	−6744.66	+	6744.66i	−0.257051	+	0.257051i	0.566803	+	0.823854i	$0.308180\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1302.99	+	1302.99i	−0.0493238	+	0.0493238i	−0.731338	−	0.682015i	$-0.761104\pi$
$887$	−1302.99	+	1302.99i	−0.0493238	+	0.0493238i	0.682015	+	0.731338i	$0.261104\pi$
$888$		0			0
$889$		−	17092.1i		−	0.644827i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15989.7	−	15989.7i	−0.599189	−	0.599189i
$894$		0			0
$895$	−39636.4	+	34911.3i	−1.48033	+	1.30386i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	21114.0			0.783306
$900$		0			0
$901$	22121.0			0.817933
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12990.6	−	11442.0i	0.477152	−	0.420271i
$906$		0			0
$907$	3070.33	+	3070.33i	0.112402	+	0.112402i	0.761071	−	0.648669i	$-0.224674\pi$
$907$	3070.33	+	3070.33i	0.112402	+	0.112402i	−0.648669	+	0.761071i	$0.724674\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	19102.6i		−	0.694727i	−0.937731	−	0.347364i	$-0.887077\pi$
$911$		−	19102.6i		−	0.694727i	0.937731	−	0.347364i	$-0.112923\pi$
$912$		0			0
$913$	17285.3	−	17285.3i	0.626572	−	0.626572i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4871.55	−	4871.55i	0.175434	−	0.175434i
$918$		0			0
$919$			29786.7i			1.06917i	0.845113	+	0.534587i	$0.179533\pi$
$919$			29786.7i			1.06917i	−0.845113	+	0.534587i	$0.820467\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−22613.0	−	22613.0i	−0.806410	−	0.806410i
$924$		0			0
$925$	−2656.45	+	2056.59i	−0.0944254	+	0.0731029i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7724.87			0.272815			0.136407	−	0.990653i	$-0.456444\pi$
$929$	7724.87			0.272815			0.136407	+	0.990653i	$0.456444\pi$
$930$		0			0
$931$	−11724.1			−0.412720
$932$		0			0
$933$		0			0
$934$		0			0
$935$	36080.8	+	2286.91i	1.26200	+	0.0799892i
$936$		0			0
$937$	−20722.5	−	20722.5i	−0.722492	−	0.722492i	0.246620	−	0.969112i	$-0.420680\pi$
$937$	−20722.5	−	20722.5i	−0.722492	−	0.722492i	−0.969112	+	0.246620i	$0.920680\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			21780.7i			0.754548i	0.926102	+	0.377274i	$0.123138\pi$
$941$			21780.7i			0.754548i	−0.926102	+	0.377274i	$0.876862\pi$
$942$		0			0
$943$	10370.3	−	10370.3i	0.358116	−	0.358116i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	29938.4	−	29938.4i	1.02731	−	1.02731i	0.0276975	−	0.999616i	$-0.491182\pi$
$947$	29938.4	−	29938.4i	1.02731	−	1.02731i	0.999616	−	0.0276975i	$-0.00881753\pi$
$948$		0			0
$949$			46257.8i			1.58229i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	29572.1	+	29572.1i	1.00518	+	1.00518i	0.999987	+	0.00519163i	$0.00165255\pi$
$953$	29572.1	+	29572.1i	1.00518	+	1.00518i	0.00519163	+	0.999987i	$0.498347\pi$
$954$		0			0
$955$	8060.12	+	9151.01i	0.273109	+	0.310073i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−20646.4			−0.695212
$960$		0			0
$961$	24814.3			0.832945
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2826.16	−	44588.7i	0.0942769	−	1.48742i
$966$		0			0
$967$	9478.37	+	9478.37i	0.315206	+	0.315206i	0.846922	−	0.531717i	$-0.178453\pi$
$967$	9478.37	+	9478.37i	0.315206	+	0.315206i	−0.531717	+	0.846922i	$0.678453\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			8382.79i			0.277051i	0.990359	+	0.138525i	$0.0442363\pi$
$971$			8382.79i			0.277051i	−0.990359	+	0.138525i	$0.955764\pi$
$972$		0			0
$973$	1224.89	−	1224.89i	0.0403577	−	0.0403577i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−24277.3	+	24277.3i	−0.794986	+	0.794986i	−0.982300	−	0.187314i	$-0.940022\pi$
$977$	−24277.3	+	24277.3i	−0.794986	+	0.794986i	0.187314	+	0.982300i	$0.440022\pi$
$978$		0			0
$979$		−	51714.8i		−	1.68826i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−14666.4	−	14666.4i	−0.475876	−	0.475876i	0.427934	−	0.903810i	$-0.359242\pi$
$983$	−14666.4	−	14666.4i	−0.475876	−	0.475876i	−0.903810	+	0.427934i	$0.859242\pi$
$984$		0			0
$985$	1164.18	−	18367.4i	0.0376588	−	0.594147i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	14014.8			0.450603
$990$		0			0
$991$	−17192.0			−0.551080			−0.275540	−	0.961290i	$-0.588857\pi$
$991$	−17192.0			−0.551080			−0.275540	+	0.961290i	$0.588857\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3844.58	−	4364.92i	−0.122494	−	0.139073i
$996$		0			0
$997$	13617.4	+	13617.4i	0.432566	+	0.432566i	0.889500	−	0.456935i	$-0.151053\pi$
$997$	13617.4	+	13617.4i	0.432566	+	0.432566i	−0.456935	+	0.889500i	$0.651053\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.4.w.d.17.2		12
3.2	odd	2	inner	720.4.w.d.17.5		12
4.3	odd	2		45.4.f.a.17.2	yes	12
5.3	odd	4	inner	720.4.w.d.593.5		12
12.11	even	2		45.4.f.a.17.5	yes	12
15.8	even	4	inner	720.4.w.d.593.2		12
20.3	even	4		45.4.f.a.8.5	yes	12
20.7	even	4		225.4.f.c.143.2		12
20.19	odd	2		225.4.f.c.107.5		12
60.23	odd	4		45.4.f.a.8.2	✓	12
60.47	odd	4		225.4.f.c.143.5		12
60.59	even	2		225.4.f.c.107.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.f.a.8.2	✓	12	60.23	odd	4
45.4.f.a.8.5	yes	12	20.3	even	4
45.4.f.a.17.2	yes	12	4.3	odd	2
45.4.f.a.17.5	yes	12	12.11	even	2
225.4.f.c.107.2		12	60.59	even	2
225.4.f.c.107.5		12	20.19	odd	2
225.4.f.c.143.2		12	20.7	even	4
225.4.f.c.143.5		12	60.47	odd	4
720.4.w.d.17.2		12	1.1	even	1	trivial
720.4.w.d.17.5		12	3.2	odd	2	inner
720.4.w.d.593.2		12	15.8	even	4	inner
720.4.w.d.593.5		12	5.3	odd	4	inner

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	720.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-8.38994 + 7.38978i) q^{5} +(10.8652 + 10.8652i) q^{7} +37.8905i q^{11} +(-48.1085 + 48.1085i) q^{13} +(-60.3458 + 60.3458i) q^{17} -109.678i q^{19} +(-39.7660 - 39.7660i) q^{23} +(15.7823 - 124.000i) q^{25} +90.3553 q^{29} +233.678 q^{31} +(-171.449 - 10.8670i) q^{35} +(-19.0042 - 19.0042i) q^{37} +260.783i q^{41} +(-176.216 + 176.216i) q^{43} +(145.788 - 145.788i) q^{47} -106.896i q^{49} +(-183.285 - 183.285i) q^{53} +(-280.002 - 317.899i) q^{55} -279.564 q^{59} -390.267 q^{61} +(48.1164 - 759.139i) q^{65} +(-150.444 - 150.444i) q^{67} +470.042i q^{71} +(480.765 - 480.765i) q^{73} +(-411.687 + 411.687i) q^{77} -1322.44i q^{79} +(-456.192 - 456.192i) q^{83} +(60.3558 - 952.240i) q^{85} -1364.85 q^{89} -1045.41 q^{91} +(810.494 + 920.190i) q^{95} +(-785.308 - 785.308i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 24 q^{7} - 108 q^{13} - 144 q^{25} + 1248 q^{31} + 828 q^{37} + 96 q^{43} + 1512 q^{55} + 96 q^{61} - 1632 q^{67} + 3972 q^{73} - 1752 q^{85} - 4752 q^{91} + 2772 q^{97}+O(q^{100}) \) 12 * q - 24 * q^7 - 108 * q^13 - 144 * q^25 + 1248 * q^31 + 828 * q^37 + 96 * q^43 + 1512 * q^55 + 96 * q^61 - 1632 * q^67 + 3972 * q^73 - 1752 * q^85 - 4752 * q^91 + 2772 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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