LMFDB - Embedded newform 576.4.l.a.143.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [576,4,Mod(143,576)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("576.143"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	576.l (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.18
Character	$\chi$	$=$	576.143
Dual form			576.4.l.a.431.18

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.31601 - 6.31601i) q^{5} -16.2772 q^{7} +(-48.1287 - 48.1287i) q^{11} +(-8.61642 + 8.61642i) q^{13} +53.2206i q^{17} +(55.5604 + 55.5604i) q^{19} +66.9842i q^{23} +45.2160i q^{25} +(126.481 + 126.481i) q^{29} +121.117i q^{31} +(-102.807 + 102.807i) q^{35} +(-250.289 - 250.289i) q^{37} +402.012 q^{41} +(-187.233 + 187.233i) q^{43} +96.1703 q^{47} -78.0518 q^{49} +(-90.3337 + 90.3337i) q^{53} -607.963 q^{55} +(488.025 + 488.025i) q^{59} +(378.467 - 378.467i) q^{61} +108.843i q^{65} +(223.231 + 223.231i) q^{67} +231.902i q^{71} +265.600i q^{73} +(783.402 + 783.402i) q^{77} +604.662i q^{79} +(-351.298 + 351.298i) q^{83} +(336.142 + 336.142i) q^{85} -1365.36 q^{89} +(140.251 - 140.251i) q^{91} +701.839 q^{95} -1854.47 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 48 q^{19} - 864 q^{43} + 2352 q^{49} + 576 q^{55} + 1824 q^{61} - 816 q^{67} - 480 q^{85} + 3600 q^{91}+O(q^{100}) $ 48 * q + 48 * q^19 - 864 * q^43 + 2352 * q^49 + 576 * q^55 + 1824 * q^61 - 816 * q^67 - 480 * q^85 + 3600 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	6.31601	−	6.31601i	0.564921	−	0.564921i	−0.365780	−	0.930701i	$-0.619198\pi$
$5$	6.31601	−	6.31601i	0.564921	−	0.564921i	0.930701	+	0.365780i	$0.119198\pi$
$6$		0			0
$7$	−16.2772			−0.878888			−0.439444	−	0.898270i	$-0.644824\pi$
$7$	−16.2772			−0.878888			−0.439444	+	0.898270i	$0.644824\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−48.1287	−	48.1287i	−1.31921	−	1.31921i	−0.914398	−	0.404816i	$-0.867335\pi$
$11$	−48.1287	−	48.1287i	−1.31921	−	1.31921i	−0.404816	−	0.914398i	$-0.632665\pi$
$12$		0			0
$13$	−8.61642	+	8.61642i	−0.183828	+	0.183828i	−0.793022	−	0.609194i	$-0.791493\pi$
$13$	−8.61642	+	8.61642i	−0.183828	+	0.183828i	0.609194	+	0.793022i	$0.291493\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			53.2206i			0.759287i	0.925133	+	0.379644i	$0.123953\pi$
$17$			53.2206i			0.759287i	−0.925133	+	0.379644i	$0.876047\pi$
$18$		0			0
$19$	55.5604	+	55.5604i	0.670864	+	0.670864i	0.957915	−	0.287051i	$-0.0926750\pi$
$19$	55.5604	+	55.5604i	0.670864	+	0.670864i	−0.287051	+	0.957915i	$0.592675\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			66.9842i			0.607269i	0.952789	+	0.303634i	$0.0982001\pi$
$23$			66.9842i			0.607269i	−0.952789	+	0.303634i	$0.901800\pi$
$24$		0			0
$25$			45.2160i			0.361728i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	126.481	+	126.481i	0.809892	+	0.809892i	0.984617	−	0.174726i	$-0.0559038\pi$
$29$	126.481	+	126.481i	0.809892	+	0.809892i	−0.174726	+	0.984617i	$0.555904\pi$
$30$		0			0
$31$			121.117i			0.701717i	0.936429	+	0.350858i	$0.114110\pi$
$31$			121.117i			0.701717i	−0.936429	+	0.350858i	$0.885890\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−102.807	+	102.807i	−0.496502	+	0.496502i
$36$		0			0
$37$	−250.289	−	250.289i	−1.11209	−	1.11209i	−0.992868	−	0.119222i	$-0.961960\pi$
$37$	−250.289	−	250.289i	−1.11209	−	1.11209i	−0.119222	−	0.992868i	$-0.538040\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	402.012			1.53131			0.765655	−	0.643252i	$-0.222415\pi$
$41$	402.012			1.53131			0.765655	+	0.643252i	$0.222415\pi$
$42$		0			0
$43$	−187.233	+	187.233i	−0.664018	+	0.664018i	−0.956325	−	0.292307i	$-0.905577\pi$
$43$	−187.233	+	187.233i	−0.664018	+	0.664018i	0.292307	+	0.956325i	$0.405577\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	96.1703			0.298465			0.149233	−	0.988802i	$-0.452320\pi$
$47$	96.1703			0.298465			0.149233	+	0.988802i	$0.452320\pi$
$48$		0			0
$49$	−78.0518			−0.227556
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−90.3337	+	90.3337i	−0.234119	+	0.234119i	−0.814409	−	0.580291i	$-0.802939\pi$
$53$	−90.3337	+	90.3337i	−0.234119	+	0.234119i	0.580291	+	0.814409i	$0.302939\pi$
$54$		0			0
$55$	−607.963			−1.49050
$56$		0			0
$57$		0			0
$58$		0			0
$59$	488.025	+	488.025i	1.07687	+	1.07687i	0.996788	+	0.0800840i	$0.0255188\pi$
$59$	488.025	+	488.025i	1.07687	+	1.07687i	0.0800840	+	0.996788i	$0.474481\pi$
$60$		0			0
$61$	378.467	−	378.467i	0.794389	−	0.794389i	−0.187815	−	0.982204i	$-0.560141\pi$
$61$	378.467	−	378.467i	0.794389	−	0.794389i	0.982204	+	0.187815i	$0.0601405\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			108.843i			0.207697i
$66$		0			0
$67$	223.231	+	223.231i	0.407045	+	0.407045i	0.880707	−	0.473662i	$-0.157068\pi$
$67$	223.231	+	223.231i	0.407045	+	0.407045i	−0.473662	+	0.880707i	$0.657068\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			231.902i			0.387630i	0.981038	+	0.193815i	$0.0620862\pi$
$71$			231.902i			0.387630i	−0.981038	+	0.193815i	$0.937914\pi$
$72$		0			0
$73$			265.600i			0.425838i	0.977070	+	0.212919i	$0.0682970\pi$
$73$			265.600i			0.425838i	−0.977070	+	0.212919i	$0.931703\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	783.402	+	783.402i	1.15944	+	1.15944i
$78$		0			0
$79$			604.662i			0.861137i	0.902558	+	0.430569i	$0.141687\pi$
$79$			604.662i			0.861137i	−0.902558	+	0.430569i	$0.858313\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−351.298	+	351.298i	−0.464577	+	0.464577i	−0.900152	−	0.435575i	$-0.856545\pi$
$83$	−351.298	+	351.298i	−0.464577	+	0.464577i	0.435575	+	0.900152i	$0.356545\pi$
$84$		0			0
$85$	336.142	+	336.142i	0.428937	+	0.428937i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1365.36			−1.62615			−0.813077	−	0.582156i	$-0.802209\pi$
$89$	−1365.36			−1.62615			−0.813077	+	0.582156i	$0.802209\pi$
$90$		0			0
$91$	140.251	−	140.251i	0.161564	−	0.161564i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	701.839			0.757971
$96$		0			0
$97$	−1854.47			−1.94117			−0.970584	−	0.240762i	$-0.922603\pi$
$97$	−1854.47			−1.94117			−0.970584	+	0.240762i	$0.922603\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	755.210	−	755.210i	0.744022	−	0.744022i	−0.229328	−	0.973349i	$-0.573653\pi$
$101$	755.210	−	755.210i	0.744022	−	0.744022i	0.973349	+	0.229328i	$0.0736527\pi$
$102$		0			0
$103$	−1428.89			−1.36692			−0.683461	−	0.729987i	$-0.739526\pi$
$103$	−1428.89			−1.36692			−0.683461	+	0.729987i	$0.739526\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1013.69	+	1013.69i	0.915859	+	0.915859i	0.996725	−	0.0808660i	$-0.0257686\pi$
$107$	1013.69	+	1013.69i	0.915859	+	0.915859i	−0.0808660	+	0.996725i	$0.525769\pi$
$108$		0			0
$109$	−1069.24	+	1069.24i	−0.939579	+	0.939579i	−0.998276	−	0.0586964i	$-0.981306\pi$
$109$	−1069.24	+	1069.24i	−0.939579	+	0.939579i	0.0586964	+	0.998276i	$0.481306\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1280.42i			1.06595i	0.846132	+	0.532974i	$0.178926\pi$
$113$			1280.42i			1.06595i	−0.846132	+	0.532974i	$0.821074\pi$
$114$		0			0
$115$	423.073	+	423.073i	0.343059	+	0.343059i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	866.283i		−	0.667328i
$120$		0			0
$121$			3301.75i			2.48065i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1075.09	+	1075.09i	0.769269	+	0.769269i
$126$		0			0
$127$		−	2778.90i		−	1.94163i	−0.239828	−	0.970815i	$-0.577091\pi$
$127$		−	2778.90i		−	1.94163i	0.239828	−	0.970815i	$-0.422909\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−938.133	+	938.133i	−0.625687	+	0.625687i	−0.946980	−	0.321293i	$-0.895883\pi$
$131$	−938.133	+	938.133i	−0.625687	+	0.625687i	0.321293	+	0.946980i	$0.395883\pi$
$132$		0			0
$133$	−904.369	−	904.369i	−0.589614	−	0.589614i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	96.6385			0.0602656			0.0301328	−	0.999546i	$-0.490407\pi$
$137$	96.6385			0.0602656			0.0301328	+	0.999546i	$0.490407\pi$
$138$		0			0
$139$	211.002	−	211.002i	0.128755	−	0.128755i	−0.639793	−	0.768548i	$-0.720980\pi$
$139$	211.002	−	211.002i	0.128755	−	0.128755i	0.768548	+	0.639793i	$0.220980\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	829.395			0.485017
$144$		0			0
$145$	1597.71			0.915050
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1113.17	+	1113.17i	−0.612042	+	0.612042i	−0.943478	−	0.331435i	$-0.892467\pi$
$149$	−1113.17	+	1113.17i	−0.612042	+	0.612042i	0.331435	+	0.943478i	$0.392467\pi$
$150$		0			0
$151$	−285.222			−0.153716			−0.0768578	−	0.997042i	$-0.524489\pi$
$151$	−285.222			−0.153716			−0.0768578	+	0.997042i	$0.524489\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	764.975	+	764.975i	0.396414	+	0.396414i
$156$		0			0
$157$	2536.76	−	2536.76i	1.28953	−	1.28953i	0.354454	−	0.935073i	$-0.384667\pi$
$157$	2536.76	−	2536.76i	1.28953	−	1.28953i	0.935073	−	0.354454i	$-0.115333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	1090.32i		−	0.533721i
$162$		0			0
$163$	1268.22	+	1268.22i	0.609414	+	0.609414i	0.942793	−	0.333379i	$-0.108189\pi$
$163$	1268.22	+	1268.22i	0.609414	+	0.609414i	−0.333379	+	0.942793i	$0.608189\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	2766.46i		−	1.28189i	−0.767588	−	0.640943i	$-0.778543\pi$
$167$		−	2766.46i		−	1.28189i	0.767588	−	0.640943i	$-0.221457\pi$
$168$		0			0
$169$			2048.51i			0.932415i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−346.606	−	346.606i	−0.152324	−	0.152324i	0.626831	−	0.779155i	$-0.284351\pi$
$173$	−346.606	−	346.606i	−0.152324	−	0.152324i	−0.779155	+	0.626831i	$0.784351\pi$
$174$		0			0
$175$		−	735.992i		−	0.317919i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	106.140	−	106.140i	0.0443198	−	0.0443198i	−0.684600	−	0.728919i	$-0.740023\pi$
$179$	106.140	−	106.140i	0.0443198	−	0.0443198i	0.728919	+	0.684600i	$0.240023\pi$
$180$		0			0
$181$	−1553.97	−	1553.97i	−0.638154	−	0.638154i	0.311946	−	0.950100i	$-0.399019\pi$
$181$	−1553.97	−	1553.97i	−0.638154	−	0.638154i	−0.950100	+	0.311946i	$0.899019\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3161.66			−1.25649
$186$		0			0
$187$	2561.44	−	2561.44i	1.00166	−	1.00166i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1311.80			0.496955			0.248478	−	0.968638i	$-0.420070\pi$
$191$	1311.80			0.496955			0.248478	+	0.968638i	$0.420070\pi$
$192$		0			0
$193$	−1692.72			−0.631319			−0.315659	−	0.948873i	$-0.602226\pi$
$193$	−1692.72			−0.631319			−0.315659	+	0.948873i	$0.602226\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2419.50	−	2419.50i	0.875037	−	0.875037i	−0.117979	−	0.993016i	$-0.537642\pi$
$197$	2419.50	−	2419.50i	0.875037	−	0.875037i	0.993016	+	0.117979i	$0.0376416\pi$
$198$		0			0
$199$	−4687.76			−1.66988			−0.834941	−	0.550339i	$-0.814498\pi$
$199$	−4687.76			−1.66988			−0.834941	+	0.550339i	$0.814498\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2058.75	−	2058.75i	−0.711804	−	0.711804i
$204$		0			0
$205$	2539.11	−	2539.11i	0.865069	−	0.865069i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	5348.10i		−	1.77003i
$210$		0			0
$211$	1994.60	+	1994.60i	0.650777	+	0.650777i	0.953180	−	0.302403i	$-0.0977888\pi$
$211$	1994.60	+	1994.60i	0.650777	+	0.650777i	−0.302403	+	0.953180i	$0.597789\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			2365.13i			0.750236i
$216$		0			0
$217$		−	1971.45i		−	0.616730i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−458.571	−	458.571i	−0.139578	−	0.139578i
$222$		0			0
$223$			3668.79i			1.10170i	0.834603	+	0.550852i	$0.185697\pi$
$223$			3668.79i			1.10170i	−0.834603	+	0.550852i	$0.814303\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2455.95	+	2455.95i	−0.718092	+	0.718092i	−0.968214	−	0.250123i	$-0.919529\pi$
$227$	−2455.95	+	2455.95i	−0.718092	+	0.718092i	0.250123	+	0.968214i	$0.419529\pi$
$228$		0			0
$229$	1201.57	+	1201.57i	0.346733	+	0.346733i	0.858891	−	0.512158i	$-0.171154\pi$
$229$	1201.57	+	1201.57i	0.346733	+	0.346733i	−0.512158	+	0.858891i	$0.671154\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1560.55			−0.438776			−0.219388	−	0.975638i	$-0.570406\pi$
$233$	−1560.55			−0.438776			−0.219388	+	0.975638i	$0.570406\pi$
$234$		0			0
$235$	607.412	−	607.412i	0.168609	−	0.168609i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3543.70			−0.959091			−0.479546	−	0.877517i	$-0.659198\pi$
$239$	−3543.70			−0.959091			−0.479546	+	0.877517i	$0.659198\pi$
$240$		0			0
$241$	1481.98			0.396110			0.198055	−	0.980191i	$-0.436538\pi$
$241$	1481.98			0.396110			0.198055	+	0.980191i	$0.436538\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−492.976	+	492.976i	−0.128551	+	0.128551i
$246$		0			0
$247$	−957.462			−0.246647
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2357.87	−	2357.87i	−0.592938	−	0.592938i	0.345486	−	0.938424i	$-0.387714\pi$
$251$	−2357.87	−	2357.87i	−0.592938	−	0.592938i	−0.938424	+	0.345486i	$0.887714\pi$
$252$		0			0
$253$	3223.87	−	3223.87i	0.801118	−	0.801118i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			2117.93i			0.514059i	0.966404	+	0.257029i	$0.0827437\pi$
$257$			2117.93i			0.514059i	−0.966404	+	0.257029i	$0.917256\pi$
$258$		0			0
$259$	4074.02	+	4074.02i	0.977402	+	0.977402i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	7741.72i		−	1.81511i	−0.419929	−	0.907557i	$-0.637945\pi$
$263$		−	7741.72i		−	1.81511i	0.419929	−	0.907557i	$-0.362055\pi$
$264$		0			0
$265$			1141.10i			0.264517i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1111.14	+	1111.14i	0.251849	+	0.251849i	0.821728	−	0.569880i	$-0.193010\pi$
$269$	1111.14	+	1111.14i	0.251849	+	0.251849i	−0.569880	+	0.821728i	$0.693010\pi$
$270$		0			0
$271$			1728.54i			0.387458i	0.981055	+	0.193729i	$0.0620583\pi$
$271$			1728.54i			0.387458i	−0.981055	+	0.193729i	$0.937942\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2176.19	−	2176.19i	0.477197	−	0.477197i
$276$		0			0
$277$	−1977.36	−	1977.36i	−0.428910	−	0.428910i	0.459347	−	0.888257i	$-0.348083\pi$
$277$	−1977.36	−	1977.36i	−0.428910	−	0.428910i	−0.888257	+	0.459347i	$0.848083\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1583.09			0.336083			0.168041	−	0.985780i	$-0.446256\pi$
$281$	1583.09			0.336083			0.168041	+	0.985780i	$0.446256\pi$
$282$		0			0
$283$	987.718	−	987.718i	0.207469	−	0.207469i	−0.595722	−	0.803191i	$-0.703134\pi$
$283$	987.718	−	987.718i	0.207469	−	0.207469i	0.803191	+	0.595722i	$0.203134\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6543.64			−1.34585
$288$		0			0
$289$	2080.57			0.423483
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−140.498	+	140.498i	−0.0280136	+	0.0280136i	−0.720975	−	0.692961i	$-0.756306\pi$
$293$	−140.498	+	140.498i	−0.0280136	+	0.0280136i	0.692961	+	0.720975i	$0.256306\pi$
$294$		0			0
$295$	6164.74			1.21670
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−577.164	−	577.164i	−0.111633	−	0.111633i
$300$		0			0
$301$	3047.64	−	3047.64i	0.583597	−	0.583597i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	4780.80i		−	0.897535i
$306$		0			0
$307$	−5197.73	−	5197.73i	−0.966288	−	0.966288i	0.0331621	−	0.999450i	$-0.489442\pi$
$307$	−5197.73	−	5197.73i	−0.966288	−	0.966288i	−0.999450	+	0.0331621i	$0.989442\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	8112.26i		−	1.47911i	−0.673094	−	0.739557i	$-0.735035\pi$
$311$		−	8112.26i		−	1.47911i	0.673094	−	0.739557i	$-0.264965\pi$
$312$		0			0
$313$		−	3307.27i		−	0.597246i	−0.954371	−	0.298623i	$-0.903473\pi$
$313$		−	3307.27i		−	0.597246i	0.954371	−	0.298623i	$-0.0965273\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2991.50	−	2991.50i	−0.530030	−	0.530030i	0.390551	−	0.920581i	$-0.372284\pi$
$317$	−2991.50	−	2991.50i	−0.530030	−	0.530030i	−0.920581	+	0.390551i	$0.872284\pi$
$318$		0			0
$319$		−	12174.7i		−	2.13684i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2956.95	+	2956.95i	−0.509379	+	0.509379i
$324$		0			0
$325$	−389.600	−	389.600i	−0.0664958	−	0.0664958i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1565.39			−0.262318
$330$		0			0
$331$	−3171.15	+	3171.15i	−0.526592	+	0.526592i	−0.919555	−	0.392962i	$-0.871450\pi$
$331$	−3171.15	+	3171.15i	−0.526592	+	0.526592i	0.392962	+	0.919555i	$0.371450\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2819.86			0.459897
$336$		0			0
$337$	−2213.85			−0.357852			−0.178926	−	0.983863i	$-0.557262\pi$
$337$	−2213.85			−0.357852			−0.178926	+	0.983863i	$0.557262\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5829.20	−	5829.20i	0.925715	−	0.925715i
$342$		0			0
$343$	6853.56			1.07888
$344$		0			0
$345$		0			0
$346$		0			0
$347$	344.315	+	344.315i	0.0532674	+	0.0532674i	0.733239	−	0.679971i	$-0.238008\pi$
$347$	344.315	+	344.315i	0.0532674	+	0.0532674i	−0.679971	+	0.733239i	$0.738008\pi$
$348$		0			0
$349$	−8129.29	+	8129.29i	−1.24685	+	1.24685i	−0.289747	+	0.957103i	$0.593571\pi$
$349$	−8129.29	+	8129.29i	−1.24685	+	1.24685i	−0.957103	+	0.289747i	$0.906429\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			3235.69i			0.487871i	0.969791	+	0.243935i	$0.0784385\pi$
$353$			3235.69i			0.487871i	−0.969791	+	0.243935i	$0.921561\pi$
$354$		0			0
$355$	1464.70	+	1464.70i	0.218980	+	0.218980i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4710.91i			0.692569i	0.938129	+	0.346285i	$0.112557\pi$
$359$			4710.91i			0.692569i	−0.938129	+	0.346285i	$0.887443\pi$
$360$		0			0
$361$		−	685.094i		−	0.0998825i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1677.53	+	1677.53i	0.240565	+	0.240565i
$366$		0			0
$367$			1585.57i			0.225521i	0.993622	+	0.112761i	$0.0359693\pi$
$367$			1585.57i			0.225521i	−0.993622	+	0.112761i	$0.964031\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1470.38	−	1470.38i	0.205764	−	0.205764i
$372$		0			0
$373$	2078.83	+	2078.83i	0.288573	+	0.288573i	0.836516	−	0.547943i	$-0.184589\pi$
$373$	2078.83	+	2078.83i	0.288573	+	0.288573i	−0.547943	+	0.836516i	$0.684589\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2179.62			−0.297762
$378$		0			0
$379$	3704.15	−	3704.15i	0.502031	−	0.502031i	−0.410038	−	0.912068i	$-0.634485\pi$
$379$	3704.15	−	3704.15i	0.502031	−	0.502031i	0.912068	+	0.410038i	$0.134485\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7560.19			1.00864			0.504318	−	0.863518i	$-0.331744\pi$
$383$	7560.19			1.00864			0.504318	+	0.863518i	$0.331744\pi$
$384$		0			0
$385$	9895.95			1.30999
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4868.51	+	4868.51i	−0.634559	+	0.634559i	−0.949208	−	0.314649i	$-0.898113\pi$
$389$	−4868.51	+	4868.51i	−0.634559	+	0.634559i	0.314649	+	0.949208i	$0.398113\pi$
$390$		0			0
$391$	−3564.94			−0.461091
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3819.05	+	3819.05i	0.486474	+	0.486474i
$396$		0			0
$397$	−2442.83	+	2442.83i	−0.308822	+	0.308822i	−0.844452	−	0.535631i	$-0.820074\pi$
$397$	−2442.83	+	2442.83i	−0.308822	+	0.308822i	0.535631	+	0.844452i	$0.320074\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			10348.7i			1.28876i	0.764707	+	0.644378i	$0.222883\pi$
$401$			10348.7i			1.28876i	−0.764707	+	0.644378i	$0.777117\pi$
$402$		0			0
$403$	−1043.59	−	1043.59i	−0.128995	−	0.128995i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			24092.2i			2.93417i
$408$		0			0
$409$		−	3378.72i		−	0.408477i	−0.978921	−	0.204238i	$-0.934528\pi$
$409$		−	3378.72i		−	0.408477i	0.978921	−	0.204238i	$-0.0654717\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7943.70	−	7943.70i	−0.946450	−	0.946450i
$414$		0			0
$415$			4437.60i			0.524899i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8945.75	−	8945.75i	1.04303	−	1.04303i	0.0439961	−	0.999032i	$-0.485991\pi$
$419$	8945.75	−	8945.75i	1.04303	−	1.04303i	0.999032	−	0.0439961i	$-0.0140089\pi$
$420$		0			0
$421$	10686.4	+	10686.4i	1.23711	+	1.23711i	0.961177	+	0.275933i	$0.0889867\pi$
$421$	10686.4	+	10686.4i	1.23711	+	1.23711i	0.275933	+	0.961177i	$0.411013\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2406.42			−0.274656
$426$		0			0
$427$	−6160.40	+	6160.40i	−0.698179	+	0.698179i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4267.29			0.476910			0.238455	−	0.971154i	$-0.423359\pi$
$431$	4267.29			0.476910			0.238455	+	0.971154i	$0.423359\pi$
$432$		0			0
$433$	−15686.4			−1.74097			−0.870483	−	0.492198i	$-0.836194\pi$
$433$	−15686.4			−1.74097			−0.870483	+	0.492198i	$0.836194\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3721.67	+	3721.67i	−0.407395	+	0.407395i
$438$		0			0
$439$	−5963.75			−0.648370			−0.324185	−	0.945994i	$-0.605090\pi$
$439$	−5963.75			−0.648370			−0.324185	+	0.945994i	$0.605090\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1512.40	−	1512.40i	−0.162204	−	0.162204i	0.621338	−	0.783542i	$-0.286589\pi$
$443$	−1512.40	−	1512.40i	−0.162204	−	0.162204i	−0.783542	+	0.621338i	$0.786589\pi$
$444$		0			0
$445$	−8623.62	+	8623.62i	−0.918649	+	0.918649i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	5201.31i		−	0.546693i	−0.961916	−	0.273346i	$-0.911869\pi$
$449$		−	5201.31i		−	0.546693i	0.961916	−	0.273346i	$-0.0881305\pi$
$450$		0			0
$451$	−19348.3	−	19348.3i	−2.02013	−	2.02013i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	1771.66i		−	0.182542i
$456$		0			0
$457$			6308.57i			0.645738i	0.946444	+	0.322869i	$0.104647\pi$
$457$			6308.57i			0.645738i	−0.946444	+	0.322869i	$0.895353\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1200.54	+	1200.54i	0.121290	+	0.121290i	0.765146	−	0.643856i	$-0.222667\pi$
$461$	1200.54	+	1200.54i	0.121290	+	0.121290i	−0.643856	+	0.765146i	$0.722667\pi$
$462$		0			0
$463$			15194.4i			1.52515i	0.646901	+	0.762574i	$0.276065\pi$
$463$			15194.4i			1.52515i	−0.646901	+	0.762574i	$0.723935\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4068.55	−	4068.55i	0.403148	−	0.403148i	−0.476193	−	0.879341i	$-0.657984\pi$
$467$	4068.55	−	4068.55i	0.403148	−	0.403148i	0.879341	+	0.476193i	$0.157984\pi$
$468$		0			0
$469$	−3633.58	−	3633.58i	−0.357747	−	0.357747i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	18022.6			1.75196
$474$		0			0
$475$	−2512.22	+	2512.22i	−0.242671	+	0.242671i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3340.15			−0.318612			−0.159306	−	0.987229i	$-0.550926\pi$
$479$	−3340.15			−0.318612			−0.159306	+	0.987229i	$0.550926\pi$
$480$		0			0
$481$	4313.20			0.408867
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11712.9	+	11712.9i	−1.09661	+	1.09661i
$486$		0			0
$487$	10788.7			1.00386			0.501932	−	0.864907i	$-0.332623\pi$
$487$	10788.7			1.00386			0.501932	+	0.864907i	$0.332623\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4711.73	+	4711.73i	0.433071	+	0.433071i	0.889672	−	0.456601i	$-0.150933\pi$
$491$	4711.73	+	4711.73i	0.433071	+	0.433071i	−0.456601	+	0.889672i	$0.650933\pi$
$492$		0			0
$493$	−6731.37	+	6731.37i	−0.614940	+	0.614940i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	3774.73i		−	0.340683i
$498$		0			0
$499$	5656.27	+	5656.27i	0.507433	+	0.507433i	0.913738	−	0.406304i	$-0.133183\pi$
$499$	5656.27	+	5656.27i	0.507433	+	0.507433i	−0.406304	+	0.913738i	$0.633183\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			4974.64i			0.440971i	0.975390	+	0.220485i	$0.0707641\pi$
$503$			4974.64i			0.440971i	−0.975390	+	0.220485i	$0.929236\pi$
$504$		0			0
$505$		−	9539.83i		−	0.840627i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5549.92	+	5549.92i	0.483292	+	0.483292i	0.906181	−	0.422889i	$-0.138984\pi$
$509$	5549.92	+	5549.92i	0.483292	+	0.483292i	−0.422889	+	0.906181i	$0.638984\pi$
$510$		0			0
$511$		−	4323.24i		−	0.374264i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9024.90	+	9024.90i	−0.772203	+	0.772203i
$516$		0			0
$517$	−4628.55	−	4628.55i	−0.393740	−	0.393740i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	21737.7			1.82792			0.913961	−	0.405802i	$-0.133008\pi$
$521$	21737.7			1.82792			0.913961	+	0.405802i	$0.133008\pi$
$522$		0			0
$523$	−2235.12	+	2235.12i	−0.186874	+	0.186874i	−0.794343	−	0.607469i	$-0.792185\pi$
$523$	−2235.12	+	2235.12i	−0.186874	+	0.186874i	0.607469	+	0.794343i	$0.292185\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6445.90			−0.532804
$528$		0			0
$529$	7680.11			0.631225
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3463.90	+	3463.90i	−0.281498	+	0.281498i
$534$		0			0
$535$	12804.9			1.03478
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3756.53	+	3756.53i	0.300196	+	0.300196i
$540$		0			0
$541$	−3371.85	+	3371.85i	−0.267961	+	0.267961i	−0.828278	−	0.560317i	$-0.810679\pi$
$541$	−3371.85	+	3371.85i	−0.267961	+	0.267961i	0.560317	+	0.828278i	$0.310679\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			13506.6i			1.06158i
$546$		0			0
$547$	−3242.62	−	3242.62i	−0.253463	−	0.253463i	0.568926	−	0.822389i	$-0.307359\pi$
$547$	−3242.62	−	3242.62i	−0.253463	−	0.253463i	−0.822389	+	0.568926i	$0.807359\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			14054.6i			1.08665i
$552$		0			0
$553$		−	9842.23i		−	0.756843i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4967.01	−	4967.01i	−0.377844	−	0.377844i	0.492480	−	0.870324i	$-0.336091\pi$
$557$	−4967.01	−	4967.01i	−0.377844	−	0.377844i	−0.870324	+	0.492480i	$0.836091\pi$
$558$		0			0
$559$		−	3226.56i		−	0.244130i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10845.8	−	10845.8i	0.811891	−	0.811891i	−0.173026	−	0.984917i	$-0.555355\pi$
$563$	10845.8	−	10845.8i	0.811891	−	0.811891i	0.984917	+	0.173026i	$0.0553545\pi$
$564$		0			0
$565$	8087.17	+	8087.17i	0.602177	+	0.602177i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18374.6			1.35378			0.676892	−	0.736082i	$-0.263326\pi$
$569$	18374.6			1.35378			0.676892	+	0.736082i	$0.263326\pi$
$570$		0			0
$571$	−4779.72	+	4779.72i	−0.350306	+	0.350306i	−0.860224	−	0.509917i	$-0.829676\pi$
$571$	−4779.72	+	4779.72i	−0.350306	+	0.350306i	0.509917	+	0.860224i	$0.329676\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3028.76			−0.219666
$576$		0			0
$577$	−14300.3			−1.03176			−0.515882	−	0.856659i	$-0.672536\pi$
$577$	−14300.3			−1.03176			−0.515882	+	0.856659i	$0.672536\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5718.15	−	5718.15i	0.408311	−	0.408311i
$582$		0			0
$583$	8695.29			0.617705
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7852.73	−	7852.73i	−0.552158	−	0.552158i	0.374905	−	0.927063i	$-0.377675\pi$
$587$	−7852.73	−	7852.73i	−0.552158	−	0.552158i	−0.927063	+	0.374905i	$0.877675\pi$
$588$		0			0
$589$	−6729.29	+	6729.29i	−0.470756	+	0.470756i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			27409.5i			1.89810i	0.315122	+	0.949051i	$0.397954\pi$
$593$			27409.5i			1.89810i	−0.315122	+	0.949051i	$0.602046\pi$
$594$		0			0
$595$	−5471.45	−	5471.45i	−0.376988	−	0.376988i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			8983.16i			0.612758i	0.951910	+	0.306379i	$0.0991174\pi$
$599$			8983.16i			0.612758i	−0.951910	+	0.306379i	$0.900883\pi$
$600$		0			0
$601$		−	1686.23i		−	0.114447i	−0.998361	−	0.0572236i	$-0.981775\pi$
$601$		−	1686.23i		−	0.114447i	0.998361	−	0.0572236i	$-0.0182248\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	20853.9	+	20853.9i	1.40137	+	1.40137i
$606$		0			0
$607$			19571.7i			1.30872i	0.756185	+	0.654358i	$0.227061\pi$
$607$			19571.7i			1.30872i	−0.756185	+	0.654358i	$0.772939\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−828.643	+	828.643i	−0.0548663	+	0.0548663i
$612$		0			0
$613$	−16131.0	−	16131.0i	−1.06285	−	1.06285i	−0.997888	−	0.0649592i	$-0.979308\pi$
$613$	−16131.0	−	16131.0i	−1.06285	−	1.06285i	−0.0649592	−	0.997888i	$-0.520692\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14419.1			−0.940829			−0.470415	−	0.882445i	$-0.655896\pi$
$617$	−14419.1			−0.940829			−0.470415	+	0.882445i	$0.655896\pi$
$618$		0			0
$619$	−14348.2	+	14348.2i	−0.931665	+	0.931665i	−0.997810	−	0.0661446i	$-0.978930\pi$
$619$	−14348.2	+	14348.2i	−0.931665	+	0.931665i	0.0661446	+	0.997810i	$0.478930\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	22224.3			1.42921
$624$		0			0
$625$	7928.50			0.507424
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13320.5	−	13320.5i	0.844395	−	0.844395i
$630$		0			0
$631$	−28650.9			−1.80757			−0.903785	−	0.427987i	$-0.859223\pi$
$631$	−28650.9			−1.80757			−0.903785	+	0.427987i	$0.859223\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−17551.5	−	17551.5i	−1.09687	−	1.09687i
$636$		0			0
$637$	672.527	−	672.527i	0.0418312	−	0.0418312i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	11361.1i		−	0.700060i	−0.936739	−	0.350030i	$-0.886171\pi$
$641$		−	11361.1i		−	0.700060i	0.936739	−	0.350030i	$-0.113829\pi$
$642$		0			0
$643$	−1801.19	−	1801.19i	−0.110470	−	0.110470i	0.649711	−	0.760181i	$-0.274890\pi$
$643$	−1801.19	−	1801.19i	−0.110470	−	0.110470i	−0.760181	+	0.649711i	$0.774890\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	3042.62i		−	0.184881i	−0.995718	−	0.0924403i	$-0.970533\pi$
$647$		−	3042.62i		−	0.184881i	0.995718	−	0.0924403i	$-0.0294667\pi$
$648$		0			0
$649$		−	46976.1i		−	2.84125i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15755.2	−	15755.2i	−0.944176	−	0.944176i	0.0543465	−	0.998522i	$-0.482692\pi$
$653$	−15755.2	−	15755.2i	−0.944176	−	0.944176i	−0.998522	+	0.0543465i	$0.982692\pi$
$654$		0			0
$655$			11850.5i			0.706928i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3650.15	−	3650.15i	0.215766	−	0.215766i	−0.590946	−	0.806711i	$-0.701245\pi$
$659$	3650.15	−	3650.15i	0.215766	−	0.215766i	0.806711	+	0.590946i	$0.201245\pi$
$660$		0			0
$661$	17498.6	+	17498.6i	1.02968	+	1.02968i	0.999546	+	0.0301299i	$0.00959208\pi$
$661$	17498.6	+	17498.6i	1.02968	+	1.02968i	0.0301299	+	0.999546i	$0.490408\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−11424.0			−0.666171
$666$		0			0
$667$	−8472.21	+	8472.21i	−0.491822	+	0.491822i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−36430.3			−2.09594
$672$		0			0
$673$	5976.34			0.342305			0.171152	−	0.985245i	$-0.445251\pi$
$673$	5976.34			0.342305			0.171152	+	0.985245i	$0.445251\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5706.93	−	5706.93i	0.323981	−	0.323981i	−0.526311	−	0.850292i	$-0.676425\pi$
$677$	5706.93	−	5706.93i	0.323981	−	0.323981i	0.850292	+	0.526311i	$0.176425\pi$
$678$		0			0
$679$	30185.7			1.70607
$680$		0			0
$681$		0			0
$682$		0			0
$683$	18585.6	+	18585.6i	1.04123	+	1.04123i	0.999113	+	0.0421122i	$0.0134087\pi$
$683$	18585.6	+	18585.6i	1.04123	+	1.04123i	0.0421122	+	0.999113i	$0.486591\pi$
$684$		0			0
$685$	610.370	−	610.370i	0.0340453	−	0.0340453i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1556.71i		−	0.0860751i
$690$		0			0
$691$	7076.12	+	7076.12i	0.389563	+	0.389563i	0.874532	−	0.484968i	$-0.161169\pi$
$691$	7076.12	+	7076.12i	0.389563	+	0.389563i	−0.484968	+	0.874532i	$0.661169\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	2665.38i		−	0.145473i
$696$		0			0
$697$			21395.3i			1.16270i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−11574.5	−	11574.5i	−0.623626	−	0.623626i	0.322831	−	0.946457i	$-0.395366\pi$
$701$	−11574.5	−	11574.5i	−0.623626	−	0.623626i	−0.946457	+	0.322831i	$0.895366\pi$
$702$		0			0
$703$		−	27812.3i		−	1.49212i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12292.7	+	12292.7i	−0.653912	+	0.653912i
$708$		0			0
$709$	11104.7	+	11104.7i	0.588216	+	0.588216i	0.937148	−	0.348932i	$-0.113456\pi$
$709$	11104.7	+	11104.7i	0.588216	+	0.588216i	−0.348932	+	0.937148i	$0.613456\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−8112.92			−0.426131
$714$		0			0
$715$	5238.46	−	5238.46i	0.273996	−	0.273996i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9398.56			0.487493			0.243746	−	0.969839i	$-0.421624\pi$
$719$	9398.56			0.487493			0.243746	+	0.969839i	$0.421624\pi$
$720$		0			0
$721$	23258.4			1.20137
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5718.95	+	5718.95i	−0.292961	+	0.292961i
$726$		0			0
$727$	16800.4			0.857071			0.428536	−	0.903525i	$-0.359030\pi$
$727$	16800.4			0.857071			0.428536	+	0.903525i	$0.359030\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9964.65	−	9964.65i	−0.504180	−	0.504180i
$732$		0			0
$733$	5536.14	−	5536.14i	0.278966	−	0.278966i	−0.553730	−	0.832696i	$-0.686796\pi$
$733$	5536.14	−	5536.14i	0.278966	−	0.278966i	0.832696	+	0.553730i	$0.186796\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	21487.7i		−	1.07396i
$738$		0			0
$739$	775.030	+	775.030i	0.0385791	+	0.0385791i	0.726133	−	0.687554i	$-0.241316\pi$
$739$	775.030	+	775.030i	0.0385791	+	0.0385791i	−0.687554	+	0.726133i	$0.741316\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	21471.2i		−	1.06016i	−0.847947	−	0.530081i	$-0.822162\pi$
$743$		−	21471.2i		−	1.06016i	0.847947	−	0.530081i	$-0.177838\pi$
$744$		0			0
$745$			14061.6i			0.691511i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−16500.0	−	16500.0i	−0.804937	−	0.804937i
$750$		0			0
$751$			2293.60i			0.111444i	0.998446	+	0.0557221i	$0.0177461\pi$
$751$			2293.60i			0.111444i	−0.998446	+	0.0557221i	$0.982254\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1801.47	+	1801.47i	−0.0868372	+	0.0868372i
$756$		0			0
$757$	−14151.4	−	14151.4i	−0.679448	−	0.679448i	0.280427	−	0.959875i	$-0.409524\pi$
$757$	−14151.4	−	14151.4i	−0.679448	−	0.679448i	−0.959875	+	0.280427i	$0.909524\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−18852.9			−0.898054			−0.449027	−	0.893518i	$-0.648229\pi$
$761$	−18852.9			−0.898054			−0.449027	+	0.893518i	$0.648229\pi$
$762$		0			0
$763$	17404.2	−	17404.2i	0.825785	−	0.825785i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8410.06			−0.395919
$768$		0			0
$769$	−12172.9			−0.570828			−0.285414	−	0.958404i	$-0.592131\pi$
$769$	−12172.9			−0.570828			−0.285414	+	0.958404i	$0.592131\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4971.12	+	4971.12i	−0.231305	+	0.231305i	−0.813237	−	0.581932i	$-0.802297\pi$
$773$	−4971.12	+	4971.12i	−0.231305	+	0.231305i	0.581932	+	0.813237i	$0.302297\pi$
$774$		0			0
$775$	−5476.42			−0.253831
$776$		0			0
$777$		0			0
$778$		0			0
$779$	22335.9	+	22335.9i	1.02730	+	1.02730i
$780$		0			0
$781$	11161.2	−	11161.2i	0.511367	−	0.511367i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	32044.5i		−	1.45696i
$786$		0			0
$787$	3776.32	+	3776.32i	0.171043	+	0.171043i	0.787438	−	0.616394i	$-0.211407\pi$
$787$	3776.32	+	3776.32i	0.171043	+	0.171043i	−0.616394	+	0.787438i	$0.711407\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	20841.8i		−	0.936849i
$792$		0			0
$793$			6522.06i			0.292062i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1305.71	−	1305.71i	−0.0580311	−	0.0580311i	0.677496	−	0.735527i	$-0.263065\pi$
$797$	−1305.71	−	1305.71i	−0.0580311	−	0.0580311i	−0.735527	+	0.677496i	$0.763065\pi$
$798$		0			0
$799$			5118.23i			0.226621i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12783.0	−	12783.0i	0.561771	−	0.561771i
$804$		0			0
$805$	−6886.46	−	6886.46i	−0.301510	−	0.301510i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	16741.4			0.727560			0.363780	−	0.931485i	$-0.381486\pi$
$809$	16741.4			0.727560			0.363780	+	0.931485i	$0.381486\pi$
$810$		0			0
$811$	17759.3	−	17759.3i	0.768946	−	0.768946i	−0.208975	−	0.977921i	$-0.567013\pi$
$811$	17759.3	−	17759.3i	0.768946	−	0.768946i	0.977921	+	0.208975i	$0.0670128\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16020.2			0.688542
$816$		0			0
$817$	−20805.5			−0.890932
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−30920.3	+	30920.3i	−1.31440	+	1.31440i	−0.396269	+	0.918135i	$0.629695\pi$
$821$	−30920.3	+	30920.3i	−1.31440	+	1.31440i	−0.918135	+	0.396269i	$0.870305\pi$
$822$		0			0
$823$	−19433.1			−0.823080			−0.411540	−	0.911392i	$-0.635009\pi$
$823$	−19433.1			−0.823080			−0.411540	+	0.911392i	$0.635009\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−22294.5	−	22294.5i	−0.937431	−	0.937431i	0.0607235	−	0.998155i	$-0.480659\pi$
$827$	−22294.5	−	22294.5i	−0.937431	−	0.937431i	−0.998155	+	0.0607235i	$0.980659\pi$
$828$		0			0
$829$	16674.7	−	16674.7i	0.698595	−	0.698595i	−0.265513	−	0.964107i	$-0.585541\pi$
$829$	16674.7	−	16674.7i	0.698595	−	0.698595i	0.964107	+	0.265513i	$0.0855412\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	4153.96i		−	0.172781i
$834$		0			0
$835$	−17473.0	−	17473.0i	−0.724164	−	0.724164i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	30412.9i		−	1.25145i	−0.780042	−	0.625727i	$-0.784802\pi$
$839$		−	30412.9i		−	1.25145i	0.780042	−	0.625727i	$-0.215198\pi$
$840$		0			0
$841$			7605.69i			0.311849i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12938.4	+	12938.4i	0.526741	+	0.526741i
$846$		0			0
$847$		−	53743.3i		−	2.18022i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	16765.4	−	16765.4i	0.675337	−	0.675337i
$852$		0			0
$853$	15673.4	+	15673.4i	0.629128	+	0.629128i	0.947849	−	0.318721i	$-0.103253\pi$
$853$	15673.4	+	15673.4i	0.629128	+	0.629128i	−0.318721	+	0.947849i	$0.603253\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7458.42			−0.297287			−0.148643	−	0.988891i	$-0.547491\pi$
$857$	−7458.42			−0.297287			−0.148643	+	0.988891i	$0.547491\pi$
$858$		0			0
$859$	22482.2	−	22482.2i	0.892996	−	0.892996i	−0.101808	−	0.994804i	$-0.532463\pi$
$859$	22482.2	−	22482.2i	0.892996	−	0.892996i	0.994804	+	0.101808i	$0.0324628\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−27444.8			−1.08254			−0.541270	−	0.840849i	$-0.682056\pi$
$863$	−27444.8			−1.08254			−0.541270	+	0.840849i	$0.682056\pi$
$864$		0			0
$865$	−4378.34			−0.172102
$866$		0			0
$867$		0			0
$868$		0			0
$869$	29101.6	−	29101.6i	1.13602	−	1.13602i
$870$		0			0
$871$	−3846.90			−0.149653
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−17499.4	−	17499.4i	−0.676101	−	0.676101i
$876$		0			0
$877$	−167.934	+	167.934i	−0.00646604	+	0.00646604i	−0.710332	−	0.703866i	$-0.751455\pi$
$877$	−167.934	+	167.934i	−0.00646604	+	0.00646604i	0.703866	+	0.710332i	$0.251455\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			20968.8i			0.801882i	0.916104	+	0.400941i	$0.131317\pi$
$881$			20968.8i			0.801882i	−0.916104	+	0.400941i	$0.868683\pi$
$882$		0			0
$883$	22968.4	+	22968.4i	0.875366	+	0.875366i	0.993051	−	0.117685i	$-0.0375474\pi$
$883$	22968.4	+	22968.4i	0.875366	+	0.875366i	−0.117685	+	0.993051i	$0.537547\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			41314.4i			1.56393i	0.623325	+	0.781963i	$0.285782\pi$
$887$			41314.4i			1.56393i	−0.623325	+	0.781963i	$0.714218\pi$
$888$		0			0
$889$			45232.7i			1.70648i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5343.25	+	5343.25i	0.200230	+	0.200230i
$894$		0			0
$895$		−	1340.76i		−	0.0500743i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−15318.9	+	15318.9i	−0.568314	+	0.568314i
$900$		0			0
$901$	−4807.61	−	4807.61i	−0.177763	−	0.177763i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−19629.8			−0.721014
$906$		0			0
$907$	28396.2	−	28396.2i	1.03956	−	1.03956i	0.0403761	−	0.999185i	$-0.487144\pi$
$907$	28396.2	−	28396.2i	1.03956	−	1.03956i	0.999185	−	0.0403761i	$-0.0128556\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	42779.4			1.55581			0.777906	−	0.628380i	$-0.216282\pi$
$911$	42779.4			1.55581			0.777906	+	0.628380i	$0.216282\pi$
$912$		0			0
$913$	33815.0			1.22575
$914$		0			0
$915$		0			0
$916$		0			0
$917$	15270.2	−	15270.2i	0.549909	−	0.549909i
$918$		0			0
$919$	−39163.0			−1.40573			−0.702866	−	0.711323i	$-0.748096\pi$
$919$	−39163.0			−1.40573			−0.702866	+	0.711323i	$0.748096\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1998.17	−	1998.17i	−0.0712573	−	0.0712573i
$924$		0			0
$925$	11317.1	−	11317.1i	0.402274	−	0.402274i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	22120.8i		−	0.781225i	−0.920555	−	0.390613i	$-0.872263\pi$
$929$		−	22120.8i		−	0.781225i	0.920555	−	0.390613i	$-0.127737\pi$
$930$		0			0
$931$	−4336.59	−	4336.59i	−0.152659	−	0.152659i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	32356.1i		−	1.13172i
$936$		0			0
$937$			13514.2i			0.471172i	0.971853	+	0.235586i	$0.0757010\pi$
$937$			13514.2i			0.471172i	−0.971853	+	0.235586i	$0.924299\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3813.59	+	3813.59i	0.132114	+	0.132114i	0.770072	−	0.637957i	$-0.220220\pi$
$941$	3813.59	+	3813.59i	0.132114	+	0.132114i	−0.637957	+	0.770072i	$0.720220\pi$
$942$		0			0
$943$			26928.5i			0.929916i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15811.7	+	15811.7i	−0.542567	+	0.542567i	−0.924281	−	0.381714i	$-0.875334\pi$
$947$	−15811.7	+	15811.7i	−0.542567	+	0.542567i	0.381714	+	0.924281i	$0.375334\pi$
$948$		0			0
$949$	−2288.52	−	2288.52i	−0.0782809	−	0.0782809i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−48996.7			−1.66544			−0.832718	−	0.553697i	$-0.813216\pi$
$953$	−48996.7			−1.66544			−0.832718	+	0.553697i	$0.813216\pi$
$954$		0			0
$955$	8285.33	−	8285.33i	0.280740	−	0.280740i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1573.01			−0.0529667
$960$		0			0
$961$	15121.7			0.507594
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10691.2	+	10691.2i	−0.356645	+	0.356645i
$966$		0			0
$967$	17758.0			0.590548			0.295274	−	0.955413i	$-0.404589\pi$
$967$	17758.0			0.590548			0.295274	+	0.955413i	$0.404589\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	278.185	+	278.185i	0.00919401	+	0.00919401i	0.711689	−	0.702495i	$-0.247931\pi$
$971$	278.185	+	278.185i	0.00919401	+	0.00919401i	−0.702495	+	0.711689i	$0.747931\pi$
$972$		0			0
$973$	−3434.52	+	3434.52i	−0.113161	+	0.113161i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			2036.82i			0.0666978i	0.999444	+	0.0333489i	$0.0106173\pi$
$977$			2036.82i			0.0666978i	−0.999444	+	0.0333489i	$0.989383\pi$
$978$		0			0
$979$	65713.0	+	65713.0i	2.14525	+	2.14525i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	4969.20i		−	0.161234i	−0.996745	−	0.0806169i	$-0.974311\pi$
$983$		−	4969.20i		−	0.161234i	0.996745	−	0.0806169i	$-0.0256890\pi$
$984$		0			0
$985$		−	30563.2i		−	0.988654i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12541.7	−	12541.7i	−0.403237	−	0.403237i
$990$		0			0
$991$		−	43195.4i		−	1.38461i	−0.721606	−	0.692304i	$-0.756596\pi$
$991$		−	43195.4i		−	1.38461i	0.721606	−	0.692304i	$-0.243404\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−29607.9	+	29607.9i	−0.943352	+	0.943352i
$996$		0			0
$997$	−22458.2	−	22458.2i	−0.713400	−	0.713400i	0.253845	−	0.967245i	$-0.418305\pi$
$997$	−22458.2	−	22458.2i	−0.713400	−	0.713400i	−0.967245	+	0.253845i	$0.918305\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.l.a.143.18		48
3.2	odd	2	inner	576.4.l.a.143.7		48
4.3	odd	2		144.4.l.a.107.18	yes	48
8.3	odd	2		1152.4.l.b.287.7		48
8.5	even	2		1152.4.l.a.287.7		48
12.11	even	2		144.4.l.a.107.7	yes	48
16.3	odd	4	inner	576.4.l.a.431.7		48
16.5	even	4		1152.4.l.b.863.18		48
16.11	odd	4		1152.4.l.a.863.18		48
16.13	even	4		144.4.l.a.35.7	✓	48
24.5	odd	2		1152.4.l.a.287.18		48
24.11	even	2		1152.4.l.b.287.18		48
48.5	odd	4		1152.4.l.b.863.7		48
48.11	even	4		1152.4.l.a.863.7		48
48.29	odd	4		144.4.l.a.35.18	yes	48
48.35	even	4	inner	576.4.l.a.431.18		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.7	✓	48	16.13	even	4
144.4.l.a.35.18	yes	48	48.29	odd	4
144.4.l.a.107.7	yes	48	12.11	even	2
144.4.l.a.107.18	yes	48	4.3	odd	2
576.4.l.a.143.7		48	3.2	odd	2	inner
576.4.l.a.143.18		48	1.1	even	1	trivial
576.4.l.a.431.7		48	16.3	odd	4	inner
576.4.l.a.431.18		48	48.35	even	4	inner
1152.4.l.a.287.7		48	8.5	even	2
1152.4.l.a.287.18		48	24.5	odd	2
1152.4.l.a.863.7		48	48.11	even	4
1152.4.l.a.863.18		48	16.11	odd	4
1152.4.l.b.287.7		48	8.3	odd	2
1152.4.l.b.287.18		48	24.11	even	2
1152.4.l.b.863.7		48	48.5	odd	4
1152.4.l.b.863.18		48	16.5	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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.31601 - 6.31601i) q^{5} -16.2772 q^{7} +(-48.1287 - 48.1287i) q^{11} +(-8.61642 + 8.61642i) q^{13} +53.2206i q^{17} +(55.5604 + 55.5604i) q^{19} +66.9842i q^{23} +45.2160i q^{25} +(126.481 + 126.481i) q^{29} +121.117i q^{31} +(-102.807 + 102.807i) q^{35} +(-250.289 - 250.289i) q^{37} +402.012 q^{41} +(-187.233 + 187.233i) q^{43} +96.1703 q^{47} -78.0518 q^{49} +(-90.3337 + 90.3337i) q^{53} -607.963 q^{55} +(488.025 + 488.025i) q^{59} +(378.467 - 378.467i) q^{61} +108.843i q^{65} +(223.231 + 223.231i) q^{67} +231.902i q^{71} +265.600i q^{73} +(783.402 + 783.402i) q^{77} +604.662i q^{79} +(-351.298 + 351.298i) q^{83} +(336.142 + 336.142i) q^{85} -1365.36 q^{89} +(140.251 - 140.251i) q^{91} +701.839 q^{95} -1854.47 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 48 q^{19} - 864 q^{43} + 2352 q^{49} + 576 q^{55} + 1824 q^{61} - 816 q^{67} - 480 q^{85} + 3600 q^{91}+O(q^{100}) \) 48 * q + 48 * q^19 - 864 * q^43 + 2352 * q^49 + 576 * q^55 + 1824 * q^61 - 816 * q^67 - 480 * q^85 + 3600 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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