LMFDB - Embedded newform 2880.2.t.e.2161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(721,2880)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2880.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.t (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2161.3
Character	$\chi$	$=$	2880.2161
Dual form			2880.2.t.e.721.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{5} +2.05446i q^{7} +O(q^{10})$	$q+(-0.707107 - 0.707107i) q^{5} +2.05446i q^{7} +(-2.89872 - 2.89872i) q^{11} +(-0.887884 + 0.887884i) q^{13} +7.70569 q^{17} +(-1.96412 + 1.96412i) q^{19} -1.75237i q^{23} +1.00000i q^{25} +(1.03909 - 1.03909i) q^{29} -1.03962 q^{31} +(1.45273 - 1.45273i) q^{35} +(-7.76676 - 7.76676i) q^{37} +1.08887i q^{41} +(4.29390 + 4.29390i) q^{43} +8.19445 q^{47} +2.77918 q^{49} +(3.76115 + 3.76115i) q^{53} +4.09941i q^{55} +(3.92403 + 3.92403i) q^{59} +(-6.18708 + 6.18708i) q^{61} +1.25566 q^{65} +(8.26972 - 8.26972i) q^{67} -6.34181i q^{71} -14.3810i q^{73} +(5.95531 - 5.95531i) q^{77} +16.1119 q^{79} +(2.72548 - 2.72548i) q^{83} +(-5.44875 - 5.44875i) q^{85} -7.96641i q^{89} +(-1.82412 - 1.82412i) q^{91} +2.77768 q^{95} +7.11484 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 8 q^{19} - 32 q^{37} - 16 q^{43} - 32 q^{49} - 16 q^{61} + 16 q^{67} + 16 q^{79} + 16 q^{85} + 80 q^{91}+O(q^{100}) $ 32 * q - 8 * q^19 - 32 * q^37 - 16 * q^43 - 32 * q^49 - 16 * q^61 + 16 * q^67 + 16 * q^79 + 16 * q^85 + 80 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$6$		0			0
$7$			2.05446i			0.776514i	0.921551	+	0.388257i	$0.126923\pi$
$7$			2.05446i			0.776514i	−0.921551	+	0.388257i	$0.873077\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.89872	−	2.89872i	−0.873997	−	0.873997i	0.118908	−	0.992905i	$-0.462061\pi$
$11$	−2.89872	−	2.89872i	−0.873997	−	0.873997i	−0.992905	+	0.118908i	$0.962061\pi$
$12$		0			0
$13$	−0.887884	+	0.887884i	−0.246255	+	0.246255i	−0.819432	−	0.573177i	$-0.805711\pi$
$13$	−0.887884	+	0.887884i	−0.246255	+	0.246255i	0.573177	+	0.819432i	$0.305711\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.70569			1.86890			0.934452	−	0.356088i	$-0.115890\pi$
$17$	7.70569			1.86890			0.934452	+	0.356088i	$0.115890\pi$
$18$		0			0
$19$	−1.96412	+	1.96412i	−0.450599	+	0.450599i	−0.895553	−	0.444954i	$-0.853220\pi$
$19$	−1.96412	+	1.96412i	−0.450599	+	0.450599i	0.444954	+	0.895553i	$0.353220\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.75237i		−	0.365395i	−0.983169	−	0.182697i	$-0.941517\pi$
$23$		−	1.75237i		−	0.365395i	0.983169	−	0.182697i	$-0.0584829\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.03909	−	1.03909i	0.192954	−	0.192954i	−0.604017	−	0.796971i	$-0.706434\pi$
$29$	1.03909	−	1.03909i	0.192954	−	0.192954i	0.796971	+	0.604017i	$0.206434\pi$
$30$		0			0
$31$	−1.03962			−0.186721			−0.0933606	−	0.995632i	$-0.529761\pi$
$31$	−1.03962			−0.186721			−0.0933606	+	0.995632i	$0.529761\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.45273	−	1.45273i	0.245555	−	0.245555i
$36$		0			0
$37$	−7.76676	−	7.76676i	−1.27685	−	1.27685i	−0.942422	−	0.334425i	$-0.891458\pi$
$37$	−7.76676	−	7.76676i	−1.27685	−	1.27685i	−0.334425	−	0.942422i	$-0.608542\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.08887i			0.170053i	0.996379	+	0.0850265i	$0.0270975\pi$
$41$			1.08887i			0.170053i	−0.996379	+	0.0850265i	$0.972903\pi$
$42$		0			0
$43$	4.29390	+	4.29390i	0.654814	+	0.654814i	0.954148	−	0.299334i	$-0.0967646\pi$
$43$	4.29390	+	4.29390i	0.654814	+	0.654814i	−0.299334	+	0.954148i	$0.596765\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.19445			1.19528			0.597641	−	0.801764i	$-0.296105\pi$
$47$	8.19445			1.19528			0.597641	+	0.801764i	$0.296105\pi$
$48$		0			0
$49$	2.77918			0.397026
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.76115	+	3.76115i	0.516634	+	0.516634i	0.916551	−	0.399917i	$-0.130961\pi$
$53$	3.76115	+	3.76115i	0.516634	+	0.516634i	−0.399917	+	0.916551i	$0.630961\pi$
$54$		0			0
$55$			4.09941i			0.552764i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.92403	+	3.92403i	0.510865	+	0.510865i	0.914791	−	0.403926i	$-0.132355\pi$
$59$	3.92403	+	3.92403i	0.510865	+	0.510865i	−0.403926	+	0.914791i	$0.632355\pi$
$60$		0			0
$61$	−6.18708	+	6.18708i	−0.792175	+	0.792175i	−0.981847	−	0.189673i	$-0.939257\pi$
$61$	−6.18708	+	6.18708i	−0.792175	+	0.792175i	0.189673	+	0.981847i	$0.439257\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.25566			0.155745
$66$		0			0
$67$	8.26972	−	8.26972i	1.01031	−	1.01031i	0.0103604	−	0.999946i	$-0.496702\pi$
$67$	8.26972	−	8.26972i	1.01031	−	1.01031i	0.999946	−	0.0103604i	$-0.00329786\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6.34181i		−	0.752634i	−0.926491	−	0.376317i	$-0.877190\pi$
$71$		−	6.34181i		−	0.752634i	0.926491	−	0.376317i	$-0.122810\pi$
$72$		0			0
$73$		−	14.3810i		−	1.68317i	−0.540121	−	0.841587i	$-0.681622\pi$
$73$		−	14.3810i		−	1.68317i	0.540121	−	0.841587i	$-0.318378\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.95531	−	5.95531i	0.678671	−	0.678671i
$78$		0			0
$79$	16.1119			1.81273			0.906363	−	0.422499i	$-0.138847\pi$
$79$	16.1119			1.81273			0.906363	+	0.422499i	$0.138847\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.72548	−	2.72548i	0.299161	−	0.299161i	−0.541524	−	0.840685i	$-0.682153\pi$
$83$	2.72548	−	2.72548i	0.299161	−	0.299161i	0.840685	+	0.541524i	$0.182153\pi$
$84$		0			0
$85$	−5.44875	−	5.44875i	−0.591000	−	0.591000i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	7.96641i		−	0.844438i	−0.906494	−	0.422219i	$-0.861251\pi$
$89$		−	7.96641i		−	0.844438i	0.906494	−	0.422219i	$-0.138749\pi$
$90$		0			0
$91$	−1.82412	−	1.82412i	−0.191220	−	0.191220i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.77768			0.284984
$96$		0			0
$97$	7.11484			0.722403			0.361201	−	0.932488i	$-0.382367\pi$
$97$	7.11484			0.722403			0.361201	+	0.932488i	$0.382367\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.8253	+	10.8253i	1.07716	+	1.07716i	0.996763	+	0.0803944i	$0.0256180\pi$
$101$	10.8253	+	10.8253i	1.07716	+	1.07716i	0.0803944	+	0.996763i	$0.474382\pi$
$102$		0			0
$103$			15.3515i			1.51263i	0.654207	+	0.756316i	$0.273003\pi$
$103$			15.3515i			1.51263i	−0.654207	+	0.756316i	$0.726997\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.85832	+	5.85832i	0.566346	+	0.566346i	0.931103	−	0.364757i	$-0.118848\pi$
$107$	5.85832	+	5.85832i	0.566346	+	0.566346i	−0.364757	+	0.931103i	$0.618848\pi$
$108$		0			0
$109$	3.48123	−	3.48123i	0.333441	−	0.333441i	−0.520451	−	0.853892i	$-0.674236\pi$
$109$	3.48123	−	3.48123i	0.333441	−	0.333441i	0.853892	+	0.520451i	$0.174236\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.23400			0.210157			0.105079	−	0.994464i	$-0.466491\pi$
$113$	2.23400			0.210157			0.105079	+	0.994464i	$0.466491\pi$
$114$		0			0
$115$	−1.23911	+	1.23911i	−0.115548	+	0.115548i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			15.8311i			1.45123i
$120$		0			0
$121$			5.80515i			0.527741i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	−	0.707107i	0.0632456	−	0.0632456i
$126$		0			0
$127$	−14.5524			−1.29131			−0.645657	−	0.763627i	$-0.723417\pi$
$127$	−14.5524			−1.29131			−0.645657	+	0.763627i	$0.723417\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.6093	−	10.6093i	0.926937	−	0.926937i	−0.0705699	−	0.997507i	$-0.522482\pi$
$131$	10.6093	−	10.6093i	0.926937	−	0.926937i	0.997507	+	0.0705699i	$0.0224818\pi$
$132$		0			0
$133$	−4.03521	−	4.03521i	−0.349897	−	0.349897i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	8.94984i		−	0.764636i	−0.924031	−	0.382318i	$-0.875126\pi$
$137$		−	8.94984i		−	0.764636i	0.924031	−	0.382318i	$-0.124874\pi$
$138$		0			0
$139$	−4.86533	−	4.86533i	−0.412672	−	0.412672i	0.469996	−	0.882668i	$-0.344255\pi$
$139$	−4.86533	−	4.86533i	−0.412672	−	0.412672i	−0.882668	+	0.469996i	$0.844255\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.14745			0.430452
$144$		0			0
$145$	−1.46949			−0.122035
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.63849	−	6.63849i	−0.543847	−	0.543847i	0.380808	−	0.924654i	$-0.375646\pi$
$149$	−6.63849	−	6.63849i	−0.543847	−	0.543847i	−0.924654	+	0.380808i	$0.875646\pi$
$150$		0			0
$151$			9.76110i			0.794347i	0.917744	+	0.397174i	$0.130009\pi$
$151$			9.76110i			0.794347i	−0.917744	+	0.397174i	$0.869991\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.735122	+	0.735122i	0.0590464	+	0.0590464i
$156$		0			0
$157$	−7.71042	+	7.71042i	−0.615358	+	0.615358i	−0.944337	−	0.328979i	$-0.893295\pi$
$157$	−7.71042	+	7.71042i	−0.615358	+	0.615358i	0.328979	+	0.944337i	$0.393295\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.60019			0.283734
$162$		0			0
$163$	6.82491	−	6.82491i	0.534568	−	0.534568i	−0.387361	−	0.921928i	$-0.626613\pi$
$163$	6.82491	−	6.82491i	0.534568	−	0.534568i	0.921928	+	0.387361i	$0.126613\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.56804i		−	0.353485i	−0.984257	−	0.176743i	$-0.943444\pi$
$167$		−	4.56804i		−	0.353485i	0.984257	−	0.176743i	$-0.0565560\pi$
$168$		0			0
$169$			11.4233i			0.878717i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.66585	−	7.66585i	0.582824	−	0.582824i	−0.352854	−	0.935678i	$-0.614789\pi$
$173$	7.66585	−	7.66585i	0.582824	−	0.582824i	0.935678	+	0.352854i	$0.114789\pi$
$174$		0			0
$175$	−2.05446			−0.155303
$176$		0			0
$177$		0			0
$178$		0			0
$179$	17.5773	−	17.5773i	1.31379	−	1.31379i	0.395196	−	0.918597i	$-0.370677\pi$
$179$	17.5773	−	17.5773i	1.31379	−	1.31379i	0.918597	−	0.395196i	$-0.129323\pi$
$180$		0			0
$181$	4.08504	+	4.08504i	0.303638	+	0.303638i	0.842436	−	0.538797i	$-0.181121\pi$
$181$	4.08504	+	4.08504i	0.303638	+	0.303638i	−0.538797	+	0.842436i	$0.681121\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			10.9839i			0.807549i
$186$		0			0
$187$	−22.3366	−	22.3366i	−1.63342	−	1.63342i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	20.6486			1.49408			0.747039	−	0.664781i	$-0.231475\pi$
$191$	20.6486			1.49408			0.747039	+	0.664781i	$0.231475\pi$
$192$		0			0
$193$	13.4703			0.969613			0.484806	−	0.874621i	$-0.338890\pi$
$193$	13.4703			0.969613			0.484806	+	0.874621i	$0.338890\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.8364	−	12.8364i	−0.914555	−	0.914555i	0.0820712	−	0.996626i	$-0.473846\pi$
$197$	−12.8364	−	12.8364i	−0.914555	−	0.914555i	−0.996626	+	0.0820712i	$0.973846\pi$
$198$		0			0
$199$			18.0460i			1.27925i	0.768688	+	0.639624i	$0.220910\pi$
$199$			18.0460i			1.27925i	−0.768688	+	0.639624i	$0.779090\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.13477	+	2.13477i	0.149831	+	0.149831i
$204$		0			0
$205$	0.769948	−	0.769948i	0.0537755	−	0.0537755i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.3868			0.787645
$210$		0			0
$211$	13.6178	−	13.6178i	0.937485	−	0.937485i	−0.0606726	−	0.998158i	$-0.519325\pi$
$211$	13.6178	−	13.6178i	0.937485	−	0.937485i	0.998158	+	0.0606726i	$0.0193246\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	6.07249i		−	0.414141i
$216$		0			0
$217$		−	2.13586i		−	0.144992i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.84176	+	6.84176i	−0.460226	+	0.460226i
$222$		0			0
$223$	2.36364			0.158281			0.0791404	−	0.996863i	$-0.474782\pi$
$223$	2.36364			0.158281			0.0791404	+	0.996863i	$0.474782\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.9760	+	19.9760i	−1.32585	+	1.32585i	−0.416901	+	0.908952i	$0.636884\pi$
$227$	−19.9760	+	19.9760i	−1.32585	+	1.32585i	−0.908952	+	0.416901i	$0.863116\pi$
$228$		0			0
$229$	−2.00826	−	2.00826i	−0.132710	−	0.132710i	0.637632	−	0.770341i	$-0.279914\pi$
$229$	−2.00826	−	2.00826i	−0.132710	−	0.132710i	−0.770341	+	0.637632i	$0.779914\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	16.3450i		−	1.07080i	−0.844599	−	0.535399i	$-0.820161\pi$
$233$		−	16.3450i		−	1.07080i	0.844599	−	0.535399i	$-0.179839\pi$
$234$		0			0
$235$	−5.79435	−	5.79435i	−0.377982	−	0.377982i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.7828			−0.697481			−0.348741	−	0.937219i	$-0.613391\pi$
$239$	−10.7828			−0.697481			−0.348741	+	0.937219i	$0.613391\pi$
$240$		0			0
$241$	20.3537			1.31110			0.655549	−	0.755152i	$-0.272437\pi$
$241$	20.3537			1.31110			0.655549	+	0.755152i	$0.272437\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.96518	−	1.96518i	−0.125551	−	0.125551i
$246$		0			0
$247$		−	3.48781i		−	0.221924i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3.85183	+	3.85183i	0.243125	+	0.243125i	0.818142	−	0.575016i	$-0.195004\pi$
$251$	3.85183	+	3.85183i	0.243125	+	0.243125i	−0.575016	+	0.818142i	$0.695004\pi$
$252$		0			0
$253$	−5.07964	+	5.07964i	−0.319354	+	0.319354i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.30480			−0.455661			−0.227831	−	0.973701i	$-0.573163\pi$
$257$	−7.30480			−0.455661			−0.227831	+	0.973701i	$0.573163\pi$
$258$		0			0
$259$	15.9565	−	15.9565i	0.991490	−	0.991490i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	7.64434i		−	0.471370i	−0.971830	−	0.235685i	$-0.924267\pi$
$263$		−	7.64434i		−	0.471370i	0.971830	−	0.235685i	$-0.0757333\pi$
$264$		0			0
$265$		−	5.31908i		−	0.326748i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.6072	−	14.6072i	0.890620	−	0.890620i	−0.103962	−	0.994581i	$-0.533152\pi$
$269$	14.6072	−	14.6072i	0.890620	−	0.890620i	0.994581	+	0.103962i	$0.0331519\pi$
$270$		0			0
$271$	−9.78522			−0.594410			−0.297205	−	0.954814i	$-0.596054\pi$
$271$	−9.78522			−0.594410			−0.297205	+	0.954814i	$0.596054\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.89872	−	2.89872i	0.174799	−	0.174799i
$276$		0			0
$277$	−3.42295	−	3.42295i	−0.205665	−	0.205665i	0.596757	−	0.802422i	$-0.296456\pi$
$277$	−3.42295	−	3.42295i	−0.205665	−	0.205665i	−0.802422	+	0.596757i	$0.796456\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			17.5315i			1.04584i	0.852381	+	0.522922i	$0.175158\pi$
$281$			17.5315i			1.04584i	−0.852381	+	0.522922i	$0.824842\pi$
$282$		0			0
$283$	3.45835	+	3.45835i	0.205577	+	0.205577i	0.802385	−	0.596807i	$-0.203564\pi$
$283$	3.45835	+	3.45835i	0.205577	+	0.205577i	−0.596807	+	0.802385i	$0.703564\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.23705			−0.132049
$288$		0			0
$289$	42.3777			2.49280
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.7674	+	14.7674i	0.862720	+	0.862720i	0.991653	−	0.128934i	$-0.0411554\pi$
$293$	14.7674	+	14.7674i	0.862720	+	0.862720i	−0.128934	+	0.991653i	$0.541155\pi$
$294$		0			0
$295$		−	5.54941i		−	0.323099i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.55590	+	1.55590i	0.0899802	+	0.0899802i
$300$		0			0
$301$	−8.82167	+	8.82167i	−0.508472	+	0.508472i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.74986			0.501015
$306$		0			0
$307$	−11.4469	+	11.4469i	−0.653309	+	0.653309i	−0.953788	−	0.300479i	$-0.902853\pi$
$307$	−11.4469	+	11.4469i	−0.653309	+	0.653309i	0.300479	+	0.953788i	$0.402853\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	33.2894i		−	1.88767i	−0.330419	−	0.943834i	$-0.607190\pi$
$311$		−	33.2894i		−	1.88767i	0.330419	−	0.943834i	$-0.392810\pi$
$312$		0			0
$313$			17.5967i			0.994627i	0.867571	+	0.497313i	$0.165680\pi$
$313$			17.5967i			0.994627i	−0.867571	+	0.497313i	$0.834320\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.81377	−	7.81377i	0.438865	−	0.438865i	−0.452765	−	0.891630i	$-0.649562\pi$
$317$	7.81377	−	7.81377i	0.438865	−	0.438865i	0.891630	+	0.452765i	$0.149562\pi$
$318$		0			0
$319$	−6.02404			−0.337282
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−15.1349	+	15.1349i	−0.842127	+	0.842127i
$324$		0			0
$325$	−0.887884	−	0.887884i	−0.0492509	−	0.0492509i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			16.8352i			0.928154i
$330$		0			0
$331$	−18.6231	−	18.6231i	−1.02362	−	1.02362i	−0.999714	−	0.0239030i	$-0.992391\pi$
$331$	−18.6231	−	18.6231i	−1.02362	−	1.02362i	−0.0239030	−	0.999714i	$-0.507609\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.6951			−0.638974
$336$		0			0
$337$	−26.2854			−1.43186			−0.715930	−	0.698172i	$-0.753997\pi$
$337$	−26.2854			−1.43186			−0.715930	+	0.698172i	$0.753997\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.01357	+	3.01357i	0.163194	+	0.163194i
$342$		0			0
$343$			20.0910i			1.08481i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.36875	−	1.36875i	−0.0734784	−	0.0734784i	0.669413	−	0.742891i	$-0.266546\pi$
$347$	−1.36875	−	1.36875i	−0.0734784	−	0.0734784i	−0.742891	+	0.669413i	$0.766546\pi$
$348$		0			0
$349$	−12.2214	+	12.2214i	−0.654198	+	0.654198i	−0.954001	−	0.299803i	$-0.903079\pi$
$349$	−12.2214	+	12.2214i	−0.654198	+	0.654198i	0.299803	+	0.954001i	$0.403079\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−17.2113			−0.916065			−0.458033	−	0.888935i	$-0.651446\pi$
$353$	−17.2113			−0.916065			−0.458033	+	0.888935i	$0.651446\pi$
$354$		0			0
$355$	−4.48433	+	4.48433i	−0.238004	+	0.238004i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			2.47446i			0.130597i	0.997866	+	0.0652984i	$0.0207999\pi$
$359$			2.47446i			0.130597i	−0.997866	+	0.0652984i	$0.979200\pi$
$360$		0			0
$361$			11.2845i			0.593921i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.1689	+	10.1689i	−0.532266	+	0.532266i
$366$		0			0
$367$	−4.04135			−0.210957			−0.105478	−	0.994422i	$-0.533637\pi$
$367$	−4.04135			−0.210957			−0.105478	+	0.994422i	$0.533637\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.72715	+	7.72715i	−0.401174	+	0.401174i
$372$		0			0
$373$	−4.19175	−	4.19175i	−0.217041	−	0.217041i	0.590209	−	0.807250i	$-0.299045\pi$
$373$	−4.19175	−	4.19175i	−0.217041	−	0.217041i	−0.807250	+	0.590209i	$0.799045\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			1.84518i			0.0950314i
$378$		0			0
$379$	13.9046	+	13.9046i	0.714233	+	0.714233i	0.967418	−	0.253185i	$-0.0814780\pi$
$379$	13.9046	+	13.9046i	0.714233	+	0.714233i	−0.253185	+	0.967418i	$0.581478\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	15.8337			0.809062			0.404531	−	0.914524i	$-0.367435\pi$
$383$	15.8337			0.809062			0.404531	+	0.914524i	$0.367435\pi$
$384$		0			0
$385$	−8.42209			−0.429229
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.60774	+	3.60774i	0.182920	+	0.182920i	0.792627	−	0.609707i	$-0.208713\pi$
$389$	3.60774	+	3.60774i	0.182920	+	0.182920i	−0.609707	+	0.792627i	$0.708713\pi$
$390$		0			0
$391$		−	13.5032i		−	0.682888i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.3928	−	11.3928i	−0.573235	−	0.573235i
$396$		0			0
$397$	−15.0930	+	15.0930i	−0.757497	+	0.757497i	−0.975866	−	0.218370i	$-0.929926\pi$
$397$	−15.0930	+	15.0930i	−0.757497	+	0.757497i	0.218370	+	0.975866i	$0.429926\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.3220			−0.914956			−0.457478	−	0.889221i	$-0.651247\pi$
$401$	−18.3220			−0.914956			−0.457478	+	0.889221i	$0.651247\pi$
$402$		0			0
$403$	0.923061	−	0.923061i	0.0459810	−	0.0459810i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			45.0273i			2.23192i
$408$		0			0
$409$			16.5356i			0.817631i	0.912617	+	0.408816i	$0.134058\pi$
$409$			16.5356i			0.817631i	−0.912617	+	0.408816i	$0.865942\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.06177	+	8.06177i	−0.396694	+	0.396694i
$414$		0			0
$415$	−3.85442			−0.189206
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.35052	−	4.35052i	0.212537	−	0.212537i	−0.592807	−	0.805344i	$-0.701980\pi$
$419$	4.35052	−	4.35052i	0.212537	−	0.212537i	0.805344	+	0.592807i	$0.201980\pi$
$420$		0			0
$421$	−7.22188	−	7.22188i	−0.351973	−	0.351973i	0.508870	−	0.860843i	$-0.330063\pi$
$421$	−7.22188	−	7.22188i	−0.351973	−	0.351973i	−0.860843	+	0.508870i	$0.830063\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			7.70569i			0.373781i
$426$		0			0
$427$	−12.7111	−	12.7111i	−0.615135	−	0.615135i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−31.9861			−1.54071			−0.770357	−	0.637612i	$-0.779922\pi$
$431$	−31.9861			−1.54071			−0.770357	+	0.637612i	$0.779922\pi$
$432$		0			0
$433$	37.5361			1.80387			0.901936	−	0.431870i	$-0.142146\pi$
$433$	37.5361			1.80387			0.901936	+	0.431870i	$0.142146\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.44186	+	3.44186i	0.164647	+	0.164647i
$438$		0			0
$439$			17.4912i			0.834810i	0.908721	+	0.417405i	$0.137060\pi$
$439$			17.4912i			0.834810i	−0.908721	+	0.417405i	$0.862940\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.51139	+	7.51139i	0.356877	+	0.356877i	0.862660	−	0.505784i	$-0.168797\pi$
$443$	7.51139	+	7.51139i	0.356877	+	0.356877i	−0.505784	+	0.862660i	$0.668797\pi$
$444$		0			0
$445$	−5.63310	+	5.63310i	−0.267035	+	0.267035i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	19.0002			0.896676			0.448338	−	0.893864i	$-0.352016\pi$
$449$	19.0002			0.896676			0.448338	+	0.893864i	$0.352016\pi$
$450$		0			0
$451$	3.15633	−	3.15633i	0.148626	−	0.148626i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			2.57970i			0.120938i
$456$		0			0
$457$		−	25.8215i		−	1.20788i	−0.797031	−	0.603939i	$-0.793597\pi$
$457$		−	25.8215i		−	1.20788i	0.797031	−	0.603939i	$-0.206403\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.0676	+	12.0676i	−0.562046	+	0.562046i	−0.929888	−	0.367842i	$-0.880097\pi$
$461$	−12.0676	+	12.0676i	−0.562046	+	0.562046i	0.367842	+	0.929888i	$0.380097\pi$
$462$		0			0
$463$	9.76913			0.454010			0.227005	−	0.973894i	$-0.427107\pi$
$463$	9.76913			0.454010			0.227005	+	0.973894i	$0.427107\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.64911	−	8.64911i	0.400233	−	0.400233i	−0.478082	−	0.878315i	$-0.658668\pi$
$467$	8.64911	−	8.64911i	0.400233	−	0.400233i	0.878315	+	0.478082i	$0.158668\pi$
$468$		0			0
$469$	16.9898	+	16.9898i	0.784518	+	0.784518i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	24.8936i		−	1.14461i
$474$		0			0
$475$	−1.96412	−	1.96412i	−0.0901199	−	0.0901199i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−35.0313			−1.60062			−0.800310	−	0.599586i	$-0.795332\pi$
$479$	−35.0313			−1.60062			−0.800310	+	0.599586i	$0.795332\pi$
$480$		0			0
$481$	13.7920			0.628859
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.03095	−	5.03095i	−0.228444	−	0.228444i
$486$		0			0
$487$		−	14.3328i		−	0.649481i	−0.945803	−	0.324740i	$-0.894723\pi$
$487$		−	14.3328i		−	0.649481i	0.945803	−	0.324740i	$-0.105277\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−30.8986	−	30.8986i	−1.39443	−	1.39443i	−0.815065	−	0.579370i	$-0.803299\pi$
$491$	−30.8986	−	30.8986i	−1.39443	−	1.39443i	−0.579370	−	0.815065i	$-0.696701\pi$
$492$		0			0
$493$	8.00688	−	8.00688i	0.360612	−	0.360612i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13.0290			0.584431
$498$		0			0
$499$	17.8903	−	17.8903i	0.800879	−	0.800879i	−0.182354	−	0.983233i	$-0.558372\pi$
$499$	17.8903	−	17.8903i	0.800879	−	0.800879i	0.983233	+	0.182354i	$0.0583717\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			5.72534i			0.255280i	0.991821	+	0.127640i	$0.0407403\pi$
$503$			5.72534i			0.255280i	−0.991821	+	0.127640i	$0.959260\pi$
$504$		0			0
$505$		−	15.3093i		−	0.681254i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−17.1742	+	17.1742i	−0.761234	+	0.761234i	−0.976546	−	0.215311i	$-0.930923\pi$
$509$	−17.1742	+	17.1742i	−0.761234	+	0.761234i	0.215311	+	0.976546i	$0.430923\pi$
$510$		0			0
$511$	29.5453			1.30701
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.8552	−	10.8552i	0.478336	−	0.478336i
$516$		0			0
$517$	−23.7534	−	23.7534i	−1.04467	−	1.04467i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	5.98934i		−	0.262398i	−0.991356	−	0.131199i	$-0.958117\pi$
$521$		−	5.98934i		−	0.262398i	0.991356	−	0.131199i	$-0.0418826\pi$
$522$		0			0
$523$	−18.0403	−	18.0403i	−0.788848	−	0.788848i	0.192458	−	0.981305i	$-0.438354\pi$
$523$	−18.0403	−	18.0403i	−0.788848	−	0.788848i	−0.981305	+	0.192458i	$0.938354\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.01099			−0.348964
$528$		0			0
$529$	19.9292			0.866487
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.966791	−	0.966791i	−0.0418763	−	0.0418763i
$534$		0			0
$535$		−	8.28492i		−	0.358188i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.05606	−	8.05606i	−0.346999	−	0.346999i
$540$		0			0
$541$	29.9297	−	29.9297i	1.28678	−	1.28678i	0.350047	−	0.936732i	$-0.386166\pi$
$541$	29.9297	−	29.9297i	1.28678	−	1.28678i	0.936732	−	0.350047i	$-0.113834\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.92320			−0.210887
$546$		0			0
$547$	−19.9093	+	19.9093i	−0.851259	+	0.851259i	−0.990288	−	0.139029i	$-0.955602\pi$
$547$	−19.9093	+	19.9093i	−0.851259	+	0.851259i	0.139029	+	0.990288i	$0.455602\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			4.08178i			0.173890i
$552$		0			0
$553$			33.1012i			1.40761i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.8066	+	10.8066i	−0.457889	+	0.457889i	−0.897962	−	0.440073i	$-0.854953\pi$
$557$	−10.8066	+	10.8066i	−0.457889	+	0.457889i	0.440073	+	0.897962i	$0.354953\pi$
$558$		0			0
$559$	−7.62497			−0.322502
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−24.5766	+	24.5766i	−1.03578	+	1.03578i	−0.0364461	+	0.999336i	$0.511604\pi$
$563$	−24.5766	+	24.5766i	−1.03578	+	1.03578i	−0.999336	+	0.0364461i	$0.988396\pi$
$564$		0			0
$565$	−1.57968	−	1.57968i	−0.0664575	−	0.0664575i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			23.5603i			0.987697i	0.869548	+	0.493849i	$0.164410\pi$
$569$			23.5603i			0.987697i	−0.869548	+	0.493849i	$0.835590\pi$
$570$		0			0
$571$	19.6252	+	19.6252i	0.821290	+	0.821290i	0.986293	−	0.165003i	$-0.0527633\pi$
$571$	19.6252	+	19.6252i	0.821290	+	0.821290i	−0.165003	+	0.986293i	$0.552763\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.75237			0.0730790
$576$		0			0
$577$	−20.8716			−0.868895			−0.434447	−	0.900697i	$-0.643056\pi$
$577$	−20.8716			−0.868895			−0.434447	+	0.900697i	$0.643056\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.59941	+	5.59941i	0.232303	+	0.232303i
$582$		0			0
$583$		−	21.8051i		−	0.903073i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.17477	+	9.17477i	0.378683	+	0.378683i	0.870627	−	0.491944i	$-0.163713\pi$
$587$	9.17477	+	9.17477i	0.378683	+	0.378683i	−0.491944	+	0.870627i	$0.663713\pi$
$588$		0			0
$589$	2.04193	−	2.04193i	0.0841364	−	0.0841364i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−47.2551			−1.94054			−0.970268	−	0.242033i	$-0.922186\pi$
$593$	−47.2551			−1.94054			−0.970268	+	0.242033i	$0.922186\pi$
$594$		0			0
$595$	11.1943	−	11.1943i	0.458920	−	0.458920i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			45.7838i			1.87067i	0.353757	+	0.935337i	$0.384904\pi$
$599$			45.7838i			1.87067i	−0.353757	+	0.935337i	$0.615096\pi$
$600$		0			0
$601$		−	39.9200i		−	1.62837i	−0.580606	−	0.814185i	$-0.697184\pi$
$601$		−	39.9200i		−	1.62837i	0.580606	−	0.814185i	$-0.302816\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.10486	−	4.10486i	0.166886	−	0.166886i
$606$		0			0
$607$	−34.7803			−1.41169			−0.705845	−	0.708367i	$-0.749432\pi$
$607$	−34.7803			−1.41169			−0.705845	+	0.708367i	$0.749432\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.27571	+	7.27571i	−0.294344	+	0.294344i
$612$		0			0
$613$	11.5495	+	11.5495i	0.466480	+	0.466480i	0.900772	−	0.434292i	$-0.143001\pi$
$613$	11.5495	+	11.5495i	0.466480	+	0.466480i	−0.434292	+	0.900772i	$0.643001\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	36.3120i		−	1.46187i	−0.682449	−	0.730934i	$-0.739085\pi$
$617$		−	36.3120i		−	1.46187i	0.682449	−	0.730934i	$-0.260915\pi$
$618$		0			0
$619$	25.6174	+	25.6174i	1.02965	+	1.02965i	0.999547	+	0.0301014i	$0.00958303\pi$
$619$	25.6174	+	25.6174i	1.02965	+	1.02965i	0.0301014	+	0.999547i	$0.490417\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	16.3667			0.655718
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−59.8482	−	59.8482i	−2.38631	−	2.38631i
$630$		0			0
$631$			33.2875i			1.32515i	0.748994	+	0.662577i	$0.230537\pi$
$631$			33.2875i			1.32515i	−0.748994	+	0.662577i	$0.769463\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10.2901	+	10.2901i	0.408350	+	0.408350i
$636$		0			0
$637$	−2.46759	+	2.46759i	−0.0977694	+	0.0977694i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−23.5209			−0.929019			−0.464509	−	0.885568i	$-0.653769\pi$
$641$	−23.5209			−0.929019			−0.464509	+	0.885568i	$0.653769\pi$
$642$		0			0
$643$	9.39620	−	9.39620i	0.370550	−	0.370550i	−0.497127	−	0.867678i	$-0.665612\pi$
$643$	9.39620	−	9.39620i	0.370550	−	0.370550i	0.867678	+	0.497127i	$0.165612\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	37.6472i		−	1.48006i	−0.672572	−	0.740032i	$-0.734810\pi$
$647$		−	37.6472i		−	1.48006i	0.672572	−	0.740032i	$-0.265190\pi$
$648$		0			0
$649$		−	22.7493i		−	0.892989i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.6303	+	11.6303i	−0.455130	+	0.455130i	−0.897053	−	0.441923i	$-0.854297\pi$
$653$	−11.6303	+	11.6303i	−0.455130	+	0.455130i	0.441923	+	0.897053i	$0.354297\pi$
$654$		0			0
$655$	−15.0038			−0.586246
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.30674	−	7.30674i	0.284630	−	0.284630i	−0.550322	−	0.834952i	$-0.685495\pi$
$659$	7.30674	−	7.30674i	0.284630	−	0.284630i	0.834952	+	0.550322i	$0.185495\pi$
$660$		0			0
$661$	−24.9364	−	24.9364i	−0.969914	−	0.969914i	0.0296469	−	0.999560i	$-0.490562\pi$
$661$	−24.9364	−	24.9364i	−0.969914	−	0.969914i	−0.999560	+	0.0296469i	$0.990562\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			5.70664i			0.221294i
$666$		0			0
$667$	−1.82087	−	1.82087i	−0.0705043	−	0.0705043i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	35.8692			1.38472
$672$		0			0
$673$	−33.9922			−1.31030			−0.655151	−	0.755498i	$-0.727395\pi$
$673$	−33.9922			−1.31030			−0.655151	+	0.755498i	$0.727395\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.55513	+	1.55513i	0.0597684	+	0.0597684i	0.736359	−	0.676591i	$-0.236543\pi$
$677$	1.55513	+	1.55513i	0.0597684	+	0.0597684i	−0.676591	+	0.736359i	$0.736543\pi$
$678$		0			0
$679$			14.6172i			0.560956i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−26.3238	−	26.3238i	−1.00725	−	1.00725i	−0.999974	−	0.00727981i	$-0.997683\pi$
$683$	−26.3238	−	26.3238i	−1.00725	−	1.00725i	−0.00727981	−	0.999974i	$-0.502317\pi$
$684$		0			0
$685$	−6.32849	+	6.32849i	−0.241799	+	0.241799i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.67893			−0.254447
$690$		0			0
$691$	−20.4472	+	20.4472i	−0.777848	+	0.777848i	−0.979465	−	0.201616i	$-0.935381\pi$
$691$	−20.4472	+	20.4472i	−0.777848	+	0.777848i	0.201616	+	0.979465i	$0.435381\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			6.88062i			0.260997i
$696$		0			0
$697$			8.39050i			0.317813i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10.7257	+	10.7257i	−0.405105	+	0.405105i	−0.880028	−	0.474922i	$-0.842476\pi$
$701$	−10.7257	+	10.7257i	−0.405105	+	0.405105i	0.474922	+	0.880028i	$0.342476\pi$
$702$		0			0
$703$	30.5096			1.15069
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.2402	+	22.2402i	−0.836428	+	0.836428i
$708$		0			0
$709$	28.0895	+	28.0895i	1.05492	+	1.05492i	0.998401	+	0.0565230i	$0.0180014\pi$
$709$	28.0895	+	28.0895i	1.05492	+	1.05492i	0.0565230	+	0.998401i	$0.481999\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			1.82180i			0.0682270i
$714$		0			0
$715$	−3.63980	−	3.63980i	−0.136121	−	0.136121i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−33.4334			−1.24686			−0.623428	−	0.781881i	$-0.714260\pi$
$719$	−33.4334			−1.24686			−0.623428	+	0.781881i	$0.714260\pi$
$720$		0			0
$721$	−31.5392			−1.17458
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.03909	+	1.03909i	0.0385907	+	0.0385907i
$726$		0			0
$727$			23.3379i			0.865556i	0.901501	+	0.432778i	$0.142467\pi$
$727$			23.3379i			0.865556i	−0.901501	+	0.432778i	$0.857533\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	33.0875	+	33.0875i	1.22378	+	1.22378i
$732$		0			0
$733$	0.485701	−	0.485701i	0.0179398	−	0.0179398i	−0.698080	−	0.716020i	$-0.745962\pi$
$733$	0.485701	−	0.485701i	0.0179398	−	0.0179398i	0.716020	+	0.698080i	$0.245962\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−47.9432			−1.76601
$738$		0			0
$739$	−15.4245	+	15.4245i	−0.567398	+	0.567398i	−0.931399	−	0.364001i	$-0.881410\pi$
$739$	−15.4245	+	15.4245i	−0.567398	+	0.567398i	0.364001	+	0.931399i	$0.381410\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	38.1386i		−	1.39917i	−0.714550	−	0.699584i	$-0.753369\pi$
$743$		−	38.1386i		−	1.39917i	0.714550	−	0.699584i	$-0.246631\pi$
$744$		0			0
$745$			9.38825i			0.343959i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.0357	+	12.0357i	−0.439775	+	0.439775i
$750$		0			0
$751$	14.5670			0.531557			0.265778	−	0.964034i	$-0.414371\pi$
$751$	14.5670			0.531557			0.265778	+	0.964034i	$0.414371\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.90214	−	6.90214i	0.251195	−	0.251195i
$756$		0			0
$757$	−31.9423	−	31.9423i	−1.16096	−	1.16096i	−0.984265	−	0.176700i	$-0.943458\pi$
$757$	−31.9423	−	31.9423i	−1.16096	−	1.16096i	−0.176700	−	0.984265i	$-0.556542\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	4.11004i		−	0.148989i	−0.997221	−	0.0744944i	$-0.976266\pi$
$761$		−	4.11004i		−	0.148989i	0.997221	−	0.0744944i	$-0.0237343\pi$
$762$		0			0
$763$	7.15205	+	7.15205i	0.258922	+	0.258922i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.96816			−0.251606
$768$		0			0
$769$	−33.2997			−1.20082			−0.600409	−	0.799693i	$-0.704995\pi$
$769$	−33.2997			−1.20082			−0.600409	+	0.799693i	$0.704995\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	11.5368	+	11.5368i	0.414948	+	0.414948i	0.883458	−	0.468510i	$-0.155209\pi$
$773$	11.5368	+	11.5368i	0.414948	+	0.414948i	−0.468510	+	0.883458i	$0.655209\pi$
$774$		0			0
$775$		−	1.03962i		−	0.0373442i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.13867	−	2.13867i	−0.0766258	−	0.0766258i
$780$		0			0
$781$	−18.3831	+	18.3831i	−0.657800	+	0.657800i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10.9042			0.389187
$786$		0			0
$787$	25.1943	−	25.1943i	0.898080	−	0.898080i	−0.0971865	−	0.995266i	$-0.530984\pi$
$787$	25.1943	−	25.1943i	0.898080	−	0.898080i	0.995266	+	0.0971865i	$0.0309843\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			4.58967i			0.163190i
$792$		0			0
$793$		−	10.9868i		−	0.390153i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.8873	−	17.8873i	0.633601	−	0.633601i	−0.315368	−	0.948969i	$-0.602128\pi$
$797$	17.8873	−	17.8873i	0.633601	−	0.633601i	0.948969	+	0.315368i	$0.102128\pi$
$798$		0			0
$799$	63.1439			2.23387
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−41.6866	+	41.6866i	−1.47109	+	1.47109i
$804$		0			0
$805$	−2.54572	−	2.54572i	−0.0897247	−	0.0897247i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			42.6682i			1.50013i	0.661362	+	0.750066i	$0.269979\pi$
$809$			42.6682i			1.50013i	−0.661362	+	0.750066i	$0.730021\pi$
$810$		0			0
$811$	23.0275	+	23.0275i	0.808606	+	0.808606i	0.984423	−	0.175817i	$-0.0562566\pi$
$811$	23.0275	+	23.0275i	0.808606	+	0.808606i	−0.175817	+	0.984423i	$0.556257\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.65187			−0.338090
$816$		0			0
$817$	−16.8675			−0.590117
$818$		0			0
$819$		0			0
$820$		0			0
$821$	20.8339	+	20.8339i	0.727107	+	0.727107i	0.970043	−	0.242935i	$-0.0781102\pi$
$821$	20.8339	+	20.8339i	0.727107	+	0.727107i	−0.242935	+	0.970043i	$0.578110\pi$
$822$		0			0
$823$		−	7.72425i		−	0.269250i	−0.990897	−	0.134625i	$-0.957017\pi$
$823$		−	7.72425i		−	0.269250i	0.990897	−	0.134625i	$-0.0429831\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.55125	−	4.55125i	−0.158262	−	0.158262i	0.623534	−	0.781796i	$-0.285696\pi$
$827$	−4.55125	−	4.55125i	−0.158262	−	0.158262i	−0.781796	+	0.623534i	$0.785696\pi$
$828$		0			0
$829$	14.9428	−	14.9428i	0.518984	−	0.518984i	−0.398280	−	0.917264i	$-0.630393\pi$
$829$	14.9428	−	14.9428i	0.518984	−	0.518984i	0.917264	+	0.398280i	$0.130393\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	21.4155			0.742003
$834$		0			0
$835$	−3.23009	+	3.23009i	−0.111782	+	0.111782i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			8.76768i			0.302694i	0.988481	+	0.151347i	$0.0483611\pi$
$839$			8.76768i			0.302694i	−0.988481	+	0.151347i	$0.951639\pi$
$840$		0			0
$841$			26.8406i			0.925538i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.07751	−	8.07751i	0.277875	−	0.277875i
$846$		0			0
$847$	−11.9265			−0.409799
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13.6103	+	13.6103i	−0.466553	+	0.466553i
$852$		0			0
$853$	21.1361	+	21.1361i	0.723686	+	0.723686i	0.969354	−	0.245668i	$-0.0790073\pi$
$853$	21.1361	+	21.1361i	0.723686	+	0.723686i	−0.245668	+	0.969354i	$0.579007\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			18.0812i			0.617643i	0.951120	+	0.308822i	$0.0999346\pi$
$857$			18.0812i			0.617643i	−0.951120	+	0.308822i	$0.900065\pi$
$858$		0			0
$859$	34.0375	+	34.0375i	1.16134	+	1.16134i	0.984182	+	0.177162i	$0.0566915\pi$
$859$	34.0375	+	34.0375i	1.16134	+	1.16134i	0.177162	+	0.984182i	$0.443308\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	12.5506			0.427228			0.213614	−	0.976918i	$-0.431477\pi$
$863$	12.5506			0.427228			0.213614	+	0.976918i	$0.431477\pi$
$864$		0			0
$865$	−10.8412			−0.368610
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−46.7038	−	46.7038i	−1.58432	−	1.58432i
$870$		0			0
$871$			14.6851i			0.497585i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.45273	+	1.45273i	0.0491111	+	0.0491111i
$876$		0			0
$877$	30.1742	−	30.1742i	1.01891	−	1.01891i	0.0190940	−	0.999818i	$-0.493922\pi$
$877$	30.1742	−	30.1742i	1.01891	−	1.01891i	0.999818	−	0.0190940i	$-0.00607817\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9.94748			0.335139			0.167570	−	0.985860i	$-0.446408\pi$
$881$	9.94748			0.335139			0.167570	+	0.985860i	$0.446408\pi$
$882$		0			0
$883$	−18.2661	+	18.2661i	−0.614703	+	0.614703i	−0.944168	−	0.329465i	$-0.893132\pi$
$883$	−18.2661	+	18.2661i	−0.614703	+	0.614703i	0.329465	+	0.944168i	$0.393132\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			10.5713i			0.354950i	0.984125	+	0.177475i	$0.0567928\pi$
$887$			10.5713i			0.354950i	−0.984125	+	0.177475i	$0.943207\pi$
$888$		0			0
$889$		−	29.8973i		−	1.00272i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−16.0948	+	16.0948i	−0.538594	+	0.538594i
$894$		0			0
$895$	−24.8581			−0.830915
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.08026	+	1.08026i	−0.0360285	+	0.0360285i
$900$		0			0
$901$	28.9823	+	28.9823i	0.965540	+	0.965540i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	5.77712i		−	0.192038i
$906$		0			0
$907$	−15.7722	−	15.7722i	−0.523709	−	0.523709i	0.394981	−	0.918689i	$-0.370751\pi$
$907$	−15.7722	−	15.7722i	−0.523709	−	0.523709i	−0.918689	+	0.394981i	$0.870751\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.4842			−0.413620			−0.206810	−	0.978381i	$-0.566308\pi$
$911$	−12.4842			−0.413620			−0.206810	+	0.978381i	$0.566308\pi$
$912$		0			0
$913$	−15.8008			−0.522931
$914$		0			0
$915$		0			0
$916$		0			0
$917$	21.7964	+	21.7964i	0.719780	+	0.719780i
$918$		0			0
$919$		−	35.5289i		−	1.17199i	−0.810314	−	0.585996i	$-0.800704\pi$
$919$		−	35.5289i		−	1.17199i	0.810314	−	0.585996i	$-0.199296\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.63079	+	5.63079i	0.185340	+	0.185340i
$924$		0			0
$925$	7.76676	−	7.76676i	0.255369	−	0.255369i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	51.3640			1.68520			0.842600	−	0.538541i	$-0.181024\pi$
$929$	51.3640			1.68520			0.842600	+	0.538541i	$0.181024\pi$
$930$		0			0
$931$	−5.45863	+	5.45863i	−0.178899	+	0.178899i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			31.5888i			1.03306i
$936$		0			0
$937$		−	10.2493i		−	0.334830i	−0.985887	−	0.167415i	$-0.946458\pi$
$937$		−	10.2493i		−	0.334830i	0.985887	−	0.167415i	$-0.0535420\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.6572	−	20.6572i	0.673404	−	0.673404i	−0.285095	−	0.958499i	$-0.592025\pi$
$941$	20.6572	−	20.6572i	0.673404	−	0.673404i	0.958499	+	0.285095i	$0.0920252\pi$
$942$		0			0
$943$	1.90811			0.0621365
$944$		0			0
$945$		0			0
$946$		0			0
$947$	30.3670	−	30.3670i	0.986794	−	0.986794i	−0.0131196	−	0.999914i	$-0.504176\pi$
$947$	30.3670	−	30.3670i	0.986794	−	0.986794i	0.999914	+	0.0131196i	$0.00417622\pi$
$948$		0			0
$949$	12.7687	+	12.7687i	0.414489	+	0.414489i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	6.04357i		−	0.195771i	−0.995198	−	0.0978853i	$-0.968792\pi$
$953$		−	6.04357i		−	0.195771i	0.995198	−	0.0978853i	$-0.0312078\pi$
$954$		0			0
$955$	−14.6007	−	14.6007i	−0.472469	−	0.472469i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	18.3871			0.593751
$960$		0			0
$961$	−29.9192			−0.965135
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.52494	−	9.52494i	−0.306619	−	0.306619i
$966$		0			0
$967$		−	21.2426i		−	0.683116i	−0.939861	−	0.341558i	$-0.889045\pi$
$967$		−	21.2426i		−	0.683116i	0.939861	−	0.341558i	$-0.110955\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−28.0932	−	28.0932i	−0.901553	−	0.901553i	0.0940180	−	0.995571i	$-0.470029\pi$
$971$	−28.0932	−	28.0932i	−0.901553	−	0.901553i	−0.995571	+	0.0940180i	$0.970029\pi$
$972$		0			0
$973$	9.99565	−	9.99565i	0.320446	−	0.320446i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.854313			0.0273319			0.0136660	−	0.999907i	$-0.495650\pi$
$977$	0.854313			0.0273319			0.0136660	+	0.999907i	$0.495650\pi$
$978$		0			0
$979$	−23.0924	+	23.0924i	−0.738036	+	0.738036i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	21.0489i		−	0.671357i	−0.941977	−	0.335678i	$-0.891034\pi$
$983$		−	21.0489i		−	0.671357i	0.941977	−	0.335678i	$-0.108966\pi$
$984$		0			0
$985$			18.1534i			0.578416i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.52452	−	7.52452i	0.239266	−	0.239266i
$990$		0			0
$991$	−31.6982			−1.00693			−0.503464	−	0.864016i	$-0.667941\pi$
$991$	−31.6982			−1.00693			−0.503464	+	0.864016i	$0.667941\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	12.7605	−	12.7605i	0.404534	−	0.404534i
$996$		0			0
$997$	−12.0541	−	12.0541i	−0.381758	−	0.381758i	0.489977	−	0.871735i	$-0.337005\pi$
$997$	−12.0541	−	12.0541i	−0.381758	−	0.381758i	−0.871735	+	0.489977i	$0.837005\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.e.2161.3		32
3.2	odd	2	inner	2880.2.t.e.2161.9		32
4.3	odd	2		720.2.t.e.181.15	yes	32
12.11	even	2		720.2.t.e.181.2	✓	32
16.3	odd	4		720.2.t.e.541.15	yes	32
16.13	even	4	inner	2880.2.t.e.721.3		32
48.29	odd	4	inner	2880.2.t.e.721.9		32
48.35	even	4		720.2.t.e.541.2	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.t.e.181.2	✓	32	12.11	even	2
720.2.t.e.181.15	yes	32	4.3	odd	2
720.2.t.e.541.2	yes	32	48.35	even	4
720.2.t.e.541.15	yes	32	16.3	odd	4
2880.2.t.e.721.3		32	16.13	even	4	inner
2880.2.t.e.721.9		32	48.29	odd	4	inner
2880.2.t.e.2161.3		32	1.1	even	1	trivial
2880.2.t.e.2161.9		32	3.2	odd	2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.t (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{5} +2.05446i q^{7} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{5} +2.05446i q^{7} +(-2.89872 - 2.89872i) q^{11} +(-0.887884 + 0.887884i) q^{13} +7.70569 q^{17} +(-1.96412 + 1.96412i) q^{19} -1.75237i q^{23} +1.00000i q^{25} +(1.03909 - 1.03909i) q^{29} -1.03962 q^{31} +(1.45273 - 1.45273i) q^{35} +(-7.76676 - 7.76676i) q^{37} +1.08887i q^{41} +(4.29390 + 4.29390i) q^{43} +8.19445 q^{47} +2.77918 q^{49} +(3.76115 + 3.76115i) q^{53} +4.09941i q^{55} +(3.92403 + 3.92403i) q^{59} +(-6.18708 + 6.18708i) q^{61} +1.25566 q^{65} +(8.26972 - 8.26972i) q^{67} -6.34181i q^{71} -14.3810i q^{73} +(5.95531 - 5.95531i) q^{77} +16.1119 q^{79} +(2.72548 - 2.72548i) q^{83} +(-5.44875 - 5.44875i) q^{85} -7.96641i q^{89} +(-1.82412 - 1.82412i) q^{91} +2.77768 q^{95} +7.11484 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 8 q^{19} - 32 q^{37} - 16 q^{43} - 32 q^{49} - 16 q^{61} + 16 q^{67} + 16 q^{79} + 16 q^{85} + 80 q^{91}+O(q^{100}) \) 32 * q - 8 * q^19 - 32 * q^37 - 16 * q^43 - 32 * q^49 - 16 * q^61 + 16 * q^67 + 16 * q^79 + 16 * q^85 + 80 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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