LMFDB - Embedded newform 2880.2.t.e.721.3

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			721.3
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.e.2161.3

$q$-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 $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−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	$-0.873077\pi$
$7$		−	2.05446i		−	0.776514i	0.921551	−	0.388257i	$-0.126923\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.89872	+	2.89872i	−0.873997	+	0.873997i	−0.992905	−	0.118908i	$-0.962061\pi$
$11$	−2.89872	+	2.89872i	−0.873997	+	0.873997i	0.118908	+	0.992905i	$0.462061\pi$
$12$		0			0
$13$	−0.887884	−	0.887884i	−0.246255	−	0.246255i	0.573177	−	0.819432i	$-0.305711\pi$
$13$	−0.887884	−	0.887884i	−0.246255	−	0.246255i	−0.819432	+	0.573177i	$0.805711\pi$
$14$		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	0.444954	−	0.895553i	$-0.353220\pi$
$19$	−1.96412	−	1.96412i	−0.450599	−	0.450599i	−0.895553	+	0.444954i	$0.853220\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.75237i			0.365395i	0.983169	+	0.182697i	$0.0584829\pi$
$23$			1.75237i			0.365395i	−0.983169	+	0.182697i	$0.941517\pi$
$24$		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	0.796971	−	0.604017i	$-0.206434\pi$
$29$	1.03909	+	1.03909i	0.192954	+	0.192954i	−0.604017	+	0.796971i	$0.706434\pi$
$30$		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	−0.334425	+	0.942422i	$0.608542\pi$
$37$	−7.76676	+	7.76676i	−1.27685	+	1.27685i	−0.942422	+	0.334425i	$0.891458\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.08887i		−	0.170053i	−0.996379	−	0.0850265i	$-0.972903\pi$
$41$		−	1.08887i		−	0.170053i	0.996379	−	0.0850265i	$-0.0270975\pi$
$42$		0			0
$43$	4.29390	−	4.29390i	0.654814	−	0.654814i	−0.299334	−	0.954148i	$-0.596765\pi$
$43$	4.29390	−	4.29390i	0.654814	−	0.654814i	0.954148	+	0.299334i	$0.0967646\pi$
$44$		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	−0.399917	−	0.916551i	$-0.630961\pi$
$53$	3.76115	−	3.76115i	0.516634	−	0.516634i	0.916551	+	0.399917i	$0.130961\pi$
$54$		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	−0.403926	−	0.914791i	$-0.632355\pi$
$59$	3.92403	−	3.92403i	0.510865	−	0.510865i	0.914791	+	0.403926i	$0.132355\pi$
$60$		0			0
$61$	−6.18708	−	6.18708i	−0.792175	−	0.792175i	0.189673	−	0.981847i	$-0.439257\pi$
$61$	−6.18708	−	6.18708i	−0.792175	−	0.792175i	−0.981847	+	0.189673i	$0.939257\pi$
$62$		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	0.999946	+	0.0103604i	$0.00329786\pi$
$67$	8.26972	+	8.26972i	1.01031	+	1.01031i	0.0103604	+	0.999946i	$0.496702\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			6.34181i			0.752634i	0.926491	+	0.376317i	$0.122810\pi$
$71$			6.34181i			0.752634i	−0.926491	+	0.376317i	$0.877190\pi$
$72$		0			0
$73$			14.3810i			1.68317i	0.540121	+	0.841587i	$0.318378\pi$
$73$			14.3810i			1.68317i	−0.540121	+	0.841587i	$0.681622\pi$
$74$		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	0.840685	−	0.541524i	$-0.182153\pi$
$83$	2.72548	+	2.72548i	0.299161	+	0.299161i	−0.541524	+	0.840685i	$0.682153\pi$
$84$		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	$0.138749\pi$
$89$			7.96641i			0.844438i	−0.906494	+	0.422219i	$0.861251\pi$
$90$		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	0.0803944	−	0.996763i	$-0.474382\pi$
$101$	10.8253	−	10.8253i	1.07716	−	1.07716i	0.996763	−	0.0803944i	$-0.0256180\pi$
$102$		0			0
$103$		−	15.3515i		−	1.51263i	−0.654207	−	0.756316i	$-0.726997\pi$
$103$		−	15.3515i		−	1.51263i	0.654207	−	0.756316i	$-0.273003\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.85832	−	5.85832i	0.566346	−	0.566346i	−0.364757	−	0.931103i	$-0.618848\pi$
$107$	5.85832	−	5.85832i	0.566346	−	0.566346i	0.931103	+	0.364757i	$0.118848\pi$
$108$		0			0
$109$	3.48123	+	3.48123i	0.333441	+	0.333441i	0.853892	−	0.520451i	$-0.174236\pi$
$109$	3.48123	+	3.48123i	0.333441	+	0.333441i	−0.520451	+	0.853892i	$0.674236\pi$
$110$		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	0.997507	−	0.0705699i	$-0.0224818\pi$
$131$	10.6093	+	10.6093i	0.926937	+	0.926937i	−0.0705699	+	0.997507i	$0.522482\pi$
$132$		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	$0.124874\pi$
$137$			8.94984i			0.764636i	−0.924031	+	0.382318i	$0.875126\pi$
$138$		0			0
$139$	−4.86533	+	4.86533i	−0.412672	+	0.412672i	−0.882668	−	0.469996i	$-0.844255\pi$
$139$	−4.86533	+	4.86533i	−0.412672	+	0.412672i	0.469996	+	0.882668i	$0.344255\pi$
$140$		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	−0.924654	−	0.380808i	$-0.875646\pi$
$149$	−6.63849	+	6.63849i	−0.543847	+	0.543847i	0.380808	+	0.924654i	$0.375646\pi$
$150$		0			0
$151$		−	9.76110i		−	0.794347i	−0.917744	−	0.397174i	$-0.869991\pi$
$151$		−	9.76110i		−	0.794347i	0.917744	−	0.397174i	$-0.130009\pi$
$152$		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	0.328979	−	0.944337i	$-0.393295\pi$
$157$	−7.71042	−	7.71042i	−0.615358	−	0.615358i	−0.944337	+	0.328979i	$0.893295\pi$
$158$		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	0.921928	−	0.387361i	$-0.126613\pi$
$163$	6.82491	+	6.82491i	0.534568	+	0.534568i	−0.387361	+	0.921928i	$0.626613\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			4.56804i			0.353485i	0.984257	+	0.176743i	$0.0565560\pi$
$167$			4.56804i			0.353485i	−0.984257	+	0.176743i	$0.943444\pi$
$168$		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	0.935678	−	0.352854i	$-0.114789\pi$
$173$	7.66585	+	7.66585i	0.582824	+	0.582824i	−0.352854	+	0.935678i	$0.614789\pi$
$174$		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	0.918597	+	0.395196i	$0.129323\pi$
$179$	17.5773	+	17.5773i	1.31379	+	1.31379i	0.395196	+	0.918597i	$0.370677\pi$
$180$		0			0
$181$	4.08504	−	4.08504i	0.303638	−	0.303638i	−0.538797	−	0.842436i	$-0.681121\pi$
$181$	4.08504	−	4.08504i	0.303638	−	0.303638i	0.842436	+	0.538797i	$0.181121\pi$
$182$		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	−0.996626	−	0.0820712i	$-0.973846\pi$
$197$	−12.8364	+	12.8364i	−0.914555	+	0.914555i	0.0820712	+	0.996626i	$0.473846\pi$
$198$		0			0
$199$		−	18.0460i		−	1.27925i	−0.768688	−	0.639624i	$-0.779090\pi$
$199$		−	18.0460i		−	1.27925i	0.768688	−	0.639624i	$-0.220910\pi$
$200$		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	0.998158	−	0.0606726i	$-0.0193246\pi$
$211$	13.6178	+	13.6178i	0.937485	+	0.937485i	−0.0606726	+	0.998158i	$0.519325\pi$
$212$		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	−0.908952	−	0.416901i	$-0.863116\pi$
$227$	−19.9760	−	19.9760i	−1.32585	−	1.32585i	−0.416901	−	0.908952i	$-0.636884\pi$
$228$		0			0
$229$	−2.00826	+	2.00826i	−0.132710	+	0.132710i	−0.770341	−	0.637632i	$-0.779914\pi$
$229$	−2.00826	+	2.00826i	−0.132710	+	0.132710i	0.637632	+	0.770341i	$0.279914\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.3450i			1.07080i	0.844599	+	0.535399i	$0.179839\pi$
$233$			16.3450i			1.07080i	−0.844599	+	0.535399i	$0.820161\pi$
$234$		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	−0.575016	−	0.818142i	$-0.695004\pi$
$251$	3.85183	−	3.85183i	0.243125	−	0.243125i	0.818142	+	0.575016i	$0.195004\pi$
$252$		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	$0.0757333\pi$
$263$			7.64434i			0.471370i	−0.971830	+	0.235685i	$0.924267\pi$
$264$		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	0.994581	−	0.103962i	$-0.0331519\pi$
$269$	14.6072	+	14.6072i	0.890620	+	0.890620i	−0.103962	+	0.994581i	$0.533152\pi$
$270$		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	−0.802422	−	0.596757i	$-0.796456\pi$
$277$	−3.42295	+	3.42295i	−0.205665	+	0.205665i	0.596757	+	0.802422i	$0.296456\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	17.5315i		−	1.04584i	−0.852381	−	0.522922i	$-0.824842\pi$
$281$		−	17.5315i		−	1.04584i	0.852381	−	0.522922i	$-0.175158\pi$
$282$		0			0
$283$	3.45835	−	3.45835i	0.205577	−	0.205577i	−0.596807	−	0.802385i	$-0.703564\pi$
$283$	3.45835	−	3.45835i	0.205577	−	0.205577i	0.802385	+	0.596807i	$0.203564\pi$
$284$		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	−0.128934	−	0.991653i	$-0.541155\pi$
$293$	14.7674	−	14.7674i	0.862720	−	0.862720i	0.991653	+	0.128934i	$0.0411554\pi$
$294$		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	0.300479	−	0.953788i	$-0.402853\pi$
$307$	−11.4469	−	11.4469i	−0.653309	−	0.653309i	−0.953788	+	0.300479i	$0.902853\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			33.2894i			1.88767i	0.330419	+	0.943834i	$0.392810\pi$
$311$			33.2894i			1.88767i	−0.330419	+	0.943834i	$0.607190\pi$
$312$		0			0
$313$		−	17.5967i		−	0.994627i	−0.867571	−	0.497313i	$-0.834320\pi$
$313$		−	17.5967i		−	0.994627i	0.867571	−	0.497313i	$-0.165680\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.81377	+	7.81377i	0.438865	+	0.438865i	0.891630	−	0.452765i	$-0.149562\pi$
$317$	7.81377	+	7.81377i	0.438865	+	0.438865i	−0.452765	+	0.891630i	$0.649562\pi$
$318$		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	−0.0239030	+	0.999714i	$0.507609\pi$
$331$	−18.6231	+	18.6231i	−1.02362	+	1.02362i	−0.999714	+	0.0239030i	$0.992391\pi$
$332$		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	−0.742891	−	0.669413i	$-0.766546\pi$
$347$	−1.36875	+	1.36875i	−0.0734784	+	0.0734784i	0.669413	+	0.742891i	$0.266546\pi$
$348$		0			0
$349$	−12.2214	−	12.2214i	−0.654198	−	0.654198i	0.299803	−	0.954001i	$-0.403079\pi$
$349$	−12.2214	−	12.2214i	−0.654198	−	0.654198i	−0.954001	+	0.299803i	$0.903079\pi$
$350$		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	$-0.979200\pi$
$359$		−	2.47446i		−	0.130597i	0.997866	−	0.0652984i	$-0.0207999\pi$
$360$		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	−0.807250	−	0.590209i	$-0.799045\pi$
$373$	−4.19175	+	4.19175i	−0.217041	+	0.217041i	0.590209	+	0.807250i	$0.299045\pi$
$374$		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	−0.253185	−	0.967418i	$-0.581478\pi$
$379$	13.9046	−	13.9046i	0.714233	−	0.714233i	0.967418	+	0.253185i	$0.0814780\pi$
$380$		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	−0.609707	−	0.792627i	$-0.708713\pi$
$389$	3.60774	−	3.60774i	0.182920	−	0.182920i	0.792627	+	0.609707i	$0.208713\pi$
$390$		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	0.218370	−	0.975866i	$-0.429926\pi$
$397$	−15.0930	−	15.0930i	−0.757497	−	0.757497i	−0.975866	+	0.218370i	$0.929926\pi$
$398$		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	$-0.865942\pi$
$409$		−	16.5356i		−	0.817631i	0.912617	−	0.408816i	$-0.134058\pi$
$410$		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	0.805344	−	0.592807i	$-0.201980\pi$
$419$	4.35052	+	4.35052i	0.212537	+	0.212537i	−0.592807	+	0.805344i	$0.701980\pi$
$420$		0			0
$421$	−7.22188	+	7.22188i	−0.351973	+	0.351973i	−0.860843	−	0.508870i	$-0.830063\pi$
$421$	−7.22188	+	7.22188i	−0.351973	+	0.351973i	0.508870	+	0.860843i	$0.330063\pi$
$422$		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	$-0.862940\pi$
$439$		−	17.4912i		−	0.834810i	0.908721	−	0.417405i	$-0.137060\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.51139	−	7.51139i	0.356877	−	0.356877i	−0.505784	−	0.862660i	$-0.668797\pi$
$443$	7.51139	−	7.51139i	0.356877	−	0.356877i	0.862660	+	0.505784i	$0.168797\pi$
$444$		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	$0.206403\pi$
$457$			25.8215i			1.20788i	−0.797031	+	0.603939i	$0.793597\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.0676	−	12.0676i	−0.562046	−	0.562046i	0.367842	−	0.929888i	$-0.380097\pi$
$461$	−12.0676	−	12.0676i	−0.562046	−	0.562046i	−0.929888	+	0.367842i	$0.880097\pi$
$462$		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	0.878315	−	0.478082i	$-0.158668\pi$
$467$	8.64911	+	8.64911i	0.400233	+	0.400233i	−0.478082	+	0.878315i	$0.658668\pi$
$468$		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	$0.105277\pi$
$487$			14.3328i			0.649481i	−0.945803	+	0.324740i	$0.894723\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−30.8986	+	30.8986i	−1.39443	+	1.39443i	−0.579370	+	0.815065i	$0.696701\pi$
$491$	−30.8986	+	30.8986i	−1.39443	+	1.39443i	−0.815065	+	0.579370i	$0.803299\pi$
$492$		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	0.983233	−	0.182354i	$-0.0583717\pi$
$499$	17.8903	+	17.8903i	0.800879	+	0.800879i	−0.182354	+	0.983233i	$0.558372\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	5.72534i		−	0.255280i	−0.991821	−	0.127640i	$-0.959260\pi$
$503$		−	5.72534i		−	0.255280i	0.991821	−	0.127640i	$-0.0407403\pi$
$504$		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	0.215311	−	0.976546i	$-0.430923\pi$
$509$	−17.1742	−	17.1742i	−0.761234	−	0.761234i	−0.976546	+	0.215311i	$0.930923\pi$
$510$		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	$0.0418826\pi$
$521$			5.98934i			0.262398i	−0.991356	+	0.131199i	$0.958117\pi$
$522$		0			0
$523$	−18.0403	+	18.0403i	−0.788848	+	0.788848i	−0.981305	−	0.192458i	$-0.938354\pi$
$523$	−18.0403	+	18.0403i	−0.788848	+	0.788848i	0.192458	+	0.981305i	$0.438354\pi$
$524$		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	0.936732	+	0.350047i	$0.113834\pi$
$541$	29.9297	+	29.9297i	1.28678	+	1.28678i	0.350047	+	0.936732i	$0.386166\pi$
$542$		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	0.139029	−	0.990288i	$-0.455602\pi$
$547$	−19.9093	−	19.9093i	−0.851259	−	0.851259i	−0.990288	+	0.139029i	$0.955602\pi$
$548$		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	0.440073	−	0.897962i	$-0.354953\pi$
$557$	−10.8066	−	10.8066i	−0.457889	−	0.457889i	−0.897962	+	0.440073i	$0.854953\pi$
$558$		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	−0.999336	−	0.0364461i	$-0.988396\pi$
$563$	−24.5766	−	24.5766i	−1.03578	−	1.03578i	−0.0364461	−	0.999336i	$-0.511604\pi$
$564$		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	$-0.835590\pi$
$569$		−	23.5603i		−	0.987697i	0.869548	−	0.493849i	$-0.164410\pi$
$570$		0			0
$571$	19.6252	−	19.6252i	0.821290	−	0.821290i	−0.165003	−	0.986293i	$-0.552763\pi$
$571$	19.6252	−	19.6252i	0.821290	−	0.821290i	0.986293	+	0.165003i	$0.0527633\pi$
$572$		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	−0.491944	−	0.870627i	$-0.663713\pi$
$587$	9.17477	−	9.17477i	0.378683	−	0.378683i	0.870627	+	0.491944i	$0.163713\pi$
$588$		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	$-0.615096\pi$
$599$		−	45.7838i		−	1.87067i	0.353757	−	0.935337i	$-0.384904\pi$
$600$		0			0
$601$			39.9200i			1.62837i	0.580606	+	0.814185i	$0.302816\pi$
$601$			39.9200i			1.62837i	−0.580606	+	0.814185i	$0.697184\pi$
$602$		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	−0.434292	−	0.900772i	$-0.643001\pi$
$613$	11.5495	−	11.5495i	0.466480	−	0.466480i	0.900772	+	0.434292i	$0.143001\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			36.3120i			1.46187i	0.682449	+	0.730934i	$0.260915\pi$
$617$			36.3120i			1.46187i	−0.682449	+	0.730934i	$0.739085\pi$
$618$		0			0
$619$	25.6174	−	25.6174i	1.02965	−	1.02965i	0.0301014	−	0.999547i	$-0.490417\pi$
$619$	25.6174	−	25.6174i	1.02965	−	1.02965i	0.999547	−	0.0301014i	$-0.00958303\pi$
$620$		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	$-0.769463\pi$
$631$		−	33.2875i		−	1.32515i	0.748994	−	0.662577i	$-0.230537\pi$
$632$		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	0.867678	−	0.497127i	$-0.165612\pi$
$643$	9.39620	+	9.39620i	0.370550	+	0.370550i	−0.497127	+	0.867678i	$0.665612\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			37.6472i			1.48006i	0.672572	+	0.740032i	$0.265190\pi$
$647$			37.6472i			1.48006i	−0.672572	+	0.740032i	$0.734810\pi$
$648$		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	0.441923	−	0.897053i	$-0.354297\pi$
$653$	−11.6303	−	11.6303i	−0.455130	−	0.455130i	−0.897053	+	0.441923i	$0.854297\pi$
$654$		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	0.834952	−	0.550322i	$-0.185495\pi$
$659$	7.30674	+	7.30674i	0.284630	+	0.284630i	−0.550322	+	0.834952i	$0.685495\pi$
$660$		0			0
$661$	−24.9364	+	24.9364i	−0.969914	+	0.969914i	−0.999560	−	0.0296469i	$-0.990562\pi$
$661$	−24.9364	+	24.9364i	−0.969914	+	0.969914i	0.0296469	+	0.999560i	$0.490562\pi$
$662$		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	−0.676591	−	0.736359i	$-0.736543\pi$
$677$	1.55513	−	1.55513i	0.0597684	−	0.0597684i	0.736359	+	0.676591i	$0.236543\pi$
$678$		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	−0.00727981	+	0.999974i	$0.502317\pi$
$683$	−26.3238	+	26.3238i	−1.00725	+	1.00725i	−0.999974	+	0.00727981i	$0.997683\pi$
$684$		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	0.201616	−	0.979465i	$-0.435381\pi$
$691$	−20.4472	−	20.4472i	−0.777848	−	0.777848i	−0.979465	+	0.201616i	$0.935381\pi$
$692$		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	0.474922	−	0.880028i	$-0.342476\pi$
$701$	−10.7257	−	10.7257i	−0.405105	−	0.405105i	−0.880028	+	0.474922i	$0.842476\pi$
$702$		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	0.0565230	−	0.998401i	$-0.481999\pi$
$709$	28.0895	−	28.0895i	1.05492	−	1.05492i	0.998401	−	0.0565230i	$-0.0180014\pi$
$710$		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	$-0.857533\pi$
$727$		−	23.3379i		−	0.865556i	0.901501	−	0.432778i	$-0.142467\pi$
$728$		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	0.716020	−	0.698080i	$-0.245962\pi$
$733$	0.485701	+	0.485701i	0.0179398	+	0.0179398i	−0.698080	+	0.716020i	$0.745962\pi$
$734$		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	0.364001	−	0.931399i	$-0.381410\pi$
$739$	−15.4245	−	15.4245i	−0.567398	−	0.567398i	−0.931399	+	0.364001i	$0.881410\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			38.1386i			1.39917i	0.714550	+	0.699584i	$0.246631\pi$
$743$			38.1386i			1.39917i	−0.714550	+	0.699584i	$0.753369\pi$
$744$		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	−0.176700	+	0.984265i	$0.556542\pi$
$757$	−31.9423	+	31.9423i	−1.16096	+	1.16096i	−0.984265	+	0.176700i	$0.943458\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			4.11004i			0.148989i	0.997221	+	0.0744944i	$0.0237343\pi$
$761$			4.11004i			0.148989i	−0.997221	+	0.0744944i	$0.976266\pi$
$762$		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	−0.468510	−	0.883458i	$-0.655209\pi$
$773$	11.5368	−	11.5368i	0.414948	−	0.414948i	0.883458	+	0.468510i	$0.155209\pi$
$774$		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	0.995266	−	0.0971865i	$-0.0309843\pi$
$787$	25.1943	+	25.1943i	0.898080	+	0.898080i	−0.0971865	+	0.995266i	$0.530984\pi$
$788$		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	0.948969	−	0.315368i	$-0.102128\pi$
$797$	17.8873	+	17.8873i	0.633601	+	0.633601i	−0.315368	+	0.948969i	$0.602128\pi$
$798$		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	$-0.730021\pi$
$809$		−	42.6682i		−	1.50013i	0.661362	−	0.750066i	$-0.269979\pi$
$810$		0			0
$811$	23.0275	−	23.0275i	0.808606	−	0.808606i	−0.175817	−	0.984423i	$-0.556257\pi$
$811$	23.0275	−	23.0275i	0.808606	−	0.808606i	0.984423	+	0.175817i	$0.0562566\pi$
$812$		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	−0.242935	−	0.970043i	$-0.578110\pi$
$821$	20.8339	−	20.8339i	0.727107	−	0.727107i	0.970043	+	0.242935i	$0.0781102\pi$
$822$		0			0
$823$			7.72425i			0.269250i	0.990897	+	0.134625i	$0.0429831\pi$
$823$			7.72425i			0.269250i	−0.990897	+	0.134625i	$0.957017\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.55125	+	4.55125i	−0.158262	+	0.158262i	−0.781796	−	0.623534i	$-0.785696\pi$
$827$	−4.55125	+	4.55125i	−0.158262	+	0.158262i	0.623534	+	0.781796i	$0.285696\pi$
$828$		0			0
$829$	14.9428	+	14.9428i	0.518984	+	0.518984i	0.917264	−	0.398280i	$-0.130393\pi$
$829$	14.9428	+	14.9428i	0.518984	+	0.518984i	−0.398280	+	0.917264i	$0.630393\pi$
$830$		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	$-0.951639\pi$
$839$		−	8.76768i		−	0.302694i	0.988481	−	0.151347i	$-0.0483611\pi$
$840$		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	−0.245668	−	0.969354i	$-0.579007\pi$
$853$	21.1361	−	21.1361i	0.723686	−	0.723686i	0.969354	+	0.245668i	$0.0790073\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	18.0812i		−	0.617643i	−0.951120	−	0.308822i	$-0.900065\pi$
$857$		−	18.0812i		−	0.617643i	0.951120	−	0.308822i	$-0.0999346\pi$
$858$		0			0
$859$	34.0375	−	34.0375i	1.16134	−	1.16134i	0.177162	−	0.984182i	$-0.443308\pi$
$859$	34.0375	−	34.0375i	1.16134	−	1.16134i	0.984182	−	0.177162i	$-0.0566915\pi$
$860$		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	0.999818	+	0.0190940i	$0.00607817\pi$
$877$	30.1742	+	30.1742i	1.01891	+	1.01891i	0.0190940	+	0.999818i	$0.493922\pi$
$878$		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	0.329465	−	0.944168i	$-0.393132\pi$
$883$	−18.2661	−	18.2661i	−0.614703	−	0.614703i	−0.944168	+	0.329465i	$0.893132\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	10.5713i		−	0.354950i	−0.984125	−	0.177475i	$-0.943207\pi$
$887$		−	10.5713i		−	0.354950i	0.984125	−	0.177475i	$-0.0567928\pi$
$888$		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	−0.918689	−	0.394981i	$-0.870751\pi$
$907$	−15.7722	+	15.7722i	−0.523709	+	0.523709i	0.394981	+	0.918689i	$0.370751\pi$
$908$		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	$0.199296\pi$
$919$			35.5289i			1.17199i	−0.810314	+	0.585996i	$0.800704\pi$
$920$		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	$0.0535420\pi$
$937$			10.2493i			0.334830i	−0.985887	+	0.167415i	$0.946458\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.6572	+	20.6572i	0.673404	+	0.673404i	0.958499	−	0.285095i	$-0.0920252\pi$
$941$	20.6572	+	20.6572i	0.673404	+	0.673404i	−0.285095	+	0.958499i	$0.592025\pi$
$942$		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	0.999914	−	0.0131196i	$-0.00417622\pi$
$947$	30.3670	+	30.3670i	0.986794	+	0.986794i	−0.0131196	+	0.999914i	$0.504176\pi$
$948$		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	$0.0312078\pi$
$953$			6.04357i			0.195771i	−0.995198	+	0.0978853i	$0.968792\pi$
$954$		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	$0.110955\pi$
$967$			21.2426i			0.683116i	−0.939861	+	0.341558i	$0.889045\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−28.0932	+	28.0932i	−0.901553	+	0.901553i	−0.995571	−	0.0940180i	$-0.970029\pi$
$971$	−28.0932	+	28.0932i	−0.901553	+	0.901553i	0.0940180	+	0.995571i	$0.470029\pi$
$972$		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	$0.108966\pi$
$983$			21.0489i			0.671357i	−0.941977	+	0.335678i	$0.891034\pi$
$984$		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	−0.871735	−	0.489977i	$-0.837005\pi$
$997$	−12.0541	+	12.0541i	−0.381758	+	0.381758i	0.489977	+	0.871735i	$0.337005\pi$
$998$		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.721.3		32
3.2	odd	2	inner	2880.2.t.e.721.9		32
4.3	odd	2		720.2.t.e.541.15	yes	32
12.11	even	2		720.2.t.e.541.2	yes	32
16.5	even	4	inner	2880.2.t.e.2161.3		32
16.11	odd	4		720.2.t.e.181.15	yes	32
48.5	odd	4	inner	2880.2.t.e.2161.9		32
48.11	even	4		720.2.t.e.181.2	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.t.e.181.2	✓	32	48.11	even	4
720.2.t.e.181.15	yes	32	16.11	odd	4
720.2.t.e.541.2	yes	32	12.11	even	2
720.2.t.e.541.15	yes	32	4.3	odd	2
2880.2.t.e.721.3		32	1.1	even	1	trivial
2880.2.t.e.721.9		32	3.2	odd	2	inner
2880.2.t.e.2161.3		32	16.5	even	4	inner
2880.2.t.e.2161.9		32	48.5	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 $\F_{q}$

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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