LMFDB - Embedded newform 2880.2.t.e.721.13

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			721.13
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.e.2161.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{5} +1.69880i q^{7} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{5} +1.69880i q^{7} +(-1.72808 + 1.72808i) q^{11} +(0.217214 + 0.217214i) q^{13} -3.27804 q^{17} +(-1.73107 - 1.73107i) q^{19} -2.93129i q^{23} -1.00000i q^{25} +(-6.42961 - 6.42961i) q^{29} -7.75081 q^{31} +(1.20123 + 1.20123i) q^{35} +(4.70155 - 4.70155i) q^{37} -4.36530i q^{41} +(4.30731 - 4.30731i) q^{43} +0.568734 q^{47} +4.11408 q^{49} +(0.749102 - 0.749102i) q^{53} +2.44387i q^{55} +(2.21126 - 2.21126i) q^{59} +(7.39554 + 7.39554i) q^{61} +0.307187 q^{65} +(-9.00587 - 9.00587i) q^{67} -2.32389i q^{71} +0.285439i q^{73} +(-2.93566 - 2.93566i) q^{77} +2.59608 q^{79} +(-10.7338 - 10.7338i) q^{83} +(-2.31792 + 2.31792i) q^{85} -10.1057i q^{89} +(-0.369003 + 0.369003i) q^{91} -2.44811 q^{95} -10.0062 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			1.69880i			0.642086i	0.947065	+	0.321043i	$0.104033\pi$
$7$			1.69880i			0.642086i	−0.947065	+	0.321043i	$0.895967\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.72808	+	1.72808i	−0.521036	+	0.521036i	−0.917884	−	0.396849i	$-0.870104\pi$
$11$	−1.72808	+	1.72808i	−0.521036	+	0.521036i	0.396849	+	0.917884i	$0.370104\pi$
$12$		0			0
$13$	0.217214	+	0.217214i	0.0602443	+	0.0602443i	0.736587	−	0.676343i	$-0.236436\pi$
$13$	0.217214	+	0.217214i	0.0602443	+	0.0602443i	−0.676343	+	0.736587i	$0.736436\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.27804			−0.795042			−0.397521	−	0.917593i	$-0.630129\pi$
$17$	−3.27804			−0.795042			−0.397521	+	0.917593i	$0.630129\pi$
$18$		0			0
$19$	−1.73107	−	1.73107i	−0.397135	−	0.397135i	0.480086	−	0.877221i	$-0.340605\pi$
$19$	−1.73107	−	1.73107i	−0.397135	−	0.397135i	−0.877221	+	0.480086i	$0.840605\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.93129i		−	0.611216i	−0.952157	−	0.305608i	$-0.901140\pi$
$23$		−	2.93129i		−	0.611216i	0.952157	−	0.305608i	$-0.0988597\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.42961	−	6.42961i	−1.19395	−	1.19395i	−0.975949	−	0.218000i	$-0.930047\pi$
$29$	−6.42961	−	6.42961i	−1.19395	−	1.19395i	−0.218000	−	0.975949i	$-0.569953\pi$
$30$		0			0
$31$	−7.75081			−1.39209			−0.696043	−	0.718000i	$-0.745058\pi$
$31$	−7.75081			−1.39209			−0.696043	+	0.718000i	$0.745058\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.20123	+	1.20123i	0.203046	+	0.203046i
$36$		0			0
$37$	4.70155	−	4.70155i	0.772930	−	0.772930i	−0.205688	−	0.978618i	$-0.565943\pi$
$37$	4.70155	−	4.70155i	0.772930	−	0.772930i	0.978618	+	0.205688i	$0.0659431\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.36530i		−	0.681745i	−0.940109	−	0.340873i	$-0.889277\pi$
$41$		−	4.36530i		−	0.681745i	0.940109	−	0.340873i	$-0.110723\pi$
$42$		0			0
$43$	4.30731	−	4.30731i	0.656858	−	0.656858i	−0.297777	−	0.954635i	$-0.596245\pi$
$43$	4.30731	−	4.30731i	0.656858	−	0.656858i	0.954635	+	0.297777i	$0.0962452\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.568734			0.0829584			0.0414792	−	0.999139i	$-0.486793\pi$
$47$	0.568734			0.0829584			0.0414792	+	0.999139i	$0.486793\pi$
$48$		0			0
$49$	4.11408			0.587725
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.749102	−	0.749102i	0.102897	−	0.102897i	−0.653784	−	0.756681i	$-0.726820\pi$
$53$	0.749102	−	0.749102i	0.102897	−	0.102897i	0.756681	+	0.653784i	$0.226820\pi$
$54$		0			0
$55$			2.44387i			0.329532i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.21126	−	2.21126i	0.287882	−	0.287882i	−0.548360	−	0.836242i	$-0.684748\pi$
$59$	2.21126	−	2.21126i	0.287882	−	0.287882i	0.836242	+	0.548360i	$0.184748\pi$
$60$		0			0
$61$	7.39554	+	7.39554i	0.946902	+	0.946902i	0.998660	−	0.0517574i	$-0.0164822\pi$
$61$	7.39554	+	7.39554i	0.946902	+	0.946902i	−0.0517574	+	0.998660i	$0.516482\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.307187			0.0381019
$66$		0			0
$67$	−9.00587	−	9.00587i	−1.10024	−	1.10024i	−0.994381	−	0.105862i	$-0.966240\pi$
$67$	−9.00587	−	9.00587i	−1.10024	−	1.10024i	−0.105862	−	0.994381i	$-0.533760\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.32389i		−	0.275795i	−0.990447	−	0.137897i	$-0.955966\pi$
$71$		−	2.32389i		−	0.275795i	0.990447	−	0.137897i	$-0.0440344\pi$
$72$		0			0
$73$			0.285439i			0.0334081i	0.999860	+	0.0167040i	$0.00531731\pi$
$73$			0.285439i			0.0334081i	−0.999860	+	0.0167040i	$0.994683\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.93566	−	2.93566i	−0.334550	−	0.334550i
$78$		0			0
$79$	2.59608			0.292082			0.146041	−	0.989279i	$-0.453347\pi$
$79$	2.59608			0.292082			0.146041	+	0.989279i	$0.453347\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.7338	−	10.7338i	−1.17819	−	1.17819i	−0.980204	−	0.197988i	$-0.936559\pi$
$83$	−10.7338	−	10.7338i	−1.17819	−	1.17819i	−0.197988	−	0.980204i	$-0.563441\pi$
$84$		0			0
$85$	−2.31792	+	2.31792i	−0.251414	+	0.251414i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	10.1057i		−	1.07121i	−0.844470	−	0.535604i	$-0.820084\pi$
$89$		−	10.1057i		−	1.07121i	0.844470	−	0.535604i	$-0.179916\pi$
$90$		0			0
$91$	−0.369003	+	0.369003i	−0.0386821	+	0.0386821i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.44811			−0.251170
$96$		0			0
$97$	−10.0062			−1.01598			−0.507990	−	0.861363i	$-0.669611\pi$
$97$	−10.0062			−1.01598			−0.507990	+	0.861363i	$0.669611\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.37039	−	8.37039i	0.832885	−	0.832885i	−0.155025	−	0.987911i	$-0.549546\pi$
$101$	8.37039	−	8.37039i	0.832885	−	0.832885i	0.987911	+	0.155025i	$0.0495458\pi$
$102$		0			0
$103$			6.53186i			0.643604i	0.946807	+	0.321802i	$0.104288\pi$
$103$			6.53186i			0.643604i	−0.946807	+	0.321802i	$0.895712\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.25677	−	5.25677i	0.508191	−	0.508191i	−0.405780	−	0.913971i	$-0.633000\pi$
$107$	5.25677	−	5.25677i	0.508191	−	0.508191i	0.913971	+	0.405780i	$0.133000\pi$
$108$		0			0
$109$	11.6006	+	11.6006i	1.11114	+	1.11114i	0.992997	+	0.118143i	$0.0376941\pi$
$109$	11.6006	+	11.6006i	1.11114	+	1.11114i	0.118143	+	0.992997i	$0.462306\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.80988			−0.358404			−0.179202	−	0.983812i	$-0.557352\pi$
$113$	−3.80988			−0.358404			−0.179202	+	0.983812i	$0.557352\pi$
$114$		0			0
$115$	−2.07273	−	2.07273i	−0.193283	−	0.193283i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	5.56874i		−	0.510485i
$120$		0			0
$121$			5.02748i			0.457044i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.0632456	−	0.0632456i
$126$		0			0
$127$	−16.4060			−1.45579			−0.727897	−	0.685687i	$-0.759502\pi$
$127$	−16.4060			−1.45579			−0.727897	+	0.685687i	$0.759502\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.7965	−	10.7965i	−0.943291	−	0.943291i	0.0551849	−	0.998476i	$-0.482425\pi$
$131$	−10.7965	−	10.7965i	−0.943291	−	0.943291i	−0.998476	+	0.0551849i	$0.982425\pi$
$132$		0			0
$133$	2.94075	−	2.94075i	0.254995	−	0.254995i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.96450i			0.424146i	0.977254	+	0.212073i	$0.0680214\pi$
$137$			4.96450i			0.424146i	−0.977254	+	0.212073i	$0.931979\pi$
$138$		0			0
$139$	5.85795	−	5.85795i	0.496865	−	0.496865i	−0.413596	−	0.910461i	$-0.635727\pi$
$139$	5.85795	−	5.85795i	0.496865	−	0.496865i	0.910461	+	0.413596i	$0.135727\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.750726			−0.0627789
$144$		0			0
$145$	−9.09284			−0.755119
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.8051	+	11.8051i	−0.967108	+	0.967108i	−0.999476	−	0.0323682i	$-0.989695\pi$
$149$	−11.8051	+	11.8051i	−0.967108	+	0.967108i	0.0323682	+	0.999476i	$0.489695\pi$
$150$		0			0
$151$			10.7013i			0.870862i	0.900222	+	0.435431i	$0.143404\pi$
$151$			10.7013i			0.870862i	−0.900222	+	0.435431i	$0.856596\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.48065	+	5.48065i	−0.440216	+	0.440216i
$156$		0			0
$157$	4.87849	+	4.87849i	0.389346	+	0.389346i	0.874454	−	0.485108i	$-0.161220\pi$
$157$	4.87849	+	4.87849i	0.389346	+	0.389346i	−0.485108	+	0.874454i	$0.661220\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.97967			0.392453
$162$		0			0
$163$	10.3627	+	10.3627i	0.811665	+	0.811665i	0.984883	−	0.173218i	$-0.0554165\pi$
$163$	10.3627	+	10.3627i	0.811665	+	0.811665i	−0.173218	+	0.984883i	$0.555417\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	22.8744i		−	1.77007i	−0.465520	−	0.885037i	$-0.654133\pi$
$167$		−	22.8744i		−	1.77007i	0.465520	−	0.885037i	$-0.345867\pi$
$168$		0			0
$169$		−	12.9056i		−	0.992741i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.15292	−	3.15292i	−0.239712	−	0.239712i	0.577019	−	0.816731i	$-0.304216\pi$
$173$	−3.15292	−	3.15292i	−0.239712	−	0.239712i	−0.816731	+	0.577019i	$0.804216\pi$
$174$		0			0
$175$	1.69880			0.128417
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.950602	−	0.950602i	−0.0710513	−	0.0710513i	0.670688	−	0.741739i	$-0.265999\pi$
$179$	−0.950602	−	0.950602i	−0.0710513	−	0.0710513i	−0.741739	+	0.670688i	$0.765999\pi$
$180$		0			0
$181$	11.0786	−	11.0786i	0.823465	−	0.823465i	−0.163139	−	0.986603i	$-0.552162\pi$
$181$	11.0786	−	11.0786i	0.823465	−	0.823465i	0.986603	+	0.163139i	$0.0521618\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	6.64899i		−	0.488844i
$186$		0			0
$187$	5.66472	−	5.66472i	0.414245	−	0.414245i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−24.3378			−1.76102			−0.880509	−	0.474029i	$-0.842799\pi$
$191$	−24.3378			−1.76102			−0.880509	+	0.474029i	$0.842799\pi$
$192$		0			0
$193$	−24.6566			−1.77482			−0.887409	−	0.460982i	$-0.847497\pi$
$193$	−24.6566			−1.77482			−0.887409	+	0.460982i	$0.847497\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.20031	+	1.20031i	−0.0855185	+	0.0855185i	−0.748572	−	0.663054i	$-0.769260\pi$
$197$	−1.20031	+	1.20031i	−0.0855185	+	0.0855185i	0.663054	+	0.748572i	$0.269260\pi$
$198$		0			0
$199$			18.1355i			1.28559i	0.766037	+	0.642797i	$0.222226\pi$
$199$			18.1355i			1.28559i	−0.766037	+	0.642797i	$0.777774\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.9226	−	10.9226i	0.766618	−	0.766618i
$204$		0			0
$205$	−3.08673	−	3.08673i	−0.215587	−	0.215587i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.98286			0.413843
$210$		0			0
$211$	−10.3307	−	10.3307i	−0.711197	−	0.711197i	0.255588	−	0.966786i	$-0.417731\pi$
$211$	−10.3307	−	10.3307i	−0.711197	−	0.711197i	−0.966786	+	0.255588i	$0.917731\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	6.09145i		−	0.415434i
$216$		0			0
$217$		−	13.1671i		−	0.893839i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.712036	−	0.712036i	−0.0478968	−	0.0478968i
$222$		0			0
$223$	10.6110			0.710567			0.355283	−	0.934759i	$-0.384384\pi$
$223$	10.6110			0.710567			0.355283	+	0.934759i	$0.384384\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.89383	+	2.89383i	0.192071	+	0.192071i	0.796590	−	0.604520i	$-0.206635\pi$
$227$	2.89383	+	2.89383i	0.192071	+	0.192071i	−0.604520	+	0.796590i	$0.706635\pi$
$228$		0			0
$229$	7.02023	−	7.02023i	0.463910	−	0.463910i	−0.436025	−	0.899935i	$-0.643614\pi$
$229$	7.02023	−	7.02023i	0.463910	−	0.463910i	0.899935	+	0.436025i	$0.143614\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1.30153i		−	0.0852663i	−0.999091	−	0.0426332i	$-0.986425\pi$
$233$		−	1.30153i		−	0.0852663i	0.999091	−	0.0426332i	$-0.0135747\pi$
$234$		0			0
$235$	0.402156	−	0.402156i	0.0262337	−	0.0262337i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.18489			0.270698			0.135349	−	0.990798i	$-0.456784\pi$
$239$	4.18489			0.270698			0.135349	+	0.990798i	$0.456784\pi$
$240$		0			0
$241$	−8.57746			−0.552523			−0.276262	−	0.961082i	$-0.589096\pi$
$241$	−8.57746			−0.552523			−0.276262	+	0.961082i	$0.589096\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.90909	−	2.90909i	0.185855	−	0.185855i
$246$		0			0
$247$		−	0.752027i		−	0.0478503i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.03798	+	8.03798i	−0.507353	+	0.507353i	−0.913713	−	0.406360i	$-0.866798\pi$
$251$	−8.03798	+	8.03798i	−0.507353	+	0.507353i	0.406360	+	0.913713i	$0.366798\pi$
$252$		0			0
$253$	5.06550	+	5.06550i	0.318465	+	0.318465i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−31.2973			−1.95227			−0.976136	−	0.217158i	$-0.930321\pi$
$257$	−31.2973			−1.95227			−0.976136	+	0.217158i	$0.930321\pi$
$258$		0			0
$259$	7.98699	+	7.98699i	0.496288	+	0.496288i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			19.6640i			1.21253i	0.795262	+	0.606266i	$0.207333\pi$
$263$			19.6640i			1.21253i	−0.795262	+	0.606266i	$0.792667\pi$
$264$		0			0
$265$		−	1.05939i		−	0.0650779i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.50053	+	3.50053i	0.213431	+	0.213431i	0.805723	−	0.592292i	$-0.201777\pi$
$269$	3.50053	+	3.50053i	0.213431	+	0.213431i	−0.592292	+	0.805723i	$0.701777\pi$
$270$		0			0
$271$	3.81899			0.231987			0.115994	−	0.993250i	$-0.462995\pi$
$271$	3.81899			0.231987			0.115994	+	0.993250i	$0.462995\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.72808	+	1.72808i	0.104207	+	0.104207i
$276$		0			0
$277$	−15.2534	+	15.2534i	−0.916487	+	0.916487i	−0.996772	−	0.0802850i	$-0.974417\pi$
$277$	−15.2534	+	15.2534i	−0.916487	+	0.916487i	0.0802850	+	0.996772i	$0.474417\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	10.1406i		−	0.604940i	−0.953159	−	0.302470i	$-0.902189\pi$
$281$		−	10.1406i		−	0.604940i	0.953159	−	0.302470i	$-0.0978113\pi$
$282$		0			0
$283$	18.9364	−	18.9364i	1.12565	−	1.12565i	0.134777	−	0.990876i	$-0.456968\pi$
$283$	18.9364	−	18.9364i	1.12565	−	1.12565i	0.990876	−	0.134777i	$-0.0430318\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.41578			0.437739
$288$		0			0
$289$	−6.25445			−0.367909
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.0806	−	11.0806i	0.647334	−	0.647334i	−0.305014	−	0.952348i	$-0.598661\pi$
$293$	11.0806	−	11.0806i	0.647334	−	0.647334i	0.952348	+	0.305014i	$0.0986610\pi$
$294$		0			0
$295$		−	3.12719i		−	0.182072i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.636717	−	0.636717i	0.0368223	−	0.0368223i
$300$		0			0
$301$	7.31726	+	7.31726i	0.421760	+	0.421760i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.4589			0.598874
$306$		0			0
$307$	−7.43194	−	7.43194i	−0.424163	−	0.424163i	0.462471	−	0.886634i	$-0.346963\pi$
$307$	−7.43194	−	7.43194i	−0.424163	−	0.424163i	−0.886634	+	0.462471i	$0.846963\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			28.5084i			1.61656i	0.588795	+	0.808282i	$0.299602\pi$
$311$			28.5084i			1.61656i	−0.588795	+	0.808282i	$0.700398\pi$
$312$		0			0
$313$		−	5.85037i		−	0.330683i	−0.986236	−	0.165341i	$-0.947127\pi$
$313$		−	5.85037i		−	0.330683i	0.986236	−	0.165341i	$-0.0528726\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.0284388	−	0.0284388i	−0.00159728	−	0.00159728i	0.706308	−	0.707905i	$-0.250360\pi$
$317$	−0.0284388	−	0.0284388i	−0.00159728	−	0.00159728i	−0.707905	+	0.706308i	$0.750360\pi$
$318$		0			0
$319$	22.2218			1.24418
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.67453	+	5.67453i	0.315739	+	0.315739i
$324$		0			0
$325$	0.217214	−	0.217214i	0.0120489	−	0.0120489i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			0.966165i			0.0532664i
$330$		0			0
$331$	14.8266	−	14.8266i	0.814945	−	0.814945i	−0.170426	−	0.985371i	$-0.554514\pi$
$331$	14.8266	−	14.8266i	0.814945	−	0.814945i	0.985371	+	0.170426i	$0.0545143\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.7362			−0.695854
$336$		0			0
$337$	−35.5373			−1.93584			−0.967920	−	0.251260i	$-0.919155\pi$
$337$	−35.5373			−1.93584			−0.967920	+	0.251260i	$0.919155\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	13.3940	−	13.3940i	0.725326	−	0.725326i
$342$		0			0
$343$			18.8806i			1.01946i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.2007	−	10.2007i	0.547600	−	0.547600i	−0.378146	−	0.925746i	$-0.623438\pi$
$347$	10.2007	−	10.2007i	0.547600	−	0.547600i	0.925746	+	0.378146i	$0.123438\pi$
$348$		0			0
$349$	12.9923	+	12.9923i	0.695460	+	0.695460i	0.963428	−	0.267968i	$-0.0863520\pi$
$349$	12.9923	+	12.9923i	0.695460	+	0.695460i	−0.267968	+	0.963428i	$0.586352\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−25.6457			−1.36498			−0.682490	−	0.730895i	$-0.739103\pi$
$353$	−25.6457			−1.36498			−0.682490	+	0.730895i	$0.739103\pi$
$354$		0			0
$355$	−1.64324	−	1.64324i	−0.0872139	−	0.0872139i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	30.5800i		−	1.61395i	−0.590584	−	0.806976i	$-0.701102\pi$
$359$		−	30.5800i		−	1.61395i	0.590584	−	0.806976i	$-0.298898\pi$
$360$		0			0
$361$		−	13.0068i		−	0.684567i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.201836	+	0.201836i	0.0105646	+	0.0105646i
$366$		0			0
$367$	−23.2750			−1.21494			−0.607472	−	0.794341i	$-0.707816\pi$
$367$	−23.2750			−1.21494			−0.607472	+	0.794341i	$0.707816\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.27258	+	1.27258i	0.0660688	+	0.0660688i
$372$		0			0
$373$	4.84829	−	4.84829i	0.251035	−	0.251035i	−0.570360	−	0.821395i	$-0.693196\pi$
$373$	4.84829	−	4.84829i	0.251035	−	0.251035i	0.821395	+	0.570360i	$0.193196\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	2.79320i		−	0.143857i
$378$		0			0
$379$	22.3646	−	22.3646i	1.14879	−	1.14879i	0.162001	−	0.986791i	$-0.448205\pi$
$379$	22.3646	−	22.3646i	1.14879	−	1.14879i	0.986791	−	0.162001i	$-0.0517947\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	21.9345			1.12080			0.560400	−	0.828222i	$-0.310647\pi$
$383$	21.9345			1.12080			0.560400	+	0.828222i	$0.310647\pi$
$384$		0			0
$385$	−4.15165			−0.211588
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−22.7795	+	22.7795i	−1.15497	+	1.15497i	−0.169423	+	0.985543i	$0.554191\pi$
$389$	−22.7795	+	22.7795i	−1.15497	+	1.15497i	−0.985543	+	0.169423i	$0.945809\pi$
$390$		0			0
$391$			9.60888i			0.485942i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.83571	−	1.83571i	0.0923645	−	0.0923645i
$396$		0			0
$397$	−26.2806	−	26.2806i	−1.31899	−	1.31899i	−0.914578	−	0.404409i	$-0.867477\pi$
$397$	−26.2806	−	26.2806i	−1.31899	−	1.31899i	−0.404409	−	0.914578i	$-0.632523\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.4543			0.821689			0.410845	−	0.911705i	$-0.365234\pi$
$401$	16.4543			0.821689			0.410845	+	0.911705i	$0.365234\pi$
$402$		0			0
$403$	−1.68358	−	1.68358i	−0.0838653	−	0.0838653i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			16.2493i			0.805448i
$408$		0			0
$409$			33.2439i			1.64380i	0.569629	+	0.821902i	$0.307087\pi$
$409$			33.2439i			1.64380i	−0.569629	+	0.821902i	$0.692913\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.75649	+	3.75649i	0.184845	+	0.184845i
$414$		0			0
$415$	−15.1800			−0.745155
$416$		0			0
$417$		0			0
$418$		0			0
$419$	26.3438	+	26.3438i	1.28698	+	1.28698i	0.936613	+	0.350366i	$0.113943\pi$
$419$	26.3438	+	26.3438i	1.28698	+	1.28698i	0.350366	+	0.936613i	$0.386057\pi$
$420$		0			0
$421$	−8.84601	+	8.84601i	−0.431128	+	0.431128i	−0.889012	−	0.457884i	$-0.848608\pi$
$421$	−8.84601	+	8.84601i	−0.431128	+	0.431128i	0.457884	+	0.889012i	$0.348608\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			3.27804i			0.159008i
$426$		0			0
$427$	−12.5636	+	12.5636i	−0.607993	+	0.607993i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	28.1304			1.35500			0.677498	−	0.735525i	$-0.263064\pi$
$431$	28.1304			1.35500			0.677498	+	0.735525i	$0.263064\pi$
$432$		0			0
$433$	14.3640			0.690288			0.345144	−	0.938550i	$-0.387830\pi$
$433$	14.3640			0.690288			0.345144	+	0.938550i	$0.387830\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.07427	+	5.07427i	−0.242735	+	0.242735i
$438$		0			0
$439$			17.0265i			0.812628i	0.913733	+	0.406314i	$0.133186\pi$
$439$			17.0265i			0.812628i	−0.913733	+	0.406314i	$0.866814\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.6944	+	13.6944i	−0.650643	+	0.650643i	−0.953148	−	0.302505i	$-0.902177\pi$
$443$	−13.6944	+	13.6944i	−0.650643	+	0.650643i	0.302505	+	0.953148i	$0.402177\pi$
$444$		0			0
$445$	−7.14584	−	7.14584i	−0.338745	−	0.338745i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	40.9793			1.93393			0.966967	−	0.254900i	$-0.0820426\pi$
$449$	40.9793			1.93393			0.966967	+	0.254900i	$0.0820426\pi$
$450$		0			0
$451$	7.54359	+	7.54359i	0.355214	+	0.355214i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			0.521850i			0.0244647i
$456$		0			0
$457$		−	4.79658i		−	0.224375i	−0.993687	−	0.112187i	$-0.964214\pi$
$457$		−	4.79658i		−	0.224375i	0.993687	−	0.112187i	$-0.0357856\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.15223	+	9.15223i	0.426262	+	0.426262i	0.887353	−	0.461091i	$-0.152542\pi$
$461$	9.15223	+	9.15223i	0.426262	+	0.426262i	−0.461091	+	0.887353i	$0.652542\pi$
$462$		0			0
$463$	13.4617			0.625620			0.312810	−	0.949816i	$-0.398730\pi$
$463$	13.4617			0.625620			0.312810	+	0.949816i	$0.398730\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.22075	+	1.22075i	0.0564894	+	0.0564894i	0.734787	−	0.678298i	$-0.237282\pi$
$467$	1.22075	+	1.22075i	0.0564894	+	0.0564894i	−0.678298	+	0.734787i	$0.737282\pi$
$468$		0			0
$469$	15.2992	−	15.2992i	0.706451	−	0.706451i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			14.8867i			0.684493i
$474$		0			0
$475$	−1.73107	+	1.73107i	−0.0794271	+	0.0794271i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−23.1159			−1.05619			−0.528097	−	0.849184i	$-0.677094\pi$
$479$	−23.1159			−1.05619			−0.528097	+	0.849184i	$0.677094\pi$
$480$		0			0
$481$	2.04248			0.0931293
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.07549	+	7.07549i	−0.321281	+	0.321281i
$486$		0			0
$487$			21.8892i			0.991895i	0.868353	+	0.495947i	$0.165179\pi$
$487$			21.8892i			0.991895i	−0.868353	+	0.495947i	$0.834821\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.64551	+	5.64551i	−0.254778	+	0.254778i	−0.822926	−	0.568148i	$-0.807660\pi$
$491$	−5.64551	+	5.64551i	−0.254778	+	0.254778i	0.568148	+	0.822926i	$0.307660\pi$
$492$		0			0
$493$	21.0765	+	21.0765i	0.949239	+	0.949239i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.94782			0.177084
$498$		0			0
$499$	9.76144	+	9.76144i	0.436982	+	0.436982i	0.890995	−	0.454013i	$-0.150008\pi$
$499$	9.76144	+	9.76144i	0.436982	+	0.436982i	−0.454013	+	0.890995i	$0.650008\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			25.8483i			1.15252i	0.817267	+	0.576259i	$0.195488\pi$
$503$			25.8483i			1.15252i	−0.817267	+	0.576259i	$0.804512\pi$
$504$		0			0
$505$		−	11.8375i		−	0.526763i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10.7719	−	10.7719i	−0.477455	−	0.477455i	0.426862	−	0.904317i	$-0.359619\pi$
$509$	−10.7719	−	10.7719i	−0.477455	−	0.477455i	−0.904317	+	0.426862i	$0.859619\pi$
$510$		0			0
$511$	−0.484903			−0.0214509
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.61872	+	4.61872i	0.203525	+	0.203525i
$516$		0			0
$517$	−0.982817	+	0.982817i	−0.0432243	+	0.0432243i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			8.73686i			0.382769i	0.981515	+	0.191384i	$0.0612977\pi$
$521$			8.73686i			0.382769i	−0.981515	+	0.191384i	$0.938702\pi$
$522$		0			0
$523$	−8.99908	+	8.99908i	−0.393502	+	0.393502i	−0.875934	−	0.482431i	$-0.839754\pi$
$523$	−8.99908	+	8.99908i	−0.393502	+	0.393502i	0.482431	+	0.875934i	$0.339754\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	25.4075			1.10677
$528$		0			0
$529$	14.4075			0.626415
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.948204	−	0.948204i	0.0410713	−	0.0410713i
$534$		0			0
$535$		−	7.43419i		−	0.321408i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7.10945	+	7.10945i	−0.306226	+	0.306226i
$540$		0			0
$541$	−5.06136	−	5.06136i	−0.217605	−	0.217605i	0.589884	−	0.807488i	$-0.299174\pi$
$541$	−5.06136	−	5.06136i	−0.217605	−	0.217605i	−0.807488	+	0.589884i	$0.799174\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	16.4058			0.702746
$546$		0			0
$547$	−17.8840	−	17.8840i	−0.764663	−	0.764663i	0.212498	−	0.977161i	$-0.431840\pi$
$547$	−17.8840	−	17.8840i	−0.764663	−	0.764663i	−0.977161	+	0.212498i	$0.931840\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			22.2603i			0.948319i
$552$		0			0
$553$			4.41023i			0.187542i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.40581	+	4.40581i	0.186680	+	0.186680i	0.794259	−	0.607579i	$-0.207859\pi$
$557$	4.40581	+	4.40581i	0.186680	+	0.186680i	−0.607579	+	0.794259i	$0.707859\pi$
$558$		0			0
$559$	1.87122			0.0791440
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−25.2174	−	25.2174i	−1.06279	−	1.06279i	−0.997892	−	0.0648946i	$-0.979329\pi$
$563$	−25.2174	−	25.2174i	−1.06279	−	1.06279i	−0.0648946	−	0.997892i	$-0.520671\pi$
$564$		0			0
$565$	−2.69399	+	2.69399i	−0.113337	+	0.113337i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			23.5000i			0.985173i	0.870264	+	0.492586i	$0.163948\pi$
$569$			23.5000i			0.985173i	−0.870264	+	0.492586i	$0.836052\pi$
$570$		0			0
$571$	8.63302	−	8.63302i	0.361281	−	0.361281i	−0.503004	−	0.864284i	$-0.667772\pi$
$571$	8.63302	−	8.63302i	0.361281	−	0.361281i	0.864284	+	0.503004i	$0.167772\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.93129			−0.122243
$576$		0			0
$577$	−17.6023			−0.732792			−0.366396	−	0.930459i	$-0.619408\pi$
$577$	−17.6023			−0.732792			−0.366396	+	0.930459i	$0.619408\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	18.2347	−	18.2347i	0.756501	−	0.756501i
$582$		0			0
$583$			2.58902i			0.107226i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.32030	+	4.32030i	−0.178318	+	0.178318i	−0.790622	−	0.612304i	$-0.790243\pi$
$587$	−4.32030	+	4.32030i	−0.178318	+	0.178318i	0.612304	+	0.790622i	$0.290243\pi$
$588$		0			0
$589$	13.4172	+	13.4172i	0.552847	+	0.552847i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7.94251			−0.326160			−0.163080	−	0.986613i	$-0.552143\pi$
$593$	−7.94251			−0.326160			−0.163080	+	0.986613i	$0.552143\pi$
$594$		0			0
$595$	−3.93769	−	3.93769i	−0.161430	−	0.161430i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	6.80370i		−	0.277992i	−0.990293	−	0.138996i	$-0.955613\pi$
$599$		−	6.80370i		−	0.277992i	0.990293	−	0.138996i	$-0.0443875\pi$
$600$		0			0
$601$		−	24.6158i		−	1.00410i	−0.864838	−	0.502050i	$-0.832579\pi$
$601$		−	24.6158i		−	1.00410i	0.864838	−	0.502050i	$-0.167421\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.55497	+	3.55497i	0.144530	+	0.144530i
$606$		0			0
$607$	−14.9281			−0.605912			−0.302956	−	0.953005i	$-0.597974\pi$
$607$	−14.9281			−0.605912			−0.302956	+	0.953005i	$0.597974\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.123537	+	0.123537i	0.00499777	+	0.00499777i
$612$		0			0
$613$	−1.32040	+	1.32040i	−0.0533306	+	0.0533306i	−0.733269	−	0.679939i	$-0.762006\pi$
$613$	−1.32040	+	1.32040i	−0.0533306	+	0.0533306i	0.679939	+	0.733269i	$0.262006\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			38.7296i			1.55920i	0.626281	+	0.779598i	$0.284576\pi$
$617$			38.7296i			1.55920i	−0.626281	+	0.779598i	$0.715424\pi$
$618$		0			0
$619$	−6.60439	+	6.60439i	−0.265453	+	0.265453i	−0.827265	−	0.561812i	$-0.810104\pi$
$619$	−6.60439	+	6.60439i	−0.265453	+	0.265453i	0.561812	+	0.827265i	$0.310104\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	17.1677			0.687807
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.4119	+	15.4119i	−0.614511	+	0.614511i
$630$		0			0
$631$		−	19.8253i		−	0.789233i	−0.918846	−	0.394617i	$-0.870877\pi$
$631$		−	19.8253i		−	0.789233i	0.918846	−	0.394617i	$-0.129123\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.6008	+	11.6008i	−0.460362	+	0.460362i
$636$		0			0
$637$	0.893635	+	0.893635i	0.0354071	+	0.0354071i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.8775			−0.745617			−0.372808	−	0.927908i	$-0.621605\pi$
$641$	−18.8775			−0.745617			−0.372808	+	0.927908i	$0.621605\pi$
$642$		0			0
$643$	−22.9375	−	22.9375i	−0.904568	−	0.904568i	0.0912588	−	0.995827i	$-0.470911\pi$
$643$	−22.9375	−	22.9375i	−0.904568	−	0.904568i	−0.995827	+	0.0912588i	$0.970911\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			6.49513i			0.255350i	0.991816	+	0.127675i	$0.0407514\pi$
$647$			6.49513i			0.255350i	−0.991816	+	0.127675i	$0.959249\pi$
$648$		0			0
$649$			7.64247i			0.299993i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	29.1452	+	29.1452i	1.14054	+	1.14054i	0.988351	+	0.152190i	$0.0486326\pi$
$653$	29.1452	+	29.1452i	1.14054	+	1.14054i	0.152190	+	0.988351i	$0.451367\pi$
$654$		0			0
$655$	−15.2685			−0.596590
$656$		0			0
$657$		0			0
$658$		0			0
$659$	17.7671	+	17.7671i	0.692107	+	0.692107i	0.962695	−	0.270589i	$-0.0872183\pi$
$659$	17.7671	+	17.7671i	0.692107	+	0.692107i	−0.270589	+	0.962695i	$0.587218\pi$
$660$		0			0
$661$	3.75307	−	3.75307i	0.145978	−	0.145978i	−0.630341	−	0.776319i	$-0.717085\pi$
$661$	3.75307	−	3.75307i	0.145978	−	0.145978i	0.776319	+	0.630341i	$0.217085\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	4.15885i		−	0.161273i
$666$		0			0
$667$	−18.8470	+	18.8470i	−0.729760	+	0.729760i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−25.5602			−0.986740
$672$		0			0
$673$	48.9037			1.88510			0.942550	−	0.334064i	$-0.108420\pi$
$673$	48.9037			1.88510			0.942550	+	0.334064i	$0.108420\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−20.0199	+	20.0199i	−0.769427	+	0.769427i	−0.978006	−	0.208579i	$-0.933116\pi$
$677$	−20.0199	+	20.0199i	−0.769427	+	0.769427i	0.208579	+	0.978006i	$0.433116\pi$
$678$		0			0
$679$		−	16.9986i		−	0.652347i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	24.3833	−	24.3833i	0.932999	−	0.932999i	−0.0648930	−	0.997892i	$-0.520671\pi$
$683$	24.3833	−	24.3833i	0.932999	−	0.932999i	0.997892	+	0.0648930i	$0.0206706\pi$
$684$		0			0
$685$	3.51043	+	3.51043i	0.134127	+	0.134127i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.325431			0.0123979
$690$		0			0
$691$	14.4576	+	14.4576i	0.549994	+	0.549994i	0.926439	−	0.376445i	$-0.122854\pi$
$691$	14.4576	+	14.4576i	0.549994	+	0.549994i	−0.376445	+	0.926439i	$0.622854\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	8.28439i		−	0.314245i
$696$		0			0
$697$			14.3096i			0.542016i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	13.2194	+	13.2194i	0.499291	+	0.499291i	0.911217	−	0.411926i	$-0.135144\pi$
$701$	13.2194	+	13.2194i	0.499291	+	0.499291i	−0.411926	+	0.911217i	$0.635144\pi$
$702$		0			0
$703$	−16.2775			−0.613916
$704$		0			0
$705$		0			0
$706$		0			0
$707$	14.2196	+	14.2196i	0.534784	+	0.534784i
$708$		0			0
$709$	5.23663	−	5.23663i	0.196666	−	0.196666i	−0.601903	−	0.798569i	$-0.705591\pi$
$709$	5.23663	−	5.23663i	0.196666	−	0.196666i	0.798569	+	0.601903i	$0.205591\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			22.7198i			0.850865i
$714$		0			0
$715$	−0.530844	+	0.530844i	−0.0198524	+	0.0198524i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.72657			0.176271			0.0881357	−	0.996108i	$-0.471909\pi$
$719$	4.72657			0.176271			0.0881357	+	0.996108i	$0.471909\pi$
$720$		0			0
$721$	−11.0963			−0.413249
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.42961	+	6.42961i	−0.238790	+	0.238790i
$726$		0			0
$727$		−	45.5233i		−	1.68837i	−0.536056	−	0.844183i	$-0.680086\pi$
$727$		−	45.5233i		−	1.68837i	0.536056	−	0.844183i	$-0.319914\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14.1195	+	14.1195i	−0.522230	+	0.522230i
$732$		0			0
$733$	6.37965	+	6.37965i	0.235638	+	0.235638i	0.815041	−	0.579403i	$-0.196714\pi$
$733$	6.37965	+	6.37965i	0.235638	+	0.235638i	−0.579403	+	0.815041i	$0.696714\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	31.1257			1.14653
$738$		0			0
$739$	25.9104	+	25.9104i	0.953129	+	0.953129i	0.998950	−	0.0458203i	$-0.0145902\pi$
$739$	25.9104	+	25.9104i	0.953129	+	0.953129i	−0.0458203	+	0.998950i	$0.514590\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			4.17917i			0.153319i	0.997057	+	0.0766595i	$0.0244254\pi$
$743$			4.17917i			0.153319i	−0.997057	+	0.0766595i	$0.975575\pi$
$744$		0			0
$745$			16.6949i			0.611653i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	8.93020	+	8.93020i	0.326302	+	0.326302i
$750$		0			0
$751$	24.2420			0.884604			0.442302	−	0.896866i	$-0.354162\pi$
$751$	24.2420			0.884604			0.442302	+	0.896866i	$0.354162\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.56699	+	7.56699i	0.275391	+	0.275391i
$756$		0			0
$757$	−2.98821	+	2.98821i	−0.108608	+	0.108608i	−0.759323	−	0.650714i	$-0.774470\pi$
$757$	−2.98821	+	2.98821i	−0.108608	+	0.108608i	0.650714	+	0.759323i	$0.274470\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			30.9665i			1.12254i	0.827634	+	0.561268i	$0.189686\pi$
$761$			30.9665i			1.12254i	−0.827634	+	0.561268i	$0.810314\pi$
$762$		0			0
$763$	−19.7072	+	19.7072i	−0.713448	+	0.713448i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.960634			0.0346865
$768$		0			0
$769$	37.1376			1.33921			0.669607	−	0.742715i	$-0.266462\pi$
$769$	37.1376			1.33921			0.669607	+	0.742715i	$0.266462\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	36.1287	−	36.1287i	1.29946	−	1.29946i	0.370712	−	0.928748i	$-0.379114\pi$
$773$	36.1287	−	36.1287i	1.29946	−	1.29946i	0.928748	−	0.370712i	$-0.120886\pi$
$774$		0			0
$775$			7.75081i			0.278417i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.55665	+	7.55665i	−0.270745	+	0.270745i
$780$		0			0
$781$	4.01586	+	4.01586i	0.143699	+	0.143699i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.89923			0.246244
$786$		0			0
$787$	−17.7811	−	17.7811i	−0.633828	−	0.633828i	0.315198	−	0.949026i	$-0.397929\pi$
$787$	−17.7811	−	17.7811i	−0.633828	−	0.633828i	−0.949026	+	0.315198i	$0.897929\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	6.47223i		−	0.230126i
$792$		0			0
$793$			3.21283i			0.114091i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	14.4674	+	14.4674i	0.512463	+	0.512463i	0.915280	−	0.402817i	$-0.131969\pi$
$797$	14.4674	+	14.4674i	0.512463	+	0.512463i	−0.402817	+	0.915280i	$0.631969\pi$
$798$		0			0
$799$	−1.86433			−0.0659554
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.493261	−	0.493261i	−0.0174068	−	0.0174068i
$804$		0			0
$805$	3.52116	−	3.52116i	0.124105	−	0.124105i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	42.6805i		−	1.50056i	−0.661117	−	0.750282i	$-0.729917\pi$
$809$		−	42.6805i		−	1.50056i	0.661117	−	0.750282i	$-0.270083\pi$
$810$		0			0
$811$	−24.8068	+	24.8068i	−0.871083	+	0.871083i	−0.992591	−	0.121507i	$-0.961227\pi$
$811$	−24.8068	+	24.8068i	−0.871083	+	0.871083i	0.121507	+	0.992591i	$0.461227\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.6550			0.513342
$816$		0			0
$817$	−14.9125			−0.521723
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6.88057	+	6.88057i	−0.240133	+	0.240133i	−0.816905	−	0.576772i	$-0.804312\pi$
$821$	−6.88057	+	6.88057i	−0.240133	+	0.240133i	0.576772	+	0.816905i	$0.304312\pi$
$822$		0			0
$823$		−	26.3873i		−	0.919802i	−0.887970	−	0.459901i	$-0.847885\pi$
$823$		−	26.3873i		−	0.919802i	0.887970	−	0.459901i	$-0.152115\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.319732	−	0.319732i	0.0111182	−	0.0111182i	−0.701526	−	0.712644i	$-0.747498\pi$
$827$	0.319732	−	0.319732i	0.0111182	−	0.0111182i	0.712644	+	0.701526i	$0.247498\pi$
$828$		0			0
$829$	3.47656	+	3.47656i	0.120746	+	0.120746i	0.764898	−	0.644152i	$-0.222790\pi$
$829$	3.47656	+	3.47656i	0.120746	+	0.120746i	−0.644152	+	0.764898i	$0.722790\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−13.4861			−0.467266
$834$		0			0
$835$	−16.1746	−	16.1746i	−0.559747	−	0.559747i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	50.6941i		−	1.75015i	−0.483983	−	0.875077i	$-0.660810\pi$
$839$		−	50.6941i		−	1.75015i	0.483983	−	0.875077i	$-0.339190\pi$
$840$		0			0
$841$			53.6798i			1.85103i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.12566	−	9.12566i	−0.313932	−	0.313932i
$846$		0			0
$847$	−8.54069			−0.293462
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13.7816	−	13.7816i	−0.472427	−	0.472427i
$852$		0			0
$853$	2.15143	−	2.15143i	0.0736635	−	0.0736635i	−0.669315	−	0.742979i	$-0.733412\pi$
$853$	2.15143	−	2.15143i	0.0736635	−	0.0736635i	0.742979	+	0.669315i	$0.233412\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.79872i		−	0.0614430i	−0.999528	−	0.0307215i	$-0.990220\pi$
$857$		−	1.79872i		−	0.0614430i	0.999528	−	0.0307215i	$-0.00978050\pi$
$858$		0			0
$859$	−3.27463	+	3.27463i	−0.111729	+	0.111729i	−0.760761	−	0.649032i	$-0.775174\pi$
$859$	−3.27463	+	3.27463i	−0.111729	+	0.111729i	0.649032	+	0.760761i	$0.275174\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−19.3753			−0.659542			−0.329771	−	0.944061i	$-0.606972\pi$
$863$	−19.3753			−0.659542			−0.329771	+	0.944061i	$0.606972\pi$
$864$		0			0
$865$	−4.45891			−0.151607
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.48624	+	4.48624i	−0.152185	+	0.152185i
$870$		0			0
$871$		−	3.91240i		−	0.132567i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.20123	−	1.20123i	0.0406091	−	0.0406091i
$876$		0			0
$877$	−20.4639	−	20.4639i	−0.691017	−	0.691017i	0.271439	−	0.962456i	$-0.412501\pi$
$877$	−20.4639	−	20.4639i	−0.691017	−	0.691017i	−0.962456	+	0.271439i	$0.912501\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−24.8698			−0.837883			−0.418942	−	0.908013i	$-0.637599\pi$
$881$	−24.8698			−0.837883			−0.418942	+	0.908013i	$0.637599\pi$
$882$		0			0
$883$	7.70378	+	7.70378i	0.259253	+	0.259253i	0.824750	−	0.565497i	$-0.191316\pi$
$883$	7.70378	+	7.70378i	0.259253	+	0.259253i	−0.565497	+	0.824750i	$0.691316\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			51.9786i			1.74527i	0.488373	+	0.872635i	$0.337591\pi$
$887$			51.9786i			1.74527i	−0.488373	+	0.872635i	$0.662409\pi$
$888$		0			0
$889$		−	27.8705i		−	0.934745i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.984520	−	0.984520i	−0.0329457	−	0.0329457i
$894$		0			0
$895$	−1.34435			−0.0449368
$896$		0			0
$897$		0			0
$898$		0			0
$899$	49.8347	+	49.8347i	1.66208	+	1.66208i
$900$		0			0
$901$	−2.45559	+	2.45559i	−0.0818075	+	0.0818075i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	15.6675i		−	0.520805i
$906$		0			0
$907$	21.6946	−	21.6946i	0.720357	−	0.720357i	−0.248321	−	0.968678i	$-0.579879\pi$
$907$	21.6946	−	21.6946i	0.720357	−	0.720357i	0.968678	+	0.248321i	$0.0798788\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	23.1996			0.768636			0.384318	−	0.923201i	$-0.374437\pi$
$911$	23.1996			0.768636			0.384318	+	0.923201i	$0.374437\pi$
$912$		0			0
$913$	37.0979			1.22776
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18.3410	−	18.3410i	0.605674	−	0.605674i
$918$		0			0
$919$		−	14.3527i		−	0.473451i	−0.971577	−	0.236726i	$-0.923926\pi$
$919$		−	14.3527i		−	0.473451i	0.971577	−	0.236726i	$-0.0760743\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.504781	−	0.504781i	0.0166151	−	0.0166151i
$924$		0			0
$925$	−4.70155	−	4.70155i	−0.154586	−	0.154586i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23.7983			0.780798			0.390399	−	0.920646i	$-0.372337\pi$
$929$	23.7983			0.780798			0.390399	+	0.920646i	$0.372337\pi$
$930$		0			0
$931$	−7.12177	−	7.12177i	−0.233406	−	0.233406i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	8.01112i		−	0.261992i
$936$		0			0
$937$		−	38.2290i		−	1.24889i	−0.781070	−	0.624444i	$-0.785326\pi$
$937$		−	38.2290i		−	1.24889i	0.781070	−	0.624444i	$-0.214674\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.41971	+	5.41971i	0.176677	+	0.176677i	0.789906	−	0.613228i	$-0.210129\pi$
$941$	5.41971	+	5.41971i	0.176677	+	0.176677i	−0.613228	+	0.789906i	$0.710129\pi$
$942$		0			0
$943$	−12.7960			−0.416694
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−40.3115	−	40.3115i	−1.30995	−	1.30995i	−0.921450	−	0.388497i	$-0.872994\pi$
$947$	−40.3115	−	40.3115i	−1.30995	−	1.30995i	−0.388497	−	0.921450i	$-0.627006\pi$
$948$		0			0
$949$	−0.0620013	+	0.0620013i	−0.00201265	+	0.00201265i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			42.2929i			1.37000i	0.728542	+	0.685001i	$0.240198\pi$
$953$			42.2929i			1.37000i	−0.728542	+	0.685001i	$0.759802\pi$
$954$		0			0
$955$	−17.2094	+	17.2094i	−0.556883	+	0.556883i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.43370			−0.272338
$960$		0			0
$961$	29.0750			0.937903
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−17.4348	+	17.4348i	−0.561247	+	0.561247i
$966$		0			0
$967$		−	18.3045i		−	0.588632i	−0.955708	−	0.294316i	$-0.904908\pi$
$967$		−	18.3045i		−	0.588632i	0.955708	−	0.294316i	$-0.0950919\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	40.3477	−	40.3477i	1.29482	−	1.29482i	0.363051	−	0.931769i	$-0.381735\pi$
$971$	40.3477	−	40.3477i	1.29482	−	1.29482i	0.931769	−	0.363051i	$-0.118265\pi$
$972$		0			0
$973$	9.95149	+	9.95149i	0.319030	+	0.319030i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	28.8993			0.924569			0.462284	−	0.886732i	$-0.347030\pi$
$977$	28.8993			0.924569			0.462284	+	0.886732i	$0.347030\pi$
$978$		0			0
$979$	17.4635	+	17.4635i	0.558137	+	0.558137i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			57.4280i			1.83167i	0.401557	+	0.915834i	$0.368469\pi$
$983$			57.4280i			1.83167i	−0.401557	+	0.915834i	$0.631531\pi$
$984$		0			0
$985$			1.69749i			0.0540866i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.6260	−	12.6260i	−0.401482	−	0.401482i
$990$		0			0
$991$	30.9239			0.982332			0.491166	−	0.871066i	$-0.336571\pi$
$991$	30.9239			0.982332			0.491166	+	0.871066i	$0.336571\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	12.8238	+	12.8238i	0.406540	+	0.406540i
$996$		0			0
$997$	−26.2473	+	26.2473i	−0.831260	+	0.831260i	−0.987689	−	0.156429i	$-0.950002\pi$
$997$	−26.2473	+	26.2473i	−0.831260	+	0.831260i	0.156429	+	0.987689i	$0.450002\pi$
$998$		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.721.13		32
3.2	odd	2	inner	2880.2.t.e.721.5		32
4.3	odd	2		720.2.t.e.541.13	yes	32
12.11	even	2		720.2.t.e.541.4	yes	32
16.5	even	4	inner	2880.2.t.e.2161.13		32
16.11	odd	4		720.2.t.e.181.13	yes	32
48.5	odd	4	inner	2880.2.t.e.2161.5		32
48.11	even	4		720.2.t.e.181.4	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.t.e.181.4	✓	32	48.11	even	4
720.2.t.e.181.13	yes	32	16.11	odd	4
720.2.t.e.541.4	yes	32	12.11	even	2
720.2.t.e.541.13	yes	32	4.3	odd	2
2880.2.t.e.721.5		32	3.2	odd	2	inner
2880.2.t.e.721.13		32	1.1	even	1	trivial
2880.2.t.e.2161.5		32	48.5	odd	4	inner
2880.2.t.e.2161.13		32	16.5	even	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)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{5} +1.69880i q^{7} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{5} +1.69880i q^{7} +(-1.72808 + 1.72808i) q^{11} +(0.217214 + 0.217214i) q^{13} -3.27804 q^{17} +(-1.73107 - 1.73107i) q^{19} -2.93129i q^{23} -1.00000i q^{25} +(-6.42961 - 6.42961i) q^{29} -7.75081 q^{31} +(1.20123 + 1.20123i) q^{35} +(4.70155 - 4.70155i) q^{37} -4.36530i q^{41} +(4.30731 - 4.30731i) q^{43} +0.568734 q^{47} +4.11408 q^{49} +(0.749102 - 0.749102i) q^{53} +2.44387i q^{55} +(2.21126 - 2.21126i) q^{59} +(7.39554 + 7.39554i) q^{61} +0.307187 q^{65} +(-9.00587 - 9.00587i) q^{67} -2.32389i q^{71} +0.285439i q^{73} +(-2.93566 - 2.93566i) q^{77} +2.59608 q^{79} +(-10.7338 - 10.7338i) q^{83} +(-2.31792 + 2.31792i) q^{85} -10.1057i q^{89} +(-0.369003 + 0.369003i) q^{91} -2.44811 q^{95} -10.0062 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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