LMFDB - Embedded newform 2880.2.bl.c.431.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(431,2880)]

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([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("2880.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bl (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			431.2
Character	$\chi$	$=$	2880.431
Dual form			2880.2.bl.c.1871.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{5} -3.83891 q^{7} +O(q^{10})$	$q+(-0.707107 - 0.707107i) q^{5} -3.83891 q^{7} +(0.214796 - 0.214796i) q^{11} +(-0.177819 - 0.177819i) q^{13} -1.29544i q^{17} +(-2.10783 + 2.10783i) q^{19} +3.78684i q^{23} +1.00000i q^{25} +(-3.15157 + 3.15157i) q^{29} -0.745073i q^{31} +(2.71452 + 2.71452i) q^{35} +(7.10346 - 7.10346i) q^{37} +10.5322 q^{41} +(2.04046 + 2.04046i) q^{43} -10.9223 q^{47} +7.73720 q^{49} +(0.467757 + 0.467757i) q^{53} -0.303768 q^{55} +(1.61894 - 1.61894i) q^{59} +(1.34166 + 1.34166i) q^{61} +0.251474i q^{65} +(6.92451 - 6.92451i) q^{67} +8.46318i q^{71} -7.24339i q^{73} +(-0.824582 + 0.824582i) q^{77} -12.5416i q^{79} +(9.42498 + 9.42498i) q^{83} +(-0.916015 + 0.916015i) q^{85} +9.41491 q^{89} +(0.682631 + 0.682631i) q^{91} +2.98093 q^{95} +18.1943 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 8 q^{19} - 32 q^{49} + 16 q^{55} + 16 q^{61} - 16 q^{67} + 16 q^{85} + 16 q^{91}+O(q^{100}) $ 32 * q + 8 * q^19 - 32 * q^49 + 16 * q^55 + 16 * q^61 - 16 * q^67 + 16 * q^85 + 16 * 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
$7$	−3.83891			−1.45097			−0.725485	−	0.688238i	$-0.758385\pi$
$7$	−3.83891			−1.45097			−0.725485	+	0.688238i	$0.758385\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.214796	−	0.214796i	0.0647635	−	0.0647635i	−0.673983	−	0.738747i	$-0.735418\pi$
$11$	0.214796	−	0.214796i	0.0647635	−	0.0647635i	0.738747	+	0.673983i	$0.235418\pi$
$12$		0			0
$13$	−0.177819	−	0.177819i	−0.0493181	−	0.0493181i	0.682018	−	0.731336i	$-0.261103\pi$
$13$	−0.177819	−	0.177819i	−0.0493181	−	0.0493181i	−0.731336	+	0.682018i	$0.761103\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.29544i		−	0.314190i	−0.987583	−	0.157095i	$-0.949787\pi$
$17$		−	1.29544i		−	0.314190i	0.987583	−	0.157095i	$-0.0502130\pi$
$18$		0			0
$19$	−2.10783	+	2.10783i	−0.483570	+	0.483570i	−0.906270	−	0.422700i	$-0.861083\pi$
$19$	−2.10783	+	2.10783i	−0.483570	+	0.483570i	0.422700	+	0.906270i	$0.361083\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.78684i			0.789611i	0.918765	+	0.394805i	$0.129188\pi$
$23$			3.78684i			0.789611i	−0.918765	+	0.394805i	$0.870812\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.15157	+	3.15157i	−0.585232	+	0.585232i	−0.936336	−	0.351105i	$-0.885806\pi$
$29$	−3.15157	+	3.15157i	−0.585232	+	0.585232i	0.351105	+	0.936336i	$0.385806\pi$
$30$		0			0
$31$		−	0.745073i		−	0.133819i	−0.997759	−	0.0669095i	$-0.978686\pi$
$31$		−	0.745073i		−	0.133819i	0.997759	−	0.0669095i	$-0.0213139\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.71452	+	2.71452i	0.458837	+	0.458837i
$36$		0			0
$37$	7.10346	−	7.10346i	1.16780	−	1.16780i	0.185078	−	0.982724i	$-0.440746\pi$
$37$	7.10346	−	7.10346i	1.16780	−	1.16780i	0.982724	−	0.185078i	$-0.0592536\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.5322			1.64486			0.822429	−	0.568867i	$-0.192618\pi$
$41$	10.5322			1.64486			0.822429	+	0.568867i	$0.192618\pi$
$42$		0			0
$43$	2.04046	+	2.04046i	0.311167	+	0.311167i	0.845362	−	0.534194i	$-0.179385\pi$
$43$	2.04046	+	2.04046i	0.311167	+	0.311167i	−0.534194	+	0.845362i	$0.679385\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−10.9223			−1.59318			−0.796588	−	0.604523i	$-0.793364\pi$
$47$	−10.9223			−1.59318			−0.796588	+	0.604523i	$0.793364\pi$
$48$		0			0
$49$	7.73720			1.10531
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.467757	+	0.467757i	0.0642514	+	0.0642514i	0.738502	−	0.674251i	$-0.235533\pi$
$53$	0.467757	+	0.467757i	0.0642514	+	0.0642514i	−0.674251	+	0.738502i	$0.735533\pi$
$54$		0			0
$55$	−0.303768			−0.0409600
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.61894	−	1.61894i	0.210768	−	0.210768i	−0.593826	−	0.804594i	$-0.702383\pi$
$59$	1.61894	−	1.61894i	0.210768	−	0.210768i	0.804594	+	0.593826i	$0.202383\pi$
$60$		0			0
$61$	1.34166	+	1.34166i	0.171782	+	0.171782i	0.787762	−	0.615980i	$-0.211240\pi$
$61$	1.34166	+	1.34166i	0.171782	+	0.171782i	−0.615980	+	0.787762i	$0.711240\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			0.251474i			0.0311915i
$66$		0			0
$67$	6.92451	−	6.92451i	0.845964	−	0.845964i	−0.143663	−	0.989627i	$-0.545888\pi$
$67$	6.92451	−	6.92451i	0.845964	−	0.845964i	0.989627	+	0.143663i	$0.0458880\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			8.46318i			1.00440i	0.864753	+	0.502198i	$0.167475\pi$
$71$			8.46318i			1.00440i	−0.864753	+	0.502198i	$0.832525\pi$
$72$		0			0
$73$		−	7.24339i		−	0.847774i	−0.905715	−	0.423887i	$-0.860665\pi$
$73$		−	7.24339i		−	0.847774i	0.905715	−	0.423887i	$-0.139335\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.824582	+	0.824582i	−0.0939699	+	0.0939699i
$78$		0			0
$79$		−	12.5416i		−	1.41104i	−0.708689	−	0.705521i	$-0.750713\pi$
$79$		−	12.5416i		−	1.41104i	0.708689	−	0.705521i	$-0.249287\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.42498	+	9.42498i	1.03453	+	1.03453i	0.999382	+	0.0351442i	$0.0111890\pi$
$83$	9.42498	+	9.42498i	1.03453	+	1.03453i	0.0351442	+	0.999382i	$0.488811\pi$
$84$		0			0
$85$	−0.916015	+	0.916015i	−0.0993557	+	0.0993557i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.41491			0.997979			0.498989	−	0.866608i	$-0.333705\pi$
$89$	9.41491			0.997979			0.498989	+	0.866608i	$0.333705\pi$
$90$		0			0
$91$	0.682631	+	0.682631i	0.0715591	+	0.0715591i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.98093			0.305837
$96$		0			0
$97$	18.1943			1.84735			0.923674	−	0.383178i	$-0.125171\pi$
$97$	18.1943			1.84735			0.923674	+	0.383178i	$0.125171\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.32696	+	7.32696i	0.729060	+	0.729060i	0.970432	−	0.241373i	$-0.0775976\pi$
$101$	7.32696	+	7.32696i	0.729060	+	0.729060i	−0.241373	+	0.970432i	$0.577598\pi$
$102$		0			0
$103$	10.1311			0.998248			0.499124	−	0.866531i	$-0.333655\pi$
$103$	10.1311			0.998248			0.499124	+	0.866531i	$0.333655\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.79195	+	7.79195i	−0.753277	+	0.753277i	−0.975089	−	0.221813i	$-0.928803\pi$
$107$	−7.79195	+	7.79195i	−0.753277	+	0.753277i	0.221813	+	0.975089i	$0.428803\pi$
$108$		0			0
$109$	8.51485	+	8.51485i	0.815574	+	0.815574i	0.985463	−	0.169889i	$-0.0543409\pi$
$109$	8.51485	+	8.51485i	0.815574	+	0.815574i	−0.169889	+	0.985463i	$0.554341\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.0710i		−	1.13554i	−0.823187	−	0.567771i	$-0.807806\pi$
$113$		−	12.0710i		−	1.13554i	0.823187	−	0.567771i	$-0.192194\pi$
$114$		0			0
$115$	2.67770	−	2.67770i	0.249697	−	0.249697i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			4.97307i			0.455881i
$120$		0			0
$121$			10.9077i			0.991611i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	−	0.707107i	0.0632456	−	0.0632456i
$126$		0			0
$127$		−	9.72800i		−	0.863221i	−0.902060	−	0.431610i	$-0.857946\pi$
$127$		−	9.72800i		−	0.863221i	0.902060	−	0.431610i	$-0.142054\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	13.6884	+	13.6884i	1.19596	+	1.19596i	0.975366	+	0.220595i	$0.0707998\pi$
$131$	13.6884	+	13.6884i	1.19596	+	1.19596i	0.220595	+	0.975366i	$0.429200\pi$
$132$		0			0
$133$	8.09178	−	8.09178i	0.701646	−	0.701646i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.3237			−1.05288			−0.526442	−	0.850211i	$-0.676474\pi$
$137$	−12.3237			−1.05288			−0.526442	+	0.850211i	$0.676474\pi$
$138$		0			0
$139$	1.97240	+	1.97240i	0.167297	+	0.167297i	0.785790	−	0.618493i	$-0.212257\pi$
$139$	1.97240	+	1.97240i	0.167297	+	0.167297i	−0.618493	+	0.785790i	$0.712257\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.0763897			−0.00638803
$144$		0			0
$145$	4.45699			0.370133
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.05302	+	6.05302i	0.495883	+	0.495883i	0.910154	−	0.414271i	$-0.135963\pi$
$149$	6.05302	+	6.05302i	0.495883	+	0.495883i	−0.414271	+	0.910154i	$0.635963\pi$
$150$		0			0
$151$	22.2195			1.80820			0.904098	−	0.427325i	$-0.140544\pi$
$151$	22.2195			1.80820			0.904098	+	0.427325i	$0.140544\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.526846	+	0.526846i	−0.0423173	+	0.0423173i
$156$		0			0
$157$	−16.1205	−	16.1205i	−1.28656	−	1.28656i	−0.936864	−	0.349693i	$-0.886286\pi$
$157$	−16.1205	−	16.1205i	−1.28656	−	1.28656i	−0.349693	−	0.936864i	$-0.613714\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	14.5373i		−	1.14570i
$162$		0			0
$163$	−10.2959	+	10.2959i	−0.806437	+	0.806437i	−0.984093	−	0.177656i	$-0.943149\pi$
$163$	−10.2959	+	10.2959i	−0.806437	+	0.806437i	0.177656	+	0.984093i	$0.443149\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	23.1651i		−	1.79257i	−0.443482	−	0.896283i	$-0.646257\pi$
$167$		−	23.1651i		−	1.79257i	0.443482	−	0.896283i	$-0.353743\pi$
$168$		0			0
$169$		−	12.9368i		−	0.995135i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.0819	+	14.0819i	−1.07063	+	1.07063i	−0.0733180	+	0.997309i	$0.523359\pi$
$173$	−14.0819	+	14.0819i	−1.07063	+	1.07063i	−0.997309	+	0.0733180i	$0.976641\pi$
$174$		0			0
$175$		−	3.83891i		−	0.290194i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.68257	+	6.68257i	0.499479	+	0.499479i	0.911276	−	0.411797i	$-0.135099\pi$
$179$	6.68257	+	6.68257i	0.499479	+	0.499479i	−0.411797	+	0.911276i	$0.635099\pi$
$180$		0			0
$181$	−16.3416	+	16.3416i	−1.21467	+	1.21467i	−0.245190	+	0.969475i	$0.578850\pi$
$181$	−16.3416	+	16.3416i	−1.21467	+	1.21467i	−0.969475	+	0.245190i	$0.921150\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10.0458			−0.738583
$186$		0			0
$187$	−0.278256	−	0.278256i	−0.0203481	−	0.0203481i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.67079			−0.482681			−0.241341	−	0.970440i	$-0.577587\pi$
$191$	−6.67079			−0.482681			−0.241341	+	0.970440i	$0.577587\pi$
$192$		0			0
$193$	8.38312			0.603430			0.301715	−	0.953398i	$-0.402441\pi$
$193$	8.38312			0.603430			0.301715	+	0.953398i	$0.402441\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.31421	+	8.31421i	0.592363	+	0.592363i	0.938269	−	0.345906i	$-0.112428\pi$
$197$	8.31421	+	8.31421i	0.592363	+	0.592363i	−0.345906	+	0.938269i	$0.612428\pi$
$198$		0			0
$199$	10.6248			0.753174			0.376587	−	0.926381i	$-0.377098\pi$
$199$	10.6248			0.753174			0.376587	+	0.926381i	$0.377098\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.0986	−	12.0986i	0.849154	−	0.849154i
$204$		0			0
$205$	−7.44741	−	7.44741i	−0.520150	−	0.520150i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			0.905509i			0.0626354i
$210$		0			0
$211$	11.6632	−	11.6632i	0.802929	−	0.802929i	−0.180623	−	0.983552i	$-0.557812\pi$
$211$	11.6632	−	11.6632i	0.802929	−	0.802929i	0.983552	+	0.180623i	$0.0578115\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	2.88565i		−	0.196800i
$216$		0			0
$217$			2.86027i			0.194167i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.230354	+	0.230354i	−0.0154953	+	0.0154953i
$222$		0			0
$223$			11.5818i			0.775574i	0.921749	+	0.387787i	$0.126760\pi$
$223$			11.5818i			0.775574i	−0.921749	+	0.387787i	$0.873240\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.349583	−	0.349583i	−0.0232026	−	0.0232026i	0.695410	−	0.718613i	$-0.255223\pi$
$227$	−0.349583	−	0.349583i	−0.0232026	−	0.0232026i	−0.718613	+	0.695410i	$0.755223\pi$
$228$		0			0
$229$	−12.3899	+	12.3899i	−0.818746	+	0.818746i	−0.985926	−	0.167181i	$-0.946534\pi$
$229$	−12.3899	+	12.3899i	−0.818746	+	0.818746i	0.167181	+	0.985926i	$0.446534\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.97429			0.391389			0.195694	−	0.980665i	$-0.437304\pi$
$233$	5.97429			0.391389			0.195694	+	0.980665i	$0.437304\pi$
$234$		0			0
$235$	7.72320	+	7.72320i	0.503806	+	0.503806i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	13.4047			0.867081			0.433540	−	0.901134i	$-0.357264\pi$
$239$	13.4047			0.867081			0.433540	+	0.901134i	$0.357264\pi$
$240$		0			0
$241$	−15.7309			−1.01332			−0.506660	−	0.862146i	$-0.669120\pi$
$241$	−15.7309			−1.01332			−0.506660	+	0.862146i	$0.669120\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.47103	−	5.47103i	−0.349531	−	0.349531i
$246$		0			0
$247$	0.749626			0.0476975
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.53586	−	1.53586i	0.0969423	−	0.0969423i	−0.656972	−	0.753915i	$-0.728163\pi$
$251$	1.53586	−	1.53586i	0.0969423	−	0.0969423i	0.753915	+	0.656972i	$0.228163\pi$
$252$		0			0
$253$	0.813399	+	0.813399i	0.0511379	+	0.0511379i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			8.67880i			0.541368i	0.962668	+	0.270684i	$0.0872499\pi$
$257$			8.67880i			0.541368i	−0.962668	+	0.270684i	$0.912750\pi$
$258$		0			0
$259$	−27.2695	+	27.2695i	−1.69445	+	1.69445i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2.30739i			0.142279i	0.997466	+	0.0711397i	$0.0226636\pi$
$263$			2.30739i			0.142279i	−0.997466	+	0.0711397i	$0.977336\pi$
$264$		0			0
$265$		−	0.661508i		−	0.0406361i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.59842	+	7.59842i	−0.463284	+	0.463284i	−0.899730	−	0.436446i	$-0.856237\pi$
$269$	−7.59842	+	7.59842i	−0.463284	+	0.463284i	0.436446	+	0.899730i	$0.356237\pi$
$270$		0			0
$271$		−	29.5935i		−	1.79768i	−0.438282	−	0.898838i	$-0.644413\pi$
$271$		−	29.5935i		−	1.79768i	0.438282	−	0.898838i	$-0.355587\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.214796	+	0.214796i	0.0129527	+	0.0129527i
$276$		0			0
$277$	14.8775	−	14.8775i	0.893902	−	0.893902i	−0.100986	−	0.994888i	$-0.532200\pi$
$277$	14.8775	−	14.8775i	0.893902	−	0.893902i	0.994888	+	0.100986i	$0.0321998\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−13.6466			−0.814086			−0.407043	−	0.913409i	$-0.633440\pi$
$281$	−13.6466			−0.814086			−0.407043	+	0.913409i	$0.633440\pi$
$282$		0			0
$283$	−8.19089	−	8.19089i	−0.486898	−	0.486898i	0.420428	−	0.907326i	$-0.361880\pi$
$283$	−8.19089	−	8.19089i	−0.486898	−	0.486898i	−0.907326	+	0.420428i	$0.861880\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−40.4323			−2.38664
$288$		0			0
$289$	15.3218			0.901284
$290$		0			0
$291$		0			0
$292$		0			0
$293$	18.3315	+	18.3315i	1.07094	+	1.07094i	0.997284	+	0.0736556i	$0.0234666\pi$
$293$	18.3315	+	18.3315i	1.07094	+	1.07094i	0.0736556	+	0.997284i	$0.476533\pi$
$294$		0			0
$295$	−2.28952			−0.133301
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.673372	−	0.673372i	0.0389421	−	0.0389421i
$300$		0			0
$301$	−7.83314	−	7.83314i	−0.451495	−	0.451495i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	1.89739i		−	0.108645i
$306$		0			0
$307$	0.859523	−	0.859523i	0.0490556	−	0.0490556i	−0.682153	−	0.731209i	$-0.738956\pi$
$307$	0.859523	−	0.859523i	0.0490556	−	0.0490556i	0.731209	+	0.682153i	$0.238956\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	2.04247i		−	0.115818i	−0.998322	−	0.0579089i	$-0.981557\pi$
$311$		−	2.04247i		−	0.115818i	0.998322	−	0.0579089i	$-0.0184433\pi$
$312$		0			0
$313$			9.51059i			0.537571i	0.963200	+	0.268785i	$0.0866222\pi$
$313$			9.51059i			0.537571i	−0.963200	+	0.268785i	$0.913378\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.7157	+	13.7157i	−0.770349	+	0.770349i	−0.978167	−	0.207818i	$-0.933364\pi$
$317$	−13.7157	+	13.7157i	−0.770349	+	0.770349i	0.207818	+	0.978167i	$0.433364\pi$
$318$		0			0
$319$			1.35389i			0.0758033i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.73057	+	2.73057i	0.151933	+	0.151933i
$324$		0			0
$325$	0.177819	−	0.177819i	0.00986362	−	0.00986362i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	41.9295			2.31165
$330$		0			0
$331$	4.51338	+	4.51338i	0.248078	+	0.248078i	0.820181	−	0.572104i	$-0.193872\pi$
$331$	4.51338	+	4.51338i	0.248078	+	0.248078i	−0.572104	+	0.820181i	$0.693872\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.79274			−0.535035
$336$		0			0
$337$	13.8857			0.756405			0.378202	−	0.925723i	$-0.376542\pi$
$337$	13.8857			0.756405			0.378202	+	0.925723i	$0.376542\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.160039	−	0.160039i	−0.00866659	−	0.00866659i
$342$		0			0
$343$	−2.83006			−0.152809
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.3358	−	17.3358i	0.930632	−	0.930632i	−0.0671132	−	0.997745i	$-0.521379\pi$
$347$	17.3358	−	17.3358i	0.930632	−	0.930632i	0.997745	+	0.0671132i	$0.0213789\pi$
$348$		0			0
$349$	−17.2056	−	17.2056i	−0.920994	−	0.920994i	0.0761058	−	0.997100i	$-0.475751\pi$
$349$	−17.2056	−	17.2056i	−0.920994	−	0.920994i	−0.997100	+	0.0761058i	$0.975751\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			4.28779i			0.228216i	0.993468	+	0.114108i	$0.0364010\pi$
$353$			4.28779i			0.228216i	−0.993468	+	0.114108i	$0.963599\pi$
$354$		0			0
$355$	5.98437	−	5.98437i	0.317618	−	0.317618i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	13.4890i		−	0.711921i	−0.934501	−	0.355960i	$-0.884154\pi$
$359$		−	13.4890i		−	0.711921i	0.934501	−	0.355960i	$-0.115846\pi$
$360$		0			0
$361$			10.1141i			0.532320i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.12185	+	5.12185i	−0.268090	+	0.268090i
$366$		0			0
$367$			5.03456i			0.262802i	0.991329	+	0.131401i	$0.0419475\pi$
$367$			5.03456i			0.262802i	−0.991329	+	0.131401i	$0.958052\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.79568	−	1.79568i	−0.0932268	−	0.0932268i
$372$		0			0
$373$	1.45718	−	1.45718i	0.0754501	−	0.0754501i	−0.668375	−	0.743825i	$-0.733010\pi$
$373$	1.45718	−	1.45718i	0.0754501	−	0.0754501i	0.743825	+	0.668375i	$0.233010\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.12082			0.0577251
$378$		0			0
$379$	26.5423	+	26.5423i	1.36339	+	1.36339i	0.869561	+	0.493826i	$0.164402\pi$
$379$	26.5423	+	26.5423i	1.36339	+	1.36339i	0.493826	+	0.869561i	$0.335598\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.12655			0.0575642			0.0287821	−	0.999586i	$-0.490837\pi$
$383$	1.12655			0.0575642			0.0287821	+	0.999586i	$0.490837\pi$
$384$		0			0
$385$	1.16614			0.0594318
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.36402	+	6.36402i	0.322668	+	0.322668i	0.849790	−	0.527121i	$-0.176729\pi$
$389$	6.36402	+	6.36402i	0.322668	+	0.322668i	−0.527121	+	0.849790i	$0.676729\pi$
$390$		0			0
$391$	4.90563			0.248088
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.86827	+	8.86827i	−0.446211	+	0.446211i
$396$		0			0
$397$	6.54196	+	6.54196i	0.328331	+	0.328331i	0.851952	−	0.523620i	$-0.175419\pi$
$397$	6.54196	+	6.54196i	0.328331	+	0.328331i	−0.523620	+	0.851952i	$0.675419\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	9.60993i		−	0.479897i	−0.970786	−	0.239949i	$-0.922869\pi$
$401$		−	9.60993i		−	0.479897i	0.970786	−	0.239949i	$-0.0771306\pi$
$402$		0			0
$403$	−0.132488	+	0.132488i	−0.00659970	+	0.00659970i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	3.05159i		−	0.151262i
$408$		0			0
$409$			12.6444i			0.625224i	0.949881	+	0.312612i	$0.101204\pi$
$409$			12.6444i			0.625224i	−0.949881	+	0.312612i	$0.898796\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.21495	+	6.21495i	−0.305818	+	0.305818i
$414$		0			0
$415$		−	13.3289i		−	0.654292i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.73617	−	3.73617i	−0.182524	−	0.182524i	0.609931	−	0.792455i	$-0.291197\pi$
$419$	−3.73617	−	3.73617i	−0.182524	−	0.182524i	−0.792455	+	0.609931i	$0.791197\pi$
$420$		0			0
$421$	16.9508	−	16.9508i	0.826130	−	0.826130i	−0.160849	−	0.986979i	$-0.551423\pi$
$421$	16.9508	−	16.9508i	0.826130	−	0.826130i	0.986979	+	0.160849i	$0.0514232\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.29544			0.0628381
$426$		0			0
$427$	−5.15051	−	5.15051i	−0.249251	−	0.249251i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.0390			−0.724402			−0.362201	−	0.932100i	$-0.617975\pi$
$431$	−15.0390			−0.724402			−0.362201	+	0.932100i	$0.617975\pi$
$432$		0			0
$433$	−27.1574			−1.30510			−0.652552	−	0.757744i	$-0.726302\pi$
$433$	−27.1574			−1.30510			−0.652552	+	0.757744i	$0.726302\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.98203	−	7.98203i	−0.381832	−	0.381832i
$438$		0			0
$439$	−16.6307			−0.793738			−0.396869	−	0.917875i	$-0.629903\pi$
$439$	−16.6307			−0.793738			−0.396869	+	0.917875i	$0.629903\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	26.2352	−	26.2352i	1.24647	−	1.24647i	0.289204	−	0.957268i	$-0.406610\pi$
$443$	26.2352	−	26.2352i	1.24647	−	1.24647i	0.957268	−	0.289204i	$-0.0933905\pi$
$444$		0			0
$445$	−6.65735	−	6.65735i	−0.315589	−	0.315589i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			27.7821i			1.31112i	0.755143	+	0.655560i	$0.227567\pi$
$449$			27.7821i			1.31112i	−0.755143	+	0.655560i	$0.772433\pi$
$450$		0			0
$451$	2.26228	−	2.26228i	0.106527	−	0.106527i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	0.965385i		−	0.0452580i
$456$		0			0
$457$		−	22.0885i		−	1.03326i	−0.856210	−	0.516628i	$-0.827187\pi$
$457$		−	22.0885i		−	1.03326i	0.856210	−	0.516628i	$-0.172813\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.9897	+	19.9897i	−0.931015	+	0.931015i	−0.997769	−	0.0667540i	$-0.978736\pi$
$461$	−19.9897	+	19.9897i	−0.931015	+	0.931015i	0.0667540	+	0.997769i	$0.478736\pi$
$462$		0			0
$463$			5.90536i			0.274445i	0.990540	+	0.137223i	$0.0438176\pi$
$463$			5.90536i			0.274445i	−0.990540	+	0.137223i	$0.956182\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13.4037	−	13.4037i	−0.620248	−	0.620248i	0.325346	−	0.945595i	$-0.394519\pi$
$467$	−13.4037	−	13.4037i	−0.620248	−	0.620248i	−0.945595	+	0.325346i	$0.894519\pi$
$468$		0			0
$469$	−26.5826	+	26.5826i	−1.22747	+	1.22747i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.876567			0.0403046
$474$		0			0
$475$	−2.10783	−	2.10783i	−0.0967140	−	0.0967140i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	18.3524			0.838544			0.419272	−	0.907861i	$-0.362285\pi$
$479$	18.3524			0.838544			0.419272	+	0.907861i	$0.362285\pi$
$480$		0			0
$481$	−2.52626			−0.115188
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.8653	−	12.8653i	−0.584183	−	0.584183i
$486$		0			0
$487$	−33.0699			−1.49854			−0.749270	−	0.662265i	$-0.769595\pi$
$487$	−33.0699			−1.49854			−0.749270	+	0.662265i	$0.769595\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8.13752	−	8.13752i	0.367241	−	0.367241i	−0.499229	−	0.866470i	$-0.666383\pi$
$491$	8.13752	−	8.13752i	0.367241	−	0.367241i	0.866470	+	0.499229i	$0.166383\pi$
$492$		0			0
$493$	4.08267	+	4.08267i	0.183874	+	0.183874i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	32.4894i		−	1.45735i
$498$		0			0
$499$	−16.1350	+	16.1350i	−0.722304	+	0.722304i	−0.969074	−	0.246770i	$-0.920631\pi$
$499$	−16.1350	+	16.1350i	−0.722304	+	0.722304i	0.246770	+	0.969074i	$0.420631\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			4.24518i			0.189283i	0.995511	+	0.0946417i	$0.0301705\pi$
$503$			4.24518i			0.189283i	−0.995511	+	0.0946417i	$0.969829\pi$
$504$		0			0
$505$		−	10.3619i		−	0.461098i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	20.1649	−	20.1649i	0.893795	−	0.893795i	−0.101083	−	0.994878i	$-0.532231\pi$
$509$	20.1649	−	20.1649i	0.893795	−	0.893795i	0.994878	+	0.101083i	$0.0322309\pi$
$510$		0			0
$511$			27.8067i			1.23010i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.16378	−	7.16378i	−0.315674	−	0.315674i
$516$		0			0
$517$	−2.34606	+	2.34606i	−0.103180	+	0.103180i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11.7970			−0.516836			−0.258418	−	0.966033i	$-0.583201\pi$
$521$	−11.7970			−0.516836			−0.258418	+	0.966033i	$0.583201\pi$
$522$		0			0
$523$	−6.70782	−	6.70782i	−0.293313	−	0.293313i	0.545075	−	0.838387i	$-0.316501\pi$
$523$	−6.70782	−	6.70782i	−0.293313	−	0.293313i	−0.838387	+	0.545075i	$0.816501\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.965198			−0.0420447
$528$		0			0
$529$	8.65984			0.376515
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.87283	−	1.87283i	−0.0811213	−	0.0811213i
$534$		0			0
$535$	11.0195			0.476414
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.66192	−	1.66192i	0.0715840	−	0.0715840i
$540$		0			0
$541$	−2.14915	−	2.14915i	−0.0923994	−	0.0923994i	0.659396	−	0.751796i	$-0.270812\pi$
$541$	−2.14915	−	2.14915i	−0.0923994	−	0.0923994i	−0.751796	+	0.659396i	$0.770812\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	12.0418i		−	0.515815i
$546$		0			0
$547$	−24.5780	+	24.5780i	−1.05088	+	1.05088i	−0.0522461	+	0.998634i	$0.516638\pi$
$547$	−24.5780	+	24.5780i	−1.05088	+	1.05088i	−0.998634	+	0.0522461i	$0.983362\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	13.2860i		−	0.566001i
$552$		0			0
$553$			48.1461i			2.04738i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.51628	+	5.51628i	−0.233732	+	0.233732i	−0.814249	−	0.580516i	$-0.802851\pi$
$557$	−5.51628	+	5.51628i	−0.233732	+	0.233732i	0.580516	+	0.814249i	$0.302851\pi$
$558$		0			0
$559$		−	0.725666i		−	0.0306924i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.2621	+	13.2621i	0.558931	+	0.558931i	0.929003	−	0.370072i	$-0.120667\pi$
$563$	13.2621	+	13.2621i	0.558931	+	0.558931i	−0.370072	+	0.929003i	$0.620667\pi$
$564$		0			0
$565$	−8.53547	+	8.53547i	−0.359090	+	0.359090i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	16.2485			0.681174			0.340587	−	0.940213i	$-0.389374\pi$
$569$	16.2485			0.681174			0.340587	+	0.940213i	$0.389374\pi$
$570$		0			0
$571$	−19.5062	−	19.5062i	−0.816307	−	0.816307i	0.169264	−	0.985571i	$-0.445861\pi$
$571$	−19.5062	−	19.5062i	−0.816307	−	0.816307i	−0.985571	+	0.169264i	$0.945861\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.78684			−0.157922
$576$		0			0
$577$	−7.61350			−0.316954			−0.158477	−	0.987363i	$-0.550658\pi$
$577$	−7.61350			−0.316954			−0.158477	+	0.987363i	$0.550658\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−36.1816	−	36.1816i	−1.50107	−	1.50107i
$582$		0			0
$583$	0.200945			0.00832228
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.8450	+	14.8450i	−0.612719	+	0.612719i	−0.943654	−	0.330935i	$-0.892636\pi$
$587$	−14.8450	+	14.8450i	−0.612719	+	0.612719i	0.330935	+	0.943654i	$0.392636\pi$
$588$		0			0
$589$	1.57049	+	1.57049i	0.0647109	+	0.0647109i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	34.5714i		−	1.41968i	−0.704364	−	0.709839i	$-0.748768\pi$
$593$		−	34.5714i		−	1.41968i	0.704364	−	0.709839i	$-0.251232\pi$
$594$		0			0
$595$	3.51649	−	3.51649i	0.144162	−	0.144162i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			1.69952i			0.0694406i	0.999397	+	0.0347203i	$0.0110540\pi$
$599$			1.69952i			0.0694406i	−0.999397	+	0.0347203i	$0.988946\pi$
$600$		0			0
$601$		−	1.54584i		−	0.0630561i	−0.999503	−	0.0315281i	$-0.989963\pi$
$601$		−	1.54584i		−	0.0630561i	0.999503	−	0.0315281i	$-0.0100374\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.71293	−	7.71293i	0.313575	−	0.313575i
$606$		0			0
$607$		−	2.11308i		−	0.0857674i	−0.999080	−	0.0428837i	$-0.986346\pi$
$607$		−	2.11308i		−	0.0857674i	0.999080	−	0.0428837i	$-0.0136545\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.94219	+	1.94219i	0.0785724	+	0.0785724i
$612$		0			0
$613$	−12.2409	+	12.2409i	−0.494405	+	0.494405i	−0.909691	−	0.415286i	$-0.863682\pi$
$613$	−12.2409	+	12.2409i	−0.494405	+	0.494405i	0.415286	+	0.909691i	$0.363682\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−21.9584			−0.884013			−0.442007	−	0.897012i	$-0.645733\pi$
$617$	−21.9584			−0.884013			−0.442007	+	0.897012i	$0.645733\pi$
$618$		0			0
$619$	24.0983	+	24.0983i	0.968593	+	0.968593i	0.999522	−	0.0309290i	$-0.00984657\pi$
$619$	24.0983	+	24.0983i	0.968593	+	0.968593i	−0.0309290	+	0.999522i	$0.509847\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−36.1430			−1.44804
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9.20211	−	9.20211i	−0.366912	−	0.366912i
$630$		0			0
$631$	−13.7381			−0.546904			−0.273452	−	0.961886i	$-0.588165\pi$
$631$	−13.7381			−0.546904			−0.273452	+	0.961886i	$0.588165\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.87874	+	6.87874i	−0.272974	+	0.272974i
$636$		0			0
$637$	−1.37582	−	1.37582i	−0.0545121	−	0.0545121i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	38.1567i		−	1.50710i	−0.657391	−	0.753549i	$-0.728340\pi$
$641$		−	38.1567i		−	1.50710i	0.657391	−	0.753549i	$-0.271660\pi$
$642$		0			0
$643$	−0.775769	+	0.775769i	−0.0305933	+	0.0305933i	−0.722238	−	0.691645i	$-0.756886\pi$
$643$	−0.775769	+	0.775769i	−0.0305933	+	0.0305933i	0.691645	+	0.722238i	$0.256886\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.67247i		−	0.0657514i	−0.999459	−	0.0328757i	$-0.989533\pi$
$647$		−	1.67247i		−	0.0657514i	0.999459	−	0.0328757i	$-0.0104666\pi$
$648$		0			0
$649$		−	0.695483i		−	0.0273001i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	24.3372	−	24.3372i	0.952390	−	0.952390i	−0.0465271	−	0.998917i	$-0.514815\pi$
$653$	24.3372	−	24.3372i	0.952390	−	0.952390i	0.998917	+	0.0465271i	$0.0148154\pi$
$654$		0			0
$655$		−	19.3583i		−	0.756392i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	22.5973	+	22.5973i	0.880265	+	0.880265i	0.993561	−	0.113296i	$-0.0361410\pi$
$659$	22.5973	+	22.5973i	0.880265	+	0.880265i	−0.113296	+	0.993561i	$0.536141\pi$
$660$		0			0
$661$	−11.4472	+	11.4472i	−0.445245	+	0.445245i	−0.893770	−	0.448525i	$-0.851950\pi$
$661$	−11.4472	+	11.4472i	−0.445245	+	0.445245i	0.448525	+	0.893770i	$0.351950\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−11.4435			−0.443760
$666$		0			0
$667$	−11.9345	−	11.9345i	−0.462105	−	0.462105i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	0.576367			0.0222504
$672$		0			0
$673$	−41.0206			−1.58123			−0.790614	−	0.612315i	$-0.790238\pi$
$673$	−41.0206			−1.58123			−0.790614	+	0.612315i	$0.790238\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.7765	−	13.7765i	−0.529473	−	0.529473i	0.390943	−	0.920415i	$-0.372149\pi$
$677$	−13.7765	−	13.7765i	−0.529473	−	0.529473i	−0.920415	+	0.390943i	$0.872149\pi$
$678$		0			0
$679$	−69.8461			−2.68045
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−25.6321	+	25.6321i	−0.980784	+	0.980784i	−0.999819	−	0.0190347i	$-0.993941\pi$
$683$	−25.6321	+	25.6321i	−0.980784	+	0.980784i	0.0190347	+	0.999819i	$0.493941\pi$
$684$		0			0
$685$	8.71416	+	8.71416i	0.332951	+	0.332951i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	0.166352i		−	0.00633751i
$690$		0			0
$691$	−6.44050	+	6.44050i	−0.245008	+	0.245008i	−0.818918	−	0.573910i	$-0.805426\pi$
$691$	−6.44050	+	6.44050i	−0.245008	+	0.245008i	0.573910	+	0.818918i	$0.305426\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	2.78939i		−	0.105808i
$696$		0			0
$697$		−	13.6439i		−	0.516799i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	13.0706	−	13.0706i	0.493670	−	0.493670i	−0.415791	−	0.909460i	$-0.636495\pi$
$701$	13.0706	−	13.0706i	0.493670	−	0.493670i	0.909460	+	0.415791i	$0.136495\pi$
$702$		0			0
$703$			29.9458i			1.12943i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−28.1275	−	28.1275i	−1.05784	−	1.05784i
$708$		0			0
$709$	−24.6678	+	24.6678i	−0.926420	+	0.926420i	−0.997473	−	0.0710527i	$-0.977364\pi$
$709$	−24.6678	+	24.6678i	−0.926420	+	0.926420i	0.0710527	+	0.997473i	$0.477364\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.82147			0.105665
$714$		0			0
$715$	0.0540157	+	0.0540157i	0.00202007	+	0.00202007i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	35.8290			1.33620			0.668099	−	0.744072i	$-0.267108\pi$
$719$	35.8290			1.33620			0.668099	+	0.744072i	$0.267108\pi$
$720$		0			0
$721$	−38.8924			−1.44843
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.15157	−	3.15157i	−0.117046	−	0.117046i
$726$		0			0
$727$	37.5183			1.39148			0.695739	−	0.718295i	$-0.255077\pi$
$727$	37.5183			1.39148			0.695739	+	0.718295i	$0.255077\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.64330	−	2.64330i	0.0977658	−	0.0977658i
$732$		0			0
$733$	20.8396	+	20.8396i	0.769727	+	0.769727i	0.978058	−	0.208331i	$-0.0668032\pi$
$733$	20.8396	+	20.8396i	0.769727	+	0.769727i	−0.208331	+	0.978058i	$0.566803\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	2.97472i		−	0.109575i
$738$		0			0
$739$	9.45480	−	9.45480i	0.347801	−	0.347801i	−0.511489	−	0.859290i	$-0.670906\pi$
$739$	9.45480	−	9.45480i	0.347801	−	0.347801i	0.859290	+	0.511489i	$0.170906\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	25.3935i		−	0.931597i	−0.884891	−	0.465799i	$-0.845767\pi$
$743$		−	25.3935i		−	0.931597i	0.884891	−	0.465799i	$-0.154233\pi$
$744$		0			0
$745$		−	8.56026i		−	0.313624i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	29.9126	−	29.9126i	1.09298	−	1.09298i
$750$		0			0
$751$			22.3845i			0.816823i	0.912798	+	0.408412i	$0.133917\pi$
$751$			22.3845i			0.816823i	−0.912798	+	0.408412i	$0.866083\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−15.7115	−	15.7115i	−0.571802	−	0.571802i
$756$		0			0
$757$	7.57143	−	7.57143i	0.275188	−	0.275188i	−0.555996	−	0.831185i	$-0.687663\pi$
$757$	7.57143	−	7.57143i	0.275188	−	0.275188i	0.831185	+	0.555996i	$0.187663\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5.42940			−0.196816			−0.0984078	−	0.995146i	$-0.531375\pi$
$761$	−5.42940			−0.196816			−0.0984078	+	0.995146i	$0.531375\pi$
$762$		0			0
$763$	−32.6877	−	32.6877i	−1.18337	−	1.18337i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.575756			−0.0207893
$768$		0			0
$769$	17.9314			0.646624			0.323312	−	0.946292i	$-0.395204\pi$
$769$	17.9314			0.646624			0.323312	+	0.946292i	$0.395204\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	32.2317	+	32.2317i	1.15930	+	1.15930i	0.984627	+	0.174668i	$0.0558852\pi$
$773$	32.2317	+	32.2317i	1.15930	+	1.15930i	0.174668	+	0.984627i	$0.444115\pi$
$774$		0			0
$775$	0.745073			0.0267638
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−22.2002	+	22.2002i	−0.795405	+	0.795405i
$780$		0			0
$781$	1.81786	+	1.81786i	0.0650481	+	0.0650481i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			22.7979i			0.813690i
$786$		0			0
$787$	14.7312	−	14.7312i	0.525109	−	0.525109i	−0.394001	−	0.919110i	$-0.628909\pi$
$787$	14.7312	−	14.7312i	0.525109	−	0.525109i	0.919110	+	0.394001i	$0.128909\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			46.3394i			1.64764i
$792$		0			0
$793$		−	0.477146i		−	0.0169439i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10.3916	+	10.3916i	−0.368089	+	0.368089i	−0.866780	−	0.498691i	$-0.833814\pi$
$797$	−10.3916	+	10.3916i	−0.368089	+	0.368089i	0.498691	+	0.866780i	$0.333814\pi$
$798$		0			0
$799$			14.1491i			0.500561i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.55585	−	1.55585i	−0.0549048	−	0.0549048i
$804$		0			0
$805$	−10.2794	+	10.2794i	−0.362303	+	0.362303i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−39.9148			−1.40333			−0.701666	−	0.712506i	$-0.747560\pi$
$809$	−39.9148			−1.40333			−0.701666	+	0.712506i	$0.747560\pi$
$810$		0			0
$811$	0.303197	+	0.303197i	0.0106467	+	0.0106467i	0.712410	−	0.701763i	$-0.247604\pi$
$811$	0.303197	+	0.303197i	0.0106467	+	0.0106467i	−0.701763	+	0.712410i	$0.747604\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.5606			0.510036
$816$		0			0
$817$	−8.60191			−0.300943
$818$		0			0
$819$		0			0
$820$		0			0
$821$	38.2299	+	38.2299i	1.33423	+	1.33423i	0.901544	+	0.432688i	$0.142435\pi$
$821$	38.2299	+	38.2299i	1.33423	+	1.33423i	0.432688	+	0.901544i	$0.357565\pi$
$822$		0			0
$823$	12.3498			0.430487			0.215243	−	0.976560i	$-0.430946\pi$
$823$	12.3498			0.430487			0.215243	+	0.976560i	$0.430946\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11.8655	+	11.8655i	−0.412605	+	0.412605i	−0.882645	−	0.470040i	$-0.844239\pi$
$827$	−11.8655	+	11.8655i	−0.412605	+	0.412605i	0.470040	+	0.882645i	$0.344239\pi$
$828$		0			0
$829$	2.36024	+	2.36024i	0.0819746	+	0.0819746i	0.746905	−	0.664931i	$-0.231539\pi$
$829$	2.36024	+	2.36024i	0.0819746	+	0.0819746i	−0.664931	+	0.746905i	$0.731539\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	10.0231i		−	0.347279i
$834$		0			0
$835$	−16.3802	+	16.3802i	−0.566859	+	0.566859i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	27.6899i		−	0.955963i	−0.878370	−	0.477982i	$-0.841369\pi$
$839$		−	27.6899i		−	0.955963i	0.878370	−	0.477982i	$-0.158631\pi$
$840$		0			0
$841$			9.13522i			0.315008i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.14767	+	9.14767i	−0.314689	+	0.314689i
$846$		0			0
$847$		−	41.8737i		−	1.43880i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	26.8997	+	26.8997i	0.922109	+	0.922109i
$852$		0			0
$853$	22.9953	−	22.9953i	0.787343	−	0.787343i	−0.193715	−	0.981058i	$-0.562054\pi$
$853$	22.9953	−	22.9953i	0.787343	−	0.787343i	0.981058	+	0.193715i	$0.0620536\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−17.9441			−0.612958			−0.306479	−	0.951877i	$-0.599151\pi$
$857$	−17.9441			−0.612958			−0.306479	+	0.951877i	$0.599151\pi$
$858$		0			0
$859$	−18.3836	−	18.3836i	−0.627242	−	0.627242i	0.320131	−	0.947373i	$-0.396273\pi$
$859$	−18.3836	−	18.3836i	−0.627242	−	0.627242i	−0.947373	+	0.320131i	$0.896273\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−34.1838			−1.16363			−0.581815	−	0.813321i	$-0.697657\pi$
$863$	−34.1838			−1.16363			−0.581815	+	0.813321i	$0.697657\pi$
$864$		0			0
$865$	19.9148			0.677124
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.69389	−	2.69389i	−0.0913840	−	0.0913840i
$870$		0			0
$871$	−2.46262			−0.0834427
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.71452	+	2.71452i	−0.0917674	+	0.0917674i
$876$		0			0
$877$	8.60130	+	8.60130i	0.290445	+	0.290445i	0.837256	−	0.546811i	$-0.184158\pi$
$877$	8.60130	+	8.60130i	0.290445	+	0.290445i	−0.546811	+	0.837256i	$0.684158\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			32.5526i			1.09673i	0.836240	+	0.548363i	$0.184749\pi$
$881$			32.5526i			1.09673i	−0.836240	+	0.548363i	$0.815251\pi$
$882$		0			0
$883$	27.9330	−	27.9330i	0.940020	−	0.940020i	−0.0582798	−	0.998300i	$-0.518562\pi$
$883$	27.9330	−	27.9330i	0.940020	−	0.940020i	0.998300	+	0.0582798i	$0.0185616\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			13.7416i			0.461397i	0.973025	+	0.230698i	$0.0741010\pi$
$887$			13.7416i			0.461397i	−0.973025	+	0.230698i	$0.925899\pi$
$888$		0			0
$889$			37.3449i			1.25251i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	23.0223	−	23.0223i	0.770412	−	0.770412i
$894$		0			0
$895$		−	9.45058i		−	0.315898i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.34815	+	2.34815i	0.0783152	+	0.0783152i
$900$		0			0
$901$	0.605951	−	0.605951i	0.0201872	−	0.0201872i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	23.1106			0.768222
$906$		0			0
$907$	−19.7473	−	19.7473i	−0.655697	−	0.655697i	0.298662	−	0.954359i	$-0.403460\pi$
$907$	−19.7473	−	19.7473i	−0.655697	−	0.655697i	−0.954359	+	0.298662i	$0.903460\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	48.0800			1.59296			0.796481	−	0.604663i	$-0.206692\pi$
$911$	48.0800			1.59296			0.796481	+	0.604663i	$0.206692\pi$
$912$		0			0
$913$	4.04890			0.133999
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−52.5485	−	52.5485i	−1.73530	−	1.73530i
$918$		0			0
$919$	−14.1967			−0.468305			−0.234152	−	0.972200i	$-0.575232\pi$
$919$	−14.1967			−0.468305			−0.234152	+	0.972200i	$0.575232\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.50491	−	1.50491i	0.0495349	−	0.0495349i
$924$		0			0
$925$	7.10346	+	7.10346i	0.233560	+	0.233560i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			19.9508i			0.654565i	0.944927	+	0.327282i	$0.106133\pi$
$929$			19.9508i			0.654565i	−0.944927	+	0.327282i	$0.893867\pi$
$930$		0			0
$931$	−16.3087	+	16.3087i	−0.534497	+	0.534497i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			0.393513i			0.0128692i
$936$		0			0
$937$		−	53.8894i		−	1.76049i	−0.474519	−	0.880245i	$-0.657378\pi$
$937$		−	53.8894i		−	1.76049i	0.474519	−	0.880245i	$-0.342622\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	16.2221	−	16.2221i	0.528825	−	0.528825i	−0.391397	−	0.920222i	$-0.628008\pi$
$941$	16.2221	−	16.2221i	0.528825	−	0.528825i	0.920222	+	0.391397i	$0.128008\pi$
$942$		0			0
$943$			39.8839i			1.29880i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21.4793	−	21.4793i	−0.697983	−	0.697983i	0.265992	−	0.963975i	$-0.414300\pi$
$947$	−21.4793	−	21.4793i	−0.697983	−	0.697983i	−0.963975	+	0.265992i	$0.914300\pi$
$948$		0			0
$949$	−1.28801	+	1.28801i	−0.0418106	+	0.0418106i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.12681			0.133680			0.0668402	−	0.997764i	$-0.478708\pi$
$953$	4.12681			0.133680			0.0668402	+	0.997764i	$0.478708\pi$
$954$		0			0
$955$	4.71696	+	4.71696i	0.152637	+	0.152637i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	47.3095			1.52770
$960$		0			0
$961$	30.4449			0.982092
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.92776	−	5.92776i	−0.190821	−	0.190821i
$966$		0			0
$967$	44.2853			1.42412			0.712060	−	0.702119i	$-0.247763\pi$
$967$	44.2853			1.42412			0.712060	+	0.702119i	$0.247763\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2.12262	+	2.12262i	−0.0681181	+	0.0681181i	−0.740345	−	0.672227i	$-0.765338\pi$
$971$	−2.12262	+	2.12262i	−0.0681181	+	0.0681181i	0.672227	+	0.740345i	$0.265338\pi$
$972$		0			0
$973$	−7.57185	−	7.57185i	−0.242742	−	0.242742i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.40556i		−	0.0449677i	−0.999747	−	0.0224839i	$-0.992843\pi$
$977$		−	1.40556i		−	0.0449677i	0.999747	−	0.0224839i	$-0.00715744\pi$
$978$		0			0
$979$	2.02229	−	2.02229i	0.0646326	−	0.0646326i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			42.6887i			1.36156i	0.732488	+	0.680780i	$0.238359\pi$
$983$			42.6887i			1.36156i	−0.732488	+	0.680780i	$0.761641\pi$
$984$		0			0
$985$		−	11.7581i		−	0.374643i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7.72690	+	7.72690i	−0.245701	+	0.245701i
$990$		0			0
$991$			52.2269i			1.65904i	0.558474	+	0.829522i	$0.311387\pi$
$991$			52.2269i			1.65904i	−0.558474	+	0.829522i	$0.688613\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7.51289	−	7.51289i	−0.238175	−	0.238175i
$996$		0			0
$997$	21.0768	−	21.0768i	0.667509	−	0.667509i	−0.289629	−	0.957139i	$-0.593532\pi$
$997$	21.0768	−	21.0768i	0.667509	−	0.667509i	0.957139	+	0.289629i	$0.0935320\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bl.c.431.2		32
3.2	odd	2	inner	2880.2.bl.c.431.11		32
4.3	odd	2		720.2.bl.c.611.4	yes	32
12.11	even	2		720.2.bl.c.611.13	yes	32
16.5	even	4		720.2.bl.c.251.13	yes	32
16.11	odd	4	inner	2880.2.bl.c.1871.11		32
48.5	odd	4		720.2.bl.c.251.4	✓	32
48.11	even	4	inner	2880.2.bl.c.1871.2		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.c.251.4	✓	32	48.5	odd	4
720.2.bl.c.251.13	yes	32	16.5	even	4
720.2.bl.c.611.4	yes	32	4.3	odd	2
720.2.bl.c.611.13	yes	32	12.11	even	2
2880.2.bl.c.431.2		32	1.1	even	1	trivial
2880.2.bl.c.431.11		32	3.2	odd	2	inner
2880.2.bl.c.1871.2		32	48.11	even	4	inner
2880.2.bl.c.1871.11		32	16.11	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.bl (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{5} -3.83891 q^{7} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{5} -3.83891 q^{7} +(0.214796 - 0.214796i) q^{11} +(-0.177819 - 0.177819i) q^{13} -1.29544i q^{17} +(-2.10783 + 2.10783i) q^{19} +3.78684i q^{23} +1.00000i q^{25} +(-3.15157 + 3.15157i) q^{29} -0.745073i q^{31} +(2.71452 + 2.71452i) q^{35} +(7.10346 - 7.10346i) q^{37} +10.5322 q^{41} +(2.04046 + 2.04046i) q^{43} -10.9223 q^{47} +7.73720 q^{49} +(0.467757 + 0.467757i) q^{53} -0.303768 q^{55} +(1.61894 - 1.61894i) q^{59} +(1.34166 + 1.34166i) q^{61} +0.251474i q^{65} +(6.92451 - 6.92451i) q^{67} +8.46318i q^{71} -7.24339i q^{73} +(-0.824582 + 0.824582i) q^{77} -12.5416i q^{79} +(9.42498 + 9.42498i) q^{83} +(-0.916015 + 0.916015i) q^{85} +9.41491 q^{89} +(0.682631 + 0.682631i) q^{91} +2.98093 q^{95} +18.1943 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 8 q^{19} - 32 q^{49} + 16 q^{55} + 16 q^{61} - 16 q^{67} + 16 q^{85} + 16 q^{91}+O(q^{100}) \) 32 * q + 8 * q^19 - 32 * q^49 + 16 * q^55 + 16 * q^61 - 16 * q^67 + 16 * q^85 + 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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