LMFDB - Embedded newform 2880.2.t.e.721.10

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			721.10
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.e.2161.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{5} +1.47784i q^{7} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{5} +1.47784i q^{7} +(1.91908 - 1.91908i) q^{11} +(4.06529 + 4.06529i) q^{13} +1.32472 q^{17} +(-3.46908 - 3.46908i) q^{19} -7.30930i q^{23} -1.00000i q^{25} +(4.04370 + 4.04370i) q^{29} -0.0828313 q^{31} +(1.04499 + 1.04499i) q^{35} +(4.52851 - 4.52851i) q^{37} +0.696426i q^{41} +(-6.70405 + 6.70405i) q^{43} +8.68232 q^{47} +4.81600 q^{49} +(-4.80896 + 4.80896i) q^{53} -2.71399i q^{55} +(0.484200 - 0.484200i) q^{59} +(-6.60785 - 6.60785i) q^{61} +5.74918 q^{65} +(-0.600483 - 0.600483i) q^{67} +7.46609i q^{71} +12.5725i q^{73} +(2.83608 + 2.83608i) q^{77} +2.25007 q^{79} +(8.28161 + 8.28161i) q^{83} +(0.936716 - 0.936716i) q^{85} -6.83920i q^{89} +(-6.00782 + 6.00782i) q^{91} -4.90602 q^{95} +6.90038 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			1.47784i			0.558569i	0.960208	+	0.279285i	$0.0900973\pi$
$7$			1.47784i			0.558569i	−0.960208	+	0.279285i	$0.909903\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.91908	−	1.91908i	0.578624	−	0.578624i	−0.355900	−	0.934524i	$-0.615826\pi$
$11$	1.91908	−	1.91908i	0.578624	−	0.578624i	0.934524	+	0.355900i	$0.115826\pi$
$12$		0			0
$13$	4.06529	+	4.06529i	1.12751	+	1.12751i	0.990581	+	0.136926i	$0.0437223\pi$
$13$	4.06529	+	4.06529i	1.12751	+	1.12751i	0.136926	+	0.990581i	$0.456278\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.32472			0.321291			0.160645	−	0.987012i	$-0.448642\pi$
$17$	1.32472			0.321291			0.160645	+	0.987012i	$0.448642\pi$
$18$		0			0
$19$	−3.46908	−	3.46908i	−0.795861	−	0.795861i	0.186579	−	0.982440i	$-0.440260\pi$
$19$	−3.46908	−	3.46908i	−0.795861	−	0.795861i	−0.982440	+	0.186579i	$0.940260\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.30930i		−	1.52409i	−0.647522	−	0.762047i	$-0.724195\pi$
$23$		−	7.30930i		−	1.52409i	0.647522	−	0.762047i	$-0.275805\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.04370	+	4.04370i	0.750897	+	0.750897i	0.974647	−	0.223750i	$-0.0718299\pi$
$29$	4.04370	+	4.04370i	0.750897	+	0.750897i	−0.223750	+	0.974647i	$0.571830\pi$
$30$		0			0
$31$	−0.0828313			−0.0148769			−0.00743847	−	0.999972i	$-0.502368\pi$
$31$	−0.0828313			−0.0148769			−0.00743847	+	0.999972i	$0.502368\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.04499	+	1.04499i	0.176635	+	0.176635i
$36$		0			0
$37$	4.52851	−	4.52851i	0.744482	−	0.744482i	−0.228955	−	0.973437i	$-0.573531\pi$
$37$	4.52851	−	4.52851i	0.744482	−	0.744482i	0.973437	+	0.228955i	$0.0735308\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			0.696426i			0.108763i	0.998520	+	0.0543817i	$0.0173188\pi$
$41$			0.696426i			0.108763i	−0.998520	+	0.0543817i	$0.982681\pi$
$42$		0			0
$43$	−6.70405	+	6.70405i	−1.02236	+	1.02236i	−0.0226131	+	0.999744i	$0.507199\pi$
$43$	−6.70405	+	6.70405i	−1.02236	+	1.02236i	−0.999744	+	0.0226131i	$0.992801\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.68232			1.26645			0.633223	−	0.773969i	$-0.281731\pi$
$47$	8.68232			1.26645			0.633223	+	0.773969i	$0.281731\pi$
$48$		0			0
$49$	4.81600			0.688000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.80896	+	4.80896i	−0.660562	+	0.660562i	−0.955512	−	0.294951i	$-0.904697\pi$
$53$	−4.80896	+	4.80896i	−0.660562	+	0.660562i	0.294951	+	0.955512i	$0.404697\pi$
$54$		0			0
$55$		−	2.71399i		−	0.365954i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.484200	−	0.484200i	0.0630375	−	0.0630375i	−0.674885	−	0.737923i	$-0.735807\pi$
$59$	0.484200	−	0.484200i	0.0630375	−	0.0630375i	0.737923	+	0.674885i	$0.235807\pi$
$60$		0			0
$61$	−6.60785	−	6.60785i	−0.846049	−	0.846049i	0.143589	−	0.989637i	$-0.454136\pi$
$61$	−6.60785	−	6.60785i	−0.846049	−	0.846049i	−0.989637	+	0.143589i	$0.954136\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.74918			0.713098
$66$		0			0
$67$	−0.600483	−	0.600483i	−0.0733607	−	0.0733607i	0.669474	−	0.742835i	$-0.266519\pi$
$67$	−0.600483	−	0.600483i	−0.0733607	−	0.0733607i	−0.742835	+	0.669474i	$0.766519\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.46609i			0.886062i	0.896506	+	0.443031i	$0.146097\pi$
$71$			7.46609i			0.886062i	−0.896506	+	0.443031i	$0.853903\pi$
$72$		0			0
$73$			12.5725i			1.47150i	0.677256	+	0.735748i	$0.263169\pi$
$73$			12.5725i			1.47150i	−0.677256	+	0.735748i	$0.736831\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.83608	+	2.83608i	0.323201	+	0.323201i
$78$		0			0
$79$	2.25007			0.253153			0.126576	−	0.991957i	$-0.459601\pi$
$79$	2.25007			0.253153			0.126576	+	0.991957i	$0.459601\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.28161	+	8.28161i	0.909025	+	0.909025i	0.996194	−	0.0871685i	$-0.0277819\pi$
$83$	8.28161	+	8.28161i	0.909025	+	0.909025i	−0.0871685	+	0.996194i	$0.527782\pi$
$84$		0			0
$85$	0.936716	−	0.936716i	0.101601	−	0.101601i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	6.83920i		−	0.724953i	−0.931993	−	0.362477i	$-0.881931\pi$
$89$		−	6.83920i		−	0.724953i	0.931993	−	0.362477i	$-0.118069\pi$
$90$		0			0
$91$	−6.00782	+	6.00782i	−0.629791	+	0.629791i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.90602			−0.503347
$96$		0			0
$97$	6.90038			0.700627			0.350314	−	0.936632i	$-0.386075\pi$
$97$	6.90038			0.700627			0.350314	+	0.936632i	$0.386075\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	13.5677	−	13.5677i	1.35003	−	1.35003i	0.464418	−	0.885616i	$-0.346264\pi$
$101$	13.5677	−	13.5677i	1.35003	−	1.35003i	0.885616	−	0.464418i	$-0.153736\pi$
$102$		0			0
$103$		−	12.5670i		−	1.23827i	−0.785286	−	0.619133i	$-0.787484\pi$
$103$		−	12.5670i		−	1.23827i	0.785286	−	0.619133i	$-0.212516\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.6977	−	12.6977i	1.22753	−	1.22753i	0.262639	−	0.964894i	$-0.415407\pi$
$107$	12.6977	−	12.6977i	1.22753	−	1.22753i	0.964894	−	0.262639i	$-0.0845929\pi$
$108$		0			0
$109$	−6.89854	−	6.89854i	−0.660760	−	0.660760i	0.294799	−	0.955559i	$-0.404747\pi$
$109$	−6.89854	−	6.89854i	−0.660760	−	0.660760i	−0.955559	+	0.294799i	$0.904747\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	15.7492			1.48156			0.740780	−	0.671748i	$-0.234456\pi$
$113$	15.7492			1.48156			0.740780	+	0.671748i	$0.234456\pi$
$114$		0			0
$115$	−5.16845	−	5.16845i	−0.481961	−	0.481961i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			1.95771i			0.179463i
$120$		0			0
$121$			3.63428i			0.330389i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.0632456	−	0.0632456i
$126$		0			0
$127$	15.2696			1.35496			0.677478	−	0.735543i	$-0.263073\pi$
$127$	15.2696			1.35496			0.677478	+	0.735543i	$0.263073\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.27469	+	1.27469i	0.111370	+	0.111370i	0.760596	−	0.649226i	$-0.224907\pi$
$131$	1.27469	+	1.27469i	0.111370	+	0.111370i	−0.649226	+	0.760596i	$0.724907\pi$
$132$		0			0
$133$	5.12673	−	5.12673i	0.444544	−	0.444544i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.09475i			0.520710i	0.965513	+	0.260355i	$0.0838396\pi$
$137$			6.09475i			0.520710i	−0.965513	+	0.260355i	$0.916160\pi$
$138$		0			0
$139$	−10.6477	+	10.6477i	−0.903129	+	0.903129i	−0.995706	−	0.0925765i	$-0.970490\pi$
$139$	−10.6477	+	10.6477i	−0.903129	+	0.903129i	0.0925765	+	0.995706i	$0.470490\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	15.6032			1.30481
$144$		0			0
$145$	5.71866			0.474909
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.19516	−	8.19516i	0.671374	−	0.671374i	−0.286659	−	0.958033i	$-0.592545\pi$
$149$	8.19516	−	8.19516i	0.671374	−	0.671374i	0.958033	+	0.286659i	$0.0925447\pi$
$150$		0			0
$151$			3.69667i			0.300830i	0.988623	+	0.150415i	$0.0480610\pi$
$151$			3.69667i			0.300830i	−0.988623	+	0.150415i	$0.951939\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.0585706	+	0.0585706i	−0.00470450	+	0.00470450i
$156$		0			0
$157$	−6.93249	−	6.93249i	−0.553273	−	0.553273i	0.374111	−	0.927384i	$-0.377948\pi$
$157$	−6.93249	−	6.93249i	−0.553273	−	0.553273i	−0.927384	+	0.374111i	$0.877948\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.8019			0.851312
$162$		0			0
$163$	−9.71740	−	9.71740i	−0.761125	−	0.761125i	0.215401	−	0.976526i	$-0.430894\pi$
$163$	−9.71740	−	9.71740i	−0.761125	−	0.761125i	−0.976526	+	0.215401i	$0.930894\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	0.150050i		−	0.0116112i	−0.999983	−	0.00580559i	$-0.998152\pi$
$167$		−	0.150050i		−	0.0116112i	0.999983	−	0.00580559i	$-0.00184799\pi$
$168$		0			0
$169$			20.0531i			1.54255i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.77943	−	6.77943i	−0.515431	−	0.515431i	0.400755	−	0.916185i	$-0.368748\pi$
$173$	−6.77943	−	6.77943i	−0.515431	−	0.515431i	−0.916185	+	0.400755i	$0.868748\pi$
$174$		0			0
$175$	1.47784			0.111714
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.0806613	−	0.0806613i	−0.00602891	−	0.00602891i	0.704086	−	0.710115i	$-0.251357\pi$
$179$	−0.0806613	−	0.0806613i	−0.00602891	−	0.00602891i	−0.710115	+	0.704086i	$0.751357\pi$
$180$		0			0
$181$	8.92028	−	8.92028i	0.663039	−	0.663039i	−0.293056	−	0.956095i	$-0.594672\pi$
$181$	8.92028	−	8.92028i	0.663039	−	0.663039i	0.956095	+	0.293056i	$0.0946722\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	6.40428i		−	0.470852i
$186$		0			0
$187$	2.54223	−	2.54223i	0.185907	−	0.185907i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.59973			0.115752			0.0578760	−	0.998324i	$-0.481567\pi$
$191$	1.59973			0.115752			0.0578760	+	0.998324i	$0.481567\pi$
$192$		0			0
$193$	3.04722			0.219344			0.109672	−	0.993968i	$-0.465020\pi$
$193$	3.04722			0.219344			0.109672	+	0.993968i	$0.465020\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.89240	+	3.89240i	−0.277322	+	0.277322i	−0.832039	−	0.554717i	$-0.812826\pi$
$197$	−3.89240	+	3.89240i	−0.277322	+	0.277322i	0.554717	+	0.832039i	$0.312826\pi$
$198$		0			0
$199$			21.6539i			1.53501i	0.641045	+	0.767503i	$0.278501\pi$
$199$			21.6539i			1.53501i	−0.641045	+	0.767503i	$0.721499\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.97593	+	5.97593i	−0.419428	+	0.419428i
$204$		0			0
$205$	0.492448	+	0.492448i	0.0343940	+	0.0343940i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−13.3149			−0.921008
$210$		0			0
$211$	1.61230	+	1.61230i	0.110995	+	0.110995i	0.760423	−	0.649428i	$-0.224992\pi$
$211$	1.61230	+	1.61230i	0.110995	+	0.110995i	−0.649428	+	0.760423i	$0.724992\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			9.48095i			0.646596i
$216$		0			0
$217$		−	0.122411i		−	0.00830980i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.38535	+	5.38535i	0.362258	+	0.362258i
$222$		0			0
$223$	−2.49314			−0.166953			−0.0834765	−	0.996510i	$-0.526602\pi$
$223$	−2.49314			−0.166953			−0.0834765	+	0.996510i	$0.526602\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.681211	+	0.681211i	0.0452136	+	0.0452136i	0.729352	−	0.684139i	$-0.239822\pi$
$227$	0.681211	+	0.681211i	0.0452136	+	0.0452136i	−0.684139	+	0.729352i	$0.739822\pi$
$228$		0			0
$229$	−13.4099	+	13.4099i	−0.886153	+	0.886153i	−0.994151	−	0.107998i	$-0.965556\pi$
$229$	−13.4099	+	13.4099i	−0.886153	+	0.886153i	0.107998	+	0.994151i	$0.465556\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			2.67624i			0.175326i	0.996150	+	0.0876631i	$0.0279399\pi$
$233$			2.67624i			0.175326i	−0.996150	+	0.0876631i	$0.972060\pi$
$234$		0			0
$235$	6.13933	−	6.13933i	0.400486	−	0.400486i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	14.5999			0.944386			0.472193	−	0.881495i	$-0.343463\pi$
$239$	14.5999			0.944386			0.472193	+	0.881495i	$0.343463\pi$
$240$		0			0
$241$	−20.9908			−1.35213			−0.676067	−	0.736840i	$-0.736317\pi$
$241$	−20.9908			−1.35213			−0.676067	+	0.736840i	$0.736317\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.40543	−	3.40543i	0.217565	−	0.217565i
$246$		0			0
$247$		−	28.2056i		−	1.79468i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.3292	+	21.3292i	−1.34629	+	1.34629i	−0.456632	+	0.889656i	$0.650944\pi$
$251$	−21.3292	+	21.3292i	−1.34629	+	1.34629i	−0.889656	+	0.456632i	$0.849056\pi$
$252$		0			0
$253$	−14.0271	−	14.0271i	−0.881877	−	0.881877i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.41831			−0.0884717			−0.0442358	−	0.999021i	$-0.514085\pi$
$257$	−1.41831			−0.0884717			−0.0442358	+	0.999021i	$0.514085\pi$
$258$		0			0
$259$	6.69239	+	6.69239i	0.415845	+	0.415845i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	17.9559i		−	1.10721i	−0.832780	−	0.553604i	$-0.813252\pi$
$263$		−	17.9559i		−	1.10721i	0.832780	−	0.553604i	$-0.186748\pi$
$264$		0			0
$265$			6.80090i			0.417776i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	20.2805	+	20.2805i	1.23652	+	1.23652i	0.961412	+	0.275111i	$0.0887147\pi$
$269$	20.2805	+	20.2805i	1.23652	+	1.23652i	0.275111	+	0.961412i	$0.411285\pi$
$270$		0			0
$271$	32.8458			1.99524			0.997619	−	0.0689711i	$-0.0219716\pi$
$271$	32.8458			1.99524			0.997619	+	0.0689711i	$0.0219716\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.91908	−	1.91908i	−0.115725	−	0.115725i
$276$		0			0
$277$	17.9214	−	17.9214i	1.07679	−	1.07679i	0.0799968	−	0.996795i	$-0.474509\pi$
$277$	17.9214	−	17.9214i	1.07679	−	1.07679i	0.996795	−	0.0799968i	$-0.0254910\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	17.9795i		−	1.07257i	−0.844038	−	0.536283i	$-0.819828\pi$
$281$		−	17.9795i		−	1.07257i	0.844038	−	0.536283i	$-0.180172\pi$
$282$		0			0
$283$	2.79537	−	2.79537i	0.166167	−	0.166167i	−0.619125	−	0.785292i	$-0.712513\pi$
$283$	2.79537	−	2.79537i	0.166167	−	0.166167i	0.785292	+	0.619125i	$0.212513\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.02920			−0.0607519
$288$		0			0
$289$	−15.2451			−0.896772
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.38479	−	9.38479i	0.548266	−	0.548266i	−0.377673	−	0.925939i	$-0.623276\pi$
$293$	9.38479	−	9.38479i	0.548266	−	0.548266i	0.925939	+	0.377673i	$0.123276\pi$
$294$		0			0
$295$		−	0.684762i		−	0.0398684i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	29.7144	−	29.7144i	1.71843	−	1.71843i
$300$		0			0
$301$	−9.90748	−	9.90748i	−0.571058	−	0.571058i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.34491			−0.535088
$306$		0			0
$307$	0.221745	+	0.221745i	0.0126556	+	0.0126556i	0.713406	−	0.700751i	$-0.247151\pi$
$307$	0.221745	+	0.221745i	0.0126556	+	0.0126556i	−0.700751	+	0.713406i	$0.747151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			0.227549i			0.0129031i	0.999979	+	0.00645155i	$0.00205361\pi$
$311$			0.227549i			0.0129031i	−0.999979	+	0.00645155i	$0.997946\pi$
$312$		0			0
$313$			21.1613i			1.19611i	0.801457	+	0.598053i	$0.204059\pi$
$313$			21.1613i			1.19611i	−0.801457	+	0.598053i	$0.795941\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−19.1043	−	19.1043i	−1.07301	−	1.07301i	−0.997116	−	0.0758899i	$-0.975820\pi$
$317$	−19.1043	−	19.1043i	−1.07301	−	1.07301i	−0.0758899	−	0.997116i	$-0.524180\pi$
$318$		0			0
$319$	15.5204			0.868973
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.59555	−	4.59555i	−0.255703	−	0.255703i
$324$		0			0
$325$	4.06529	−	4.06529i	0.225501	−	0.225501i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			12.8310i			0.707398i
$330$		0			0
$331$	−9.72129	+	9.72129i	−0.534330	+	0.534330i	−0.921858	−	0.387528i	$-0.873329\pi$
$331$	−9.72129	+	9.72129i	−0.534330	+	0.534330i	0.387528	+	0.921858i	$0.373329\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.849212			−0.0463974
$336$		0			0
$337$	7.73476			0.421339			0.210670	−	0.977557i	$-0.432436\pi$
$337$	7.73476			0.421339			0.210670	+	0.977557i	$0.432436\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.158960	+	0.158960i	−0.00860815	+	0.00860815i
$342$		0			0
$343$			17.4621i			0.942865i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.27232	−	4.27232i	0.229350	−	0.229350i	−0.583071	−	0.812421i	$-0.698149\pi$
$347$	4.27232	−	4.27232i	0.229350	−	0.229350i	0.812421	+	0.583071i	$0.198149\pi$
$348$		0			0
$349$	−0.390826	−	0.390826i	−0.0209204	−	0.0209204i	0.696569	−	0.717490i	$-0.254709\pi$
$349$	−0.390826	−	0.390826i	−0.0209204	−	0.0209204i	−0.717490	+	0.696569i	$0.754709\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.49816			−0.505536			−0.252768	−	0.967527i	$-0.581341\pi$
$353$	−9.49816			−0.505536			−0.252768	+	0.967527i	$0.581341\pi$
$354$		0			0
$355$	5.27932	+	5.27932i	0.280197	+	0.280197i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			2.76642i			0.146006i	0.997332	+	0.0730031i	$0.0232583\pi$
$359$			2.76642i			0.146006i	−0.997332	+	0.0730031i	$0.976742\pi$
$360$		0			0
$361$			5.06901i			0.266790i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.89007	+	8.89007i	0.465328	+	0.465328i
$366$		0			0
$367$	35.3081			1.84307			0.921535	−	0.388295i	$-0.126936\pi$
$367$	35.3081			1.84307			0.921535	+	0.388295i	$0.126936\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.10685	−	7.10685i	−0.368969	−	0.368969i
$372$		0			0
$373$	−18.0924	+	18.0924i	−0.936789	+	0.936789i	−0.998118	−	0.0613290i	$-0.980466\pi$
$373$	−18.0924	+	18.0924i	−0.936789	+	0.936789i	0.0613290	+	0.998118i	$0.480466\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			32.8776i			1.69328i
$378$		0			0
$379$	−23.6281	+	23.6281i	−1.21370	+	1.21370i	−0.243893	+	0.969802i	$0.578425\pi$
$379$	−23.6281	+	23.6281i	−1.21370	+	1.21370i	−0.969802	+	0.243893i	$0.921575\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8.05256			−0.411466			−0.205733	−	0.978608i	$-0.565958\pi$
$383$	−8.05256			−0.411466			−0.205733	+	0.978608i	$0.565958\pi$
$384$		0			0
$385$	4.01083			0.204411
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−20.3343	+	20.3343i	−1.03099	+	1.03099i	−0.0314860	+	0.999504i	$0.510024\pi$
$389$	−20.3343	+	20.3343i	−1.03099	+	1.03099i	−0.999504	+	0.0314860i	$0.989976\pi$
$390$		0			0
$391$		−	9.68274i		−	0.489677i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.59104	−	1.59104i	0.0800539	−	0.0800539i
$396$		0			0
$397$	−20.7059	−	20.7059i	−1.03920	−	1.03920i	−0.999200	−	0.0399980i	$-0.987265\pi$
$397$	−20.7059	−	20.7059i	−1.03920	−	1.03920i	−0.0399980	−	0.999200i	$-0.512735\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	36.1805			1.80677			0.903385	−	0.428830i	$-0.141074\pi$
$401$	36.1805			1.80677			0.903385	+	0.428830i	$0.141074\pi$
$402$		0			0
$403$	−0.336733	−	0.336733i	−0.0167739	−	0.0167739i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	17.3811i		−	0.861550i
$408$		0			0
$409$			22.1669i			1.09608i	0.836451	+	0.548042i	$0.184627\pi$
$409$			22.1669i			1.09608i	−0.836451	+	0.548042i	$0.815373\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.715568	+	0.715568i	0.0352108	+	0.0352108i
$414$		0			0
$415$	11.7120			0.574918
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.20112	+	4.20112i	0.205238	+	0.205238i	0.802240	−	0.597002i	$-0.203641\pi$
$419$	4.20112	+	4.20112i	0.205238	+	0.205238i	−0.597002	+	0.802240i	$0.703641\pi$
$420$		0			0
$421$	−12.5767	+	12.5767i	−0.612950	+	0.612950i	−0.943714	−	0.330763i	$-0.892694\pi$
$421$	−12.5767	+	12.5767i	−0.612950	+	0.612950i	0.330763	+	0.943714i	$0.392694\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	1.32472i		−	0.0642582i
$426$		0			0
$427$	9.76532	−	9.76532i	0.472577	−	0.472577i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.0263			−0.771961			−0.385981	−	0.922507i	$-0.626137\pi$
$431$	−16.0263			−0.771961			−0.385981	+	0.922507i	$0.626137\pi$
$432$		0			0
$433$	1.04321			0.0501334			0.0250667	−	0.999686i	$-0.492020\pi$
$433$	1.04321			0.0501334			0.0250667	+	0.999686i	$0.492020\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−25.3565	+	25.3565i	−1.21297	+	1.21297i
$438$		0			0
$439$		−	16.0865i		−	0.767767i	−0.923382	−	0.383883i	$-0.874586\pi$
$439$		−	16.0865i		−	0.767767i	0.923382	−	0.383883i	$-0.125414\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.9667	+	13.9667i	−0.663576	+	0.663576i	−0.956221	−	0.292645i	$-0.905465\pi$
$443$	−13.9667	+	13.9667i	−0.663576	+	0.663576i	0.292645	+	0.956221i	$0.405465\pi$
$444$		0			0
$445$	−4.83604	−	4.83604i	−0.229250	−	0.229250i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−40.5215			−1.91233			−0.956164	−	0.292833i	$-0.905402\pi$
$449$	−40.5215			−1.91233			−0.956164	+	0.292833i	$0.905402\pi$
$450$		0			0
$451$	1.33650	+	1.33650i	0.0629331	+	0.0629331i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			8.49635i			0.398315i
$456$		0			0
$457$		−	28.8900i		−	1.35142i	−0.737168	−	0.675709i	$-0.763838\pi$
$457$		−	28.8900i		−	1.35142i	0.737168	−	0.675709i	$-0.236162\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−20.4212	−	20.4212i	−0.951109	−	0.951109i	0.0477499	−	0.998859i	$-0.484795\pi$
$461$	−20.4212	−	20.4212i	−0.951109	−	0.951109i	−0.998859	+	0.0477499i	$0.984795\pi$
$462$		0			0
$463$	24.4554			1.13654			0.568269	−	0.822843i	$-0.307613\pi$
$463$	24.4554			1.13654			0.568269	+	0.822843i	$0.307613\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.05946	−	3.05946i	−0.141575	−	0.141575i	0.632767	−	0.774342i	$-0.281919\pi$
$467$	−3.05946	−	3.05946i	−0.141575	−	0.141575i	−0.774342	+	0.632767i	$0.781919\pi$
$468$		0			0
$469$	0.887416	−	0.887416i	0.0409771	−	0.0409771i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			25.7312i			1.18312i
$474$		0			0
$475$	−3.46908	+	3.46908i	−0.159172	+	0.159172i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−33.4104			−1.52656			−0.763280	−	0.646068i	$-0.776412\pi$
$479$	−33.4104			−1.52656			−0.763280	+	0.646068i	$0.776412\pi$
$480$		0			0
$481$	36.8194			1.67882
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.87931	−	4.87931i	0.221558	−	0.221558i
$486$		0			0
$487$		−	7.53883i		−	0.341617i	−0.985304	−	0.170808i	$-0.945362\pi$
$487$		−	7.53883i		−	0.341617i	0.985304	−	0.170808i	$-0.0546379\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−27.0773	+	27.0773i	−1.22198	+	1.22198i	−0.255052	+	0.966927i	$0.582093\pi$
$491$	−27.0773	+	27.0773i	−1.22198	+	1.22198i	−0.966927	+	0.255052i	$0.917907\pi$
$492$		0			0
$493$	5.35676	+	5.35676i	0.241256	+	0.241256i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11.0337			−0.494927
$498$		0			0
$499$	−18.0706	−	18.0706i	−0.808953	−	0.808953i	0.175522	−	0.984475i	$-0.443839\pi$
$499$	−18.0706	−	18.0706i	−0.808953	−	0.808953i	−0.984475	+	0.175522i	$0.943839\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			27.8224i			1.24054i	0.784389	+	0.620269i	$0.212977\pi$
$503$			27.8224i			1.24054i	−0.784389	+	0.620269i	$0.787023\pi$
$504$		0			0
$505$		−	19.1876i		−	0.853836i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.33288	−	5.33288i	−0.236376	−	0.236376i	0.578972	−	0.815348i	$-0.303454\pi$
$509$	−5.33288	−	5.33288i	−0.236376	−	0.236376i	−0.815348	+	0.578972i	$0.803454\pi$
$510$		0			0
$511$	−18.5800			−0.821932
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.88624	−	8.88624i	−0.391574	−	0.391574i
$516$		0			0
$517$	16.6620	−	16.6620i	0.732796	−	0.732796i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	11.7002i		−	0.512596i	−0.966598	−	0.256298i	$-0.917497\pi$
$521$		−	11.7002i		−	0.512596i	0.966598	−	0.256298i	$-0.0825028\pi$
$522$		0			0
$523$	−30.6923	+	30.6923i	−1.34208	+	1.34208i	−0.448091	+	0.893988i	$0.647896\pi$
$523$	−30.6923	+	30.6923i	−1.34208	+	1.34208i	−0.893988	+	0.448091i	$0.852104\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.109728			−0.00477982
$528$		0			0
$529$	−30.4258			−1.32286
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.83117	+	2.83117i	−0.122632	+	0.122632i
$534$		0			0
$535$		−	17.9573i		−	0.776360i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.24228	−	9.24228i	0.398093	−	0.398093i
$540$		0			0
$541$	12.2821	+	12.2821i	0.528050	+	0.528050i	0.919991	−	0.391940i	$-0.128196\pi$
$541$	12.2821	+	12.2821i	0.528050	+	0.528050i	−0.391940	+	0.919991i	$0.628196\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.75601			−0.417901
$546$		0			0
$547$	−22.4128	−	22.4128i	−0.958304	−	0.958304i	0.0408610	−	0.999165i	$-0.486990\pi$
$547$	−22.4128	−	22.4128i	−0.958304	−	0.958304i	−0.999165	+	0.0408610i	$0.986990\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	28.0558i		−	1.19522i
$552$		0			0
$553$			3.32523i			0.141403i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.03965	+	7.03965i	0.298279	+	0.298279i	0.840340	−	0.542060i	$-0.182356\pi$
$557$	7.03965	+	7.03965i	0.298279	+	0.298279i	−0.542060	+	0.840340i	$0.682356\pi$
$558$		0			0
$559$	−54.5077			−2.30543
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−21.5425	−	21.5425i	−0.907910	−	0.907910i	0.0881933	−	0.996103i	$-0.471891\pi$
$563$	−21.5425	−	21.5425i	−0.907910	−	0.907910i	−0.996103	+	0.0881933i	$0.971891\pi$
$564$		0			0
$565$	11.1364	−	11.1364i	0.468510	−	0.468510i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	42.5645i		−	1.78440i	−0.451643	−	0.892199i	$-0.649162\pi$
$569$		−	42.5645i		−	1.78440i	0.451643	−	0.892199i	$-0.350838\pi$
$570$		0			0
$571$	8.71612	−	8.71612i	0.364758	−	0.364758i	−0.500803	−	0.865561i	$-0.666962\pi$
$571$	8.71612	−	8.71612i	0.364758	−	0.364758i	0.865561	+	0.500803i	$0.166962\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−7.30930			−0.304819
$576$		0			0
$577$	15.1260			0.629703			0.314852	−	0.949141i	$-0.398045\pi$
$577$	15.1260			0.629703			0.314852	+	0.949141i	$0.398045\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.2389	+	12.2389i	−0.507754	+	0.507754i
$582$		0			0
$583$			18.4575i			0.764433i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.58349	+	6.58349i	−0.271730	+	0.271730i	−0.829796	−	0.558067i	$-0.811543\pi$
$587$	−6.58349	+	6.58349i	−0.271730	+	0.271730i	0.558067	+	0.829796i	$0.311543\pi$
$588$		0			0
$589$	0.287348	+	0.287348i	0.0118400	+	0.0118400i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	17.2532			0.708503			0.354252	−	0.935150i	$-0.384736\pi$
$593$	17.2532			0.708503			0.354252	+	0.935150i	$0.384736\pi$
$594$		0			0
$595$	1.38431	+	1.38431i	0.0567513	+	0.0567513i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			41.4646i			1.69420i	0.531437	+	0.847098i	$0.321652\pi$
$599$			41.4646i			1.69420i	−0.531437	+	0.847098i	$0.678348\pi$
$600$		0			0
$601$			28.8962i			1.17870i	0.807878	+	0.589350i	$0.200616\pi$
$601$			28.8962i			1.17870i	−0.807878	+	0.589350i	$0.799384\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.56982	+	2.56982i	0.104478	+	0.104478i
$606$		0			0
$607$	−13.0235			−0.528607			−0.264304	−	0.964440i	$-0.585142\pi$
$607$	−13.0235			−0.528607			−0.264304	+	0.964440i	$0.585142\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	35.2961	+	35.2961i	1.42793	+	1.42793i
$612$		0			0
$613$	3.17675	−	3.17675i	0.128308	−	0.128308i	−0.640037	−	0.768344i	$-0.721081\pi$
$613$	3.17675	−	3.17675i	0.128308	−	0.128308i	0.768344	+	0.640037i	$0.221081\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	35.0141i		−	1.40961i	−0.709399	−	0.704807i	$-0.751034\pi$
$617$		−	35.0141i		−	1.40961i	0.709399	−	0.704807i	$-0.248966\pi$
$618$		0			0
$619$	5.58826	−	5.58826i	0.224611	−	0.224611i	−0.585826	−	0.810437i	$-0.699230\pi$
$619$	5.58826	−	5.58826i	0.224611	−	0.224611i	0.810437	+	0.585826i	$0.199230\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.1072			0.404937
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.99899	−	5.99899i	0.239195	−	0.239195i
$630$		0			0
$631$			24.5661i			0.977962i	0.872295	+	0.488981i	$0.162631\pi$
$631$			24.5661i			0.977962i	−0.872295	+	0.488981i	$0.837369\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10.7972	−	10.7972i	0.428474	−	0.428474i
$636$		0			0
$637$	19.5784	+	19.5784i	0.775725	+	0.775725i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.27928			−0.248017			−0.124008	−	0.992281i	$-0.539575\pi$
$641$	−6.27928			−0.248017			−0.124008	+	0.992281i	$0.539575\pi$
$642$		0			0
$643$	10.1173	+	10.1173i	0.398988	+	0.398988i	0.877876	−	0.478888i	$-0.158960\pi$
$643$	10.1173	+	10.1173i	0.398988	+	0.398988i	−0.478888	+	0.877876i	$0.658960\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	21.8855i		−	0.860406i	−0.902732	−	0.430203i	$-0.858442\pi$
$647$		−	21.8855i		−	0.860406i	0.902732	−	0.430203i	$-0.141558\pi$
$648$		0			0
$649$		−	1.85843i		−	0.0729499i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−33.1470	−	33.1470i	−1.29714	−	1.29714i	−0.930273	−	0.366867i	$-0.880430\pi$
$653$	−33.1470	−	33.1470i	−1.29714	−	1.29714i	−0.366867	−	0.930273i	$-0.619570\pi$
$654$		0			0
$655$	1.80269			0.0704368
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−31.3303	−	31.3303i	−1.22045	−	1.22045i	−0.967471	−	0.252982i	$-0.918589\pi$
$659$	−31.3303	−	31.3303i	−1.22045	−	1.22045i	−0.252982	−	0.967471i	$-0.581411\pi$
$660$		0			0
$661$	−0.749418	+	0.749418i	−0.0291490	+	0.0291490i	−0.721531	−	0.692382i	$-0.756561\pi$
$661$	−0.749418	+	0.749418i	−0.0291490	+	0.0291490i	0.692382	+	0.721531i	$0.256561\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	7.25029i		−	0.281154i
$666$		0			0
$667$	29.5566	−	29.5566i	1.14444	−	1.14444i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−25.3620			−0.979088
$672$		0			0
$673$	−35.0268			−1.35018			−0.675092	−	0.737734i	$-0.735896\pi$
$673$	−35.0268			−1.35018			−0.675092	+	0.737734i	$0.735896\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.22294	−	8.22294i	0.316033	−	0.316033i	−0.531208	−	0.847241i	$-0.678262\pi$
$677$	8.22294	−	8.22294i	0.316033	−	0.316033i	0.847241	+	0.531208i	$0.178262\pi$
$678$		0			0
$679$			10.1976i			0.391349i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−16.4450	+	16.4450i	−0.629250	+	0.629250i	−0.947879	−	0.318629i	$-0.896778\pi$
$683$	−16.4450	+	16.4450i	−0.629250	+	0.629250i	0.318629	+	0.947879i	$0.396778\pi$
$684$		0			0
$685$	4.30964	+	4.30964i	0.164663	+	0.164663i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−39.0996			−1.48958
$690$		0			0
$691$	−15.7291	−	15.7291i	−0.598364	−	0.598364i	0.341513	−	0.939877i	$-0.389061\pi$
$691$	−15.7291	−	15.7291i	−0.598364	−	0.598364i	−0.939877	+	0.341513i	$0.889061\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			15.0582i			0.571189i
$696$		0			0
$697$			0.922567i			0.0349447i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−23.0479	−	23.0479i	−0.870508	−	0.870508i	0.122019	−	0.992528i	$-0.461063\pi$
$701$	−23.0479	−	23.0479i	−0.870508	−	0.870508i	−0.992528	+	0.122019i	$0.961063\pi$
$702$		0			0
$703$	−31.4195			−1.18501
$704$		0			0
$705$		0			0
$706$		0			0
$707$	20.0508	+	20.0508i	0.754088	+	0.754088i
$708$		0			0
$709$	3.02110	−	3.02110i	0.113460	−	0.113460i	−0.648097	−	0.761557i	$-0.724435\pi$
$709$	3.02110	−	3.02110i	0.113460	−	0.113460i	0.761557	+	0.648097i	$0.224435\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			0.605438i			0.0226738i
$714$		0			0
$715$	11.0331	−	11.0331i	0.412616	−	0.412616i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−36.5630			−1.36357			−0.681784	−	0.731554i	$-0.738796\pi$
$719$	−36.5630			−1.36357			−0.681784	+	0.731554i	$0.738796\pi$
$720$		0			0
$721$	18.5720			0.691658
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.04370	−	4.04370i	0.150179	−	0.150179i
$726$		0			0
$727$		−	10.5961i		−	0.392986i	−0.980505	−	0.196493i	$-0.937045\pi$
$727$		−	10.5961i		−	0.392986i	0.980505	−	0.196493i	$-0.0629553\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.88096	+	8.88096i	−0.328474	+	0.328474i
$732$		0			0
$733$	−0.303476	−	0.303476i	−0.0112091	−	0.0112091i	0.701480	−	0.712689i	$-0.252523\pi$
$733$	−0.303476	−	0.303476i	−0.0112091	−	0.0112091i	−0.712689	+	0.701480i	$0.752523\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.30475			−0.0848965
$738$		0			0
$739$	30.6819	+	30.6819i	1.12865	+	1.12865i	0.990397	+	0.138254i	$0.0441491\pi$
$739$	30.6819	+	30.6819i	1.12865	+	1.12865i	0.138254	+	0.990397i	$0.455851\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	51.3124i		−	1.88247i	−0.337755	−	0.941234i	$-0.609667\pi$
$743$		−	51.3124i		−	1.88247i	0.337755	−	0.941234i	$-0.390333\pi$
$744$		0			0
$745$		−	11.5897i		−	0.424614i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.7651	+	18.7651i	0.685663	+	0.685663i
$750$		0			0
$751$	−38.7490			−1.41397			−0.706986	−	0.707228i	$-0.749945\pi$
$751$	−38.7490			−1.41397			−0.706986	+	0.707228i	$0.749945\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.61394	+	2.61394i	0.0951309	+	0.0951309i
$756$		0			0
$757$	7.55414	−	7.55414i	0.274560	−	0.274560i	−0.556373	−	0.830933i	$-0.687807\pi$
$757$	7.55414	−	7.55414i	0.274560	−	0.274560i	0.830933	+	0.556373i	$0.187807\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			43.1563i			1.56441i	0.623019	+	0.782206i	$0.285906\pi$
$761$			43.1563i			1.56441i	−0.623019	+	0.782206i	$0.714094\pi$
$762$		0			0
$763$	10.1949	−	10.1949i	0.369080	−	0.369080i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.93682			0.142150
$768$		0			0
$769$	4.21794			0.152103			0.0760513	−	0.997104i	$-0.475769\pi$
$769$	4.21794			0.152103			0.0760513	+	0.997104i	$0.475769\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−26.7499	+	26.7499i	−0.962127	+	0.962127i	−0.999309	−	0.0371818i	$-0.988162\pi$
$773$	−26.7499	+	26.7499i	−0.962127	+	0.962127i	0.0371818	+	0.999309i	$0.488162\pi$
$774$		0			0
$775$			0.0828313i			0.00297539i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.41596	−	2.41596i	0.0865606	−	0.0865606i
$780$		0			0
$781$	14.3280	+	14.3280i	0.512697	+	0.512697i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.80402			−0.349921
$786$		0			0
$787$	7.19345	+	7.19345i	0.256419	+	0.256419i	0.823596	−	0.567177i	$-0.191965\pi$
$787$	7.19345	+	7.19345i	0.256419	+	0.256419i	−0.567177	+	0.823596i	$0.691965\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			23.2747i			0.827554i
$792$		0			0
$793$		−	53.7256i		−	1.90785i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15.2501	+	15.2501i	0.540185	+	0.540185i	0.923583	−	0.383398i	$-0.125246\pi$
$797$	15.2501	+	15.2501i	0.540185	+	0.540185i	−0.383398	+	0.923583i	$0.625246\pi$
$798$		0			0
$799$	11.5016			0.406898
$800$		0			0
$801$		0			0
$802$		0			0
$803$	24.1275	+	24.1275i	0.851442	+	0.851442i
$804$		0			0
$805$	7.63812	−	7.63812i	0.269208	−	0.269208i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			18.6393i			0.655324i	0.944795	+	0.327662i	$0.106261\pi$
$809$			18.6393i			0.655324i	−0.944795	+	0.327662i	$0.893739\pi$
$810$		0			0
$811$	−4.78280	+	4.78280i	−0.167947	+	0.167947i	−0.786076	−	0.618129i	$-0.787891\pi$
$811$	−4.78280	+	4.78280i	−0.167947	+	0.167947i	0.618129	+	0.786076i	$0.287891\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−13.7425			−0.481378
$816$		0			0
$817$	46.5137			1.62731
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.35011	−	1.35011i	0.0471190	−	0.0471190i	−0.683155	−	0.730274i	$-0.739393\pi$
$821$	1.35011	−	1.35011i	0.0471190	−	0.0471190i	0.730274	+	0.683155i	$0.239393\pi$
$822$		0			0
$823$			24.3508i			0.848816i	0.905471	+	0.424408i	$0.139518\pi$
$823$			24.3508i			0.848816i	−0.905471	+	0.424408i	$0.860482\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.3558	+	36.3558i	−1.26422	+	1.26422i	−0.315186	+	0.949030i	$0.602067\pi$
$827$	−36.3558	+	36.3558i	−1.26422	+	1.26422i	−0.949030	+	0.315186i	$0.897933\pi$
$828$		0			0
$829$	14.7913	+	14.7913i	0.513724	+	0.513724i	0.915665	−	0.401941i	$-0.131664\pi$
$829$	14.7913	+	14.7913i	0.513724	+	0.513724i	−0.401941	+	0.915665i	$0.631664\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.37984			0.221048
$834$		0			0
$835$	−0.106101	−	0.106101i	−0.00367178	−	0.00367178i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			25.9479i			0.895822i	0.894078	+	0.447911i	$0.147832\pi$
$839$			25.9479i			0.895822i	−0.894078	+	0.447911i	$0.852168\pi$
$840$		0			0
$841$			3.70305i			0.127691i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	14.1797	+	14.1797i	0.487796	+	0.487796i
$846$		0			0
$847$	−5.37087			−0.184545
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−33.1002	−	33.1002i	−1.13466	−	1.13466i
$852$		0			0
$853$	−6.15380	+	6.15380i	−0.210702	+	0.210702i	−0.804566	−	0.593864i	$-0.797602\pi$
$853$	−6.15380	+	6.15380i	−0.210702	+	0.210702i	0.593864	+	0.804566i	$0.297602\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	19.1175i		−	0.653040i	−0.945190	−	0.326520i	$-0.894124\pi$
$857$		−	19.1175i		−	0.653040i	0.945190	−	0.326520i	$-0.105876\pi$
$858$		0			0
$859$	−36.0000	+	36.0000i	−1.22830	+	1.22830i	−0.263698	+	0.964605i	$0.584942\pi$
$859$	−36.0000	+	36.0000i	−1.22830	+	1.22830i	−0.964605	+	0.263698i	$0.915058\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.16967			0.141937			0.0709686	−	0.997479i	$-0.477391\pi$
$863$	4.16967			0.141937			0.0709686	+	0.997479i	$0.477391\pi$
$864$		0			0
$865$	−9.58757			−0.325987
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.31806	−	4.31806i	0.146480	−	0.146480i
$870$		0			0
$871$		−	4.88227i		−	0.165430i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.04499	−	1.04499i	0.0353270	−	0.0353270i
$876$		0			0
$877$	13.4372	+	13.4372i	0.453742	+	0.453742i	0.896594	−	0.442853i	$-0.146034\pi$
$877$	13.4372	+	13.4372i	0.453742	+	0.453742i	−0.442853	+	0.896594i	$0.646034\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−12.5561			−0.423027			−0.211514	−	0.977375i	$-0.567839\pi$
$881$	−12.5561			−0.423027			−0.211514	+	0.977375i	$0.567839\pi$
$882$		0			0
$883$	19.5353	+	19.5353i	0.657414	+	0.657414i	0.954767	−	0.297354i	$-0.0961040\pi$
$883$	19.5353	+	19.5353i	0.657414	+	0.657414i	−0.297354	+	0.954767i	$0.596104\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			25.2176i			0.846723i	0.905961	+	0.423361i	$0.139150\pi$
$887$			25.2176i			0.846723i	−0.905961	+	0.423361i	$0.860850\pi$
$888$		0			0
$889$			22.5659i			0.756837i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−30.1197	−	30.1197i	−1.00792	−	1.00792i
$894$		0			0
$895$	−0.114072			−0.00381302
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.334945	−	0.334945i	−0.0111710	−	0.0111710i
$900$		0			0
$901$	−6.37051	+	6.37051i	−0.212232	+	0.212232i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	12.6152i		−	0.419343i
$906$		0			0
$907$	20.5394	−	20.5394i	0.682000	−	0.682000i	−0.278450	−	0.960451i	$-0.589821\pi$
$907$	20.5394	−	20.5394i	0.682000	−	0.682000i	0.960451	+	0.278450i	$0.0898207\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	25.6356			0.849344			0.424672	−	0.905347i	$-0.360389\pi$
$911$	25.6356			0.849344			0.424672	+	0.905347i	$0.360389\pi$
$912$		0			0
$913$	31.7861			1.05197
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.88379	+	1.88379i	−0.0622081	+	0.0622081i
$918$		0			0
$919$			29.5813i			0.975797i	0.872900	+	0.487898i	$0.162236\pi$
$919$			29.5813i			0.975797i	−0.872900	+	0.487898i	$0.837764\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−30.3518	+	30.3518i	−0.999042	+	0.999042i
$924$		0			0
$925$	−4.52851	−	4.52851i	−0.148896	−	0.148896i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	15.3227			0.502722			0.251361	−	0.967893i	$-0.419122\pi$
$929$	15.3227			0.502722			0.251361	+	0.967893i	$0.419122\pi$
$930$		0			0
$931$	−16.7071	−	16.7071i	−0.547553	−	0.547553i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	3.59526i		−	0.117578i
$936$		0			0
$937$		−	17.5250i		−	0.572516i	−0.958153	−	0.286258i	$-0.907589\pi$
$937$		−	17.5250i		−	0.572516i	0.958153	−	0.286258i	$-0.0924115\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	2.76741	+	2.76741i	0.0902150	+	0.0902150i	0.750774	−	0.660559i	$-0.229681\pi$
$941$	2.76741	+	2.76741i	0.0902150	+	0.0902150i	−0.660559	+	0.750774i	$0.729681\pi$
$942$		0			0
$943$	5.09038			0.165766
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−31.4180	−	31.4180i	−1.02095	−	1.02095i	−0.999776	−	0.0211726i	$-0.993260\pi$
$947$	−31.4180	−	31.4180i	−1.02095	−	1.02095i	−0.0211726	−	0.999776i	$-0.506740\pi$
$948$		0			0
$949$	−51.1107	+	51.1107i	−1.65912	+	1.65912i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			17.4802i			0.566238i	0.959085	+	0.283119i	$0.0913692\pi$
$953$			17.4802i			0.566238i	−0.959085	+	0.283119i	$0.908631\pi$
$954$		0			0
$955$	1.13118	−	1.13118i	0.0366040	−	0.0366040i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9.00704			−0.290853
$960$		0			0
$961$	−30.9931			−0.999779
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.15471	−	2.15471i	0.0693626	−	0.0693626i
$966$		0			0
$967$		−	45.8737i		−	1.47520i	−0.675238	−	0.737600i	$-0.735959\pi$
$967$		−	45.8737i		−	1.47520i	0.675238	−	0.737600i	$-0.264041\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	38.2110	−	38.2110i	1.22625	−	1.22625i	0.260877	−	0.965372i	$-0.415988\pi$
$971$	38.2110	−	38.2110i	1.22625	−	1.22625i	0.965372	−	0.260877i	$-0.0840118\pi$
$972$		0			0
$973$	−15.7356	−	15.7356i	−0.504460	−	0.504460i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25.2632			−0.808242			−0.404121	−	0.914706i	$-0.632423\pi$
$977$	−25.2632			−0.808242			−0.404121	+	0.914706i	$0.632423\pi$
$978$		0			0
$979$	−13.1249	−	13.1249i	−0.419475	−	0.419475i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	43.3662i		−	1.38317i	−0.722297	−	0.691583i	$-0.756914\pi$
$983$		−	43.3662i		−	1.38317i	0.722297	−	0.691583i	$-0.243086\pi$
$984$		0			0
$985$			5.50468i			0.175394i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	49.0019	+	49.0019i	1.55817	+	1.55817i
$990$		0			0
$991$	−4.12013			−0.130880			−0.0654402	−	0.997856i	$-0.520845\pi$
$991$	−4.12013			−0.130880			−0.0654402	+	0.997856i	$0.520845\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	15.3116	+	15.3116i	0.485412	+	0.485412i
$996$		0			0
$997$	20.9812	−	20.9812i	0.664483	−	0.664483i	−0.291951	−	0.956433i	$-0.594304\pi$
$997$	20.9812	−	20.9812i	0.664483	−	0.664483i	0.956433	+	0.291951i	$0.0943043\pi$
$998$		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.721.10		32
3.2	odd	2	inner	2880.2.t.e.721.2		32
4.3	odd	2		720.2.t.e.541.9	yes	32
12.11	even	2		720.2.t.e.541.8	yes	32
16.5	even	4	inner	2880.2.t.e.2161.10		32
16.11	odd	4		720.2.t.e.181.9	yes	32
48.5	odd	4	inner	2880.2.t.e.2161.2		32
48.11	even	4		720.2.t.e.181.8	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.t.e.181.8	✓	32	48.11	even	4
720.2.t.e.181.9	yes	32	16.11	odd	4
720.2.t.e.541.8	yes	32	12.11	even	2
720.2.t.e.541.9	yes	32	4.3	odd	2
2880.2.t.e.721.2		32	3.2	odd	2	inner
2880.2.t.e.721.10		32	1.1	even	1	trivial
2880.2.t.e.2161.2		32	48.5	odd	4	inner
2880.2.t.e.2161.10		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)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{5} +1.47784i q^{7} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{5} +1.47784i q^{7} +(1.91908 - 1.91908i) q^{11} +(4.06529 + 4.06529i) q^{13} +1.32472 q^{17} +(-3.46908 - 3.46908i) q^{19} -7.30930i q^{23} -1.00000i q^{25} +(4.04370 + 4.04370i) q^{29} -0.0828313 q^{31} +(1.04499 + 1.04499i) q^{35} +(4.52851 - 4.52851i) q^{37} +0.696426i q^{41} +(-6.70405 + 6.70405i) q^{43} +8.68232 q^{47} +4.81600 q^{49} +(-4.80896 + 4.80896i) q^{53} -2.71399i q^{55} +(0.484200 - 0.484200i) q^{59} +(-6.60785 - 6.60785i) q^{61} +5.74918 q^{65} +(-0.600483 - 0.600483i) q^{67} +7.46609i q^{71} +12.5725i q^{73} +(2.83608 + 2.83608i) q^{77} +2.25007 q^{79} +(8.28161 + 8.28161i) q^{83} +(0.936716 - 0.936716i) q^{85} -6.83920i q^{89} +(-6.00782 + 6.00782i) q^{91} -4.90602 q^{95} +6.90038 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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