LMFDB - Embedded newform 2880.2.t.e.721.15

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			721.15
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.e.2161.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{5} +4.06749i q^{7} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{5} +4.06749i q^{7} +(-1.15766 + 1.15766i) q^{11} +(-2.48659 - 2.48659i) q^{13} +3.58496 q^{17} +(-4.19898 - 4.19898i) q^{19} +4.42705i q^{23} -1.00000i q^{25} +(2.76466 + 2.76466i) q^{29} +10.7364 q^{31} +(2.87615 + 2.87615i) q^{35} +(-4.11129 + 4.11129i) q^{37} +10.9739i q^{41} +(-0.217624 + 0.217624i) q^{43} -7.21909 q^{47} -9.54448 q^{49} +(-4.93200 + 4.93200i) q^{53} +1.63718i q^{55} +(-9.20765 + 9.20765i) q^{59} +(1.03762 + 1.03762i) q^{61} -3.51657 q^{65} +(0.201641 + 0.201641i) q^{67} -6.73032i q^{71} +1.83598i q^{73} +(-4.70878 - 4.70878i) q^{77} +6.37885 q^{79} +(-3.17238 - 3.17238i) q^{83} +(2.53495 - 2.53495i) q^{85} -13.9938i q^{89} +(10.1142 - 10.1142i) q^{91} -5.93826 q^{95} -13.9543 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			4.06749i			1.53737i	0.639629	+	0.768683i	$0.279088\pi$
$7$			4.06749i			1.53737i	−0.639629	+	0.768683i	$0.720912\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.15766	+	1.15766i	−0.349049	+	0.349049i	−0.859755	−	0.510707i	$-0.829384\pi$
$11$	−1.15766	+	1.15766i	−0.349049	+	0.349049i	0.510707	+	0.859755i	$0.329384\pi$
$12$		0			0
$13$	−2.48659	−	2.48659i	−0.689655	−	0.689655i	0.272500	−	0.962156i	$-0.412149\pi$
$13$	−2.48659	−	2.48659i	−0.689655	−	0.689655i	−0.962156	+	0.272500i	$0.912149\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.58496			0.869481			0.434741	−	0.900556i	$-0.356840\pi$
$17$	3.58496			0.869481			0.434741	+	0.900556i	$0.356840\pi$
$18$		0			0
$19$	−4.19898	−	4.19898i	−0.963313	−	0.963313i	0.0360378	−	0.999350i	$-0.488526\pi$
$19$	−4.19898	−	4.19898i	−0.963313	−	0.963313i	−0.999350	+	0.0360378i	$0.988526\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.42705i			0.923104i	0.887113	+	0.461552i	$0.152707\pi$
$23$			4.42705i			0.923104i	−0.887113	+	0.461552i	$0.847293\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.76466	+	2.76466i	0.513385	+	0.513385i	0.915562	−	0.402177i	$-0.131746\pi$
$29$	2.76466	+	2.76466i	0.513385	+	0.513385i	−0.402177	+	0.915562i	$0.631746\pi$
$30$		0			0
$31$	10.7364			1.92831			0.964154	−	0.265342i	$-0.0854848\pi$
$31$	10.7364			1.92831			0.964154	+	0.265342i	$0.0854848\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.87615	+	2.87615i	0.486158	+	0.486158i
$36$		0			0
$37$	−4.11129	+	4.11129i	−0.675893	+	0.675893i	−0.959068	−	0.283175i	$-0.908612\pi$
$37$	−4.11129	+	4.11129i	−0.675893	+	0.675893i	0.283175	+	0.959068i	$0.408612\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.9739i			1.71383i	0.515459	+	0.856914i	$0.327621\pi$
$41$			10.9739i			1.71383i	−0.515459	+	0.856914i	$0.672379\pi$
$42$		0			0
$43$	−0.217624	+	0.217624i	−0.0331874	+	0.0331874i	−0.723506	−	0.690318i	$-0.757470\pi$
$43$	−0.217624	+	0.217624i	−0.0331874	+	0.0331874i	0.690318	+	0.723506i	$0.257470\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.21909			−1.05301			−0.526506	−	0.850171i	$-0.676498\pi$
$47$	−7.21909			−1.05301			−0.526506	+	0.850171i	$0.676498\pi$
$48$		0			0
$49$	−9.54448			−1.36350
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.93200	+	4.93200i	−0.677462	+	0.677462i	−0.959425	−	0.281963i	$-0.909014\pi$
$53$	−4.93200	+	4.93200i	−0.677462	+	0.677462i	0.281963	+	0.959425i	$0.409014\pi$
$54$		0			0
$55$			1.63718i			0.220758i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.20765	+	9.20765i	−1.19873	+	1.19873i	−0.224187	+	0.974546i	$0.571973\pi$
$59$	−9.20765	+	9.20765i	−1.19873	+	1.19873i	−0.974546	+	0.224187i	$0.928027\pi$
$60$		0			0
$61$	1.03762	+	1.03762i	0.132853	+	0.132853i	0.770406	−	0.637553i	$-0.220053\pi$
$61$	1.03762	+	1.03762i	0.132853	+	0.132853i	−0.637553	+	0.770406i	$0.720053\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.51657			−0.436176
$66$		0			0
$67$	0.201641	+	0.201641i	0.0246344	+	0.0246344i	0.719317	−	0.694682i	$-0.244455\pi$
$67$	0.201641	+	0.201641i	0.0246344	+	0.0246344i	−0.694682	+	0.719317i	$0.744455\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6.73032i		−	0.798742i	−0.916789	−	0.399371i	$-0.869228\pi$
$71$		−	6.73032i		−	0.798742i	0.916789	−	0.399371i	$-0.130772\pi$
$72$		0			0
$73$			1.83598i			0.214885i	0.994211	+	0.107442i	$0.0342662\pi$
$73$			1.83598i			0.214885i	−0.994211	+	0.107442i	$0.965734\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.70878	−	4.70878i	−0.536616	−	0.536616i
$78$		0			0
$79$	6.37885			0.717676			0.358838	−	0.933400i	$-0.383173\pi$
$79$	6.37885			0.717676			0.358838	+	0.933400i	$0.383173\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.17238	−	3.17238i	−0.348214	−	0.348214i	0.511230	−	0.859444i	$-0.329190\pi$
$83$	−3.17238	−	3.17238i	−0.348214	−	0.348214i	−0.859444	+	0.511230i	$0.829190\pi$
$84$		0			0
$85$	2.53495	−	2.53495i	0.274954	−	0.274954i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	13.9938i		−	1.48334i	−0.670763	−	0.741672i	$-0.734033\pi$
$89$		−	13.9938i		−	1.48334i	0.670763	−	0.741672i	$-0.265967\pi$
$90$		0			0
$91$	10.1142	−	10.1142i	1.06025	−	1.06025i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.93826			−0.609252
$96$		0			0
$97$	−13.9543			−1.41684			−0.708422	−	0.705790i	$-0.750592\pi$
$97$	−13.9543			−1.41684			−0.708422	+	0.705790i	$0.750592\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.91660	+	7.91660i	−0.787731	+	0.787731i	−0.981122	−	0.193391i	$-0.938052\pi$
$101$	−7.91660	+	7.91660i	−0.787731	+	0.787731i	0.193391	+	0.981122i	$0.438052\pi$
$102$		0			0
$103$			12.9114i			1.27220i	0.771606	+	0.636101i	$0.219454\pi$
$103$			12.9114i			1.27220i	−0.771606	+	0.636101i	$0.780546\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.55299	+	3.55299i	−0.343480	+	0.343480i	−0.857674	−	0.514194i	$-0.828091\pi$
$107$	−3.55299	+	3.55299i	−0.343480	+	0.343480i	0.514194	+	0.857674i	$0.328091\pi$
$108$		0			0
$109$	−6.14657	−	6.14657i	−0.588735	−	0.588735i	0.348554	−	0.937289i	$-0.386673\pi$
$109$	−6.14657	−	6.14657i	−0.588735	−	0.588735i	−0.937289	+	0.348554i	$0.886673\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.28166			−0.685001			−0.342500	−	0.939518i	$-0.611274\pi$
$113$	−7.28166			−0.685001			−0.342500	+	0.939518i	$0.611274\pi$
$114$		0			0
$115$	3.13040	+	3.13040i	0.291911	+	0.291911i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			14.5818i			1.33671i
$120$		0			0
$121$			8.31963i			0.756330i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.0632456	−	0.0632456i
$126$		0			0
$127$	1.36335			0.120978			0.0604890	−	0.998169i	$-0.480734\pi$
$127$	1.36335			0.120978			0.0604890	+	0.998169i	$0.480734\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.39120	+	3.39120i	0.296290	+	0.296290i	0.839559	−	0.543269i	$-0.182813\pi$
$131$	3.39120	+	3.39120i	0.296290	+	0.296290i	−0.543269	+	0.839559i	$0.682813\pi$
$132$		0			0
$133$	17.0793	−	17.0793i	1.48096	−	1.48096i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			20.9080i			1.78629i	0.449765	+	0.893147i	$0.351508\pi$
$137$			20.9080i			1.78629i	−0.449765	+	0.893147i	$0.648492\pi$
$138$		0			0
$139$	−2.85353	+	2.85353i	−0.242033	+	0.242033i	−0.817691	−	0.575658i	$-0.804746\pi$
$139$	−2.85353	+	2.85353i	−0.242033	+	0.242033i	0.575658	+	0.817691i	$0.304746\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.75726			0.481446
$144$		0			0
$145$	3.90982			0.324693
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−15.6683	+	15.6683i	−1.28359	+	1.28359i	−0.344988	+	0.938607i	$0.612117\pi$
$149$	−15.6683	+	15.6683i	−1.28359	+	1.28359i	−0.938607	+	0.344988i	$0.887883\pi$
$150$		0			0
$151$		−	2.89464i		−	0.235563i	−0.993040	−	0.117781i	$-0.962422\pi$
$151$		−	2.89464i		−	0.235563i	0.993040	−	0.117781i	$-0.0375782\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.59176	−	7.59176i	0.609785	−	0.609785i
$156$		0			0
$157$	6.50220	+	6.50220i	0.518932	+	0.518932i	0.917248	−	0.398316i	$-0.130405\pi$
$157$	6.50220	+	6.50220i	0.518932	+	0.518932i	−0.398316	+	0.917248i	$0.630405\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−18.0070			−1.41915
$162$		0			0
$163$	−8.26099	−	8.26099i	−0.647051	−	0.647051i	0.305228	−	0.952279i	$-0.401267\pi$
$163$	−8.26099	−	8.26099i	−0.647051	−	0.647051i	−0.952279	+	0.305228i	$0.901267\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			14.9706i			1.15846i	0.815165	+	0.579229i	$0.196646\pi$
$167$			14.9706i			1.15846i	−0.815165	+	0.579229i	$0.803354\pi$
$168$		0			0
$169$		−	0.633769i		−	0.0487514i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.0898	+	11.0898i	0.843145	+	0.843145i	0.989267	−	0.146122i	$-0.0466792\pi$
$173$	11.0898	+	11.0898i	0.843145	+	0.843145i	−0.146122	+	0.989267i	$0.546679\pi$
$174$		0			0
$175$	4.06749			0.307473
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.84174	−	1.84174i	−0.137658	−	0.137658i	0.634920	−	0.772578i	$-0.281033\pi$
$179$	−1.84174	−	1.84174i	−0.137658	−	0.137658i	−0.772578	+	0.634920i	$0.781033\pi$
$180$		0			0
$181$	11.0086	−	11.0086i	0.818261	−	0.818261i	−0.167595	−	0.985856i	$-0.553600\pi$
$181$	11.0086	−	11.0086i	0.818261	−	0.818261i	0.985856	+	0.167595i	$0.0536001\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			5.81425i			0.427472i
$186$		0			0
$187$	−4.15018	+	4.15018i	−0.303491	+	0.303491i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	23.2648			1.68338			0.841691	−	0.539960i	$-0.181561\pi$
$191$	23.2648			1.68338			0.841691	+	0.539960i	$0.181561\pi$
$192$		0			0
$193$	−6.30211			−0.453636			−0.226818	−	0.973937i	$-0.572832\pi$
$193$	−6.30211			−0.453636			−0.226818	+	0.973937i	$0.572832\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.2016	−	17.2016i	1.22557	−	1.22557i	0.259943	−	0.965624i	$-0.416296\pi$
$197$	17.2016	−	17.2016i	1.22557	−	1.22557i	0.965624	−	0.259943i	$-0.0837036\pi$
$198$		0			0
$199$		−	7.91932i		−	0.561386i	−0.959798	−	0.280693i	$-0.909436\pi$
$199$		−	7.91932i		−	0.561386i	0.959798	−	0.280693i	$-0.0905643\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−11.2452	+	11.2452i	−0.789261	+	0.789261i
$204$		0			0
$205$	7.75969	+	7.75969i	0.541960	+	0.541960i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.72201			0.672486
$210$		0			0
$211$	−11.0637	−	11.0637i	−0.761659	−	0.761659i	0.214963	−	0.976622i	$-0.431037\pi$
$211$	−11.0637	−	11.0637i	−0.761659	−	0.761659i	−0.976622	+	0.214963i	$0.931037\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			0.307768i			0.0209896i
$216$		0			0
$217$			43.6701i			2.96452i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.91432	−	8.91432i	−0.599642	−	0.599642i
$222$		0			0
$223$	−3.36837			−0.225563			−0.112781	−	0.993620i	$-0.535976\pi$
$223$	−3.36837			−0.225563			−0.112781	+	0.993620i	$0.535976\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.3341	+	15.3341i	1.01776	+	1.01776i	0.999839	+	0.0179176i	$0.00570366\pi$
$227$	15.3341	+	15.3341i	1.01776	+	1.01776i	0.0179176	+	0.999839i	$0.494296\pi$
$228$		0			0
$229$	19.6656	−	19.6656i	1.29954	−	1.29954i	0.370845	−	0.928695i	$-0.379068\pi$
$229$	19.6656	−	19.6656i	1.29954	−	1.29954i	0.928695	−	0.370845i	$-0.120932\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			12.2611i			0.803254i	0.915803	+	0.401627i	$0.131555\pi$
$233$			12.2611i			0.803254i	−0.915803	+	0.401627i	$0.868445\pi$
$234$		0			0
$235$	−5.10467	+	5.10467i	−0.332992	+	0.332992i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.92121			−0.447696			−0.223848	−	0.974624i	$-0.571862\pi$
$239$	−6.92121			−0.447696			−0.223848	+	0.974624i	$0.571862\pi$
$240$		0			0
$241$	−24.1877			−1.55807			−0.779034	−	0.626982i	$-0.784290\pi$
$241$	−24.1877			−1.55807			−0.779034	+	0.626982i	$0.784290\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.74896	+	6.74896i	−0.431175	+	0.431175i
$246$		0			0
$247$			20.8823i			1.32871i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.15690	−	4.15690i	0.262381	−	0.262381i	−0.563640	−	0.826021i	$-0.690599\pi$
$251$	4.15690	−	4.15690i	0.262381	−	0.262381i	0.826021	+	0.563640i	$0.190599\pi$
$252$		0			0
$253$	−5.12503	−	5.12503i	−0.322208	−	0.322208i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.81626			−0.175674			−0.0878368	−	0.996135i	$-0.527995\pi$
$257$	−2.81626			−0.175674			−0.0878368	+	0.996135i	$0.527995\pi$
$258$		0			0
$259$	−16.7226	−	16.7226i	−1.03909	−	1.03909i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	3.20397i		−	0.197565i	−0.995109	−	0.0987827i	$-0.968505\pi$
$263$		−	3.20397i		−	0.197565i	0.995109	−	0.0987827i	$-0.0314949\pi$
$264$		0			0
$265$			6.97490i			0.428465i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	17.6924	+	17.6924i	1.07873	+	1.07873i	0.996624	+	0.0821039i	$0.0261639\pi$
$269$	17.6924	+	17.6924i	1.07873	+	1.07873i	0.0821039	+	0.996624i	$0.473836\pi$
$270$		0			0
$271$	−6.38167			−0.387659			−0.193829	−	0.981035i	$-0.562091\pi$
$271$	−6.38167			−0.387659			−0.193829	+	0.981035i	$0.562091\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.15766	+	1.15766i	0.0698097	+	0.0698097i
$276$		0			0
$277$	9.11390	−	9.11390i	0.547601	−	0.547601i	−0.378145	−	0.925746i	$-0.623438\pi$
$277$	9.11390	−	9.11390i	0.547601	−	0.547601i	0.925746	+	0.378145i	$0.123438\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3.33476i			0.198935i	0.995041	+	0.0994676i	$0.0317140\pi$
$281$			3.33476i			0.198935i	−0.995041	+	0.0994676i	$0.968286\pi$
$282$		0			0
$283$	−10.2900	+	10.2900i	−0.611676	+	0.611676i	−0.943383	−	0.331706i	$-0.892376\pi$
$283$	−10.2900	+	10.2900i	−0.611676	+	0.611676i	0.331706	+	0.943383i	$0.392376\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−44.6360			−2.63478
$288$		0			0
$289$	−4.14804			−0.244002
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.9244	−	12.9244i	0.755053	−	0.755053i	−0.220365	−	0.975418i	$-0.570725\pi$
$293$	12.9244	−	12.9244i	0.755053	−	0.755053i	0.975418	+	0.220365i	$0.0707247\pi$
$294$		0			0
$295$			13.0216i			0.758146i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.0082	−	11.0082i	0.636623	−	0.636623i
$300$		0			0
$301$	−0.885185	−	0.885185i	−0.0510212	−	0.0510212i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.46741			0.0840236
$306$		0			0
$307$	−3.25732	−	3.25732i	−0.185905	−	0.185905i	0.608018	−	0.793923i	$-0.291965\pi$
$307$	−3.25732	−	3.25732i	−0.185905	−	0.185905i	−0.793923	+	0.608018i	$0.791965\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			12.4952i			0.708540i	0.935143	+	0.354270i	$0.115271\pi$
$311$			12.4952i			0.708540i	−0.935143	+	0.354270i	$0.884729\pi$
$312$		0			0
$313$		−	22.7473i		−	1.28575i	−0.765970	−	0.642876i	$-0.777741\pi$
$313$		−	22.7473i		−	1.28575i	0.765970	−	0.642876i	$-0.222259\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.13539	−	3.13539i	−0.176101	−	0.176101i	0.613553	−	0.789654i	$-0.289740\pi$
$317$	−3.13539	−	3.13539i	−0.176101	−	0.176101i	−0.789654	+	0.613553i	$0.789740\pi$
$318$		0			0
$319$	−6.40109			−0.358392
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−15.0532	−	15.0532i	−0.837582	−	0.837582i
$324$		0			0
$325$	−2.48659	+	2.48659i	−0.137931	+	0.137931i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	29.3636i		−	1.61887i
$330$		0			0
$331$	−6.10294	+	6.10294i	−0.335448	+	0.335448i	−0.854651	−	0.519203i	$-0.826229\pi$
$331$	−6.10294	+	6.10294i	−0.335448	+	0.335448i	0.519203	+	0.854651i	$0.326229\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.285164			0.0155802
$336$		0			0
$337$	26.2215			1.42837			0.714187	−	0.699955i	$-0.246796\pi$
$337$	26.2215			1.42837			0.714187	+	0.699955i	$0.246796\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−12.4291	+	12.4291i	−0.673073	+	0.673073i
$342$		0			0
$343$		−	10.3496i		−	0.558828i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.96386	+	8.96386i	−0.481205	+	0.481205i	−0.905516	−	0.424311i	$-0.860516\pi$
$347$	−8.96386	+	8.96386i	−0.481205	+	0.481205i	0.424311	+	0.905516i	$0.360516\pi$
$348$		0			0
$349$	8.24296	+	8.24296i	0.441236	+	0.441236i	0.892427	−	0.451191i	$-0.149001\pi$
$349$	8.24296	+	8.24296i	0.441236	+	0.441236i	−0.451191	+	0.892427i	$0.649001\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.8638			0.684670			0.342335	−	0.939578i	$-0.388782\pi$
$353$	12.8638			0.684670			0.342335	+	0.939578i	$0.388782\pi$
$354$		0			0
$355$	−4.75906	−	4.75906i	−0.252585	−	0.252585i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	21.6213i		−	1.14113i	−0.821254	−	0.570563i	$-0.806725\pi$
$359$		−	21.6213i		−	1.14113i	0.821254	−	0.570563i	$-0.193275\pi$
$360$		0			0
$361$			16.2629i			0.855943i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.29823	+	1.29823i	0.0679526	+	0.0679526i
$366$		0			0
$367$	35.8418			1.87092			0.935462	−	0.353427i	$-0.114984\pi$
$367$	35.8418			1.87092			0.935462	+	0.353427i	$0.114984\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−20.0609	−	20.0609i	−1.04151	−	1.04151i
$372$		0			0
$373$	−2.63845	+	2.63845i	−0.136614	+	0.136614i	−0.772107	−	0.635493i	$-0.780797\pi$
$373$	−2.63845	+	2.63845i	−0.136614	+	0.136614i	0.635493	+	0.772107i	$0.280797\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	13.7491i		−	0.708117i
$378$		0			0
$379$	24.9844	−	24.9844i	1.28336	−	1.28336i	0.344624	−	0.938741i	$-0.388006\pi$
$379$	24.9844	−	24.9844i	1.28336	−	1.28336i	0.938741	−	0.344624i	$-0.111994\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	19.8329			1.01341			0.506706	−	0.862119i	$-0.330863\pi$
$383$	19.8329			1.01341			0.506706	+	0.862119i	$0.330863\pi$
$384$		0			0
$385$	−6.65923			−0.339386
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.01597	−	7.01597i	0.355724	−	0.355724i	−0.506510	−	0.862234i	$-0.669065\pi$
$389$	7.01597	−	7.01597i	0.355724	−	0.355724i	0.862234	+	0.506510i	$0.169065\pi$
$390$		0			0
$391$			15.8708i			0.802622i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.51052	−	4.51052i	0.226949	−	0.226949i
$396$		0			0
$397$	18.4893	+	18.4893i	0.927951	+	0.927951i	0.997573	−	0.0696220i	$-0.0221793\pi$
$397$	18.4893	+	18.4893i	0.927951	+	0.927951i	−0.0696220	+	0.997573i	$0.522179\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.74098			0.236753			0.118377	−	0.992969i	$-0.462231\pi$
$401$	4.74098			0.236753			0.118377	+	0.992969i	$0.462231\pi$
$402$		0			0
$403$	−26.6969	−	26.6969i	−1.32987	−	1.32987i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	9.51899i		−	0.471839i
$408$		0			0
$409$		−	21.4939i		−	1.06281i	−0.847119	−	0.531403i	$-0.821665\pi$
$409$		−	21.4939i		−	1.06281i	0.847119	−	0.531403i	$-0.178335\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−37.4520	−	37.4520i	−1.84289	−	1.84289i
$414$		0			0
$415$	−4.48642			−0.220230
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9.55376	+	9.55376i	0.466732	+	0.466732i	0.900854	−	0.434122i	$-0.142941\pi$
$419$	9.55376	+	9.55376i	0.466732	+	0.466732i	−0.434122	+	0.900854i	$0.642941\pi$
$420$		0			0
$421$	−16.8731	+	16.8731i	−0.822346	+	0.822346i	−0.986444	−	0.164098i	$-0.947529\pi$
$421$	−16.8731	+	16.8731i	−0.822346	+	0.822346i	0.164098	+	0.986444i	$0.447529\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	3.58496i		−	0.173896i
$426$		0			0
$427$	−4.22049	+	4.22049i	−0.204244	+	0.204244i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	15.9103			0.766374			0.383187	−	0.923671i	$-0.374827\pi$
$431$	15.9103			0.766374			0.383187	+	0.923671i	$0.374827\pi$
$432$		0			0
$433$	5.49432			0.264040			0.132020	−	0.991247i	$-0.457854\pi$
$433$	5.49432			0.264040			0.132020	+	0.991247i	$0.457854\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	18.5891	−	18.5891i	0.889238	−	0.889238i
$438$		0			0
$439$			2.88419i			0.137655i	0.997629	+	0.0688274i	$0.0219258\pi$
$439$			2.88419i			0.137655i	−0.997629	+	0.0688274i	$0.978074\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−20.9109	+	20.9109i	−0.993509	+	0.993509i	−0.999979	−	0.00646993i	$-0.997941\pi$
$443$	−20.9109	+	20.9109i	−0.993509	+	0.993509i	0.00646993	+	0.999979i	$0.497941\pi$
$444$		0			0
$445$	−9.89513	−	9.89513i	−0.469074	−	0.469074i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13.7807			−0.650351			−0.325175	−	0.945654i	$-0.605423\pi$
$449$	−13.7807			−0.650351			−0.325175	+	0.945654i	$0.605423\pi$
$450$		0			0
$451$	−12.7040	−	12.7040i	−0.598209	−	0.598209i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	14.3036i		−	0.670563i
$456$		0			0
$457$		−	8.34960i		−	0.390578i	−0.980746	−	0.195289i	$-0.937436\pi$
$457$		−	8.34960i		−	0.390578i	0.980746	−	0.195289i	$-0.0625644\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	11.9059	+	11.9059i	0.554513	+	0.554513i	0.927740	−	0.373227i	$-0.121749\pi$
$461$	11.9059	+	11.9059i	0.554513	+	0.554513i	−0.373227	+	0.927740i	$0.621749\pi$
$462$		0			0
$463$	5.77643			0.268453			0.134227	−	0.990951i	$-0.457145\pi$
$463$	5.77643			0.268453			0.134227	+	0.990951i	$0.457145\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.0096	−	14.0096i	−0.648289	−	0.648289i	0.304290	−	0.952579i	$-0.401581\pi$
$467$	−14.0096	−	14.0096i	−0.648289	−	0.648289i	−0.952579	+	0.304290i	$0.901581\pi$
$468$		0			0
$469$	−0.820174	+	0.820174i	−0.0378721	+	0.0378721i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	0.503872i		−	0.0231680i
$474$		0			0
$475$	−4.19898	+	4.19898i	−0.192663	+	0.192663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−36.6587			−1.67498			−0.837489	−	0.546454i	$-0.815977\pi$
$479$	−36.6587			−1.67498			−0.837489	+	0.546454i	$0.815977\pi$
$480$		0			0
$481$	20.4462			0.932266
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.86717	+	9.86717i	−0.448045	+	0.448045i
$486$		0			0
$487$			38.6546i			1.75160i	0.482670	+	0.875802i	$0.339667\pi$
$487$			38.6546i			1.75160i	−0.482670	+	0.875802i	$0.660333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	23.0511	−	23.0511i	1.04028	−	1.04028i	0.0411259	−	0.999154i	$-0.486906\pi$
$491$	23.0511	−	23.0511i	1.04028	−	1.04028i	0.999154	−	0.0411259i	$-0.0130945\pi$
$492$		0			0
$493$	9.91121	+	9.91121i	0.446378	+	0.446378i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	27.3755			1.22796
$498$		0			0
$499$	−18.0132	−	18.0132i	−0.806380	−	0.806380i	0.177704	−	0.984084i	$-0.443133\pi$
$499$	−18.0132	−	18.0132i	−0.806380	−	0.806380i	−0.984084	+	0.177704i	$0.943133\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			18.3208i			0.816885i	0.912784	+	0.408443i	$0.133928\pi$
$503$			18.3208i			0.816885i	−0.912784	+	0.408443i	$0.866072\pi$
$504$		0			0
$505$			11.1958i			0.498205i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.7423	+	13.7423i	0.609115	+	0.609115i	0.942715	−	0.333600i	$-0.108263\pi$
$509$	13.7423	+	13.7423i	0.609115	+	0.609115i	−0.333600	+	0.942715i	$0.608263\pi$
$510$		0			0
$511$	−7.46782			−0.330357
$512$		0			0
$513$		0			0
$514$		0			0
$515$	9.12976	+	9.12976i	0.402305	+	0.402305i
$516$		0			0
$517$	8.35727	−	8.35727i	0.367552	−	0.367552i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			5.23969i			0.229555i	0.993391	+	0.114778i	$0.0366155\pi$
$521$			5.23969i			0.229555i	−0.993391	+	0.114778i	$0.963384\pi$
$522$		0			0
$523$	22.8819	−	22.8819i	1.00056	−	1.00056i	0.000555243	−	1.00000i	$-0.499823\pi$
$523$	22.8819	−	22.8819i	1.00056	−	1.00056i	1.00000		0.000555243i	$-0.000176739\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	38.4895			1.67663
$528$		0			0
$529$	3.40122			0.147879
$530$		0			0
$531$		0			0
$532$		0			0
$533$	27.2874	−	27.2874i	1.18195	−	1.18195i
$534$		0			0
$535$			5.02468i			0.217236i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	11.0493	−	11.0493i	0.475927	−	0.475927i
$540$		0			0
$541$	−9.69583	−	9.69583i	−0.416856	−	0.416856i	0.467262	−	0.884119i	$-0.345240\pi$
$541$	−9.69583	−	9.69583i	−0.416856	−	0.416856i	−0.884119	+	0.467262i	$0.845240\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.69257			−0.372349
$546$		0			0
$547$	27.5330	+	27.5330i	1.17722	+	1.17722i	0.980449	+	0.196776i	$0.0630471\pi$
$547$	27.5330	+	27.5330i	1.17722	+	1.17722i	0.196776	+	0.980449i	$0.436953\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	23.2175i		−	0.989100i
$552$		0			0
$553$			25.9459i			1.10333i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.47916	+	9.47916i	0.401645	+	0.401645i	0.878812	−	0.477168i	$-0.158337\pi$
$557$	9.47916	+	9.47916i	0.401645	+	0.401645i	−0.477168	+	0.878812i	$0.658337\pi$
$558$		0			0
$559$	1.08228			0.0457758
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.0130303	−	0.0130303i	−0.000549161	−	0.000549161i	0.706832	−	0.707381i	$-0.250124\pi$
$563$	−0.0130303	−	0.0130303i	−0.000549161	−	0.000549161i	−0.707381	+	0.706832i	$0.750124\pi$
$564$		0			0
$565$	−5.14891	+	5.14891i	−0.216616	+	0.216616i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	10.2013i		−	0.427661i	−0.976871	−	0.213831i	$-0.931406\pi$
$569$		−	10.2013i		−	0.427661i	0.976871	−	0.213831i	$-0.0685941\pi$
$570$		0			0
$571$	−24.1385	+	24.1385i	−1.01017	+	1.01017i	−0.0102174	+	0.999948i	$0.503252\pi$
$571$	−24.1385	+	24.1385i	−1.01017	+	1.01017i	−0.999948	+	0.0102174i	$0.996748\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.42705			0.184621
$576$		0			0
$577$	−3.96268			−0.164969			−0.0824843	−	0.996592i	$-0.526285\pi$
$577$	−3.96268			−0.164969			−0.0824843	+	0.996592i	$0.526285\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.9036	−	12.9036i	0.535332	−	0.535332i
$582$		0			0
$583$		−	11.4192i		−	0.472934i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.1865	−	13.1865i	0.544267	−	0.544267i	−0.380510	−	0.924777i	$-0.624252\pi$
$587$	13.1865	−	13.1865i	0.544267	−	0.544267i	0.924777	+	0.380510i	$0.124252\pi$
$588$		0			0
$589$	−45.0818	−	45.0818i	−1.85756	−	1.85756i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	21.3086			0.875041			0.437520	−	0.899208i	$-0.355857\pi$
$593$	21.3086			0.875041			0.437520	+	0.899208i	$0.355857\pi$
$594$		0			0
$595$	10.3109	+	10.3109i	0.422705	+	0.422705i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	18.9806i		−	0.775528i	−0.921759	−	0.387764i	$-0.873248\pi$
$599$		−	18.9806i		−	0.775528i	0.921759	−	0.387764i	$-0.126752\pi$
$600$		0			0
$601$		−	35.0852i		−	1.43116i	−0.698533	−	0.715578i	$-0.746164\pi$
$601$		−	35.0852i		−	1.43116i	0.698533	−	0.715578i	$-0.253836\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.88287	+	5.88287i	0.239173	+	0.239173i
$606$		0			0
$607$	6.91269			0.280578			0.140289	−	0.990111i	$-0.455197\pi$
$607$	6.91269			0.280578			0.140289	+	0.990111i	$0.455197\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	17.9509	+	17.9509i	0.726215	+	0.726215i
$612$		0			0
$613$	7.96288	−	7.96288i	0.321618	−	0.321618i	−0.527770	−	0.849388i	$-0.676972\pi$
$613$	7.96288	−	7.96288i	0.321618	−	0.321618i	0.849388	+	0.527770i	$0.176972\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	28.8070i		−	1.15973i	−0.814714	−	0.579863i	$-0.803106\pi$
$617$		−	28.8070i		−	1.15973i	0.814714	−	0.579863i	$-0.196894\pi$
$618$		0			0
$619$	−0.195139	+	0.195139i	−0.00784331	+	0.00784331i	−0.711018	−	0.703174i	$-0.751765\pi$
$619$	−0.195139	+	0.195139i	−0.00784331	+	0.00784331i	0.703174	+	0.711018i	$0.251765\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	56.9198			2.28044
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−14.7388	+	14.7388i	−0.587676	+	0.587676i
$630$		0			0
$631$		−	5.50835i		−	0.219284i	−0.993971	−	0.109642i	$-0.965030\pi$
$631$		−	5.50835i		−	0.219284i	0.993971	−	0.109642i	$-0.0349704\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.964037	−	0.964037i	0.0382566	−	0.0382566i
$636$		0			0
$637$	23.7332	+	23.7332i	0.940342	+	0.940342i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.90493			−0.0752402			−0.0376201	−	0.999292i	$-0.511978\pi$
$641$	−1.90493			−0.0752402			−0.0376201	+	0.999292i	$0.511978\pi$
$642$		0			0
$643$	29.7470	+	29.7470i	1.17311	+	1.17311i	0.981465	+	0.191640i	$0.0613806\pi$
$643$	29.7470	+	29.7470i	1.17311	+	1.17311i	0.191640	+	0.981465i	$0.438619\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	10.0394i		−	0.394691i	−0.980334	−	0.197345i	$-0.936768\pi$
$647$		−	10.0394i		−	0.394691i	0.980334	−	0.197345i	$-0.0632320\pi$
$648$		0			0
$649$		−	21.3187i		−	0.836832i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−24.5692	−	24.5692i	−0.961468	−	0.961468i	0.0378171	−	0.999285i	$-0.487960\pi$
$653$	−24.5692	−	24.5692i	−0.961468	−	0.961468i	−0.999285	+	0.0378171i	$0.987960\pi$
$654$		0			0
$655$	4.79588			0.187390
$656$		0			0
$657$		0			0
$658$		0			0
$659$	31.0086	+	31.0086i	1.20792	+	1.20792i	0.971698	+	0.236225i	$0.0759104\pi$
$659$	31.0086	+	31.0086i	1.20792	+	1.20792i	0.236225	+	0.971698i	$0.424090\pi$
$660$		0			0
$661$	26.3563	−	26.3563i	1.02514	−	1.02514i	0.0254663	−	0.999676i	$-0.491893\pi$
$661$	26.3563	−	26.3563i	1.02514	−	1.02514i	0.999676	−	0.0254663i	$-0.00810705\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	24.1538i		−	0.936644i
$666$		0			0
$667$	−12.2393	+	12.2393i	−0.473907	+	0.473907i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.40242			−0.0927443
$672$		0			0
$673$	−40.3923			−1.55701			−0.778504	−	0.627640i	$-0.784021\pi$
$673$	−40.3923			−1.55701			−0.778504	+	0.627640i	$0.784021\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−19.6158	+	19.6158i	−0.753895	+	0.753895i	−0.975204	−	0.221309i	$-0.928967\pi$
$677$	−19.6158	+	19.6158i	−0.753895	+	0.753895i	0.221309	+	0.975204i	$0.428967\pi$
$678$		0			0
$679$		−	56.7589i		−	2.17821i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	17.8339	−	17.8339i	0.682394	−	0.682394i	−0.278145	−	0.960539i	$-0.589720\pi$
$683$	17.8339	−	17.8339i	0.682394	−	0.682394i	0.960539	+	0.278145i	$0.0897196\pi$
$684$		0			0
$685$	14.7842	+	14.7842i	0.564876	+	0.564876i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	24.5277			0.934430
$690$		0			0
$691$	4.01123	+	4.01123i	0.152594	+	0.152594i	0.779276	−	0.626681i	$-0.215587\pi$
$691$	4.01123	+	4.01123i	0.152594	+	0.152594i	−0.626681	+	0.779276i	$0.715587\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			4.03549i			0.153075i
$696$		0			0
$697$			39.3409i			1.49014i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−19.5604	−	19.5604i	−0.738787	−	0.738787i	0.233556	−	0.972343i	$-0.424964\pi$
$701$	−19.5604	−	19.5604i	−0.738787	−	0.738787i	−0.972343	+	0.233556i	$0.924964\pi$
$702$		0			0
$703$	34.5265			1.30219
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−32.2007	−	32.2007i	−1.21103	−	1.21103i
$708$		0			0
$709$	8.97994	−	8.97994i	0.337249	−	0.337249i	−0.518082	−	0.855331i	$-0.673354\pi$
$709$	8.97994	−	8.97994i	0.337249	−	0.337249i	0.855331	+	0.518082i	$0.173354\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			47.5305i			1.78003i
$714$		0			0
$715$	4.07100	−	4.07100i	0.152247	−	0.152247i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−35.6493			−1.32950			−0.664748	−	0.747068i	$-0.731461\pi$
$719$	−35.6493			−1.32950			−0.664748	+	0.747068i	$0.731461\pi$
$720$		0			0
$721$	−52.5171			−1.95584
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.76466	−	2.76466i	0.102677	−	0.102677i
$726$		0			0
$727$			2.56198i			0.0950188i	0.998871	+	0.0475094i	$0.0151284\pi$
$727$			2.56198i			0.0950188i	−0.998871	+	0.0475094i	$0.984872\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.780176	+	0.780176i	−0.0288558	+	0.0288558i
$732$		0			0
$733$	−17.0422	−	17.0422i	−0.629468	−	0.629468i	0.318466	−	0.947934i	$-0.396832\pi$
$733$	−17.0422	−	17.0422i	−0.629468	−	0.629468i	−0.947934	+	0.318466i	$0.896832\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.466865			−0.0171972
$738$		0			0
$739$	3.77826	+	3.77826i	0.138986	+	0.138986i	0.773176	−	0.634191i	$-0.218667\pi$
$739$	3.77826	+	3.77826i	0.138986	+	0.138986i	−0.634191	+	0.773176i	$0.718667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	48.5423i		−	1.78084i	−0.455135	−	0.890422i	$-0.650409\pi$
$743$		−	48.5423i		−	1.78084i	0.455135	−	0.890422i	$-0.349591\pi$
$744$		0			0
$745$			22.1583i			0.811817i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−14.4517	−	14.4517i	−0.528055	−	0.528055i
$750$		0			0
$751$	8.62147			0.314602			0.157301	−	0.987551i	$-0.449721\pi$
$751$	8.62147			0.314602			0.157301	+	0.987551i	$0.449721\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.04682	−	2.04682i	−0.0744915	−	0.0744915i
$756$		0			0
$757$	−36.1396	+	36.1396i	−1.31351	+	1.31351i	−0.394708	+	0.918807i	$0.629154\pi$
$757$	−36.1396	+	36.1396i	−1.31351	+	1.31351i	−0.918807	+	0.394708i	$0.870846\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			17.6198i			0.638716i	0.947634	+	0.319358i	$0.103467\pi$
$761$			17.6198i			0.638716i	−0.947634	+	0.319358i	$0.896533\pi$
$762$		0			0
$763$	25.0011	−	25.0011i	0.905101	−	0.905101i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	45.7912			1.65343
$768$		0			0
$769$	−13.3510			−0.481451			−0.240725	−	0.970593i	$-0.577385\pi$
$769$	−13.3510			−0.481451			−0.240725	+	0.970593i	$0.577385\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.54275	−	5.54275i	0.199359	−	0.199359i	−0.600366	−	0.799725i	$-0.704979\pi$
$773$	5.54275	−	5.54275i	0.199359	−	0.199359i	0.799725	+	0.600366i	$0.204979\pi$
$774$		0			0
$775$		−	10.7364i		−	0.385662i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	46.0790	−	46.0790i	1.65095	−	1.65095i
$780$		0			0
$781$	7.79145	+	7.79145i	0.278800	+	0.278800i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	9.19550			0.328201
$786$		0			0
$787$	35.0086	+	35.0086i	1.24792	+	1.24792i	0.956636	+	0.291285i	$0.0940828\pi$
$787$	35.0086	+	35.0086i	1.24792	+	1.24792i	0.291285	+	0.956636i	$0.405917\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	29.6181i		−	1.05310i
$792$		0			0
$793$		−	5.16024i		−	0.183246i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.36356	+	5.36356i	0.189987	+	0.189987i	0.795690	−	0.605703i	$-0.207108\pi$
$797$	5.36356	+	5.36356i	0.189987	+	0.189987i	−0.605703	+	0.795690i	$0.707108\pi$
$798$		0			0
$799$	−25.8802			−0.915574
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.12544	−	2.12544i	−0.0750053	−	0.0750053i
$804$		0			0
$805$	−12.7329	+	12.7329i	−0.448774	+	0.448774i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	47.6926i		−	1.67678i	−0.545068	−	0.838392i	$-0.683496\pi$
$809$		−	47.6926i		−	1.67678i	0.545068	−	0.838392i	$-0.316504\pi$
$810$		0			0
$811$	8.62830	−	8.62830i	0.302980	−	0.302980i	−0.539198	−	0.842179i	$-0.681273\pi$
$811$	8.62830	−	8.62830i	0.302980	−	0.302980i	0.842179	+	0.539198i	$0.181273\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−11.6828			−0.409231
$816$		0			0
$817$	1.82760			0.0639397
$818$		0			0
$819$		0			0
$820$		0			0
$821$	35.0291	−	35.0291i	1.22253	−	1.22253i	0.255794	−	0.966731i	$-0.417663\pi$
$821$	35.0291	−	35.0291i	1.22253	−	1.22253i	0.966731	−	0.255794i	$-0.0823370\pi$
$822$		0			0
$823$		−	35.4922i		−	1.23718i	−0.785714	−	0.618591i	$-0.787704\pi$
$823$		−	35.4922i		−	1.23718i	0.785714	−	0.618591i	$-0.212296\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.2765	+	31.2765i	−1.08759	+	1.08759i	−0.0918130	+	0.995776i	$0.529266\pi$
$827$	−31.2765	+	31.2765i	−1.08759	+	1.08759i	−0.995776	+	0.0918130i	$0.970734\pi$
$828$		0			0
$829$	−15.0595	−	15.0595i	−0.523039	−	0.523039i	0.395449	−	0.918488i	$-0.370589\pi$
$829$	−15.0595	−	15.0595i	−0.523039	−	0.523039i	−0.918488	+	0.395449i	$0.870589\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−34.2166			−1.18553
$834$		0			0
$835$	10.5858	+	10.5858i	0.366336	+	0.366336i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			39.9784i			1.38021i	0.723711	+	0.690103i	$0.242435\pi$
$839$			39.9784i			1.38021i	−0.723711	+	0.690103i	$0.757565\pi$
$840$		0			0
$841$		−	13.7133i		−	0.472872i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.448142	−	0.448142i	−0.0154166	−	0.0154166i
$846$		0			0
$847$	−33.8400			−1.16276
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−18.2009	−	18.2009i	−0.623919	−	0.623919i
$852$		0			0
$853$	−25.0838	+	25.0838i	−0.858853	+	0.858853i	−0.991203	−	0.132350i	$-0.957748\pi$
$853$	−25.0838	+	25.0838i	−0.858853	+	0.858853i	0.132350	+	0.991203i	$0.457748\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	40.8755i		−	1.39628i	−0.715961	−	0.698140i	$-0.754011\pi$
$857$		−	40.8755i		−	1.39628i	0.715961	−	0.698140i	$-0.245989\pi$
$858$		0			0
$859$	1.90446	−	1.90446i	0.0649792	−	0.0649792i	−0.673870	−	0.738850i	$-0.735369\pi$
$859$	1.90446	−	1.90446i	0.0649792	−	0.0649792i	0.738850	+	0.673870i	$0.235369\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−33.4074			−1.13720			−0.568601	−	0.822614i	$-0.692515\pi$
$863$	−33.4074			−1.13720			−0.568601	+	0.822614i	$0.692515\pi$
$864$		0			0
$865$	15.6834			0.533251
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.38455	+	7.38455i	−0.250504	+	0.250504i
$870$		0			0
$871$		−	1.00280i		−	0.0339785i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.87615	−	2.87615i	0.0972316	−	0.0972316i
$876$		0			0
$877$	−21.2090	−	21.2090i	−0.716178	−	0.716178i	0.251642	−	0.967820i	$-0.419029\pi$
$877$	−21.2090	−	21.2090i	−0.716178	−	0.716178i	−0.967820	+	0.251642i	$0.919029\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−44.9504			−1.51442			−0.757208	−	0.653173i	$-0.773437\pi$
$881$	−44.9504			−1.51442			−0.757208	+	0.653173i	$0.773437\pi$
$882$		0			0
$883$	2.28813	+	2.28813i	0.0770018	+	0.0770018i	0.744559	−	0.667557i	$-0.232660\pi$
$883$	2.28813	+	2.28813i	0.0770018	+	0.0770018i	−0.667557	+	0.744559i	$0.732660\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			34.5561i			1.16028i	0.814517	+	0.580140i	$0.197002\pi$
$887$			34.5561i			1.16028i	−0.814517	+	0.580140i	$0.802998\pi$
$888$		0			0
$889$			5.54543i			0.185988i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	30.3128	+	30.3128i	1.01438	+	1.01438i
$894$		0			0
$895$	−2.60461			−0.0870625
$896$		0			0
$897$		0			0
$898$		0			0
$899$	29.6824	+	29.6824i	0.989964	+	0.989964i
$900$		0			0
$901$	−17.6810	+	17.6810i	−0.589041	+	0.589041i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	15.5685i		−	0.517514i
$906$		0			0
$907$	1.98178	−	1.98178i	0.0658039	−	0.0658039i	−0.673439	−	0.739243i	$-0.735184\pi$
$907$	1.98178	−	1.98178i	0.0658039	−	0.0658039i	0.739243	+	0.673439i	$0.235184\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	19.9443			0.660784			0.330392	−	0.943844i	$-0.392819\pi$
$911$	19.9443			0.660784			0.330392	+	0.943844i	$0.392819\pi$
$912$		0			0
$913$	7.34509			0.243087
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−13.7937	+	13.7937i	−0.455507	+	0.455507i
$918$		0			0
$919$			16.8647i			0.556314i	0.960536	+	0.278157i	$0.0897236\pi$
$919$			16.8647i			0.556314i	−0.960536	+	0.278157i	$0.910276\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−16.7355	+	16.7355i	−0.550857	+	0.550857i
$924$		0			0
$925$	4.11129	+	4.11129i	0.135179	+	0.135179i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	34.1650			1.12092			0.560458	−	0.828183i	$-0.310625\pi$
$929$	34.1650			1.12092			0.560458	+	0.828183i	$0.310625\pi$
$930$		0			0
$931$	40.0771	+	40.0771i	1.31347	+	1.31347i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			5.86924i			0.191945i
$936$		0			0
$937$		−	19.2914i		−	0.630224i	−0.949055	−	0.315112i	$-0.897958\pi$
$937$		−	19.2914i		−	0.630224i	0.949055	−	0.315112i	$-0.102042\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−26.1176	−	26.1176i	−0.851409	−	0.851409i	0.138898	−	0.990307i	$-0.455644\pi$
$941$	−26.1176	−	26.1176i	−0.851409	−	0.851409i	−0.990307	+	0.138898i	$0.955644\pi$
$942$		0			0
$943$	−48.5818			−1.58204
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.00965410	−	0.00965410i	−0.000313716	−	0.000313716i	0.706950	−	0.707264i	$-0.250071\pi$
$947$	−0.00965410	−	0.00965410i	−0.000313716	−	0.000313716i	−0.707264	+	0.706950i	$0.750071\pi$
$948$		0			0
$949$	4.56532	−	4.56532i	0.148197	−	0.148197i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			54.0003i			1.74924i	0.484809	+	0.874620i	$0.338889\pi$
$953$			54.0003i			1.74924i	−0.484809	+	0.874620i	$0.661111\pi$
$954$		0			0
$955$	16.4507	−	16.4507i	0.532332	−	0.532332i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−85.0432			−2.74619
$960$		0			0
$961$	84.2696			2.71838
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.45627	+	4.45627i	−0.143452	+	0.143452i
$966$		0			0
$967$			13.1472i			0.422785i	0.977401	+	0.211393i	$0.0677998\pi$
$967$			13.1472i			0.422785i	−0.977401	+	0.211393i	$0.932200\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−33.3062	+	33.3062i	−1.06885	+	1.06885i	−0.0713986	+	0.997448i	$0.522746\pi$
$971$	−33.3062	+	33.3062i	−1.06885	+	1.06885i	−0.997448	+	0.0713986i	$0.977254\pi$
$972$		0			0
$973$	−11.6067	−	11.6067i	−0.372093	−	0.372093i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−12.2576			−0.392154			−0.196077	−	0.980589i	$-0.562820\pi$
$977$	−12.2576			−0.392154			−0.196077	+	0.980589i	$0.562820\pi$
$978$		0			0
$979$	16.2001	+	16.2001i	0.517759	+	0.517759i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	8.51469i		−	0.271577i	−0.990738	−	0.135788i	$-0.956643\pi$
$983$		−	8.51469i		−	0.271577i	0.990738	−	0.135788i	$-0.0433567\pi$
$984$		0			0
$985$		−	24.3268i		−	0.775116i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.963435	−	0.963435i	−0.0306354	−	0.0306354i
$990$		0			0
$991$	−14.7339			−0.468037			−0.234019	−	0.972232i	$-0.575188\pi$
$991$	−14.7339			−0.468037			−0.234019	+	0.972232i	$0.575188\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.59981	−	5.59981i	−0.177526	−	0.177526i
$996$		0			0
$997$	−22.6956	+	22.6956i	−0.718777	+	0.718777i	−0.968355	−	0.249577i	$-0.919708\pi$
$997$	−22.6956	+	22.6956i	−0.718777	+	0.718777i	0.249577	+	0.968355i	$0.419708\pi$
$998$		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.721.15		32
3.2	odd	2	inner	2880.2.t.e.721.8		32
4.3	odd	2		720.2.t.e.541.16	yes	32
12.11	even	2		720.2.t.e.541.1	yes	32
16.5	even	4	inner	2880.2.t.e.2161.15		32
16.11	odd	4		720.2.t.e.181.16	yes	32
48.5	odd	4	inner	2880.2.t.e.2161.8		32
48.11	even	4		720.2.t.e.181.1	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.t.e.181.1	✓	32	48.11	even	4
720.2.t.e.181.16	yes	32	16.11	odd	4
720.2.t.e.541.1	yes	32	12.11	even	2
720.2.t.e.541.16	yes	32	4.3	odd	2
2880.2.t.e.721.8		32	3.2	odd	2	inner
2880.2.t.e.721.15		32	1.1	even	1	trivial
2880.2.t.e.2161.8		32	48.5	odd	4	inner
2880.2.t.e.2161.15		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)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{5} +4.06749i q^{7} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{5} +4.06749i q^{7} +(-1.15766 + 1.15766i) q^{11} +(-2.48659 - 2.48659i) q^{13} +3.58496 q^{17} +(-4.19898 - 4.19898i) q^{19} +4.42705i q^{23} -1.00000i q^{25} +(2.76466 + 2.76466i) q^{29} +10.7364 q^{31} +(2.87615 + 2.87615i) q^{35} +(-4.11129 + 4.11129i) q^{37} +10.9739i q^{41} +(-0.217624 + 0.217624i) q^{43} -7.21909 q^{47} -9.54448 q^{49} +(-4.93200 + 4.93200i) q^{53} +1.63718i q^{55} +(-9.20765 + 9.20765i) q^{59} +(1.03762 + 1.03762i) q^{61} -3.51657 q^{65} +(0.201641 + 0.201641i) q^{67} -6.73032i q^{71} +1.83598i q^{73} +(-4.70878 - 4.70878i) q^{77} +6.37885 q^{79} +(-3.17238 - 3.17238i) q^{83} +(2.53495 - 2.53495i) q^{85} -13.9938i q^{89} +(10.1142 - 10.1142i) q^{91} -5.93826 q^{95} -13.9543 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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