LMFDB - Embedded newform 2880.2.bl.c.431.14

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			431.14
Character	$\chi$	$=$	2880.431
Dual form			2880.2.bl.c.1871.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +1.80170 q^{7} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{5} +1.80170 q^{7} +(0.135807 - 0.135807i) q^{11} +(-4.41906 - 4.41906i) q^{13} +3.38137i q^{17} +(-3.50481 + 3.50481i) q^{19} +5.73431i q^{23} +1.00000i q^{25} +(6.69453 - 6.69453i) q^{29} +10.6117i q^{31} +(1.27399 + 1.27399i) q^{35} +(-2.24159 + 2.24159i) q^{37} -2.15455 q^{41} +(8.06179 + 8.06179i) q^{43} +0.779727 q^{47} -3.75388 q^{49} +(5.61091 + 5.61091i) q^{53} +0.192059 q^{55} +(4.77223 - 4.77223i) q^{59} +(4.75279 + 4.75279i) q^{61} -6.24949i q^{65} +(-1.44728 + 1.44728i) q^{67} +9.77099i q^{71} +1.76739i q^{73} +(0.244682 - 0.244682i) q^{77} -8.26499i q^{79} +(1.69729 + 1.69729i) q^{83} +(-2.39099 + 2.39099i) q^{85} +7.74792 q^{89} +(-7.96181 - 7.96181i) q^{91} -4.95655 q^{95} -8.62272 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	1.80170			0.680978			0.340489	−	0.940248i	$-0.389407\pi$
$7$	1.80170			0.680978			0.340489	+	0.940248i	$0.389407\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.135807	−	0.135807i	0.0409472	−	0.0409472i	−0.686337	−	0.727284i	$-0.740782\pi$
$11$	0.135807	−	0.135807i	0.0409472	−	0.0409472i	0.727284	+	0.686337i	$0.240782\pi$
$12$		0			0
$13$	−4.41906	−	4.41906i	−1.22563	−	1.22563i	−0.965602	−	0.260023i	$-0.916270\pi$
$13$	−4.41906	−	4.41906i	−1.22563	−	1.22563i	−0.260023	−	0.965602i	$-0.583730\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.38137i			0.820102i	0.912063	+	0.410051i	$0.134489\pi$
$17$			3.38137i			0.820102i	−0.912063	+	0.410051i	$0.865511\pi$
$18$		0			0
$19$	−3.50481	+	3.50481i	−0.804059	+	0.804059i	−0.983727	−	0.179668i	$-0.942498\pi$
$19$	−3.50481	+	3.50481i	−0.804059	+	0.804059i	0.179668	+	0.983727i	$0.442498\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.73431i			1.19569i	0.801613	+	0.597843i	$0.203975\pi$
$23$			5.73431i			1.19569i	−0.801613	+	0.597843i	$0.796025\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.69453	−	6.69453i	1.24314	−	1.24314i	0.284454	−	0.958690i	$-0.408188\pi$
$29$	6.69453	−	6.69453i	1.24314	−	1.24314i	0.958690	−	0.284454i	$-0.0918123\pi$
$30$		0			0
$31$			10.6117i			1.90593i	0.303088	+	0.952963i	$0.401982\pi$
$31$			10.6117i			1.90593i	−0.303088	+	0.952963i	$0.598018\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.27399	+	1.27399i	0.215344	+	0.215344i
$36$		0			0
$37$	−2.24159	+	2.24159i	−0.368516	+	0.368516i	−0.866936	−	0.498420i	$-0.833914\pi$
$37$	−2.24159	+	2.24159i	−0.368516	+	0.368516i	0.498420	+	0.866936i	$0.333914\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.15455			−0.336484			−0.168242	−	0.985746i	$-0.553809\pi$
$41$	−2.15455			−0.336484			−0.168242	+	0.985746i	$0.553809\pi$
$42$		0			0
$43$	8.06179	+	8.06179i	1.22941	+	1.22941i	0.964187	+	0.265224i	$0.0854459\pi$
$43$	8.06179	+	8.06179i	1.22941	+	1.22941i	0.265224	+	0.964187i	$0.414554\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.779727			0.113735			0.0568674	−	0.998382i	$-0.481889\pi$
$47$	0.779727			0.113735			0.0568674	+	0.998382i	$0.481889\pi$
$48$		0			0
$49$	−3.75388			−0.536269
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.61091	+	5.61091i	0.770717	+	0.770717i	0.978232	−	0.207515i	$-0.0665375\pi$
$53$	5.61091	+	5.61091i	0.770717	+	0.770717i	−0.207515	+	0.978232i	$0.566538\pi$
$54$		0			0
$55$	0.192059			0.0258973
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.77223	−	4.77223i	0.621291	−	0.621291i	−0.324571	−	0.945861i	$-0.605220\pi$
$59$	4.77223	−	4.77223i	0.621291	−	0.621291i	0.945861	+	0.324571i	$0.105220\pi$
$60$		0			0
$61$	4.75279	+	4.75279i	0.608533	+	0.608533i	0.942563	−	0.334030i	$-0.108409\pi$
$61$	4.75279	+	4.75279i	0.608533	+	0.608533i	−0.334030	+	0.942563i	$0.608409\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	6.24949i		−	0.775154i
$66$		0			0
$67$	−1.44728	+	1.44728i	−0.176814	+	0.176814i	−0.789965	−	0.613151i	$-0.789902\pi$
$67$	−1.44728	+	1.44728i	−0.176814	+	0.176814i	0.613151	+	0.789965i	$0.289902\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			9.77099i			1.15960i	0.814758	+	0.579802i	$0.196870\pi$
$71$			9.77099i			1.15960i	−0.814758	+	0.579802i	$0.803130\pi$
$72$		0			0
$73$			1.76739i			0.206857i	0.994637	+	0.103429i	$0.0329813\pi$
$73$			1.76739i			0.206857i	−0.994637	+	0.103429i	$0.967019\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.244682	−	0.244682i	0.0278842	−	0.0278842i
$78$		0			0
$79$		−	8.26499i		−	0.929884i	−0.885341	−	0.464942i	$-0.846075\pi$
$79$		−	8.26499i		−	0.929884i	0.885341	−	0.464942i	$-0.153925\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.69729	+	1.69729i	0.186302	+	0.186302i	0.794095	−	0.607794i	$-0.207945\pi$
$83$	1.69729	+	1.69729i	0.186302	+	0.186302i	−0.607794	+	0.794095i	$0.707945\pi$
$84$		0			0
$85$	−2.39099	+	2.39099i	−0.259339	+	0.259339i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.74792			0.821278			0.410639	−	0.911798i	$-0.365306\pi$
$89$	7.74792			0.821278			0.410639	+	0.911798i	$0.365306\pi$
$90$		0			0
$91$	−7.96181	−	7.96181i	−0.834624	−	0.834624i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.95655			−0.508532
$96$		0			0
$97$	−8.62272			−0.875505			−0.437752	−	0.899096i	$-0.644225\pi$
$97$	−8.62272			−0.875505			−0.437752	+	0.899096i	$0.644225\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.58032	+	1.58032i	0.157248	+	0.157248i	0.781346	−	0.624098i	$-0.214534\pi$
$101$	1.58032	+	1.58032i	0.157248	+	0.157248i	−0.624098	+	0.781346i	$0.714534\pi$
$102$		0			0
$103$	6.96053			0.685842			0.342921	−	0.939364i	$-0.388584\pi$
$103$	6.96053			0.685842			0.342921	+	0.939364i	$0.388584\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.01128	−	2.01128i	0.194438	−	0.194438i	−0.603173	−	0.797610i	$-0.706097\pi$
$107$	2.01128	−	2.01128i	0.194438	−	0.194438i	0.797610	+	0.603173i	$0.206097\pi$
$108$		0			0
$109$	−7.13166	−	7.13166i	−0.683089	−	0.683089i	0.277606	−	0.960695i	$-0.410459\pi$
$109$	−7.13166	−	7.13166i	−0.683089	−	0.683089i	−0.960695	+	0.277606i	$0.910459\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	8.53120i		−	0.802548i	−0.915958	−	0.401274i	$-0.868568\pi$
$113$		−	8.53120i		−	0.802548i	0.915958	−	0.401274i	$-0.131432\pi$
$114$		0			0
$115$	−4.05477	+	4.05477i	−0.378109	+	0.378109i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			6.09221i			0.558472i
$120$		0			0
$121$			10.9631i			0.996647i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$			16.7664i			1.48778i	0.668302	+	0.743890i	$0.267021\pi$
$127$			16.7664i			1.48778i	−0.668302	+	0.743890i	$0.732979\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.87209	−	5.87209i	−0.513047	−	0.513047i	0.402412	−	0.915459i	$-0.368172\pi$
$131$	−5.87209	−	5.87209i	−0.513047	−	0.513047i	−0.915459	+	0.402412i	$0.868172\pi$
$132$		0			0
$133$	−6.31462	+	6.31462i	−0.547547	+	0.547547i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.62788			−0.822565			−0.411283	−	0.911508i	$-0.634919\pi$
$137$	−9.62788			−0.822565			−0.411283	+	0.911508i	$0.634919\pi$
$138$		0			0
$139$	−2.84863	−	2.84863i	−0.241618	−	0.241618i	0.575901	−	0.817519i	$-0.304651\pi$
$139$	−2.84863	−	2.84863i	−0.241618	−	0.241618i	−0.817519	+	0.575901i	$0.804651\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.20027			−0.100372
$144$		0			0
$145$	9.46750			0.786233
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.74511	+	8.74511i	0.716428	+	0.716428i	0.967872	−	0.251444i	$-0.0809055\pi$
$149$	8.74511	+	8.74511i	0.716428	+	0.716428i	−0.251444	+	0.967872i	$0.580905\pi$
$150$		0			0
$151$	−12.1195			−0.986271			−0.493136	−	0.869952i	$-0.664149\pi$
$151$	−12.1195			−0.986271			−0.493136	+	0.869952i	$0.664149\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.50364	+	7.50364i	−0.602706	+	0.602706i
$156$		0			0
$157$	4.04707	+	4.04707i	0.322991	+	0.322991i	0.849913	−	0.526922i	$-0.176654\pi$
$157$	4.04707	+	4.04707i	0.322991	+	0.322991i	−0.526922	+	0.849913i	$0.676654\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			10.3315i			0.814236i
$162$		0			0
$163$	−9.66580	+	9.66580i	−0.757084	+	0.757084i	−0.975791	−	0.218707i	$-0.929816\pi$
$163$	−9.66580	+	9.66580i	−0.757084	+	0.757084i	0.218707	+	0.975791i	$0.429816\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			1.61403i			0.124897i	0.998048	+	0.0624487i	$0.0198910\pi$
$167$			1.61403i			0.124897i	−0.998048	+	0.0624487i	$0.980109\pi$
$168$		0			0
$169$			26.0561i			2.00432i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	18.1130	−	18.1130i	1.37710	−	1.37710i	0.527628	−	0.849475i	$-0.323081\pi$
$173$	18.1130	−	18.1130i	1.37710	−	1.37710i	0.849475	−	0.527628i	$-0.176919\pi$
$174$		0			0
$175$			1.80170i			0.136196i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.17290	+	1.17290i	0.0876665	+	0.0876665i	0.749580	−	0.661914i	$-0.230255\pi$
$179$	1.17290	+	1.17290i	0.0876665	+	0.0876665i	−0.661914	+	0.749580i	$0.730255\pi$
$180$		0			0
$181$	11.2921	−	11.2921i	0.839336	−	0.839336i	−0.149435	−	0.988772i	$-0.547746\pi$
$181$	11.2921	−	11.2921i	0.839336	−	0.839336i	0.988772	+	0.149435i	$0.0477456\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.17009			−0.233070
$186$		0			0
$187$	0.459212	+	0.459212i	0.0335809	+	0.0335809i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	20.6809			1.49641			0.748207	−	0.663465i	$-0.230915\pi$
$191$	20.6809			1.49641			0.748207	+	0.663465i	$0.230915\pi$
$192$		0			0
$193$	−16.8866			−1.21552			−0.607760	−	0.794121i	$-0.707932\pi$
$193$	−16.8866			−1.21552			−0.607760	+	0.794121i	$0.707932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.5832	+	18.5832i	1.32400	+	1.32400i	0.910508	+	0.413492i	$0.135691\pi$
$197$	18.5832	+	18.5832i	1.32400	+	1.32400i	0.413492	+	0.910508i	$0.364309\pi$
$198$		0			0
$199$	7.35180			0.521155			0.260578	−	0.965453i	$-0.416087\pi$
$199$	7.35180			0.521155			0.260578	+	0.965453i	$0.416087\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.0615	−	12.0615i	0.846554	−	0.846554i
$204$		0			0
$205$	−1.52350	−	1.52350i	−0.106406	−	0.106406i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			0.951953i			0.0658480i
$210$		0			0
$211$	−13.4823	+	13.4823i	−0.928159	+	0.928159i	−0.997587	−	0.0694281i	$-0.977883\pi$
$211$	−13.4823	+	13.4823i	−0.928159	+	0.928159i	0.0694281	+	0.997587i	$0.477883\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			11.4011i			0.777548i
$216$		0			0
$217$			19.1192i			1.29789i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	14.9425	−	14.9425i	1.00514	−	1.00514i
$222$		0			0
$223$			0.478098i			0.0320158i	0.999872	+	0.0160079i	$0.00509569\pi$
$223$			0.478098i			0.0320158i	−0.999872	+	0.0160079i	$0.994904\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−14.9140	−	14.9140i	−0.989878	−	0.989878i	0.0100711	−	0.999949i	$-0.496794\pi$
$227$	−14.9140	−	14.9140i	−0.989878	−	0.989878i	−0.999949	+	0.0100711i	$0.996794\pi$
$228$		0			0
$229$	8.54035	−	8.54035i	0.564362	−	0.564362i	−0.366181	−	0.930544i	$-0.619335\pi$
$229$	8.54035	−	8.54035i	0.564362	−	0.564362i	0.930544	+	0.366181i	$0.119335\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.54514			−0.232250			−0.116125	−	0.993235i	$-0.537047\pi$
$233$	−3.54514			−0.232250			−0.116125	+	0.993235i	$0.537047\pi$
$234$		0			0
$235$	0.551350	+	0.551350i	0.0359661	+	0.0359661i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	14.5228			0.939404			0.469702	−	0.882825i	$-0.344361\pi$
$239$	14.5228			0.939404			0.469702	+	0.882825i	$0.344361\pi$
$240$		0			0
$241$	5.72577			0.368829			0.184415	−	0.982849i	$-0.440961\pi$
$241$	5.72577			0.368829			0.184415	+	0.982849i	$0.440961\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.65440	−	2.65440i	−0.169583	−	0.169583i
$246$		0			0
$247$	30.9759			1.97095
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.40367	−	5.40367i	0.341076	−	0.341076i	−0.515696	−	0.856772i	$-0.672466\pi$
$251$	5.40367	−	5.40367i	0.341076	−	0.341076i	0.856772	+	0.515696i	$0.172466\pi$
$252$		0			0
$253$	0.778757	+	0.778757i	0.0489600	+	0.0489600i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			12.2752i			0.765709i	0.923809	+	0.382854i	$0.125059\pi$
$257$			12.2752i			0.765709i	−0.923809	+	0.382854i	$0.874941\pi$
$258$		0			0
$259$	−4.03867	+	4.03867i	−0.250951	+	0.250951i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			24.2541i			1.49557i	0.663940	+	0.747786i	$0.268883\pi$
$263$			24.2541i			1.49557i	−0.663940	+	0.747786i	$0.731117\pi$
$264$		0			0
$265$			7.93502i			0.487444i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.2347	+	15.2347i	−0.928874	+	0.928874i	−0.997633	−	0.0687597i	$-0.978096\pi$
$269$	−15.2347	+	15.2347i	−0.928874	+	0.928874i	0.0687597	+	0.997633i	$0.478096\pi$
$270$		0			0
$271$		−	21.4799i		−	1.30481i	−0.757870	−	0.652406i	$-0.773760\pi$
$271$		−	21.4799i		−	1.30481i	0.757870	−	0.652406i	$-0.226240\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.135807	+	0.135807i	0.00818944	+	0.00818944i
$276$		0			0
$277$	−13.4495	+	13.4495i	−0.808100	+	0.808100i	−0.984346	−	0.176246i	$-0.943605\pi$
$277$	−13.4495	+	13.4495i	−0.808100	+	0.808100i	0.176246	+	0.984346i	$0.443605\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−21.1292			−1.26046			−0.630231	−	0.776408i	$-0.717040\pi$
$281$	−21.1292			−1.26046			−0.630231	+	0.776408i	$0.717040\pi$
$282$		0			0
$283$	−20.8350	−	20.8350i	−1.23851	−	1.23851i	−0.960609	−	0.277905i	$-0.910360\pi$
$283$	−20.8350	−	20.8350i	−1.23851	−	1.23851i	−0.277905	−	0.960609i	$-0.589640\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.88185			−0.229138
$288$		0			0
$289$	5.56635			0.327432
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.87664	−	4.87664i	−0.284896	−	0.284896i	0.550162	−	0.835058i	$-0.314566\pi$
$293$	−4.87664	−	4.87664i	−0.284896	−	0.284896i	−0.835058	+	0.550162i	$0.814566\pi$
$294$		0			0
$295$	6.74895			0.392939
$296$		0			0
$297$		0			0
$298$		0			0
$299$	25.3402	−	25.3402i	1.46546	−	1.46546i
$300$		0			0
$301$	14.5249	+	14.5249i	0.837202	+	0.837202i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			6.72146i			0.384870i
$306$		0			0
$307$	1.10222	−	1.10222i	0.0629069	−	0.0629069i	−0.674953	−	0.737860i	$-0.735836\pi$
$307$	1.10222	−	1.10222i	0.0629069	−	0.0629069i	0.737860	+	0.674953i	$0.235836\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			0.998041i			0.0565937i	0.999600	+	0.0282969i	$0.00900837\pi$
$311$			0.998041i			0.0565937i	−0.999600	+	0.0282969i	$0.990992\pi$
$312$		0			0
$313$		−	6.95852i		−	0.393319i	−0.980472	−	0.196659i	$-0.936991\pi$
$313$		−	6.95852i		−	0.393319i	0.980472	−	0.196659i	$-0.0630093\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−21.3658	+	21.3658i	−1.20002	+	1.20002i	−0.225866	+	0.974158i	$0.572521\pi$
$317$	−21.3658	+	21.3658i	−1.20002	+	1.20002i	−0.974158	+	0.225866i	$0.927479\pi$
$318$		0			0
$319$		−	1.81832i		−	0.101807i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−11.8511	−	11.8511i	−0.659411	−	0.659411i
$324$		0			0
$325$	4.41906	−	4.41906i	0.245125	−	0.245125i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.40483			0.0774510
$330$		0			0
$331$	−9.07140	−	9.07140i	−0.498609	−	0.498609i	0.412396	−	0.911005i	$-0.364692\pi$
$331$	−9.07140	−	9.07140i	−0.498609	−	0.498609i	−0.911005	+	0.412396i	$0.864692\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.04677			−0.111827
$336$		0			0
$337$	10.5978			0.577300			0.288650	−	0.957435i	$-0.406794\pi$
$337$	10.5978			0.577300			0.288650	+	0.957435i	$0.406794\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.44114	+	1.44114i	0.0780423	+	0.0780423i
$342$		0			0
$343$	−19.3753			−1.04617
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.35030	+	6.35030i	−0.340902	+	0.340902i	−0.856706	−	0.515804i	$-0.827493\pi$
$347$	−6.35030	+	6.35030i	−0.340902	+	0.340902i	0.515804	+	0.856706i	$0.327493\pi$
$348$		0			0
$349$	−26.0640	−	26.0640i	−1.39518	−	1.39518i	−0.813220	−	0.581956i	$-0.802288\pi$
$349$	−26.0640	−	26.0640i	−1.39518	−	1.39518i	−0.581956	−	0.813220i	$-0.697712\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.8153i			1.00144i	0.865610	+	0.500719i	$0.166931\pi$
$353$			18.8153i			1.00144i	−0.865610	+	0.500719i	$0.833069\pi$
$354$		0			0
$355$	−6.90913	+	6.90913i	−0.366699	+	0.366699i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	20.0443i		−	1.05790i	−0.848654	−	0.528949i	$-0.822586\pi$
$359$		−	20.0443i		−	1.05790i	0.848654	−	0.528949i	$-0.177414\pi$
$360$		0			0
$361$		−	5.56742i		−	0.293022i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.24973	+	1.24973i	−0.0654140	+	0.0654140i
$366$		0			0
$367$			9.40339i			0.490853i	0.969415	+	0.245426i	$0.0789280\pi$
$367$			9.40339i			0.490853i	−0.969415	+	0.245426i	$0.921072\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.1092	+	10.1092i	0.524841	+	0.524841i
$372$		0			0
$373$	23.0609	−	23.0609i	1.19405	−	1.19405i	0.218126	−	0.975921i	$-0.430006\pi$
$373$	23.0609	−	23.0609i	1.19405	−	1.19405i	0.975921	−	0.218126i	$-0.0699943\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−59.1670			−3.04726
$378$		0			0
$379$	2.68669	+	2.68669i	0.138006	+	0.138006i	0.772735	−	0.634729i	$-0.218888\pi$
$379$	2.68669	+	2.68669i	0.138006	+	0.138006i	−0.634729	+	0.772735i	$0.718888\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	26.8426			1.37159			0.685797	−	0.727793i	$-0.259454\pi$
$383$	26.8426			1.37159			0.685797	+	0.727793i	$0.259454\pi$
$384$		0			0
$385$	0.346033			0.0176355
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.01600	+	8.01600i	0.406427	+	0.406427i	0.880491	−	0.474063i	$-0.157213\pi$
$389$	8.01600	+	8.01600i	0.406427	+	0.406427i	−0.474063	+	0.880491i	$0.657213\pi$
$390$		0			0
$391$	−19.3898			−0.980585
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.84423	−	5.84423i	0.294055	−	0.294055i
$396$		0			0
$397$	−12.2426	−	12.2426i	−0.614437	−	0.614437i	0.329662	−	0.944099i	$-0.393065\pi$
$397$	−12.2426	−	12.2426i	−0.614437	−	0.614437i	−0.944099	+	0.329662i	$0.893065\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	27.5344i		−	1.37500i	−0.726183	−	0.687501i	$-0.758708\pi$
$401$		−	27.5344i		−	1.37500i	0.726183	−	0.687501i	$-0.241292\pi$
$402$		0			0
$403$	46.8939	−	46.8939i	2.33595	−	2.33595i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.608846i			0.0301794i
$408$		0			0
$409$		−	27.4931i		−	1.35944i	−0.733470	−	0.679722i	$-0.762100\pi$
$409$		−	27.4931i		−	1.35944i	0.733470	−	0.679722i	$-0.237900\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.59811	−	8.59811i	0.423085	−	0.423085i
$414$		0			0
$415$			2.40033i			0.117827i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	21.3366	+	21.3366i	1.04236	+	1.04236i	0.999062	+	0.0432975i	$0.0137863\pi$
$419$	21.3366	+	21.3366i	1.04236	+	1.04236i	0.0432975	+	0.999062i	$0.486214\pi$
$420$		0			0
$421$	6.91218	−	6.91218i	0.336879	−	0.336879i	−0.518312	−	0.855191i	$-0.673439\pi$
$421$	6.91218	−	6.91218i	0.336879	−	0.336879i	0.855191	+	0.518312i	$0.173439\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.38137			−0.164020
$426$		0			0
$427$	8.56310	+	8.56310i	0.414398	+	0.414398i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−30.5137			−1.46980			−0.734898	−	0.678178i	$-0.762770\pi$
$431$	−30.5137			−1.46980			−0.734898	+	0.678178i	$0.762770\pi$
$432$		0			0
$433$	−7.83127			−0.376347			−0.188173	−	0.982136i	$-0.560257\pi$
$433$	−7.83127			−0.376347			−0.188173	+	0.982136i	$0.560257\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−20.0977	−	20.0977i	−0.961402	−	0.961402i
$438$		0			0
$439$	40.2979			1.92331			0.961657	−	0.274254i	$-0.0884308\pi$
$439$	40.2979			1.92331			0.961657	+	0.274254i	$0.0884308\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.82049	−	1.82049i	0.0864942	−	0.0864942i	−0.662536	−	0.749030i	$-0.730520\pi$
$443$	1.82049	−	1.82049i	0.0864942	−	0.0864942i	0.749030	+	0.662536i	$0.230520\pi$
$444$		0			0
$445$	5.47861	+	5.47861i	0.259711	+	0.259711i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	11.6096i		−	0.547891i	−0.961745	−	0.273946i	$-0.911671\pi$
$449$		−	11.6096i		−	0.547891i	0.961745	−	0.273946i	$-0.0883288\pi$
$450$		0			0
$451$	−0.292602	+	0.292602i	−0.0137781	+	0.0137781i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	11.2597i		−	0.527863i
$456$		0			0
$457$			13.3520i			0.624580i	0.949987	+	0.312290i	$0.101096\pi$
$457$			13.3520i			0.624580i	−0.949987	+	0.312290i	$0.898904\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.58299	−	2.58299i	0.120302	−	0.120302i	−0.644393	−	0.764695i	$-0.722890\pi$
$461$	2.58299	−	2.58299i	0.120302	−	0.120302i	0.764695	+	0.644393i	$0.222890\pi$
$462$		0			0
$463$			29.0147i			1.34843i	0.738536	+	0.674214i	$0.235517\pi$
$463$			29.0147i			1.34843i	−0.738536	+	0.674214i	$0.764483\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.0947	−	20.0947i	−0.929873	−	0.929873i	0.0678242	−	0.997697i	$-0.478394\pi$
$467$	−20.0947	−	20.0947i	−0.929873	−	0.929873i	−0.997697	+	0.0678242i	$0.978394\pi$
$468$		0			0
$469$	−2.60757	+	2.60757i	−0.120406	+	0.120406i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.18969			0.100682
$474$		0			0
$475$	−3.50481	−	3.50481i	−0.160812	−	0.160812i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−24.7810			−1.13227			−0.566136	−	0.824312i	$-0.691562\pi$
$479$	−24.7810			−1.13227			−0.566136	+	0.824312i	$0.691562\pi$
$480$		0			0
$481$	19.8114			0.903324
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.09718	−	6.09718i	−0.276859	−	0.276859i
$486$		0			0
$487$	−1.19412			−0.0541108			−0.0270554	−	0.999634i	$-0.508613\pi$
$487$	−1.19412			−0.0541108			−0.0270554	+	0.999634i	$0.508613\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13.5737	−	13.5737i	0.612574	−	0.612574i	−0.331042	−	0.943616i	$-0.607400\pi$
$491$	13.5737	−	13.5737i	0.612574	−	0.612574i	0.943616	+	0.331042i	$0.107400\pi$
$492$		0			0
$493$	22.6367	+	22.6367i	1.01950	+	1.01950i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			17.6044i			0.789664i
$498$		0			0
$499$	−14.4047	+	14.4047i	−0.644843	+	0.644843i	−0.951742	−	0.306899i	$-0.900709\pi$
$499$	−14.4047	+	14.4047i	−0.644843	+	0.644843i	0.306899	+	0.951742i	$0.400709\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	14.9448i		−	0.666355i	−0.942864	−	0.333178i	$-0.891879\pi$
$503$		−	14.9448i		−	0.666355i	0.942864	−	0.333178i	$-0.108121\pi$
$504$		0			0
$505$			2.23491i			0.0994521i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.7194	−	16.7194i	0.741075	−	0.741075i	−0.231710	−	0.972785i	$-0.574432\pi$
$509$	16.7194	−	16.7194i	0.741075	−	0.741075i	0.972785	+	0.231710i	$0.0744321\pi$
$510$		0			0
$511$			3.18430i			0.140865i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.92184	+	4.92184i	0.216882	+	0.216882i
$516$		0			0
$517$	0.105892	−	0.105892i	0.00465713	−	0.00465713i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	20.4012			0.893793			0.446896	−	0.894586i	$-0.352529\pi$
$521$	20.4012			0.893793			0.446896	+	0.894586i	$0.352529\pi$
$522$		0			0
$523$	0.495665	+	0.495665i	0.0216739	+	0.0216739i	0.717861	−	0.696187i	$-0.245121\pi$
$523$	0.495665	+	0.495665i	0.0216739	+	0.0216739i	−0.696187	+	0.717861i	$0.745121\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−35.8822			−1.56305
$528$		0			0
$529$	−9.88231			−0.429666
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9.52107	+	9.52107i	0.412404	+	0.412404i
$534$		0			0
$535$	2.84438			0.122973
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.509802	+	0.509802i	−0.0219587	+	0.0219587i
$540$		0			0
$541$	−14.8245	−	14.8245i	−0.637357	−	0.637357i	0.312546	−	0.949903i	$-0.398818\pi$
$541$	−14.8245	−	14.8245i	−0.637357	−	0.637357i	−0.949903	+	0.312546i	$0.898818\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	10.0857i		−	0.432023i
$546$		0			0
$547$	−10.4634	+	10.4634i	−0.447381	+	0.447381i	−0.894483	−	0.447102i	$-0.852456\pi$
$547$	−10.4634	+	10.4634i	−0.447381	+	0.447381i	0.447102	+	0.894483i	$0.352456\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			46.9262i			1.99912i
$552$		0			0
$553$		−	14.8910i		−	0.633231i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.67655	+	4.67655i	−0.198152	+	0.198152i	−0.799207	−	0.601056i	$-0.794747\pi$
$557$	−4.67655	+	4.67655i	−0.198152	+	0.198152i	0.601056	+	0.799207i	$0.294747\pi$
$558$		0			0
$559$		−	71.2510i		−	3.01359i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−26.6235	−	26.6235i	−1.12205	−	1.12205i	−0.991433	−	0.130613i	$-0.958305\pi$
$563$	−26.6235	−	26.6235i	−1.12205	−	1.12205i	−0.130613	−	0.991433i	$-0.541695\pi$
$564$		0			0
$565$	6.03247	−	6.03247i	0.253788	−	0.253788i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	25.5025			1.06912			0.534560	−	0.845131i	$-0.320478\pi$
$569$	25.5025			1.06912			0.534560	+	0.845131i	$0.320478\pi$
$570$		0			0
$571$	−5.18948	−	5.18948i	−0.217173	−	0.217173i	0.590133	−	0.807306i	$-0.299075\pi$
$571$	−5.18948	−	5.18948i	−0.217173	−	0.217173i	−0.807306	+	0.590133i	$0.799075\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.73431			−0.239137
$576$		0			0
$577$	17.6237			0.733684			0.366842	−	0.930283i	$-0.380439\pi$
$577$	17.6237			0.733684			0.366842	+	0.930283i	$0.380439\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.05800	+	3.05800i	0.126867	+	0.126867i
$582$		0			0
$583$	1.52400			0.0631174
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.1455	+	16.1455i	−0.666397	+	0.666397i	−0.956880	−	0.290483i	$-0.906184\pi$
$587$	−16.1455	+	16.1455i	−0.666397	+	0.666397i	0.290483	+	0.956880i	$0.406184\pi$
$588$		0			0
$589$	−37.1922	−	37.1922i	−1.53248	−	1.53248i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			22.0031i			0.903559i	0.892130	+	0.451780i	$0.149211\pi$
$593$			22.0031i			0.903559i	−0.892130	+	0.451780i	$0.850789\pi$
$594$		0			0
$595$	−4.30784	+	4.30784i	−0.176604	+	0.176604i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	36.4005i		−	1.48728i	−0.668578	−	0.743642i	$-0.733097\pi$
$599$		−	36.4005i		−	1.48728i	0.668578	−	0.743642i	$-0.266903\pi$
$600$		0			0
$601$		−	14.1980i		−	0.579149i	−0.957155	−	0.289575i	$-0.906486\pi$
$601$		−	14.1980i		−	0.579149i	0.957155	−	0.289575i	$-0.0935139\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.75209	+	7.75209i	−0.315167	+	0.315167i
$606$		0			0
$607$		−	17.7146i		−	0.719014i	−0.933142	−	0.359507i	$-0.882945\pi$
$607$		−	17.7146i		−	0.719014i	0.933142	−	0.359507i	$-0.117055\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.44566	−	3.44566i	−0.139396	−	0.139396i
$612$		0			0
$613$	−4.85062	+	4.85062i	−0.195915	+	0.195915i	−0.798246	−	0.602331i	$-0.794239\pi$
$613$	−4.85062	+	4.85062i	−0.195915	+	0.195915i	0.602331	+	0.798246i	$0.294239\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.01154			0.0809814			0.0404907	−	0.999180i	$-0.487108\pi$
$617$	2.01154			0.0809814			0.0404907	+	0.999180i	$0.487108\pi$
$618$		0			0
$619$	−22.1640	−	22.1640i	−0.890845	−	0.890845i	0.103758	−	0.994603i	$-0.466913\pi$
$619$	−22.1640	−	22.1640i	−0.890845	−	0.890845i	−0.994603	+	0.103758i	$0.966913\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	13.9594			0.559272
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7.57965	−	7.57965i	−0.302220	−	0.302220i
$630$		0			0
$631$	7.44812			0.296505			0.148252	−	0.988950i	$-0.452635\pi$
$631$	7.44812			0.296505			0.148252	+	0.988950i	$0.452635\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.8557	+	11.8557i	−0.470477	+	0.470477i
$636$		0			0
$637$	16.5886	+	16.5886i	0.657265	+	0.657265i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	30.2636i		−	1.19534i	−0.801741	−	0.597671i	$-0.796093\pi$
$641$		−	30.2636i		−	1.19534i	0.801741	−	0.597671i	$-0.203907\pi$
$642$		0			0
$643$	2.10433	−	2.10433i	0.0829868	−	0.0829868i	−0.664395	−	0.747382i	$-0.731311\pi$
$643$	2.10433	−	2.10433i	0.0829868	−	0.0829868i	0.747382	+	0.664395i	$0.231311\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			47.1268i			1.85274i	0.376610	+	0.926372i	$0.377090\pi$
$647$			47.1268i			1.85274i	−0.376610	+	0.926372i	$0.622910\pi$
$648$		0			0
$649$		−	1.29620i		−	0.0508803i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.1607	−	10.1607i	0.397618	−	0.397618i	−0.479774	−	0.877392i	$-0.659281\pi$
$653$	10.1607	−	10.1607i	0.397618	−	0.397618i	0.877392	+	0.479774i	$0.159281\pi$
$654$		0			0
$655$		−	8.30438i		−	0.324479i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−21.5774	−	21.5774i	−0.840537	−	0.840537i	0.148392	−	0.988929i	$-0.452590\pi$
$659$	−21.5774	−	21.5774i	−0.840537	−	0.840537i	−0.988929	+	0.148392i	$0.952590\pi$
$660$		0			0
$661$	0.359781	−	0.359781i	0.0139939	−	0.0139939i	−0.700075	−	0.714069i	$-0.746850\pi$
$661$	0.359781	−	0.359781i	0.0139939	−	0.0139939i	0.714069	+	0.700075i	$0.246850\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.93021			−0.346299
$666$		0			0
$667$	38.3885	+	38.3885i	1.48641	+	1.48641i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.29092			0.0498354
$672$		0			0
$673$	−23.6899			−0.913180			−0.456590	−	0.889677i	$-0.650929\pi$
$673$	−23.6899			−0.913180			−0.456590	+	0.889677i	$0.650929\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9.07478	−	9.07478i	−0.348772	−	0.348772i	0.510880	−	0.859652i	$-0.329320\pi$
$677$	−9.07478	−	9.07478i	−0.348772	−	0.348772i	−0.859652	+	0.510880i	$0.829320\pi$
$678$		0			0
$679$	−15.5355			−0.596199
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−23.9837	+	23.9837i	−0.917709	+	0.917709i	−0.996862	−	0.0791534i	$-0.974778\pi$
$683$	−23.9837	+	23.9837i	−0.917709	+	0.917709i	0.0791534	+	0.996862i	$0.474778\pi$
$684$		0			0
$685$	−6.80794	−	6.80794i	−0.260118	−	0.260118i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	49.5898i		−	1.88922i
$690$		0			0
$691$	31.8778	−	31.8778i	1.21269	−	1.21269i	0.242552	−	0.970139i	$-0.422016\pi$
$691$	31.8778	−	31.8778i	1.21269	−	1.21269i	0.970139	−	0.242552i	$-0.0779844\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	4.02858i		−	0.152813i
$696$		0			0
$697$		−	7.28532i		−	0.275951i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	15.3601	−	15.3601i	0.580142	−	0.580142i	−0.354800	−	0.934942i	$-0.615451\pi$
$701$	15.3601	−	15.3601i	0.580142	−	0.580142i	0.934942	+	0.354800i	$0.115451\pi$
$702$		0			0
$703$		−	15.7127i		−	0.592617i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.84726	+	2.84726i	0.107082	+	0.107082i
$708$		0			0
$709$	2.33501	−	2.33501i	0.0876930	−	0.0876930i	−0.661900	−	0.749593i	$-0.730249\pi$
$709$	2.33501	−	2.33501i	0.0876930	−	0.0876930i	0.749593	+	0.661900i	$0.230249\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−60.8510			−2.27889
$714$		0			0
$715$	−0.848722	−	0.848722i	−0.0317404	−	0.0317404i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	27.1177			1.01132			0.505659	−	0.862733i	$-0.331249\pi$
$719$	27.1177			1.01132			0.505659	+	0.862733i	$0.331249\pi$
$720$		0			0
$721$	12.5408			0.467043
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.69453	+	6.69453i	0.248629	+	0.248629i
$726$		0			0
$727$	18.2784			0.677908			0.338954	−	0.940803i	$-0.389927\pi$
$727$	18.2784			0.677908			0.338954	+	0.940803i	$0.389927\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−27.2599	+	27.2599i	−1.00824	+	1.00824i
$732$		0			0
$733$	18.1495	+	18.1495i	0.670366	+	0.670366i	0.957800	−	0.287434i	$-0.0928022\pi$
$733$	18.1495	+	18.1495i	0.670366	+	0.670366i	−0.287434	+	0.957800i	$0.592802\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			0.393101i			0.0144801i
$738$		0			0
$739$	−9.07996	+	9.07996i	−0.334012	+	0.334012i	−0.854108	−	0.520096i	$-0.825896\pi$
$739$	−9.07996	+	9.07996i	−0.334012	+	0.334012i	0.520096	+	0.854108i	$0.325896\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			5.32424i			0.195327i	0.995219	+	0.0976636i	$0.0311369\pi$
$743$			5.32424i			0.195327i	−0.995219	+	0.0976636i	$0.968863\pi$
$744$		0			0
$745$			12.3675i			0.453109i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.62372	−	3.62372i	0.132408	−	0.132408i
$750$		0			0
$751$			2.63762i			0.0962481i	0.998841	+	0.0481241i	$0.0153243\pi$
$751$			2.63762i			0.0962481i	−0.998841	+	0.0481241i	$0.984676\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.56979	−	8.56979i	−0.311886	−	0.311886i
$756$		0			0
$757$	3.86568	−	3.86568i	0.140501	−	0.140501i	−0.633358	−	0.773859i	$-0.718324\pi$
$757$	3.86568	−	3.86568i	0.140501	−	0.140501i	0.773859	+	0.633358i	$0.218324\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.50777			−0.163406			−0.0817032	−	0.996657i	$-0.526036\pi$
$761$	−4.50777			−0.163406			−0.0817032	+	0.996657i	$0.526036\pi$
$762$		0			0
$763$	−12.8491	−	12.8491i	−0.465168	−	0.465168i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−42.1775			−1.52294
$768$		0			0
$769$	50.3163			1.81445			0.907225	−	0.420645i	$-0.138196\pi$
$769$	50.3163			1.81445			0.907225	+	0.420645i	$0.138196\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7.10457	+	7.10457i	0.255533	+	0.255533i	0.823235	−	0.567701i	$-0.192167\pi$
$773$	7.10457	+	7.10457i	0.255533	+	0.255533i	−0.567701	+	0.823235i	$0.692167\pi$
$774$		0			0
$775$	−10.6117			−0.381185
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7.55129	−	7.55129i	0.270553	−	0.270553i
$780$		0			0
$781$	1.32696	+	1.32696i	0.0474825	+	0.0474825i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			5.72342i			0.204278i
$786$		0			0
$787$	38.5553	−	38.5553i	1.37435	−	1.37435i	0.520466	−	0.853883i	$-0.325758\pi$
$787$	38.5553	−	38.5553i	1.37435	−	1.37435i	0.853883	−	0.520466i	$-0.174242\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	15.3707i		−	0.546518i
$792$		0			0
$793$		−	42.0057i		−	1.49167i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.19532	+	4.19532i	−0.148606	+	0.148606i	−0.777495	−	0.628889i	$-0.783510\pi$
$797$	−4.19532	+	4.19532i	−0.148606	+	0.148606i	0.628889	+	0.777495i	$0.283510\pi$
$798$		0			0
$799$			2.63654i			0.0932742i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.240023	+	0.240023i	0.00847023	+	0.00847023i
$804$		0			0
$805$	−7.30547	+	7.30547i	−0.257484	+	0.257484i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	47.4807			1.66933			0.834667	−	0.550755i	$-0.185660\pi$
$809$	47.4807			1.66933			0.834667	+	0.550755i	$0.185660\pi$
$810$		0			0
$811$	22.9138	+	22.9138i	0.804614	+	0.804614i	0.983813	−	0.179199i	$-0.0573506\pi$
$811$	22.9138	+	22.9138i	0.804614	+	0.804614i	−0.179199	+	0.983813i	$0.557351\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−13.6695			−0.478822
$816$		0			0
$817$	−56.5101			−1.97704
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−17.6060	−	17.6060i	−0.614452	−	0.614452i	0.329651	−	0.944103i	$-0.393069\pi$
$821$	−17.6060	−	17.6060i	−0.614452	−	0.614452i	−0.944103	+	0.329651i	$0.893069\pi$
$822$		0			0
$823$	−30.7477			−1.07180			−0.535899	−	0.844282i	$-0.680027\pi$
$823$	−30.7477			−1.07180			−0.535899	+	0.844282i	$0.680027\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.8874	+	15.8874i	−0.552458	+	0.552458i	−0.927149	−	0.374692i	$-0.877749\pi$
$827$	−15.8874	+	15.8874i	−0.552458	+	0.552458i	0.374692	+	0.927149i	$0.377749\pi$
$828$		0			0
$829$	−9.73720	−	9.73720i	−0.338187	−	0.338187i	0.517498	−	0.855685i	$-0.326864\pi$
$829$	−9.73720	−	9.73720i	−0.338187	−	0.338187i	−0.855685	+	0.517498i	$0.826864\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	12.6933i		−	0.439795i
$834$		0			0
$835$	−1.14129	+	1.14129i	−0.0394960	+	0.0394960i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			5.49164i			0.189593i	0.995497	+	0.0947963i	$0.0302200\pi$
$839$			5.49164i			0.189593i	−0.995497	+	0.0947963i	$0.969780\pi$
$840$		0			0
$841$		−	60.6336i		−	2.09081i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−18.4245	+	18.4245i	−0.633820	+	0.633820i
$846$		0			0
$847$			19.7522i			0.678695i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12.8540	−	12.8540i	−0.440629	−	0.440629i
$852$		0			0
$853$	13.4809	−	13.4809i	0.461579	−	0.461579i	−0.437594	−	0.899173i	$-0.644169\pi$
$853$	13.4809	−	13.4809i	0.461579	−	0.461579i	0.899173	+	0.437594i	$0.144169\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	47.6620			1.62810			0.814051	−	0.580794i	$-0.197258\pi$
$857$	47.6620			1.62810			0.814051	+	0.580794i	$0.197258\pi$
$858$		0			0
$859$	−9.14211	−	9.14211i	−0.311925	−	0.311925i	0.533730	−	0.845655i	$-0.320790\pi$
$859$	−9.14211	−	9.14211i	−0.311925	−	0.311925i	−0.845655	+	0.533730i	$0.820790\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	16.6653			0.567294			0.283647	−	0.958929i	$-0.408456\pi$
$863$	16.6653			0.567294			0.283647	+	0.958929i	$0.408456\pi$
$864$		0			0
$865$	25.6156			0.870957
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.12244	−	1.12244i	−0.0380762	−	0.0380762i
$870$		0			0
$871$	12.7913			0.433415
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.27399	+	1.27399i	−0.0430688	+	0.0430688i
$876$		0			0
$877$	26.3127	+	26.3127i	0.888516	+	0.888516i	0.994381	−	0.105865i	$-0.0337611\pi$
$877$	26.3127	+	26.3127i	0.888516	+	0.888516i	−0.105865	+	0.994381i	$0.533761\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			39.2980i			1.32398i	0.749511	+	0.661991i	$0.230288\pi$
$881$			39.2980i			1.32398i	−0.749511	+	0.661991i	$0.769712\pi$
$882$		0			0
$883$	5.21473	−	5.21473i	0.175490	−	0.175490i	−0.613897	−	0.789386i	$-0.710399\pi$
$883$	5.21473	−	5.21473i	0.175490	−	0.175490i	0.789386	+	0.613897i	$0.210399\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	36.1199i		−	1.21279i	−0.795164	−	0.606395i	$-0.792615\pi$
$887$		−	36.1199i		−	1.21279i	0.795164	−	0.606395i	$-0.207385\pi$
$888$		0			0
$889$			30.2081i			1.01315i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.73280	+	2.73280i	−0.0914496	+	0.0914496i
$894$		0			0
$895$			1.65873i			0.0554452i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	71.0407	+	71.0407i	2.36934	+	2.36934i
$900$		0			0
$901$	−18.9725	+	18.9725i	−0.632067	+	0.632067i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	15.9695			0.530843
$906$		0			0
$907$	11.3864	+	11.3864i	0.378080	+	0.378080i	0.870409	−	0.492329i	$-0.163854\pi$
$907$	11.3864	+	11.3864i	0.378080	+	0.378080i	−0.492329	+	0.870409i	$0.663854\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	24.9794			0.827604			0.413802	−	0.910367i	$-0.364201\pi$
$911$	24.9794			0.827604			0.413802	+	0.910367i	$0.364201\pi$
$912$		0			0
$913$	0.461006			0.0152571
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.5797	−	10.5797i	−0.349373	−	0.349373i
$918$		0			0
$919$	50.2103			1.65629			0.828143	−	0.560517i	$-0.189398\pi$
$919$	50.2103			1.65629			0.828143	+	0.560517i	$0.189398\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	43.1786	−	43.1786i	1.42124	−	1.42124i
$924$		0			0
$925$	−2.24159	−	2.24159i	−0.0737031	−	0.0737031i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	14.5279i		−	0.476646i	−0.971186	−	0.238323i	$-0.923402\pi$
$929$		−	14.5279i		−	0.476646i	0.971186	−	0.238323i	$-0.0765977\pi$
$930$		0			0
$931$	13.1567	−	13.1567i	0.431192	−	0.431192i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			0.649424i			0.0212384i
$936$		0			0
$937$		−	13.8611i		−	0.452823i	−0.974032	−	0.226411i	$-0.927301\pi$
$937$		−	13.8611i		−	0.452823i	0.974032	−	0.226411i	$-0.0726994\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.989738	−	0.989738i	0.0322645	−	0.0322645i	−0.690790	−	0.723055i	$-0.742737\pi$
$941$	0.989738	−	0.989738i	0.0322645	−	0.0322645i	0.723055	+	0.690790i	$0.242737\pi$
$942$		0			0
$943$		−	12.3549i		−	0.402329i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	22.3927	+	22.3927i	0.727665	+	0.727665i	0.970154	−	0.242489i	$-0.0779640\pi$
$947$	22.3927	+	22.3927i	0.727665	+	0.727665i	−0.242489	+	0.970154i	$0.577964\pi$
$948$		0			0
$949$	7.81019	−	7.81019i	0.253530	−	0.253530i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	42.6367			1.38114			0.690568	−	0.723267i	$-0.257360\pi$
$953$	42.6367			1.38114			0.690568	+	0.723267i	$0.257360\pi$
$954$		0			0
$955$	14.6236	+	14.6236i	0.473208	+	0.473208i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−17.3465			−0.560149
$960$		0			0
$961$	−81.6091			−2.63255
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−11.9406	−	11.9406i	−0.384381	−	0.384381i
$966$		0			0
$967$	−43.6818			−1.40471			−0.702357	−	0.711825i	$-0.747869\pi$
$967$	−43.6818			−1.40471			−0.702357	+	0.711825i	$0.747869\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−42.0104	+	42.0104i	−1.34818	+	1.34818i	−0.460539	+	0.887640i	$0.652344\pi$
$971$	−42.0104	+	42.0104i	−1.34818	+	1.34818i	−0.887640	+	0.460539i	$0.847656\pi$
$972$		0			0
$973$	−5.13238	−	5.13238i	−0.164537	−	0.164537i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	34.3074i		−	1.09759i	−0.835957	−	0.548795i	$-0.815087\pi$
$977$		−	34.3074i		−	1.09759i	0.835957	−	0.548795i	$-0.184913\pi$
$978$		0			0
$979$	1.05222	−	1.05222i	0.0336290	−	0.0336290i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	56.2669i		−	1.79464i	−0.441385	−	0.897318i	$-0.645513\pi$
$983$		−	56.2669i		−	1.79464i	0.441385	−	0.897318i	$-0.354487\pi$
$984$		0			0
$985$			26.2806i			0.837371i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−46.2288	+	46.2288i	−1.46999	+	1.46999i
$990$		0			0
$991$			36.4248i			1.15707i	0.815657	+	0.578536i	$0.196376\pi$
$991$			36.4248i			1.15707i	−0.815657	+	0.578536i	$0.803624\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.19851	+	5.19851i	0.164804	+	0.164804i
$996$		0			0
$997$	26.9327	−	26.9327i	0.852967	−	0.852967i	−0.137531	−	0.990497i	$-0.543917\pi$
$997$	26.9327	−	26.9327i	0.852967	−	0.852967i	0.990497	+	0.137531i	$0.0439166\pi$
$998$		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.431.14		32
3.2	odd	2	inner	2880.2.bl.c.431.5		32
4.3	odd	2		720.2.bl.c.611.6	yes	32
12.11	even	2		720.2.bl.c.611.11	yes	32
16.5	even	4		720.2.bl.c.251.11	yes	32
16.11	odd	4	inner	2880.2.bl.c.1871.5		32
48.5	odd	4		720.2.bl.c.251.6	✓	32
48.11	even	4	inner	2880.2.bl.c.1871.13		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.c.251.6	✓	32	48.5	odd	4
720.2.bl.c.251.11	yes	32	16.5	even	4
720.2.bl.c.611.6	yes	32	4.3	odd	2
720.2.bl.c.611.11	yes	32	12.11	even	2
2880.2.bl.c.431.5		32	3.2	odd	2	inner
2880.2.bl.c.431.14		32	1.1	even	1	trivial
2880.2.bl.c.1871.5		32	16.11	odd	4	inner
2880.2.bl.c.1871.13		32	48.11	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.bl (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +1.80170 q^{7} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{5} +1.80170 q^{7} +(0.135807 - 0.135807i) q^{11} +(-4.41906 - 4.41906i) q^{13} +3.38137i q^{17} +(-3.50481 + 3.50481i) q^{19} +5.73431i q^{23} +1.00000i q^{25} +(6.69453 - 6.69453i) q^{29} +10.6117i q^{31} +(1.27399 + 1.27399i) q^{35} +(-2.24159 + 2.24159i) q^{37} -2.15455 q^{41} +(8.06179 + 8.06179i) q^{43} +0.779727 q^{47} -3.75388 q^{49} +(5.61091 + 5.61091i) q^{53} +0.192059 q^{55} +(4.77223 - 4.77223i) q^{59} +(4.75279 + 4.75279i) q^{61} -6.24949i q^{65} +(-1.44728 + 1.44728i) q^{67} +9.77099i q^{71} +1.76739i q^{73} +(0.244682 - 0.244682i) q^{77} -8.26499i q^{79} +(1.69729 + 1.69729i) q^{83} +(-2.39099 + 2.39099i) q^{85} +7.74792 q^{89} +(-7.96181 - 7.96181i) q^{91} -4.95655 q^{95} -8.62272 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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