LMFDB - Embedded newform 2880.2.w.p.2753.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(2177,2880)]

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, 0, 2, 1]))

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

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

chi := DirichletCharacter("2880.2177");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.w (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:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 27x^{8} + 107x^{4} + 1 $ x^12 + 27x^8 + 107x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2753.1
Root			$-1.53448 + 1.53448i$ of defining polynomial
Character	$\chi$	$=$	2880.2753
Dual form			2880.2.w.p.2177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.91575 - 1.15322i) q^{5} +(1.70928 - 1.70928i) q^{7} +O(q^{10})$	$q+(-1.91575 - 1.15322i) q^{5} +(1.70928 - 1.70928i) q^{7} +0.892224i q^{11} +(2.70928 + 2.70928i) q^{13} +(-1.65475 - 1.65475i) q^{17} +7.41855i q^{19} +(6.13793 - 6.13793i) q^{23} +(2.34017 + 4.41855i) q^{25} +0.521990 q^{29} -5.26180 q^{31} +(-5.24571 + 1.30337i) q^{35} +(-1.78765 + 1.78765i) q^{37} +0.110843i q^{41} +(1.26180 + 1.26180i) q^{43} +(6.77076 + 6.77076i) q^{47} +1.15676i q^{49} +(4.07203 - 4.07203i) q^{53} +(1.02893 - 1.70928i) q^{55} -10.5613 q^{59} +12.0989 q^{61} +(-2.06590 - 8.31467i) q^{65} +(5.26180 - 5.26180i) q^{67} +3.05011i q^{71} +(7.34017 + 7.34017i) q^{73} +(1.52506 + 1.52506i) q^{77} -13.2618i q^{79} +(-6.54908 + 6.54908i) q^{83} +(1.26180 + 5.07838i) q^{85} +13.4307 q^{89} +9.26180 q^{91} +(8.55521 - 14.2121i) q^{95} +(8.60197 - 8.60197i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{7}+O(q^{10}) $ 12 * q - 8 * q^7	$ 12 q - 8 q^{7} + 4 q^{13} - 16 q^{25} - 32 q^{31} + 20 q^{37} - 16 q^{43} + 72 q^{55} + 32 q^{67} + 44 q^{73} - 16 q^{85} + 80 q^{91} + 28 q^{97}+O(q^{100}) $ 12 * q - 8 * q^7 + 4 * q^13 - 16 * q^25 - 32 * q^31 + 20 * q^37 - 16 * q^43 + 72 * q^55 + 32 * q^67 + 44 * q^73 - 16 * q^85 + 80 * q^91 + 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$1$	$1$

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
$5$	−1.91575	−	1.15322i	−0.856748	−	0.515735i
$5$	−1.91575	−	1.15322i	−0.856748	−	0.515735i
$6$		0			0
$7$	1.70928	−	1.70928i	0.646045	−	0.646045i	−0.305990	−	0.952035i	$-0.598987\pi$
$7$	1.70928	−	1.70928i	0.646045	−	0.646045i	0.952035	+	0.305990i	$0.0989873\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.892224i			0.269016i	0.990913	+	0.134508i	$0.0429453\pi$
$11$			0.892224i			0.269016i	−0.990913	+	0.134508i	$0.957055\pi$
$12$		0			0
$13$	2.70928	+	2.70928i	0.751418	+	0.751418i	0.974744	−	0.223326i	$-0.0716915\pi$
$13$	2.70928	+	2.70928i	0.751418	+	0.751418i	−0.223326	+	0.974744i	$0.571691\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.65475	−	1.65475i	−0.401336	−	0.401336i	0.477367	−	0.878704i	$-0.341591\pi$
$17$	−1.65475	−	1.65475i	−0.401336	−	0.401336i	−0.878704	+	0.477367i	$0.841591\pi$
$18$		0			0
$19$			7.41855i			1.70193i	0.525221	+	0.850966i	$0.323983\pi$
$19$			7.41855i			1.70193i	−0.525221	+	0.850966i	$0.676017\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.13793	−	6.13793i	1.27985	−	1.27985i	0.339095	−	0.940752i	$-0.389879\pi$
$23$	6.13793	−	6.13793i	1.27985	−	1.27985i	0.940752	−	0.339095i	$-0.110121\pi$
$24$		0			0
$25$	2.34017	+	4.41855i	0.468035	+	0.883710i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.521990			0.0969310			0.0484655	−	0.998825i	$-0.484567\pi$
$29$	0.521990			0.0969310			0.0484655	+	0.998825i	$0.484567\pi$
$30$		0			0
$31$	−5.26180			−0.945046			−0.472523	−	0.881318i	$-0.656657\pi$
$31$	−5.26180			−0.945046			−0.472523	+	0.881318i	$0.656657\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.24571	+	1.30337i	−0.886686	+	0.220310i
$36$		0			0
$37$	−1.78765	+	1.78765i	−0.293888	+	0.293888i	−0.838614	−	0.544726i	$-0.816634\pi$
$37$	−1.78765	+	1.78765i	−0.293888	+	0.293888i	0.544726	+	0.838614i	$0.316634\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			0.110843i			0.0173107i	0.999963	+	0.00865537i	$0.00275513\pi$
$41$			0.110843i			0.0173107i	−0.999963	+	0.00865537i	$0.997245\pi$
$42$		0			0
$43$	1.26180	+	1.26180i	0.192422	+	0.192422i	0.796742	−	0.604320i	$-0.206555\pi$
$43$	1.26180	+	1.26180i	0.192422	+	0.192422i	−0.604320	+	0.796742i	$0.706555\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.77076	+	6.77076i	0.987617	+	0.987617i	0.999924	−	0.0123068i	$-0.00391749\pi$
$47$	6.77076	+	6.77076i	0.987617	+	0.987617i	−0.0123068	+	0.999924i	$0.503917\pi$
$48$		0			0
$49$			1.15676i			0.165251i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.07203	−	4.07203i	0.559337	−	0.559337i	−0.369782	−	0.929119i	$-0.620568\pi$
$53$	4.07203	−	4.07203i	0.559337	−	0.559337i	0.929119	+	0.369782i	$0.120568\pi$
$54$		0			0
$55$	1.02893	−	1.70928i	0.138741	−	0.230479i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.5613			−1.37497			−0.687485	−	0.726199i	$-0.741285\pi$
$59$	−10.5613			−1.37497			−0.687485	+	0.726199i	$0.741285\pi$
$60$		0			0
$61$	12.0989			1.54910			0.774552	−	0.632510i	$-0.217975\pi$
$61$	12.0989			1.54910			0.774552	+	0.632510i	$0.217975\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.06590	−	8.31467i	−0.256243	−	1.03131i
$66$		0			0
$67$	5.26180	−	5.26180i	0.642831	−	0.642831i	−0.308420	−	0.951250i	$-0.599800\pi$
$67$	5.26180	−	5.26180i	0.642831	−	0.642831i	0.951250	+	0.308420i	$0.0998001\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.05011i			0.361982i	0.983485	+	0.180991i	$0.0579305\pi$
$71$			3.05011i			0.361982i	−0.983485	+	0.180991i	$0.942070\pi$
$72$		0			0
$73$	7.34017	+	7.34017i	0.859102	+	0.859102i	0.991232	−	0.132130i	$-0.0421817\pi$
$73$	7.34017	+	7.34017i	0.859102	+	0.859102i	−0.132130	+	0.991232i	$0.542182\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.52506	+	1.52506i	0.173796	+	0.173796i
$78$		0			0
$79$		−	13.2618i		−	1.49207i	−0.665908	−	0.746034i	$-0.731955\pi$
$79$		−	13.2618i		−	1.49207i	0.665908	−	0.746034i	$-0.268045\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.54908	+	6.54908i	−0.718855	+	0.718855i	−0.968371	−	0.249516i	$-0.919728\pi$
$83$	−6.54908	+	6.54908i	−0.718855	+	0.718855i	0.249516	+	0.968371i	$0.419728\pi$
$84$		0			0
$85$	1.26180	+	5.07838i	0.136861	+	0.550827i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.4307			1.42365			0.711825	−	0.702357i	$-0.247869\pi$
$89$	13.4307			1.42365			0.711825	+	0.702357i	$0.247869\pi$
$90$		0			0
$91$	9.26180			0.970900
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.55521	−	14.2121i	0.877746	−	1.45813i
$96$		0			0
$97$	8.60197	−	8.60197i	0.873398	−	0.873398i	−0.119443	−	0.992841i	$-0.538111\pi$
$97$	8.60197	−	8.60197i	0.873398	−	0.873398i	0.992841	+	0.119443i	$0.0381110\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	11.2728i		−	1.12169i	−0.827922	−	0.560843i	$-0.810477\pi$
$101$		−	11.2728i		−	1.12169i	0.827922	−	0.560843i	$-0.189523\pi$
$102$		0			0
$103$	−5.12783	−	5.12783i	−0.505260	−	0.505260i	0.407808	−	0.913068i	$-0.366293\pi$
$103$	−5.12783	−	5.12783i	−0.505260	−	0.505260i	−0.913068	+	0.407808i	$0.866293\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.76463	−	4.76463i	−0.460614	−	0.460614i	0.438243	−	0.898857i	$-0.355601\pi$
$107$	−4.76463	−	4.76463i	−0.460614	−	0.460614i	−0.898857	+	0.438243i	$0.855601\pi$
$108$		0			0
$109$		−	3.10504i		−	0.297409i	−0.988882	−	0.148704i	$-0.952490\pi$
$109$		−	3.10504i		−	0.297409i	0.988882	−	0.148704i	$-0.0475103\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.07510	−	5.07510i	0.477425	−	0.477425i	−0.426882	−	0.904307i	$-0.640388\pi$
$113$	5.07510	−	5.07510i	0.477425	−	0.477425i	0.904307	+	0.426882i	$0.140388\pi$
$114$		0			0
$115$	−18.8371	+	4.68035i	−1.75657	+	0.436445i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.65685			−0.518563
$120$		0			0
$121$	10.2039			0.927631
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.612376	−	11.1636i	0.0547726	−	0.998499i
$126$		0			0
$127$	−6.29072	+	6.29072i	−0.558212	+	0.558212i	−0.928798	−	0.370586i	$-0.879157\pi$
$127$	−6.29072	+	6.29072i	−0.558212	+	0.558212i	0.370586	+	0.928798i	$0.379157\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			0.0699305i			0.00610986i	0.999995	+	0.00305493i	$0.000972415\pi$
$131$			0.0699305i			0.00610986i	−0.999995	+	0.00305493i	$0.999028\pi$
$132$		0			0
$133$	12.6803	+	12.6803i	1.09953	+	1.09953i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.59402	+	4.59402i	0.392494	+	0.392494i	0.875575	−	0.483082i	$-0.160483\pi$
$137$	4.59402	+	4.59402i	0.392494	+	0.392494i	−0.483082	+	0.875575i	$0.660483\pi$
$138$		0			0
$139$			14.1568i			1.20076i	0.799715	+	0.600380i	$0.204984\pi$
$139$			14.1568i			1.20076i	−0.799715	+	0.600380i	$0.795016\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.41728	+	2.41728i	−0.202143	+	0.202143i
$144$		0			0
$145$	−1.00000	−	0.601968i	−0.0830455	−	0.0499907i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	23.7703			1.94734			0.973671	−	0.227957i	$-0.0732045\pi$
$149$	23.7703			1.94734			0.973671	+	0.227957i	$0.0732045\pi$
$150$		0			0
$151$	7.51745			0.611761			0.305881	−	0.952070i	$-0.401049\pi$
$151$	7.51745			0.611761			0.305881	+	0.952070i	$0.401049\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	10.0803	+	6.06800i	0.809667	+	0.487394i
$156$		0			0
$157$	15.7298	−	15.7298i	1.25537	−	1.25537i	0.302097	−	0.953277i	$-0.402313\pi$
$157$	15.7298	−	15.7298i	1.25537	−	1.25537i	0.953277	−	0.302097i	$-0.0976867\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	20.9828i		−	1.65368i
$162$		0			0
$163$	−5.26180	−	5.26180i	−0.412136	−	0.412136i	0.470346	−	0.882482i	$-0.344129\pi$
$163$	−5.26180	−	5.26180i	−0.412136	−	0.412136i	−0.882482	+	0.470346i	$0.844129\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.30337	+	1.30337i	0.100858	+	0.100858i	0.755735	−	0.654877i	$-0.227280\pi$
$167$	1.30337	+	1.30337i	0.100858	+	0.100858i	−0.654877	+	0.755735i	$0.727280\pi$
$168$		0			0
$169$			1.68035i			0.129257i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.20076	+	7.20076i	−0.547464	+	0.547464i	−0.925707	−	0.378243i	$-0.876529\pi$
$173$	−7.20076	+	7.20076i	−0.547464	+	0.547464i	0.378243	+	0.925707i	$0.376529\pi$
$174$		0			0
$175$	11.5525	+	3.55252i	0.873288	+	0.268545i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	22.6973			1.69648			0.848240	−	0.529613i	$-0.177663\pi$
$179$	22.6973			1.69648			0.848240	+	0.529613i	$0.177663\pi$
$180$		0			0
$181$	−11.7321			−0.872037			−0.436019	−	0.899938i	$-0.643612\pi$
$181$	−11.7321			−0.872037			−0.436019	+	0.899938i	$0.643612\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.48625	−	1.36314i	0.403357	−	0.100220i
$186$		0			0
$187$	1.47641	−	1.47641i	0.107966	−	0.107966i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			8.26360i			0.597933i	0.954264	+	0.298966i	$0.0966419\pi$
$191$			8.26360i			0.597933i	−0.954264	+	0.298966i	$0.903358\pi$
$192$		0			0
$193$	10.5753	+	10.5753i	0.761227	+	0.761227i	0.976544	−	0.215317i	$-0.0690785\pi$
$193$	10.5753	+	10.5753i	0.761227	+	0.761227i	−0.215317	+	0.976544i	$0.569078\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.10681	+	2.10681i	0.150104	+	0.150104i	0.778165	−	0.628060i	$-0.216151\pi$
$197$	2.10681	+	2.10681i	0.150104	+	0.150104i	−0.628060	+	0.778165i	$0.716151\pi$
$198$		0			0
$199$		−	7.31965i		−	0.518877i	−0.965760	−	0.259438i	$-0.916463\pi$
$199$		−	7.31965i		−	0.518877i	0.965760	−	0.259438i	$-0.0835374\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.892224	−	0.892224i	0.0626218	−	0.0626218i
$204$		0			0
$205$	0.127826	−	0.212347i	0.00892776	−	0.0148310i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.61901			−0.457846
$210$		0			0
$211$	20.9939			1.44528			0.722638	−	0.691226i	$-0.242929\pi$
$211$	20.9939			1.44528			0.722638	+	0.691226i	$0.242929\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.962154	−	3.87241i	−0.0656184	−	0.264096i
$216$		0			0
$217$	−8.99386	+	8.99386i	−0.610543	+	0.610543i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	8.96636i		−	0.603143i
$222$		0			0
$223$	19.2846	+	19.2846i	1.29139	+	1.29139i	0.933927	+	0.357464i	$0.116359\pi$
$223$	19.2846	+	19.2846i	1.29139	+	1.29139i	0.357464	+	0.933927i	$0.383641\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.44130	−	7.44130i	−0.493897	−	0.493897i	0.415635	−	0.909532i	$-0.363559\pi$
$227$	−7.44130	−	7.44130i	−0.493897	−	0.493897i	−0.909532	+	0.415635i	$0.863559\pi$
$228$		0			0
$229$			18.1978i			1.20254i	0.799044	+	0.601272i	$0.205339\pi$
$229$			18.1978i			1.20254i	−0.799044	+	0.601272i	$0.794661\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.4935	+	14.4935i	−0.949502	+	0.949502i	−0.998785	−	0.0492830i	$-0.984306\pi$
$233$	−14.4935	+	14.4935i	−0.949502	+	0.949502i	0.0492830	+	0.998785i	$0.484306\pi$
$234$		0			0
$235$	−5.16290	−	20.7792i	−0.336790	−	1.35549i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−26.6397			−1.72318			−0.861589	−	0.507607i	$-0.830530\pi$
$239$	−26.6397			−1.72318			−0.861589	+	0.507607i	$0.830530\pi$
$240$		0			0
$241$	6.99386			0.450514			0.225257	−	0.974299i	$-0.427678\pi$
$241$	6.99386			0.450514			0.225257	+	0.974299i	$0.427678\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.33399	−	2.21605i	0.0852256	−	0.141578i
$246$		0			0
$247$	−20.0989	+	20.0989i	−1.27886	+	1.27886i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	14.9525i		−	0.943796i	−0.881653	−	0.471898i	$-0.843569\pi$
$251$		−	14.9525i		−	0.943796i	0.881653	−	0.471898i	$-0.156431\pi$
$252$		0			0
$253$	5.47641	+	5.47641i	0.344299	+	0.344299i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.76980	+	9.76980i	0.609423	+	0.609423i	0.942795	−	0.333372i	$-0.108187\pi$
$257$	9.76980	+	9.76980i	0.609423	+	0.609423i	−0.333372	+	0.942795i	$0.608187\pi$
$258$		0			0
$259$			6.11118i			0.379730i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.56904	−	2.56904i	0.158414	−	0.158414i	−0.623450	−	0.781863i	$-0.714269\pi$
$263$	2.56904	−	2.56904i	0.158414	−	0.158414i	0.781863	+	0.623450i	$0.214269\pi$
$264$		0			0
$265$	−12.4969	+	3.10504i	−0.767680	+	0.190741i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.2411			−0.868294			−0.434147	−	0.900842i	$-0.642950\pi$
$269$	−14.2411			−0.868294			−0.434147	+	0.900842i	$0.642950\pi$
$270$		0			0
$271$	−27.8310			−1.69061			−0.845305	−	0.534284i	$-0.820581\pi$
$271$	−27.8310			−1.69061			−0.845305	+	0.534284i	$0.820581\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.94234	+	2.08796i	−0.237732	+	0.125909i
$276$		0			0
$277$	−0.709275	+	0.709275i	−0.0426162	+	0.0426162i	−0.728094	−	0.685478i	$-0.759593\pi$
$277$	−0.709275	+	0.709275i	−0.0426162	+	0.0426162i	0.685478	+	0.728094i	$0.259593\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	14.1712i		−	0.845380i	−0.906274	−	0.422690i	$-0.861086\pi$
$281$		−	14.1712i		−	0.845380i	0.906274	−	0.422690i	$-0.138914\pi$
$282$		0			0
$283$	−14.5236	−	14.5236i	−0.863338	−	0.863338i	0.128386	−	0.991724i	$-0.459020\pi$
$283$	−14.5236	−	14.5236i	−0.863338	−	0.863338i	−0.991724	+	0.128386i	$0.959020\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.189461	+	0.189461i	0.0111835	+	0.0111835i
$288$		0			0
$289$		−	11.5236i		−	0.677858i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−14.8260	+	14.8260i	−0.866147	+	0.866147i	−0.992043	−	0.125897i	$-0.959819\pi$
$293$	−14.8260	+	14.8260i	−0.866147	+	0.866147i	0.125897	+	0.992043i	$0.459819\pi$
$294$		0			0
$295$	20.2329	+	12.1795i	1.17800	+	0.709120i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	33.2587			1.92340
$300$		0			0
$301$	4.31351			0.248627
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−23.1784	−	13.9527i	−1.32719	−	0.798928i
$306$		0			0
$307$	15.4186	−	15.4186i	0.879983	−	0.879983i	−0.113549	−	0.993532i	$-0.536222\pi$
$307$	15.4186	−	15.4186i	0.879983	−	0.879983i	0.993532	+	0.113549i	$0.0362220\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			25.3740i			1.43883i	0.694581	+	0.719414i	$0.255590\pi$
$311$			25.3740i			1.43883i	−0.694581	+	0.719414i	$0.744410\pi$
$312$		0			0
$313$	20.9421	+	20.9421i	1.18372	+	1.18372i	0.978773	+	0.204947i	$0.0657021\pi$
$313$	20.9421	+	20.9421i	1.18372	+	1.18372i	0.204947	+	0.978773i	$0.434298\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.59709	−	5.59709i	−0.314364	−	0.314364i	0.532234	−	0.846598i	$-0.321353\pi$
$317$	−5.59709	−	5.59709i	−0.314364	−	0.314364i	−0.846598	+	0.532234i	$0.821353\pi$
$318$		0			0
$319$			0.465732i			0.0260760i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	12.2759	−	12.2759i	0.683047	−	0.683047i
$324$		0			0
$325$	−5.63090	+	18.3112i	−0.312346	+	1.01573i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	23.1462			1.27609
$330$		0			0
$331$	−23.6163			−1.29807			−0.649036	−	0.760758i	$-0.724827\pi$
$331$	−23.6163			−1.29807			−0.649036	+	0.760758i	$0.724827\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−16.1483	+	4.01227i	−0.882274	+	0.219214i
$336$		0			0
$337$	21.6537	−	21.6537i	1.17955	−	1.17955i	0.199693	−	0.979859i	$-0.436006\pi$
$337$	21.6537	−	21.6537i	1.17955	−	1.17955i	0.979859	−	0.199693i	$-0.0639944\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	4.69470i		−	0.254232i
$342$		0			0
$343$	13.9421	+	13.9421i	0.752805	+	0.752805i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.66912	+	9.66912i	0.519066	+	0.519066i	0.917289	−	0.398223i	$-0.130373\pi$
$347$	9.66912	+	9.66912i	0.519066	+	0.519066i	−0.398223	+	0.917289i	$0.630373\pi$
$348$		0			0
$349$		−	28.0989i		−	1.50410i	−0.659106	−	0.752050i	$-0.729065\pi$
$349$		−	28.0989i		−	1.50410i	0.659106	−	0.752050i	$-0.270935\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−18.0624	+	18.0624i	−0.961365	+	0.961365i	−0.999281	−	0.0379157i	$-0.987928\pi$
$353$	−18.0624	+	18.0624i	−0.961365	+	0.961365i	0.0379157	+	0.999281i	$0.487928\pi$
$354$		0			0
$355$	3.51745	−	5.84324i	0.186687	−	0.310127i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−17.7929			−0.939071			−0.469536	−	0.882914i	$-0.655579\pi$
$359$	−17.7929			−0.939071			−0.469536	+	0.882914i	$0.655579\pi$
$360$		0			0
$361$	−36.0349			−1.89657
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.59709	−	22.5267i	−0.292965	−	1.17910i
$366$		0			0
$367$	9.49466	−	9.49466i	0.495617	−	0.495617i	−0.414453	−	0.910071i	$-0.636027\pi$
$367$	9.49466	−	9.49466i	0.495617	−	0.495617i	0.910071	+	0.414453i	$0.136027\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	13.9204i		−	0.722714i
$372$		0			0
$373$	2.73594	+	2.73594i	0.141661	+	0.141661i	0.774381	−	0.632720i	$-0.218061\pi$
$373$	2.73594	+	2.73594i	0.141661	+	0.141661i	−0.632720	+	0.774381i	$0.718061\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.41421	+	1.41421i	0.0728357	+	0.0728357i
$378$		0			0
$379$			14.1568i			0.727184i	0.931558	+	0.363592i	$0.118450\pi$
$379$			14.1568i			0.727184i	−0.931558	+	0.363592i	$0.881550\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.51737	+	9.51737i	−0.486315	+	0.486315i	−0.907141	−	0.420826i	$-0.861740\pi$
$383$	−9.51737	+	9.51737i	−0.486315	+	0.486315i	0.420826	+	0.907141i	$0.361740\pi$
$384$		0			0
$385$	−1.16290	−	4.68035i	−0.0592668	−	0.238533i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.47608			0.277648			0.138824	−	0.990317i	$-0.455668\pi$
$389$	5.47608			0.277648			0.138824	+	0.990317i	$0.455668\pi$
$390$		0			0
$391$	−20.3135			−1.02730
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−15.2938	+	25.4062i	−0.769512	+	1.27833i
$396$		0			0
$397$	1.10731	−	1.10731i	0.0555742	−	0.0555742i	−0.678774	−	0.734348i	$-0.737488\pi$
$397$	1.10731	−	1.10731i	0.0555742	−	0.0555742i	0.734348	+	0.678774i	$0.237488\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			16.8801i			0.842949i	0.906840	+	0.421475i	$0.138487\pi$
$401$			16.8801i			0.842949i	−0.906840	+	0.421475i	$0.861513\pi$
$402$		0			0
$403$	−14.2557	−	14.2557i	−0.710125	−	0.710125i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.59499	−	1.59499i	−0.0790606	−	0.0790606i
$408$		0			0
$409$			4.68035i			0.231428i	0.993283	+	0.115714i	$0.0369156\pi$
$409$			4.68035i			0.231428i	−0.993283	+	0.115714i	$0.963084\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−18.0522	+	18.0522i	−0.888293	+	0.888293i
$414$		0			0
$415$	20.0989	−	4.99386i	0.986616	−	0.245139i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	21.0528			1.02849			0.514247	−	0.857642i	$-0.328071\pi$
$419$	21.0528			1.02849			0.514247	+	0.857642i	$0.328071\pi$
$420$		0			0
$421$	−20.8371			−1.01554			−0.507769	−	0.861493i	$-0.669530\pi$
$421$	−20.8371			−1.01554			−0.507769	+	0.861493i	$0.669530\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.43920	−	11.1840i	0.166826	−	0.542504i
$426$		0			0
$427$	20.6803	−	20.6803i	1.00079	−	1.00079i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	30.6520i		−	1.47645i	−0.674553	−	0.738226i	$-0.735664\pi$
$431$		−	30.6520i		−	1.47645i	0.674553	−	0.738226i	$-0.264336\pi$
$432$		0			0
$433$	−17.7792	−	17.7792i	−0.854416	−	0.854416i	0.136258	−	0.990673i	$-0.456493\pi$
$433$	−17.7792	−	17.7792i	−0.854416	−	0.854416i	−0.990673	+	0.136258i	$0.956493\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	45.5346	+	45.5346i	2.17821	+	2.17821i
$438$		0			0
$439$		−	36.5646i		−	1.74513i	−0.488494	−	0.872567i	$-0.662454\pi$
$439$		−	36.5646i		−	1.74513i	0.488494	−	0.872567i	$-0.337546\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.4914	+	10.4914i	−0.498462	+	0.498462i	−0.910959	−	0.412497i	$-0.864657\pi$
$443$	−10.4914	+	10.4914i	−0.498462	+	0.498462i	0.412497	+	0.910959i	$0.364657\pi$
$444$		0			0
$445$	−25.7298	−	15.4885i	−1.21971	−	0.734226i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.6461			−0.596806			−0.298403	−	0.954440i	$-0.596454\pi$
$449$	−12.6461			−0.596806			−0.298403	+	0.954440i	$0.596454\pi$
$450$		0			0
$451$	−0.0988967			−0.00465686
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−17.7433	−	10.6809i	−0.831817	−	0.500727i
$456$		0			0
$457$	−21.1256	+	21.1256i	−0.988212	+	0.988212i	−0.999931	−	0.0117194i	$-0.996270\pi$
$457$	−21.1256	+	21.1256i	−0.988212	+	0.988212i	0.0117194	+	0.999931i	$0.496270\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			3.38812i			0.157801i	0.996883	+	0.0789003i	$0.0251409\pi$
$461$			3.38812i			0.157801i	−0.996883	+	0.0789003i	$0.974859\pi$
$462$		0			0
$463$	−11.2846	−	11.2846i	−0.524439	−	0.524439i	0.394470	−	0.918909i	$-0.370928\pi$
$463$	−11.2846	−	11.2846i	−0.524439	−	0.524439i	−0.918909	+	0.394470i	$0.870928\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.59919	−	9.59919i	−0.444198	−	0.444198i	0.449222	−	0.893420i	$-0.351701\pi$
$467$	−9.59919	−	9.59919i	−0.444198	−	0.444198i	−0.893420	+	0.449222i	$0.851701\pi$
$468$		0			0
$469$		−	17.9877i		−	0.830595i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.12580	+	1.12580i	−0.0517645	+	0.0517645i
$474$		0			0
$475$	−32.7792	+	17.3607i	−1.50401	+	0.796563i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.61901			0.302430			0.151215	−	0.988501i	$-0.451681\pi$
$479$	6.61901			0.302430			0.151215	+	0.988501i	$0.451681\pi$
$480$		0			0
$481$	−9.68649			−0.441666
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−26.3991	+	6.55924i	−1.19872	+	0.297840i
$486$		0			0
$487$	−15.6514	+	15.6514i	−0.709233	+	0.709233i	−0.966374	−	0.257141i	$-0.917220\pi$
$487$	−15.6514	+	15.6514i	−0.709233	+	0.709233i	0.257141	+	0.966374i	$0.417220\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	29.7597i		−	1.34304i	−0.740987	−	0.671519i	$-0.765642\pi$
$491$		−	29.7597i		−	1.34304i	0.740987	−	0.671519i	$-0.234358\pi$
$492$		0			0
$493$	−0.863763	−	0.863763i	−0.0389019	−	0.0389019i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.21348	+	5.21348i	0.233857	+	0.233857i
$498$		0			0
$499$		−	11.3028i		−	0.505984i	−0.967468	−	0.252992i	$-0.918585\pi$
$499$		−	11.3028i		−	0.505984i	0.967468	−	0.252992i	$-0.0814147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5.80861	+	5.80861i	−0.258993	+	0.258993i	−0.824644	−	0.565651i	$-0.808625\pi$
$503$	−5.80861	+	5.80861i	−0.258993	+	0.258993i	0.565651	+	0.824644i	$0.308625\pi$
$504$		0			0
$505$	−13.0000	+	21.5958i	−0.578492	+	0.961002i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	35.3434			1.56657			0.783285	−	0.621662i	$-0.213542\pi$
$509$	35.3434			1.56657			0.783285	+	0.621662i	$0.213542\pi$
$510$		0			0
$511$	25.0928			1.11004
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.91011	+	15.7371i	0.172300	+	0.693460i
$516$		0			0
$517$	−6.04104	+	6.04104i	−0.265685	+	0.265685i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			24.9220i			1.09185i	0.837834	+	0.545925i	$0.183822\pi$
$521$			24.9220i			1.09185i	−0.837834	+	0.545925i	$0.816178\pi$
$522$		0			0
$523$	18.3545	+	18.3545i	0.802588	+	0.802588i	0.983499	−	0.180911i	$-0.0579046\pi$
$523$	18.3545	+	18.3545i	0.802588	+	0.802588i	−0.180911	+	0.983499i	$0.557905\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.70697	+	8.70697i	0.379281	+	0.379281i
$528$		0			0
$529$		−	52.3484i		−	2.27602i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.300304	+	0.300304i	−0.0130076	+	0.0130076i
$534$		0			0
$535$	3.63317	+	14.6225i	0.157075	+	0.632185i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.03208			−0.0444550
$540$		0			0
$541$	−13.6286			−0.585941			−0.292970	−	0.956122i	$-0.594644\pi$
$541$	−13.6286			−0.585941			−0.292970	+	0.956122i	$0.594644\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.58079	+	5.94847i	−0.153384	+	0.254804i
$546$		0			0
$547$	−24.1978	+	24.1978i	−1.03462	+	1.03462i	−0.0352442	+	0.999379i	$0.511221\pi$
$547$	−24.1978	+	24.1978i	−1.03462	+	1.03462i	−0.999379	+	0.0352442i	$0.988779\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			3.87241i			0.164970i
$552$		0			0
$553$	−22.6681	−	22.6681i	−0.963944	−	0.963944i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−23.2704	−	23.2704i	−0.985999	−	0.985999i	0.0139042	−	0.999903i	$-0.495574\pi$
$557$	−23.2704	−	23.2704i	−0.985999	−	0.985999i	−0.999903	+	0.0139042i	$0.995574\pi$
$558$		0			0
$559$			6.83710i			0.289179i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.60674	+	2.60674i	−0.109861	+	0.109861i	−0.759900	−	0.650039i	$-0.774752\pi$
$563$	−2.60674	+	2.60674i	−0.109861	+	0.109861i	0.650039	+	0.759900i	$0.274752\pi$
$564$		0			0
$565$	−15.5753	+	3.86991i	−0.655258	+	0.162808i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	35.7169			1.49733			0.748665	−	0.662949i	$-0.230695\pi$
$569$	35.7169			1.49733			0.748665	+	0.662949i	$0.230695\pi$
$570$		0			0
$571$	42.1399			1.76350			0.881751	−	0.471716i	$-0.156365\pi$
$571$	42.1399			1.76350			0.881751	+	0.471716i	$0.156365\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	41.4846	+	12.7569i	1.73003	+	0.532001i
$576$		0			0
$577$	28.6742	−	28.6742i	1.19372	−	1.19372i	0.217709	−	0.976014i	$-0.430142\pi$
$577$	28.6742	−	28.6742i	1.19372	−	1.19372i	0.976014	−	0.217709i	$-0.0698584\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			22.3884i			0.928826i
$582$		0			0
$583$	3.63317	+	3.63317i	0.150470	+	0.150470i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15.3959	−	15.3959i	−0.635457	−	0.635457i	0.313974	−	0.949431i	$-0.398339\pi$
$587$	−15.3959	−	15.3959i	−0.635457	−	0.635457i	−0.949431	+	0.313974i	$0.898339\pi$
$588$		0			0
$589$		−	39.0349i		−	1.60840i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−26.0956	+	26.0956i	−1.07162	+	1.07162i	−0.0743901	+	0.997229i	$0.523701\pi$
$593$	−26.0956	+	26.0956i	−1.07162	+	1.07162i	−0.997229	+	0.0743901i	$0.976299\pi$
$594$		0			0
$595$	10.8371	+	6.52359i	0.444278	+	0.267441i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	27.0831			1.10658			0.553292	−	0.832987i	$-0.313371\pi$
$599$	27.0831			1.10658			0.553292	+	0.832987i	$0.313371\pi$
$600$		0			0
$601$	−7.83096			−0.319431			−0.159716	−	0.987163i	$-0.551058\pi$
$601$	−7.83096			−0.319431			−0.159716	+	0.987163i	$0.551058\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−19.5482	−	11.7674i	−0.794746	−	0.478412i
$606$		0			0
$607$	−23.6514	+	23.6514i	−0.959981	+	0.959981i	−0.999229	−	0.0392481i	$-0.987504\pi$
$607$	−23.6514	+	23.6514i	−0.959981	+	0.959981i	0.0392481	+	0.999229i	$0.487504\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			36.6877i			1.48423i
$612$		0			0
$613$	−3.94441	−	3.94441i	−0.159313	−	0.159313i	0.622949	−	0.782262i	$-0.285934\pi$
$613$	−3.94441	−	3.94441i	−0.159313	−	0.159313i	−0.782262	+	0.622949i	$0.785934\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.3218	+	18.3218i	0.737608	+	0.737608i	0.972115	−	0.234506i	$-0.0753474\pi$
$617$	18.3218	+	18.3218i	0.737608	+	0.737608i	−0.234506	+	0.972115i	$0.575347\pi$
$618$		0			0
$619$			8.05332i			0.323690i	0.986816	+	0.161845i	$0.0517445\pi$
$619$			8.05332i			0.323690i	−0.986816	+	0.161845i	$0.948255\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	22.9567	−	22.9567i	0.919742	−	0.919742i
$624$		0			0
$625$	−14.0472	+	20.6803i	−0.561887	+	0.827214i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.91625			0.235896
$630$		0			0
$631$	−17.1461			−0.682575			−0.341287	−	0.939959i	$-0.610863\pi$
$631$	−17.1461			−0.682575			−0.341287	+	0.939959i	$0.610863\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	19.3060	−	4.79686i	0.766136	−	0.190357i
$636$		0			0
$637$	−3.13397	+	3.13397i	−0.124172	+	0.124172i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	26.0101i		−	1.02734i	−0.857989	−	0.513668i	$-0.828286\pi$
$641$		−	26.0101i		−	1.02734i	0.857989	−	0.513668i	$-0.171714\pi$
$642$		0			0
$643$	13.3607	+	13.3607i	0.526894	+	0.526894i	0.919645	−	0.392751i	$-0.128476\pi$
$643$	13.3607	+	13.3607i	0.526894	+	0.526894i	−0.392751	+	0.919645i	$0.628476\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	19.0466	+	19.0466i	0.748800	+	0.748800i	0.974254	−	0.225454i	$-0.0723864\pi$
$647$	19.0466	+	19.0466i	0.748800	+	0.748800i	−0.225454	+	0.974254i	$0.572386\pi$
$648$		0			0
$649$		−	9.42309i		−	0.369888i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.1263	+	15.1263i	−0.591940	+	0.591940i	−0.938155	−	0.346215i	$-0.887467\pi$
$653$	−15.1263	+	15.1263i	−0.591940	+	0.591940i	0.346215	+	0.938155i	$0.387467\pi$
$654$		0			0
$655$	0.0806452	−	0.133969i	0.00315107	−	0.00523461i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−44.6423			−1.73902			−0.869509	−	0.493917i	$-0.835565\pi$
$659$	−44.6423			−1.73902			−0.869509	+	0.493917i	$0.835565\pi$
$660$		0			0
$661$	−14.4247			−0.561056			−0.280528	−	0.959846i	$-0.590509\pi$
$661$	−14.4247			−0.561056			−0.280528	+	0.959846i	$0.590509\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−9.66912	−	38.9155i	−0.374952	−	1.50908i
$666$		0			0
$667$	3.20394	−	3.20394i	0.124057	−	0.124057i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			10.7949i			0.416733i
$672$		0			0
$673$	1.21461	+	1.21461i	0.0468199	+	0.0468199i	0.730129	−	0.683309i	$-0.239460\pi$
$673$	1.21461	+	1.21461i	0.0468199	+	0.0468199i	−0.683309	+	0.730129i	$0.739460\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	23.3899	+	23.3899i	0.898949	+	0.898949i	0.995343	−	0.0963946i	$-0.0307311\pi$
$677$	23.3899	+	23.3899i	0.898949	+	0.898949i	−0.0963946	+	0.995343i	$0.530731\pi$
$678$		0			0
$679$		−	29.4063i		−	1.12851i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−25.8873	+	25.8873i	−0.990551	+	0.990551i	−0.999956	−	0.00940495i	$-0.997006\pi$
$683$	−25.8873	+	25.8873i	−0.990551	+	0.990551i	0.00940495	+	0.999956i	$0.497006\pi$
$684$		0			0
$685$	−3.50307	−	14.0989i	−0.133845	−	0.538691i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	22.0645			0.840591
$690$		0			0
$691$	1.94214			0.0738825			0.0369413	−	0.999317i	$-0.488239\pi$
$691$	1.94214			0.0738825			0.0369413	+	0.999317i	$0.488239\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	16.3258	−	27.1208i	0.619274	−	1.02875i
$696$		0			0
$697$	0.183417	−	0.183417i	0.00694743	−	0.00694743i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	8.62835i		−	0.325888i	−0.986635	−	0.162944i	$-0.947901\pi$
$701$		−	8.62835i		−	0.325888i	0.986635	−	0.162944i	$-0.0520990\pi$
$702$		0			0
$703$	−13.2618	−	13.2618i	−0.500178	−	0.500178i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−19.2683	−	19.2683i	−0.724659	−	0.724659i
$708$		0			0
$709$			9.88428i			0.371212i	0.982624	+	0.185606i	$0.0594248\pi$
$709$			9.88428i			0.371212i	−0.982624	+	0.185606i	$0.940575\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−32.2965	+	32.2965i	−1.20951	+	1.20951i
$714$		0			0
$715$	7.41855	−	1.84324i	0.277438	−	0.0689334i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.60674			−0.0972150			−0.0486075	−	0.998818i	$-0.515478\pi$
$719$	−2.60674			−0.0972150			−0.0486075	+	0.998818i	$0.515478\pi$
$720$		0			0
$721$	−17.5297			−0.652841
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.22155	+	2.30644i	0.0453671	+	0.0856589i
$726$		0			0
$727$	25.0121	−	25.0121i	0.927648	−	0.927648i	−0.0699058	−	0.997554i	$-0.522270\pi$
$727$	25.0121	−	25.0121i	0.927648	−	0.927648i	0.997554	+	0.0699058i	$0.0222699\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	4.17592i		−	0.154452i
$732$		0			0
$733$	−6.06997	−	6.06997i	−0.224199	−	0.224199i	0.586065	−	0.810264i	$-0.300676\pi$
$733$	−6.06997	−	6.06997i	−0.224199	−	0.224199i	−0.810264	+	0.586065i	$0.800676\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.69470	+	4.69470i	0.172931	+	0.172931i
$738$		0			0
$739$			22.8827i			0.841753i	0.907118	+	0.420876i	$0.138277\pi$
$739$			22.8827i			0.841753i	−0.907118	+	0.420876i	$0.861723\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.9965	−	15.9965i	0.586855	−	0.586855i	−0.349923	−	0.936778i	$-0.613792\pi$
$743$	15.9965	−	15.9965i	0.586855	−	0.586855i	0.936778	+	0.349923i	$0.113792\pi$
$744$		0			0
$745$	−45.5380	−	27.4124i	−1.66838	−	1.00431i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−16.2881			−0.595155
$750$		0			0
$751$	−7.78539			−0.284093			−0.142046	−	0.989860i	$-0.545368\pi$
$751$	−7.78539			−0.284093			−0.142046	+	0.989860i	$0.545368\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−14.4015	−	8.66926i	−0.524125	−	0.315507i
$756$		0			0
$757$	4.02893	−	4.02893i	0.146434	−	0.146434i	−0.630089	−	0.776523i	$-0.716982\pi$
$757$	4.02893	−	4.02893i	0.146434	−	0.146434i	0.776523	+	0.630089i	$0.216982\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			33.5535i			1.21631i	0.793817	+	0.608157i	$0.208091\pi$
$761$			33.5535i			1.21631i	−0.793817	+	0.608157i	$0.791909\pi$
$762$		0			0
$763$	−5.30737	−	5.30737i	−0.192140	−	0.192140i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−28.6136	−	28.6136i	−1.03318	−	1.03318i
$768$		0			0
$769$			8.99386i			0.324327i	0.986764	+	0.162163i	$0.0518472\pi$
$769$			8.99386i			0.324327i	−0.986764	+	0.162163i	$0.948153\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20.4797	+	20.4797i	−0.736603	+	0.736603i	−0.971919	−	0.235316i	$-0.924388\pi$
$773$	−20.4797	+	20.4797i	−0.736603	+	0.736603i	0.235316	+	0.971919i	$0.424388\pi$
$774$		0			0
$775$	−12.3135	−	23.2495i	−0.442314	−	0.835147i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.822293			−0.0294617
$780$		0			0
$781$	−2.72138			−0.0973788
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−48.2742	+	11.9944i	−1.72298	+	0.428099i
$786$		0			0
$787$	−15.7321	+	15.7321i	−0.560787	+	0.560787i	−0.929531	−	0.368744i	$-0.879788\pi$
$787$	−15.7321	+	15.7321i	−0.560787	+	0.560787i	0.368744	+	0.929531i	$0.379788\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	17.3495i		−	0.616877i
$792$		0			0
$793$	32.7792	+	32.7792i	1.16403	+	1.16403i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.11294	+	4.11294i	0.145688	+	0.145688i	0.776189	−	0.630501i	$-0.217150\pi$
$797$	4.11294	+	4.11294i	0.145688	+	0.145688i	−0.630501	+	0.776189i	$0.717150\pi$
$798$		0			0
$799$		−	22.4079i		−	0.792734i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.54908	+	6.54908i	−0.231112	+	0.231112i
$804$		0			0
$805$	−24.1978	+	40.1978i	−0.852860	+	1.41679i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−25.3276			−0.890472			−0.445236	−	0.895413i	$-0.646880\pi$
$809$	−25.3276			−0.890472			−0.445236	+	0.895413i	$0.646880\pi$
$810$		0			0
$811$	−14.3545			−0.504056			−0.252028	−	0.967720i	$-0.581098\pi$
$811$	−14.3545			−0.504056			−0.252028	+	0.967720i	$0.581098\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.01227	+	16.1483i	0.140544	+	0.565649i
$816$		0			0
$817$	−9.36069	+	9.36069i	−0.327489	+	0.327489i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	14.4424i		−	0.504045i	−0.967721	−	0.252022i	$-0.918904\pi$
$821$		−	14.4424i		−	0.504045i	0.967721	−	0.252022i	$-0.0810956\pi$
$822$		0			0
$823$	2.65756	+	2.65756i	0.0926367	+	0.0926367i	0.751906	−	0.659270i	$-0.229134\pi$
$823$	2.65756	+	2.65756i	0.0926367	+	0.0926367i	−0.659270	+	0.751906i	$0.729134\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.139861	+	0.139861i	0.00486344	+	0.00486344i	0.709534	−	0.704671i	$-0.248905\pi$
$827$	0.139861	+	0.139861i	0.00486344	+	0.00486344i	−0.704671	+	0.709534i	$0.748905\pi$
$828$		0			0
$829$		−	15.2208i		−	0.528639i	−0.964435	−	0.264319i	$-0.914853\pi$
$829$		−	15.2208i		−	0.528639i	0.964435	−	0.264319i	$-0.0851473\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.91414	−	1.91414i	0.0663211	−	0.0663211i
$834$		0			0
$835$	−0.993857	−	4.00000i	−0.0343938	−	0.138426i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21.3618			0.737490			0.368745	−	0.929531i	$-0.379788\pi$
$839$	21.3618			0.737490			0.368745	+	0.929531i	$0.379788\pi$
$840$		0			0
$841$	−28.7275			−0.990604
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.93781	−	3.21912i	0.0666626	−	0.110741i
$846$		0			0
$847$	17.4413	−	17.4413i	0.599291	−	0.599291i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			21.9450i			0.752264i
$852$		0			0
$853$	−3.88655	−	3.88655i	−0.133073	−	0.133073i	0.637433	−	0.770506i	$-0.279996\pi$
$853$	−3.88655	−	3.88655i	−0.133073	−	0.133073i	−0.770506	+	0.637433i	$0.779996\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−13.4872	−	13.4872i	−0.460715	−	0.460715i	0.438174	−	0.898890i	$-0.355625\pi$
$857$	−13.4872	−	13.4872i	−0.460715	−	0.460715i	−0.898890	+	0.438174i	$0.855625\pi$
$858$		0			0
$859$			10.6681i			0.363990i	0.983300	+	0.181995i	$0.0582554\pi$
$859$			10.6681i			0.363990i	−0.983300	+	0.181995i	$0.941745\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.620938	−	0.620938i	0.0211370	−	0.0211370i	−0.696459	−	0.717596i	$-0.745242\pi$
$863$	0.620938	−	0.620938i	0.0211370	−	0.0211370i	0.717596	+	0.696459i	$0.245242\pi$
$864$		0			0
$865$	22.0989	−	5.49079i	0.751385	−	0.186692i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	11.8325			0.401390
$870$		0			0
$871$	28.5113			0.966069
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−18.0349	−	20.1283i	−0.609690	−	0.680461i
$876$		0			0
$877$	15.3051	−	15.3051i	0.516817	−	0.516817i	−0.399790	−	0.916607i	$-0.630917\pi$
$877$	15.3051	−	15.3051i	0.516817	−	0.516817i	0.916607	+	0.399790i	$0.130917\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	35.5974i		−	1.19931i	−0.800260	−	0.599653i	$-0.795305\pi$
$881$		−	35.5974i		−	1.19931i	0.800260	−	0.599653i	$-0.204695\pi$
$882$		0			0
$883$	4.09890	+	4.09890i	0.137939	+	0.137939i	0.772705	−	0.634766i	$-0.218903\pi$
$883$	4.09890	+	4.09890i	0.137939	+	0.137939i	−0.634766	+	0.772705i	$0.718903\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.5291	−	10.5291i	−0.353533	−	0.353533i	0.507889	−	0.861422i	$-0.330426\pi$
$887$	−10.5291	−	10.5291i	−0.353533	−	0.353533i	−0.861422	+	0.507889i	$0.830426\pi$
$888$		0			0
$889$			21.5052i			0.721260i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−50.2293	+	50.2293i	−1.68086	+	1.68086i
$894$		0			0
$895$	−43.4824	−	26.1750i	−1.45346	−	0.874934i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.74660			−0.0916043
$900$		0			0
$901$	−13.4764			−0.448964
$902$		0			0
$903$		0			0
$904$		0			0
$905$	22.4757	+	13.5296i	0.747116	+	0.449740i
$906$		0			0
$907$	−14.2557	+	14.2557i	−0.473351	+	0.473351i	−0.902997	−	0.429646i	$-0.858638\pi$
$907$	−14.2557	+	14.2557i	−0.473351	+	0.473351i	0.429646	+	0.902997i	$0.358638\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			38.0933i			1.26209i	0.775748	+	0.631043i	$0.217373\pi$
$911$			38.0933i			1.26209i	−0.775748	+	0.631043i	$0.782627\pi$
$912$		0			0
$913$	−5.84324	−	5.84324i	−0.193383	−	0.193383i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.119530	+	0.119530i	0.00394724	+	0.00394724i
$918$		0			0
$919$			29.4596i			0.971782i	0.874019	+	0.485891i	$0.161505\pi$
$919$			29.4596i			0.971782i	−0.874019	+	0.485891i	$0.838495\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.26360	+	8.26360i	−0.272000	+	0.272000i
$924$		0			0
$925$	−12.0823	−	3.71542i	−0.397262	−	0.122162i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−25.3653			−0.832210			−0.416105	−	0.909317i	$-0.636605\pi$
$929$	−25.3653			−0.832210			−0.416105	+	0.909317i	$0.636605\pi$
$930$		0			0
$931$	−8.58145			−0.281246
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.53105	+	1.12580i	−0.148181	+	0.0368177i
$936$		0			0
$937$	−12.7275	+	12.7275i	−0.415790	+	0.415790i	−0.883750	−	0.467960i	$-0.844989\pi$
$937$	−12.7275	+	12.7275i	−0.415790	+	0.415790i	0.467960	+	0.883750i	$0.344989\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			14.5446i			0.474140i	0.971493	+	0.237070i	$0.0761871\pi$
$941$			14.5446i			0.474140i	−0.971493	+	0.237070i	$0.923813\pi$
$942$		0			0
$943$	0.680346	+	0.680346i	0.0221551	+	0.0221551i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	38.0933	+	38.0933i	1.23786	+	1.23786i	0.960872	+	0.276992i	$0.0893376\pi$
$947$	38.0933	+	38.0933i	1.23786	+	1.23786i	0.276992	+	0.960872i	$0.410662\pi$
$948$		0			0
$949$			39.7731i			1.29109i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.619461	+	0.619461i	−0.0200663	+	0.0200663i	−0.717069	−	0.697002i	$-0.754517\pi$
$953$	−0.619461	+	0.619461i	−0.0200663	+	0.0200663i	0.697002	+	0.717069i	$0.254517\pi$
$954$		0			0
$955$	9.52973	−	15.8310i	0.308375	−	0.512278i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	15.7049			0.507138
$960$		0			0
$961$	−3.31351			−0.106887
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.06397	−	32.4553i	−0.259588	−	1.04477i
$966$		0			0
$967$	23.8660	−	23.8660i	0.767480	−	0.767480i	−0.210182	−	0.977662i	$-0.567406\pi$
$967$	23.8660	−	23.8660i	0.767480	−	0.767480i	0.977662	+	0.210182i	$0.0674058\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			43.5403i			1.39728i	0.715476	+	0.698638i	$0.246210\pi$
$971$			43.5403i			1.39728i	−0.715476	+	0.698638i	$0.753790\pi$
$972$		0			0
$973$	24.1978	+	24.1978i	0.775746	+	0.775746i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	30.6418	+	30.6418i	0.980318	+	0.980318i	0.999810	−	0.0194924i	$-0.00620502\pi$
$977$	30.6418	+	30.6418i	0.980318	+	0.980318i	−0.0194924	+	0.999810i	$0.506205\pi$
$978$		0			0
$979$			11.9832i			0.382984i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.07606	+	2.07606i	−0.0662162	+	0.0662162i	−0.739439	−	0.673223i	$-0.764909\pi$
$983$	−2.07606	+	2.07606i	−0.0662162	+	0.0662162i	0.673223	+	0.739439i	$0.264909\pi$
$984$		0			0
$985$	−1.60650	−	6.46573i	−0.0511874	−	0.206015i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	15.4896			0.492541
$990$		0			0
$991$	−24.1445			−0.766974			−0.383487	−	0.923546i	$-0.625277\pi$
$991$	−24.1445			−0.766974			−0.383487	+	0.923546i	$0.625277\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.44116	+	14.0226i	−0.267603	+	0.444546i
$996$		0			0
$997$	12.4680	−	12.4680i	0.394865	−	0.394865i	−0.481552	−	0.876418i	$-0.659927\pi$
$997$	12.4680	−	12.4680i	0.394865	−	0.394865i	0.876418	+	0.481552i	$0.159927\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.w.p.2753.1		12
3.2	odd	2	inner	2880.2.w.p.2753.6		12
4.3	odd	2		2880.2.w.q.2753.1		12
5.2	odd	4	inner	2880.2.w.p.2177.6		12
8.3	odd	2		1440.2.w.g.1313.6	yes	12
8.5	even	2		1440.2.w.f.1313.6	yes	12
12.11	even	2		2880.2.w.q.2753.6		12
15.2	even	4	inner	2880.2.w.p.2177.1		12
20.7	even	4		2880.2.w.q.2177.6		12
24.5	odd	2		1440.2.w.f.1313.1	yes	12
24.11	even	2		1440.2.w.g.1313.1	yes	12
40.27	even	4		1440.2.w.g.737.1	yes	12
40.37	odd	4		1440.2.w.f.737.1	✓	12
60.47	odd	4		2880.2.w.q.2177.1		12
120.77	even	4		1440.2.w.f.737.6	yes	12
120.107	odd	4		1440.2.w.g.737.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.w.f.737.1	✓	12	40.37	odd	4
1440.2.w.f.737.6	yes	12	120.77	even	4
1440.2.w.f.1313.1	yes	12	24.5	odd	2
1440.2.w.f.1313.6	yes	12	8.5	even	2
1440.2.w.g.737.1	yes	12	40.27	even	4
1440.2.w.g.737.6	yes	12	120.107	odd	4
1440.2.w.g.1313.1	yes	12	24.11	even	2
1440.2.w.g.1313.6	yes	12	8.3	odd	2
2880.2.w.p.2177.1		12	15.2	even	4	inner
2880.2.w.p.2177.6		12	5.2	odd	4	inner
2880.2.w.p.2753.1		12	1.1	even	1	trivial
2880.2.w.p.2753.6		12	3.2	odd	2	inner
2880.2.w.q.2177.1		12	60.47	odd	4
2880.2.w.q.2177.6		12	20.7	even	4
2880.2.w.q.2753.1		12	4.3	odd	2
2880.2.w.q.2753.6		12	12.11	even	2

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.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.91575 - 1.15322i) q^{5} +(1.70928 - 1.70928i) q^{7} +O(q^{10})\)	\(q+(-1.91575 - 1.15322i) q^{5} +(1.70928 - 1.70928i) q^{7} +0.892224i q^{11} +(2.70928 + 2.70928i) q^{13} +(-1.65475 - 1.65475i) q^{17} +7.41855i q^{19} +(6.13793 - 6.13793i) q^{23} +(2.34017 + 4.41855i) q^{25} +0.521990 q^{29} -5.26180 q^{31} +(-5.24571 + 1.30337i) q^{35} +(-1.78765 + 1.78765i) q^{37} +0.110843i q^{41} +(1.26180 + 1.26180i) q^{43} +(6.77076 + 6.77076i) q^{47} +1.15676i q^{49} +(4.07203 - 4.07203i) q^{53} +(1.02893 - 1.70928i) q^{55} -10.5613 q^{59} +12.0989 q^{61} +(-2.06590 - 8.31467i) q^{65} +(5.26180 - 5.26180i) q^{67} +3.05011i q^{71} +(7.34017 + 7.34017i) q^{73} +(1.52506 + 1.52506i) q^{77} -13.2618i q^{79} +(-6.54908 + 6.54908i) q^{83} +(1.26180 + 5.07838i) q^{85} +13.4307 q^{89} +9.26180 q^{91} +(8.55521 - 14.2121i) q^{95} +(8.60197 - 8.60197i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{7}+O(q^{10}) \) 12 * q - 8 * q^7	\( 12 q - 8 q^{7} + 4 q^{13} - 16 q^{25} - 32 q^{31} + 20 q^{37} - 16 q^{43} + 72 q^{55} + 32 q^{67} + 44 q^{73} - 16 q^{85} + 80 q^{91} + 28 q^{97}+O(q^{100}) \) 12 * q - 8 * q^7 + 4 * q^13 - 16 * q^25 - 32 * q^31 + 20 * q^37 - 16 * q^43 + 72 * q^55 + 32 * q^67 + 44 * q^73 - 16 * q^85 + 80 * q^91 + 28 * q^97

Introduction

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