LMFDB - Embedded newform 2880.2.w.p.2177.1

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			2177.1
Root			$-1.53448 - 1.53448i$ of defining polynomial
Character	$\chi$	$=$	2880.2177
Dual form			2880.2.w.p.2753.1

$q$-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 $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

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

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	0.952035	−	0.305990i	$-0.0989873\pi$
$7$	1.70928	+	1.70928i	0.646045	+	0.646045i	−0.305990	+	0.952035i	$0.598987\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.892224i		−	0.269016i	−0.990913	−	0.134508i	$-0.957055\pi$
$11$		−	0.892224i		−	0.269016i	0.990913	−	0.134508i	$-0.0429453\pi$
$12$		0			0
$13$	2.70928	−	2.70928i	0.751418	−	0.751418i	−0.223326	−	0.974744i	$-0.571691\pi$
$13$	2.70928	−	2.70928i	0.751418	−	0.751418i	0.974744	+	0.223326i	$0.0716915\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.65475	+	1.65475i	−0.401336	+	0.401336i	−0.878704	−	0.477367i	$-0.841591\pi$
$17$	−1.65475	+	1.65475i	−0.401336	+	0.401336i	0.477367	+	0.878704i	$0.341591\pi$
$18$		0			0
$19$		−	7.41855i		−	1.70193i	−0.525221	−	0.850966i	$-0.676017\pi$
$19$		−	7.41855i		−	1.70193i	0.525221	−	0.850966i	$-0.323983\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.13793	+	6.13793i	1.27985	+	1.27985i	0.940752	+	0.339095i	$0.110121\pi$
$23$	6.13793	+	6.13793i	1.27985	+	1.27985i	0.339095	+	0.940752i	$0.389879\pi$
$24$		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	0.544726	−	0.838614i	$-0.316634\pi$
$37$	−1.78765	−	1.78765i	−0.293888	−	0.293888i	−0.838614	+	0.544726i	$0.816634\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	0.110843i		−	0.0173107i	−0.999963	−	0.00865537i	$-0.997245\pi$
$41$		−	0.110843i		−	0.0173107i	0.999963	−	0.00865537i	$-0.00275513\pi$
$42$		0			0
$43$	1.26180	−	1.26180i	0.192422	−	0.192422i	−0.604320	−	0.796742i	$-0.706555\pi$
$43$	1.26180	−	1.26180i	0.192422	−	0.192422i	0.796742	+	0.604320i	$0.206555\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.77076	−	6.77076i	0.987617	−	0.987617i	−0.0123068	−	0.999924i	$-0.503917\pi$
$47$	6.77076	−	6.77076i	0.987617	−	0.987617i	0.999924	+	0.0123068i	$0.00391749\pi$
$48$		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	0.929119	−	0.369782i	$-0.120568\pi$
$53$	4.07203	+	4.07203i	0.559337	+	0.559337i	−0.369782	+	0.929119i	$0.620568\pi$
$54$		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	0.951250	−	0.308420i	$-0.0998001\pi$
$67$	5.26180	+	5.26180i	0.642831	+	0.642831i	−0.308420	+	0.951250i	$0.599800\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.05011i		−	0.361982i	−0.983485	−	0.180991i	$-0.942070\pi$
$71$		−	3.05011i		−	0.361982i	0.983485	−	0.180991i	$-0.0579305\pi$
$72$		0			0
$73$	7.34017	−	7.34017i	0.859102	−	0.859102i	−0.132130	−	0.991232i	$-0.542182\pi$
$73$	7.34017	−	7.34017i	0.859102	−	0.859102i	0.991232	+	0.132130i	$0.0421817\pi$
$74$		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	$0.268045\pi$
$79$			13.2618i			1.49207i	−0.665908	+	0.746034i	$0.731955\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.54908	−	6.54908i	−0.718855	−	0.718855i	0.249516	−	0.968371i	$-0.419728\pi$
$83$	−6.54908	−	6.54908i	−0.718855	−	0.718855i	−0.968371	+	0.249516i	$0.919728\pi$
$84$		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	0.992841	−	0.119443i	$-0.0381110\pi$
$97$	8.60197	+	8.60197i	0.873398	+	0.873398i	−0.119443	+	0.992841i	$0.538111\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			11.2728i			1.12169i	0.827922	+	0.560843i	$0.189523\pi$
$101$			11.2728i			1.12169i	−0.827922	+	0.560843i	$0.810477\pi$
$102$		0			0
$103$	−5.12783	+	5.12783i	−0.505260	+	0.505260i	−0.913068	−	0.407808i	$-0.866293\pi$
$103$	−5.12783	+	5.12783i	−0.505260	+	0.505260i	0.407808	+	0.913068i	$0.366293\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.76463	+	4.76463i	−0.460614	+	0.460614i	−0.898857	−	0.438243i	$-0.855601\pi$
$107$	−4.76463	+	4.76463i	−0.460614	+	0.460614i	0.438243	+	0.898857i	$0.355601\pi$
$108$		0			0
$109$			3.10504i			0.297409i	0.988882	+	0.148704i	$0.0475103\pi$
$109$			3.10504i			0.297409i	−0.988882	+	0.148704i	$0.952490\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.07510	+	5.07510i	0.477425	+	0.477425i	0.904307	−	0.426882i	$-0.140388\pi$
$113$	5.07510	+	5.07510i	0.477425	+	0.477425i	−0.426882	+	0.904307i	$0.640388\pi$
$114$		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	0.370586	−	0.928798i	$-0.379157\pi$
$127$	−6.29072	−	6.29072i	−0.558212	−	0.558212i	−0.928798	+	0.370586i	$0.879157\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	0.0699305i		−	0.00610986i	−0.999995	−	0.00305493i	$-0.999028\pi$
$131$		−	0.0699305i		−	0.00610986i	0.999995	−	0.00305493i	$-0.000972415\pi$
$132$		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	−0.483082	−	0.875575i	$-0.660483\pi$
$137$	4.59402	−	4.59402i	0.392494	−	0.392494i	0.875575	+	0.483082i	$0.160483\pi$
$138$		0			0
$139$		−	14.1568i		−	1.20076i	−0.799715	−	0.600380i	$-0.795016\pi$
$139$		−	14.1568i		−	1.20076i	0.799715	−	0.600380i	$-0.204984\pi$
$140$		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	0.953277	+	0.302097i	$0.0976867\pi$
$157$	15.7298	+	15.7298i	1.25537	+	1.25537i	0.302097	+	0.953277i	$0.402313\pi$
$158$		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	−0.882482	−	0.470346i	$-0.844129\pi$
$163$	−5.26180	+	5.26180i	−0.412136	+	0.412136i	0.470346	+	0.882482i	$0.344129\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.30337	−	1.30337i	0.100858	−	0.100858i	−0.654877	−	0.755735i	$-0.727280\pi$
$167$	1.30337	−	1.30337i	0.100858	−	0.100858i	0.755735	+	0.654877i	$0.227280\pi$
$168$		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	0.378243	−	0.925707i	$-0.376529\pi$
$173$	−7.20076	−	7.20076i	−0.547464	−	0.547464i	−0.925707	+	0.378243i	$0.876529\pi$
$174$		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	$-0.903358\pi$
$191$		−	8.26360i		−	0.597933i	0.954264	−	0.298966i	$-0.0966419\pi$
$192$		0			0
$193$	10.5753	−	10.5753i	0.761227	−	0.761227i	−0.215317	−	0.976544i	$-0.569078\pi$
$193$	10.5753	−	10.5753i	0.761227	−	0.761227i	0.976544	+	0.215317i	$0.0690785\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.10681	−	2.10681i	0.150104	−	0.150104i	−0.628060	−	0.778165i	$-0.716151\pi$
$197$	2.10681	−	2.10681i	0.150104	−	0.150104i	0.778165	+	0.628060i	$0.216151\pi$
$198$		0			0
$199$			7.31965i			0.518877i	0.965760	+	0.259438i	$0.0835374\pi$
$199$			7.31965i			0.518877i	−0.965760	+	0.259438i	$0.916463\pi$
$200$		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	0.357464	−	0.933927i	$-0.383641\pi$
$223$	19.2846	−	19.2846i	1.29139	−	1.29139i	0.933927	−	0.357464i	$-0.116359\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.44130	+	7.44130i	−0.493897	+	0.493897i	−0.909532	−	0.415635i	$-0.863559\pi$
$227$	−7.44130	+	7.44130i	−0.493897	+	0.493897i	0.415635	+	0.909532i	$0.363559\pi$
$228$		0			0
$229$		−	18.1978i		−	1.20254i	−0.799044	−	0.601272i	$-0.794661\pi$
$229$		−	18.1978i		−	1.20254i	0.799044	−	0.601272i	$-0.205339\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.4935	−	14.4935i	−0.949502	−	0.949502i	0.0492830	−	0.998785i	$-0.484306\pi$
$233$	−14.4935	−	14.4935i	−0.949502	−	0.949502i	−0.998785	+	0.0492830i	$0.984306\pi$
$234$		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	$0.156431\pi$
$251$			14.9525i			0.943796i	−0.881653	+	0.471898i	$0.843569\pi$
$252$		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	−0.333372	−	0.942795i	$-0.608187\pi$
$257$	9.76980	−	9.76980i	0.609423	−	0.609423i	0.942795	+	0.333372i	$0.108187\pi$
$258$		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	0.781863	−	0.623450i	$-0.214269\pi$
$263$	2.56904	+	2.56904i	0.158414	+	0.158414i	−0.623450	+	0.781863i	$0.714269\pi$
$264$		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	0.685478	−	0.728094i	$-0.259593\pi$
$277$	−0.709275	−	0.709275i	−0.0426162	−	0.0426162i	−0.728094	+	0.685478i	$0.759593\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			14.1712i			0.845380i	0.906274	+	0.422690i	$0.138914\pi$
$281$			14.1712i			0.845380i	−0.906274	+	0.422690i	$0.861086\pi$
$282$		0			0
$283$	−14.5236	+	14.5236i	−0.863338	+	0.863338i	−0.991724	−	0.128386i	$-0.959020\pi$
$283$	−14.5236	+	14.5236i	−0.863338	+	0.863338i	0.128386	+	0.991724i	$0.459020\pi$
$284$		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	0.125897	−	0.992043i	$-0.459819\pi$
$293$	−14.8260	−	14.8260i	−0.866147	−	0.866147i	−0.992043	+	0.125897i	$0.959819\pi$
$294$		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	0.993532	−	0.113549i	$-0.0362220\pi$
$307$	15.4186	+	15.4186i	0.879983	+	0.879983i	−0.113549	+	0.993532i	$0.536222\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	25.3740i		−	1.43883i	−0.694581	−	0.719414i	$-0.744410\pi$
$311$		−	25.3740i		−	1.43883i	0.694581	−	0.719414i	$-0.255590\pi$
$312$		0			0
$313$	20.9421	−	20.9421i	1.18372	−	1.18372i	0.204947	−	0.978773i	$-0.434298\pi$
$313$	20.9421	−	20.9421i	1.18372	−	1.18372i	0.978773	−	0.204947i	$-0.0657021\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.59709	+	5.59709i	−0.314364	+	0.314364i	−0.846598	−	0.532234i	$-0.821353\pi$
$317$	−5.59709	+	5.59709i	−0.314364	+	0.314364i	0.532234	+	0.846598i	$0.321353\pi$
$318$		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	0.979859	+	0.199693i	$0.0639944\pi$
$337$	21.6537	+	21.6537i	1.17955	+	1.17955i	0.199693	+	0.979859i	$0.436006\pi$
$338$		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	−0.398223	−	0.917289i	$-0.630373\pi$
$347$	9.66912	−	9.66912i	0.519066	−	0.519066i	0.917289	+	0.398223i	$0.130373\pi$
$348$		0			0
$349$			28.0989i			1.50410i	0.659106	+	0.752050i	$0.270935\pi$
$349$			28.0989i			1.50410i	−0.659106	+	0.752050i	$0.729065\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−18.0624	−	18.0624i	−0.961365	−	0.961365i	0.0379157	−	0.999281i	$-0.487928\pi$
$353$	−18.0624	−	18.0624i	−0.961365	−	0.961365i	−0.999281	+	0.0379157i	$0.987928\pi$
$354$		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	0.910071	−	0.414453i	$-0.136027\pi$
$367$	9.49466	+	9.49466i	0.495617	+	0.495617i	−0.414453	+	0.910071i	$0.636027\pi$
$368$		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	−0.632720	−	0.774381i	$-0.718061\pi$
$373$	2.73594	−	2.73594i	0.141661	−	0.141661i	0.774381	+	0.632720i	$0.218061\pi$
$374$		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	$-0.881550\pi$
$379$		−	14.1568i		−	0.727184i	0.931558	−	0.363592i	$-0.118450\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.51737	−	9.51737i	−0.486315	−	0.486315i	0.420826	−	0.907141i	$-0.361740\pi$
$383$	−9.51737	−	9.51737i	−0.486315	−	0.486315i	−0.907141	+	0.420826i	$0.861740\pi$
$384$		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	0.734348	−	0.678774i	$-0.237488\pi$
$397$	1.10731	+	1.10731i	0.0555742	+	0.0555742i	−0.678774	+	0.734348i	$0.737488\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	16.8801i		−	0.842949i	−0.906840	−	0.421475i	$-0.861513\pi$
$401$		−	16.8801i		−	0.842949i	0.906840	−	0.421475i	$-0.138487\pi$
$402$		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	$-0.963084\pi$
$409$		−	4.68035i		−	0.231428i	0.993283	−	0.115714i	$-0.0369156\pi$
$410$		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	$0.264336\pi$
$431$			30.6520i			1.47645i	−0.674553	+	0.738226i	$0.735664\pi$
$432$		0			0
$433$	−17.7792	+	17.7792i	−0.854416	+	0.854416i	−0.990673	−	0.136258i	$-0.956493\pi$
$433$	−17.7792	+	17.7792i	−0.854416	+	0.854416i	0.136258	+	0.990673i	$0.456493\pi$
$434$		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	$0.337546\pi$
$439$			36.5646i			1.74513i	−0.488494	+	0.872567i	$0.662454\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.4914	−	10.4914i	−0.498462	−	0.498462i	0.412497	−	0.910959i	$-0.364657\pi$
$443$	−10.4914	−	10.4914i	−0.498462	−	0.498462i	−0.910959	+	0.412497i	$0.864657\pi$
$444$		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	0.0117194	−	0.999931i	$-0.496270\pi$
$457$	−21.1256	−	21.1256i	−0.988212	−	0.988212i	−0.999931	+	0.0117194i	$0.996270\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	3.38812i		−	0.157801i	−0.996883	−	0.0789003i	$-0.974859\pi$
$461$		−	3.38812i		−	0.157801i	0.996883	−	0.0789003i	$-0.0251409\pi$
$462$		0			0
$463$	−11.2846	+	11.2846i	−0.524439	+	0.524439i	−0.918909	−	0.394470i	$-0.870928\pi$
$463$	−11.2846	+	11.2846i	−0.524439	+	0.524439i	0.394470	+	0.918909i	$0.370928\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.59919	+	9.59919i	−0.444198	+	0.444198i	−0.893420	−	0.449222i	$-0.851701\pi$
$467$	−9.59919	+	9.59919i	−0.444198	+	0.444198i	0.449222	+	0.893420i	$0.351701\pi$
$468$		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	0.257141	−	0.966374i	$-0.417220\pi$
$487$	−15.6514	−	15.6514i	−0.709233	−	0.709233i	−0.966374	+	0.257141i	$0.917220\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			29.7597i			1.34304i	0.740987	+	0.671519i	$0.234358\pi$
$491$			29.7597i			1.34304i	−0.740987	+	0.671519i	$0.765642\pi$
$492$		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	$0.0814147\pi$
$499$			11.3028i			0.505984i	−0.967468	+	0.252992i	$0.918585\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5.80861	−	5.80861i	−0.258993	−	0.258993i	0.565651	−	0.824644i	$-0.308625\pi$
$503$	−5.80861	−	5.80861i	−0.258993	−	0.258993i	−0.824644	+	0.565651i	$0.808625\pi$
$504$		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	$-0.816178\pi$
$521$		−	24.9220i		−	1.09185i	0.837834	−	0.545925i	$-0.183822\pi$
$522$		0			0
$523$	18.3545	−	18.3545i	0.802588	−	0.802588i	−0.180911	−	0.983499i	$-0.557905\pi$
$523$	18.3545	−	18.3545i	0.802588	−	0.802588i	0.983499	+	0.180911i	$0.0579046\pi$
$524$		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	−0.999379	−	0.0352442i	$-0.988779\pi$
$547$	−24.1978	−	24.1978i	−1.03462	−	1.03462i	−0.0352442	−	0.999379i	$-0.511221\pi$
$548$		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	−0.999903	−	0.0139042i	$-0.995574\pi$
$557$	−23.2704	+	23.2704i	−0.985999	+	0.985999i	0.0139042	+	0.999903i	$0.495574\pi$
$558$		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	0.650039	−	0.759900i	$-0.274752\pi$
$563$	−2.60674	−	2.60674i	−0.109861	−	0.109861i	−0.759900	+	0.650039i	$0.774752\pi$
$564$		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	0.976014	+	0.217709i	$0.0698584\pi$
$577$	28.6742	+	28.6742i	1.19372	+	1.19372i	0.217709	+	0.976014i	$0.430142\pi$
$578$		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	−0.949431	−	0.313974i	$-0.898339\pi$
$587$	−15.3959	+	15.3959i	−0.635457	+	0.635457i	0.313974	+	0.949431i	$0.398339\pi$
$588$		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	−0.997229	−	0.0743901i	$-0.976299\pi$
$593$	−26.0956	−	26.0956i	−1.07162	−	1.07162i	−0.0743901	−	0.997229i	$-0.523701\pi$
$594$		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	0.0392481	−	0.999229i	$-0.487504\pi$
$607$	−23.6514	−	23.6514i	−0.959981	−	0.959981i	−0.999229	+	0.0392481i	$0.987504\pi$
$608$		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	−0.782262	−	0.622949i	$-0.785934\pi$
$613$	−3.94441	+	3.94441i	−0.159313	+	0.159313i	0.622949	+	0.782262i	$0.285934\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.3218	−	18.3218i	0.737608	−	0.737608i	−0.234506	−	0.972115i	$-0.575347\pi$
$617$	18.3218	−	18.3218i	0.737608	−	0.737608i	0.972115	+	0.234506i	$0.0753474\pi$
$618$		0			0
$619$		−	8.05332i		−	0.323690i	−0.986816	−	0.161845i	$-0.948255\pi$
$619$		−	8.05332i		−	0.323690i	0.986816	−	0.161845i	$-0.0517445\pi$
$620$		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	$0.171714\pi$
$641$			26.0101i			1.02734i	−0.857989	+	0.513668i	$0.828286\pi$
$642$		0			0
$643$	13.3607	−	13.3607i	0.526894	−	0.526894i	−0.392751	−	0.919645i	$-0.628476\pi$
$643$	13.3607	−	13.3607i	0.526894	−	0.526894i	0.919645	+	0.392751i	$0.128476\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	19.0466	−	19.0466i	0.748800	−	0.748800i	−0.225454	−	0.974254i	$-0.572386\pi$
$647$	19.0466	−	19.0466i	0.748800	−	0.748800i	0.974254	+	0.225454i	$0.0723864\pi$
$648$		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	0.346215	−	0.938155i	$-0.387467\pi$
$653$	−15.1263	−	15.1263i	−0.591940	−	0.591940i	−0.938155	+	0.346215i	$0.887467\pi$
$654$		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	−0.683309	−	0.730129i	$-0.739460\pi$
$673$	1.21461	−	1.21461i	0.0468199	−	0.0468199i	0.730129	+	0.683309i	$0.239460\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	23.3899	−	23.3899i	0.898949	−	0.898949i	−0.0963946	−	0.995343i	$-0.530731\pi$
$677$	23.3899	−	23.3899i	0.898949	−	0.898949i	0.995343	+	0.0963946i	$0.0307311\pi$
$678$		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	0.00940495	−	0.999956i	$-0.497006\pi$
$683$	−25.8873	−	25.8873i	−0.990551	−	0.990551i	−0.999956	+	0.00940495i	$0.997006\pi$
$684$		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	$0.0520990\pi$
$701$			8.62835i			0.325888i	−0.986635	+	0.162944i	$0.947901\pi$
$702$		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	$-0.940575\pi$
$709$		−	9.88428i		−	0.371212i	0.982624	−	0.185606i	$-0.0594248\pi$
$710$		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	0.997554	−	0.0699058i	$-0.0222699\pi$
$727$	25.0121	+	25.0121i	0.927648	+	0.927648i	−0.0699058	+	0.997554i	$0.522270\pi$
$728$		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	−0.810264	−	0.586065i	$-0.800676\pi$
$733$	−6.06997	+	6.06997i	−0.224199	+	0.224199i	0.586065	+	0.810264i	$0.300676\pi$
$734$		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	$-0.861723\pi$
$739$		−	22.8827i		−	0.841753i	0.907118	−	0.420876i	$-0.138277\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.9965	+	15.9965i	0.586855	+	0.586855i	0.936778	−	0.349923i	$-0.113792\pi$
$743$	15.9965	+	15.9965i	0.586855	+	0.586855i	−0.349923	+	0.936778i	$0.613792\pi$
$744$		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	0.776523	−	0.630089i	$-0.216982\pi$
$757$	4.02893	+	4.02893i	0.146434	+	0.146434i	−0.630089	+	0.776523i	$0.716982\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	33.5535i		−	1.21631i	−0.793817	−	0.608157i	$-0.791909\pi$
$761$		−	33.5535i		−	1.21631i	0.793817	−	0.608157i	$-0.208091\pi$
$762$		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	$-0.948153\pi$
$769$		−	8.99386i		−	0.324327i	0.986764	−	0.162163i	$-0.0518472\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20.4797	−	20.4797i	−0.736603	−	0.736603i	0.235316	−	0.971919i	$-0.424388\pi$
$773$	−20.4797	−	20.4797i	−0.736603	−	0.736603i	−0.971919	+	0.235316i	$0.924388\pi$
$774$		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	0.368744	−	0.929531i	$-0.379788\pi$
$787$	−15.7321	−	15.7321i	−0.560787	−	0.560787i	−0.929531	+	0.368744i	$0.879788\pi$
$788$		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	−0.630501	−	0.776189i	$-0.717150\pi$
$797$	4.11294	−	4.11294i	0.145688	−	0.145688i	0.776189	+	0.630501i	$0.217150\pi$
$798$		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	$0.0810956\pi$
$821$			14.4424i			0.504045i	−0.967721	+	0.252022i	$0.918904\pi$
$822$		0			0
$823$	2.65756	−	2.65756i	0.0926367	−	0.0926367i	−0.659270	−	0.751906i	$-0.729134\pi$
$823$	2.65756	−	2.65756i	0.0926367	−	0.0926367i	0.751906	+	0.659270i	$0.229134\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.139861	−	0.139861i	0.00486344	−	0.00486344i	−0.704671	−	0.709534i	$-0.748905\pi$
$827$	0.139861	−	0.139861i	0.00486344	−	0.00486344i	0.709534	+	0.704671i	$0.248905\pi$
$828$		0			0
$829$			15.2208i			0.528639i	0.964435	+	0.264319i	$0.0851473\pi$
$829$			15.2208i			0.528639i	−0.964435	+	0.264319i	$0.914853\pi$
$830$		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	−0.770506	−	0.637433i	$-0.779996\pi$
$853$	−3.88655	+	3.88655i	−0.133073	+	0.133073i	0.637433	+	0.770506i	$0.279996\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−13.4872	+	13.4872i	−0.460715	+	0.460715i	−0.898890	−	0.438174i	$-0.855625\pi$
$857$	−13.4872	+	13.4872i	−0.460715	+	0.460715i	0.438174	+	0.898890i	$0.355625\pi$
$858$		0			0
$859$		−	10.6681i		−	0.363990i	−0.983300	−	0.181995i	$-0.941745\pi$
$859$		−	10.6681i		−	0.363990i	0.983300	−	0.181995i	$-0.0582554\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.620938	+	0.620938i	0.0211370	+	0.0211370i	0.717596	−	0.696459i	$-0.245242\pi$
$863$	0.620938	+	0.620938i	0.0211370	+	0.0211370i	−0.696459	+	0.717596i	$0.745242\pi$
$864$		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	0.916607	−	0.399790i	$-0.130917\pi$
$877$	15.3051	+	15.3051i	0.516817	+	0.516817i	−0.399790	+	0.916607i	$0.630917\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			35.5974i			1.19931i	0.800260	+	0.599653i	$0.204695\pi$
$881$			35.5974i			1.19931i	−0.800260	+	0.599653i	$0.795305\pi$
$882$		0			0
$883$	4.09890	−	4.09890i	0.137939	−	0.137939i	−0.634766	−	0.772705i	$-0.718903\pi$
$883$	4.09890	−	4.09890i	0.137939	−	0.137939i	0.772705	+	0.634766i	$0.218903\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.5291	+	10.5291i	−0.353533	+	0.353533i	−0.861422	−	0.507889i	$-0.830426\pi$
$887$	−10.5291	+	10.5291i	−0.353533	+	0.353533i	0.507889	+	0.861422i	$0.330426\pi$
$888$		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	0.429646	−	0.902997i	$-0.358638\pi$
$907$	−14.2557	−	14.2557i	−0.473351	−	0.473351i	−0.902997	+	0.429646i	$0.858638\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	38.0933i		−	1.26209i	−0.775748	−	0.631043i	$-0.782627\pi$
$911$		−	38.0933i		−	1.26209i	0.775748	−	0.631043i	$-0.217373\pi$
$912$		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	$-0.838495\pi$
$919$		−	29.4596i		−	0.971782i	0.874019	−	0.485891i	$-0.161505\pi$
$920$		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	0.467960	−	0.883750i	$-0.344989\pi$
$937$	−12.7275	−	12.7275i	−0.415790	−	0.415790i	−0.883750	+	0.467960i	$0.844989\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	14.5446i		−	0.474140i	−0.971493	−	0.237070i	$-0.923813\pi$
$941$		−	14.5446i		−	0.474140i	0.971493	−	0.237070i	$-0.0761871\pi$
$942$		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	0.276992	−	0.960872i	$-0.410662\pi$
$947$	38.0933	−	38.0933i	1.23786	−	1.23786i	0.960872	−	0.276992i	$-0.0893376\pi$
$948$		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	0.697002	−	0.717069i	$-0.254517\pi$
$953$	−0.619461	−	0.619461i	−0.0200663	−	0.0200663i	−0.717069	+	0.697002i	$0.754517\pi$
$954$		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	0.977662	−	0.210182i	$-0.0674058\pi$
$967$	23.8660	+	23.8660i	0.767480	+	0.767480i	−0.210182	+	0.977662i	$0.567406\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	43.5403i		−	1.39728i	−0.715476	−	0.698638i	$-0.753790\pi$
$971$		−	43.5403i		−	1.39728i	0.715476	−	0.698638i	$-0.246210\pi$
$972$		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	−0.0194924	−	0.999810i	$-0.506205\pi$
$977$	30.6418	−	30.6418i	0.980318	−	0.980318i	0.999810	+	0.0194924i	$0.00620502\pi$
$978$		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	0.673223	−	0.739439i	$-0.264909\pi$
$983$	−2.07606	−	2.07606i	−0.0662162	−	0.0662162i	−0.739439	+	0.673223i	$0.764909\pi$
$984$		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	0.876418	−	0.481552i	$-0.159927\pi$
$997$	12.4680	+	12.4680i	0.394865	+	0.394865i	−0.481552	+	0.876418i	$0.659927\pi$
$998$		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.2177.1		12
3.2	odd	2	inner	2880.2.w.p.2177.6		12
4.3	odd	2		2880.2.w.q.2177.1		12
5.3	odd	4	inner	2880.2.w.p.2753.6		12
8.3	odd	2		1440.2.w.g.737.6	yes	12
8.5	even	2		1440.2.w.f.737.6	yes	12
12.11	even	2		2880.2.w.q.2177.6		12
15.8	even	4	inner	2880.2.w.p.2753.1		12
20.3	even	4		2880.2.w.q.2753.6		12
24.5	odd	2		1440.2.w.f.737.1	✓	12
24.11	even	2		1440.2.w.g.737.1	yes	12
40.3	even	4		1440.2.w.g.1313.1	yes	12
40.13	odd	4		1440.2.w.f.1313.1	yes	12
60.23	odd	4		2880.2.w.q.2753.1		12
120.53	even	4		1440.2.w.f.1313.6	yes	12
120.83	odd	4		1440.2.w.g.1313.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.w.f.737.1	✓	12	24.5	odd	2
1440.2.w.f.737.6	yes	12	8.5	even	2
1440.2.w.f.1313.1	yes	12	40.13	odd	4
1440.2.w.f.1313.6	yes	12	120.53	even	4
1440.2.w.g.737.1	yes	12	24.11	even	2
1440.2.w.g.737.6	yes	12	8.3	odd	2
1440.2.w.g.1313.1	yes	12	40.3	even	4
1440.2.w.g.1313.6	yes	12	120.83	odd	4
2880.2.w.p.2177.1		12	1.1	even	1	trivial
2880.2.w.p.2177.6		12	3.2	odd	2	inner
2880.2.w.p.2753.1		12	15.8	even	4	inner
2880.2.w.p.2753.6		12	5.3	odd	4	inner
2880.2.w.q.2177.1		12	4.3	odd	2
2880.2.w.q.2177.6		12	12.11	even	2
2880.2.w.q.2753.1		12	60.23	odd	4
2880.2.w.q.2753.6		12	20.3	even	4

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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