LMFDB - Embedded newform 2880.2.f.a.1729.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(1729,2880)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("2880.1729");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.f (of order $2$, degree $1$, 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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1729.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2880.1729
Dual form			2880.2.f.a.1729.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 1.00000i) q^{5} +2.00000i q^{7} +O(q^{10})$	$q+(-2.00000 - 1.00000i) q^{5} +2.00000i q^{7} -6.00000 q^{11} -2.00000i q^{13} +6.00000i q^{17} +4.00000 q^{19} -8.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -8.00000 q^{31} +(2.00000 - 4.00000i) q^{35} -2.00000i q^{37} +6.00000 q^{41} +4.00000i q^{43} +4.00000i q^{47} +3.00000 q^{49} -6.00000i q^{53} +(12.0000 + 6.00000i) q^{55} +6.00000 q^{59} +6.00000 q^{61} +(-2.00000 + 4.00000i) q^{65} +4.00000 q^{71} -12.0000i q^{73} -12.0000i q^{77} +8.00000 q^{79} -12.0000i q^{83} +(6.00000 - 12.0000i) q^{85} +14.0000 q^{89} +4.00000 q^{91} +(-8.00000 - 4.00000i) q^{95} +8.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} - 12 q^{11} + 8 q^{19} + 6 q^{25} - 16 q^{31} + 4 q^{35} + 12 q^{41} + 6 q^{49} + 24 q^{55} + 12 q^{59} + 12 q^{61} - 4 q^{65} + 8 q^{71} + 16 q^{79} + 12 q^{85} + 28 q^{89} + 8 q^{91} - 16 q^{95}+O(q^{100}) $ 2 * q - 4 * q^5 - 12 * q^11 + 8 * q^19 + 6 * q^25 - 16 * q^31 + 4 * q^35 + 12 * q^41 + 6 * q^49 + 24 * q^55 + 12 * q^59 + 12 * q^61 - 4 * q^65 + 8 * q^71 + 16 * q^79 + 12 * q^85 + 28 * q^89 + 8 * q^91 - 16 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.00000			−1.80907			−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000			−1.80907			−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.00000i		−	1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$		−	8.00000i		−	1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000	−	4.00000i	0.338062	−	0.676123i
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			4.00000i			0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$			4.00000i			0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$	12.0000	+	6.00000i	1.61808	+	0.809040i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.00000			0.781133			0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000			0.781133			0.390567	+	0.920575i	$0.372279\pi$
$60$		0			0
$61$	6.00000			0.768221			0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000			0.768221			0.384111	+	0.923287i	$0.374508\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000	+	4.00000i	−0.248069	+	0.496139i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.00000			0.474713			0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000			0.474713			0.237356	+	0.971423i	$0.423719\pi$
$72$		0			0
$73$		−	12.0000i		−	1.40449i	−0.711934	−	0.702247i	$-0.752180\pi$
$73$		−	12.0000i		−	1.40449i	0.711934	−	0.702247i	$-0.247820\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	12.0000i		−	1.36753i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$	6.00000	−	12.0000i	0.650791	−	1.30158i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.0000			1.48400			0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000			1.48400			0.741999	+	0.670402i	$0.233878\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.00000	−	4.00000i	−0.820783	−	0.410391i
$96$		0			0
$97$			8.00000i			0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$			8.00000i			0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.00000			0.796030			0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000			0.796030			0.398015	+	0.917379i	$0.369699\pi$
$102$		0			0
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2.00000i		−	0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$		−	2.00000i		−	0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$		0			0
$115$	−8.00000	+	16.0000i	−0.746004	+	1.49201i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−12.0000			−1.10004
$120$		0			0
$121$	25.0000			2.27273
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$126$		0			0
$127$		−	22.0000i		−	1.95218i	−0.217357	−	0.976092i	$-0.569744\pi$
$127$		−	22.0000i		−	1.95218i	0.217357	−	0.976092i	$-0.430256\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.0000			−0.873704			−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000			−0.873704			−0.436852	+	0.899533i	$0.643907\pi$
$132$		0			0
$133$			8.00000i			0.693688i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	6.00000i		−	0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$		−	6.00000i		−	0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$		0			0
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			12.0000i			1.00349i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.00000			0.327693			0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000			0.327693			0.163846	+	0.986486i	$0.447610\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.0000	+	8.00000i	1.28515	+	0.642575i
$156$		0			0
$157$		−	22.0000i		−	1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$		−	22.0000i		−	1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	16.0000			1.26098
$162$		0			0
$163$		−	8.00000i		−	0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$		−	8.00000i		−	0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			16.0000i			1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$			16.0000i			1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$			14.0000i			1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$			14.0000i			1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$		0			0
$175$	−8.00000	+	6.00000i	−0.604743	+	0.453557i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.00000			−0.149487			−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000			−0.149487			−0.0747435	+	0.997203i	$0.523814\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.00000	+	4.00000i	−0.147043	+	0.294086i
$186$		0			0
$187$		−	36.0000i		−	2.63258i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	20.0000			1.44715			0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000			1.44715			0.723575	+	0.690246i	$0.242498\pi$
$192$		0			0
$193$		−	12.0000i		−	0.863779i	−0.901927	−	0.431889i	$-0.857847\pi$
$193$		−	12.0000i		−	0.863779i	0.901927	−	0.431889i	$-0.142153\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−12.0000	−	6.00000i	−0.838116	−	0.419058i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−24.0000			−1.66011
$210$		0			0
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000	−	8.00000i	0.272798	−	0.545595i
$216$		0			0
$217$		−	16.0000i		−	1.08615i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$		−	10.0000i		−	0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$		−	10.0000i		−	0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			20.0000i			1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$			20.0000i			1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$		0			0
$229$	14.0000			0.925146			0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000			0.925146			0.462573	+	0.886581i	$0.346926\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	22.0000i		−	1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$		−	22.0000i		−	1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$		0			0
$235$	4.00000	−	8.00000i	0.260931	−	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	20.0000			1.29369			0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000			1.29369			0.646846	+	0.762620i	$0.276088\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.00000	−	3.00000i	−0.383326	−	0.191663i
$246$		0			0
$247$		−	8.00000i		−	0.509028i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	14.0000			0.883672			0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000			0.883672			0.441836	+	0.897096i	$0.354327\pi$
$252$		0			0
$253$			48.0000i			3.01773i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	6.00000i		−	0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$		−	6.00000i		−	0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$		0			0
$259$	4.00000			0.248548
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$	−6.00000	+	12.0000i	−0.368577	+	0.737154i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−18.0000	−	24.0000i	−1.08544	−	1.44725i
$276$		0			0
$277$		−	2.00000i		−	0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$		−	2.00000i		−	0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.00000			0.119310			0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000			0.119310			0.0596550	+	0.998219i	$0.481000\pi$
$282$		0			0
$283$			8.00000i			0.475551i	0.971320	+	0.237775i	$0.0764182\pi$
$283$			8.00000i			0.475551i	−0.971320	+	0.237775i	$0.923582\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			12.0000i			0.708338i
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$			26.0000i			1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$			26.0000i			1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$		0			0
$295$	−12.0000	−	6.00000i	−0.698667	−	0.349334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.0000			−0.925304
$300$		0			0
$301$	−8.00000			−0.461112
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.0000	−	6.00000i	−0.687118	−	0.343559i
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$		−	28.0000i		−	1.58265i	−0.611393	−	0.791327i	$-0.709391\pi$
$313$		−	28.0000i		−	1.58265i	0.611393	−	0.791327i	$-0.290609\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$			24.0000i			1.33540i
$324$		0			0
$325$	8.00000	−	6.00000i	0.443760	−	0.332820i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.00000			−0.441054
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$			20.0000i			1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$			20.0000i			1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	48.0000			2.59935
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			4.00000i			0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$			4.00000i			0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	6.00000i		−	0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$		−	6.00000i		−	0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$		0			0
$355$	−8.00000	−	4.00000i	−0.424596	−	0.212298i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12.0000	+	24.0000i	−0.628109	+	1.25622i
$366$		0			0
$367$			10.0000i			0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$			10.0000i			0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.0000			0.623009
$372$		0			0
$373$		−	10.0000i		−	0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$		−	10.0000i		−	0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−8.00000			−0.410932			−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000			−0.410932			−0.205466	+	0.978664i	$0.565871\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			12.0000i			0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$			12.0000i			0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$		0			0
$385$	−12.0000	+	24.0000i	−0.611577	+	1.22315i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−36.0000			−1.82527			−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000			−1.82527			−0.912636	+	0.408773i	$0.865957\pi$
$390$		0			0
$391$	48.0000			2.42746
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.0000	−	8.00000i	−0.805047	−	0.402524i
$396$		0			0
$397$			38.0000i			1.90717i	0.301131	+	0.953583i	$0.402636\pi$
$397$			38.0000i			1.90717i	−0.301131	+	0.953583i	$0.597364\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$		0			0
$403$			16.0000i			0.797017i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			12.0000i			0.594818i
$408$		0			0
$409$	26.0000			1.28562			0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000			1.28562			0.642809	+	0.766027i	$0.277769\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			12.0000i			0.590481i
$414$		0			0
$415$	−12.0000	+	24.0000i	−0.589057	+	1.17811i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	26.0000			1.27018			0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000			1.27018			0.635092	+	0.772437i	$0.280962\pi$
$420$		0			0
$421$	2.00000			0.0974740			0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000			0.0974740			0.0487370	+	0.998812i	$0.484480\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−24.0000	+	18.0000i	−1.16417	+	0.873128i
$426$		0			0
$427$			12.0000i			0.580721i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−32.0000			−1.54139			−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000			−1.54139			−0.770693	+	0.637207i	$0.780090\pi$
$432$		0			0
$433$		−	28.0000i		−	1.34559i	−0.739827	−	0.672797i	$-0.765093\pi$
$433$		−	28.0000i		−	1.34559i	0.739827	−	0.672797i	$-0.234907\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	32.0000i		−	1.53077i
$438$		0			0
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	20.0000i		−	0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$		−	20.0000i		−	0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$		0			0
$445$	−28.0000	−	14.0000i	−1.32733	−	0.663664i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	−36.0000			−1.69517
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.00000	−	4.00000i	−0.375046	−	0.187523i
$456$		0			0
$457$			16.0000i			0.748448i	0.927338	+	0.374224i	$0.122091\pi$
$457$			16.0000i			0.748448i	−0.927338	+	0.374224i	$0.877909\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−20.0000			−0.931493			−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000			−0.931493			−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$		−	14.0000i		−	0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$		−	14.0000i		−	0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	28.0000i		−	1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$		−	28.0000i		−	1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	24.0000i		−	1.10352i
$474$		0			0
$475$	12.0000	+	16.0000i	0.550598	+	0.734130i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−28.0000			−1.27935			−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000			−1.27935			−0.639676	+	0.768644i	$0.720932\pi$
$480$		0			0
$481$	−4.00000			−0.182384
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.00000	−	16.0000i	0.363261	−	0.726523i
$486$		0			0
$487$		−	10.0000i		−	0.453143i	−0.973995	−	0.226572i	$-0.927248\pi$
$487$		−	10.0000i		−	0.453143i	0.973995	−	0.226572i	$-0.0727517\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	14.0000			0.631811			0.315906	−	0.948791i	$-0.397692\pi$
$491$	14.0000			0.631811			0.315906	+	0.948791i	$0.397692\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			8.00000i			0.358849i
$498$		0			0
$499$	36.0000			1.61158			0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000			1.61158			0.805791	+	0.592200i	$0.201741\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	4.00000i		−	0.178351i	−0.996016	−	0.0891756i	$-0.971577\pi$
$503$		−	4.00000i		−	0.178351i	0.996016	−	0.0891756i	$-0.0284232\pi$
$504$		0			0
$505$	−16.0000	−	8.00000i	−0.711991	−	0.355995i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	24.0000			1.06170
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.00000	−	12.0000i	0.264392	−	0.528783i
$516$		0			0
$517$		−	24.0000i		−	1.05552i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	26.0000			1.13908			0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000			1.13908			0.569540	+	0.821963i	$0.307121\pi$
$522$		0			0
$523$			24.0000i			1.04945i	0.851273	+	0.524723i	$0.175831\pi$
$523$			24.0000i			1.04945i	−0.851273	+	0.524723i	$0.824169\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	48.0000i		−	2.09091i
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	12.0000i		−	0.519778i
$534$		0			0
$535$	12.0000	−	24.0000i	0.518805	−	1.03761i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−18.0000			−0.775315
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.00000	−	2.00000i	−0.171341	−	0.0856706i
$546$		0			0
$547$		−	12.0000i		−	0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$		−	12.0000i		−	0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			16.0000i			0.680389i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			14.0000i			0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$			14.0000i			0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$			4.00000i			0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$			4.00000i			0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$		0			0
$565$	−2.00000	+	4.00000i	−0.0841406	+	0.168281i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.00000			−0.0838444			−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000			−0.0838444			−0.0419222	+	0.999121i	$0.513348\pi$
$570$		0			0
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	32.0000	−	24.0000i	1.33449	−	1.00087i
$576$		0			0
$577$		−	16.0000i		−	0.666089i	−0.942911	−	0.333044i	$-0.891924\pi$
$577$		−	16.0000i		−	0.666089i	0.942911	−	0.333044i	$-0.108076\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	24.0000			0.995688
$582$		0			0
$583$			36.0000i			1.49097i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	20.0000i		−	0.825488i	−0.910847	−	0.412744i	$-0.864570\pi$
$587$		−	20.0000i		−	0.825488i	0.910847	−	0.412744i	$-0.135430\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	2.00000i		−	0.0821302i	−0.999156	−	0.0410651i	$-0.986925\pi$
$593$		−	2.00000i		−	0.0821302i	0.999156	−	0.0410651i	$-0.0130751\pi$
$594$		0			0
$595$	24.0000	+	12.0000i	0.983904	+	0.491952i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	32.0000			1.30748			0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000			1.30748			0.653742	+	0.756717i	$0.273198\pi$
$600$		0			0
$601$	34.0000			1.38689			0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000			1.38689			0.693444	+	0.720510i	$0.256092\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−50.0000	−	25.0000i	−2.03279	−	1.01639i
$606$		0			0
$607$			22.0000i			0.892952i	0.894795	+	0.446476i	$0.147321\pi$
$607$			22.0000i			0.892952i	−0.894795	+	0.446476i	$0.852679\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.00000			0.323645
$612$		0			0
$613$			34.0000i			1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$			34.0000i			1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			38.0000i			1.52982i	0.644136	+	0.764911i	$0.277217\pi$
$617$			38.0000i			1.52982i	−0.644136	+	0.764911i	$0.722783\pi$
$618$		0			0
$619$	24.0000			0.964641			0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000			0.964641			0.482321	+	0.875995i	$0.339794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			28.0000i			1.12180i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12.0000			0.478471
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−22.0000	+	44.0000i	−0.873043	+	1.74609i
$636$		0			0
$637$		−	6.00000i		−	0.237729i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$		−	48.0000i		−	1.89294i	−0.322799	−	0.946468i	$-0.604624\pi$
$643$		−	48.0000i		−	1.89294i	0.322799	−	0.946468i	$-0.395376\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	12.0000i		−	0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$		−	12.0000i		−	0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$		0			0
$649$	−36.0000			−1.41312
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	10.0000i		−	0.391330i	−0.980671	−	0.195665i	$-0.937313\pi$
$653$		−	10.0000i		−	0.391330i	0.980671	−	0.195665i	$-0.0626866\pi$
$654$		0			0
$655$	20.0000	+	10.0000i	0.781465	+	0.390732i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$		0			0
$661$	6.00000			0.233373			0.116686	−	0.993169i	$-0.462773\pi$
$661$	6.00000			0.233373			0.116686	+	0.993169i	$0.462773\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.00000	−	16.0000i	0.310227	−	0.620453i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−36.0000			−1.38976
$672$		0			0
$673$			28.0000i			1.07932i	0.841883	+	0.539660i	$0.181447\pi$
$673$			28.0000i			1.07932i	−0.841883	+	0.539660i	$0.818553\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			18.0000i			0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$			18.0000i			0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$		0			0
$679$	−16.0000			−0.614024
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	36.0000i		−	1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$		−	36.0000i		−	1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$		0			0
$685$	−6.00000	+	12.0000i	−0.229248	+	0.458496i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12.0000			−0.457164
$690$		0			0
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	32.0000	+	16.0000i	1.21383	+	0.606915i
$696$		0			0
$697$			36.0000i			1.36360i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		−	8.00000i		−	0.301726i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			16.0000i			0.601742i
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			64.0000i			2.39682i
$714$		0			0
$715$	12.0000	−	24.0000i	0.448775	−	0.897549i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.00000			0.298350			0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		0			0
$721$	−12.0000			−0.446903
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	26.0000i		−	0.964287i	−0.876092	−	0.482143i	$-0.839858\pi$
$727$		−	26.0000i		−	0.964287i	0.876092	−	0.482143i	$-0.160142\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−24.0000			−0.887672
$732$		0			0
$733$			2.00000i			0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$			2.00000i			0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			12.0000i			0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$			12.0000i			0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$		0			0
$745$	−8.00000	−	4.00000i	−0.293097	−	0.146549i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−24.0000			−0.876941
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.0000	+	8.00000i	0.582300	+	0.291150i
$756$		0			0
$757$		−	18.0000i		−	0.654221i	−0.944986	−	0.327111i	$-0.893925\pi$
$757$		−	18.0000i		−	0.654221i	0.944986	−	0.327111i	$-0.106075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$			4.00000i			0.144810i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	12.0000i		−	0.433295i
$768$		0			0
$769$	−2.00000			−0.0721218			−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000			−0.0721218			−0.0360609	+	0.999350i	$0.511481\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	26.0000i		−	0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$		−	26.0000i		−	0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$		0			0
$775$	−24.0000	−	32.0000i	−0.862105	−	1.14947i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	24.0000			0.859889
$780$		0			0
$781$	−24.0000			−0.858788
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−22.0000	+	44.0000i	−0.785214	+	1.57043i
$786$		0			0
$787$			24.0000i			0.855508i	0.903895	+	0.427754i	$0.140695\pi$
$787$			24.0000i			0.855508i	−0.903895	+	0.427754i	$0.859305\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4.00000			0.142224
$792$		0			0
$793$		−	12.0000i		−	0.426132i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	30.0000i		−	1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$		−	30.0000i		−	1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$		0			0
$802$		0			0
$803$			72.0000i			2.54082i
$804$		0			0
$805$	−32.0000	−	16.0000i	−1.12785	−	0.563926i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	34.0000			1.19538			0.597688	−	0.801729i	$-0.296086\pi$
$809$	34.0000			1.19538			0.597688	+	0.801729i	$0.296086\pi$
$810$		0			0
$811$	16.0000			0.561836			0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000			0.561836			0.280918	+	0.959732i	$0.409361\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8.00000	+	16.0000i	−0.280228	+	0.560456i
$816$		0			0
$817$			16.0000i			0.559769i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.0000			−0.837606			−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000			−0.837606			−0.418803	+	0.908077i	$0.637550\pi$
$822$		0			0
$823$		−	22.0000i		−	0.766872i	−0.923567	−	0.383436i	$-0.874741\pi$
$823$		−	22.0000i		−	0.766872i	0.923567	−	0.383436i	$-0.125259\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			12.0000i			0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$			12.0000i			0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$		0			0
$829$	−38.0000			−1.31979			−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000			−1.31979			−0.659897	+	0.751356i	$0.729400\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			18.0000i			0.623663i
$834$		0			0
$835$	16.0000	−	32.0000i	0.553703	−	1.10741i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−8.00000			−0.276191			−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000			−0.276191			−0.138095	+	0.990419i	$0.544098\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−18.0000	−	9.00000i	−0.619219	−	0.309609i
$846$		0			0
$847$			50.0000i			1.71802i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−16.0000			−0.548473
$852$		0			0
$853$		−	6.00000i		−	0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$		−	6.00000i		−	0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	18.0000i		−	0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$		−	18.0000i		−	0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$		0			0
$859$	8.00000			0.272956			0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000			0.272956			0.136478	+	0.990643i	$0.456422\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	24.0000i		−	0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$		−	24.0000i		−	0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$		0			0
$865$	14.0000	−	28.0000i	0.476014	−	0.952029i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−48.0000			−1.62829
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	22.0000	−	4.00000i	0.743736	−	0.135225i
$876$		0			0
$877$			10.0000i			0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$			10.0000i			0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.0000			0.875962			0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000			0.875962			0.437981	+	0.898984i	$0.355694\pi$
$882$		0			0
$883$		−	32.0000i		−	1.07689i	−0.842662	−	0.538443i	$-0.819013\pi$
$883$		−	32.0000i		−	1.07689i	0.842662	−	0.538443i	$-0.180987\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			48.0000i			1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$			48.0000i			1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$		0			0
$889$	44.0000			1.47571
$890$		0			0
$891$		0			0
$892$		0			0
$893$			16.0000i			0.535420i
$894$		0			0
$895$	4.00000	+	2.00000i	0.133705	+	0.0668526i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	36.0000			1.19933
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−44.0000	−	22.0000i	−1.46261	−	0.731305i
$906$		0			0
$907$			44.0000i			1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$			44.0000i			1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	32.0000			1.06021			0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000			1.06021			0.530104	+	0.847933i	$0.322153\pi$
$912$		0			0
$913$			72.0000i			2.38285i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	20.0000i		−	0.660458i
$918$		0			0
$919$	48.0000			1.58337			0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000			1.58337			0.791687	+	0.610927i	$0.209203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	8.00000i		−	0.263323i
$924$		0			0
$925$	8.00000	−	6.00000i	0.263038	−	0.197279i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−34.0000			−1.11550			−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000			−1.11550			−0.557752	+	0.830008i	$0.688336\pi$
$930$		0			0
$931$	12.0000			0.393284
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−36.0000	+	72.0000i	−1.17733	+	2.35465i
$936$		0			0
$937$		−	8.00000i		−	0.261349i	−0.991425	−	0.130674i	$-0.958286\pi$
$937$		−	8.00000i		−	0.261349i	0.991425	−	0.130674i	$-0.0417142\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.0000			0.651981			0.325991	−	0.945373i	$-0.394302\pi$
$941$	20.0000			0.651981			0.325991	+	0.945373i	$0.394302\pi$
$942$		0			0
$943$		−	48.0000i		−	1.56310i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	20.0000i		−	0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$		−	20.0000i		−	0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$		0			0
$949$	−24.0000			−0.779073
$950$		0			0
$951$		0			0
$952$		0			0
$953$			6.00000i			0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$			6.00000i			0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$		0			0
$955$	−40.0000	−	20.0000i	−1.29437	−	0.647185i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	12.0000			0.387500
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12.0000	+	24.0000i	−0.386294	+	0.772587i
$966$		0			0
$967$			42.0000i			1.35063i	0.737530	+	0.675314i	$0.235992\pi$
$967$			42.0000i			1.35063i	−0.737530	+	0.675314i	$0.764008\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	14.0000			0.449281			0.224641	−	0.974442i	$-0.427879\pi$
$971$	14.0000			0.449281			0.224641	+	0.974442i	$0.427879\pi$
$972$		0			0
$973$		−	32.0000i		−	1.02587i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			30.0000i			0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$			30.0000i			0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$		0			0
$979$	−84.0000			−2.68465
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	12.0000i		−	0.382741i	−0.981518	−	0.191370i	$-0.938707\pi$
$983$		−	12.0000i		−	0.382741i	0.981518	−	0.191370i	$-0.0612931\pi$
$984$		0			0
$985$	6.00000	−	12.0000i	0.191176	−	0.382352i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	32.0000			1.01754
$990$		0			0
$991$	48.0000			1.52477			0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000			1.52477			0.762385	+	0.647124i	$0.224028\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16.0000	+	8.00000i	0.507234	+	0.253617i
$996$		0			0
$997$			10.0000i			0.316703i	0.987383	+	0.158352i	$0.0506179\pi$
$997$			10.0000i			0.316703i	−0.987383	+	0.158352i	$0.949382\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.f.a.1729.1		2
3.2	odd	2		960.2.f.j.769.2		2
4.3	odd	2		2880.2.f.g.1729.1		2
5.4	even	2	inner	2880.2.f.a.1729.2		2
8.3	odd	2		1440.2.f.e.289.2		2
8.5	even	2		1440.2.f.g.289.2		2
12.11	even	2		960.2.f.g.769.1		2
15.2	even	4		4800.2.a.bu.1.1		1
15.8	even	4		4800.2.a.ba.1.1		1
15.14	odd	2		960.2.f.j.769.1		2
20.19	odd	2		2880.2.f.g.1729.2		2
24.5	odd	2		480.2.f.a.289.1	✓	2
24.11	even	2		480.2.f.b.289.2	yes	2
40.3	even	4		7200.2.a.j.1.1		1
40.13	odd	4		7200.2.a.br.1.1		1
40.19	odd	2		1440.2.f.e.289.1		2
40.27	even	4		7200.2.a.bl.1.1		1
40.29	even	2		1440.2.f.g.289.1		2
40.37	odd	4		7200.2.a.p.1.1		1
48.5	odd	4		3840.2.d.u.2689.2		2
48.11	even	4		3840.2.d.e.2689.2		2
48.29	odd	4		3840.2.d.k.2689.1		2
48.35	even	4		3840.2.d.bc.2689.1		2
60.23	odd	4		4800.2.a.bt.1.1		1
60.47	odd	4		4800.2.a.z.1.1		1
60.59	even	2		960.2.f.g.769.2		2
120.29	odd	2		480.2.f.a.289.2	yes	2
120.53	even	4		2400.2.a.be.1.1		1
120.59	even	2		480.2.f.b.289.1	yes	2
120.77	even	4		2400.2.a.c.1.1		1
120.83	odd	4		2400.2.a.d.1.1		1
120.107	odd	4		2400.2.a.bf.1.1		1
240.29	odd	4		3840.2.d.u.2689.1		2
240.59	even	4		3840.2.d.bc.2689.2		2
240.149	odd	4		3840.2.d.k.2689.2		2
240.179	even	4		3840.2.d.e.2689.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.f.a.289.1	✓	2	24.5	odd	2
480.2.f.a.289.2	yes	2	120.29	odd	2
480.2.f.b.289.1	yes	2	120.59	even	2
480.2.f.b.289.2	yes	2	24.11	even	2
960.2.f.g.769.1		2	12.11	even	2
960.2.f.g.769.2		2	60.59	even	2
960.2.f.j.769.1		2	15.14	odd	2
960.2.f.j.769.2		2	3.2	odd	2
1440.2.f.e.289.1		2	40.19	odd	2
1440.2.f.e.289.2		2	8.3	odd	2
1440.2.f.g.289.1		2	40.29	even	2
1440.2.f.g.289.2		2	8.5	even	2
2400.2.a.c.1.1		1	120.77	even	4
2400.2.a.d.1.1		1	120.83	odd	4
2400.2.a.be.1.1		1	120.53	even	4
2400.2.a.bf.1.1		1	120.107	odd	4
2880.2.f.a.1729.1		2	1.1	even	1	trivial
2880.2.f.a.1729.2		2	5.4	even	2	inner
2880.2.f.g.1729.1		2	4.3	odd	2
2880.2.f.g.1729.2		2	20.19	odd	2
3840.2.d.e.2689.1		2	240.179	even	4
3840.2.d.e.2689.2		2	48.11	even	4
3840.2.d.k.2689.1		2	48.29	odd	4
3840.2.d.k.2689.2		2	240.149	odd	4
3840.2.d.u.2689.1		2	240.29	odd	4
3840.2.d.u.2689.2		2	48.5	odd	4
3840.2.d.bc.2689.1		2	48.35	even	4
3840.2.d.bc.2689.2		2	240.59	even	4
4800.2.a.z.1.1		1	60.47	odd	4
4800.2.a.ba.1.1		1	15.8	even	4
4800.2.a.bt.1.1		1	60.23	odd	4
4800.2.a.bu.1.1		1	15.2	even	4
7200.2.a.j.1.1		1	40.3	even	4
7200.2.a.p.1.1		1	40.37	odd	4
7200.2.a.bl.1.1		1	40.27	even	4
7200.2.a.br.1.1		1	40.13	odd	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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 1.00000i) q^{5} +2.00000i q^{7} +O(q^{10})\)	\(q+(-2.00000 - 1.00000i) q^{5} +2.00000i q^{7} -6.00000 q^{11} -2.00000i q^{13} +6.00000i q^{17} +4.00000 q^{19} -8.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -8.00000 q^{31} +(2.00000 - 4.00000i) q^{35} -2.00000i q^{37} +6.00000 q^{41} +4.00000i q^{43} +4.00000i q^{47} +3.00000 q^{49} -6.00000i q^{53} +(12.0000 + 6.00000i) q^{55} +6.00000 q^{59} +6.00000 q^{61} +(-2.00000 + 4.00000i) q^{65} +4.00000 q^{71} -12.0000i q^{73} -12.0000i q^{77} +8.00000 q^{79} -12.0000i q^{83} +(6.00000 - 12.0000i) q^{85} +14.0000 q^{89} +4.00000 q^{91} +(-8.00000 - 4.00000i) q^{95} +8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} - 12 q^{11} + 8 q^{19} + 6 q^{25} - 16 q^{31} + 4 q^{35} + 12 q^{41} + 6 q^{49} + 24 q^{55} + 12 q^{59} + 12 q^{61} - 4 q^{65} + 8 q^{71} + 16 q^{79} + 12 q^{85} + 28 q^{89} + 8 q^{91} - 16 q^{95}+O(q^{100}) \) 2 * q - 4 * q^5 - 12 * q^11 + 8 * q^19 + 6 * q^25 - 16 * q^31 + 4 * q^35 + 12 * q^41 + 6 * q^49 + 24 * q^55 + 12 * q^59 + 12 * q^61 - 4 * q^65 + 8 * q^71 + 16 * q^79 + 12 * q^85 + 28 * q^89 + 8 * q^91 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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