LMFDB - Embedded newform 480.2.k.a.241.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,2,Mod(241,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("480.241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	480.k (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:	$3.83281929702$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	480.241
Dual form			480.2.k.a.241.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} -1.00000 q^{15} -6.00000 q^{17} +4.00000i q^{19} -2.00000i q^{21} +4.00000 q^{23} -1.00000 q^{25} -1.00000i q^{27} +6.00000i q^{29} -10.0000 q^{31} -4.00000 q^{33} -2.00000i q^{35} -4.00000i q^{37} +10.0000 q^{41} -4.00000i q^{43} -1.00000i q^{45} +4.00000 q^{47} -3.00000 q^{49} -6.00000i q^{51} +10.0000i q^{53} -4.00000 q^{55} -4.00000 q^{57} +8.00000i q^{59} +8.00000i q^{61} +2.00000 q^{63} -12.0000i q^{67} +4.00000i q^{69} +4.00000 q^{71} +10.0000 q^{73} -1.00000i q^{75} -8.00000i q^{77} +14.0000 q^{79} +1.00000 q^{81} -6.00000i q^{85} -6.00000 q^{87} +14.0000 q^{89} -10.0000i q^{93} -4.00000 q^{95} -10.0000 q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^7 - 2 * q^9	$ 2 q - 4 q^{7} - 2 q^{9} - 2 q^{15} - 12 q^{17} + 8 q^{23} - 2 q^{25} - 20 q^{31} - 8 q^{33} + 20 q^{41} + 8 q^{47} - 6 q^{49} - 8 q^{55} - 8 q^{57} + 4 q^{63} + 8 q^{71} + 20 q^{73} + 28 q^{79} + 2 q^{81} - 12 q^{87} + 28 q^{89} - 8 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 - 2 * q^9 - 2 * q^15 - 12 * q^17 + 8 * q^23 - 2 * q^25 - 20 * q^31 - 8 * q^33 + 20 * q^41 + 8 * q^47 - 6 * q^49 - 8 * q^55 - 8 * q^57 + 4 * q^63 + 8 * q^71 + 20 * q^73 + 28 * q^79 + 2 * q^81 - 12 * q^87 + 28 * q^89 - 8 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	− 2.00000i	− 0.436436i
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	− 2.00000i	− 0.338062i
$36$	0	0
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	− 6.00000i	− 0.840168i
$52$	0	0
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	8.00000i	1.02430i	0.858898	+	0.512148i	$0.171150\pi$
$61$	8.00000i	1.02430i	−0.858898	+	0.512148i	$0.828850\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	4.00000i	0.481543i
$70$	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
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	− 8.00000i	− 0.911685i
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	− 6.00000i	− 0.650791i
$86$	0	0
$87$	−6.00000	−0.643268
$88$	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
$91$	0	0
$92$	0	0
$93$	− 10.0000i	− 1.03695i
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	− 4.00000i	− 0.402015i
$100$	0	0
$101$	− 14.0000i	− 1.39305i	−0.717532	−	0.696526i	$-0.754728\pi$
$101$	− 14.0000i	− 1.39305i	0.717532	−	0.696526i	$-0.245272\pi$
$102$	0	0
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	− 4.00000i	− 0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$	− 4.00000i	− 0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	4.00000i	0.373002i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	10.0000i	0.901670i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	− 4.00000i	− 0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$	− 4.00000i	− 0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$	0	0
$141$	4.00000i	0.336861i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	− 10.0000i	− 0.803219i
$156$	0	0
$157$	− 20.0000i	− 1.59617i	−0.602542	−	0.798087i	$-0.705846\pi$
$157$	− 20.0000i	− 1.59617i	0.602542	−	0.798087i	$-0.294154\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	0	0
$165$	− 4.00000i	− 0.311400i
$166$	0	0
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	10.0000i	0.760286i	0.924928	+	0.380143i	$0.124125\pi$
$173$	10.0000i	0.760286i	−0.924928	+	0.380143i	$0.875875\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	20.0000i	1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$	20.0000i	1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	− 24.0000i	− 1.75505i
$188$	0	0
$189$	2.00000i	0.145479i
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	0	0
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	10.0000i	0.698430i
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	12.0000i	0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$	12.0000i	0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$	0	0
$213$	4.00000i	0.274075i
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	20.0000	1.35769
$218$	0	0
$219$	10.0000i	0.675737i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	4.00000i	0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.00000i	0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	4.00000i	0.260931i
$236$	0	0
$237$	14.0000i	0.909398i
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000i	0.757433i	0.925513	+	0.378717i	$0.123635\pi$
$251$	12.0000i	0.757433i	−0.925513	+	0.378717i	$0.876365\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	0	0
$255$	6.00000	0.375735
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	8.00000i	0.497096i
$260$	0	0
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	0	0
$267$	14.0000i	0.856786i
$268$	0	0
$269$	− 10.0000i	− 0.609711i	−0.952399	−	0.304855i	$-0.901392\pi$
$269$	− 10.0000i	− 0.609711i	0.952399	−	0.304855i	$-0.0986081\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.00000i	− 0.241209i
$276$	0	0
$277$	16.0000i	0.961347i	0.876900	+	0.480673i	$0.159608\pi$
$277$	16.0000i	0.961347i	−0.876900	+	0.480673i	$0.840392\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	− 4.00000i	− 0.236940i
$286$	0	0
$287$	−20.0000	−1.18056
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 10.0000i	− 0.586210i
$292$	0	0
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.00000i	0.461112i
$302$	0	0
$303$	14.0000	0.804279
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	28.0000i	1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$	28.0000i	1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$	0	0
$309$	2.00000i	0.113776i
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	2.00000i	0.112687i
$316$	0	0
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.00000	0.221201
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	4.00000i	0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$	4.00000i	0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	0	0
$333$	4.00000i	0.219199i
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	0	0
$339$	6.00000i	0.325875i
$340$	0	0
$341$	− 40.0000i	− 2.16612i
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	16.0000i	0.858925i	0.903085	+	0.429463i	$0.141297\pi$
$347$	16.0000i	0.858925i	−0.903085	+	0.429463i	$0.858703\pi$
$348$	0	0
$349$	− 8.00000i	− 0.428230i	−0.976808	−	0.214115i	$-0.931313\pi$
$349$	− 8.00000i	− 0.428230i	0.976808	−	0.214115i	$-0.0686868\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	4.00000i	0.212298i
$356$	0	0
$357$	12.0000i	0.635107i
$358$	0	0
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	10.0000i	0.523424i
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	− 20.0000i	− 1.03835i
$372$	0	0
$373$	− 20.0000i	− 1.03556i	−0.855514	−	0.517780i	$-0.826758\pi$
$373$	− 20.0000i	− 1.03556i	0.855514	−	0.517780i	$-0.173242\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 36.0000i	− 1.84920i	−0.380945	−	0.924598i	$-0.624401\pi$
$379$	− 36.0000i	− 1.84920i	0.380945	−	0.924598i	$-0.375599\pi$
$380$	0	0
$381$	6.00000i	0.307389i
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	− 10.0000i	− 0.507020i	−0.967333	−	0.253510i	$-0.918415\pi$
$389$	− 10.0000i	− 0.507020i	0.967333	−	0.253510i	$-0.0815851\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14.0000i	0.704416i
$396$	0	0
$397$	− 20.0000i	− 1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$	− 20.0000i	− 1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	16.0000	0.793091
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	− 2.00000i	− 0.0986527i
$412$	0	0
$413$	− 16.0000i	− 0.787309i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	− 4.00000i	− 0.195413i	−0.995215	−	0.0977064i	$-0.968849\pi$
$419$	− 4.00000i	− 0.195413i	0.995215	−	0.0977064i	$-0.0311506\pi$
$420$	0	0
$421$	− 20.0000i	− 0.974740i	−0.873195	−	0.487370i	$-0.837956\pi$
$421$	− 20.0000i	− 0.974740i	0.873195	−	0.487370i	$-0.162044\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	− 16.0000i	− 0.774294i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	− 6.00000i	− 0.287678i
$436$	0	0
$437$	16.0000i	0.765384i
$438$	0	0
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	16.0000i	0.760183i	0.924949	+	0.380091i	$0.124107\pi$
$443$	16.0000i	0.760183i	−0.924949	+	0.380091i	$0.875893\pi$
$444$	0	0
$445$	14.0000i	0.663664i
$446$	0	0
$447$	−6.00000	−0.283790
$448$	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
$451$	40.0000i	1.88353i
$452$	0	0
$453$	2.00000i	0.0939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	6.00000i	0.280056i
$460$	0	0
$461$	34.0000i	1.58354i	0.610821	+	0.791769i	$0.290840\pi$
$461$	34.0000i	1.58354i	−0.610821	+	0.791769i	$0.709160\pi$
$462$	0	0
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	0	0
$465$	10.0000	0.463739
$466$	0	0
$467$	− 8.00000i	− 0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$	− 8.00000i	− 0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$	0	0
$469$	24.0000i	1.10822i
$470$	0	0
$471$	20.0000	0.921551
$472$	0	0
$473$	16.0000	0.735681
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	− 10.0000i	− 0.457869i
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 8.00000i	− 0.364013i
$484$	0	0
$485$	− 10.0000i	− 0.454077i
$486$	0	0
$487$	30.0000	1.35943	0.679715	−	0.733476i	$-0.262104\pi$
$487$	30.0000	1.35943	0.679715	+	0.733476i	$0.262104\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	− 40.0000i	− 1.80517i	−0.430507	−	0.902587i	$-0.641665\pi$
$491$	− 40.0000i	− 1.80517i	0.430507	−	0.902587i	$-0.358335\pi$
$492$	0	0
$493$	− 36.0000i	− 1.62136i
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	− 20.0000i	− 0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$	− 20.0000i	− 0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$	0	0
$501$	− 24.0000i	− 1.07224i
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	13.0000i	0.577350i
$508$	0	0
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	2.00000i	0.0881305i
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	0	0
$525$	2.00000i	0.0872872i
$526$	0	0
$527$	60.0000	2.61364
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 8.00000i	− 0.347170i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 12.0000i	− 0.516877i
$540$	0	0
$541$	44.0000i	1.89171i	0.324593	+	0.945854i	$0.394773\pi$
$541$	44.0000i	1.89171i	−0.324593	+	0.945854i	$0.605227\pi$
$542$	0	0
$543$	−20.0000	−0.858282
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	0	0
$549$	− 8.00000i	− 0.341432i
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	−28.0000	−1.19068
$554$	0	0
$555$	4.00000i	0.169791i
$556$	0	0
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	24.0000	1.01328
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	6.00000i	0.252422i
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	2.00000	0.0838444	0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000	0.0838444	0.0419222	+	0.999121i	$0.486652\pi$
$570$	0	0
$571$	− 20.0000i	− 0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$	− 20.0000i	− 0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	− 14.0000i	− 0.581820i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−40.0000	−1.65663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	− 40.0000i	− 1.64817i
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	0	0
$595$	12.0000i	0.491952i
$596$	0	0
$597$	6.00000i	0.245564i
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	0	0
$605$	− 5.00000i	− 0.203279i
$606$	0	0
$607$	−2.00000	−0.0811775	−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000	−0.0811775	−0.0405887	+	0.999176i	$0.512923\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	20.0000i	0.807792i	0.914805	+	0.403896i	$0.132344\pi$
$613$	20.0000i	0.807792i	−0.914805	+	0.403896i	$0.867656\pi$
$614$	0	0
$615$	−10.0000	−0.403239
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	20.0000i	0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$	20.0000i	0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$	0	0
$621$	− 4.00000i	− 0.160514i
$622$	0	0
$623$	−28.0000	−1.12180
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 16.0000i	− 0.638978i
$628$	0	0
$629$	24.0000i	0.956943i
$630$	0	0
$631$	18.0000	0.716569	0.358284	−	0.933613i	$-0.383362\pi$
$631$	18.0000	0.716569	0.358284	+	0.933613i	$0.383362\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	6.00000i	0.238103i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	4.00000i	0.157500i
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−32.0000	−1.25611
$650$	0	0
$651$	20.0000i	0.783862i
$652$	0	0
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	− 32.0000i	− 1.24466i	−0.782757	−	0.622328i	$-0.786187\pi$
$661$	− 32.0000i	− 1.24466i	0.782757	−	0.622328i	$-0.213813\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	24.0000i	0.929284i
$668$	0	0
$669$	− 10.0000i	− 0.386622i
$670$	0	0
$671$	−32.0000	−1.23535
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	0	0
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	38.0000i	1.46046i	0.683202	+	0.730229i	$0.260587\pi$
$677$	38.0000i	1.46046i	−0.683202	+	0.730229i	$0.739413\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	48.0000i	1.83667i	0.395805	+	0.918334i	$0.370466\pi$
$683$	48.0000i	1.83667i	−0.395805	+	0.918334i	$0.629534\pi$
$684$	0	0
$685$	− 2.00000i	− 0.0764161i
$686$	0	0
$687$	−4.00000	−0.152610
$688$	0	0
$689$	0	0
$690$	0	0
$691$	12.0000i	0.456502i	0.973602	+	0.228251i	$0.0733006\pi$
$691$	12.0000i	0.456502i	−0.973602	+	0.228251i	$0.926699\pi$
$692$	0	0
$693$	8.00000i	0.303895i
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	−60.0000	−2.27266
$698$	0	0
$699$	− 6.00000i	− 0.226941i
$700$	0	0
$701$	22.0000i	0.830929i	0.909610	+	0.415464i	$0.136381\pi$
$701$	22.0000i	0.830929i	−0.909610	+	0.415464i	$0.863619\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	0	0
$705$	−4.00000	−0.150649
$706$	0	0
$707$	28.0000i	1.05305i
$708$	0	0
$709$	− 44.0000i	− 1.65245i	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	− 44.0000i	− 1.65245i	0.563337	−	0.826227i	$-0.309517\pi$
$710$	0	0
$711$	−14.0000	−0.525041
$712$	0	0
$713$	−40.0000	−1.49801
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 16.0000i	− 0.597531i
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	− 30.0000i	− 1.11571i
$724$	0	0
$725$	− 6.00000i	− 0.222834i
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000i	0.887672i
$732$	0	0
$733$	24.0000i	0.886460i	0.896408	+	0.443230i	$0.146168\pi$
$733$	24.0000i	0.886460i	−0.896408	+	0.443230i	$0.853832\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	48.0000	1.76810
$738$	0	0
$739$	12.0000i	0.441427i	0.975339	+	0.220714i	$0.0708386\pi$
$739$	12.0000i	0.441427i	−0.975339	+	0.220714i	$0.929161\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 8.00000i	− 0.292314i
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	2.00000i	0.0727875i
$756$	0	0
$757$	8.00000i	0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$	8.00000i	0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$	0	0
$759$	−16.0000	−0.580763
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	0	0
$765$	6.00000i	0.216930i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	− 2.00000i	− 0.0720282i
$772$	0	0
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	10.0000	0.359211
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	40.0000i	1.43315i
$780$	0	0
$781$	16.0000i	0.572525i
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	20.0000	0.713831
$786$	0	0
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 10.0000i	− 0.354663i
$796$	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
$801$	−14.0000	−0.494666
$802$	0	0
$803$	40.0000i	1.41157i
$804$	0	0
$805$	− 8.00000i	− 0.281963i
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	− 4.00000i	− 0.140459i	−0.997531	−	0.0702295i	$-0.977627\pi$
$811$	− 4.00000i	− 0.140459i	0.997531	−	0.0702295i	$-0.0223732\pi$
$812$	0	0
$813$	2.00000i	0.0701431i
$814$	0	0
$815$	−20.0000	−0.700569
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.0000i	0.349002i	0.984657	+	0.174501i	$0.0558313\pi$
$821$	10.0000i	0.349002i	−0.984657	+	0.174501i	$0.944169\pi$
$822$	0	0
$823$	−42.0000	−1.46403	−0.732014	−	0.681290i	$-0.761419\pi$
$823$	−42.0000	−1.46403	−0.732014	+	0.681290i	$0.761419\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	− 4.00000i	− 0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$	− 4.00000i	− 0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$	0	0
$829$	44.0000i	1.52818i	0.645108	+	0.764092i	$0.276812\pi$
$829$	44.0000i	1.52818i	−0.645108	+	0.764092i	$0.723188\pi$
$830$	0	0
$831$	−16.0000	−0.555034
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	− 24.0000i	− 0.830554i
$836$	0	0
$837$	10.0000i	0.345651i
$838$	0	0
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	− 10.0000i	− 0.344418i
$844$	0	0
$845$	13.0000i	0.447214i
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	− 16.0000i	− 0.548473i
$852$	0	0
$853$	44.0000i	1.50653i	0.657716	+	0.753266i	$0.271523\pi$
$853$	44.0000i	1.50653i	−0.657716	+	0.753266i	$0.728477\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	− 44.0000i	− 1.50126i	−0.660722	−	0.750630i	$-0.729750\pi$
$859$	− 44.0000i	− 1.50126i	0.660722	−	0.750630i	$-0.270250\pi$
$860$	0	0
$861$	− 20.0000i	− 0.681598i
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	19.0000i	0.645274i
$868$	0	0
$869$	56.0000i	1.89967i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	2.00000i	0.0676123i
$876$	0	0
$877$	− 44.0000i	− 1.48577i	−0.669417	−	0.742887i	$-0.733456\pi$
$877$	− 44.0000i	− 1.48577i	0.669417	−	0.742887i	$-0.266544\pi$
$878$	0	0
$879$	−2.00000	−0.0674583
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	0	0
$885$	− 8.00000i	− 0.268917i
$886$	0	0
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	4.00000i	0.134005i
$892$	0	0
$893$	16.0000i	0.535420i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 60.0000i	− 2.00111i
$900$	0	0
$901$	− 60.0000i	− 1.99889i
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	−20.0000	−0.664822
$906$	0	0
$907$	− 52.0000i	− 1.72663i	−0.504664	−	0.863316i	$-0.668384\pi$
$907$	− 52.0000i	− 1.72663i	0.504664	−	0.863316i	$-0.331616\pi$
$908$	0	0
$909$	14.0000i	0.464351i
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 8.00000i	− 0.264472i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	26.0000	0.857661	0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000	0.857661	0.428830	+	0.903385i	$0.358926\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	0	0
$923$	0	0
$924$	0	0
$925$	4.00000i	0.131519i
$926$	0	0
$927$	−2.00000	−0.0656886
$928$	0	0
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	0	0
$931$	− 12.0000i	− 0.393284i
$932$	0	0
$933$	− 8.00000i	− 0.261908i
$934$	0	0
$935$	24.0000	0.784884
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	6.00000i	0.195803i
$940$	0	0
$941$	26.0000i	0.847576i	0.905761	+	0.423788i	$0.139300\pi$
$941$	26.0000i	0.847576i	−0.905761	+	0.423788i	$0.860700\pi$
$942$	0	0
$943$	40.0000	1.30258
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	8.00000i	0.258874i
$956$	0	0
$957$	− 24.0000i	− 0.775810i
$958$	0	0
$959$	4.00000	0.129167
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	− 4.00000i	− 0.128898i
$964$	0	0
$965$	− 14.0000i	− 0.450676i
$966$	0	0
$967$	−46.0000	−1.47926	−0.739630	−	0.673014i	$-0.765000\pi$
$967$	−46.0000	−1.47926	−0.739630	+	0.673014i	$0.765000\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	12.0000i	0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$971$	12.0000i	0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$972$	0	0
$973$	8.00000i	0.256468i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−22.0000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$978$	0	0
$979$	56.0000i	1.78977i
$980$	0	0
$981$	4.00000i	0.127710i
$982$	0	0
$983$	32.0000	1.02064	0.510321	−	0.859984i	$-0.329527\pi$
$983$	32.0000	1.02064	0.510321	+	0.859984i	$0.329527\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	− 16.0000i	− 0.508770i
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	6.00000i	0.190213i
$996$	0	0
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	0	0
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.2.k.a.241.2		2
3.2	odd	2		1440.2.k.a.721.1		2
4.3	odd	2		120.2.k.a.61.2	yes	2
5.2	odd	4		2400.2.d.d.49.1		2
5.3	odd	4		2400.2.d.a.49.2		2
5.4	even	2		2400.2.k.b.1201.1		2
8.3	odd	2		120.2.k.a.61.1	✓	2
8.5	even	2	inner	480.2.k.a.241.1		2
12.11	even	2		360.2.k.b.181.1		2
15.2	even	4		7200.2.d.f.2449.1		2
15.8	even	4		7200.2.d.e.2449.2		2
15.14	odd	2		7200.2.k.f.3601.1		2
16.3	odd	4		3840.2.a.w.1.1		1
16.5	even	4		3840.2.a.r.1.1		1
16.11	odd	4		3840.2.a.d.1.1		1
16.13	even	4		3840.2.a.m.1.1		1
20.3	even	4		600.2.d.d.349.1		2
20.7	even	4		600.2.d.a.349.2		2
20.19	odd	2		600.2.k.a.301.1		2
24.5	odd	2		1440.2.k.a.721.2		2
24.11	even	2		360.2.k.b.181.2		2
40.3	even	4		600.2.d.a.349.1		2
40.13	odd	4		2400.2.d.d.49.2		2
40.19	odd	2		600.2.k.a.301.2		2
40.27	even	4		600.2.d.d.349.2		2
40.29	even	2		2400.2.k.b.1201.2		2
40.37	odd	4		2400.2.d.a.49.1		2
60.23	odd	4		1800.2.d.c.1549.2		2
60.47	odd	4		1800.2.d.h.1549.1		2
60.59	even	2		1800.2.k.g.901.2		2
120.29	odd	2		7200.2.k.f.3601.2		2
120.53	even	4		7200.2.d.f.2449.2		2
120.59	even	2		1800.2.k.g.901.1		2
120.77	even	4		7200.2.d.e.2449.1		2
120.83	odd	4		1800.2.d.h.1549.2		2
120.107	odd	4		1800.2.d.c.1549.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.a.61.1	✓	2	8.3	odd	2
120.2.k.a.61.2	yes	2	4.3	odd	2
360.2.k.b.181.1		2	12.11	even	2
360.2.k.b.181.2		2	24.11	even	2
480.2.k.a.241.1		2	8.5	even	2	inner
480.2.k.a.241.2		2	1.1	even	1	trivial
600.2.d.a.349.1		2	40.3	even	4
600.2.d.a.349.2		2	20.7	even	4
600.2.d.d.349.1		2	20.3	even	4
600.2.d.d.349.2		2	40.27	even	4
600.2.k.a.301.1		2	20.19	odd	2
600.2.k.a.301.2		2	40.19	odd	2
1440.2.k.a.721.1		2	3.2	odd	2
1440.2.k.a.721.2		2	24.5	odd	2
1800.2.d.c.1549.1		2	120.107	odd	4
1800.2.d.c.1549.2		2	60.23	odd	4
1800.2.d.h.1549.1		2	60.47	odd	4
1800.2.d.h.1549.2		2	120.83	odd	4
1800.2.k.g.901.1		2	120.59	even	2
1800.2.k.g.901.2		2	60.59	even	2
2400.2.d.a.49.1		2	40.37	odd	4
2400.2.d.a.49.2		2	5.3	odd	4
2400.2.d.d.49.1		2	5.2	odd	4
2400.2.d.d.49.2		2	40.13	odd	4
2400.2.k.b.1201.1		2	5.4	even	2
2400.2.k.b.1201.2		2	40.29	even	2
3840.2.a.d.1.1		1	16.11	odd	4
3840.2.a.m.1.1		1	16.13	even	4
3840.2.a.r.1.1		1	16.5	even	4
3840.2.a.w.1.1		1	16.3	odd	4
7200.2.d.e.2449.1		2	120.77	even	4
7200.2.d.e.2449.2		2	15.8	even	4
7200.2.d.f.2449.1		2	15.2	even	4
7200.2.d.f.2449.2		2	120.53	even	4
7200.2.k.f.3601.1		2	15.14	odd	2
7200.2.k.f.3601.2		2	120.29	odd	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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	480.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} -1.00000 q^{15} -6.00000 q^{17} +4.00000i q^{19} -2.00000i q^{21} +4.00000 q^{23} -1.00000 q^{25} -1.00000i q^{27} +6.00000i q^{29} -10.0000 q^{31} -4.00000 q^{33} -2.00000i q^{35} -4.00000i q^{37} +10.0000 q^{41} -4.00000i q^{43} -1.00000i q^{45} +4.00000 q^{47} -3.00000 q^{49} -6.00000i q^{51} +10.0000i q^{53} -4.00000 q^{55} -4.00000 q^{57} +8.00000i q^{59} +8.00000i q^{61} +2.00000 q^{63} -12.0000i q^{67} +4.00000i q^{69} +4.00000 q^{71} +10.0000 q^{73} -1.00000i q^{75} -8.00000i q^{77} +14.0000 q^{79} +1.00000 q^{81} -6.00000i q^{85} -6.00000 q^{87} +14.0000 q^{89} -10.0000i q^{93} -4.00000 q^{95} -10.0000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^7 - 2 * q^9	\( 2 q - 4 q^{7} - 2 q^{9} - 2 q^{15} - 12 q^{17} + 8 q^{23} - 2 q^{25} - 20 q^{31} - 8 q^{33} + 20 q^{41} + 8 q^{47} - 6 q^{49} - 8 q^{55} - 8 q^{57} + 4 q^{63} + 8 q^{71} + 20 q^{73} + 28 q^{79} + 2 q^{81} - 12 q^{87} + 28 q^{89} - 8 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 - 2 * q^9 - 2 * q^15 - 12 * q^17 + 8 * q^23 - 2 * q^25 - 20 * q^31 - 8 * q^33 + 20 * q^41 + 8 * q^47 - 6 * q^49 - 8 * q^55 - 8 * q^57 + 4 * q^63 + 8 * q^71 + 20 * q^73 + 28 * q^79 + 2 * q^81 - 12 * q^87 + 28 * q^89 - 8 * q^95 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more