LMFDB - Embedded newform 5280.2.w.b.2641.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5280,2,Mod(2641,5280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 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("5280.2641");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5280 = 2^{5} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5280.w (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:	$42.1610122672$
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 1320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2641.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5280.2641
Dual form			5280.2.w.b.2641.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} +1.00000i q^{11} -4.00000i q^{13} +1.00000 q^{15} +2.00000 q^{17} +4.00000i q^{19} -2.00000i q^{21} -1.00000 q^{25} +1.00000i q^{27} -6.00000i q^{29} +6.00000 q^{31} +1.00000 q^{33} +2.00000i q^{35} -4.00000i q^{37} -4.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} -1.00000i q^{45} -3.00000 q^{49} -2.00000i q^{51} -14.0000i q^{53} -1.00000 q^{55} +4.00000 q^{57} +12.0000i q^{61} -2.00000 q^{63} +4.00000 q^{65} +4.00000i q^{67} +8.00000 q^{71} +10.0000 q^{73} +1.00000i q^{75} +2.00000i q^{77} -6.00000 q^{79} +1.00000 q^{81} +2.00000i q^{85} -6.00000 q^{87} +6.00000 q^{89} -8.00000i q^{91} -6.00000i q^{93} -4.00000 q^{95} +6.00000 q^{97} -1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7} - 2 q^{9} + 2 q^{15} + 4 q^{17} - 2 q^{25} + 12 q^{31} + 2 q^{33} - 8 q^{39} + 4 q^{41} - 6 q^{49} - 2 q^{55} + 8 q^{57} - 4 q^{63} + 8 q^{65} + 16 q^{71} + 20 q^{73} - 12 q^{79} + 2 q^{81}+ \cdots + 12 q^{97}+O(q^{100}) $ 2 * q + 4 * q^7 - 2 * q^9 + 2 * q^15 + 4 * q^17 - 2 * q^25 + 12 * q^31 + 2 * q^33 - 8 * q^39 + 4 * q^41 - 6 * q^49 - 2 * q^55 + 8 * q^57 - 4 * q^63 + 8 * q^65 + 16 * q^71 + 20 * q^73 - 12 * q^79 + 2 * q^81 - 12 * q^87 + 12 * q^89 - 8 * q^95 + 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5280\mathbb{Z}\right)^\times$.

$n$	$991$	$1057$	$3301$	$3521$	$3841$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000i	0.301511i
$11$	1.00000i	0.301511i
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	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
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	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	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	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
$39$	−4.00000	−0.640513
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	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
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	− 2.00000i	− 0.280056i
$52$	0	0
$53$	− 14.0000i	− 1.92305i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	− 14.0000i	− 1.92305i	0.274721	−	0.961524i	$-0.411414\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	12.0000i	1.53644i	0.640184	+	0.768221i	$0.278858\pi$
$61$	12.0000i	1.53644i	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	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
$77$	2.00000i	0.227921i
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	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
$85$	2.00000i	0.216930i
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	− 8.00000i	− 0.838628i
$92$	0	0
$93$	− 6.00000i	− 0.622171i
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	− 1.00000i	− 0.100504i
$100$	0	0
$101$	− 2.00000i	− 0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	− 2.00000i	− 0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	− 20.0000i	− 1.93347i	−0.255774	−	0.966736i	$-0.582330\pi$
$107$	− 20.0000i	− 1.93347i	0.255774	−	0.966736i	$-0.417670\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00000i	0.369800i
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	− 2.00000i	− 0.180334i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	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
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	− 12.0000i	− 1.01783i	−0.860818	−	0.508913i	$-0.830047\pi$
$139$	− 12.0000i	− 1.01783i	0.860818	−	0.508913i	$-0.169953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	3.00000i	0.247436i
$148$	0	0
$149$	2.00000i	0.163846i	0.996639	+	0.0819232i	$0.0261062\pi$
$149$	2.00000i	0.163846i	−0.996639	+	0.0819232i	$0.973894\pi$
$150$	0	0
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	6.00000i	0.481932i
$156$	0	0
$157$	4.00000i	0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$	4.00000i	0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$	0	0
$159$	−14.0000	−1.11027
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	1.00000i	0.0778499i
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.0000i	1.79384i	0.442189	+	0.896922i	$0.354202\pi$
$179$	24.0000i	1.79384i	−0.442189	+	0.896922i	$0.645798\pi$
$180$	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
$183$	12.0000	0.887066
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	2.00000i	0.146254i
$188$	0	0
$189$	2.00000i	0.145479i
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	0	0
$195$	− 4.00000i	− 0.286446i
$196$	0	0
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	2.00000i	0.139686i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	− 20.0000i	− 1.37686i	−0.725304	−	0.688428i	$-0.758301\pi$
$211$	− 20.0000i	− 1.37686i	0.725304	−	0.688428i	$-0.241699\pi$
$212$	0	0
$213$	− 8.00000i	− 0.548151i
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	12.0000	0.814613
$218$	0	0
$219$	− 10.0000i	− 0.675737i
$220$	0	0
$221$	− 8.00000i	− 0.538138i
$222$	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
$229$	20.0000i	1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$	20.0000i	1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	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
$235$	0	0
$236$	0	0
$237$	6.00000i	0.389742i
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.00000i	0.252478i	0.992000	+	0.126239i	$0.0402906\pi$
$251$	4.00000i	0.252478i	−0.992000	+	0.126239i	$0.959709\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	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
$265$	14.0000	0.860013
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	0	0
$269$	− 18.0000i	− 1.09748i	−0.835993	−	0.548740i	$-0.815108\pi$
$269$	− 18.0000i	− 1.09748i	0.835993	−	0.548740i	$-0.184892\pi$
$270$	0	0
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	− 1.00000i	− 0.0603023i
$276$	0	0
$277$	12.0000i	0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$	12.0000i	0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	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
$287$	4.00000	0.236113
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	− 6.00000i	− 0.351726i
$292$	0	0
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.00000i	0.461112i
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	− 10.0000i	− 0.568880i
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	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
$317$	− 34.0000i	− 1.90963i	−0.297200	−	0.954815i	$-0.596053\pi$
$317$	− 34.0000i	− 1.90963i	0.297200	−	0.954815i	$-0.403947\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	−20.0000	−1.11629
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	4.00000i	0.221880i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 12.0000i	− 0.659580i	−0.944054	−	0.329790i	$-0.893022\pi$
$331$	− 12.0000i	− 0.659580i	0.944054	−	0.329790i	$-0.106978\pi$
$332$	0	0
$333$	4.00000i	0.219199i
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	2.00000i	0.108625i
$340$	0	0
$341$	6.00000i	0.324918i
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	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
$349$	− 28.0000i	− 1.49881i	−0.662114	−	0.749403i	$-0.730341\pi$
$349$	− 28.0000i	− 1.49881i	0.662114	−	0.749403i	$-0.269659\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	8.00000i	0.424596i
$356$	0	0
$357$	− 4.00000i	− 0.211702i
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	1.00000i	0.0524864i
$364$	0	0
$365$	10.0000i	0.523424i
$366$	0	0
$367$	34.0000	1.77479	0.887393	−	0.461014i	$-0.152514\pi$
$367$	34.0000	1.77479	0.887393	+	0.461014i	$0.152514\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	− 28.0000i	− 1.45369i
$372$	0	0
$373$	− 24.0000i	− 1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$	− 24.0000i	− 1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−24.0000	−1.23606
$378$	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
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	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
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 6.00000i	− 0.301893i
$396$	0	0
$397$	− 28.0000i	− 1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$	− 28.0000i	− 1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−34.0000	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$402$	0	0
$403$	− 24.0000i	− 1.19553i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	− 6.00000i	− 0.295958i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	4.00000i	0.195413i	0.995215	+	0.0977064i	$0.0311506\pi$
$419$	4.00000i	0.195413i	−0.995215	+	0.0977064i	$0.968849\pi$
$420$	0	0
$421$	− 12.0000i	− 0.584844i	−0.956289	−	0.292422i	$-0.905539\pi$
$421$	− 12.0000i	− 0.584844i	0.956289	−	0.292422i	$-0.0944612\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	24.0000i	1.16144i
$428$	0	0
$429$	− 4.00000i	− 0.193122i
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	− 6.00000i	− 0.287678i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	8.00000i	0.380091i	0.981775	+	0.190046i	$0.0608636\pi$
$443$	8.00000i	0.380091i	−0.981775	+	0.190046i	$0.939136\pi$
$444$	0	0
$445$	6.00000i	0.284427i
$446$	0	0
$447$	2.00000	0.0945968
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	2.00000i	0.0941763i
$452$	0	0
$453$	− 22.0000i	− 1.03365i
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	0	0
$459$	2.00000i	0.0933520i
$460$	0	0
$461$	− 10.0000i	− 0.465746i	−0.972507	−	0.232873i	$-0.925187\pi$
$461$	− 10.0000i	− 0.465746i	0.972507	−	0.232873i	$-0.0748127\pi$
$462$	0	0
$463$	−2.00000	−0.0929479	−0.0464739	−	0.998920i	$-0.514798\pi$
$463$	−2.00000	−0.0929479	−0.0464739	+	0.998920i	$0.514798\pi$
$464$	0	0
$465$	6.00000	0.278243
$466$	0	0
$467$	32.0000i	1.48078i	0.672176	+	0.740392i	$0.265360\pi$
$467$	32.0000i	1.48078i	−0.672176	+	0.740392i	$0.734640\pi$
$468$	0	0
$469$	8.00000i	0.369406i
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	14.0000i	0.641016i
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.00000i	0.272446i
$486$	0	0
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	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
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	16.0000	0.717698
$498$	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
$501$	0	0
$502$	0	0
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	3.00000i	0.133235i
$508$	0	0
$509$	− 42.0000i	− 1.86162i	−0.365507	−	0.930809i	$-0.619104\pi$
$509$	− 42.0000i	− 1.86162i	0.365507	−	0.930809i	$-0.380896\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	10.0000i	0.440653i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	0	0
$525$	2.00000i	0.0872872i
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 8.00000i	− 0.346518i
$534$	0	0
$535$	20.0000	0.864675
$536$	0	0
$537$	24.0000	1.03568
$538$	0	0
$539$	− 3.00000i	− 0.129219i
$540$	0	0
$541$	16.0000i	0.687894i	0.938989	+	0.343947i	$0.111764\pi$
$541$	16.0000i	0.687894i	−0.938989	+	0.343947i	$0.888236\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	0	0
$546$	0	0
$547$	44.0000i	1.88130i	0.339372	+	0.940652i	$0.389785\pi$
$547$	44.0000i	1.88130i	−0.339372	+	0.940652i	$0.610215\pi$
$548$	0	0
$549$	− 12.0000i	− 0.512148i
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	−12.0000	−0.510292
$554$	0	0
$555$	− 4.00000i	− 0.169791i
$556$	0	0
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	16.0000i	0.674320i	0.941447	+	0.337160i	$0.109466\pi$
$563$	16.0000i	0.674320i	−0.941447	+	0.337160i	$0.890534\pi$
$564$	0	0
$565$	− 2.00000i	− 0.0841406i
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	− 28.0000i	− 1.17176i	−0.810397	−	0.585882i	$-0.800748\pi$
$571$	− 28.0000i	− 1.17176i	0.810397	−	0.585882i	$-0.199252\pi$
$572$	0	0
$573$	4.00000i	0.167102i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	0	0
$579$	− 18.0000i	− 0.748054i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	14.0000	0.579821
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	24.0000i	0.988903i
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	4.00000i	0.163984i
$596$	0	0
$597$	2.00000i	0.0818546i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	− 1.00000i	− 0.0406558i
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	8.00000i	0.323117i	0.986863	+	0.161558i	$0.0516520\pi$
$613$	8.00000i	0.323117i	−0.986863	+	0.161558i	$0.948348\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	44.0000i	1.76851i	0.467005	+	0.884255i	$0.345333\pi$
$619$	44.0000i	1.76851i	−0.467005	+	0.884255i	$0.654667\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.0000	0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.00000i	0.159745i
$628$	0	0
$629$	− 8.00000i	− 0.318981i
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	− 6.00000i	− 0.238103i
$636$	0	0
$637$	12.0000i	0.475457i
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	− 28.0000i	− 1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$	− 28.0000i	− 1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	0	0
$645$	4.00000i	0.157500i
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 12.0000i	− 0.470317i
$652$	0	0
$653$	26.0000i	1.01746i	0.860927	+	0.508729i	$0.169885\pi$
$653$	26.0000i	1.01746i	−0.860927	+	0.508729i	$0.830115\pi$
$654$	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
$661$	8.00000i	0.311164i	0.987823	+	0.155582i	$0.0497253\pi$
$661$	8.00000i	0.311164i	−0.987823	+	0.155582i	$0.950275\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	0	0
$668$	0	0
$669$	10.0000i	0.386622i
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	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
$679$	12.0000	0.460518
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 8.00000i	− 0.306111i	−0.988218	−	0.153056i	$-0.951089\pi$
$683$	− 8.00000i	− 0.306111i	0.988218	−	0.153056i	$-0.0489114\pi$
$684$	0	0
$685$	6.00000i	0.229248i
$686$	0	0
$687$	20.0000	0.763048
$688$	0	0
$689$	−56.0000	−2.13343
$690$	0	0
$691$	36.0000i	1.36950i	0.728776	+	0.684752i	$0.240090\pi$
$691$	36.0000i	1.36950i	−0.728776	+	0.684752i	$0.759910\pi$
$692$	0	0
$693$	− 2.00000i	− 0.0759737i
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	6.00000i	0.226941i
$700$	0	0
$701$	34.0000i	1.28416i	0.766637	+	0.642081i	$0.221929\pi$
$701$	34.0000i	1.28416i	−0.766637	+	0.642081i	$0.778071\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.00000i	− 0.150435i
$708$	0	0
$709$	52.0000i	1.95290i	0.215742	+	0.976450i	$0.430783\pi$
$709$	52.0000i	1.95290i	−0.215742	+	0.976450i	$0.569217\pi$
$710$	0	0
$711$	6.00000	0.225018
$712$	0	0
$713$	0	0
$714$	0	0
$715$	4.00000i	0.149592i
$716$	0	0
$717$	8.00000i	0.298765i
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	0	0
$723$	− 10.0000i	− 0.371904i
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	6.00000	0.222528	0.111264	−	0.993791i	$-0.464510\pi$
$727$	6.00000	0.222528	0.111264	+	0.993791i	$0.464510\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.00000i	0.295891i
$732$	0	0
$733$	20.0000i	0.738717i	0.929287	+	0.369358i	$0.120423\pi$
$733$	20.0000i	0.738717i	−0.929287	+	0.369358i	$0.879577\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	28.0000i	1.03000i	0.857191	+	0.514998i	$0.172207\pi$
$739$	28.0000i	1.03000i	−0.857191	+	0.514998i	$0.827793\pi$
$740$	0	0
$741$	− 16.0000i	− 0.587775i
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 40.0000i	− 1.46157i
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	0	0
$753$	4.00000	0.145768
$754$	0	0
$755$	22.0000i	0.800662i
$756$	0	0
$757$	− 16.0000i	− 0.581530i	−0.956795	−	0.290765i	$-0.906090\pi$
$757$	− 16.0000i	− 0.581530i	0.956795	−	0.290765i	$-0.0939098\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 2.00000i	− 0.0723102i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	− 22.0000i	− 0.792311i
$772$	0	0
$773$	− 46.0000i	− 1.65451i	−0.561830	−	0.827253i	$-0.689903\pi$
$773$	− 46.0000i	− 1.65451i	0.561830	−	0.827253i	$-0.310097\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	8.00000i	0.286630i
$780$	0	0
$781$	8.00000i	0.286263i
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	−4.00000	−0.142766
$786$	0	0
$787$	52.0000i	1.85360i	0.375555	+	0.926800i	$0.377452\pi$
$787$	52.0000i	1.85360i	−0.375555	+	0.926800i	$0.622548\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	48.0000	1.70453
$794$	0	0
$795$	− 14.0000i	− 0.496529i
$796$	0	0
$797$	− 14.0000i	− 0.495905i	−0.968772	−	0.247953i	$-0.920242\pi$
$797$	− 14.0000i	− 0.495905i	0.968772	−	0.247953i	$-0.0797578\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	10.0000i	0.352892i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	4.00000i	0.140459i	0.997531	+	0.0702295i	$0.0223732\pi$
$811$	4.00000i	0.140459i	−0.997531	+	0.0702295i	$0.977627\pi$
$812$	0	0
$813$	− 22.0000i	− 0.771574i
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	8.00000i	0.279543i
$820$	0	0
$821$	54.0000i	1.88461i	0.334751	+	0.942306i	$0.391348\pi$
$821$	54.0000i	1.88461i	−0.334751	+	0.942306i	$0.608652\pi$
$822$	0	0
$823$	−10.0000	−0.348578	−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000	−0.348578	−0.174289	+	0.984695i	$0.555763\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	0	0
$829$	− 28.0000i	− 0.972480i	−0.873825	−	0.486240i	$-0.838368\pi$
$829$	− 28.0000i	− 0.972480i	0.873825	−	0.486240i	$-0.161632\pi$
$830$	0	0
$831$	12.0000	0.416275
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.00000i	0.207390i
$838$	0	0
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	− 22.0000i	− 0.757720i
$844$	0	0
$845$	− 3.00000i	− 0.103203i
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 16.0000i	− 0.547830i	−0.961754	−	0.273915i	$-0.911681\pi$
$853$	− 16.0000i	− 0.547830i	0.961754	−	0.273915i	$-0.0883186\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	20.0000i	0.682391i	0.939992	+	0.341196i	$0.110832\pi$
$859$	20.0000i	0.682391i	−0.939992	+	0.341196i	$0.889168\pi$
$860$	0	0
$861$	− 4.00000i	− 0.136320i
$862$	0	0
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	− 6.00000i	− 0.203536i
$870$	0	0
$871$	16.0000	0.542139
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	− 2.00000i	− 0.0676123i
$876$	0	0
$877$	56.0000i	1.89099i	0.325643	+	0.945493i	$0.394419\pi$
$877$	56.0000i	1.89099i	−0.325643	+	0.945493i	$0.605581\pi$
$878$	0	0
$879$	14.0000	0.472208
$880$	0	0
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	1.00000i	0.0335013i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 36.0000i	− 1.20067i
$900$	0	0
$901$	− 28.0000i	− 0.932815i
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	−20.0000	−0.664822
$906$	0	0
$907$	− 20.0000i	− 0.664089i	−0.943264	−	0.332045i	$-0.892262\pi$
$907$	− 20.0000i	− 0.664089i	0.943264	−	0.332045i	$-0.107738\pi$
$908$	0	0
$909$	2.00000i	0.0663358i
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	12.0000i	0.396708i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	− 32.0000i	− 1.05329i
$924$	0	0
$925$	4.00000i	0.131519i
$926$	0	0
$927$	−10.0000	−0.328443
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	− 12.0000i	− 0.393284i
$932$	0	0
$933$	20.0000i	0.654771i
$934$	0	0
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	0	0
$939$	− 6.00000i	− 0.195803i
$940$	0	0
$941$	6.00000i	0.195594i	0.995206	+	0.0977972i	$0.0311797\pi$
$941$	6.00000i	0.195594i	−0.995206	+	0.0977972i	$0.968820\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	0	0
$949$	− 40.0000i	− 1.29845i
$950$	0	0
$951$	−34.0000	−1.10253
$952$	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
$955$	− 4.00000i	− 0.129437i
$956$	0	0
$957$	− 6.00000i	− 0.193952i
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	20.0000i	0.644491i
$964$	0	0
$965$	18.0000i	0.579441i
$966$	0	0
$967$	−18.0000	−0.578841	−0.289420	−	0.957202i	$-0.593463\pi$
$967$	−18.0000	−0.578841	−0.289420	+	0.957202i	$0.593463\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	− 12.0000i	− 0.385098i	−0.981287	−	0.192549i	$-0.938325\pi$
$971$	− 12.0000i	− 0.385098i	0.981287	−	0.192549i	$-0.0616755\pi$
$972$	0	0
$973$	− 24.0000i	− 0.769405i
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	58.0000	1.85558	0.927792	−	0.373097i	$-0.121704\pi$
$977$	58.0000	1.85558	0.927792	+	0.373097i	$0.121704\pi$
$978$	0	0
$979$	6.00000i	0.191761i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	14.0000	0.444725	0.222362	−	0.974964i	$-0.428623\pi$
$991$	14.0000	0.444725	0.222362	+	0.974964i	$0.428623\pi$
$992$	0	0
$993$	−12.0000	−0.380808
$994$	0	0
$995$	− 2.00000i	− 0.0634043i
$996$	0	0
$997$	40.0000i	1.26681i	0.773819	+	0.633406i	$0.218344\pi$
$997$	40.0000i	1.26681i	−0.773819	+	0.633406i	$0.781656\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5280.2.w.b.2641.1		2
4.3	odd	2		1320.2.w.a.661.1	✓	2
8.3	odd	2		1320.2.w.a.661.2	yes	2
8.5	even	2	inner	5280.2.w.b.2641.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.w.a.661.1	✓	2	4.3	odd	2
1320.2.w.a.661.2	yes	2	8.3	odd	2
5280.2.w.b.2641.1		2	1.1	even	1	trivial
5280.2.w.b.2641.2		2	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5280 = 2^{5} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5280.w (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} +1.00000i q^{11} -4.00000i q^{13} +1.00000 q^{15} +2.00000 q^{17} +4.00000i q^{19} -2.00000i q^{21} -1.00000 q^{25} +1.00000i q^{27} -6.00000i q^{29} +6.00000 q^{31} +1.00000 q^{33} +2.00000i q^{35} -4.00000i q^{37} -4.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} -1.00000i q^{45} -3.00000 q^{49} -2.00000i q^{51} -14.0000i q^{53} -1.00000 q^{55} +4.00000 q^{57} +12.0000i q^{61} -2.00000 q^{63} +4.00000 q^{65} +4.00000i q^{67} +8.00000 q^{71} +10.0000 q^{73} +1.00000i q^{75} +2.00000i q^{77} -6.00000 q^{79} +1.00000 q^{81} +2.00000i q^{85} -6.00000 q^{87} +6.00000 q^{89} -8.00000i q^{91} -6.00000i q^{93} -4.00000 q^{95} +6.00000 q^{97} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7} - 2 q^{9} + 2 q^{15} + 4 q^{17} - 2 q^{25} + 12 q^{31} + 2 q^{33} - 8 q^{39} + 4 q^{41} - 6 q^{49} - 2 q^{55} + 8 q^{57} - 4 q^{63} + 8 q^{65} + 16 q^{71} + 20 q^{73} - 12 q^{79} + 2 q^{81}+ \cdots + 12 q^{97}+O(q^{100}) \) 2 * q + 4 * q^7 - 2 * q^9 + 2 * q^15 + 4 * q^17 - 2 * q^25 + 12 * q^31 + 2 * q^33 - 8 * q^39 + 4 * q^41 - 6 * q^49 - 2 * q^55 + 8 * q^57 - 4 * q^63 + 8 * q^65 + 16 * q^71 + 20 * q^73 - 12 * q^79 + 2 * q^81 - 12 * q^87 + 12 * q^89 - 8 * q^95 + 12 * q^97

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