LMFDB - Embedded newform 5780.2.c.a.5201.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5780,2,Mod(5201,5780)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5780.5201"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5780, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5780 = 2^{2} \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5780.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$46.1535323683$
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 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5201.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5780.5201
Dual form			5780.2.c.a.5201.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} -1.00000i q^{5} -2.00000i q^{7} -1.00000 q^{9} +2.00000 q^{13} -2.00000 q^{15} +4.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} -1.00000 q^{25} -4.00000i q^{27} +6.00000i q^{29} -4.00000i q^{31} -2.00000 q^{35} +2.00000i q^{37} -4.00000i q^{39} -6.00000i q^{41} +10.0000 q^{43} +1.00000i q^{45} -6.00000 q^{47} +3.00000 q^{49} +6.00000 q^{53} -8.00000i q^{57} -12.0000 q^{59} -2.00000i q^{61} +2.00000i q^{63} -2.00000i q^{65} +2.00000 q^{67} -12.0000 q^{69} -12.0000i q^{71} +2.00000i q^{73} +2.00000i q^{75} -8.00000i q^{79} -11.0000 q^{81} -6.00000 q^{83} +12.0000 q^{87} -6.00000 q^{89} -4.00000i q^{91} -8.00000 q^{93} -4.00000i q^{95} +2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} + 4 q^{13} - 4 q^{15} + 8 q^{19} - 8 q^{21} - 2 q^{25} - 4 q^{35} + 20 q^{43} - 12 q^{47} + 6 q^{49} + 12 q^{53} - 24 q^{59} + 4 q^{67} - 24 q^{69} - 22 q^{81} - 12 q^{83} + 24 q^{87} - 12 q^{89}+ \cdots - 16 q^{93}+O(q^{100}) $ 2 * q - 2 * q^9 + 4 * q^13 - 4 * q^15 + 8 * q^19 - 8 * q^21 - 2 * q^25 - 4 * q^35 + 20 * q^43 - 12 * q^47 + 6 * q^49 + 12 * q^53 - 24 * q^59 + 4 * q^67 - 24 * q^69 - 22 * q^81 - 12 * q^83 + 24 * q^87 - 12 * q^89 - 16 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5780\mathbb{Z}\right)^\times$.

$n$	$581$	$1157$	$2891$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	0	0
$17$	0	0
$18$	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
$21$	−4.00000	−0.872872
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	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
$31$	− 4.00000i	− 0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$	− 4.00000i	− 0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	− 4.00000i	− 0.640513i
$40$	0	0
$41$	− 6.00000i	− 0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	− 6.00000i	− 0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	1.00000i	0.149071i
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 8.00000i	− 1.05963i
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	− 2.00000i	− 0.256074i	−0.991769	−	0.128037i	$-0.959132\pi$
$61$	− 2.00000i	− 0.256074i	0.991769	−	0.128037i	$-0.0408676\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	0	0
$65$	− 2.00000i	− 0.248069i
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	−12.0000	−1.44463
$70$	0	0
$71$	− 12.0000i	− 1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$	− 12.0000i	− 1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	2.00000i	0.230940i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 8.00000i	− 0.900070i	−0.893011	−	0.450035i	$-0.851411\pi$
$79$	− 8.00000i	− 0.900070i	0.893011	−	0.450035i	$-0.148589\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	12.0000	1.28654
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	− 4.00000i	− 0.419314i
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	− 4.00000i	− 0.410391i
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	4.00000i	0.390360i
$106$	0	0
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	−12.0000	−1.08200
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	− 20.0000i	− 1.76090i
$130$	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
$135$	−4.00000	−0.344265
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	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
$141$	12.0000i	1.01058i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	− 6.00000i	− 0.494872i
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	− 12.0000i	− 0.951662i
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	10.0000i	0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$	10.0000i	0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.0000i	1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$167$	18.0000i	1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	2.00000i	0.151186i
$176$	0	0
$177$	24.0000i	1.80395i
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	10.0000i	0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$	10.0000i	0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−8.00000	−0.581914
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	− 26.0000i	− 1.87152i	−0.352636	−	0.935760i	$-0.614715\pi$
$193$	− 26.0000i	− 1.87152i	0.352636	−	0.935760i	$-0.385285\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	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
$199$	8.00000i	0.567105i	0.958957	+	0.283552i	$0.0915130\pi$
$199$	8.00000i	0.567105i	−0.958957	+	0.283552i	$0.908487\pi$
$200$	0	0
$201$	− 4.00000i	− 0.282138i
$202$	0	0
$203$	12.0000	0.842235
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	6.00000i	0.417029i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	16.0000i	1.10149i	0.834675	+	0.550743i	$0.185655\pi$
$211$	16.0000i	1.10149i	−0.834675	+	0.550743i	$0.814345\pi$
$212$	0	0
$213$	−24.0000	−1.64445
$214$	0	0
$215$	− 10.0000i	− 0.681994i
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	6.00000i	0.398234i	0.979976	+	0.199117i	$0.0638074\pi$
$227$	6.00000i	0.398234i	−0.979976	+	0.199117i	$0.936193\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	6.00000i	0.391397i
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	14.0000i	0.901819i	0.892570	+	0.450910i	$0.148900\pi$
$241$	14.0000i	0.901819i	−0.892570	+	0.450910i	$0.851100\pi$
$242$	0	0
$243$	10.0000i	0.641500i
$244$	0	0
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	12.0000i	0.760469i
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	− 6.00000i	− 0.368577i
$266$	0	0
$267$	12.0000i	0.734388i
$268$	0	0
$269$	18.0000i	1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$	18.0000i	1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	0	0
$279$	4.00000i	0.239474i
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	0	0
$290$	0	0
$291$	4.00000	0.234484
$292$	0	0
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	0	0
$295$	12.0000i	0.698667i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 12.0000i	− 0.693978i
$300$	0	0
$301$	− 20.0000i	− 1.15278i
$302$	0	0
$303$	− 12.0000i	− 0.689382i
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	− 28.0000i	− 1.59286i
$310$	0	0
$311$	12.0000i	0.680458i	0.940343	+	0.340229i	$0.110505\pi$
$311$	12.0000i	0.680458i	−0.940343	+	0.340229i	$0.889495\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	12.0000i	0.661581i
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	− 2.00000i	− 0.109272i
$336$	0	0
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	12.0000i	0.646058i
$346$	0	0
$347$	30.0000i	1.61048i	0.592946	+	0.805242i	$0.297965\pi$
$347$	30.0000i	1.61048i	−0.592946	+	0.805242i	$0.702035\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	− 8.00000i	− 0.427008i
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	− 22.0000i	− 1.15470i
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	22.0000i	1.14839i	0.818718	+	0.574195i	$0.194685\pi$
$367$	22.0000i	1.14839i	−0.818718	+	0.574195i	$0.805315\pi$
$368$	0	0
$369$	6.00000i	0.312348i
$370$	0	0
$371$	− 12.0000i	− 0.623009i
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	2.00000	0.103280
$376$	0	0
$377$	12.0000i	0.618031i
$378$	0	0
$379$	− 28.0000i	− 1.43826i	−0.694874	−	0.719132i	$-0.744540\pi$
$379$	− 28.0000i	− 1.43826i	0.694874	−	0.719132i	$-0.255460\pi$
$380$	0	0
$381$	4.00000i	0.204926i
$382$	0	0
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.0000	−0.508329
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	−16.0000	−0.801002
$400$	0	0
$401$	30.0000i	1.49813i	0.662497	+	0.749064i	$0.269497\pi$
$401$	30.0000i	1.49813i	−0.662497	+	0.749064i	$0.730503\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	0	0
$405$	11.0000i	0.546594i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	− 36.0000i	− 1.77575i
$412$	0	0
$413$	24.0000i	1.18096i
$414$	0	0
$415$	6.00000i	0.294528i
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	− 36.0000i	− 1.75872i	−0.476162	−	0.879358i	$-0.657972\pi$
$419$	− 36.0000i	− 1.75872i	0.476162	−	0.879358i	$-0.342028\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.00000	−0.193574
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 36.0000i	− 1.73406i	−0.498257	−	0.867029i	$-0.666026\pi$
$431$	− 36.0000i	− 1.73406i	0.498257	−	0.867029i	$-0.333974\pi$
$432$	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
$435$	− 12.0000i	− 0.575356i
$436$	0	0
$437$	− 24.0000i	− 1.14808i
$438$	0	0
$439$	8.00000i	0.381819i	0.981608	+	0.190910i	$0.0611437\pi$
$439$	8.00000i	0.381819i	−0.981608	+	0.190910i	$0.938856\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	0	0
$445$	6.00000i	0.284427i
$446$	0	0
$447$	12.0000i	0.567581i
$448$	0	0
$449$	− 6.00000i	− 0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$	− 6.00000i	− 0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	40.0000i	1.87936i
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	0	0
$465$	8.00000i	0.370991i
$466$	0	0
$467$	30.0000	1.38823	0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000	1.38823	0.694117	+	0.719862i	$0.255795\pi$
$468$	0	0
$469$	− 4.00000i	− 0.184703i
$470$	0	0
$471$	44.0000i	2.02741i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	− 24.0000i	− 1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$	− 24.0000i	− 1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$	0	0
$481$	4.00000i	0.182384i
$482$	0	0
$483$	24.0000i	1.09204i
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	− 26.0000i	− 1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$	− 26.0000i	− 1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.0000	−1.07655
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	36.0000	1.60836
$502$	0	0
$503$	18.0000i	0.802580i	0.915951	+	0.401290i	$0.131438\pi$
$503$	18.0000i	0.802580i	−0.915951	+	0.401290i	$0.868562\pi$
$504$	0	0
$505$	− 6.00000i	− 0.266996i
$506$	0	0
$507$	18.0000i	0.799408i
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	− 16.0000i	− 0.706417i
$514$	0	0
$515$	− 14.0000i	− 0.616914i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	6.00000i	0.262865i	0.991325	+	0.131432i	$0.0419576\pi$
$521$	6.00000i	0.262865i	−0.991325	+	0.131432i	$0.958042\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	− 24.0000i	− 1.03568i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14.0000i	0.601907i	0.953639	+	0.300954i	$0.0973049\pi$
$541$	14.0000i	0.601907i	−0.953639	+	0.300954i	$0.902695\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	26.0000i	1.11168i	0.831289	+	0.555840i	$0.187603\pi$
$547$	26.0000i	1.11168i	−0.831289	+	0.555840i	$0.812397\pi$
$548$	0	0
$549$	2.00000i	0.0853579i
$550$	0	0
$551$	24.0000i	1.02243i
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	− 4.00000i	− 0.169791i
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	20.0000	0.845910
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	22.0000i	0.923913i
$568$	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
$571$	− 8.00000i	− 0.334790i	−0.985890	−	0.167395i	$-0.946465\pi$
$571$	− 8.00000i	− 0.334790i	0.985890	−	0.167395i	$-0.0535355\pi$
$572$	0	0
$573$	24.0000i	1.00261i
$574$	0	0
$575$	6.00000i	0.250217i
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	0	0
$579$	−52.0000	−2.16105
$580$	0	0
$581$	12.0000i	0.497844i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2.00000i	0.0826898i
$586$	0	0
$587$	6.00000	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$588$	0	0
$589$	− 16.0000i	− 0.659269i
$590$	0	0
$591$	−36.0000	−1.48084
$592$	0	0
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	10.0000i	0.407909i	0.978980	+	0.203954i	$0.0653794\pi$
$601$	10.0000i	0.407909i	−0.978980	+	0.203954i	$0.934621\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	0	0
$605$	− 11.0000i	− 0.447214i
$606$	0	0
$607$	− 22.0000i	− 0.892952i	−0.894795	−	0.446476i	$-0.852679\pi$
$607$	− 22.0000i	− 0.892952i	0.894795	−	0.446476i	$-0.147321\pi$
$608$	0	0
$609$	− 24.0000i	− 0.972529i
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	12.0000i	0.483887i
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	− 20.0000i	− 0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$	− 20.0000i	− 0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	0	0
$623$	12.0000i	0.480770i
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	0	0
$633$	32.0000	1.27189
$634$	0	0
$635$	2.00000i	0.0793676i
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	12.0000i	0.474713i
$640$	0	0
$641$	− 18.0000i	− 0.710957i	−0.934684	−	0.355479i	$-0.884318\pi$
$641$	− 18.0000i	− 0.710957i	0.934684	−	0.355479i	$-0.115682\pi$
$642$	0	0
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	0	0
$645$	−20.0000	−0.787499
$646$	0	0
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	16.0000i	0.627089i
$652$	0	0
$653$	− 42.0000i	− 1.64359i	−0.569785	−	0.821794i	$-0.692974\pi$
$653$	− 42.0000i	− 1.64359i	0.569785	−	0.821794i	$-0.307026\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	36.0000	1.39393
$668$	0	0
$669$	− 20.0000i	− 0.773245i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	46.0000i	1.77317i	0.462566	+	0.886585i	$0.346929\pi$
$673$	46.0000i	1.77317i	−0.462566	+	0.886585i	$0.653071\pi$
$674$	0	0
$675$	4.00000i	0.153960i
$676$	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
$679$	4.00000	0.153506
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	− 42.0000i	− 1.60709i	−0.595247	−	0.803543i	$-0.702946\pi$
$683$	− 42.0000i	− 1.60709i	0.595247	−	0.803543i	$-0.297054\pi$
$684$	0	0
$685$	− 18.0000i	− 0.687745i
$686$	0	0
$687$	28.0000i	1.06827i
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	− 8.00000i	− 0.304334i	−0.988355	−	0.152167i	$-0.951375\pi$
$691$	− 8.00000i	− 0.304334i	0.988355	−	0.152167i	$-0.0486252\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	8.00000i	0.301726i
$704$	0	0
$705$	12.0000	0.451946
$706$	0	0
$707$	− 12.0000i	− 0.451306i
$708$	0	0
$709$	− 34.0000i	− 1.27690i	−0.769665	−	0.638448i	$-0.779577\pi$
$709$	− 34.0000i	− 1.27690i	0.769665	−	0.638448i	$-0.220423\pi$
$710$	0	0
$711$	8.00000i	0.300023i
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.0000i	1.79259i
$718$	0	0
$719$	− 24.0000i	− 0.895049i	−0.894272	−	0.447524i	$-0.852306\pi$
$719$	− 24.0000i	− 0.895049i	0.894272	−	0.447524i	$-0.147694\pi$
$720$	0	0
$721$	− 28.0000i	− 1.04277i
$722$	0	0
$723$	28.0000	1.04133
$724$	0	0
$725$	− 6.00000i	− 0.222834i
$726$	0	0
$727$	−46.0000	−1.70605	−0.853023	−	0.521874i	$-0.825233\pi$
$727$	−46.0000	−1.70605	−0.853023	+	0.521874i	$0.825233\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	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
$741$	− 16.0000i	− 0.587775i
$742$	0	0
$743$	6.00000i	0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$	6.00000i	0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$	0	0
$745$	6.00000i	0.219823i
$746$	0	0
$747$	6.00000	0.219529
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	− 4.00000i	− 0.145962i	−0.997333	−	0.0729810i	$-0.976749\pi$
$751$	− 4.00000i	− 0.145962i	0.997333	−	0.0729810i	$-0.0232513\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	20.0000i	0.727875i
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	− 12.0000i	− 0.432169i
$772$	0	0
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	0	0
$775$	4.00000i	0.143684i
$776$	0	0
$777$	− 8.00000i	− 0.286998i
$778$	0	0
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	24.0000	0.857690
$784$	0	0
$785$	22.0000i	0.785214i
$786$	0	0
$787$	26.0000i	0.926800i	0.886149	+	0.463400i	$0.153371\pi$
$787$	26.0000i	0.926800i	−0.886149	+	0.463400i	$0.846629\pi$
$788$	0	0
$789$	− 36.0000i	− 1.28163i
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	0	0
$795$	−12.0000	−0.425596
$796$	0	0
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	12.0000i	0.422944i
$806$	0	0
$807$	36.0000	1.26726
$808$	0	0
$809$	6.00000i	0.210949i	0.994422	+	0.105474i	$0.0336361\pi$
$809$	6.00000i	0.210949i	−0.994422	+	0.105474i	$0.966364\pi$
$810$	0	0
$811$	− 16.0000i	− 0.561836i	−0.959732	−	0.280918i	$-0.909361\pi$
$811$	− 16.0000i	− 0.561836i	0.959732	−	0.280918i	$-0.0906389\pi$
$812$	0	0
$813$	− 40.0000i	− 1.40286i
$814$	0	0
$815$	10.0000	0.350285
$816$	0	0
$817$	40.0000	1.39942
$818$	0	0
$819$	4.00000i	0.139771i
$820$	0	0
$821$	− 54.0000i	− 1.88461i	−0.334751	−	0.942306i	$-0.608652\pi$
$821$	− 54.0000i	− 1.88461i	0.334751	−	0.942306i	$-0.391348\pi$
$822$	0	0
$823$	− 38.0000i	− 1.32460i	−0.749240	−	0.662298i	$-0.769581\pi$
$823$	− 38.0000i	− 1.32460i	0.749240	−	0.662298i	$-0.230419\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.0000i	1.04320i	0.853189	+	0.521601i	$0.174665\pi$
$827$	30.0000i	1.04320i	−0.853189	+	0.521601i	$0.825335\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	52.0000	1.80386
$832$	0	0
$833$	0	0
$834$	0	0
$835$	18.0000	0.622916
$836$	0	0
$837$	−16.0000	−0.553041
$838$	0	0
$839$	48.0000i	1.65714i	0.559883	+	0.828572i	$0.310846\pi$
$839$	48.0000i	1.65714i	−0.559883	+	0.828572i	$0.689154\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	12.0000i	0.413302i
$844$	0	0
$845$	9.00000i	0.309609i
$846$	0	0
$847$	− 22.0000i	− 0.755929i
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	50.0000i	1.71197i	0.517003	+	0.855984i	$0.327048\pi$
$853$	50.0000i	1.71197i	−0.517003	+	0.855984i	$0.672952\pi$
$854$	0	0
$855$	4.00000i	0.136797i
$856$	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
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	24.0000i	0.817918i
$862$	0	0
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	− 26.0000i	− 0.877958i	−0.898497	−	0.438979i	$-0.855340\pi$
$877$	− 26.0000i	− 0.877958i	0.898497	−	0.438979i	$-0.144660\pi$
$878$	0	0
$879$	60.0000i	2.02375i
$880$	0	0
$881$	− 18.0000i	− 0.606435i	−0.952921	−	0.303218i	$-0.901939\pi$
$881$	− 18.0000i	− 0.606435i	0.952921	−	0.303218i	$-0.0980609\pi$
$882$	0	0
$883$	14.0000	0.471138	0.235569	−	0.971858i	$-0.424305\pi$
$883$	14.0000	0.471138	0.235569	+	0.971858i	$0.424305\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	0	0
$887$	18.0000i	0.604381i	0.953248	+	0.302190i	$0.0977178\pi$
$887$	18.0000i	0.604381i	−0.953248	+	0.302190i	$0.902282\pi$
$888$	0	0
$889$	4.00000i	0.134156i
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−24.0000	−0.803129
$894$	0	0
$895$	− 12.0000i	− 0.401116i
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−40.0000	−1.33112
$904$	0	0
$905$	10.0000	0.332411
$906$	0	0
$907$	46.0000i	1.52740i	0.645568	+	0.763702i	$0.276621\pi$
$907$	46.0000i	1.52740i	−0.645568	+	0.763702i	$0.723379\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	− 12.0000i	− 0.397578i	−0.980042	−	0.198789i	$-0.936299\pi$
$911$	− 12.0000i	− 0.397578i	0.980042	−	0.198789i	$-0.0637008\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	4.00000i	0.132236i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	− 4.00000i	− 0.131804i
$922$	0	0
$923$	− 24.0000i	− 0.789970i
$924$	0	0
$925$	− 2.00000i	− 0.0657596i
$926$	0	0
$927$	−14.0000	−0.459820
$928$	0	0
$929$	42.0000i	1.37798i	0.724773	+	0.688988i	$0.241945\pi$
$929$	42.0000i	1.37798i	−0.724773	+	0.688988i	$0.758055\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	0	0
$936$	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
$939$	44.0000	1.43589
$940$	0	0
$941$	18.0000i	0.586783i	0.955992	+	0.293392i	$0.0947840\pi$
$941$	18.0000i	0.586783i	−0.955992	+	0.293392i	$0.905216\pi$
$942$	0	0
$943$	−36.0000	−1.17232
$944$	0	0
$945$	8.00000i	0.260240i
$946$	0	0
$947$	18.0000i	0.584921i	0.956278	+	0.292461i	$0.0944741\pi$
$947$	18.0000i	0.584921i	−0.956278	+	0.292461i	$0.905526\pi$
$948$	0	0
$949$	4.00000i	0.129845i
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	12.0000i	0.388311i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 36.0000i	− 1.16250i
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	6.00000i	0.193347i
$964$	0	0
$965$	−26.0000	−0.836970
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	0	0
$975$	4.00000i	0.128103i
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	2.00000i	0.0638551i
$982$	0	0
$983$	− 18.0000i	− 0.574111i	−0.957914	−	0.287055i	$-0.907324\pi$
$983$	− 18.0000i	− 0.574111i	0.957914	−	0.287055i	$-0.0926764\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	24.0000	0.763928
$988$	0	0
$989$	− 60.0000i	− 1.90789i
$990$	0	0
$991$	− 4.00000i	− 0.127064i	−0.997980	−	0.0635321i	$-0.979763\pi$
$991$	− 4.00000i	− 0.127064i	0.997980	−	0.0635321i	$-0.0202365\pi$
$992$	0	0
$993$	16.0000i	0.507745i
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5780.2.c.a.5201.1		2
17.4	even	4		20.2.a.a.1.1	✓	1
17.13	even	4		5780.2.a.f.1.1		1
17.16	even	2	inner	5780.2.c.a.5201.2		2
51.38	odd	4		180.2.a.a.1.1		1
68.55	odd	4		80.2.a.b.1.1		1
85.4	even	4		100.2.a.a.1.1		1
85.38	odd	4		100.2.c.a.49.1		2
85.72	odd	4		100.2.c.a.49.2		2
119.4	even	12		980.2.i.i.961.1		2
119.38	odd	12		980.2.i.c.961.1		2
119.55	odd	4		980.2.a.h.1.1		1
119.72	even	12		980.2.i.i.361.1		2
119.89	odd	12		980.2.i.c.361.1		2
136.21	even	4		320.2.a.f.1.1		1
136.123	odd	4		320.2.a.a.1.1		1
153.4	even	12		1620.2.i.h.541.1		2
153.38	odd	12		1620.2.i.b.1081.1		2
153.106	even	12		1620.2.i.h.1081.1		2
153.140	odd	12		1620.2.i.b.541.1		2
187.21	odd	4		2420.2.a.a.1.1		1
204.191	even	4		720.2.a.h.1.1		1
221.21	odd	4		3380.2.f.b.3041.1		2
221.38	even	4		3380.2.a.c.1.1		1
221.174	odd	4		3380.2.f.b.3041.2		2
255.38	even	4		900.2.d.c.649.1		2
255.89	odd	4		900.2.a.b.1.1		1
255.242	even	4		900.2.d.c.649.2		2
272.21	even	4		1280.2.d.c.641.2		2
272.123	odd	4		1280.2.d.g.641.1		2
272.157	even	4		1280.2.d.c.641.1		2
272.259	odd	4		1280.2.d.g.641.2		2
323.208	odd	4		7220.2.a.f.1.1		1
340.123	even	4		400.2.c.b.49.2		2
340.259	odd	4		400.2.a.c.1.1		1
340.327	even	4		400.2.c.b.49.1		2
357.293	even	4		8820.2.a.g.1.1		1
408.293	odd	4		2880.2.a.m.1.1		1
408.395	even	4		2880.2.a.f.1.1		1
476.55	even	4		3920.2.a.h.1.1		1
595.174	odd	4		4900.2.a.e.1.1		1
595.293	even	4		4900.2.e.f.2549.2		2
595.412	even	4		4900.2.e.f.2549.1		2
680.123	even	4		1600.2.c.e.449.1		2
680.157	odd	4		1600.2.c.d.449.1		2
680.259	odd	4		1600.2.a.w.1.1		1
680.293	odd	4		1600.2.c.d.449.2		2
680.429	even	4		1600.2.a.c.1.1		1
680.667	even	4		1600.2.c.e.449.2		2
748.395	even	4		9680.2.a.ba.1.1		1
1020.599	even	4		3600.2.a.be.1.1		1
1020.803	odd	4		3600.2.f.j.2449.2		2
1020.1007	odd	4		3600.2.f.j.2449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.2.a.a.1.1	✓	1	17.4	even	4
80.2.a.b.1.1		1	68.55	odd	4
100.2.a.a.1.1		1	85.4	even	4
100.2.c.a.49.1		2	85.38	odd	4
100.2.c.a.49.2		2	85.72	odd	4
180.2.a.a.1.1		1	51.38	odd	4
320.2.a.a.1.1		1	136.123	odd	4
320.2.a.f.1.1		1	136.21	even	4
400.2.a.c.1.1		1	340.259	odd	4
400.2.c.b.49.1		2	340.327	even	4
400.2.c.b.49.2		2	340.123	even	4
720.2.a.h.1.1		1	204.191	even	4
900.2.a.b.1.1		1	255.89	odd	4
900.2.d.c.649.1		2	255.38	even	4
900.2.d.c.649.2		2	255.242	even	4
980.2.a.h.1.1		1	119.55	odd	4
980.2.i.c.361.1		2	119.89	odd	12
980.2.i.c.961.1		2	119.38	odd	12
980.2.i.i.361.1		2	119.72	even	12
980.2.i.i.961.1		2	119.4	even	12
1280.2.d.c.641.1		2	272.157	even	4
1280.2.d.c.641.2		2	272.21	even	4
1280.2.d.g.641.1		2	272.123	odd	4
1280.2.d.g.641.2		2	272.259	odd	4
1600.2.a.c.1.1		1	680.429	even	4
1600.2.a.w.1.1		1	680.259	odd	4
1600.2.c.d.449.1		2	680.157	odd	4
1600.2.c.d.449.2		2	680.293	odd	4
1600.2.c.e.449.1		2	680.123	even	4
1600.2.c.e.449.2		2	680.667	even	4
1620.2.i.b.541.1		2	153.140	odd	12
1620.2.i.b.1081.1		2	153.38	odd	12
1620.2.i.h.541.1		2	153.4	even	12
1620.2.i.h.1081.1		2	153.106	even	12
2420.2.a.a.1.1		1	187.21	odd	4
2880.2.a.f.1.1		1	408.395	even	4
2880.2.a.m.1.1		1	408.293	odd	4
3380.2.a.c.1.1		1	221.38	even	4
3380.2.f.b.3041.1		2	221.21	odd	4
3380.2.f.b.3041.2		2	221.174	odd	4
3600.2.a.be.1.1		1	1020.599	even	4
3600.2.f.j.2449.1		2	1020.1007	odd	4
3600.2.f.j.2449.2		2	1020.803	odd	4
3920.2.a.h.1.1		1	476.55	even	4
4900.2.a.e.1.1		1	595.174	odd	4
4900.2.e.f.2549.1		2	595.412	even	4
4900.2.e.f.2549.2		2	595.293	even	4
5780.2.a.f.1.1		1	17.13	even	4
5780.2.c.a.5201.1		2	1.1	even	1	trivial
5780.2.c.a.5201.2		2	17.16	even	2	inner
7220.2.a.f.1.1		1	323.208	odd	4
8820.2.a.g.1.1		1	357.293	even	4
9680.2.a.ba.1.1		1	748.395	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5780.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} -1.00000i q^{5} -2.00000i q^{7} -1.00000 q^{9} +2.00000 q^{13} -2.00000 q^{15} +4.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} -1.00000 q^{25} -4.00000i q^{27} +6.00000i q^{29} -4.00000i q^{31} -2.00000 q^{35} +2.00000i q^{37} -4.00000i q^{39} -6.00000i q^{41} +10.0000 q^{43} +1.00000i q^{45} -6.00000 q^{47} +3.00000 q^{49} +6.00000 q^{53} -8.00000i q^{57} -12.0000 q^{59} -2.00000i q^{61} +2.00000i q^{63} -2.00000i q^{65} +2.00000 q^{67} -12.0000 q^{69} -12.0000i q^{71} +2.00000i q^{73} +2.00000i q^{75} -8.00000i q^{79} -11.0000 q^{81} -6.00000 q^{83} +12.0000 q^{87} -6.00000 q^{89} -4.00000i q^{91} -8.00000 q^{93} -4.00000i q^{95} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} + 4 q^{13} - 4 q^{15} + 8 q^{19} - 8 q^{21} - 2 q^{25} - 4 q^{35} + 20 q^{43} - 12 q^{47} + 6 q^{49} + 12 q^{53} - 24 q^{59} + 4 q^{67} - 24 q^{69} - 22 q^{81} - 12 q^{83} + 24 q^{87} - 12 q^{89}+ \cdots - 16 q^{93}+O(q^{100}) \) 2 * q - 2 * q^9 + 4 * q^13 - 4 * q^15 + 8 * q^19 - 8 * q^21 - 2 * q^25 - 4 * q^35 + 20 * q^43 - 12 * q^47 + 6 * q^49 + 12 * q^53 - 24 * q^59 + 4 * q^67 - 24 * q^69 - 22 * q^81 - 12 * q^83 + 24 * q^87 - 12 * q^89 - 16 * q^93

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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