LMFDB - Embedded newform 3380.2.f.e.3041.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(3041,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$26.9894358832$
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 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.e.3041.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000i q^{5} -1.00000i q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000i q^{5} -1.00000i q^{7} -2.00000 q^{9} +3.00000i q^{11} -1.00000i q^{15} +3.00000 q^{17} +7.00000i q^{19} -1.00000i q^{21} +3.00000 q^{23} -1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +4.00000i q^{31} +3.00000i q^{33} -1.00000 q^{35} -7.00000i q^{37} +9.00000i q^{41} -11.0000 q^{43} +2.00000i q^{45} +6.00000 q^{49} +3.00000 q^{51} -6.00000 q^{53} +3.00000 q^{55} +7.00000i q^{57} -3.00000i q^{59} +11.0000 q^{61} +2.00000i q^{63} +7.00000i q^{67} +3.00000 q^{69} +3.00000i q^{71} +2.00000i q^{73} -1.00000 q^{75} +3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -3.00000i q^{85} +3.00000 q^{87} +15.0000i q^{89} +4.00000i q^{93} +7.00000 q^{95} +7.00000i q^{97} -6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^9	$ 2 q + 2 q^{3} - 4 q^{9} + 6 q^{17} + 6 q^{23} - 2 q^{25} - 10 q^{27} + 6 q^{29} - 2 q^{35} - 22 q^{43} + 12 q^{49} + 6 q^{51} - 12 q^{53} + 6 q^{55} + 22 q^{61} + 6 q^{69} - 2 q^{75} + 6 q^{77} + 16 q^{79} + 2 q^{81} + 6 q^{87} + 14 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^9 + 6 * q^17 + 6 * q^23 - 2 * q^25 - 10 * q^27 + 6 * q^29 - 2 * q^35 - 22 * q^43 + 12 * q^49 + 6 * q^51 - 12 * q^53 + 6 * q^55 + 22 * q^61 + 6 * q^69 - 2 * q^75 + 6 * q^77 + 16 * q^79 + 2 * q^81 + 6 * q^87 + 14 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 1.00000i	− 0.258199i
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	7.00000i	1.60591i	0.596040	+	0.802955i	$0.296740\pi$
$19$	7.00000i	1.60591i	−0.596040	+	0.802955i	$0.703260\pi$
$20$	0	0
$21$	− 1.00000i	− 0.218218i
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	0	0
$33$	3.00000i	0.522233i
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.00000i	1.40556i	0.711405	+	0.702782i	$0.248059\pi$
$41$	9.00000i	1.40556i	−0.711405	+	0.702782i	$0.751941\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	7.00000i	0.927173i
$58$	0	0
$59$	− 3.00000i	− 0.390567i	−0.980747	−	0.195283i	$-0.937437\pi$
$59$	− 3.00000i	− 0.390567i	0.980747	−	0.195283i	$-0.0625627\pi$
$60$	0	0
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.00000i	0.855186i	0.903971	+	0.427593i	$0.140638\pi$
$67$	7.00000i	0.855186i	−0.903971	+	0.427593i	$0.859362\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	3.00000i	0.356034i	0.984027	+	0.178017i	$0.0569683\pi$
$71$	3.00000i	0.356034i	−0.984027	+	0.178017i	$0.943032\pi$
$72$	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
$75$	−1.00000	−0.115470
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	− 3.00000i	− 0.325396i
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	15.0000i	1.59000i	0.606612	+	0.794998i	$0.292528\pi$
$89$	15.0000i	1.59000i	−0.606612	+	0.794998i	$0.707472\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	7.00000	0.718185
$96$	0	0
$97$	7.00000i	0.710742i	0.934725	+	0.355371i	$0.115646\pi$
$97$	7.00000i	0.710742i	−0.934725	+	0.355371i	$0.884354\pi$
$98$	0	0
$99$	− 6.00000i	− 0.603023i
$100$	0	0
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	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
$111$	− 7.00000i	− 0.664411i
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	− 3.00000i	− 0.279751i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 3.00000i	− 0.275010i
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	9.00000i	0.811503i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	19.0000	1.68598	0.842989	−	0.537931i	$-0.180794\pi$
$127$	19.0000	1.68598	0.842989	+	0.537931i	$0.180794\pi$
$128$	0	0
$129$	−11.0000	−0.968496
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	7.00000	0.606977
$134$	0	0
$135$	5.00000i	0.430331i
$136$	0	0
$137$	− 15.0000i	− 1.28154i	−0.767734	−	0.640768i	$-0.778616\pi$
$137$	− 15.0000i	− 1.28154i	0.767734	−	0.640768i	$-0.221384\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 3.00000i	− 0.249136i
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	21.0000i	1.72039i	0.509968	+	0.860194i	$0.329657\pi$
$149$	21.0000i	1.72039i	−0.509968	+	0.860194i	$0.670343\pi$
$150$	0	0
$151$	8.00000i	0.651031i	0.945537	+	0.325515i	$0.105538\pi$
$151$	8.00000i	0.651031i	−0.945537	+	0.325515i	$0.894462\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	− 3.00000i	− 0.236433i
$162$	0	0
$163$	− 1.00000i	− 0.0783260i	−0.999233	−	0.0391630i	$-0.987531\pi$
$163$	− 1.00000i	− 0.0783260i	0.999233	−	0.0391630i	$-0.0124692\pi$
$164$	0	0
$165$	3.00000	0.233550
$166$	0	0
$167$	− 3.00000i	− 0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$	− 3.00000i	− 0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 14.0000i	− 1.07061i
$172$	0	0
$173$	3.00000	0.228086	0.114043	−	0.993476i	$-0.463620\pi$
$173$	3.00000	0.228086	0.114043	+	0.993476i	$0.463620\pi$
$174$	0	0
$175$	1.00000i	0.0755929i
$176$	0	0
$177$	− 3.00000i	− 0.225494i
$178$	0	0
$179$	21.0000	1.56961	0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000	1.56961	0.784807	+	0.619740i	$0.212762\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	11.0000	0.813143
$184$	0	0
$185$	−7.00000	−0.514650
$186$	0	0
$187$	9.00000i	0.658145i
$188$	0	0
$189$	5.00000i	0.363696i
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	5.00000i	0.359908i	0.983675	+	0.179954i	$0.0575949\pi$
$193$	5.00000i	0.359908i	−0.983675	+	0.179954i	$0.942405\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 21.0000i	− 1.49619i	−0.663593	−	0.748094i	$-0.730969\pi$
$197$	− 21.0000i	− 1.49619i	0.663593	−	0.748094i	$-0.269031\pi$
$198$	0	0
$199$	−17.0000	−1.20510	−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000	−1.20510	−0.602549	+	0.798082i	$0.705848\pi$
$200$	0	0
$201$	7.00000i	0.493742i
$202$	0	0
$203$	− 3.00000i	− 0.210559i
$204$	0	0
$205$	9.00000	0.628587
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−21.0000	−1.45260
$210$	0	0
$211$	11.0000	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$212$	0	0
$213$	3.00000i	0.205557i
$214$	0	0
$215$	11.0000i	0.750194i
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	2.00000i	0.135147i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	19.0000i	1.27233i	0.771551	+	0.636167i	$0.219481\pi$
$223$	19.0000i	1.27233i	−0.771551	+	0.636167i	$0.780519\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	− 27.0000i	− 1.79205i	−0.444001	−	0.896026i	$-0.646441\pi$
$227$	− 27.0000i	− 1.79205i	0.444001	−	0.896026i	$-0.353559\pi$
$228$	0	0
$229$	− 22.0000i	− 1.45380i	−0.686743	−	0.726900i	$-0.740960\pi$
$229$	− 22.0000i	− 1.45380i	0.686743	−	0.726900i	$-0.259040\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	− 1.00000i	− 0.0644157i	−0.999481	−	0.0322078i	$-0.989746\pi$
$241$	− 1.00000i	− 0.0644157i	0.999481	−	0.0322078i	$-0.0102538\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	− 6.00000i	− 0.383326i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	12.0000i	0.760469i
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	9.00000i	0.565825i
$254$	0	0
$255$	− 3.00000i	− 0.187867i
$256$	0	0
$257$	−9.00000	−0.561405	−0.280702	−	0.959795i	$-0.590567\pi$
$257$	−9.00000	−0.561405	−0.280702	+	0.959795i	$0.590567\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−3.00000	−0.184988	−0.0924940	−	0.995713i	$-0.529484\pi$
$263$	−3.00000	−0.184988	−0.0924940	+	0.995713i	$0.529484\pi$
$264$	0	0
$265$	6.00000i	0.368577i
$266$	0	0
$267$	15.0000i	0.917985i
$268$	0	0
$269$	27.0000	1.64622	0.823110	−	0.567883i	$-0.192237\pi$
$269$	27.0000	1.64622	0.823110	+	0.567883i	$0.192237\pi$
$270$	0	0
$271$	23.0000i	1.39715i	0.715537	+	0.698575i	$0.246182\pi$
$271$	23.0000i	1.39715i	−0.715537	+	0.698575i	$0.753818\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.00000i	− 0.180907i
$276$	0	0
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	0	0
$279$	− 8.00000i	− 0.478947i
$280$	0	0
$281$	6.00000i	0.357930i	0.983855	+	0.178965i	$0.0572749\pi$
$281$	6.00000i	0.357930i	−0.983855	+	0.178965i	$0.942725\pi$
$282$	0	0
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	0	0
$285$	7.00000	0.414644
$286$	0	0
$287$	9.00000	0.531253
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	7.00000i	0.410347i
$292$	0	0
$293$	− 27.0000i	− 1.57736i	−0.614806	−	0.788678i	$-0.710766\pi$
$293$	− 27.0000i	− 1.57736i	0.614806	−	0.788678i	$-0.289234\pi$
$294$	0	0
$295$	−3.00000	−0.174667
$296$	0	0
$297$	− 15.0000i	− 0.870388i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	11.0000i	0.634029i
$302$	0	0
$303$	9.00000	0.517036
$304$	0	0
$305$	− 11.0000i	− 0.629858i
$306$	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
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	9.00000i	0.503903i
$320$	0	0
$321$	9.00000	0.502331
$322$	0	0
$323$	21.0000i	1.16847i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.00000i	− 0.110600i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	19.0000i	1.04433i	0.852843	+	0.522167i	$0.174876\pi$
$331$	19.0000i	1.04433i	−0.852843	+	0.522167i	$0.825124\pi$
$332$	0	0
$333$	14.0000i	0.767195i
$334$	0	0
$335$	7.00000	0.382451
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	0	0
$339$	9.00000	0.488813
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	0	0
$345$	− 3.00000i	− 0.161515i
$346$	0	0
$347$	−33.0000	−1.77153	−0.885766	−	0.464131i	$-0.846367\pi$
$347$	−33.0000	−1.77153	−0.885766	+	0.464131i	$0.846367\pi$
$348$	0	0
$349$	− 1.00000i	− 0.0535288i	−0.999642	−	0.0267644i	$-0.991480\pi$
$349$	− 1.00000i	− 0.0535288i	0.999642	−	0.0267644i	$-0.00852039\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 9.00000i	− 0.479022i	−0.970894	−	0.239511i	$-0.923013\pi$
$353$	− 9.00000i	− 0.479022i	0.970894	−	0.239511i	$-0.0769871\pi$
$354$	0	0
$355$	3.00000	0.159223
$356$	0	0
$357$	− 3.00000i	− 0.158777i
$358$	0	0
$359$	− 24.0000i	− 1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$	− 24.0000i	− 1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$	0	0
$361$	−30.0000	−1.57895
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	5.00000	0.260998	0.130499	−	0.991448i	$-0.458342\pi$
$367$	5.00000	0.260998	0.130499	+	0.991448i	$0.458342\pi$
$368$	0	0
$369$	− 18.0000i	− 0.937043i
$370$	0	0
$371$	6.00000i	0.311504i
$372$	0	0
$373$	−31.0000	−1.60512	−0.802560	−	0.596572i	$-0.796529\pi$
$373$	−31.0000	−1.60512	−0.802560	+	0.596572i	$0.796529\pi$
$374$	0	0
$375$	1.00000i	0.0516398i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	1.00000i	0.0513665i	0.999670	+	0.0256833i	$0.00817614\pi$
$379$	1.00000i	0.0513665i	−0.999670	+	0.0256833i	$0.991824\pi$
$380$	0	0
$381$	19.0000	0.973399
$382$	0	0
$383$	9.00000i	0.459879i	0.973205	+	0.229939i	$0.0738528\pi$
$383$	9.00000i	0.459879i	−0.973205	+	0.229939i	$0.926147\pi$
$384$	0	0
$385$	− 3.00000i	− 0.152894i
$386$	0	0
$387$	22.0000	1.11832
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	9.00000	0.455150
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	− 8.00000i	− 0.402524i
$396$	0	0
$397$	5.00000i	0.250943i	0.992097	+	0.125471i	$0.0400443\pi$
$397$	5.00000i	0.250943i	−0.992097	+	0.125471i	$0.959956\pi$
$398$	0	0
$399$	7.00000	0.350438
$400$	0	0
$401$	− 33.0000i	− 1.64794i	−0.566632	−	0.823971i	$-0.691754\pi$
$401$	− 33.0000i	− 1.64794i	0.566632	−	0.823971i	$-0.308246\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	21.0000	1.04093
$408$	0	0
$409$	25.0000i	1.23617i	0.786111	+	0.618085i	$0.212091\pi$
$409$	25.0000i	1.23617i	−0.786111	+	0.618085i	$0.787909\pi$
$410$	0	0
$411$	− 15.0000i	− 0.739895i
$412$	0	0
$413$	−3.00000	−0.147620
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	5.00000	0.244851
$418$	0	0
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	− 2.00000i	− 0.0974740i	−0.998812	−	0.0487370i	$-0.984480\pi$
$421$	− 2.00000i	− 0.0974740i	0.998812	−	0.0487370i	$-0.0155196\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	− 11.0000i	− 0.532327i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.00000i	0.433515i	0.976226	+	0.216757i	$0.0695480\pi$
$431$	9.00000i	0.433515i	−0.976226	+	0.216757i	$0.930452\pi$
$432$	0	0
$433$	−29.0000	−1.39365	−0.696826	−	0.717241i	$-0.745405\pi$
$433$	−29.0000	−1.39365	−0.696826	+	0.717241i	$0.745405\pi$
$434$	0	0
$435$	− 3.00000i	− 0.143839i
$436$	0	0
$437$	21.0000i	1.00457i
$438$	0	0
$439$	−11.0000	−0.525001	−0.262501	−	0.964932i	$-0.584547\pi$
$439$	−11.0000	−0.525001	−0.262501	+	0.964932i	$0.584547\pi$
$440$	0	0
$441$	−12.0000	−0.571429
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	15.0000	0.711068
$446$	0	0
$447$	21.0000i	0.993266i
$448$	0	0
$449$	3.00000i	0.141579i	0.997491	+	0.0707894i	$0.0225518\pi$
$449$	3.00000i	0.141579i	−0.997491	+	0.0707894i	$0.977448\pi$
$450$	0	0
$451$	−27.0000	−1.27138
$452$	0	0
$453$	8.00000i	0.375873i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 5.00000i	− 0.233890i	−0.993138	−	0.116945i	$-0.962690\pi$
$457$	− 5.00000i	− 0.233890i	0.993138	−	0.116945i	$-0.0373101\pi$
$458$	0	0
$459$	−15.0000	−0.700140
$460$	0	0
$461$	− 27.0000i	− 1.25752i	−0.777601	−	0.628758i	$-0.783564\pi$
$461$	− 27.0000i	− 1.25752i	0.777601	−	0.628758i	$-0.216436\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	− 33.0000i	− 1.51734i
$474$	0	0
$475$	− 7.00000i	− 0.321182i
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	39.0000i	1.78196i	0.454047	+	0.890978i	$0.349980\pi$
$479$	39.0000i	1.78196i	−0.454047	+	0.890978i	$0.650020\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 3.00000i	− 0.136505i
$484$	0	0
$485$	7.00000	0.317854
$486$	0	0
$487$	− 11.0000i	− 0.498458i	−0.968445	−	0.249229i	$-0.919823\pi$
$487$	− 11.0000i	− 0.498458i	0.968445	−	0.249229i	$-0.0801771\pi$
$488$	0	0
$489$	− 1.00000i	− 0.0452216i
$490$	0	0
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	3.00000	0.134568
$498$	0	0
$499$	− 32.0000i	− 1.43252i	−0.697835	−	0.716258i	$-0.745853\pi$
$499$	− 32.0000i	− 1.43252i	0.697835	−	0.716258i	$-0.254147\pi$
$500$	0	0
$501$	− 3.00000i	− 0.134030i
$502$	0	0
$503$	15.0000	0.668817	0.334408	−	0.942428i	$-0.391463\pi$
$503$	15.0000	0.668817	0.334408	+	0.942428i	$0.391463\pi$
$504$	0	0
$505$	− 9.00000i	− 0.400495i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.00000i	0.398918i	0.979906	+	0.199459i	$0.0639185\pi$
$509$	9.00000i	0.398918i	−0.979906	+	0.199459i	$0.936082\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	− 35.0000i	− 1.54529i
$514$	0	0
$515$	8.00000i	0.352522i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	3.00000	0.131685
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	29.0000	1.26808	0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000	1.26808	0.634041	+	0.773300i	$0.281395\pi$
$524$	0	0
$525$	1.00000i	0.0436436i
$526$	0	0
$527$	12.0000i	0.522728i
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	6.00000i	0.260378i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 9.00000i	− 0.389104i
$536$	0	0
$537$	21.0000	0.906217
$538$	0	0
$539$	18.0000i	0.775315i
$540$	0	0
$541$	− 22.0000i	− 0.945854i	−0.881102	−	0.472927i	$-0.843197\pi$
$541$	− 22.0000i	− 0.945854i	0.881102	−	0.472927i	$-0.156803\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	0	0
$549$	−22.0000	−0.938937
$550$	0	0
$551$	21.0000i	0.894630i
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	0	0
$555$	−7.00000	−0.297133
$556$	0	0
$557$	− 39.0000i	− 1.65248i	−0.563316	−	0.826242i	$-0.690475\pi$
$557$	− 39.0000i	− 1.65248i	0.563316	−	0.826242i	$-0.309525\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	9.00000i	0.379980i
$562$	0	0
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	0	0
$565$	− 9.00000i	− 0.378633i
$566$	0	0
$567$	− 1.00000i	− 0.0419961i
$568$	0	0
$569$	−27.0000	−1.13190	−0.565949	−	0.824440i	$-0.691490\pi$
$569$	−27.0000	−1.13190	−0.565949	+	0.824440i	$0.691490\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	−3.00000	−0.125327
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	0	0
$579$	5.00000i	0.207793i
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	− 18.0000i	− 0.745484i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 33.0000i	− 1.36206i	−0.732257	−	0.681028i	$-0.761533\pi$
$587$	− 33.0000i	− 1.36206i	0.732257	−	0.681028i	$-0.238467\pi$
$588$	0	0
$589$	−28.0000	−1.15372
$590$	0	0
$591$	− 21.0000i	− 0.863825i
$592$	0	0
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	0	0
$597$	−17.0000	−0.695764
$598$	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
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	− 14.0000i	− 0.570124i
$604$	0	0
$605$	− 2.00000i	− 0.0813116i
$606$	0	0
$607$	−13.0000	−0.527654	−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000	−0.527654	−0.263827	+	0.964570i	$0.584985\pi$
$608$	0	0
$609$	− 3.00000i	− 0.121566i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	43.0000i	1.73675i	0.495905	+	0.868377i	$0.334836\pi$
$613$	43.0000i	1.73675i	−0.495905	+	0.868377i	$0.665164\pi$
$614$	0	0
$615$	9.00000	0.362915
$616$	0	0
$617$	− 33.0000i	− 1.32853i	−0.747497	−	0.664265i	$-0.768745\pi$
$617$	− 33.0000i	− 1.32853i	0.747497	−	0.664265i	$-0.231255\pi$
$618$	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
$621$	−15.0000	−0.601929
$622$	0	0
$623$	15.0000	0.600962
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−21.0000	−0.838659
$628$	0	0
$629$	− 21.0000i	− 0.837325i
$630$	0	0
$631$	17.0000i	0.676759i	0.941010	+	0.338380i	$0.109879\pi$
$631$	17.0000i	0.676759i	−0.941010	+	0.338380i	$0.890121\pi$
$632$	0	0
$633$	11.0000	0.437211
$634$	0	0
$635$	− 19.0000i	− 0.753992i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 6.00000i	− 0.237356i
$640$	0	0
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	0	0
$643$	− 11.0000i	− 0.433798i	−0.976194	−	0.216899i	$-0.930406\pi$
$643$	− 11.0000i	− 0.433798i	0.976194	−	0.216899i	$-0.0695942\pi$
$644$	0	0
$645$	11.0000i	0.433125i
$646$	0	0
$647$	−45.0000	−1.76913	−0.884566	−	0.466415i	$-0.845546\pi$
$647$	−45.0000	−1.76913	−0.884566	+	0.466415i	$0.845546\pi$
$648$	0	0
$649$	9.00000	0.353281
$650$	0	0
$651$	4.00000	0.156772
$652$	0	0
$653$	−39.0000	−1.52619	−0.763094	−	0.646288i	$-0.776321\pi$
$653$	−39.0000	−1.52619	−0.763094	+	0.646288i	$0.776321\pi$
$654$	0	0
$655$	12.0000i	0.468879i
$656$	0	0
$657$	− 4.00000i	− 0.156055i
$658$	0	0
$659$	−39.0000	−1.51922	−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000	−1.51922	−0.759612	+	0.650376i	$0.774611\pi$
$660$	0	0
$661$	− 1.00000i	− 0.0388955i	−0.999811	−	0.0194477i	$-0.993809\pi$
$661$	− 1.00000i	− 0.0388955i	0.999811	−	0.0194477i	$-0.00619080\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 7.00000i	− 0.271448i
$666$	0	0
$667$	9.00000	0.348481
$668$	0	0
$669$	19.0000i	0.734582i
$670$	0	0
$671$	33.0000i	1.27395i
$672$	0	0
$673$	−17.0000	−0.655302	−0.327651	−	0.944799i	$-0.606257\pi$
$673$	−17.0000	−0.655302	−0.327651	+	0.944799i	$0.606257\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	7.00000	0.268635
$680$	0	0
$681$	− 27.0000i	− 1.03464i
$682$	0	0
$683$	51.0000i	1.95146i	0.218975	+	0.975730i	$0.429729\pi$
$683$	51.0000i	1.95146i	−0.218975	+	0.975730i	$0.570271\pi$
$684$	0	0
$685$	−15.0000	−0.573121
$686$	0	0
$687$	− 22.0000i	− 0.839352i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 23.0000i	− 0.874961i	−0.899228	−	0.437481i	$-0.855871\pi$
$691$	− 23.0000i	− 0.874961i	0.899228	−	0.437481i	$-0.144129\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	− 5.00000i	− 0.189661i
$696$	0	0
$697$	27.0000i	1.02270i
$698$	0	0
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	49.0000	1.84807
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.00000i	− 0.338480i
$708$	0	0
$709$	− 37.0000i	− 1.38956i	−0.719220	−	0.694782i	$-0.755501\pi$
$709$	− 37.0000i	− 1.38956i	0.719220	−	0.694782i	$-0.244499\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	12.0000i	0.449404i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9.00000	−0.335643	−0.167822	−	0.985817i	$-0.553673\pi$
$719$	−9.00000	−0.335643	−0.167822	+	0.985817i	$0.553673\pi$
$720$	0	0
$721$	8.00000i	0.297936i
$722$	0	0
$723$	− 1.00000i	− 0.0371904i
$724$	0	0
$725$	−3.00000	−0.111417
$726$	0	0
$727$	52.0000	1.92857	0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000	1.92857	0.964287	+	0.264861i	$0.0853260\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−33.0000	−1.22055
$732$	0	0
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	0	0
$735$	− 6.00000i	− 0.221313i
$736$	0	0
$737$	−21.0000	−0.773545
$738$	0	0
$739$	47.0000i	1.72892i	0.502699	+	0.864461i	$0.332340\pi$
$739$	47.0000i	1.72892i	−0.502699	+	0.864461i	$0.667660\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 21.0000i	− 0.770415i	−0.922830	−	0.385208i	$-0.874130\pi$
$743$	− 21.0000i	− 0.770415i	0.922830	−	0.385208i	$-0.125870\pi$
$744$	0	0
$745$	21.0000	0.769380
$746$	0	0
$747$	− 24.0000i	− 0.878114i
$748$	0	0
$749$	− 9.00000i	− 0.328853i
$750$	0	0
$751$	13.0000	0.474377	0.237188	−	0.971464i	$-0.423774\pi$
$751$	13.0000	0.474377	0.237188	+	0.971464i	$0.423774\pi$
$752$	0	0
$753$	−21.0000	−0.765283
$754$	0	0
$755$	8.00000	0.291150
$756$	0	0
$757$	29.0000	1.05402	0.527011	−	0.849858i	$-0.323312\pi$
$757$	29.0000	1.05402	0.527011	+	0.849858i	$0.323312\pi$
$758$	0	0
$759$	9.00000i	0.326679i
$760$	0	0
$761$	3.00000i	0.108750i	0.998521	+	0.0543750i	$0.0173166\pi$
$761$	3.00000i	0.108750i	−0.998521	+	0.0543750i	$0.982683\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	6.00000i	0.216930i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	13.0000i	0.468792i	0.972141	+	0.234396i	$0.0753112\pi$
$769$	13.0000i	0.468792i	−0.972141	+	0.234396i	$0.924689\pi$
$770$	0	0
$771$	−9.00000	−0.324127
$772$	0	0
$773$	27.0000i	0.971123i	0.874203	+	0.485561i	$0.161385\pi$
$773$	27.0000i	0.971123i	−0.874203	+	0.485561i	$0.838615\pi$
$774$	0	0
$775$	− 4.00000i	− 0.143684i
$776$	0	0
$777$	−7.00000	−0.251124
$778$	0	0
$779$	−63.0000	−2.25721
$780$	0	0
$781$	−9.00000	−0.322045
$782$	0	0
$783$	−15.0000	−0.536056
$784$	0	0
$785$	10.0000i	0.356915i
$786$	0	0
$787$	− 37.0000i	− 1.31891i	−0.751745	−	0.659454i	$-0.770788\pi$
$787$	− 37.0000i	− 1.31891i	0.751745	−	0.659454i	$-0.229212\pi$
$788$	0	0
$789$	−3.00000	−0.106803
$790$	0	0
$791$	− 9.00000i	− 0.320003i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	6.00000i	0.212798i
$796$	0	0
$797$	51.0000	1.80651	0.903256	−	0.429101i	$-0.141170\pi$
$797$	51.0000	1.80651	0.903256	+	0.429101i	$0.141170\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	− 30.0000i	− 1.06000i
$802$	0	0
$803$	−6.00000	−0.211735
$804$	0	0
$805$	−3.00000	−0.105736
$806$	0	0
$807$	27.0000	0.950445
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	− 32.0000i	− 1.12367i	−0.827249	−	0.561836i	$-0.810095\pi$
$811$	− 32.0000i	− 1.12367i	0.827249	−	0.561836i	$-0.189905\pi$
$812$	0	0
$813$	23.0000i	0.806645i
$814$	0	0
$815$	−1.00000	−0.0350285
$816$	0	0
$817$	− 77.0000i	− 2.69389i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 15.0000i	− 0.523504i	−0.965135	−	0.261752i	$-0.915700\pi$
$821$	− 15.0000i	− 0.523504i	0.965135	−	0.261752i	$-0.0843002\pi$
$822$	0	0
$823$	13.0000	0.453152	0.226576	−	0.973994i	$-0.427247\pi$
$823$	13.0000	0.453152	0.226576	+	0.973994i	$0.427247\pi$
$824$	0	0
$825$	− 3.00000i	− 0.104447i
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	−11.0000	−0.382046	−0.191023	−	0.981586i	$-0.561180\pi$
$829$	−11.0000	−0.382046	−0.191023	+	0.981586i	$0.561180\pi$
$830$	0	0
$831$	19.0000	0.659103
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	−3.00000	−0.103819
$836$	0	0
$837$	− 20.0000i	− 0.691301i
$838$	0	0
$839$	21.0000i	0.725001i	0.931984	+	0.362500i	$0.118077\pi$
$839$	21.0000i	0.725001i	−0.931984	+	0.362500i	$0.881923\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	0	0
$849$	−5.00000	−0.171600
$850$	0	0
$851$	− 21.0000i	− 0.719871i
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	−14.0000	−0.478790
$856$	0	0
$857$	30.0000	1.02478	0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000	1.02478	0.512390	+	0.858753i	$0.328760\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	9.00000	0.306719
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	− 3.00000i	− 0.102003i
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	24.0000i	0.814144i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 14.0000i	− 0.473828i
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	− 41.0000i	− 1.38447i	−0.721671	−	0.692236i	$-0.756626\pi$
$877$	− 41.0000i	− 1.38447i	0.721671	−	0.692236i	$-0.243374\pi$
$878$	0	0
$879$	− 27.0000i	− 0.910687i
$880$	0	0
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	0	0
$885$	−3.00000	−0.100844
$886$	0	0
$887$	9.00000	0.302190	0.151095	−	0.988519i	$-0.451720\pi$
$887$	9.00000	0.302190	0.151095	+	0.988519i	$0.451720\pi$
$888$	0	0
$889$	− 19.0000i	− 0.637240i
$890$	0	0
$891$	3.00000i	0.100504i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 21.0000i	− 0.701953i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.0000i	0.400222i
$900$	0	0
$901$	−18.0000	−0.599667
$902$	0	0
$903$	11.0000i	0.366057i
$904$	0	0
$905$	2.00000i	0.0664822i
$906$	0	0
$907$	19.0000	0.630885	0.315442	−	0.948945i	$-0.397847\pi$
$907$	19.0000	0.630885	0.315442	+	0.948945i	$0.397847\pi$
$908$	0	0
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	0	0
$915$	− 11.0000i	− 0.363649i
$916$	0	0
$917$	12.0000i	0.396275i
$918$	0	0
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	0	0
$921$	20.0000i	0.659022i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	7.00000i	0.230159i
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	− 51.0000i	− 1.67326i	−0.547772	−	0.836628i	$-0.684524\pi$
$929$	− 51.0000i	− 1.67326i	0.547772	−	0.836628i	$-0.315476\pi$
$930$	0	0
$931$	42.0000i	1.37649i
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	9.00000	0.294331
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	0	0
$941$	− 42.0000i	− 1.36916i	−0.728937	−	0.684580i	$-0.759985\pi$
$941$	− 42.0000i	− 1.36916i	0.728937	−	0.684580i	$-0.240015\pi$
$942$	0	0
$943$	27.0000i	0.879241i
$944$	0	0
$945$	5.00000	0.162650
$946$	0	0
$947$	21.0000i	0.682408i	0.939989	+	0.341204i	$0.110835\pi$
$947$	21.0000i	0.682408i	−0.939989	+	0.341204i	$0.889165\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	18.0000i	0.583690i
$952$	0	0
$953$	51.0000	1.65205	0.826026	−	0.563632i	$-0.190596\pi$
$953$	51.0000	1.65205	0.826026	+	0.563632i	$0.190596\pi$
$954$	0	0
$955$	3.00000i	0.0970777i
$956$	0	0
$957$	9.00000i	0.290929i
$958$	0	0
$959$	−15.0000	−0.484375
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	−18.0000	−0.580042
$964$	0	0
$965$	5.00000	0.160956
$966$	0	0
$967$	40.0000i	1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$	40.0000i	1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$	0	0
$969$	21.0000i	0.674617i
$970$	0	0
$971$	21.0000	0.673922	0.336961	−	0.941519i	$-0.390601\pi$
$971$	21.0000	0.673922	0.336961	+	0.941519i	$0.390601\pi$
$972$	0	0
$973$	− 5.00000i	− 0.160293i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 9.00000i	− 0.287936i	−0.989582	−	0.143968i	$-0.954014\pi$
$977$	− 9.00000i	− 0.287936i	0.989582	−	0.143968i	$-0.0459862\pi$
$978$	0	0
$979$	−45.0000	−1.43821
$980$	0	0
$981$	4.00000i	0.127710i
$982$	0	0
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	0	0
$985$	−21.0000	−0.669116
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−33.0000	−1.04934
$990$	0	0
$991$	−61.0000	−1.93773	−0.968864	−	0.247592i	$-0.920361\pi$
$991$	−61.0000	−1.93773	−0.968864	+	0.247592i	$0.920361\pi$
$992$	0	0
$993$	19.0000i	0.602947i
$994$	0	0
$995$	17.0000i	0.538936i
$996$	0	0
$997$	−31.0000	−0.981780	−0.490890	−	0.871222i	$-0.663328\pi$
$997$	−31.0000	−0.981780	−0.490890	+	0.871222i	$0.663328\pi$
$998$	0	0
$999$	35.0000i	1.10735i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.e.3041.1		2
13.5	odd	4		3380.2.a.g.1.1		1
13.7	odd	12		260.2.i.b.81.1	yes	2
13.8	odd	4		3380.2.a.h.1.1		1
13.11	odd	12		260.2.i.b.61.1	✓	2
13.12	even	2	inner	3380.2.f.e.3041.2		2
39.11	even	12		2340.2.q.b.1621.1		2
39.20	even	12		2340.2.q.b.2161.1		2
52.7	even	12		1040.2.q.j.81.1		2
52.11	even	12		1040.2.q.j.321.1		2
65.7	even	12		1300.2.bb.a.549.1		4
65.24	odd	12		1300.2.i.e.1101.1		2
65.33	even	12		1300.2.bb.a.549.2		4
65.37	even	12		1300.2.bb.a.1049.2		4
65.59	odd	12		1300.2.i.e.601.1		2
65.63	even	12		1300.2.bb.a.1049.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.i.b.61.1	✓	2	13.11	odd	12
260.2.i.b.81.1	yes	2	13.7	odd	12
1040.2.q.j.81.1		2	52.7	even	12
1040.2.q.j.321.1		2	52.11	even	12
1300.2.i.e.601.1		2	65.59	odd	12
1300.2.i.e.1101.1		2	65.24	odd	12
1300.2.bb.a.549.1		4	65.7	even	12
1300.2.bb.a.549.2		4	65.33	even	12
1300.2.bb.a.1049.1		4	65.63	even	12
1300.2.bb.a.1049.2		4	65.37	even	12
2340.2.q.b.1621.1		2	39.11	even	12
2340.2.q.b.2161.1		2	39.20	even	12
3380.2.a.g.1.1		1	13.5	odd	4
3380.2.a.h.1.1		1	13.8	odd	4
3380.2.f.e.3041.1		2	1.1	even	1	trivial
3380.2.f.e.3041.2		2	13.12	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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000i q^{5} -1.00000i q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000i q^{5} -1.00000i q^{7} -2.00000 q^{9} +3.00000i q^{11} -1.00000i q^{15} +3.00000 q^{17} +7.00000i q^{19} -1.00000i q^{21} +3.00000 q^{23} -1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +4.00000i q^{31} +3.00000i q^{33} -1.00000 q^{35} -7.00000i q^{37} +9.00000i q^{41} -11.0000 q^{43} +2.00000i q^{45} +6.00000 q^{49} +3.00000 q^{51} -6.00000 q^{53} +3.00000 q^{55} +7.00000i q^{57} -3.00000i q^{59} +11.0000 q^{61} +2.00000i q^{63} +7.00000i q^{67} +3.00000 q^{69} +3.00000i q^{71} +2.00000i q^{73} -1.00000 q^{75} +3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -3.00000i q^{85} +3.00000 q^{87} +15.0000i q^{89} +4.00000i q^{93} +7.00000 q^{95} +7.00000i q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^9	\( 2 q + 2 q^{3} - 4 q^{9} + 6 q^{17} + 6 q^{23} - 2 q^{25} - 10 q^{27} + 6 q^{29} - 2 q^{35} - 22 q^{43} + 12 q^{49} + 6 q^{51} - 12 q^{53} + 6 q^{55} + 22 q^{61} + 6 q^{69} - 2 q^{75} + 6 q^{77} + 16 q^{79} + 2 q^{81} + 6 q^{87} + 14 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^9 + 6 * q^17 + 6 * q^23 - 2 * q^25 - 10 * q^27 + 6 * q^29 - 2 * q^35 - 22 * q^43 + 12 * q^49 + 6 * q^51 - 12 * q^53 + 6 * q^55 + 22 * q^61 + 6 * q^69 - 2 * q^75 + 6 * q^77 + 16 * q^79 + 2 * q^81 + 6 * q^87 + 14 * q^95

Introduction

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