LMFDB - Embedded newform 3380.2.f.d.3041.2

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			3041.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.d.3041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000i q^{5} -1.00000i q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000i q^{5} -1.00000i q^{7} -2.00000 q^{9} +3.00000i q^{11} +1.00000i q^{15} +3.00000 q^{17} -5.00000i q^{19} -1.00000i q^{21} -9.00000 q^{23} -1.00000 q^{25} -5.00000 q^{27} -9.00000 q^{29} -8.00000i q^{31} +3.00000i q^{33} +1.00000 q^{35} -7.00000i q^{37} -3.00000i q^{41} +1.00000 q^{43} -2.00000i q^{45} +6.00000 q^{49} +3.00000 q^{51} +6.00000 q^{53} -3.00000 q^{55} -5.00000i q^{57} +9.00000i q^{59} -1.00000 q^{61} +2.00000i q^{63} -5.00000i q^{67} -9.00000 q^{69} -9.00000i q^{71} +2.00000i q^{73} -1.00000 q^{75} +3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +3.00000i q^{85} -9.00000 q^{87} +3.00000i q^{89} -8.00000i q^{93} +5.00000 q^{95} -17.0000i q^{97} -6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 18 q^{23} - 2 q^{25} - 10 q^{27} - 18 q^{29} + 2 q^{35} + 2 q^{43} + 12 q^{49} + 6 q^{51} + 12 q^{53} - 6 q^{55} - 2 q^{61} - 18 q^{69} - 2 q^{75} + 6 q^{77} + 16 q^{79} + 2 q^{81} - 18 q^{87} + 10 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^9 + 6 * q^17 - 18 * q^23 - 2 * q^25 - 10 * q^27 - 18 * q^29 + 2 * q^35 + 2 * q^43 + 12 * q^49 + 6 * q^51 + 12 * q^53 - 6 * q^55 - 2 * q^61 - 18 * q^69 - 2 * q^75 + 6 * q^77 + 16 * q^79 + 2 * q^81 - 18 * q^87 + 10 * 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
$19$	− 5.00000i	− 1.14708i	−0.819178	−	0.573539i	$-0.805570\pi$
$19$	− 5.00000i	− 1.14708i	0.819178	−	0.573539i	$-0.194430\pi$
$20$	0	0
$21$	− 1.00000i	− 0.218218i
$22$	0	0
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	− 8.00000i	− 1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$	− 8.00000i	− 1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$	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
$41$	− 3.00000i	− 0.468521i	−0.972174	−	0.234261i	$-0.924733\pi$
$41$	− 3.00000i	− 0.468521i	0.972174	−	0.234261i	$-0.0752669\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	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	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	− 5.00000i	− 0.662266i
$58$	0	0
$59$	9.00000i	1.17170i	0.810419	+	0.585850i	$0.199239\pi$
$59$	9.00000i	1.17170i	−0.810419	+	0.585850i	$0.800761\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	0	0
$69$	−9.00000	−1.08347
$70$	0	0
$71$	− 9.00000i	− 1.06810i	−0.845452	−	0.534052i	$-0.820669\pi$
$71$	− 9.00000i	− 1.06810i	0.845452	−	0.534052i	$-0.179331\pi$
$72$	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
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	3.00000i	0.325396i
$86$	0	0
$87$	−9.00000	−0.964901
$88$	0	0
$89$	3.00000i	0.317999i	0.987279	+	0.159000i	$0.0508269\pi$
$89$	3.00000i	0.317999i	−0.987279	+	0.159000i	$0.949173\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	5.00000	0.512989
$96$	0	0
$97$	− 17.0000i	− 1.72609i	−0.505128	−	0.863044i	$-0.668555\pi$
$97$	− 17.0000i	− 1.72609i	0.505128	−	0.863044i	$-0.331445\pi$
$98$	0	0
$99$	− 6.00000i	− 0.603023i
$100$	0	0
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	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
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	− 14.0000i	− 1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$	− 14.0000i	− 1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$	0	0
$111$	− 7.00000i	− 0.664411i
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	− 9.00000i	− 0.839254i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 3.00000i	− 0.275010i
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	− 3.00000i	− 0.270501i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−17.0000	−1.50851	−0.754253	−	0.656584i	$-0.772001\pi$
$127$	−17.0000	−1.50851	−0.754253	+	0.656584i	$0.772001\pi$
$128$	0	0
$129$	1.00000	0.0880451
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−5.00000	−0.433555
$134$	0	0
$135$	− 5.00000i	− 0.430331i
$136$	0	0
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	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
$145$	− 9.00000i	− 0.747409i
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	9.00000i	0.737309i	0.929567	+	0.368654i	$0.120181\pi$
$149$	9.00000i	0.737309i	−0.929567	+	0.368654i	$0.879819\pi$
$150$	0	0
$151$	20.0000i	1.62758i	0.581161	+	0.813788i	$0.302599\pi$
$151$	20.0000i	1.62758i	−0.581161	+	0.813788i	$0.697401\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	9.00000i	0.709299i
$162$	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
$167$	− 15.0000i	− 1.16073i	−0.814355	−	0.580367i	$-0.802909\pi$
$167$	− 15.0000i	− 1.16073i	0.814355	−	0.580367i	$-0.197091\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	10.0000i	0.764719i
$172$	0	0
$173$	−21.0000	−1.59660	−0.798300	−	0.602260i	$-0.794267\pi$
$173$	−21.0000	−1.59660	−0.798300	+	0.602260i	$0.794267\pi$
$174$	0	0
$175$	1.00000i	0.0755929i
$176$	0	0
$177$	9.00000i	0.676481i
$178$	0	0
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	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
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	− 5.00000i	− 0.352673i
$202$	0	0
$203$	9.00000i	0.631676i
$204$	0	0
$205$	3.00000	0.209529
$206$	0	0
$207$	18.0000	1.25109
$208$	0	0
$209$	15.0000	1.03757
$210$	0	0
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	0	0
$213$	− 9.00000i	− 0.616670i
$214$	0	0
$215$	1.00000i	0.0681994i
$216$	0	0
$217$	−8.00000	−0.543075
$218$	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
$227$	− 15.0000i	− 0.995585i	−0.867296	−	0.497792i	$-0.834144\pi$
$227$	− 15.0000i	− 0.995585i	0.867296	−	0.497792i	$-0.165856\pi$
$228$	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
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	− 24.0000i	− 1.55243i	−0.630468	−	0.776215i	$-0.717137\pi$
$239$	− 24.0000i	− 1.55243i	0.630468	−	0.776215i	$-0.282863\pi$
$240$	0	0
$241$	23.0000i	1.48156i	0.671748	+	0.740780i	$0.265544\pi$
$241$	23.0000i	1.48156i	−0.671748	+	0.740780i	$0.734456\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	6.00000i	0.383326i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	− 27.0000i	− 1.69748i
$254$	0	0
$255$	3.00000i	0.187867i
$256$	0	0
$257$	27.0000	1.68421	0.842107	−	0.539311i	$-0.181315\pi$
$257$	27.0000	1.68421	0.842107	+	0.539311i	$0.181315\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	0	0
$261$	18.0000	1.11417
$262$	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
$267$	3.00000i	0.183597i
$268$	0	0
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	− 13.0000i	− 0.789694i	−0.918747	−	0.394847i	$-0.870798\pi$
$271$	− 13.0000i	− 0.789694i	0.918747	−	0.394847i	$-0.129202\pi$
$272$	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
$279$	16.0000i	0.957895i
$280$	0	0
$281$	− 18.0000i	− 1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$	− 18.0000i	− 1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$	0	0
$283$	−17.0000	−1.01055	−0.505273	−	0.862960i	$-0.668608\pi$
$283$	−17.0000	−1.01055	−0.505273	+	0.862960i	$0.668608\pi$
$284$	0	0
$285$	5.00000	0.296174
$286$	0	0
$287$	−3.00000	−0.177084
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	− 17.0000i	− 0.996558i
$292$	0	0
$293$	− 3.00000i	− 0.175262i	−0.996153	−	0.0876309i	$-0.972070\pi$
$293$	− 3.00000i	− 0.175262i	0.996153	−	0.0876309i	$-0.0279296\pi$
$294$	0	0
$295$	−9.00000	−0.524000
$296$	0	0
$297$	− 15.0000i	− 0.870388i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 1.00000i	− 0.0576390i
$302$	0	0
$303$	−15.0000	−0.861727
$304$	0	0
$305$	− 1.00000i	− 0.0572598i
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	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
$319$	− 27.0000i	− 1.51171i
$320$	0	0
$321$	−3.00000	−0.167444
$322$	0	0
$323$	− 15.0000i	− 0.834622i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 14.0000i	− 0.774202i
$328$	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
$335$	5.00000	0.273179
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	−15.0000	−0.814688
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	0	0
$345$	− 9.00000i	− 0.484544i
$346$	0	0
$347$	−9.00000	−0.483145	−0.241573	−	0.970383i	$-0.577663\pi$
$347$	−9.00000	−0.483145	−0.241573	+	0.970383i	$0.577663\pi$
$348$	0	0
$349$	35.0000i	1.87351i	0.349990	+	0.936754i	$0.386185\pi$
$349$	35.0000i	1.87351i	−0.349990	+	0.936754i	$0.613815\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	27.0000i	1.43706i	0.695493	+	0.718532i	$0.255186\pi$
$353$	27.0000i	1.43706i	−0.695493	+	0.718532i	$0.744814\pi$
$354$	0	0
$355$	9.00000	0.477670
$356$	0	0
$357$	− 3.00000i	− 0.158777i
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−6.00000	−0.315789
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	0	0
$369$	6.00000i	0.312348i
$370$	0	0
$371$	− 6.00000i	− 0.311504i
$372$	0	0
$373$	5.00000	0.258890	0.129445	−	0.991587i	$-0.458680\pi$
$373$	5.00000	0.258890	0.129445	+	0.991587i	$0.458680\pi$
$374$	0	0
$375$	− 1.00000i	− 0.0516398i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 11.0000i	− 0.565032i	−0.959263	−	0.282516i	$-0.908831\pi$
$379$	− 11.0000i	− 0.565032i	0.959263	−	0.282516i	$-0.0911690\pi$
$380$	0	0
$381$	−17.0000	−0.870936
$382$	0	0
$383$	21.0000i	1.07305i	0.843884	+	0.536525i	$0.180263\pi$
$383$	21.0000i	1.07305i	−0.843884	+	0.536525i	$0.819737\pi$
$384$	0	0
$385$	3.00000i	0.152894i
$386$	0	0
$387$	−2.00000	−0.101666
$388$	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
$391$	−27.0000	−1.36545
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.00000i	0.402524i
$396$	0	0
$397$	29.0000i	1.45547i	0.685859	+	0.727734i	$0.259427\pi$
$397$	29.0000i	1.45547i	−0.685859	+	0.727734i	$0.740573\pi$
$398$	0	0
$399$	−5.00000	−0.250313
$400$	0	0
$401$	15.0000i	0.749064i	0.927214	+	0.374532i	$0.122197\pi$
$401$	15.0000i	0.749064i	−0.927214	+	0.374532i	$0.877803\pi$
$402$	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
$411$	− 3.00000i	− 0.147979i
$412$	0	0
$413$	9.00000	0.442861
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.00000	0.244851
$418$	0	0
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	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
$427$	1.00000i	0.0483934i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 3.00000i	− 0.144505i	−0.997386	−	0.0722525i	$-0.976981\pi$
$431$	− 3.00000i	− 0.144505i	0.997386	−	0.0722525i	$-0.0230187\pi$
$432$	0	0
$433$	−5.00000	−0.240285	−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000	−0.240285	−0.120142	+	0.992757i	$0.538335\pi$
$434$	0	0
$435$	− 9.00000i	− 0.431517i
$436$	0	0
$437$	45.0000i	2.15264i
$438$	0	0
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	0	0
$441$	−12.0000	−0.571429
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	0	0
$447$	9.00000i	0.425685i
$448$	0	0
$449$	15.0000i	0.707894i	0.935266	+	0.353947i	$0.115161\pi$
$449$	15.0000i	0.707894i	−0.935266	+	0.353947i	$0.884839\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	0	0
$453$	20.0000i	0.939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 41.0000i	− 1.91790i	−0.283577	−	0.958950i	$-0.591521\pi$
$457$	− 41.0000i	− 1.91790i	0.283577	−	0.958950i	$-0.408479\pi$
$458$	0	0
$459$	−15.0000	−0.700140
$460$	0	0
$461$	− 15.0000i	− 0.698620i	−0.937007	−	0.349310i	$-0.886416\pi$
$461$	− 15.0000i	− 0.698620i	0.937007	−	0.349310i	$-0.113584\pi$
$462$	0	0
$463$	− 40.0000i	− 1.85896i	−0.368875	−	0.929479i	$-0.620257\pi$
$463$	− 40.0000i	− 1.85896i	0.368875	−	0.929479i	$-0.379743\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	−5.00000	−0.230879
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	3.00000i	0.137940i
$474$	0	0
$475$	5.00000i	0.229416i
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	15.0000i	0.685367i	0.939451	+	0.342684i	$0.111336\pi$
$479$	15.0000i	0.685367i	−0.939451	+	0.342684i	$0.888664\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	9.00000i	0.409514i
$484$	0	0
$485$	17.0000	0.771930
$486$	0	0
$487$	− 35.0000i	− 1.58600i	−0.609221	−	0.793001i	$-0.708518\pi$
$487$	− 35.0000i	− 1.58600i	0.609221	−	0.793001i	$-0.291482\pi$
$488$	0	0
$489$	− 1.00000i	− 0.0452216i
$490$	0	0
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	0	0
$493$	−27.0000	−1.21602
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	−9.00000	−0.403705
$498$	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
$501$	− 15.0000i	− 0.670151i
$502$	0	0
$503$	39.0000	1.73892	0.869462	−	0.494000i	$-0.164466\pi$
$503$	39.0000	1.73892	0.869462	+	0.494000i	$0.164466\pi$
$504$	0	0
$505$	− 15.0000i	− 0.667491i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.0000i	1.46270i	0.682003	+	0.731350i	$0.261109\pi$
$509$	33.0000i	1.46270i	−0.682003	+	0.731350i	$0.738891\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	25.0000i	1.10378i
$514$	0	0
$515$	− 8.00000i	− 0.352522i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−21.0000	−0.921798
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	5.00000	0.218635	0.109317	−	0.994007i	$-0.465134\pi$
$523$	5.00000	0.218635	0.109317	+	0.994007i	$0.465134\pi$
$524$	0	0
$525$	1.00000i	0.0436436i
$526$	0	0
$527$	− 24.0000i	− 1.04546i
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	− 18.0000i	− 0.781133i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 3.00000i	− 0.129701i
$536$	0	0
$537$	−15.0000	−0.647298
$538$	0	0
$539$	18.0000i	0.775315i
$540$	0	0
$541$	2.00000i	0.0859867i	0.999075	+	0.0429934i	$0.0136894\pi$
$541$	2.00000i	0.0859867i	−0.999075	+	0.0429934i	$0.986311\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	45.0000i	1.91706i
$552$	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
$563$	33.0000	1.39078	0.695392	−	0.718631i	$-0.255231\pi$
$563$	33.0000	1.39078	0.695392	+	0.718631i	$0.255231\pi$
$564$	0	0
$565$	− 15.0000i	− 0.631055i
$566$	0	0
$567$	− 1.00000i	− 0.0419961i
$568$	0	0
$569$	33.0000	1.38343	0.691716	−	0.722170i	$-0.256855\pi$
$569$	33.0000	1.38343	0.691716	+	0.722170i	$0.256855\pi$
$570$	0	0
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	0	0
$573$	−3.00000	−0.125327
$574$	0	0
$575$	9.00000	0.375326
$576$	0	0
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	0	0
$579$	5.00000i	0.207793i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	18.0000i	0.745484i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.0000i	1.11441i	0.830375	+	0.557205i	$0.188126\pi$
$587$	27.0000i	1.11441i	−0.830375	+	0.557205i	$0.811874\pi$
$588$	0	0
$589$	−40.0000	−1.64817
$590$	0	0
$591$	3.00000i	0.123404i
$592$	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
$597$	7.00000	0.286491
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−1.00000	−0.0407909	−0.0203954	−	0.999792i	$-0.506493\pi$
$601$	−1.00000	−0.0407909	−0.0203954	+	0.999792i	$0.506493\pi$
$602$	0	0
$603$	10.0000i	0.407231i
$604$	0	0
$605$	2.00000i	0.0813116i
$606$	0	0
$607$	23.0000	0.933541	0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000	0.933541	0.466771	+	0.884378i	$0.345417\pi$
$608$	0	0
$609$	9.00000i	0.364698i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 29.0000i	− 1.17130i	−0.810564	−	0.585649i	$-0.800840\pi$
$613$	− 29.0000i	− 1.17130i	0.810564	−	0.585649i	$-0.199160\pi$
$614$	0	0
$615$	3.00000	0.120972
$616$	0	0
$617$	27.0000i	1.08698i	0.839416	+	0.543490i	$0.182897\pi$
$617$	27.0000i	1.08698i	−0.839416	+	0.543490i	$0.817103\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	0	0
$621$	45.0000	1.80579
$622$	0	0
$623$	3.00000	0.120192
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	15.0000	0.599042
$628$	0	0
$629$	− 21.0000i	− 0.837325i
$630$	0	0
$631$	− 7.00000i	− 0.278666i	−0.990246	−	0.139333i	$-0.955504\pi$
$631$	− 7.00000i	− 0.278666i	0.990246	−	0.139333i	$-0.0444958\pi$
$632$	0	0
$633$	−25.0000	−0.993661
$634$	0	0
$635$	− 17.0000i	− 0.674624i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	18.0000i	0.712069i
$640$	0	0
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	0	0
$643$	25.0000i	0.985904i	0.870057	+	0.492952i	$0.164082\pi$
$643$	25.0000i	0.985904i	−0.870057	+	0.492952i	$0.835918\pi$
$644$	0	0
$645$	1.00000i	0.0393750i
$646$	0	0
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	0	0
$649$	−27.0000	−1.05984
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	−27.0000	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 4.00000i	− 0.156055i
$658$	0	0
$659$	45.0000	1.75295	0.876476	−	0.481446i	$-0.159888\pi$
$659$	45.0000	1.75295	0.876476	+	0.481446i	$0.159888\pi$
$660$	0	0
$661$	35.0000i	1.36134i	0.732589	+	0.680671i	$0.238312\pi$
$661$	35.0000i	1.36134i	−0.732589	+	0.680671i	$0.761688\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 5.00000i	− 0.193892i
$666$	0	0
$667$	81.0000	3.13633
$668$	0	0
$669$	19.0000i	0.734582i
$670$	0	0
$671$	− 3.00000i	− 0.115814i
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−17.0000	−0.652400
$680$	0	0
$681$	− 15.0000i	− 0.574801i
$682$	0	0
$683$	− 21.0000i	− 0.803543i	−0.915740	−	0.401771i	$-0.868395\pi$
$683$	− 21.0000i	− 0.803543i	0.915740	−	0.401771i	$-0.131605\pi$
$684$	0	0
$685$	3.00000	0.114624
$686$	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
$697$	− 9.00000i	− 0.340899i
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−35.0000	−1.32005
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.0000i	0.564133i
$708$	0	0
$709$	35.0000i	1.31445i	0.753693	+	0.657226i	$0.228270\pi$
$709$	35.0000i	1.31445i	−0.753693	+	0.657226i	$0.771730\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	72.0000i	2.69642i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 24.0000i	− 0.896296i
$718$	0	0
$719$	−45.0000	−1.67822	−0.839108	−	0.543964i	$-0.816923\pi$
$719$	−45.0000	−1.67822	−0.839108	+	0.543964i	$0.816923\pi$
$720$	0	0
$721$	8.00000i	0.297936i
$722$	0	0
$723$	23.0000i	0.855379i
$724$	0	0
$725$	9.00000	0.334252
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	3.00000	0.110959
$732$	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
$737$	15.0000	0.552532
$738$	0	0
$739$	11.0000i	0.404642i	0.979319	+	0.202321i	$0.0648484\pi$
$739$	11.0000i	0.404642i	−0.979319	+	0.202321i	$0.935152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 33.0000i	− 1.21065i	−0.795977	−	0.605326i	$-0.793043\pi$
$743$	− 33.0000i	− 1.21065i	0.795977	−	0.605326i	$-0.206957\pi$
$744$	0	0
$745$	−9.00000	−0.329734
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.00000i	0.109618i
$750$	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
$753$	−9.00000	−0.327978
$754$	0	0
$755$	−20.0000	−0.727875
$756$	0	0
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	0	0
$759$	− 27.0000i	− 0.980038i
$760$	0	0
$761$	− 9.00000i	− 0.326250i	−0.986605	−	0.163125i	$-0.947843\pi$
$761$	− 9.00000i	− 0.326250i	0.986605	−	0.163125i	$-0.0521573\pi$
$762$	0	0
$763$	−14.0000	−0.506834
$764$	0	0
$765$	− 6.00000i	− 0.216930i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 11.0000i	− 0.396670i	−0.980134	−	0.198335i	$-0.936447\pi$
$769$	− 11.0000i	− 0.396670i	0.980134	−	0.198335i	$-0.0635534\pi$
$770$	0	0
$771$	27.0000	0.972381
$772$	0	0
$773$	15.0000i	0.539513i	0.962929	+	0.269756i	$0.0869431\pi$
$773$	15.0000i	0.539513i	−0.962929	+	0.269756i	$0.913057\pi$
$774$	0	0
$775$	8.00000i	0.287368i
$776$	0	0
$777$	−7.00000	−0.251124
$778$	0	0
$779$	−15.0000	−0.537431
$780$	0	0
$781$	27.0000	0.966136
$782$	0	0
$783$	45.0000	1.60817
$784$	0	0
$785$	14.0000i	0.499681i
$786$	0	0
$787$	− 49.0000i	− 1.74666i	−0.487128	−	0.873331i	$-0.661955\pi$
$787$	− 49.0000i	− 1.74666i	0.487128	−	0.873331i	$-0.338045\pi$
$788$	0	0
$789$	−3.00000	−0.106803
$790$	0	0
$791$	15.0000i	0.533339i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	6.00000i	0.212798i
$796$	0	0
$797$	27.0000	0.956389	0.478195	−	0.878254i	$-0.341291\pi$
$797$	27.0000	0.956389	0.478195	+	0.878254i	$0.341291\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	− 6.00000i	− 0.212000i
$802$	0	0
$803$	−6.00000	−0.211735
$804$	0	0
$805$	−9.00000	−0.317208
$806$	0	0
$807$	−21.0000	−0.739235
$808$	0	0
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	16.0000i	0.561836i	0.959732	+	0.280918i	$0.0906389\pi$
$811$	16.0000i	0.561836i	−0.959732	+	0.280918i	$0.909361\pi$
$812$	0	0
$813$	− 13.0000i	− 0.455930i
$814$	0	0
$815$	1.00000	0.0350285
$816$	0	0
$817$	− 5.00000i	− 0.174928i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.0000i	0.732905i	0.930437	+	0.366453i	$0.119428\pi$
$821$	21.0000i	0.732905i	−0.930437	+	0.366453i	$0.880572\pi$
$822$	0	0
$823$	−47.0000	−1.63832	−0.819159	−	0.573567i	$-0.805559\pi$
$823$	−47.0000	−1.63832	−0.819159	+	0.573567i	$0.805559\pi$
$824$	0	0
$825$	− 3.00000i	− 0.104447i
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	−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
$835$	15.0000	0.519096
$836$	0	0
$837$	40.0000i	1.38260i
$838$	0	0
$839$	− 15.0000i	− 0.517858i	−0.965896	−	0.258929i	$-0.916631\pi$
$839$	− 15.0000i	− 0.517858i	0.965896	−	0.258929i	$-0.0833695\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	0	0
$849$	−17.0000	−0.583438
$850$	0	0
$851$	63.0000i	2.15961i
$852$	0	0
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	0	0
$855$	−10.0000	−0.341993
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	−3.00000	−0.102240
$862$	0	0
$863$	− 48.0000i	− 1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$	− 48.0000i	− 1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$	0	0
$865$	− 21.0000i	− 0.714021i
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	24.0000i	0.814144i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	34.0000i	1.15073i
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	− 17.0000i	− 0.574049i	−0.957923	−	0.287025i	$-0.907334\pi$
$877$	− 17.0000i	− 0.574049i	0.957923	−	0.287025i	$-0.0926662\pi$
$878$	0	0
$879$	− 3.00000i	− 0.101187i
$880$	0	0
$881$	−3.00000	−0.101073	−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000	−0.101073	−0.0505363	+	0.998722i	$0.516093\pi$
$882$	0	0
$883$	−56.0000	−1.88455	−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−56.0000	−1.88455	−0.942275	+	0.334840i	$0.891318\pi$
$884$	0	0
$885$	−9.00000	−0.302532
$886$	0	0
$887$	33.0000	1.10803	0.554016	−	0.832506i	$-0.313095\pi$
$887$	33.0000	1.10803	0.554016	+	0.832506i	$0.313095\pi$
$888$	0	0
$889$	17.0000i	0.570162i
$890$	0	0
$891$	3.00000i	0.100504i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 15.0000i	− 0.501395i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	72.0000i	2.40133i
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	− 1.00000i	− 0.0332779i
$904$	0	0
$905$	10.0000i	0.332411i
$906$	0	0
$907$	55.0000	1.82625	0.913123	−	0.407685i	$-0.133664\pi$
$907$	55.0000	1.82625	0.913123	+	0.407685i	$0.133664\pi$
$908$	0	0
$909$	30.0000	0.995037
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 1.00000i	− 0.0330590i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−7.00000	−0.230909	−0.115454	−	0.993313i	$-0.536832\pi$
$919$	−7.00000	−0.230909	−0.115454	+	0.993313i	$0.536832\pi$
$920$	0	0
$921$	− 4.00000i	− 0.131804i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	7.00000i	0.230159i
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	− 3.00000i	− 0.0984268i	−0.998788	−	0.0492134i	$-0.984329\pi$
$929$	− 3.00000i	− 0.0984268i	0.998788	−	0.0492134i	$-0.0156714\pi$
$930$	0	0
$931$	− 30.0000i	− 0.983210i
$932$	0	0
$933$	−12.0000	−0.392862
$934$	0	0
$935$	−9.00000	−0.294331
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	− 18.0000i	− 0.586783i	−0.955992	−	0.293392i	$-0.905216\pi$
$941$	− 18.0000i	− 0.586783i	0.955992	−	0.293392i	$-0.0947840\pi$
$942$	0	0
$943$	27.0000i	0.879241i
$944$	0	0
$945$	−5.00000	−0.162650
$946$	0	0
$947$	− 51.0000i	− 1.65728i	−0.559784	−	0.828639i	$-0.689116\pi$
$947$	− 51.0000i	− 1.65728i	0.559784	−	0.828639i	$-0.310884\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	18.0000i	0.583690i
$952$	0	0
$953$	−21.0000	−0.680257	−0.340128	−	0.940379i	$-0.610471\pi$
$953$	−21.0000	−0.680257	−0.340128	+	0.940379i	$0.610471\pi$
$954$	0	0
$955$	− 3.00000i	− 0.0970777i
$956$	0	0
$957$	− 27.0000i	− 0.872786i
$958$	0	0
$959$	−3.00000	−0.0968751
$960$	0	0
$961$	−33.0000	−1.06452
$962$	0	0
$963$	6.00000	0.193347
$964$	0	0
$965$	−5.00000	−0.160956
$966$	0	0
$967$	4.00000i	0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$	4.00000i	0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$	0	0
$969$	− 15.0000i	− 0.481869i
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	0	0
$973$	− 5.00000i	− 0.160293i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.00000i	0.0959785i	0.998848	+	0.0479893i	$0.0152813\pi$
$977$	3.00000i	0.0959785i	−0.998848	+	0.0479893i	$0.984719\pi$
$978$	0	0
$979$	−9.00000	−0.287641
$980$	0	0
$981$	28.0000i	0.893971i
$982$	0	0
$983$	− 48.0000i	− 1.53096i	−0.643458	−	0.765481i	$-0.722501\pi$
$983$	− 48.0000i	− 1.53096i	0.643458	−	0.765481i	$-0.277499\pi$
$984$	0	0
$985$	−3.00000	−0.0955879
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.00000	−0.286183
$990$	0	0
$991$	−49.0000	−1.55654	−0.778268	−	0.627932i	$-0.783902\pi$
$991$	−49.0000	−1.55654	−0.778268	+	0.627932i	$0.783902\pi$
$992$	0	0
$993$	19.0000i	0.602947i
$994$	0	0
$995$	7.00000i	0.221915i
$996$	0	0
$997$	5.00000	0.158352	0.0791758	−	0.996861i	$-0.474771\pi$
$997$	5.00000	0.158352	0.0791758	+	0.996861i	$0.474771\pi$
$998$	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.d.3041.2		2
13.5	odd	4		3380.2.a.i.1.1		1
13.7	odd	12		260.2.i.a.81.1	yes	2
13.8	odd	4		3380.2.a.f.1.1		1
13.11	odd	12		260.2.i.a.61.1	✓	2
13.12	even	2	inner	3380.2.f.d.3041.1		2
39.11	even	12		2340.2.q.f.1621.1		2
39.20	even	12		2340.2.q.f.2161.1		2
52.7	even	12		1040.2.q.i.81.1		2
52.11	even	12		1040.2.q.i.321.1		2
65.7	even	12		1300.2.bb.b.549.1		4
65.24	odd	12		1300.2.i.d.1101.1		2
65.33	even	12		1300.2.bb.b.549.2		4
65.37	even	12		1300.2.bb.b.1049.2		4
65.59	odd	12		1300.2.i.d.601.1		2
65.63	even	12		1300.2.bb.b.1049.1		4

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

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000i q^{5} -1.00000i q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000i q^{5} -1.00000i q^{7} -2.00000 q^{9} +3.00000i q^{11} +1.00000i q^{15} +3.00000 q^{17} -5.00000i q^{19} -1.00000i q^{21} -9.00000 q^{23} -1.00000 q^{25} -5.00000 q^{27} -9.00000 q^{29} -8.00000i q^{31} +3.00000i q^{33} +1.00000 q^{35} -7.00000i q^{37} -3.00000i q^{41} +1.00000 q^{43} -2.00000i q^{45} +6.00000 q^{49} +3.00000 q^{51} +6.00000 q^{53} -3.00000 q^{55} -5.00000i q^{57} +9.00000i q^{59} -1.00000 q^{61} +2.00000i q^{63} -5.00000i q^{67} -9.00000 q^{69} -9.00000i q^{71} +2.00000i q^{73} -1.00000 q^{75} +3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +3.00000i q^{85} -9.00000 q^{87} +3.00000i q^{89} -8.00000i q^{93} +5.00000 q^{95} -17.0000i q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^9	\( 2 q + 2 q^{3} - 4 q^{9} + 6 q^{17} - 18 q^{23} - 2 q^{25} - 10 q^{27} - 18 q^{29} + 2 q^{35} + 2 q^{43} + 12 q^{49} + 6 q^{51} + 12 q^{53} - 6 q^{55} - 2 q^{61} - 18 q^{69} - 2 q^{75} + 6 q^{77} + 16 q^{79} + 2 q^{81} - 18 q^{87} + 10 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^9 + 6 * q^17 - 18 * q^23 - 2 * q^25 - 10 * q^27 - 18 * q^29 + 2 * q^35 + 2 * q^43 + 12 * q^49 + 6 * q^51 + 12 * q^53 - 6 * q^55 - 2 * q^61 - 18 * q^69 - 2 * q^75 + 6 * q^77 + 16 * q^79 + 2 * q^81 - 18 * q^87 + 10 * q^95

Introduction

L-functions

Modular forms

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