LMFDB - Embedded newform 3380.2.f.a.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.a.3041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +1.00000i q^{5} -3.00000i q^{7} +6.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +1.00000i q^{5} -3.00000i q^{7} +6.00000 q^{9} -3.00000i q^{11} -3.00000i q^{15} +7.00000 q^{17} +1.00000i q^{19} +9.00000i q^{21} +7.00000 q^{23} -1.00000 q^{25} -9.00000 q^{27} -5.00000 q^{29} -4.00000i q^{31} +9.00000i q^{33} +3.00000 q^{35} +3.00000i q^{37} +7.00000i q^{41} +9.00000 q^{43} +6.00000i q^{45} -8.00000i q^{47} -2.00000 q^{49} -21.0000 q^{51} -6.00000 q^{53} +3.00000 q^{55} -3.00000i q^{57} -5.00000i q^{59} -5.00000 q^{61} -18.0000i q^{63} +13.0000i q^{67} -21.0000 q^{69} -3.00000i q^{71} +14.0000i q^{73} +3.00000 q^{75} -9.00000 q^{77} -8.00000 q^{79} +9.00000 q^{81} +12.0000i q^{83} +7.00000i q^{85} +15.0000 q^{87} -7.00000i q^{89} +12.0000i q^{93} -1.00000 q^{95} -11.0000i q^{97} -18.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 12 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 12 * q^9	$ 2 q - 6 q^{3} + 12 q^{9} + 14 q^{17} + 14 q^{23} - 2 q^{25} - 18 q^{27} - 10 q^{29} + 6 q^{35} + 18 q^{43} - 4 q^{49} - 42 q^{51} - 12 q^{53} + 6 q^{55} - 10 q^{61} - 42 q^{69} + 6 q^{75} - 18 q^{77} - 16 q^{79} + 18 q^{81} + 30 q^{87} - 2 q^{95}+O(q^{100}) $ 2 * q - 6 * q^3 + 12 * q^9 + 14 * q^17 + 14 * q^23 - 2 * q^25 - 18 * q^27 - 10 * q^29 + 6 * q^35 + 18 * q^43 - 4 * q^49 - 42 * q^51 - 12 * q^53 + 6 * q^55 - 10 * q^61 - 42 * q^69 + 6 * q^75 - 18 * q^77 - 16 * q^79 + 18 * q^81 + 30 * q^87 - 2 * 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
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	− 3.00000i	− 0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$	− 3.00000i	− 0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 3.00000i	− 0.774597i
$16$	0	0
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	0	0
$19$	1.00000i	0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$	1.00000i	0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$	0	0
$21$	9.00000i	1.96396i
$22$	0	0
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−9.00000	−1.73205
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	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
$33$	9.00000i	1.56670i
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.00000i	1.09322i	0.837389	+	0.546608i	$0.184081\pi$
$41$	7.00000i	1.09322i	−0.837389	+	0.546608i	$0.815919\pi$
$42$	0	0
$43$	9.00000	1.37249	0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000	1.37249	0.686244	+	0.727372i	$0.259258\pi$
$44$	0	0
$45$	6.00000i	0.894427i
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−21.0000	−2.94059
$52$	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
$57$	− 3.00000i	− 0.397360i
$58$	0	0
$59$	− 5.00000i	− 0.650945i	−0.945552	−	0.325472i	$-0.894477\pi$
$59$	− 5.00000i	− 0.650945i	0.945552	−	0.325472i	$-0.105523\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	− 18.0000i	− 2.26779i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.0000i	1.58820i	0.607785	+	0.794101i	$0.292058\pi$
$67$	13.0000i	1.58820i	−0.607785	+	0.794101i	$0.707942\pi$
$68$	0	0
$69$	−21.0000	−2.52810
$70$	0	0
$71$	− 3.00000i	− 0.356034i	−0.984027	−	0.178017i	$-0.943032\pi$
$71$	− 3.00000i	− 0.356034i	0.984027	−	0.178017i	$-0.0569683\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	0	0
$75$	3.00000	0.346410
$76$	0	0
$77$	−9.00000	−1.02565
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	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
$85$	7.00000i	0.759257i
$86$	0	0
$87$	15.0000	1.60817
$88$	0	0
$89$	− 7.00000i	− 0.741999i	−0.928633	−	0.370999i	$-0.879015\pi$
$89$	− 7.00000i	− 0.741999i	0.928633	−	0.370999i	$-0.120985\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	12.0000i	1.24434i
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	− 11.0000i	− 1.11688i	−0.829545	−	0.558440i	$-0.811400\pi$
$97$	− 11.0000i	− 1.11688i	0.829545	−	0.558440i	$-0.188600\pi$
$98$	0	0
$99$	− 18.0000i	− 1.80907i
$100$	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
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	−9.00000	−0.878310
$106$	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
$111$	− 9.00000i	− 0.854242i
$112$	0	0
$113$	13.0000	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$113$	13.0000	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$114$	0	0
$115$	7.00000i	0.652753i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 21.0000i	− 1.92507i
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	− 21.0000i	− 1.89351i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−1.00000	−0.0887357	−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000	−0.0887357	−0.0443678	+	0.999015i	$0.514127\pi$
$128$	0	0
$129$	−27.0000	−2.37722
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	0	0
$135$	− 9.00000i	− 0.774597i
$136$	0	0
$137$	3.00000i	0.256307i	0.991754	+	0.128154i	$0.0409051\pi$
$137$	3.00000i	0.256307i	−0.991754	+	0.128154i	$0.959095\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	24.0000i	2.02116i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 5.00000i	− 0.415227i
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	3.00000i	0.245770i	0.992421	+	0.122885i	$0.0392146\pi$
$149$	3.00000i	0.245770i	−0.992421	+	0.122885i	$0.960785\pi$
$150$	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
$153$	42.0000	3.39550
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	0	0
$159$	18.0000	1.42749
$160$	0	0
$161$	− 21.0000i	− 1.65503i
$162$	0	0
$163$	− 11.0000i	− 0.861586i	−0.902451	−	0.430793i	$-0.858234\pi$
$163$	− 11.0000i	− 0.861586i	0.902451	−	0.430793i	$-0.141766\pi$
$164$	0	0
$165$	−9.00000	−0.700649
$166$	0	0
$167$	− 1.00000i	− 0.0773823i	−0.999251	−	0.0386912i	$-0.987681\pi$
$167$	− 1.00000i	− 0.0773823i	0.999251	−	0.0386912i	$-0.0123189\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	6.00000i	0.458831i
$172$	0	0
$173$	15.0000	1.14043	0.570214	−	0.821496i	$-0.306860\pi$
$173$	15.0000	1.14043	0.570214	+	0.821496i	$0.306860\pi$
$174$	0	0
$175$	3.00000i	0.226779i
$176$	0	0
$177$	15.0000i	1.12747i
$178$	0	0
$179$	−19.0000	−1.42013	−0.710063	−	0.704138i	$-0.751334\pi$
$179$	−19.0000	−1.42013	−0.710063	+	0.704138i	$0.751334\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	15.0000	1.10883
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	− 21.0000i	− 1.53567i
$188$	0	0
$189$	27.0000i	1.96396i
$190$	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
$193$	15.0000i	1.07972i	0.841754	+	0.539862i	$0.181524\pi$
$193$	15.0000i	1.07972i	−0.841754	+	0.539862i	$0.818476\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 23.0000i	− 1.63868i	−0.573306	−	0.819341i	$-0.694340\pi$
$197$	− 23.0000i	− 1.63868i	0.573306	−	0.819341i	$-0.305660\pi$
$198$	0	0
$199$	−9.00000	−0.637993	−0.318997	−	0.947756i	$-0.603346\pi$
$199$	−9.00000	−0.637993	−0.318997	+	0.947756i	$0.603346\pi$
$200$	0	0
$201$	− 39.0000i	− 2.75085i
$202$	0	0
$203$	15.0000i	1.05279i
$204$	0	0
$205$	−7.00000	−0.488901
$206$	0	0
$207$	42.0000	2.91920
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	9.00000i	0.616670i
$214$	0	0
$215$	9.00000i	0.613795i
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	− 42.0000i	− 2.83810i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 23.0000i	− 1.54019i	−0.637927	−	0.770097i	$-0.720208\pi$
$223$	− 23.0000i	− 1.54019i	0.637927	−	0.770097i	$-0.279792\pi$
$224$	0	0
$225$	−6.00000	−0.400000
$226$	0	0
$227$	− 1.00000i	− 0.0663723i	−0.999449	−	0.0331862i	$-0.989435\pi$
$227$	− 1.00000i	− 0.0663723i	0.999449	−	0.0331862i	$-0.0105654\pi$
$228$	0	0
$229$	− 26.0000i	− 1.71813i	−0.511868	−	0.859064i	$-0.671046\pi$
$229$	− 26.0000i	− 1.71813i	0.511868	−	0.859064i	$-0.328954\pi$
$230$	0	0
$231$	27.0000	1.77647
$232$	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
$235$	8.00000	0.521862
$236$	0	0
$237$	24.0000	1.55897
$238$	0	0
$239$	− 16.0000i	− 1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$239$	− 16.0000i	− 1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$	0	0
$241$	1.00000i	0.0644157i	0.999481	+	0.0322078i	$0.0102538\pi$
$241$	1.00000i	0.0644157i	−0.999481	+	0.0322078i	$0.989746\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.00000i	− 0.127775i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 36.0000i	− 2.28141i
$250$	0	0
$251$	−5.00000	−0.315597	−0.157799	−	0.987471i	$-0.550440\pi$
$251$	−5.00000	−0.315597	−0.157799	+	0.987471i	$0.550440\pi$
$252$	0	0
$253$	− 21.0000i	− 1.32026i
$254$	0	0
$255$	− 21.0000i	− 1.31507i
$256$	0	0
$257$	19.0000	1.18519	0.592594	−	0.805502i	$-0.298104\pi$
$257$	19.0000	1.18519	0.592594	+	0.805502i	$0.298104\pi$
$258$	0	0
$259$	9.00000	0.559233
$260$	0	0
$261$	−30.0000	−1.85695
$262$	0	0
$263$	−7.00000	−0.431638	−0.215819	−	0.976433i	$-0.569242\pi$
$263$	−7.00000	−0.431638	−0.215819	+	0.976433i	$0.569242\pi$
$264$	0	0
$265$	− 6.00000i	− 0.368577i
$266$	0	0
$267$	21.0000i	1.28518i
$268$	0	0
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	− 23.0000i	− 1.39715i	−0.715537	−	0.698575i	$-0.753818\pi$
$271$	− 23.0000i	− 1.39715i	0.715537	−	0.698575i	$-0.246182\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.00000i	0.180907i
$276$	0	0
$277$	23.0000	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	0	0
$279$	− 24.0000i	− 1.43684i
$280$	0	0
$281$	10.0000i	0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$	10.0000i	0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$	0	0
$283$	−1.00000	−0.0594438	−0.0297219	−	0.999558i	$-0.509462\pi$
$283$	−1.00000	−0.0594438	−0.0297219	+	0.999558i	$0.509462\pi$
$284$	0	0
$285$	3.00000	0.177705
$286$	0	0
$287$	21.0000	1.23959
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	33.0000i	1.93449i
$292$	0	0
$293$	− 9.00000i	− 0.525786i	−0.964825	−	0.262893i	$-0.915323\pi$
$293$	− 9.00000i	− 0.525786i	0.964825	−	0.262893i	$-0.0846766\pi$
$294$	0	0
$295$	5.00000	0.291111
$296$	0	0
$297$	27.0000i	1.56670i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 27.0000i	− 1.55625i
$302$	0	0
$303$	−27.0000	−1.55111
$304$	0	0
$305$	− 5.00000i	− 0.286299i
$306$	0	0
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	−48.0000	−2.73062
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	18.0000	1.01419
$316$	0	0
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	0	0
$319$	15.0000i	0.839839i
$320$	0	0
$321$	9.00000	0.502331
$322$	0	0
$323$	7.00000i	0.389490i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	42.0000i	2.32261i
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	13.0000i	0.714545i	0.934000	+	0.357272i	$0.116293\pi$
$331$	13.0000i	0.714545i	−0.934000	+	0.357272i	$0.883707\pi$
$332$	0	0
$333$	18.0000i	0.986394i
$334$	0	0
$335$	−13.0000	−0.710266
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	−39.0000	−2.11819
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	0	0
$345$	− 21.0000i	− 1.13060i
$346$	0	0
$347$	−13.0000	−0.697877	−0.348938	−	0.937146i	$-0.613458\pi$
$347$	−13.0000	−0.697877	−0.348938	+	0.937146i	$0.613458\pi$
$348$	0	0
$349$	25.0000i	1.33822i	0.743164	+	0.669110i	$0.233324\pi$
$349$	25.0000i	1.33822i	−0.743164	+	0.669110i	$0.766676\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.0000i	1.11772i	0.829263	+	0.558859i	$0.188761\pi$
$353$	21.0000i	1.11772i	−0.829263	+	0.558859i	$0.811239\pi$
$354$	0	0
$355$	3.00000	0.159223
$356$	0	0
$357$	63.0000i	3.33431i
$358$	0	0
$359$	8.00000i	0.422224i	0.977462	+	0.211112i	$0.0677085\pi$
$359$	8.00000i	0.422224i	−0.977462	+	0.211112i	$0.932292\pi$
$360$	0	0
$361$	18.0000	0.947368
$362$	0	0
$363$	−6.00000	−0.314918
$364$	0	0
$365$	−14.0000	−0.732793
$366$	0	0
$367$	9.00000	0.469796	0.234898	−	0.972020i	$-0.424524\pi$
$367$	9.00000	0.469796	0.234898	+	0.972020i	$0.424524\pi$
$368$	0	0
$369$	42.0000i	2.18643i
$370$	0	0
$371$	18.0000i	0.934513i
$372$	0	0
$373$	−27.0000	−1.39801	−0.699004	−	0.715118i	$-0.746373\pi$
$373$	−27.0000	−1.39801	−0.699004	+	0.715118i	$0.746373\pi$
$374$	0	0
$375$	3.00000i	0.154919i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 9.00000i	− 0.462299i	−0.972918	−	0.231149i	$-0.925751\pi$
$379$	− 9.00000i	− 0.462299i	0.972918	−	0.231149i	$-0.0742486\pi$
$380$	0	0
$381$	3.00000	0.153695
$382$	0	0
$383$	− 13.0000i	− 0.664269i	−0.943232	−	0.332134i	$-0.892231\pi$
$383$	− 13.0000i	− 0.664269i	0.943232	−	0.332134i	$-0.107769\pi$
$384$	0	0
$385$	− 9.00000i	− 0.458682i
$386$	0	0
$387$	54.0000	2.74497
$388$	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
$391$	49.0000	2.47804
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	− 8.00000i	− 0.402524i
$396$	0	0
$397$	− 33.0000i	− 1.65622i	−0.560564	−	0.828111i	$-0.689416\pi$
$397$	− 33.0000i	− 1.65622i	0.560564	−	0.828111i	$-0.310584\pi$
$398$	0	0
$399$	−9.00000	−0.450564
$400$	0	0
$401$	− 15.0000i	− 0.749064i	−0.927214	−	0.374532i	$-0.877803\pi$
$401$	− 15.0000i	− 0.749064i	0.927214	−	0.374532i	$-0.122197\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	9.00000i	0.447214i
$406$	0	0
$407$	9.00000	0.446113
$408$	0	0
$409$	− 1.00000i	− 0.0494468i	−0.999694	−	0.0247234i	$-0.992129\pi$
$409$	− 1.00000i	− 0.0494468i	0.999694	−	0.0247234i	$-0.00787051\pi$
$410$	0	0
$411$	− 9.00000i	− 0.443937i
$412$	0	0
$413$	−15.0000	−0.738102
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	−39.0000	−1.90984
$418$	0	0
$419$	−33.0000	−1.61216	−0.806078	−	0.591810i	$-0.798414\pi$
$419$	−33.0000	−1.61216	−0.806078	+	0.591810i	$0.798414\pi$
$420$	0	0
$421$	34.0000i	1.65706i	0.559946	+	0.828529i	$0.310822\pi$
$421$	34.0000i	1.65706i	−0.559946	+	0.828529i	$0.689178\pi$
$422$	0	0
$423$	− 48.0000i	− 2.33384i
$424$	0	0
$425$	−7.00000	−0.339550
$426$	0	0
$427$	15.0000i	0.725901i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 9.00000i	− 0.433515i	−0.976226	−	0.216757i	$-0.930452\pi$
$431$	− 9.00000i	− 0.433515i	0.976226	−	0.216757i	$-0.0695480\pi$
$432$	0	0
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	0	0
$435$	15.0000i	0.719195i
$436$	0	0
$437$	7.00000i	0.334855i
$438$	0	0
$439$	−3.00000	−0.143182	−0.0715911	−	0.997434i	$-0.522808\pi$
$439$	−3.00000	−0.143182	−0.0715911	+	0.997434i	$0.522808\pi$
$440$	0	0
$441$	−12.0000	−0.571429
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	7.00000	0.331832
$446$	0	0
$447$	− 9.00000i	− 0.425685i
$448$	0	0
$449$	21.0000i	0.991051i	0.868593	+	0.495526i	$0.165025\pi$
$449$	21.0000i	0.991051i	−0.868593	+	0.495526i	$0.834975\pi$
$450$	0	0
$451$	21.0000	0.988851
$452$	0	0
$453$	− 24.0000i	− 1.12762i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 23.0000i	− 1.07589i	−0.842978	−	0.537947i	$-0.819200\pi$
$457$	− 23.0000i	− 1.07589i	0.842978	−	0.537947i	$-0.180800\pi$
$458$	0	0
$459$	−63.0000	−2.94059
$460$	0	0
$461$	11.0000i	0.512321i	0.966634	+	0.256161i	$0.0824576\pi$
$461$	11.0000i	0.512321i	−0.966634	+	0.256161i	$0.917542\pi$
$462$	0	0
$463$	28.0000i	1.30127i	0.759390	+	0.650635i	$0.225497\pi$
$463$	28.0000i	1.30127i	−0.759390	+	0.650635i	$0.774503\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	39.0000	1.80085
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	− 27.0000i	− 1.24146i
$474$	0	0
$475$	− 1.00000i	− 0.0458831i
$476$	0	0
$477$	−36.0000	−1.64833
$478$	0	0
$479$	1.00000i	0.0456912i	0.999739	+	0.0228456i	$0.00727261\pi$
$479$	1.00000i	0.0456912i	−0.999739	+	0.0228456i	$0.992727\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	63.0000i	2.86660i
$484$	0	0
$485$	11.0000	0.499484
$486$	0	0
$487$	− 17.0000i	− 0.770344i	−0.922845	−	0.385172i	$-0.874142\pi$
$487$	− 17.0000i	− 0.770344i	0.922845	−	0.385172i	$-0.125858\pi$
$488$	0	0
$489$	33.0000i	1.49231i
$490$	0	0
$491$	−23.0000	−1.03798	−0.518988	−	0.854782i	$-0.673691\pi$
$491$	−23.0000	−1.03798	−0.518988	+	0.854782i	$0.673691\pi$
$492$	0	0
$493$	−35.0000	−1.57632
$494$	0	0
$495$	18.0000	0.809040
$496$	0	0
$497$	−9.00000	−0.403705
$498$	0	0
$499$	− 16.0000i	− 0.716258i	−0.933672	−	0.358129i	$-0.883415\pi$
$499$	− 16.0000i	− 0.716258i	0.933672	−	0.358129i	$-0.116585\pi$
$500$	0	0
$501$	3.00000i	0.134030i
$502$	0	0
$503$	11.0000	0.490466	0.245233	−	0.969464i	$-0.421136\pi$
$503$	11.0000	0.490466	0.245233	+	0.969464i	$0.421136\pi$
$504$	0	0
$505$	9.00000i	0.400495i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.0000i	0.664863i	0.943127	+	0.332432i	$0.107869\pi$
$509$	15.0000i	0.664863i	−0.943127	+	0.332432i	$0.892131\pi$
$510$	0	0
$511$	42.0000	1.85797
$512$	0	0
$513$	− 9.00000i	− 0.397360i
$514$	0	0
$515$	16.0000i	0.705044i
$516$	0	0
$517$	−24.0000	−1.05552
$518$	0	0
$519$	−45.0000	−1.97528
$520$	0	0
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	0	0
$523$	−23.0000	−1.00572	−0.502860	−	0.864368i	$-0.667719\pi$
$523$	−23.0000	−1.00572	−0.502860	+	0.864368i	$0.667719\pi$
$524$	0	0
$525$	− 9.00000i	− 0.392792i
$526$	0	0
$527$	− 28.0000i	− 1.21970i
$528$	0	0
$529$	26.0000	1.13043
$530$	0	0
$531$	− 30.0000i	− 1.30189i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 3.00000i	− 0.129701i
$536$	0	0
$537$	57.0000	2.45973
$538$	0	0
$539$	6.00000i	0.258438i
$540$	0	0
$541$	− 10.0000i	− 0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$	− 10.0000i	− 0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$	0	0
$543$	−42.0000	−1.80239
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	−30.0000	−1.28037
$550$	0	0
$551$	− 5.00000i	− 0.213007i
$552$	0	0
$553$	24.0000i	1.02058i
$554$	0	0
$555$	9.00000	0.382029
$556$	0	0
$557$	19.0000i	0.805056i	0.915408	+	0.402528i	$0.131868\pi$
$557$	19.0000i	0.805056i	−0.915408	+	0.402528i	$0.868132\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	63.0000i	2.65986i
$562$	0	0
$563$	45.0000	1.89652	0.948262	−	0.317489i	$-0.102840\pi$
$563$	45.0000	1.89652	0.948262	+	0.317489i	$0.102840\pi$
$564$	0	0
$565$	13.0000i	0.546914i
$566$	0	0
$567$	− 27.0000i	− 1.13389i
$568$	0	0
$569$	−19.0000	−0.796521	−0.398261	−	0.917272i	$-0.630386\pi$
$569$	−19.0000	−0.796521	−0.398261	+	0.917272i	$0.630386\pi$
$570$	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
$573$	9.00000	0.375980
$574$	0	0
$575$	−7.00000	−0.291920
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	0	0
$579$	− 45.0000i	− 1.87014i
$580$	0	0
$581$	36.0000	1.49353
$582$	0	0
$583$	18.0000i	0.745484i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.0000i	1.52715i	0.645717	+	0.763577i	$0.276559\pi$
$587$	37.0000i	1.52715i	−0.645717	+	0.763577i	$0.723441\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	69.0000i	2.83828i
$592$	0	0
$593$	− 26.0000i	− 1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$	− 26.0000i	− 1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$	0	0
$595$	21.0000	0.860916
$596$	0	0
$597$	27.0000	1.10504
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	0	0
$603$	78.0000i	3.17641i
$604$	0	0
$605$	2.00000i	0.0813116i
$606$	0	0
$607$	−9.00000	−0.365299	−0.182649	−	0.983178i	$-0.558467\pi$
$607$	−9.00000	−0.365299	−0.182649	+	0.983178i	$0.558467\pi$
$608$	0	0
$609$	− 45.0000i	− 1.82349i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 31.0000i	− 1.25208i	−0.779792	−	0.626039i	$-0.784675\pi$
$613$	− 31.0000i	− 1.25208i	0.779792	−	0.626039i	$-0.215325\pi$
$614$	0	0
$615$	21.0000	0.846802
$616$	0	0
$617$	29.0000i	1.16750i	0.811935	+	0.583748i	$0.198414\pi$
$617$	29.0000i	1.16750i	−0.811935	+	0.583748i	$0.801586\pi$
$618$	0	0
$619$	− 12.0000i	− 0.482321i	−0.970485	−	0.241160i	$-0.922472\pi$
$619$	− 12.0000i	− 0.482321i	0.970485	−	0.241160i	$-0.0775280\pi$
$620$	0	0
$621$	−63.0000	−2.52810
$622$	0	0
$623$	−21.0000	−0.841347
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−9.00000	−0.359425
$628$	0	0
$629$	21.0000i	0.837325i
$630$	0	0
$631$	15.0000i	0.597141i	0.954388	+	0.298570i	$0.0965097\pi$
$631$	15.0000i	0.597141i	−0.954388	+	0.298570i	$0.903490\pi$
$632$	0	0
$633$	15.0000	0.596196
$634$	0	0
$635$	− 1.00000i	− 0.0396838i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 18.0000i	− 0.712069i
$640$	0	0
$641$	21.0000	0.829450	0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000	0.829450	0.414725	+	0.909947i	$0.363878\pi$
$642$	0	0
$643$	7.00000i	0.276053i	0.990429	+	0.138027i	$0.0440759\pi$
$643$	7.00000i	0.276053i	−0.990429	+	0.138027i	$0.955924\pi$
$644$	0	0
$645$	− 27.0000i	− 1.06312i
$646$	0	0
$647$	−17.0000	−0.668339	−0.334169	−	0.942513i	$-0.608456\pi$
$647$	−17.0000	−0.668339	−0.334169	+	0.942513i	$0.608456\pi$
$648$	0	0
$649$	−15.0000	−0.588802
$650$	0	0
$651$	36.0000	1.41095
$652$	0	0
$653$	−11.0000	−0.430463	−0.215232	−	0.976563i	$-0.569051\pi$
$653$	−11.0000	−0.430463	−0.215232	+	0.976563i	$0.569051\pi$
$654$	0	0
$655$	4.00000i	0.156293i
$656$	0	0
$657$	84.0000i	3.27715i
$658$	0	0
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	17.0000i	0.661223i	0.943767	+	0.330612i	$0.107255\pi$
$661$	17.0000i	0.661223i	−0.943767	+	0.330612i	$0.892745\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.00000i	0.116335i
$666$	0	0
$667$	−35.0000	−1.35521
$668$	0	0
$669$	69.0000i	2.66769i
$670$	0	0
$671$	15.0000i	0.579069i
$672$	0	0
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	0	0
$675$	9.00000	0.346410
$676$	0	0
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	−33.0000	−1.26642
$680$	0	0
$681$	3.00000i	0.114960i
$682$	0	0
$683$	− 31.0000i	− 1.18618i	−0.805135	−	0.593091i	$-0.797907\pi$
$683$	− 31.0000i	− 1.18618i	0.805135	−	0.593091i	$-0.202093\pi$
$684$	0	0
$685$	−3.00000	−0.114624
$686$	0	0
$687$	78.0000i	2.97589i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 25.0000i	− 0.951045i	−0.879704	−	0.475522i	$-0.842259\pi$
$691$	− 25.0000i	− 0.951045i	0.879704	−	0.475522i	$-0.157741\pi$
$692$	0	0
$693$	−54.0000	−2.05129
$694$	0	0
$695$	13.0000i	0.493118i
$696$	0	0
$697$	49.0000i	1.85601i
$698$	0	0
$699$	54.0000	2.04247
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	−3.00000	−0.113147
$704$	0	0
$705$	−24.0000	−0.903892
$706$	0	0
$707$	− 27.0000i	− 1.01544i
$708$	0	0
$709$	− 35.0000i	− 1.31445i	−0.753693	−	0.657226i	$-0.771730\pi$
$709$	− 35.0000i	− 1.31445i	0.753693	−	0.657226i	$-0.228270\pi$
$710$	0	0
$711$	−48.0000	−1.80014
$712$	0	0
$713$	− 28.0000i	− 1.04861i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.0000i	1.79259i
$718$	0	0
$719$	−1.00000	−0.0372937	−0.0186469	−	0.999826i	$-0.505936\pi$
$719$	−1.00000	−0.0372937	−0.0186469	+	0.999826i	$0.505936\pi$
$720$	0	0
$721$	− 48.0000i	− 1.78761i
$722$	0	0
$723$	− 3.00000i	− 0.111571i
$724$	0	0
$725$	5.00000	0.185695
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	63.0000	2.33014
$732$	0	0
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	0	0
$735$	6.00000i	0.221313i
$736$	0	0
$737$	39.0000	1.43658
$738$	0	0
$739$	− 39.0000i	− 1.43464i	−0.696745	−	0.717319i	$-0.745369\pi$
$739$	− 39.0000i	− 1.43464i	0.696745	−	0.717319i	$-0.254631\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.00000i	0.0366864i	0.999832	+	0.0183432i	$0.00583916\pi$
$743$	1.00000i	0.0366864i	−0.999832	+	0.0183432i	$0.994161\pi$
$744$	0	0
$745$	−3.00000	−0.109911
$746$	0	0
$747$	72.0000i	2.63434i
$748$	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
$753$	15.0000	0.546630
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	1.00000	0.0363456	0.0181728	−	0.999835i	$-0.494215\pi$
$757$	1.00000	0.0363456	0.0181728	+	0.999835i	$0.494215\pi$
$758$	0	0
$759$	63.0000i	2.28676i
$760$	0	0
$761$	− 51.0000i	− 1.84875i	−0.381487	−	0.924374i	$-0.624588\pi$
$761$	− 51.0000i	− 1.84875i	0.381487	−	0.924374i	$-0.375412\pi$
$762$	0	0
$763$	−42.0000	−1.52050
$764$	0	0
$765$	42.0000i	1.51851i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 5.00000i	− 0.180305i	−0.995928	−	0.0901523i	$-0.971265\pi$
$769$	− 5.00000i	− 0.180305i	0.995928	−	0.0901523i	$-0.0287354\pi$
$770$	0	0
$771$	−57.0000	−2.05280
$772$	0	0
$773$	25.0000i	0.899188i	0.893233	+	0.449594i	$0.148431\pi$
$773$	25.0000i	0.899188i	−0.893233	+	0.449594i	$0.851569\pi$
$774$	0	0
$775$	4.00000i	0.143684i
$776$	0	0
$777$	−27.0000	−0.968620
$778$	0	0
$779$	−7.00000	−0.250801
$780$	0	0
$781$	−9.00000	−0.322045
$782$	0	0
$783$	45.0000	1.60817
$784$	0	0
$785$	6.00000i	0.214149i
$786$	0	0
$787$	1.00000i	0.0356462i	0.999841	+	0.0178231i	$0.00567356\pi$
$787$	1.00000i	0.0356462i	−0.999841	+	0.0178231i	$0.994326\pi$
$788$	0	0
$789$	21.0000	0.747620
$790$	0	0
$791$	− 39.0000i	− 1.38668i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	18.0000i	0.638394i
$796$	0	0
$797$	7.00000	0.247953	0.123976	−	0.992285i	$-0.460435\pi$
$797$	7.00000	0.247953	0.123976	+	0.992285i	$0.460435\pi$
$798$	0	0
$799$	− 56.0000i	− 1.98114i
$800$	0	0
$801$	− 42.0000i	− 1.48400i
$802$	0	0
$803$	42.0000	1.48215
$804$	0	0
$805$	21.0000	0.740153
$806$	0	0
$807$	−9.00000	−0.316815
$808$	0	0
$809$	−25.0000	−0.878953	−0.439477	−	0.898254i	$-0.644836\pi$
$809$	−25.0000	−0.878953	−0.439477	+	0.898254i	$0.644836\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	69.0000i	2.41994i
$814$	0	0
$815$	11.0000	0.385313
$816$	0	0
$817$	9.00000i	0.314870i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.0000i	0.523504i	0.965135	+	0.261752i	$0.0843002\pi$
$821$	15.0000i	0.523504i	−0.965135	+	0.261752i	$0.915700\pi$
$822$	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
$825$	− 9.00000i	− 0.313340i
$826$	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
$829$	13.0000	0.451509	0.225754	−	0.974184i	$-0.427515\pi$
$829$	13.0000	0.451509	0.225754	+	0.974184i	$0.427515\pi$
$830$	0	0
$831$	−69.0000	−2.39358
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	1.00000	0.0346064
$836$	0	0
$837$	36.0000i	1.24434i
$838$	0	0
$839$	− 13.0000i	− 0.448810i	−0.974496	−	0.224405i	$-0.927956\pi$
$839$	− 13.0000i	− 0.448810i	0.974496	−	0.224405i	$-0.0720438\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	− 30.0000i	− 1.03325i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 6.00000i	− 0.206162i
$848$	0	0
$849$	3.00000	0.102960
$850$	0	0
$851$	21.0000i	0.719871i
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	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
$861$	−63.0000	−2.14703
$862$	0	0
$863$	16.0000i	0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$	16.0000i	0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$	0	0
$865$	15.0000i	0.510015i
$866$	0	0
$867$	−96.0000	−3.26033
$868$	0	0
$869$	24.0000i	0.814144i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 66.0000i	− 2.23376i
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	5.00000i	0.168838i	0.996430	+	0.0844190i	$0.0269034\pi$
$877$	5.00000i	0.168838i	−0.996430	+	0.0844190i	$0.973097\pi$
$878$	0	0
$879$	27.0000i	0.910687i
$880$	0	0
$881$	−11.0000	−0.370599	−0.185300	−	0.982682i	$-0.559326\pi$
$881$	−11.0000	−0.370599	−0.185300	+	0.982682i	$0.559326\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	−15.0000	−0.504219
$886$	0	0
$887$	−19.0000	−0.637958	−0.318979	−	0.947762i	$-0.603340\pi$
$887$	−19.0000	−0.637958	−0.318979	+	0.947762i	$0.603340\pi$
$888$	0	0
$889$	3.00000i	0.100617i
$890$	0	0
$891$	− 27.0000i	− 0.904534i
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	− 19.0000i	− 0.635100i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	20.0000i	0.667037i
$900$	0	0
$901$	−42.0000	−1.39922
$902$	0	0
$903$	81.0000i	2.69551i
$904$	0	0
$905$	14.0000i	0.465376i
$906$	0	0
$907$	−17.0000	−0.564476	−0.282238	−	0.959344i	$-0.591077\pi$
$907$	−17.0000	−0.564476	−0.282238	+	0.959344i	$0.591077\pi$
$908$	0	0
$909$	54.0000	1.79107
$910$	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
$915$	15.0000i	0.495885i
$916$	0	0
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	−43.0000	−1.41844	−0.709220	−	0.704988i	$-0.750953\pi$
$919$	−43.0000	−1.41844	−0.709220	+	0.704988i	$0.750953\pi$
$920$	0	0
$921$	84.0000i	2.76789i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 3.00000i	− 0.0986394i
$926$	0	0
$927$	96.0000	3.15305
$928$	0	0
$929$	− 53.0000i	− 1.73887i	−0.494044	−	0.869437i	$-0.664482\pi$
$929$	− 53.0000i	− 1.73887i	0.494044	−	0.869437i	$-0.335518\pi$
$930$	0	0
$931$	− 2.00000i	− 0.0655474i
$932$	0	0
$933$	−72.0000	−2.35717
$934$	0	0
$935$	21.0000	0.686773
$936$	0	0
$937$	50.0000	1.63343	0.816714	−	0.577042i	$-0.195793\pi$
$937$	50.0000	1.63343	0.816714	+	0.577042i	$0.195793\pi$
$938$	0	0
$939$	18.0000	0.587408
$940$	0	0
$941$	42.0000i	1.36916i	0.728937	+	0.684580i	$0.240015\pi$
$941$	42.0000i	1.36916i	−0.728937	+	0.684580i	$0.759985\pi$
$942$	0	0
$943$	49.0000i	1.59566i
$944$	0	0
$945$	−27.0000	−0.878310
$946$	0	0
$947$	55.0000i	1.78726i	0.448805	+	0.893630i	$0.351850\pi$
$947$	55.0000i	1.78726i	−0.448805	+	0.893630i	$0.648150\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	6.00000i	0.194563i
$952$	0	0
$953$	31.0000	1.00419	0.502094	−	0.864813i	$-0.332563\pi$
$953$	31.0000	1.00419	0.502094	+	0.864813i	$0.332563\pi$
$954$	0	0
$955$	− 3.00000i	− 0.0970777i
$956$	0	0
$957$	− 45.0000i	− 1.45464i
$958$	0	0
$959$	9.00000	0.290625
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	−18.0000	−0.580042
$964$	0	0
$965$	−15.0000	−0.482867
$966$	0	0
$967$	− 48.0000i	− 1.54358i	−0.635880	−	0.771788i	$-0.719363\pi$
$967$	− 48.0000i	− 1.54358i	0.635880	−	0.771788i	$-0.280637\pi$
$968$	0	0
$969$	− 21.0000i	− 0.674617i
$970$	0	0
$971$	−43.0000	−1.37994	−0.689968	−	0.723840i	$-0.742375\pi$
$971$	−43.0000	−1.37994	−0.689968	+	0.723840i	$0.742375\pi$
$972$	0	0
$973$	− 39.0000i	− 1.25028i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 3.00000i	− 0.0959785i	−0.998848	−	0.0479893i	$-0.984719\pi$
$977$	− 3.00000i	− 0.0959785i	0.998848	−	0.0479893i	$-0.0152813\pi$
$978$	0	0
$979$	−21.0000	−0.671163
$980$	0	0
$981$	− 84.0000i	− 2.68191i
$982$	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
$985$	23.0000	0.732841
$986$	0	0
$987$	72.0000	2.29179
$988$	0	0
$989$	63.0000	2.00328
$990$	0	0
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	0	0
$993$	− 39.0000i	− 1.23763i
$994$	0	0
$995$	− 9.00000i	− 0.285319i
$996$	0	0
$997$	13.0000	0.411714	0.205857	−	0.978582i	$-0.434002\pi$
$997$	13.0000	0.411714	0.205857	+	0.978582i	$0.434002\pi$
$998$	0	0
$999$	− 27.0000i	− 0.854242i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.a.3041.2		2
13.2	odd	12		260.2.i.d.61.1	✓	2
13.5	odd	4		3380.2.a.b.1.1		1
13.6	odd	12		260.2.i.d.81.1	yes	2
13.8	odd	4		3380.2.a.a.1.1		1
13.12	even	2	inner	3380.2.f.a.3041.1		2
39.2	even	12		2340.2.q.a.1621.1		2
39.32	even	12		2340.2.q.a.2161.1		2
52.15	even	12		1040.2.q.b.321.1		2
52.19	even	12		1040.2.q.b.81.1		2
65.2	even	12		1300.2.bb.e.1049.1		4
65.19	odd	12		1300.2.i.a.601.1		2
65.28	even	12		1300.2.bb.e.1049.2		4
65.32	even	12		1300.2.bb.e.549.2		4
65.54	odd	12		1300.2.i.a.1101.1		2
65.58	even	12		1300.2.bb.e.549.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.i.d.61.1	✓	2	13.2	odd	12
260.2.i.d.81.1	yes	2	13.6	odd	12
1040.2.q.b.81.1		2	52.19	even	12
1040.2.q.b.321.1		2	52.15	even	12
1300.2.i.a.601.1		2	65.19	odd	12
1300.2.i.a.1101.1		2	65.54	odd	12
1300.2.bb.e.549.1		4	65.58	even	12
1300.2.bb.e.549.2		4	65.32	even	12
1300.2.bb.e.1049.1		4	65.2	even	12
1300.2.bb.e.1049.2		4	65.28	even	12
2340.2.q.a.1621.1		2	39.2	even	12
2340.2.q.a.2161.1		2	39.32	even	12
3380.2.a.a.1.1		1	13.8	odd	4
3380.2.a.b.1.1		1	13.5	odd	4
3380.2.f.a.3041.1		2	13.12	even	2	inner
3380.2.f.a.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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +1.00000i q^{5} -3.00000i q^{7} +6.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +1.00000i q^{5} -3.00000i q^{7} +6.00000 q^{9} -3.00000i q^{11} -3.00000i q^{15} +7.00000 q^{17} +1.00000i q^{19} +9.00000i q^{21} +7.00000 q^{23} -1.00000 q^{25} -9.00000 q^{27} -5.00000 q^{29} -4.00000i q^{31} +9.00000i q^{33} +3.00000 q^{35} +3.00000i q^{37} +7.00000i q^{41} +9.00000 q^{43} +6.00000i q^{45} -8.00000i q^{47} -2.00000 q^{49} -21.0000 q^{51} -6.00000 q^{53} +3.00000 q^{55} -3.00000i q^{57} -5.00000i q^{59} -5.00000 q^{61} -18.0000i q^{63} +13.0000i q^{67} -21.0000 q^{69} -3.00000i q^{71} +14.0000i q^{73} +3.00000 q^{75} -9.00000 q^{77} -8.00000 q^{79} +9.00000 q^{81} +12.0000i q^{83} +7.00000i q^{85} +15.0000 q^{87} -7.00000i q^{89} +12.0000i q^{93} -1.00000 q^{95} -11.0000i q^{97} -18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 12 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 12 * q^9	\( 2 q - 6 q^{3} + 12 q^{9} + 14 q^{17} + 14 q^{23} - 2 q^{25} - 18 q^{27} - 10 q^{29} + 6 q^{35} + 18 q^{43} - 4 q^{49} - 42 q^{51} - 12 q^{53} + 6 q^{55} - 10 q^{61} - 42 q^{69} + 6 q^{75} - 18 q^{77} - 16 q^{79} + 18 q^{81} + 30 q^{87} - 2 q^{95}+O(q^{100}) \) 2 * q - 6 * q^3 + 12 * q^9 + 14 * q^17 + 14 * q^23 - 2 * q^25 - 18 * q^27 - 10 * q^29 + 6 * q^35 + 18 * q^43 - 4 * q^49 - 42 * q^51 - 12 * q^53 + 6 * q^55 - 10 * q^61 - 42 * q^69 + 6 * q^75 - 18 * q^77 - 16 * q^79 + 18 * q^81 + 30 * q^87 - 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more