LMFDB - Embedded newform 3380.2.f.f.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.f.3041.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{3} -1.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+2.00000 q^{3} -1.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{15} -2.00000 q^{17} -4.00000i q^{21} +6.00000 q^{23} -1.00000 q^{25} -4.00000 q^{27} -10.0000 q^{29} -8.00000i q^{33} -2.00000 q^{35} -10.0000i q^{37} -2.00000i q^{41} -2.00000 q^{43} -1.00000i q^{45} +6.00000i q^{47} +3.00000 q^{49} -4.00000 q^{51} +2.00000 q^{53} -4.00000 q^{55} +8.00000i q^{59} +2.00000 q^{61} -2.00000i q^{63} -6.00000i q^{67} +12.0000 q^{69} -8.00000i q^{71} -10.0000i q^{73} -2.00000 q^{75} -8.00000 q^{77} -16.0000 q^{79} -11.0000 q^{81} +6.00000i q^{83} +2.00000i q^{85} -20.0000 q^{87} -10.0000i q^{89} +2.00000i q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 + 2 * q^9	$ 2 q + 4 q^{3} + 2 q^{9} - 4 q^{17} + 12 q^{23} - 2 q^{25} - 8 q^{27} - 20 q^{29} - 4 q^{35} - 4 q^{43} + 6 q^{49} - 8 q^{51} + 4 q^{53} - 8 q^{55} + 4 q^{61} + 24 q^{69} - 4 q^{75} - 16 q^{77} - 32 q^{79} - 22 q^{81} - 40 q^{87}+O(q^{100}) $ 2 * q + 4 * q^3 + 2 * q^9 - 4 * q^17 + 12 * q^23 - 2 * q^25 - 8 * q^27 - 20 * q^29 - 4 * q^35 - 4 * q^43 + 6 * q^49 - 8 * q^51 + 4 * q^53 - 8 * q^55 + 4 * q^61 + 24 * q^69 - 4 * q^75 - 16 * q^77 - 32 * q^79 - 22 * q^81 - 40 * q^87

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 4.00000i	− 1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 2.00000i	− 0.516398i
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	− 4.00000i	− 0.872872i
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	− 8.00000i	− 1.39262i
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.00000i	− 0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
$41$	− 2.00000i	− 0.312348i	0.987730	−	0.156174i	$-0.0499160\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	− 2.00000i	− 0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 6.00000i	− 0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$	− 6.00000i	− 0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$	0	0
$69$	12.0000	1.44463
$70$	0	0
$71$	− 8.00000i	− 0.949425i	−0.880141	−	0.474713i	$-0.842552\pi$
$71$	− 8.00000i	− 0.949425i	0.880141	−	0.474713i	$-0.157448\pi$
$72$	0	0
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	0	0
$75$	−2.00000	−0.230940
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	2.00000i	0.216930i
$86$	0	0
$87$	−20.0000	−2.14423
$88$	0	0
$89$	− 10.0000i	− 1.06000i	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	− 10.0000i	− 1.06000i	0.847998	−	0.529999i	$-0.177808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	− 4.00000i	− 0.402015i
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−18.0000	−1.77359	−0.886796	−	0.462160i	$-0.847074\pi$
$103$	−18.0000	−1.77359	−0.886796	+	0.462160i	$0.847074\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	18.0000i	1.72409i	0.506834	+	0.862044i	$0.330816\pi$
$109$	18.0000i	1.72409i	−0.506834	+	0.862044i	$0.669184\pi$
$110$	0	0
$111$	− 20.0000i	− 1.89832i
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	− 6.00000i	− 0.559503i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.00000i	0.366679i
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 4.00000i	− 0.360668i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.00000i	0.344265i
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	12.0000i	1.01058i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	10.0000i	0.830455i
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	− 22.0000i	− 1.80231i	−0.433497	−	0.901155i	$-0.642720\pi$
$149$	− 22.0000i	− 1.80231i	0.433497	−	0.901155i	$-0.357280\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	− 12.0000i	− 0.945732i
$162$	0	0
$163$	− 14.0000i	− 1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$	− 14.0000i	− 1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$	0	0
$165$	−8.00000	−0.622799
$166$	0	0
$167$	− 2.00000i	− 0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$	− 2.00000i	− 0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	2.00000i	0.151186i
$176$	0	0
$177$	16.0000i	1.20263i
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	−10.0000	−0.735215
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	8.00000i	0.581914i
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	22.0000i	1.58359i	0.610784	+	0.791797i	$0.290854\pi$
$193$	22.0000i	1.58359i	−0.610784	+	0.791797i	$0.709146\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	− 12.0000i	− 0.846415i
$202$	0	0
$203$	20.0000i	1.40372i
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	− 16.0000i	− 1.09630i
$214$	0	0
$215$	2.00000i	0.136399i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 20.0000i	− 1.35147i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 26.0000i	− 1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$	− 26.0000i	− 1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	2.00000i	0.132745i	0.997795	+	0.0663723i	$0.0211425\pi$
$227$	2.00000i	0.132745i	−0.997795	+	0.0663723i	$0.978857\pi$
$228$	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
$231$	−16.0000	−1.05272
$232$	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
$235$	6.00000	0.391397
$236$	0	0
$237$	−32.0000	−2.07862
$238$	0	0
$239$	4.00000i	0.258738i	0.991596	+	0.129369i	$0.0412952\pi$
$239$	4.00000i	0.258738i	−0.991596	+	0.129369i	$0.958705\pi$
$240$	0	0
$241$	10.0000i	0.644157i	0.946713	+	0.322078i	$0.104381\pi$
$241$	10.0000i	0.644157i	−0.946713	+	0.322078i	$0.895619\pi$
$242$	0	0
$243$	−10.0000	−0.641500
$244$	0	0
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	12.0000i	0.760469i
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	− 24.0000i	− 1.50887i
$254$	0	0
$255$	4.00000i	0.250490i
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−20.0000	−1.24274
$260$	0	0
$261$	−10.0000	−0.618984
$262$	0	0
$263$	26.0000	1.60323	0.801614	−	0.597841i	$-0.203975\pi$
$263$	26.0000	1.60323	0.801614	+	0.597841i	$0.203975\pi$
$264$	0	0
$265$	− 2.00000i	− 0.122859i
$266$	0	0
$267$	− 20.0000i	− 1.22398i
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	− 16.0000i	− 0.971931i	−0.873978	−	0.485965i	$-0.838468\pi$
$271$	− 16.0000i	− 0.971931i	0.873978	−	0.485965i	$-0.161532\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.00000i	0.241209i
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.0000i	1.07379i	0.843649	+	0.536895i	$0.180403\pi$
$281$	18.0000i	1.07379i	−0.843649	+	0.536895i	$0.819597\pi$
$282$	0	0
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.00000	−0.236113
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	4.00000i	0.234484i
$292$	0	0
$293$	− 10.0000i	− 0.584206i	−0.956387	−	0.292103i	$-0.905645\pi$
$293$	− 10.0000i	− 0.584206i	0.956387	−	0.292103i	$-0.0943550\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	16.0000i	0.928414i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.00000i	0.230556i
$302$	0	0
$303$	20.0000	1.14897
$304$	0	0
$305$	− 2.00000i	− 0.114520i
$306$	0	0
$307$	30.0000i	1.71219i	0.516818	+	0.856095i	$0.327116\pi$
$307$	30.0000i	1.71219i	−0.516818	+	0.856095i	$0.672884\pi$
$308$	0	0
$309$	−36.0000	−2.04797
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	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
$319$	40.0000i	2.23957i
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	36.0000i	1.99080i
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	4.00000i	0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$	4.00000i	0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	0	0
$333$	− 10.0000i	− 0.547997i
$334$	0	0
$335$	−6.00000	−0.327815
$336$	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
$339$	4.00000	0.217250
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	− 12.0000i	− 0.646058i
$346$	0	0
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	− 14.0000i	− 0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$	− 14.0000i	− 0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.0000i	0.532246i	0.963939	+	0.266123i	$0.0857428\pi$
$353$	10.0000i	0.532246i	−0.963939	+	0.266123i	$0.914257\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	8.00000i	0.423405i
$358$	0	0
$359$	− 12.0000i	− 0.633336i	−0.948536	−	0.316668i	$-0.897436\pi$
$359$	− 12.0000i	− 0.633336i	0.948536	−	0.316668i	$-0.102564\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	−10.0000	−0.524864
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	0	0
$369$	− 2.00000i	− 0.104116i
$370$	0	0
$371$	− 4.00000i	− 0.207670i
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	2.00000i	0.103280i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 24.0000i	− 1.23280i	−0.787434	−	0.616399i	$-0.788591\pi$
$379$	− 24.0000i	− 1.23280i	0.787434	−	0.616399i	$-0.211409\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	0	0
$385$	8.00000i	0.407718i
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	32.0000	1.61419
$394$	0	0
$395$	16.0000i	0.805047i
$396$	0	0
$397$	− 26.0000i	− 1.30490i	−0.757831	−	0.652451i	$-0.773741\pi$
$397$	− 26.0000i	− 1.30490i	0.757831	−	0.652451i	$-0.226259\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0000i	1.49813i	0.662497	+	0.749064i	$0.269497\pi$
$401$	30.0000i	1.49813i	−0.662497	+	0.749064i	$0.730503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	11.0000i	0.546594i
$406$	0	0
$407$	−40.0000	−1.98273
$408$	0	0
$409$	22.0000i	1.08783i	0.839140	+	0.543915i	$0.183059\pi$
$409$	22.0000i	1.08783i	−0.839140	+	0.543915i	$0.816941\pi$
$410$	0	0
$411$	− 36.0000i	− 1.77575i
$412$	0	0
$413$	16.0000	0.787309
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	6.00000i	0.291730i
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 24.0000i	− 1.15604i	−0.816023	−	0.578020i	$-0.803826\pi$
$431$	− 24.0000i	− 1.15604i	0.816023	−	0.578020i	$-0.196174\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	20.0000i	0.958927i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	10.0000	0.475114	0.237557	−	0.971374i	$-0.423653\pi$
$443$	10.0000	0.475114	0.237557	+	0.971374i	$0.423653\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	− 44.0000i	− 2.08113i
$448$	0	0
$449$	18.0000i	0.849473i	0.905317	+	0.424736i	$0.139633\pi$
$449$	18.0000i	0.849473i	−0.905317	+	0.424736i	$0.860367\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	6.00000i	0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$	6.00000i	0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$	0	0
$463$	− 22.0000i	− 1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$	− 22.0000i	− 1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.0000	−1.01804	−0.509019	−	0.860755i	$-0.669992\pi$
$467$	−22.0000	−1.01804	−0.509019	+	0.860755i	$0.669992\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	20.0000	0.921551
$472$	0	0
$473$	8.00000i	0.367840i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	4.00000i	0.182765i	0.995816	+	0.0913823i	$0.0291285\pi$
$479$	4.00000i	0.182765i	−0.995816	+	0.0913823i	$0.970871\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 24.0000i	− 1.09204i
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	0	0
$489$	− 28.0000i	− 1.26620i
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	−16.0000	−0.717698
$498$	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
$501$	− 4.00000i	− 0.178707i
$502$	0	0
$503$	10.0000	0.445878	0.222939	−	0.974832i	$-0.428435\pi$
$503$	10.0000	0.445878	0.222939	+	0.974832i	$0.428435\pi$
$504$	0	0
$505$	− 10.0000i	− 0.444994i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18.0000i	0.793175i
$516$	0	0
$517$	24.0000	1.05552
$518$	0	0
$519$	28.0000	1.22906
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	−30.0000	−1.31181	−0.655904	−	0.754844i	$-0.727712\pi$
$523$	−30.0000	−1.31181	−0.655904	+	0.754844i	$0.727712\pi$
$524$	0	0
$525$	4.00000i	0.174574i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	8.00000i	0.347170i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 6.00000i	− 0.259403i
$536$	0	0
$537$	−40.0000	−1.72613
$538$	0	0
$539$	− 12.0000i	− 0.516877i
$540$	0	0
$541$	26.0000i	1.11783i	0.829226	+	0.558914i	$0.188782\pi$
$541$	26.0000i	1.11783i	−0.829226	+	0.558914i	$0.811218\pi$
$542$	0	0
$543$	−44.0000	−1.88822
$544$	0	0
$545$	18.0000	0.771035
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	32.0000i	1.36078i
$554$	0	0
$555$	−20.0000	−0.848953
$556$	0	0
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	16.0000i	0.675521i
$562$	0	0
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	0	0
$565$	− 2.00000i	− 0.0841406i
$566$	0	0
$567$	22.0000i	0.923913i
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	0	0
$575$	−6.00000	−0.250217
$576$	0	0
$577$	18.0000i	0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$	18.0000i	0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	0	0
$579$	44.0000i	1.82858i
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	− 8.00000i	− 0.331326i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.0000i	0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$	18.0000i	0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	36.0000i	1.48084i
$592$	0	0
$593$	− 34.0000i	− 1.39621i	−0.715994	−	0.698106i	$-0.754026\pi$
$593$	− 34.0000i	− 1.39621i	0.715994	−	0.698106i	$-0.245974\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0	0
$597$	48.0000	1.96451
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	− 6.00000i	− 0.244339i
$604$	0	0
$605$	5.00000i	0.203279i
$606$	0	0
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	0	0
$609$	40.0000i	1.62088i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	0	0
$619$	− 32.0000i	− 1.28619i	−0.765787	−	0.643094i	$-0.777650\pi$
$619$	− 32.0000i	− 1.28619i	0.765787	−	0.643094i	$-0.222350\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	0	0
$623$	−20.0000	−0.801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.0000i	0.797452i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	32.0000	1.27189
$634$	0	0
$635$	− 2.00000i	− 0.0793676i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 8.00000i	− 0.316475i
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	0	0
$645$	4.00000i	0.157500i
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	− 16.0000i	− 0.625172i
$656$	0	0
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	− 42.0000i	− 1.63361i	−0.576913	−	0.816805i	$-0.695743\pi$
$661$	− 42.0000i	− 1.63361i	0.576913	−	0.816805i	$-0.304257\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−60.0000	−2.32321
$668$	0	0
$669$	− 52.0000i	− 2.01044i
$670$	0	0
$671$	− 8.00000i	− 0.308837i
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	4.00000i	0.153280i
$682$	0	0
$683$	50.0000i	1.91320i	0.291409	+	0.956598i	$0.405876\pi$
$683$	50.0000i	1.91320i	−0.291409	+	0.956598i	$0.594124\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	− 44.0000i	− 1.67870i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 44.0000i	− 1.67384i	−0.547326	−	0.836919i	$-0.684354\pi$
$691$	− 44.0000i	− 1.67384i	0.547326	−	0.836919i	$-0.315646\pi$
$692$	0	0
$693$	−8.00000	−0.303895
$694$	0	0
$695$	− 4.00000i	− 0.151729i
$696$	0	0
$697$	4.00000i	0.151511i
$698$	0	0
$699$	12.0000	0.453882
$700$	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
$703$	0	0
$704$	0	0
$705$	12.0000	0.451946
$706$	0	0
$707$	− 20.0000i	− 0.752177i
$708$	0	0
$709$	10.0000i	0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$	10.0000i	0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.00000i	0.298765i
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	36.0000i	1.34071i
$722$	0	0
$723$	20.0000i	0.743808i
$724$	0	0
$725$	10.0000	0.371391
$726$	0	0
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	4.00000	0.147945
$732$	0	0
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	0	0
$735$	− 6.00000i	− 0.221313i
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.00000i	0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$	6.00000i	0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$	0	0
$745$	−22.0000	−0.806018
$746$	0	0
$747$	6.00000i	0.219529i
$748$	0	0
$749$	− 12.0000i	− 0.438470i
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	0	0
$753$	48.0000	1.74922
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30.0000	−1.09037	−0.545184	−	0.838316i	$-0.683540\pi$
$757$	−30.0000	−1.09037	−0.545184	+	0.838316i	$0.683540\pi$
$758$	0	0
$759$	− 48.0000i	− 1.74229i
$760$	0	0
$761$	38.0000i	1.37750i	0.724999	+	0.688749i	$0.241840\pi$
$761$	38.0000i	1.37750i	−0.724999	+	0.688749i	$0.758160\pi$
$762$	0	0
$763$	36.0000	1.30329
$764$	0	0
$765$	2.00000i	0.0723102i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 14.0000i	− 0.504853i	−0.967616	−	0.252426i	$-0.918771\pi$
$769$	− 14.0000i	− 0.504853i	0.967616	−	0.252426i	$-0.0812286\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	0	0
$773$	26.0000i	0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$	26.0000i	0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−40.0000	−1.43499
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	40.0000	1.42948
$784$	0	0
$785$	− 10.0000i	− 0.356915i
$786$	0	0
$787$	38.0000i	1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$	38.0000i	1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$	0	0
$789$	52.0000	1.85125
$790$	0	0
$791$	− 4.00000i	− 0.142224i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 4.00000i	− 0.141865i
$796$	0	0
$797$	22.0000	0.779280	0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000	0.779280	0.389640	+	0.920967i	$0.372599\pi$
$798$	0	0
$799$	− 12.0000i	− 0.424529i
$800$	0	0
$801$	− 10.0000i	− 0.353333i
$802$	0	0
$803$	−40.0000	−1.41157
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	36.0000	1.26726
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	28.0000i	0.983213i	0.870817	+	0.491606i	$0.163590\pi$
$811$	28.0000i	0.983213i	−0.870817	+	0.491606i	$0.836410\pi$
$812$	0	0
$813$	− 32.0000i	− 1.12229i
$814$	0	0
$815$	−14.0000	−0.490399
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.0000i	0.907406i	0.891153	+	0.453703i	$0.149897\pi$
$821$	26.0000i	0.907406i	−0.891153	+	0.453703i	$0.850103\pi$
$822$	0	0
$823$	−26.0000	−0.906303	−0.453152	−	0.891434i	$-0.649700\pi$
$823$	−26.0000	−0.906303	−0.453152	+	0.891434i	$0.649700\pi$
$824$	0	0
$825$	8.00000i	0.278524i
$826$	0	0
$827$	30.0000i	1.04320i	0.853189	+	0.521601i	$0.174665\pi$
$827$	30.0000i	1.04320i	−0.853189	+	0.521601i	$0.825335\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 20.0000i	− 0.690477i	−0.938515	−	0.345238i	$-0.887798\pi$
$839$	− 20.0000i	− 0.690477i	0.938515	−	0.345238i	$-0.112202\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	36.0000i	1.23991i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.0000i	0.343604i
$848$	0	0
$849$	44.0000	1.51008
$850$	0	0
$851$	− 60.0000i	− 2.05677i
$852$	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
$855$	0	0
$856$	0	0
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	0	0
$865$	− 14.0000i	− 0.476014i
$866$	0	0
$867$	−26.0000	−0.883006
$868$	0	0
$869$	64.0000i	2.17105i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	58.0000i	1.95852i	0.202606	+	0.979260i	$0.435059\pi$
$877$	58.0000i	1.95852i	−0.202606	+	0.979260i	$0.564941\pi$
$878$	0	0
$879$	− 20.0000i	− 0.674583i
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	0	0
$885$	16.0000	0.537834
$886$	0	0
$887$	22.0000	0.738688	0.369344	−	0.929293i	$-0.379582\pi$
$887$	22.0000	0.738688	0.369344	+	0.929293i	$0.379582\pi$
$888$	0	0
$889$	− 4.00000i	− 0.134156i
$890$	0	0
$891$	44.0000i	1.47406i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	20.0000i	0.668526i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	8.00000i	0.266223i
$904$	0	0
$905$	22.0000i	0.731305i
$906$	0	0
$907$	26.0000	0.863316	0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000	0.863316	0.431658	+	0.902037i	$0.357929\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	24.0000	0.794284
$914$	0	0
$915$	− 4.00000i	− 0.132236i
$916$	0	0
$917$	− 32.0000i	− 1.05673i
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	60.0000i	1.97707i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	10.0000i	0.328798i
$926$	0	0
$927$	−18.0000	−0.591198
$928$	0	0
$929$	30.0000i	0.984268i	0.870519	+	0.492134i	$0.163783\pi$
$929$	30.0000i	0.984268i	−0.870519	+	0.492134i	$0.836217\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	−54.0000	−1.76410	−0.882052	−	0.471153i	$-0.843838\pi$
$937$	−54.0000	−1.76410	−0.882052	+	0.471153i	$0.843838\pi$
$938$	0	0
$939$	36.0000	1.17482
$940$	0	0
$941$	− 58.0000i	− 1.89075i	−0.325991	−	0.945373i	$-0.605698\pi$
$941$	− 58.0000i	− 1.89075i	0.325991	−	0.945373i	$-0.394302\pi$
$942$	0	0
$943$	− 12.0000i	− 0.390774i
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	14.0000i	0.454939i	0.973785	+	0.227469i	$0.0730452\pi$
$947$	14.0000i	0.454939i	−0.973785	+	0.227469i	$0.926955\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	36.0000i	1.16738i
$952$	0	0
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	0	0
$955$	12.0000i	0.388311i
$956$	0	0
$957$	80.0000i	2.58603i
$958$	0	0
$959$	−36.0000	−1.16250
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	6.00000	0.193347
$964$	0	0
$965$	22.0000	0.708205
$966$	0	0
$967$	− 14.0000i	− 0.450210i	−0.974335	−	0.225105i	$-0.927728\pi$
$967$	− 14.0000i	− 0.450210i	0.974335	−	0.225105i	$-0.0722725\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.00000	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$972$	0	0
$973$	− 8.00000i	− 0.256468i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	−40.0000	−1.27841
$980$	0	0
$981$	18.0000i	0.574696i
$982$	0	0
$983$	− 38.0000i	− 1.21201i	−0.795460	−	0.606006i	$-0.792771\pi$
$983$	− 38.0000i	− 1.21201i	0.795460	−	0.606006i	$-0.207229\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	24.0000	0.763928
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−36.0000	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$992$	0	0
$993$	8.00000i	0.253872i
$994$	0	0
$995$	− 24.0000i	− 0.760851i
$996$	0	0
$997$	−62.0000	−1.96356	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−62.0000	−1.96356	−0.981780	+	0.190022i	$0.939144\pi$
$998$	0	0
$999$	40.0000i	1.26554i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.f.3041.1		2
13.5	odd	4		260.2.a.a.1.1	✓	1
13.8	odd	4		3380.2.a.j.1.1		1
13.12	even	2	inner	3380.2.f.f.3041.2		2
39.5	even	4		2340.2.a.i.1.1		1
52.31	even	4		1040.2.a.a.1.1		1
65.18	even	4		1300.2.c.a.1249.2		2
65.44	odd	4		1300.2.a.a.1.1		1
65.57	even	4		1300.2.c.a.1249.1		2
104.5	odd	4		4160.2.a.d.1.1		1
104.83	even	4		4160.2.a.s.1.1		1
156.83	odd	4		9360.2.a.bi.1.1		1
260.239	even	4		5200.2.a.bh.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.a.a.1.1	✓	1	13.5	odd	4
1040.2.a.a.1.1		1	52.31	even	4
1300.2.a.a.1.1		1	65.44	odd	4
1300.2.c.a.1249.1		2	65.57	even	4
1300.2.c.a.1249.2		2	65.18	even	4
2340.2.a.i.1.1		1	39.5	even	4
3380.2.a.j.1.1		1	13.8	odd	4
3380.2.f.f.3041.1		2	1.1	even	1	trivial
3380.2.f.f.3041.2		2	13.12	even	2	inner
4160.2.a.d.1.1		1	104.5	odd	4
4160.2.a.s.1.1		1	104.83	even	4
5200.2.a.bh.1.1		1	260.239	even	4
9360.2.a.bi.1.1		1	156.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{3} -1.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{3} -1.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{15} -2.00000 q^{17} -4.00000i q^{21} +6.00000 q^{23} -1.00000 q^{25} -4.00000 q^{27} -10.0000 q^{29} -8.00000i q^{33} -2.00000 q^{35} -10.0000i q^{37} -2.00000i q^{41} -2.00000 q^{43} -1.00000i q^{45} +6.00000i q^{47} +3.00000 q^{49} -4.00000 q^{51} +2.00000 q^{53} -4.00000 q^{55} +8.00000i q^{59} +2.00000 q^{61} -2.00000i q^{63} -6.00000i q^{67} +12.0000 q^{69} -8.00000i q^{71} -10.0000i q^{73} -2.00000 q^{75} -8.00000 q^{77} -16.0000 q^{79} -11.0000 q^{81} +6.00000i q^{83} +2.00000i q^{85} -20.0000 q^{87} -10.0000i q^{89} +2.00000i q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 + 2 * q^9	\( 2 q + 4 q^{3} + 2 q^{9} - 4 q^{17} + 12 q^{23} - 2 q^{25} - 8 q^{27} - 20 q^{29} - 4 q^{35} - 4 q^{43} + 6 q^{49} - 8 q^{51} + 4 q^{53} - 8 q^{55} + 4 q^{61} + 24 q^{69} - 4 q^{75} - 16 q^{77} - 32 q^{79} - 22 q^{81} - 40 q^{87}+O(q^{100}) \) 2 * q + 4 * q^3 + 2 * q^9 - 4 * q^17 + 12 * q^23 - 2 * q^25 - 8 * q^27 - 20 * q^29 - 4 * q^35 - 4 * q^43 + 6 * q^49 - 8 * q^51 + 4 * q^53 - 8 * q^55 + 4 * q^61 + 24 * q^69 - 4 * q^75 - 16 * q^77 - 32 * q^79 - 22 * q^81 - 40 * q^87

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