LMFDB - Embedded newform 3328.2.b.r.1665.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3328,2,Mod(1665,3328)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3328, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3328.1665");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.b (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:	$26.5742137927$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 416)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1665.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3328.1665
Dual form			3328.2.b.r.1665.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} +3.00000 q^{7} +2.00000 q^{9} -2.00000i q^{11} -1.00000i q^{13} -1.00000 q^{15} -3.00000 q^{17} +2.00000i q^{19} +3.00000i q^{21} +4.00000 q^{23} +4.00000 q^{25} +5.00000i q^{27} -2.00000i q^{29} -4.00000 q^{31} +2.00000 q^{33} +3.00000i q^{35} +5.00000i q^{37} +1.00000 q^{39} +12.0000 q^{41} -7.00000i q^{43} +2.00000i q^{45} +9.00000 q^{47} +2.00000 q^{49} -3.00000i q^{51} +4.00000i q^{53} +2.00000 q^{55} -2.00000 q^{57} -6.00000i q^{59} +4.00000i q^{61} +6.00000 q^{63} +1.00000 q^{65} -10.0000i q^{67} +4.00000i q^{69} -15.0000 q^{71} +2.00000 q^{73} +4.00000i q^{75} -6.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{83} -3.00000i q^{85} +2.00000 q^{87} -2.00000 q^{89} -3.00000i q^{91} -4.00000i q^{93} -2.00000 q^{95} +10.0000 q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{7} + 4 q^{9} - 2 q^{15} - 6 q^{17} + 8 q^{23} + 8 q^{25} - 8 q^{31} + 4 q^{33} + 2 q^{39} + 24 q^{41} + 18 q^{47} + 4 q^{49} + 4 q^{55} - 4 q^{57} + 12 q^{63} + 2 q^{65} - 30 q^{71} + 4 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) $ 2 * q + 6 * q^7 + 4 * q^9 - 2 * q^15 - 6 * q^17 + 8 * q^23 + 8 * q^25 - 8 * q^31 + 4 * q^33 + 2 * q^39 + 24 * q^41 + 18 * q^47 + 4 * q^49 + 4 * q^55 - 4 * q^57 + 12 * q^63 + 2 * q^65 - 30 * q^71 + 4 * q^73 + 16 * q^79 + 2 * q^81 + 4 * q^87 - 4 * q^89 - 4 * q^95 + 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	1.00000i	0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$	1.00000i	0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	0	0
$21$	3.00000i	0.654654i
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	3.00000i	0.507093i
$36$	0	0
$37$	5.00000i	0.821995i	0.911636	+	0.410997i	$0.134819\pi$
$37$	5.00000i	0.821995i	−0.911636	+	0.410997i	$0.865181\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	− 7.00000i	− 1.06749i	−0.845645	−	0.533745i	$-0.820784\pi$
$43$	− 7.00000i	− 1.06749i	0.845645	−	0.533745i	$-0.179216\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	− 3.00000i	− 0.420084i
$52$	0	0
$53$	4.00000i	0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$	4.00000i	0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	− 6.00000i	− 0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$	− 6.00000i	− 0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$	0	0
$61$	4.00000i	0.512148i	0.966657	+	0.256074i	$0.0824290\pi$
$61$	4.00000i	0.512148i	−0.966657	+	0.256074i	$0.917571\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	− 10.0000i	− 1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$	− 10.0000i	− 1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$	0	0
$69$	4.00000i	0.481543i
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	4.00000i	0.461880i
$76$	0	0
$77$	− 6.00000i	− 0.683763i
$78$	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
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	− 3.00000i	− 0.325396i
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	− 3.00000i	− 0.314485i
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	− 4.00000i	− 0.402015i
$100$	0	0
$101$	4.00000i	0.398015i	0.979998	+	0.199007i	$0.0637718\pi$
$101$	4.00000i	0.398015i	−0.979998	+	0.199007i	$0.936228\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	19.0000i	1.81987i	0.414751	+	0.909935i	$0.363869\pi$
$109$	19.0000i	1.81987i	−0.414751	+	0.909935i	$0.636131\pi$
$110$	0	0
$111$	−5.00000	−0.474579
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	4.00000i	0.373002i
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−9.00000	−0.825029
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	12.0000i	1.08200i
$124$	0	0
$125$	9.00000i	0.804984i
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	7.00000	0.616316
$130$	0	0
$131$	− 21.0000i	− 1.83478i	−0.397991	−	0.917389i	$-0.630293\pi$
$131$	− 21.0000i	− 1.83478i	0.397991	−	0.917389i	$-0.369707\pi$
$132$	0	0
$133$	6.00000i	0.520266i
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	0	0
$139$	− 19.0000i	− 1.61156i	−0.592216	−	0.805779i	$-0.701747\pi$
$139$	− 19.0000i	− 1.61156i	0.592216	−	0.805779i	$-0.298253\pi$
$140$	0	0
$141$	9.00000i	0.757937i
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	2.00000i	0.164957i
$148$	0	0
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	− 4.00000i	− 0.321288i
$156$	0	0
$157$	22.0000i	1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$	22.0000i	1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	0	0
$165$	2.00000i	0.155700i
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	− 16.0000i	− 1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$	− 16.0000i	− 1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	11.0000i	0.822179i	0.911595	+	0.411089i	$0.134852\pi$
$179$	11.0000i	0.822179i	−0.911595	+	0.411089i	$0.865148\pi$
$180$	0	0
$181$	20.0000i	1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$	20.0000i	1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	−5.00000	−0.367607
$186$	0	0
$187$	6.00000i	0.438763i
$188$	0	0
$189$	15.0000i	1.09109i
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	0	0
$195$	1.00000i	0.0716115i
$196$	0	0
$197$	15.0000i	1.06871i	0.845262	+	0.534353i	$0.179445\pi$
$197$	15.0000i	1.06871i	−0.845262	+	0.534353i	$0.820555\pi$
$198$	0	0
$199$	26.0000	1.84309	0.921546	−	0.388270i	$-0.126927\pi$
$199$	26.0000	1.84309	0.921546	+	0.388270i	$0.126927\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	0	0
$203$	− 6.00000i	− 0.421117i
$204$	0	0
$205$	12.0000i	0.838116i
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	27.0000i	1.85876i	0.369129	+	0.929378i	$0.379656\pi$
$211$	27.0000i	1.85876i	−0.369129	+	0.929378i	$0.620344\pi$
$212$	0	0
$213$	− 15.0000i	− 1.02778i
$214$	0	0
$215$	7.00000	0.477396
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	2.00000i	0.135147i
$220$	0	0
$221$	3.00000i	0.201802i
$222$	0	0
$223$	15.0000	1.00447	0.502237	−	0.864730i	$-0.332510\pi$
$223$	15.0000	1.00447	0.502237	+	0.864730i	$0.332510\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	− 9.00000i	− 0.594737i	−0.954763	−	0.297368i	$-0.903891\pi$
$229$	− 9.00000i	− 0.594737i	0.954763	−	0.297368i	$-0.0961089\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	27.0000	1.76883	0.884414	−	0.466702i	$-0.154558\pi$
$233$	27.0000	1.76883	0.884414	+	0.466702i	$0.154558\pi$
$234$	0	0
$235$	9.00000i	0.587095i
$236$	0	0
$237$	8.00000i	0.519656i
$238$	0	0
$239$	−27.0000	−1.74648	−0.873242	−	0.487286i	$-0.837987\pi$
$239$	−27.0000	−1.74648	−0.873242	+	0.487286i	$0.837987\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	2.00000i	0.127775i
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	8.00000i	0.504956i	0.967603	+	0.252478i	$0.0812455\pi$
$251$	8.00000i	0.504956i	−0.967603	+	0.252478i	$0.918755\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	0	0
$259$	15.0000i	0.932055i
$260$	0	0
$261$	− 4.00000i	− 0.247594i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	− 2.00000i	− 0.122398i
$268$	0	0
$269$	4.00000i	0.243884i	0.992537	+	0.121942i	$0.0389122\pi$
$269$	4.00000i	0.243884i	−0.992537	+	0.121942i	$0.961088\pi$
$270$	0	0
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	− 8.00000i	− 0.482418i
$276$	0	0
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	0	0
$285$	− 2.00000i	− 0.118470i
$286$	0	0
$287$	36.0000	2.12501
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	10.0000i	0.586210i
$292$	0	0
$293$	− 7.00000i	− 0.408944i	−0.978872	−	0.204472i	$-0.934452\pi$
$293$	− 7.00000i	− 0.408944i	0.978872	−	0.204472i	$-0.0655478\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	10.0000	0.580259
$298$	0	0
$299$	− 4.00000i	− 0.231326i
$300$	0	0
$301$	− 21.0000i	− 1.21042i
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	− 26.0000i	− 1.48390i	−0.670456	−	0.741949i	$-0.733902\pi$
$307$	− 26.0000i	− 1.48390i	0.670456	−	0.741949i	$-0.266098\pi$
$308$	0	0
$309$	4.00000i	0.227552i
$310$	0	0
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	0	0
$313$	−23.0000	−1.30004	−0.650018	−	0.759918i	$-0.725239\pi$
$313$	−23.0000	−1.30004	−0.650018	+	0.759918i	$0.725239\pi$
$314$	0	0
$315$	6.00000i	0.338062i
$316$	0	0
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	− 6.00000i	− 0.333849i
$324$	0	0
$325$	− 4.00000i	− 0.221880i
$326$	0	0
$327$	−19.0000	−1.05070
$328$	0	0
$329$	27.0000	1.48856
$330$	0	0
$331$	16.0000i	0.879440i	0.898135	+	0.439720i	$0.144922\pi$
$331$	16.0000i	0.879440i	−0.898135	+	0.439720i	$0.855078\pi$
$332$	0	0
$333$	10.0000i	0.547997i
$334$	0	0
$335$	10.0000	0.546358
$336$	0	0
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	0	0
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	8.00000i	0.433224i
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	− 11.0000i	− 0.590511i	−0.955418	−	0.295255i	$-0.904595\pi$
$347$	− 11.0000i	− 0.590511i	0.955418	−	0.295255i	$-0.0954048\pi$
$348$	0	0
$349$	7.00000i	0.374701i	0.982293	+	0.187351i	$0.0599901\pi$
$349$	7.00000i	0.374701i	−0.982293	+	0.187351i	$0.940010\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	− 15.0000i	− 0.796117i
$356$	0	0
$357$	− 9.00000i	− 0.476331i
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	7.00000i	0.367405i
$364$	0	0
$365$	2.00000i	0.104685i
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	0	0
$369$	24.0000	1.24939
$370$	0	0
$371$	12.0000i	0.623009i
$372$	0	0
$373$	36.0000i	1.86401i	0.362446	+	0.932005i	$0.381942\pi$
$373$	36.0000i	1.86401i	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	8.00000i	0.410932i	0.978664	+	0.205466i	$0.0658711\pi$
$379$	8.00000i	0.410932i	−0.978664	+	0.205466i	$0.934129\pi$
$380$	0	0
$381$	− 16.0000i	− 0.819705i
$382$	0	0
$383$	7.00000	0.357683	0.178842	−	0.983878i	$-0.442765\pi$
$383$	7.00000	0.357683	0.178842	+	0.983878i	$0.442765\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	− 14.0000i	− 0.711660i
$388$	0	0
$389$	6.00000i	0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$	6.00000i	0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	21.0000	1.05931
$394$	0	0
$395$	8.00000i	0.402524i
$396$	0	0
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	10.0000	0.495682
$408$	0	0
$409$	−12.0000	−0.593362	−0.296681	−	0.954977i	$-0.595880\pi$
$409$	−12.0000	−0.593362	−0.296681	+	0.954977i	$0.595880\pi$
$410$	0	0
$411$	4.00000i	0.197305i
$412$	0	0
$413$	− 18.0000i	− 0.885722i
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	19.0000	0.930434
$418$	0	0
$419$	1.00000i	0.0488532i	0.999702	+	0.0244266i	$0.00777600\pi$
$419$	1.00000i	0.0488532i	−0.999702	+	0.0244266i	$0.992224\pi$
$420$	0	0
$421$	− 3.00000i	− 0.146211i	−0.997324	−	0.0731055i	$-0.976709\pi$
$421$	− 3.00000i	− 0.146211i	0.997324	−	0.0731055i	$-0.0232910\pi$
$422$	0	0
$423$	18.0000	0.875190
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	12.0000i	0.580721i
$428$	0	0
$429$	− 2.00000i	− 0.0965609i
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	0	0
$435$	2.00000i	0.0958927i
$436$	0	0
$437$	8.00000i	0.382692i
$438$	0	0
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	0	0
$441$	4.00000	0.190476
$442$	0	0
$443$	11.0000i	0.522626i	0.965254	+	0.261313i	$0.0841554\pi$
$443$	11.0000i	0.522626i	−0.965254	+	0.261313i	$0.915845\pi$
$444$	0	0
$445$	− 2.00000i	− 0.0948091i
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	− 24.0000i	− 1.13012i
$452$	0	0
$453$	5.00000i	0.234920i
$454$	0	0
$455$	3.00000	0.140642
$456$	0	0
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	0	0
$459$	− 15.0000i	− 0.700140i
$460$	0	0
$461$	− 21.0000i	− 0.978068i	−0.872265	−	0.489034i	$-0.837349\pi$
$461$	− 21.0000i	− 0.978068i	0.872265	−	0.489034i	$-0.162651\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	0	0
$469$	− 30.0000i	− 1.38527i
$470$	0	0
$471$	−22.0000	−1.01371
$472$	0	0
$473$	−14.0000	−0.643721
$474$	0	0
$475$	8.00000i	0.367065i
$476$	0	0
$477$	8.00000i	0.366295i
$478$	0	0
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	5.00000	0.227980
$482$	0	0
$483$	12.0000i	0.546019i
$484$	0	0
$485$	10.0000i	0.454077i
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	− 31.0000i	− 1.39901i	−0.714628	−	0.699505i	$-0.753404\pi$
$491$	− 31.0000i	− 1.39901i	0.714628	−	0.699505i	$-0.246596\pi$
$492$	0	0
$493$	6.00000i	0.270226i
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	−45.0000	−2.01853
$498$	0	0
$499$	− 4.00000i	− 0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$	− 4.00000i	− 0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$	0	0
$501$	− 8.00000i	− 0.357414i
$502$	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
$505$	−4.00000	−0.177998
$506$	0	0
$507$	− 1.00000i	− 0.0444116i
$508$	0	0
$509$	− 30.0000i	− 1.32973i	−0.746965	−	0.664863i	$-0.768490\pi$
$509$	− 30.0000i	− 1.32973i	0.746965	−	0.664863i	$-0.231510\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	−10.0000	−0.441511
$514$	0	0
$515$	4.00000i	0.176261i
$516$	0	0
$517$	− 18.0000i	− 0.791639i
$518$	0	0
$519$	16.0000	0.702322
$520$	0	0
$521$	1.00000	0.0438108	0.0219054	−	0.999760i	$-0.493027\pi$
$521$	1.00000	0.0438108	0.0219054	+	0.999760i	$0.493027\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	0	0
$525$	12.0000i	0.523723i
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 12.0000i	− 0.520756i
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	−11.0000	−0.474685
$538$	0	0
$539$	− 4.00000i	− 0.172292i
$540$	0	0
$541$	17.0000i	0.730887i	0.930834	+	0.365444i	$0.119083\pi$
$541$	17.0000i	0.730887i	−0.930834	+	0.365444i	$0.880917\pi$
$542$	0	0
$543$	−20.0000	−0.858282
$544$	0	0
$545$	−19.0000	−0.813871
$546$	0	0
$547$	17.0000i	0.726868i	0.931620	+	0.363434i	$0.118396\pi$
$547$	17.0000i	0.726868i	−0.931620	+	0.363434i	$0.881604\pi$
$548$	0	0
$549$	8.00000i	0.341432i
$550$	0	0
$551$	4.00000	0.170406
$552$	0	0
$553$	24.0000	1.02058
$554$	0	0
$555$	− 5.00000i	− 0.212238i
$556$	0	0
$557$	− 7.00000i	− 0.296600i	−0.988942	−	0.148300i	$-0.952620\pi$
$557$	− 7.00000i	− 0.296600i	0.988942	−	0.148300i	$-0.0473800\pi$
$558$	0	0
$559$	−7.00000	−0.296068
$560$	0	0
$561$	−6.00000	−0.253320
$562$	0	0
$563$	39.0000i	1.64365i	0.569737	+	0.821827i	$0.307045\pi$
$563$	39.0000i	1.64365i	−0.569737	+	0.821827i	$0.692955\pi$
$564$	0	0
$565$	− 6.00000i	− 0.252422i
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	0	0
$571$	− 29.0000i	− 1.21361i	−0.794850	−	0.606806i	$-0.792450\pi$
$571$	− 29.0000i	− 1.21361i	0.794850	−	0.606806i	$-0.207550\pi$
$572$	0	0
$573$	− 18.0000i	− 0.751961i
$574$	0	0
$575$	16.0000	0.667246
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	0	0
$579$	12.0000i	0.498703i
$580$	0	0
$581$	− 12.0000i	− 0.497844i
$582$	0	0
$583$	8.00000	0.331326
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	− 20.0000i	− 0.825488i	−0.910847	−	0.412744i	$-0.864570\pi$
$587$	− 20.0000i	− 0.825488i	0.910847	−	0.412744i	$-0.135430\pi$
$588$	0	0
$589$	− 8.00000i	− 0.329634i
$590$	0	0
$591$	−15.0000	−0.617018
$592$	0	0
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	− 9.00000i	− 0.368964i
$596$	0	0
$597$	26.0000i	1.06411i
$598$	0	0
$599$	42.0000	1.71607	0.858037	−	0.513588i	$-0.171684\pi$
$599$	42.0000	1.71607	0.858037	+	0.513588i	$0.171684\pi$
$600$	0	0
$601$	−21.0000	−0.856608	−0.428304	−	0.903635i	$-0.640889\pi$
$601$	−21.0000	−0.856608	−0.428304	+	0.903635i	$0.640889\pi$
$602$	0	0
$603$	− 20.0000i	− 0.814463i
$604$	0	0
$605$	7.00000i	0.284590i
$606$	0	0
$607$	38.0000	1.54237	0.771186	−	0.636610i	$-0.219664\pi$
$607$	38.0000	1.54237	0.771186	+	0.636610i	$0.219664\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	− 9.00000i	− 0.364101i
$612$	0	0
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	0	0
$615$	−12.0000	−0.483887
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	20.0000i	0.802572i
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	4.00000i	0.159745i
$628$	0	0
$629$	− 15.0000i	− 0.598089i
$630$	0	0
$631$	25.0000	0.995234	0.497617	−	0.867397i	$-0.334208\pi$
$631$	25.0000	0.995234	0.497617	+	0.867397i	$0.334208\pi$
$632$	0	0
$633$	−27.0000	−1.07315
$634$	0	0
$635$	− 16.0000i	− 0.634941i
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	0	0
$639$	−30.0000	−1.18678
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	− 2.00000i	− 0.0788723i	−0.999222	−	0.0394362i	$-0.987444\pi$
$643$	− 2.00000i	− 0.0788723i	0.999222	−	0.0394362i	$-0.0125562\pi$
$644$	0	0
$645$	7.00000i	0.275625i
$646$	0	0
$647$	−26.0000	−1.02217	−0.511083	−	0.859532i	$-0.670755\pi$
$647$	−26.0000	−1.02217	−0.511083	+	0.859532i	$0.670755\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	− 12.0000i	− 0.470317i
$652$	0	0
$653$	− 32.0000i	− 1.25226i	−0.779720	−	0.626128i	$-0.784639\pi$
$653$	− 32.0000i	− 1.25226i	0.779720	−	0.626128i	$-0.215361\pi$
$654$	0	0
$655$	21.0000	0.820538
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	20.0000i	0.779089i	0.921008	+	0.389545i	$0.127368\pi$
$659$	20.0000i	0.779089i	−0.921008	+	0.389545i	$0.872632\pi$
$660$	0	0
$661$	− 14.0000i	− 0.544537i	−0.962221	−	0.272268i	$-0.912226\pi$
$661$	− 14.0000i	− 0.544537i	0.962221	−	0.272268i	$-0.0877739\pi$
$662$	0	0
$663$	−3.00000	−0.116510
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	− 8.00000i	− 0.309761i
$668$	0	0
$669$	15.0000i	0.579934i
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	13.0000	0.501113	0.250557	−	0.968102i	$-0.419386\pi$
$673$	13.0000	0.501113	0.250557	+	0.968102i	$0.419386\pi$
$674$	0	0
$675$	20.0000i	0.769800i
$676$	0	0
$677$	− 40.0000i	− 1.53732i	−0.639655	−	0.768662i	$-0.720923\pi$
$677$	− 40.0000i	− 1.53732i	0.639655	−	0.768662i	$-0.279077\pi$
$678$	0	0
$679$	30.0000	1.15129
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	16.0000i	0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$	16.0000i	0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$	0	0
$685$	4.00000i	0.152832i
$686$	0	0
$687$	9.00000	0.343371
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	− 32.0000i	− 1.21734i	−0.793424	−	0.608669i	$-0.791704\pi$
$691$	− 32.0000i	− 1.21734i	0.793424	−	0.608669i	$-0.208296\pi$
$692$	0	0
$693$	− 12.0000i	− 0.455842i
$694$	0	0
$695$	19.0000	0.720711
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	27.0000i	1.02123i
$700$	0	0
$701$	− 20.0000i	− 0.755390i	−0.925930	−	0.377695i	$-0.876717\pi$
$701$	− 20.0000i	− 0.755390i	0.925930	−	0.377695i	$-0.123283\pi$
$702$	0	0
$703$	−10.0000	−0.377157
$704$	0	0
$705$	−9.00000	−0.338960
$706$	0	0
$707$	12.0000i	0.451306i
$708$	0	0
$709$	− 30.0000i	− 1.12667i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	− 30.0000i	− 1.12667i	0.826227	−	0.563337i	$-0.190483\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	− 2.00000i	− 0.0747958i
$716$	0	0
$717$	− 27.0000i	− 1.00833i
$718$	0	0
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	22.0000i	0.818189i
$724$	0	0
$725$	− 8.00000i	− 0.297113i
$726$	0	0
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	21.0000i	0.776713i
$732$	0	0
$733$	29.0000i	1.07114i	0.844491	+	0.535570i	$0.179903\pi$
$733$	29.0000i	1.07114i	−0.844491	+	0.535570i	$0.820097\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	−20.0000	−0.736709
$738$	0	0
$739$	16.0000i	0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$	16.0000i	0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$	0	0
$741$	2.00000i	0.0734718i
$742$	0	0
$743$	51.0000	1.87101	0.935504	−	0.353315i	$-0.114946\pi$
$743$	51.0000	1.87101	0.935504	+	0.353315i	$0.114946\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	− 8.00000i	− 0.292705i
$748$	0	0
$749$	36.0000i	1.31541i
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	5.00000i	0.181969i
$756$	0	0
$757$	20.0000i	0.726912i	0.931611	+	0.363456i	$0.118403\pi$
$757$	20.0000i	0.726912i	−0.931611	+	0.363456i	$0.881597\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	0	0
$763$	57.0000i	2.06354i
$764$	0	0
$765$	− 6.00000i	− 0.216930i
$766$	0	0
$767$	−6.00000	−0.216647
$768$	0	0
$769$	8.00000	0.288487	0.144244	−	0.989542i	$-0.453925\pi$
$769$	8.00000	0.288487	0.144244	+	0.989542i	$0.453925\pi$
$770$	0	0
$771$	− 7.00000i	− 0.252099i
$772$	0	0
$773$	− 51.0000i	− 1.83434i	−0.398493	−	0.917171i	$-0.630467\pi$
$773$	− 51.0000i	− 1.83434i	0.398493	−	0.917171i	$-0.369533\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	−15.0000	−0.538122
$778$	0	0
$779$	24.0000i	0.859889i
$780$	0	0
$781$	30.0000i	1.07348i
$782$	0	0
$783$	10.0000	0.357371
$784$	0	0
$785$	−22.0000	−0.785214
$786$	0	0
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	− 4.00000i	− 0.141865i
$796$	0	0
$797$	− 54.0000i	− 1.91278i	−0.292096	−	0.956389i	$-0.594353\pi$
$797$	− 54.0000i	− 1.91278i	0.292096	−	0.956389i	$-0.405647\pi$
$798$	0	0
$799$	−27.0000	−0.955191
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	− 4.00000i	− 0.141157i
$804$	0	0
$805$	12.0000i	0.422944i
$806$	0	0
$807$	−4.00000	−0.140807
$808$	0	0
$809$	−47.0000	−1.65243	−0.826216	−	0.563353i	$-0.809511\pi$
$809$	−47.0000	−1.65243	−0.826216	+	0.563353i	$0.809511\pi$
$810$	0	0
$811$	4.00000i	0.140459i	0.997531	+	0.0702295i	$0.0223732\pi$
$811$	4.00000i	0.140459i	−0.997531	+	0.0702295i	$0.977627\pi$
$812$	0	0
$813$	− 15.0000i	− 0.526073i
$814$	0	0
$815$	−20.0000	−0.700569
$816$	0	0
$817$	14.0000	0.489798
$818$	0	0
$819$	− 6.00000i	− 0.209657i
$820$	0	0
$821$	− 15.0000i	− 0.523504i	−0.965135	−	0.261752i	$-0.915700\pi$
$821$	− 15.0000i	− 0.523504i	0.965135	−	0.261752i	$-0.0843002\pi$
$822$	0	0
$823$	−46.0000	−1.60346	−0.801730	−	0.597687i	$-0.796087\pi$
$823$	−46.0000	−1.60346	−0.801730	+	0.597687i	$0.796087\pi$
$824$	0	0
$825$	8.00000	0.278524
$826$	0	0
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	0	0
$829$	− 14.0000i	− 0.486240i	−0.969996	−	0.243120i	$-0.921829\pi$
$829$	− 14.0000i	− 0.486240i	0.969996	−	0.243120i	$-0.0781709\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	− 8.00000i	− 0.276851i
$836$	0	0
$837$	− 20.0000i	− 0.691301i
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	− 1.00000i	− 0.0344010i
$846$	0	0
$847$	21.0000	0.721569
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	20.0000i	0.685591i
$852$	0	0
$853$	− 9.00000i	− 0.308154i	−0.988059	−	0.154077i	$-0.950760\pi$
$853$	− 9.00000i	− 0.308154i	0.988059	−	0.154077i	$-0.0492404\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	0	0
$859$	− 4.00000i	− 0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$	− 4.00000i	− 0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$	0	0
$861$	36.0000i	1.22688i
$862$	0	0
$863$	9.00000	0.306364	0.153182	−	0.988198i	$-0.451048\pi$
$863$	9.00000	0.306364	0.153182	+	0.988198i	$0.451048\pi$
$864$	0	0
$865$	16.0000	0.544016
$866$	0	0
$867$	− 8.00000i	− 0.271694i
$868$	0	0
$869$	− 16.0000i	− 0.542763i
$870$	0	0
$871$	−10.0000	−0.338837
$872$	0	0
$873$	20.0000	0.676897
$874$	0	0
$875$	27.0000i	0.912767i
$876$	0	0
$877$	− 23.0000i	− 0.776655i	−0.921521	−	0.388327i	$-0.873053\pi$
$877$	− 23.0000i	− 0.776655i	0.921521	−	0.388327i	$-0.126947\pi$
$878$	0	0
$879$	7.00000	0.236104
$880$	0	0
$881$	−35.0000	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$881$	−35.0000	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$882$	0	0
$883$	− 11.0000i	− 0.370179i	−0.982722	−	0.185090i	$-0.940742\pi$
$883$	− 11.0000i	− 0.370179i	0.982722	−	0.185090i	$-0.0592576\pi$
$884$	0	0
$885$	6.00000i	0.201688i
$886$	0	0
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	− 2.00000i	− 0.0670025i
$892$	0	0
$893$	18.0000i	0.602347i
$894$	0	0
$895$	−11.0000	−0.367689
$896$	0	0
$897$	4.00000	0.133556
$898$	0	0
$899$	8.00000i	0.266815i
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	21.0000	0.698836
$904$	0	0
$905$	−20.0000	−0.664822
$906$	0	0
$907$	5.00000i	0.166022i	0.996549	+	0.0830111i	$0.0264537\pi$
$907$	5.00000i	0.166022i	−0.996549	+	0.0830111i	$0.973546\pi$
$908$	0	0
$909$	8.00000i	0.265343i
$910$	0	0
$911$	−26.0000	−0.861418	−0.430709	−	0.902491i	$-0.641737\pi$
$911$	−26.0000	−0.861418	−0.430709	+	0.902491i	$0.641737\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	− 4.00000i	− 0.132236i
$916$	0	0
$917$	− 63.0000i	− 2.08044i
$918$	0	0
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	26.0000	0.856729
$922$	0	0
$923$	15.0000i	0.493731i
$924$	0	0
$925$	20.0000i	0.657596i
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	4.00000i	0.131095i
$932$	0	0
$933$	2.00000i	0.0654771i
$934$	0	0
$935$	−6.00000	−0.196221
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	− 23.0000i	− 0.750577i
$940$	0	0
$941$	1.00000i	0.0325991i	0.999867	+	0.0162995i	$0.00518853\pi$
$941$	1.00000i	0.0325991i	−0.999867	+	0.0162995i	$0.994811\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	0	0
$945$	−15.0000	−0.487950
$946$	0	0
$947$	− 46.0000i	− 1.49480i	−0.664375	−	0.747400i	$-0.731302\pi$
$947$	− 46.0000i	− 1.49480i	0.664375	−	0.747400i	$-0.268698\pi$
$948$	0	0
$949$	− 2.00000i	− 0.0649227i
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	33.0000	1.06897	0.534487	−	0.845176i	$-0.320505\pi$
$953$	33.0000	1.06897	0.534487	+	0.845176i	$0.320505\pi$
$954$	0	0
$955$	− 18.0000i	− 0.582466i
$956$	0	0
$957$	− 4.00000i	− 0.129302i
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	24.0000i	0.773389i
$964$	0	0
$965$	12.0000i	0.386294i
$966$	0	0
$967$	−43.0000	−1.38279	−0.691393	−	0.722478i	$-0.743003\pi$
$967$	−43.0000	−1.38279	−0.691393	+	0.722478i	$0.743003\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	51.0000i	1.63667i	0.574743	+	0.818334i	$0.305102\pi$
$971$	51.0000i	1.63667i	−0.574743	+	0.818334i	$0.694898\pi$
$972$	0	0
$973$	− 57.0000i	− 1.82734i
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−14.0000	−0.447900	−0.223950	−	0.974601i	$-0.571895\pi$
$977$	−14.0000	−0.447900	−0.223950	+	0.974601i	$0.571895\pi$
$978$	0	0
$979$	4.00000i	0.127841i
$980$	0	0
$981$	38.0000i	1.21325i
$982$	0	0
$983$	27.0000	0.861166	0.430583	−	0.902551i	$-0.358308\pi$
$983$	27.0000	0.861166	0.430583	+	0.902551i	$0.358308\pi$
$984$	0	0
$985$	−15.0000	−0.477940
$986$	0	0
$987$	27.0000i	0.859419i
$988$	0	0
$989$	− 28.0000i	− 0.890348i
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	0	0
$993$	−16.0000	−0.507745
$994$	0	0
$995$	26.0000i	0.824255i
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	0	0
$999$	−25.0000	−0.790965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.b.r.1665.2		2
4.3	odd	2		3328.2.b.c.1665.1		2
8.3	odd	2		3328.2.b.c.1665.2		2
8.5	even	2	inner	3328.2.b.r.1665.1		2
16.3	odd	4		416.2.a.b.1.1	yes	1
16.5	even	4		832.2.a.g.1.1		1
16.11	odd	4		832.2.a.b.1.1		1
16.13	even	4		416.2.a.a.1.1	✓	1
48.5	odd	4		7488.2.a.bi.1.1		1
48.11	even	4		7488.2.a.bj.1.1		1
48.29	odd	4		3744.2.a.e.1.1		1
48.35	even	4		3744.2.a.f.1.1		1
208.51	odd	4		5408.2.a.i.1.1		1
208.77	even	4		5408.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.a.a.1.1	✓	1	16.13	even	4
416.2.a.b.1.1	yes	1	16.3	odd	4
832.2.a.b.1.1		1	16.11	odd	4
832.2.a.g.1.1		1	16.5	even	4
3328.2.b.c.1665.1		2	4.3	odd	2
3328.2.b.c.1665.2		2	8.3	odd	2
3328.2.b.r.1665.1		2	8.5	even	2	inner
3328.2.b.r.1665.2		2	1.1	even	1	trivial
3744.2.a.e.1.1		1	48.29	odd	4
3744.2.a.f.1.1		1	48.35	even	4
5408.2.a.e.1.1		1	208.77	even	4
5408.2.a.i.1.1		1	208.51	odd	4
7488.2.a.bi.1.1		1	48.5	odd	4
7488.2.a.bj.1.1		1	48.11	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} +3.00000 q^{7} +2.00000 q^{9} -2.00000i q^{11} -1.00000i q^{13} -1.00000 q^{15} -3.00000 q^{17} +2.00000i q^{19} +3.00000i q^{21} +4.00000 q^{23} +4.00000 q^{25} +5.00000i q^{27} -2.00000i q^{29} -4.00000 q^{31} +2.00000 q^{33} +3.00000i q^{35} +5.00000i q^{37} +1.00000 q^{39} +12.0000 q^{41} -7.00000i q^{43} +2.00000i q^{45} +9.00000 q^{47} +2.00000 q^{49} -3.00000i q^{51} +4.00000i q^{53} +2.00000 q^{55} -2.00000 q^{57} -6.00000i q^{59} +4.00000i q^{61} +6.00000 q^{63} +1.00000 q^{65} -10.0000i q^{67} +4.00000i q^{69} -15.0000 q^{71} +2.00000 q^{73} +4.00000i q^{75} -6.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{83} -3.00000i q^{85} +2.00000 q^{87} -2.00000 q^{89} -3.00000i q^{91} -4.00000i q^{93} -2.00000 q^{95} +10.0000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{7} + 4 q^{9} - 2 q^{15} - 6 q^{17} + 8 q^{23} + 8 q^{25} - 8 q^{31} + 4 q^{33} + 2 q^{39} + 24 q^{41} + 18 q^{47} + 4 q^{49} + 4 q^{55} - 4 q^{57} + 12 q^{63} + 2 q^{65} - 30 q^{71} + 4 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) \) 2 * q + 6 * q^7 + 4 * q^9 - 2 * q^15 - 6 * q^17 + 8 * q^23 + 8 * q^25 - 8 * q^31 + 4 * q^33 + 2 * q^39 + 24 * q^41 + 18 * q^47 + 4 * q^49 + 4 * q^55 - 4 * q^57 + 12 * q^63 + 2 * q^65 - 30 * q^71 + 4 * q^73 + 16 * q^79 + 2 * q^81 + 4 * q^87 - 4 * q^89 - 4 * q^95 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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