LMFDB - Embedded newform 4732.2.g.e.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4732,2,Mod(337,4732)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4732, 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("4732.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.g (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:	$37.7852102365$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 25 $ x^4 + 11*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.79129i$ of defining polynomial
Character	$\chi$	$=$	4732.337
Dual form			4732.2.g.e.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79129 q^{3} -0.791288i q^{5} +1.00000i q^{7} +4.79129 q^{9} +0.791288i q^{11} +2.20871i q^{15} -3.00000 q^{17} +6.37386i q^{19} -2.79129i q^{21} -6.00000 q^{23} +4.37386 q^{25} -5.00000 q^{27} -6.79129 q^{29} +1.00000i q^{31} -2.20871i q^{33} +0.791288 q^{35} -4.00000i q^{37} -7.58258i q^{41} +9.37386 q^{43} -3.79129i q^{45} +6.16515i q^{47} -1.00000 q^{49} +8.37386 q^{51} -1.41742 q^{53} +0.626136 q^{55} -17.7913i q^{57} -9.00000i q^{59} +2.00000 q^{61} +4.79129i q^{63} +7.00000i q^{67} +16.7477 q^{69} +1.58258i q^{71} +14.0000i q^{73} -12.2087 q^{75} -0.791288 q^{77} -4.00000 q^{79} -0.417424 q^{81} +9.00000i q^{83} +2.37386i q^{85} +18.9564 q^{87} +8.37386i q^{89} -2.79129i q^{93} +5.04356 q^{95} +4.62614i q^{97} +3.79129i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 10 q^{9} - 12 q^{17} - 24 q^{23} - 10 q^{25} - 20 q^{27} - 18 q^{29} - 6 q^{35} + 10 q^{43} - 4 q^{49} + 6 q^{51} - 24 q^{53} + 30 q^{55} + 8 q^{61} + 12 q^{69} - 58 q^{75} + 6 q^{77} - 16 q^{79}+ \cdots + 66 q^{95}+O(q^{100}) $ 4 * q - 2 * q^3 + 10 * q^9 - 12 * q^17 - 24 * q^23 - 10 * q^25 - 20 * q^27 - 18 * q^29 - 6 * q^35 + 10 * q^43 - 4 * q^49 + 6 * q^51 - 24 * q^53 + 30 * q^55 + 8 * q^61 + 12 * q^69 - 58 * q^75 + 6 * q^77 - 16 * q^79 - 20 * q^81 + 30 * q^87 + 66 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4732\mathbb{Z}\right)^\times$.

$n$	$2367$	$2705$	$4565$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−2.79129	−1.61155	−0.805775	−	0.592221i	$-0.798251\pi$
$3$	−2.79129	−1.61155	−0.805775	+	0.592221i	$0.798251\pi$
$4$	0	0
$5$	− 0.791288i	− 0.353875i	−0.984222	−	0.176937i	$-0.943381\pi$
$5$	− 0.791288i	− 0.353875i	0.984222	−	0.176937i	$-0.0566190\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	4.79129	1.59710
$10$	0	0
$11$	0.791288i	0.238582i	0.992859	+	0.119291i	$0.0380622\pi$
$11$	0.791288i	0.238582i	−0.992859	+	0.119291i	$0.961938\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.20871i	0.570287i
$16$	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
$19$	6.37386i	1.46226i	0.682236	+	0.731132i	$0.261008\pi$
$19$	6.37386i	1.46226i	−0.682236	+	0.731132i	$0.738992\pi$
$20$	0	0
$21$	− 2.79129i	− 0.609109i
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	4.37386	0.874773
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−6.79129	−1.26111	−0.630555	−	0.776144i	$-0.717173\pi$
$29$	−6.79129	−1.26111	−0.630555	+	0.776144i	$0.717173\pi$
$30$	0	0
$31$	1.00000i	0.179605i	0.995960	+	0.0898027i	$0.0286236\pi$
$31$	1.00000i	0.179605i	−0.995960	+	0.0898027i	$0.971376\pi$
$32$	0	0
$33$	− 2.20871i	− 0.384487i
$34$	0	0
$35$	0.791288	0.133752
$36$	0	0
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 7.58258i	− 1.18420i	−0.805865	−	0.592100i	$-0.798299\pi$
$41$	− 7.58258i	− 1.18420i	0.805865	−	0.592100i	$-0.201701\pi$
$42$	0	0
$43$	9.37386	1.42950	0.714750	−	0.699380i	$-0.246540\pi$
$43$	9.37386	1.42950	0.714750	+	0.699380i	$0.246540\pi$
$44$	0	0
$45$	− 3.79129i	− 0.565172i
$46$	0	0
$47$	6.16515i	0.899280i	0.893210	+	0.449640i	$0.148448\pi$
$47$	6.16515i	0.899280i	−0.893210	+	0.449640i	$0.851552\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	8.37386	1.17258
$52$	0	0
$53$	−1.41742	−0.194698	−0.0973491	−	0.995250i	$-0.531036\pi$
$53$	−1.41742	−0.194698	−0.0973491	+	0.995250i	$0.531036\pi$
$54$	0	0
$55$	0.626136	0.0844282
$56$	0	0
$57$	− 17.7913i	− 2.35651i
$58$	0	0
$59$	− 9.00000i	− 1.17170i	−0.810419	−	0.585850i	$-0.800761\pi$
$59$	− 9.00000i	− 1.17170i	0.810419	−	0.585850i	$-0.199239\pi$
$60$	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
$63$	4.79129i	0.603646i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.00000i	0.855186i	0.903971	+	0.427593i	$0.140638\pi$
$67$	7.00000i	0.855186i	−0.903971	+	0.427593i	$0.859362\pi$
$68$	0	0
$69$	16.7477	2.01619
$70$	0	0
$71$	1.58258i	0.187817i	0.995581	+	0.0939086i	$0.0299361\pi$
$71$	1.58258i	0.187817i	−0.995581	+	0.0939086i	$0.970064\pi$
$72$	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
$75$	−12.2087	−1.40974
$76$	0	0
$77$	−0.791288	−0.0901756
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−0.417424	−0.0463805
$82$	0	0
$83$	9.00000i	0.987878i	0.869496	+	0.493939i	$0.164443\pi$
$83$	9.00000i	0.987878i	−0.869496	+	0.493939i	$0.835557\pi$
$84$	0	0
$85$	2.37386i	0.257482i
$86$	0	0
$87$	18.9564	2.03234
$88$	0	0
$89$	8.37386i	0.887628i	0.896119	+	0.443814i	$0.146375\pi$
$89$	8.37386i	0.887628i	−0.896119	+	0.443814i	$0.853625\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 2.79129i	− 0.289443i
$94$	0	0
$95$	5.04356	0.517458
$96$	0	0
$97$	4.62614i	0.469713i	0.972030	+	0.234856i	$0.0754620\pi$
$97$	4.62614i	0.469713i	−0.972030	+	0.234856i	$0.924538\pi$
$98$	0	0
$99$	3.79129i	0.381039i
$100$	0	0
$101$	−17.5390	−1.74520	−0.872599	−	0.488438i	$-0.837567\pi$
$101$	−17.5390	−1.74520	−0.872599	+	0.488438i	$0.837567\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	−2.20871	−0.215548
$106$	0	0
$107$	14.3739	1.38957	0.694787	−	0.719216i	$-0.255499\pi$
$107$	14.3739	1.38957	0.694787	+	0.719216i	$0.255499\pi$
$108$	0	0
$109$	− 9.74773i	− 0.933663i	−0.884346	−	0.466831i	$-0.845395\pi$
$109$	− 9.74773i	− 0.933663i	0.884346	−	0.466831i	$-0.154605\pi$
$110$	0	0
$111$	11.1652i	1.05975i
$112$	0	0
$113$	6.16515	0.579969	0.289984	−	0.957031i	$-0.406350\pi$
$113$	6.16515	0.579969	0.289984	+	0.957031i	$0.406350\pi$
$114$	0	0
$115$	4.74773i	0.442728i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 3.00000i	− 0.275010i
$120$	0	0
$121$	10.3739	0.943079
$122$	0	0
$123$	21.1652i	1.90840i
$124$	0	0
$125$	− 7.41742i	− 0.663435i
$126$	0	0
$127$	15.3739	1.36421	0.682105	−	0.731254i	$-0.261065\pi$
$127$	15.3739	1.36421	0.682105	+	0.731254i	$0.261065\pi$
$128$	0	0
$129$	−26.1652	−2.30371
$130$	0	0
$131$	−18.9564	−1.65623	−0.828116	−	0.560557i	$-0.810587\pi$
$131$	−18.9564	−1.65623	−0.828116	+	0.560557i	$0.810587\pi$
$132$	0	0
$133$	−6.37386	−0.552684
$134$	0	0
$135$	3.95644i	0.340516i
$136$	0	0
$137$	− 2.37386i	− 0.202813i	−0.994845	−	0.101406i	$-0.967666\pi$
$137$	− 2.37386i	− 0.202813i	0.994845	−	0.101406i	$-0.0323342\pi$
$138$	0	0
$139$	13.3739	1.13436	0.567178	−	0.823595i	$-0.308035\pi$
$139$	13.3739	1.13436	0.567178	+	0.823595i	$0.308035\pi$
$140$	0	0
$141$	− 17.2087i	− 1.44923i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.37386i	0.446275i
$146$	0	0
$147$	2.79129	0.230222
$148$	0	0
$149$	− 21.9564i	− 1.79874i	−0.437187	−	0.899371i	$-0.644025\pi$
$149$	− 21.9564i	− 1.79874i	0.437187	−	0.899371i	$-0.355975\pi$
$150$	0	0
$151$	− 7.00000i	− 0.569652i	−0.958579	−	0.284826i	$-0.908064\pi$
$151$	− 7.00000i	− 0.569652i	0.958579	−	0.284826i	$-0.0919358\pi$
$152$	0	0
$153$	−14.3739	−1.16206
$154$	0	0
$155$	0.791288	0.0635578
$156$	0	0
$157$	−4.62614	−0.369206	−0.184603	−	0.982813i	$-0.559100\pi$
$157$	−4.62614	−0.369206	−0.184603	+	0.982813i	$0.559100\pi$
$158$	0	0
$159$	3.95644	0.313766
$160$	0	0
$161$	− 6.00000i	− 0.472866i
$162$	0	0
$163$	− 20.7477i	− 1.62509i	−0.582900	−	0.812544i	$-0.698082\pi$
$163$	− 20.7477i	− 1.62509i	0.582900	−	0.812544i	$-0.301918\pi$
$164$	0	0
$165$	−1.74773	−0.136060
$166$	0	0
$167$	− 18.1652i	− 1.40566i	−0.711357	−	0.702831i	$-0.751919\pi$
$167$	− 18.1652i	− 1.40566i	0.711357	−	0.702831i	$-0.248081\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	30.5390i	2.33538i
$172$	0	0
$173$	−15.3303	−1.16554	−0.582771	−	0.812637i	$-0.698031\pi$
$173$	−15.3303	−1.16554	−0.582771	+	0.812637i	$0.698031\pi$
$174$	0	0
$175$	4.37386i	0.330633i
$176$	0	0
$177$	25.1216i	1.88825i
$178$	0	0
$179$	4.58258	0.342518	0.171259	−	0.985226i	$-0.445217\pi$
$179$	4.58258	0.342518	0.171259	+	0.985226i	$0.445217\pi$
$180$	0	0
$181$	2.74773	0.204237	0.102118	−	0.994772i	$-0.467438\pi$
$181$	2.74773	0.204237	0.102118	+	0.994772i	$0.467438\pi$
$182$	0	0
$183$	−5.58258	−0.412676
$184$	0	0
$185$	−3.16515	−0.232707
$186$	0	0
$187$	− 2.37386i	− 0.173594i
$188$	0	0
$189$	− 5.00000i	− 0.363696i
$190$	0	0
$191$	−11.5390	−0.834934	−0.417467	−	0.908692i	$-0.637082\pi$
$191$	−11.5390	−0.834934	−0.417467	+	0.908692i	$0.637082\pi$
$192$	0	0
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.791288i	0.0563769i	0.999603	+	0.0281885i	$0.00897385\pi$
$197$	0.791288i	0.0563769i	−0.999603	+	0.0281885i	$0.991026\pi$
$198$	0	0
$199$	−15.7477	−1.11633	−0.558163	−	0.829731i	$-0.688494\pi$
$199$	−15.7477	−1.11633	−0.558163	+	0.829731i	$0.688494\pi$
$200$	0	0
$201$	− 19.5390i	− 1.37818i
$202$	0	0
$203$	− 6.79129i	− 0.476655i
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	−28.7477	−1.99811
$208$	0	0
$209$	−5.04356	−0.348870
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	− 4.41742i	− 0.302677i
$214$	0	0
$215$	− 7.41742i	− 0.505864i
$216$	0	0
$217$	−1.00000	−0.0678844
$218$	0	0
$219$	− 39.0780i	− 2.64065i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 8.62614i	− 0.577649i	−0.957382	−	0.288824i	$-0.906736\pi$
$223$	− 8.62614i	− 0.577649i	0.957382	−	0.288824i	$-0.0932644\pi$
$224$	0	0
$225$	20.9564	1.39710
$226$	0	0
$227$	13.7477i	0.912469i	0.889860	+	0.456234i	$0.150802\pi$
$227$	13.7477i	0.912469i	−0.889860	+	0.456234i	$0.849198\pi$
$228$	0	0
$229$	− 26.7477i	− 1.76754i	−0.467922	−	0.883770i	$-0.654997\pi$
$229$	− 26.7477i	− 1.76754i	0.467922	−	0.883770i	$-0.345003\pi$
$230$	0	0
$231$	2.20871	0.145323
$232$	0	0
$233$	27.9564	1.83149	0.915744	−	0.401763i	$-0.131602\pi$
$233$	27.9564	1.83149	0.915744	+	0.401763i	$0.131602\pi$
$234$	0	0
$235$	4.87841	0.318232
$236$	0	0
$237$	11.1652	0.725255
$238$	0	0
$239$	24.4955i	1.58448i	0.610210	+	0.792240i	$0.291085\pi$
$239$	24.4955i	1.58448i	−0.610210	+	0.792240i	$0.708915\pi$
$240$	0	0
$241$	− 3.37386i	− 0.217330i	−0.994078	−	0.108665i	$-0.965342\pi$
$241$	− 3.37386i	− 0.217330i	0.994078	−	0.108665i	$-0.0346575\pi$
$242$	0	0
$243$	16.1652	1.03699
$244$	0	0
$245$	0.791288i	0.0505535i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 25.1216i	− 1.59202i
$250$	0	0
$251$	−4.41742	−0.278825	−0.139413	−	0.990234i	$-0.544521\pi$
$251$	−4.41742	−0.278825	−0.139413	+	0.990234i	$0.544521\pi$
$252$	0	0
$253$	− 4.74773i	− 0.298487i
$254$	0	0
$255$	− 6.62614i	− 0.414945i
$256$	0	0
$257$	0.626136	0.0390573	0.0195287	−	0.999809i	$-0.493783\pi$
$257$	0.626136	0.0390573	0.0195287	+	0.999809i	$0.493783\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	−32.5390	−2.01411
$262$	0	0
$263$	−10.5826	−0.652550	−0.326275	−	0.945275i	$-0.605793\pi$
$263$	−10.5826	−0.652550	−0.326275	+	0.945275i	$0.605793\pi$
$264$	0	0
$265$	1.12159i	0.0688988i
$266$	0	0
$267$	− 23.3739i	− 1.43046i
$268$	0	0
$269$	−23.3739	−1.42513	−0.712565	−	0.701606i	$-0.752467\pi$
$269$	−23.3739	−1.42513	−0.712565	+	0.701606i	$0.752467\pi$
$270$	0	0
$271$	3.74773i	0.227658i	0.993500	+	0.113829i	$0.0363116\pi$
$271$	3.74773i	0.227658i	−0.993500	+	0.113829i	$0.963688\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.46099i	0.208705i
$276$	0	0
$277$	−32.4955	−1.95246	−0.976231	−	0.216731i	$-0.930460\pi$
$277$	−32.4955	−1.95246	−0.976231	+	0.216731i	$0.930460\pi$
$278$	0	0
$279$	4.79129i	0.286847i
$280$	0	0
$281$	− 22.7477i	− 1.35702i	−0.734593	−	0.678508i	$-0.762627\pi$
$281$	− 22.7477i	− 1.35702i	0.734593	−	0.678508i	$-0.237373\pi$
$282$	0	0
$283$	−26.0000	−1.54554	−0.772770	−	0.634686i	$-0.781129\pi$
$283$	−26.0000	−1.54554	−0.772770	+	0.634686i	$0.781129\pi$
$284$	0	0
$285$	−14.0780	−0.833911
$286$	0	0
$287$	7.58258	0.447585
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	− 12.9129i	− 0.756966i
$292$	0	0
$293$	− 13.5826i	− 0.793503i	−0.917926	−	0.396751i	$-0.870138\pi$
$293$	− 13.5826i	− 0.793503i	0.917926	−	0.396751i	$-0.129862\pi$
$294$	0	0
$295$	−7.12159	−0.414635
$296$	0	0
$297$	− 3.95644i	− 0.229576i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	9.37386i	0.540301i
$302$	0	0
$303$	48.9564	2.81247
$304$	0	0
$305$	− 1.58258i	− 0.0906180i
$306$	0	0
$307$	− 14.1216i	− 0.805962i	−0.915208	−	0.402981i	$-0.867974\pi$
$307$	− 14.1216i	− 0.805962i	0.915208	−	0.402981i	$-0.132026\pi$
$308$	0	0
$309$	−2.79129	−0.158791
$310$	0	0
$311$	−18.9564	−1.07492	−0.537461	−	0.843289i	$-0.680616\pi$
$311$	−18.9564	−1.07492	−0.537461	+	0.843289i	$0.680616\pi$
$312$	0	0
$313$	6.74773	0.381404	0.190702	−	0.981648i	$-0.438924\pi$
$313$	6.74773	0.381404	0.190702	+	0.981648i	$0.438924\pi$
$314$	0	0
$315$	3.79129	0.213615
$316$	0	0
$317$	16.5826i	0.931370i	0.884950	+	0.465685i	$0.154192\pi$
$317$	16.5826i	0.931370i	−0.884950	+	0.465685i	$0.845808\pi$
$318$	0	0
$319$	− 5.37386i	− 0.300879i
$320$	0	0
$321$	−40.1216	−2.23937
$322$	0	0
$323$	− 19.1216i	− 1.06395i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	27.2087i	1.50465i
$328$	0	0
$329$	−6.16515	−0.339896
$330$	0	0
$331$	− 27.1216i	− 1.49074i	−0.666652	−	0.745369i	$-0.732273\pi$
$331$	− 27.1216i	− 1.49074i	0.666652	−	0.745369i	$-0.267727\pi$
$332$	0	0
$333$	− 19.1652i	− 1.05024i
$334$	0	0
$335$	5.53901	0.302629
$336$	0	0
$337$	6.37386	0.347206	0.173603	−	0.984816i	$-0.444459\pi$
$337$	6.37386	0.347206	0.173603	+	0.984816i	$0.444459\pi$
$338$	0	0
$339$	−17.2087	−0.934649
$340$	0	0
$341$	−0.791288	−0.0428506
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	− 13.2523i	− 0.713478i
$346$	0	0
$347$	4.74773	0.254871	0.127436	−	0.991847i	$-0.459325\pi$
$347$	4.74773	0.254871	0.127436	+	0.991847i	$0.459325\pi$
$348$	0	0
$349$	− 29.7477i	− 1.59236i	−0.605060	−	0.796180i	$-0.706851\pi$
$349$	− 29.7477i	− 1.59236i	0.605060	−	0.796180i	$-0.293149\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.7042i	0.782624i	0.920258	+	0.391312i	$0.127979\pi$
$353$	14.7042i	0.782624i	−0.920258	+	0.391312i	$0.872021\pi$
$354$	0	0
$355$	1.25227	0.0664637
$356$	0	0
$357$	8.37386i	0.443192i
$358$	0	0
$359$	− 24.9564i	− 1.31715i	−0.752515	−	0.658575i	$-0.771159\pi$
$359$	− 24.9564i	− 1.31715i	0.752515	−	0.658575i	$-0.228841\pi$
$360$	0	0
$361$	−21.6261	−1.13822
$362$	0	0
$363$	−28.9564	−1.51982
$364$	0	0
$365$	11.0780	0.579851
$366$	0	0
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	0	0
$369$	− 36.3303i	− 1.89128i
$370$	0	0
$371$	− 1.41742i	− 0.0735890i
$372$	0	0
$373$	−15.3739	−0.796028	−0.398014	−	0.917379i	$-0.630301\pi$
$373$	−15.3739	−0.796028	−0.398014	+	0.917379i	$0.630301\pi$
$374$	0	0
$375$	20.7042i	1.06916i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 5.62614i	− 0.288995i	−0.989505	−	0.144498i	$-0.953843\pi$
$379$	− 5.62614i	− 0.288995i	0.989505	−	0.144498i	$-0.0461566\pi$
$380$	0	0
$381$	−42.9129	−2.19849
$382$	0	0
$383$	38.5390i	1.96925i	0.174677	+	0.984626i	$0.444112\pi$
$383$	38.5390i	1.96925i	−0.174677	+	0.984626i	$0.555888\pi$
$384$	0	0
$385$	0.626136i	0.0319109i
$386$	0	0
$387$	44.9129	2.28305
$388$	0	0
$389$	−24.3303	−1.23360	−0.616798	−	0.787122i	$-0.711570\pi$
$389$	−24.3303	−1.23360	−0.616798	+	0.787122i	$0.711570\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	52.9129	2.66910
$394$	0	0
$395$	3.16515i	0.159256i
$396$	0	0
$397$	− 31.4955i	− 1.58071i	−0.612648	−	0.790356i	$-0.709896\pi$
$397$	− 31.4955i	− 1.58071i	0.612648	−	0.790356i	$-0.290104\pi$
$398$	0	0
$399$	17.7913	0.890678
$400$	0	0
$401$	− 25.1216i	− 1.25451i	−0.778813	−	0.627256i	$-0.784178\pi$
$401$	− 25.1216i	− 1.25451i	0.778813	−	0.627256i	$-0.215822\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.330303i	0.0164129i
$406$	0	0
$407$	3.16515	0.156891
$408$	0	0
$409$	29.1216i	1.43997i	0.693990	+	0.719985i	$0.255851\pi$
$409$	29.1216i	1.43997i	−0.693990	+	0.719985i	$0.744149\pi$
$410$	0	0
$411$	6.62614i	0.326843i
$412$	0	0
$413$	9.00000	0.442861
$414$	0	0
$415$	7.12159	0.349585
$416$	0	0
$417$	−37.3303	−1.82807
$418$	0	0
$419$	−1.41742	−0.0692457	−0.0346229	−	0.999400i	$-0.511023\pi$
$419$	−1.41742	−0.0692457	−0.0346229	+	0.999400i	$0.511023\pi$
$420$	0	0
$421$	− 23.4955i	− 1.14510i	−0.819870	−	0.572549i	$-0.805955\pi$
$421$	− 23.4955i	− 1.14510i	0.819870	−	0.572549i	$-0.194045\pi$
$422$	0	0
$423$	29.5390i	1.43624i
$424$	0	0
$425$	−13.1216	−0.636491
$426$	0	0
$427$	2.00000i	0.0967868i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 8.53901i	− 0.411310i	−0.978625	−	0.205655i	$-0.934068\pi$
$431$	− 8.53901i	− 0.411310i	0.978625	−	0.205655i	$-0.0659325\pi$
$432$	0	0
$433$	22.4955	1.08106	0.540531	−	0.841324i	$-0.318223\pi$
$433$	22.4955	1.08106	0.540531	+	0.841324i	$0.318223\pi$
$434$	0	0
$435$	− 15.0000i	− 0.719195i
$436$	0	0
$437$	− 38.2432i	− 1.82942i
$438$	0	0
$439$	9.37386	0.447390	0.223695	−	0.974659i	$-0.428188\pi$
$439$	9.37386	0.447390	0.223695	+	0.974659i	$0.428188\pi$
$440$	0	0
$441$	−4.79129	−0.228157
$442$	0	0
$443$	−13.5826	−0.645328	−0.322664	−	0.946514i	$-0.604578\pi$
$443$	−13.5826	−0.645328	−0.322664	+	0.946514i	$0.604578\pi$
$444$	0	0
$445$	6.62614	0.314109
$446$	0	0
$447$	61.2867i	2.89876i
$448$	0	0
$449$	− 2.83485i	− 0.133785i	−0.997760	−	0.0668924i	$-0.978692\pi$
$449$	− 2.83485i	− 0.133785i	0.997760	−	0.0668924i	$-0.0213084\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	19.5390i	0.918023i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 17.4955i	− 0.818403i	−0.912444	−	0.409201i	$-0.865807\pi$
$457$	− 17.4955i	− 0.818403i	0.912444	−	0.409201i	$-0.134193\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	0	0
$461$	− 24.7913i	− 1.15465i	−0.816516	−	0.577323i	$-0.804097\pi$
$461$	− 24.7913i	− 1.15465i	0.816516	−	0.577323i	$-0.195903\pi$
$462$	0	0
$463$	5.00000i	0.232370i	0.993228	+	0.116185i	$0.0370665\pi$
$463$	5.00000i	0.232370i	−0.993228	+	0.116185i	$0.962933\pi$
$464$	0	0
$465$	−2.20871	−0.102427
$466$	0	0
$467$	−42.1652	−1.95117	−0.975585	−	0.219621i	$-0.929518\pi$
$467$	−42.1652	−1.95117	−0.975585	+	0.219621i	$0.929518\pi$
$468$	0	0
$469$	−7.00000	−0.323230
$470$	0	0
$471$	12.9129	0.594994
$472$	0	0
$473$	7.41742i	0.341054i
$474$	0	0
$475$	27.8784i	1.27915i
$476$	0	0
$477$	−6.79129	−0.310952
$478$	0	0
$479$	22.2867i	1.01831i	0.860676	+	0.509154i	$0.170041\pi$
$479$	22.2867i	1.01831i	−0.860676	+	0.509154i	$0.829959\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	16.7477i	0.762048i
$484$	0	0
$485$	3.66061	0.166220
$486$	0	0
$487$	− 18.1216i	− 0.821168i	−0.911823	−	0.410584i	$-0.865325\pi$
$487$	− 18.1216i	− 0.821168i	0.911823	−	0.410584i	$-0.134675\pi$
$488$	0	0
$489$	57.9129i	2.61891i
$490$	0	0
$491$	9.79129	0.441875	0.220937	−	0.975288i	$-0.429088\pi$
$491$	9.79129	0.441875	0.220937	+	0.975288i	$0.429088\pi$
$492$	0	0
$493$	20.3739	0.917593
$494$	0	0
$495$	3.00000	0.134840
$496$	0	0
$497$	−1.58258	−0.0709882
$498$	0	0
$499$	20.1216i	0.900766i	0.892835	+	0.450383i	$0.148713\pi$
$499$	20.1216i	0.900766i	−0.892835	+	0.450383i	$0.851287\pi$
$500$	0	0
$501$	50.7042i	2.26530i
$502$	0	0
$503$	−17.2087	−0.767299	−0.383649	−	0.923479i	$-0.625333\pi$
$503$	−17.2087	−0.767299	−0.383649	+	0.923479i	$0.625333\pi$
$504$	0	0
$505$	13.8784i	0.617581i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1.74773i	− 0.0774666i	−0.999250	−	0.0387333i	$-0.987668\pi$
$509$	− 1.74773i	− 0.0774666i	0.999250	−	0.0387333i	$-0.0123323\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	0	0
$513$	− 31.8693i	− 1.40706i
$514$	0	0
$515$	− 0.791288i	− 0.0348683i
$516$	0	0
$517$	−4.87841	−0.214552
$518$	0	0
$519$	42.7913	1.87833
$520$	0	0
$521$	31.5826	1.38366	0.691829	−	0.722061i	$-0.256805\pi$
$521$	31.5826	1.38366	0.691829	+	0.722061i	$0.256805\pi$
$522$	0	0
$523$	−26.7477	−1.16960	−0.584798	−	0.811179i	$-0.698826\pi$
$523$	−26.7477	−1.16960	−0.584798	+	0.811179i	$0.698826\pi$
$524$	0	0
$525$	− 12.2087i	− 0.532832i
$526$	0	0
$527$	− 3.00000i	− 0.130682i
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	− 43.1216i	− 1.87132i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 11.3739i	− 0.491735i
$536$	0	0
$537$	−12.7913	−0.551985
$538$	0	0
$539$	− 0.791288i	− 0.0340832i
$540$	0	0
$541$	39.1216i	1.68197i	0.541060	+	0.840984i	$0.318023\pi$
$541$	39.1216i	1.68197i	−0.541060	+	0.840984i	$0.681977\pi$
$542$	0	0
$543$	−7.66970	−0.329138
$544$	0	0
$545$	−7.71326	−0.330400
$546$	0	0
$547$	−41.7477	−1.78500	−0.892502	−	0.451044i	$-0.851052\pi$
$547$	−41.7477	−1.78500	−0.892502	+	0.451044i	$0.851052\pi$
$548$	0	0
$549$	9.58258	0.408974
$550$	0	0
$551$	− 43.2867i	− 1.84408i
$552$	0	0
$553$	− 4.00000i	− 0.170097i
$554$	0	0
$555$	8.83485	0.375018
$556$	0	0
$557$	− 20.2087i	− 0.856271i	−0.903715	−	0.428135i	$-0.859171\pi$
$557$	− 20.2087i	− 0.856271i	0.903715	−	0.428135i	$-0.140829\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	6.62614i	0.279756i
$562$	0	0
$563$	39.4955	1.66453	0.832267	−	0.554374i	$-0.187042\pi$
$563$	39.4955	1.66453	0.832267	+	0.554374i	$0.187042\pi$
$564$	0	0
$565$	− 4.87841i	− 0.205236i
$566$	0	0
$567$	− 0.417424i	− 0.0175302i
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	−15.1216	−0.632819	−0.316409	−	0.948623i	$-0.602477\pi$
$571$	−15.1216	−0.632819	−0.316409	+	0.948623i	$0.602477\pi$
$572$	0	0
$573$	32.2087	1.34554
$574$	0	0
$575$	−26.2432	−1.09442
$576$	0	0
$577$	− 26.4955i	− 1.10302i	−0.834168	−	0.551510i	$-0.814052\pi$
$577$	− 26.4955i	− 1.10302i	0.834168	−	0.551510i	$-0.185948\pi$
$578$	0	0
$579$	27.9129i	1.16002i
$580$	0	0
$581$	−9.00000	−0.373383
$582$	0	0
$583$	− 1.12159i	− 0.0464515i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 17.7042i	− 0.730729i	−0.930865	−	0.365365i	$-0.880944\pi$
$587$	− 17.7042i	− 0.730729i	0.930865	−	0.365365i	$-0.119056\pi$
$588$	0	0
$589$	−6.37386	−0.262630
$590$	0	0
$591$	− 2.20871i	− 0.0908543i
$592$	0	0
$593$	− 6.95644i	− 0.285667i	−0.989747	−	0.142833i	$-0.954379\pi$
$593$	− 6.95644i	− 0.285667i	0.989747	−	0.142833i	$-0.0456213\pi$
$594$	0	0
$595$	−2.37386	−0.0973189
$596$	0	0
$597$	43.9564	1.79902
$598$	0	0
$599$	36.9564	1.51000	0.755000	−	0.655725i	$-0.227637\pi$
$599$	36.9564	1.51000	0.755000	+	0.655725i	$0.227637\pi$
$600$	0	0
$601$	−12.3739	−0.504740	−0.252370	−	0.967631i	$-0.581210\pi$
$601$	−12.3739	−0.504740	−0.252370	+	0.967631i	$0.581210\pi$
$602$	0	0
$603$	33.5390i	1.36581i
$604$	0	0
$605$	− 8.20871i	− 0.333732i
$606$	0	0
$607$	21.7477	0.882713	0.441357	−	0.897332i	$-0.354497\pi$
$607$	21.7477	0.882713	0.441357	+	0.897332i	$0.354497\pi$
$608$	0	0
$609$	18.9564i	0.768154i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	20.1216i	0.812703i	0.913717	+	0.406352i	$0.133199\pi$
$613$	20.1216i	0.812703i	−0.913717	+	0.406352i	$0.866801\pi$
$614$	0	0
$615$	16.7477	0.675334
$616$	0	0
$617$	− 2.66970i	− 0.107478i	−0.998555	−	0.0537390i	$-0.982886\pi$
$617$	− 2.66970i	− 0.107478i	0.998555	−	0.0537390i	$-0.0171139\pi$
$618$	0	0
$619$	− 34.0000i	− 1.36658i	−0.730149	−	0.683288i	$-0.760549\pi$
$619$	− 34.0000i	− 1.36658i	0.730149	−	0.683288i	$-0.239451\pi$
$620$	0	0
$621$	30.0000	1.20386
$622$	0	0
$623$	−8.37386	−0.335492
$624$	0	0
$625$	16.0000	0.640000
$626$	0	0
$627$	14.0780	0.562222
$628$	0	0
$629$	12.0000i	0.478471i
$630$	0	0
$631$	− 29.1216i	− 1.15931i	−0.814861	−	0.579656i	$-0.803187\pi$
$631$	− 29.1216i	− 1.15931i	0.814861	−	0.579656i	$-0.196813\pi$
$632$	0	0
$633$	36.2867	1.44227
$634$	0	0
$635$	− 12.1652i	− 0.482759i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	7.58258i	0.299962i
$640$	0	0
$641$	−40.1216	−1.58471	−0.792354	−	0.610062i	$-0.791145\pi$
$641$	−40.1216	−1.58471	−0.792354	+	0.610062i	$0.791145\pi$
$642$	0	0
$643$	23.7477i	0.936519i	0.883591	+	0.468259i	$0.155119\pi$
$643$	23.7477i	0.936519i	−0.883591	+	0.468259i	$0.844881\pi$
$644$	0	0
$645$	20.7042i	0.815226i
$646$	0	0
$647$	19.5826	0.769870	0.384935	−	0.922944i	$-0.374224\pi$
$647$	19.5826	0.769870	0.384935	+	0.922944i	$0.374224\pi$
$648$	0	0
$649$	7.12159	0.279547
$650$	0	0
$651$	2.79129	0.109399
$652$	0	0
$653$	−0.626136	−0.0245026	−0.0122513	−	0.999925i	$-0.503900\pi$
$653$	−0.626136	−0.0245026	−0.0122513	+	0.999925i	$0.503900\pi$
$654$	0	0
$655$	15.0000i	0.586098i
$656$	0	0
$657$	67.0780i	2.61696i
$658$	0	0
$659$	21.1652	0.824477	0.412239	−	0.911076i	$-0.364747\pi$
$659$	21.1652	0.824477	0.412239	+	0.911076i	$0.364747\pi$
$660$	0	0
$661$	18.7477i	0.729202i	0.931164	+	0.364601i	$0.118795\pi$
$661$	18.7477i	0.729202i	−0.931164	+	0.364601i	$0.881205\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.04356i	0.195581i
$666$	0	0
$667$	40.7477	1.57776
$668$	0	0
$669$	24.0780i	0.930910i
$670$	0	0
$671$	1.58258i	0.0610947i
$672$	0	0
$673$	−20.4955	−0.790042	−0.395021	−	0.918672i	$-0.629263\pi$
$673$	−20.4955	−0.790042	−0.395021	+	0.918672i	$0.629263\pi$
$674$	0	0
$675$	−21.8693	−0.841750
$676$	0	0
$677$	14.0436	0.539738	0.269869	−	0.962897i	$-0.413020\pi$
$677$	14.0436	0.539738	0.269869	+	0.962897i	$0.413020\pi$
$678$	0	0
$679$	−4.62614	−0.177535
$680$	0	0
$681$	− 38.3739i	− 1.47049i
$682$	0	0
$683$	− 31.7477i	− 1.21479i	−0.794399	−	0.607397i	$-0.792214\pi$
$683$	− 31.7477i	− 1.21479i	0.794399	−	0.607397i	$-0.207786\pi$
$684$	0	0
$685$	−1.87841	−0.0717703
$686$	0	0
$687$	74.6606i	2.84848i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 5.62614i	− 0.214028i	−0.994257	−	0.107014i	$-0.965871\pi$
$691$	− 5.62614i	− 0.214028i	0.994257	−	0.107014i	$-0.0341290\pi$
$692$	0	0
$693$	−3.79129	−0.144019
$694$	0	0
$695$	− 10.5826i	− 0.401420i
$696$	0	0
$697$	22.7477i	0.861632i
$698$	0	0
$699$	−78.0345	−2.95153
$700$	0	0
$701$	34.7477	1.31240	0.656202	−	0.754585i	$-0.272162\pi$
$701$	34.7477	1.31240	0.656202	+	0.754585i	$0.272162\pi$
$702$	0	0
$703$	25.4955	0.961579
$704$	0	0
$705$	−13.6170	−0.512848
$706$	0	0
$707$	− 17.5390i	− 0.659623i
$708$	0	0
$709$	− 25.6261i	− 0.962410i	−0.876608	−	0.481205i	$-0.840199\pi$
$709$	− 25.6261i	− 0.962410i	0.876608	−	0.481205i	$-0.159801\pi$
$710$	0	0
$711$	−19.1652	−0.718749
$712$	0	0
$713$	− 6.00000i	− 0.224702i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 68.3739i	− 2.55347i
$718$	0	0
$719$	−48.8258	−1.82089	−0.910447	−	0.413626i	$-0.864262\pi$
$719$	−48.8258	−1.82089	−0.910447	+	0.413626i	$0.864262\pi$
$720$	0	0
$721$	1.00000i	0.0372419i
$722$	0	0
$723$	9.41742i	0.350238i
$724$	0	0
$725$	−29.7042	−1.10319
$726$	0	0
$727$	28.4955	1.05684	0.528419	−	0.848984i	$-0.322785\pi$
$727$	28.4955	1.05684	0.528419	+	0.848984i	$0.322785\pi$
$728$	0	0
$729$	−43.8693	−1.62479
$730$	0	0
$731$	−28.1216	−1.04011
$732$	0	0
$733$	30.8693i	1.14018i	0.821581	+	0.570092i	$0.193093\pi$
$733$	30.8693i	1.14018i	−0.821581	+	0.570092i	$0.806907\pi$
$734$	0	0
$735$	− 2.20871i	− 0.0814696i
$736$	0	0
$737$	−5.53901	−0.204032
$738$	0	0
$739$	42.7477i	1.57250i	0.617908	+	0.786250i	$0.287980\pi$
$739$	42.7477i	1.57250i	−0.617908	+	0.786250i	$0.712020\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.7913i	1.01956i	0.860304	+	0.509782i	$0.170274\pi$
$743$	27.7913i	1.01956i	−0.860304	+	0.509782i	$0.829726\pi$
$744$	0	0
$745$	−17.3739	−0.636529
$746$	0	0
$747$	43.1216i	1.57774i
$748$	0	0
$749$	14.3739i	0.525210i
$750$	0	0
$751$	−5.00000	−0.182453	−0.0912263	−	0.995830i	$-0.529079\pi$
$751$	−5.00000	−0.182453	−0.0912263	+	0.995830i	$0.529079\pi$
$752$	0	0
$753$	12.3303	0.449341
$754$	0	0
$755$	−5.53901	−0.201585
$756$	0	0
$757$	−4.00000	−0.145382	−0.0726912	−	0.997354i	$-0.523159\pi$
$757$	−4.00000	−0.145382	−0.0726912	+	0.997354i	$0.523159\pi$
$758$	0	0
$759$	13.2523i	0.481027i
$760$	0	0
$761$	− 20.5390i	− 0.744539i	−0.928125	−	0.372269i	$-0.878580\pi$
$761$	− 20.5390i	− 0.744539i	0.928125	−	0.372269i	$-0.121420\pi$
$762$	0	0
$763$	9.74773	0.352891
$764$	0	0
$765$	11.3739i	0.411223i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 35.0000i	− 1.26213i	−0.775729	−	0.631066i	$-0.782618\pi$
$769$	− 35.0000i	− 1.26213i	0.775729	−	0.631066i	$-0.217382\pi$
$770$	0	0
$771$	−1.74773	−0.0629429
$772$	0	0
$773$	42.4955i	1.52846i	0.644947	+	0.764228i	$0.276880\pi$
$773$	42.4955i	1.52846i	−0.644947	+	0.764228i	$0.723120\pi$
$774$	0	0
$775$	4.37386i	0.157114i
$776$	0	0
$777$	−11.1652	−0.400548
$778$	0	0
$779$	48.3303	1.73161
$780$	0	0
$781$	−1.25227	−0.0448098
$782$	0	0
$783$	33.9564	1.21350
$784$	0	0
$785$	3.66061i	0.130653i
$786$	0	0
$787$	14.0000i	0.499046i	0.968369	+	0.249523i	$0.0802738\pi$
$787$	14.0000i	0.499046i	−0.968369	+	0.249523i	$0.919726\pi$
$788$	0	0
$789$	29.5390	1.05162
$790$	0	0
$791$	6.16515i	0.219208i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 3.13068i	− 0.111034i
$796$	0	0
$797$	−49.9129	−1.76800	−0.884002	−	0.467482i	$-0.845161\pi$
$797$	−49.9129	−1.76800	−0.884002	+	0.467482i	$0.845161\pi$
$798$	0	0
$799$	− 18.4955i	− 0.654322i
$800$	0	0
$801$	40.1216i	1.41763i
$802$	0	0
$803$	−11.0780	−0.390935
$804$	0	0
$805$	−4.74773	−0.167335
$806$	0	0
$807$	65.2432	2.29667
$808$	0	0
$809$	6.16515	0.216755	0.108378	−	0.994110i	$-0.465434\pi$
$809$	6.16515	0.216755	0.108378	+	0.994110i	$0.465434\pi$
$810$	0	0
$811$	− 25.8693i	− 0.908395i	−0.890901	−	0.454197i	$-0.849926\pi$
$811$	− 25.8693i	− 0.908395i	0.890901	−	0.454197i	$-0.150074\pi$
$812$	0	0
$813$	− 10.4610i	− 0.366883i
$814$	0	0
$815$	−16.4174	−0.575077
$816$	0	0
$817$	59.7477i	2.09031i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 33.0000i	− 1.15171i	−0.817553	−	0.575854i	$-0.804670\pi$
$821$	− 33.0000i	− 1.15171i	0.817553	−	0.575854i	$-0.195330\pi$
$822$	0	0
$823$	−18.2523	−0.636234	−0.318117	−	0.948051i	$-0.603051\pi$
$823$	−18.2523	−0.636234	−0.318117	+	0.948051i	$0.603051\pi$
$824$	0	0
$825$	− 9.66061i	− 0.336339i
$826$	0	0
$827$	4.12159i	0.143322i	0.997429	+	0.0716609i	$0.0228299\pi$
$827$	4.12159i	0.143322i	−0.997429	+	0.0716609i	$0.977170\pi$
$828$	0	0
$829$	42.3739	1.47171	0.735853	−	0.677142i	$-0.236782\pi$
$829$	42.3739	1.47171	0.735853	+	0.677142i	$0.236782\pi$
$830$	0	0
$831$	90.7042	3.14649
$832$	0	0
$833$	3.00000	0.103944
$834$	0	0
$835$	−14.3739	−0.497428
$836$	0	0
$837$	− 5.00000i	− 0.172825i
$838$	0	0
$839$	26.0436i	0.899124i	0.893249	+	0.449562i	$0.148420\pi$
$839$	26.0436i	0.899124i	−0.893249	+	0.449562i	$0.851580\pi$
$840$	0	0
$841$	17.1216	0.590400
$842$	0	0
$843$	63.4955i	2.18690i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.3739i	0.356450i
$848$	0	0
$849$	72.5735	2.49072
$850$	0	0
$851$	24.0000i	0.822709i
$852$	0	0
$853$	9.74773i	0.333756i	0.985978	+	0.166878i	$0.0533686\pi$
$853$	9.74773i	0.333756i	−0.985978	+	0.166878i	$0.946631\pi$
$854$	0	0
$855$	24.1652	0.826431
$856$	0	0
$857$	−20.2087	−0.690316	−0.345158	−	0.938545i	$-0.612175\pi$
$857$	−20.2087	−0.690316	−0.345158	+	0.938545i	$0.612175\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	−21.1652	−0.721306
$862$	0	0
$863$	2.83485i	0.0964994i	0.998835	+	0.0482497i	$0.0153643\pi$
$863$	2.83485i	0.0964994i	−0.998835	+	0.0482497i	$0.984636\pi$
$864$	0	0
$865$	12.1307i	0.412456i
$866$	0	0
$867$	22.3303	0.758377
$868$	0	0
$869$	− 3.16515i	− 0.107370i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	22.1652i	0.750177i
$874$	0	0
$875$	7.41742	0.250755
$876$	0	0
$877$	23.7477i	0.801904i	0.916099	+	0.400952i	$0.131321\pi$
$877$	23.7477i	0.801904i	−0.916099	+	0.400952i	$0.868679\pi$
$878$	0	0
$879$	37.9129i	1.27877i
$880$	0	0
$881$	7.41742	0.249899	0.124950	−	0.992163i	$-0.460123\pi$
$881$	7.41742	0.249899	0.124950	+	0.992163i	$0.460123\pi$
$882$	0	0
$883$	29.7477	1.00109	0.500545	−	0.865710i	$-0.333133\pi$
$883$	29.7477	1.00109	0.500545	+	0.865710i	$0.333133\pi$
$884$	0	0
$885$	19.8784	0.668205
$886$	0	0
$887$	−25.2523	−0.847888	−0.423944	−	0.905688i	$-0.639355\pi$
$887$	−25.2523	−0.847888	−0.423944	+	0.905688i	$0.639355\pi$
$888$	0	0
$889$	15.3739i	0.515623i
$890$	0	0
$891$	− 0.330303i	− 0.0110656i
$892$	0	0
$893$	−39.2958	−1.31498
$894$	0	0
$895$	− 3.62614i	− 0.121208i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 6.79129i	− 0.226502i
$900$	0	0
$901$	4.25227	0.141664
$902$	0	0
$903$	− 26.1652i	− 0.870722i
$904$	0	0
$905$	− 2.17424i	− 0.0722743i
$906$	0	0
$907$	−50.9909	−1.69313	−0.846563	−	0.532289i	$-0.821332\pi$
$907$	−50.9909	−1.69313	−0.846563	+	0.532289i	$0.821332\pi$
$908$	0	0
$909$	−84.0345	−2.78725
$910$	0	0
$911$	32.3739	1.07259	0.536297	−	0.844029i	$-0.319823\pi$
$911$	32.3739	1.07259	0.536297	+	0.844029i	$0.319823\pi$
$912$	0	0
$913$	−7.12159	−0.235690
$914$	0	0
$915$	4.41742i	0.146036i
$916$	0	0
$917$	− 18.9564i	− 0.625997i
$918$	0	0
$919$	11.4955	0.379200	0.189600	−	0.981861i	$-0.439281\pi$
$919$	11.4955	0.379200	0.189600	+	0.981861i	$0.439281\pi$
$920$	0	0
$921$	39.4174i	1.29885i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 17.4955i	− 0.575247i
$926$	0	0
$927$	4.79129	0.157367
$928$	0	0
$929$	− 46.5826i	− 1.52832i	−0.645024	−	0.764162i	$-0.723153\pi$
$929$	− 46.5826i	− 1.52832i	0.645024	−	0.764162i	$-0.276847\pi$
$930$	0	0
$931$	− 6.37386i	− 0.208895i
$932$	0	0
$933$	52.9129	1.73229
$934$	0	0
$935$	−1.87841	−0.0614306
$936$	0	0
$937$	−17.7477	−0.579793	−0.289896	−	0.957058i	$-0.593621\pi$
$937$	−17.7477	−0.579793	−0.289896	+	0.957058i	$0.593621\pi$
$938$	0	0
$939$	−18.8348	−0.614652
$940$	0	0
$941$	− 38.0780i	− 1.24131i	−0.784084	−	0.620654i	$-0.786867\pi$
$941$	− 38.0780i	− 1.24131i	0.784084	−	0.620654i	$-0.213133\pi$
$942$	0	0
$943$	45.4955i	1.48154i
$944$	0	0
$945$	−3.95644	−0.128703
$946$	0	0
$947$	− 32.3739i	− 1.05201i	−0.850482	−	0.526005i	$-0.823689\pi$
$947$	− 32.3739i	− 1.05201i	0.850482	−	0.526005i	$-0.176311\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 46.2867i	− 1.50095i
$952$	0	0
$953$	−8.66970	−0.280839	−0.140420	−	0.990092i	$-0.544845\pi$
$953$	−8.66970	−0.280839	−0.140420	+	0.990092i	$0.544845\pi$
$954$	0	0
$955$	9.13068i	0.295462i
$956$	0	0
$957$	15.0000i	0.484881i
$958$	0	0
$959$	2.37386	0.0766561
$960$	0	0
$961$	30.0000	0.967742
$962$	0	0
$963$	68.8693	2.21928
$964$	0	0
$965$	−7.91288	−0.254725
$966$	0	0
$967$	− 3.74773i	− 0.120519i	−0.998183	−	0.0602594i	$-0.980807\pi$
$967$	− 3.74773i	− 0.120519i	0.998183	−	0.0602594i	$-0.0191928\pi$
$968$	0	0
$969$	53.3739i	1.71462i
$970$	0	0
$971$	−31.4174	−1.00823	−0.504117	−	0.863636i	$-0.668182\pi$
$971$	−31.4174	−1.00823	−0.504117	+	0.863636i	$0.668182\pi$
$972$	0	0
$973$	13.3739i	0.428746i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.8258i	1.56207i	0.624485	+	0.781037i	$0.285309\pi$
$977$	48.8258i	1.56207i	−0.624485	+	0.781037i	$0.714691\pi$
$978$	0	0
$979$	−6.62614	−0.211772
$980$	0	0
$981$	− 46.7042i	− 1.49115i
$982$	0	0
$983$	− 14.5390i	− 0.463723i	−0.972749	−	0.231861i	$-0.925518\pi$
$983$	− 14.5390i	− 0.463723i	0.972749	−	0.231861i	$-0.0744815\pi$
$984$	0	0
$985$	0.626136	0.0199504
$986$	0	0
$987$	17.2087	0.547759
$988$	0	0
$989$	−56.2432	−1.78843
$990$	0	0
$991$	23.6261	0.750509	0.375254	−	0.926922i	$-0.377555\pi$
$991$	23.6261	0.750509	0.375254	+	0.926922i	$0.377555\pi$
$992$	0	0
$993$	75.7042i	2.40240i
$994$	0	0
$995$	12.4610i	0.395040i
$996$	0	0
$997$	35.0000	1.10846	0.554231	−	0.832363i	$-0.313013\pi$
$997$	35.0000	1.10846	0.554231	+	0.832363i	$0.313013\pi$
$998$	0	0
$999$	20.0000i	0.632772i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.g.e.337.1		4
13.5	odd	4		4732.2.a.h.1.1		2
13.7	odd	12		364.2.k.d.29.2	✓	4
13.8	odd	4		4732.2.a.g.1.1		2
13.11	odd	12		364.2.k.d.113.2	yes	4
13.12	even	2	inner	4732.2.g.e.337.2		4
39.11	even	12		3276.2.z.e.3025.1		4
39.20	even	12		3276.2.z.e.757.1		4
52.7	even	12		1456.2.s.k.1121.1		4
52.11	even	12		1456.2.s.k.113.1		4
91.11	odd	12		2548.2.l.j.373.1		4
91.20	even	12		2548.2.k.e.393.1		4
91.24	even	12		2548.2.l.l.373.2		4
91.33	even	12		2548.2.l.l.1537.2		4
91.37	odd	12		2548.2.i.k.165.2		4
91.46	odd	12		2548.2.i.k.1745.2		4
91.59	even	12		2548.2.i.i.1745.1		4
91.72	odd	12		2548.2.l.j.1537.1		4
91.76	even	12		2548.2.k.e.1569.1		4
91.89	even	12		2548.2.i.i.165.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.k.d.29.2	✓	4	13.7	odd	12
364.2.k.d.113.2	yes	4	13.11	odd	12
1456.2.s.k.113.1		4	52.11	even	12
1456.2.s.k.1121.1		4	52.7	even	12
2548.2.i.i.165.1		4	91.89	even	12
2548.2.i.i.1745.1		4	91.59	even	12
2548.2.i.k.165.2		4	91.37	odd	12
2548.2.i.k.1745.2		4	91.46	odd	12
2548.2.k.e.393.1		4	91.20	even	12
2548.2.k.e.1569.1		4	91.76	even	12
2548.2.l.j.373.1		4	91.11	odd	12
2548.2.l.j.1537.1		4	91.72	odd	12
2548.2.l.l.373.2		4	91.24	even	12
2548.2.l.l.1537.2		4	91.33	even	12
3276.2.z.e.757.1		4	39.20	even	12
3276.2.z.e.3025.1		4	39.11	even	12
4732.2.a.g.1.1		2	13.8	odd	4
4732.2.a.h.1.1		2	13.5	odd	4
4732.2.g.e.337.1		4	1.1	even	1	trivial
4732.2.g.e.337.2		4	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.79129 q^{3} -0.791288i q^{5} +1.00000i q^{7} +4.79129 q^{9} +0.791288i q^{11} +2.20871i q^{15} -3.00000 q^{17} +6.37386i q^{19} -2.79129i q^{21} -6.00000 q^{23} +4.37386 q^{25} -5.00000 q^{27} -6.79129 q^{29} +1.00000i q^{31} -2.20871i q^{33} +0.791288 q^{35} -4.00000i q^{37} -7.58258i q^{41} +9.37386 q^{43} -3.79129i q^{45} +6.16515i q^{47} -1.00000 q^{49} +8.37386 q^{51} -1.41742 q^{53} +0.626136 q^{55} -17.7913i q^{57} -9.00000i q^{59} +2.00000 q^{61} +4.79129i q^{63} +7.00000i q^{67} +16.7477 q^{69} +1.58258i q^{71} +14.0000i q^{73} -12.2087 q^{75} -0.791288 q^{77} -4.00000 q^{79} -0.417424 q^{81} +9.00000i q^{83} +2.37386i q^{85} +18.9564 q^{87} +8.37386i q^{89} -2.79129i q^{93} +5.04356 q^{95} +4.62614i q^{97} +3.79129i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 10 q^{9} - 12 q^{17} - 24 q^{23} - 10 q^{25} - 20 q^{27} - 18 q^{29} - 6 q^{35} + 10 q^{43} - 4 q^{49} + 6 q^{51} - 24 q^{53} + 30 q^{55} + 8 q^{61} + 12 q^{69} - 58 q^{75} + 6 q^{77} - 16 q^{79}+ \cdots + 66 q^{95}+O(q^{100}) \) 4 * q - 2 * q^3 + 10 * q^9 - 12 * q^17 - 24 * q^23 - 10 * q^25 - 20 * q^27 - 18 * q^29 - 6 * q^35 + 10 * q^43 - 4 * q^49 + 6 * q^51 - 24 * q^53 + 30 * q^55 + 8 * q^61 + 12 * q^69 - 58 * q^75 + 6 * q^77 - 16 * q^79 - 20 * q^81 + 30 * q^87 + 66 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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