LMFDB - Embedded newform 4732.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4732,2,Mod(1,4732)]

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, 0]))

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

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

chi := DirichletCharacter("4732.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.a (trivial)

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:	yes
Analytic conductor:	$37.7852102365$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 364)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	4732.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79129 q^{3} -0.791288 q^{5} -1.00000 q^{7} +4.79129 q^{9} -0.791288 q^{11} +2.20871 q^{15} +3.00000 q^{17} +6.37386 q^{19} +2.79129 q^{21} +6.00000 q^{23} -4.37386 q^{25} -5.00000 q^{27} -6.79129 q^{29} +1.00000 q^{31} +2.20871 q^{33} +0.791288 q^{35} +4.00000 q^{37} -7.58258 q^{41} -9.37386 q^{43} -3.79129 q^{45} -6.16515 q^{47} +1.00000 q^{49} -8.37386 q^{51} -1.41742 q^{53} +0.626136 q^{55} -17.7913 q^{57} +9.00000 q^{59} +2.00000 q^{61} -4.79129 q^{63} +7.00000 q^{67} -16.7477 q^{69} +1.58258 q^{71} -14.0000 q^{73} +12.2087 q^{75} +0.791288 q^{77} -4.00000 q^{79} -0.417424 q^{81} +9.00000 q^{83} -2.37386 q^{85} +18.9564 q^{87} -8.37386 q^{89} -2.79129 q^{93} -5.04356 q^{95} +4.62614 q^{97} -3.79129 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 3 q^{5} - 2 q^{7} + 5 q^{9} + 3 q^{11} + 9 q^{15} + 6 q^{17} - q^{19} + q^{21} + 12 q^{23} + 5 q^{25} - 10 q^{27} - 9 q^{29} + 2 q^{31} + 9 q^{33} - 3 q^{35} + 8 q^{37} - 6 q^{41} - 5 q^{43}+ \cdots - 3 q^{99}+O(q^{100}) $ 2 * q - q^3 + 3 * q^5 - 2 * q^7 + 5 * q^9 + 3 * q^11 + 9 * q^15 + 6 * q^17 - q^19 + q^21 + 12 * q^23 + 5 * q^25 - 10 * q^27 - 9 * q^29 + 2 * q^31 + 9 * q^33 - 3 * q^35 + 8 * q^37 - 6 * q^41 - 5 * q^43 - 3 * q^45 + 6 * q^47 + 2 * q^49 - 3 * q^51 - 12 * q^53 + 15 * q^55 - 31 * q^57 + 18 * q^59 + 4 * q^61 - 5 * q^63 + 14 * q^67 - 6 * q^69 - 6 * q^71 - 28 * q^73 + 29 * q^75 - 3 * q^77 - 8 * q^79 - 10 * q^81 + 18 * q^83 + 9 * q^85 + 15 * q^87 - 3 * q^89 - q^93 - 33 * q^95 + 23 * q^97 - 3 * q^99

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
$5$	−0.791288	−0.353875	−0.176937	−	0.984222i	$-0.556619\pi$
$5$	−0.791288	−0.353875	−0.176937	+	0.984222i	$0.556619\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.79129	1.59710
$10$	0	0
$11$	−0.791288	−0.238582	−0.119291	−	0.992859i	$-0.538062\pi$
$11$	−0.791288	−0.238582	−0.119291	+	0.992859i	$0.538062\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.20871	0.570287
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	6.37386	1.46226	0.731132	−	0.682236i	$-0.238992\pi$
$19$	6.37386	1.46226	0.731132	+	0.682236i	$0.238992\pi$
$20$	0	0
$21$	2.79129	0.609109
$22$	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
$25$	−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
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	2.20871	0.384487
$34$	0	0
$35$	0.791288	0.133752
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.58258	−1.18420	−0.592100	−	0.805865i	$-0.701701\pi$
$41$	−7.58258	−1.18420	−0.592100	+	0.805865i	$0.701701\pi$
$42$	0	0
$43$	−9.37386	−1.42950	−0.714750	−	0.699380i	$-0.753460\pi$
$43$	−9.37386	−1.42950	−0.714750	+	0.699380i	$0.753460\pi$
$44$	0	0
$45$	−3.79129	−0.565172
$46$	0	0
$47$	−6.16515	−0.899280	−0.449640	−	0.893210i	$-0.648448\pi$
$47$	−6.16515	−0.899280	−0.449640	+	0.893210i	$0.648448\pi$
$48$	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
$57$	−17.7913	−2.35651
$58$	0	0
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	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
$63$	−4.79129	−0.603646
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	−16.7477	−2.01619
$70$	0	0
$71$	1.58258	0.187817	0.0939086	−	0.995581i	$-0.470064\pi$
$71$	1.58258	0.187817	0.0939086	+	0.995581i	$0.470064\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	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
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	−2.37386	−0.257482
$86$	0	0
$87$	18.9564	2.03234
$88$	0	0
$89$	−8.37386	−0.887628	−0.443814	−	0.896119i	$-0.646375\pi$
$89$	−8.37386	−0.887628	−0.443814	+	0.896119i	$0.646375\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.79129	−0.289443
$94$	0	0
$95$	−5.04356	−0.517458
$96$	0	0
$97$	4.62614	0.469713	0.234856	−	0.972030i	$-0.424538\pi$
$97$	4.62614	0.469713	0.234856	+	0.972030i	$0.424538\pi$
$98$	0	0
$99$	−3.79129	−0.381039
$100$	0	0
$101$	17.5390	1.74520	0.872599	−	0.488438i	$-0.162433\pi$
$101$	17.5390	1.74520	0.872599	+	0.488438i	$0.162433\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	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
$109$	−9.74773	−0.933663	−0.466831	−	0.884346i	$-0.654605\pi$
$109$	−9.74773	−0.933663	−0.466831	+	0.884346i	$0.654605\pi$
$110$	0	0
$111$	−11.1652	−1.05975
$112$	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
$115$	−4.74773	−0.442728
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−10.3739	−0.943079
$122$	0	0
$123$	21.1652	1.90840
$124$	0	0
$125$	7.41742	0.663435
$126$	0	0
$127$	−15.3739	−1.36421	−0.682105	−	0.731254i	$-0.738935\pi$
$127$	−15.3739	−1.36421	−0.682105	+	0.731254i	$0.738935\pi$
$128$	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
$135$	3.95644	0.340516
$136$	0	0
$137$	2.37386	0.202813	0.101406	−	0.994845i	$-0.467666\pi$
$137$	2.37386	0.202813	0.101406	+	0.994845i	$0.467666\pi$
$138$	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
$141$	17.2087	1.44923
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.37386	0.446275
$146$	0	0
$147$	−2.79129	−0.230222
$148$	0	0
$149$	−21.9564	−1.79874	−0.899371	−	0.437187i	$-0.855975\pi$
$149$	−21.9564	−1.79874	−0.899371	+	0.437187i	$0.855975\pi$
$150$	0	0
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	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
$161$	−6.00000	−0.472866
$162$	0	0
$163$	20.7477	1.62509	0.812544	−	0.582900i	$-0.198082\pi$
$163$	20.7477	1.62509	0.812544	+	0.582900i	$0.198082\pi$
$164$	0	0
$165$	−1.74773	−0.136060
$166$	0	0
$167$	18.1652	1.40566	0.702831	−	0.711357i	$-0.251919\pi$
$167$	18.1652	1.40566	0.702831	+	0.711357i	$0.251919\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	30.5390	2.33538
$172$	0	0
$173$	15.3303	1.16554	0.582771	−	0.812637i	$-0.301969\pi$
$173$	15.3303	1.16554	0.582771	+	0.812637i	$0.301969\pi$
$174$	0	0
$175$	4.37386	0.330633
$176$	0	0
$177$	−25.1216	−1.88825
$178$	0	0
$179$	−4.58258	−0.342518	−0.171259	−	0.985226i	$-0.554783\pi$
$179$	−4.58258	−0.342518	−0.171259	+	0.985226i	$0.554783\pi$
$180$	0	0
$181$	−2.74773	−0.204237	−0.102118	−	0.994772i	$-0.532562\pi$
$181$	−2.74773	−0.204237	−0.102118	+	0.994772i	$0.532562\pi$
$182$	0	0
$183$	−5.58258	−0.412676
$184$	0	0
$185$	−3.16515	−0.232707
$186$	0	0
$187$	−2.37386	−0.173594
$188$	0	0
$189$	5.00000	0.363696
$190$	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
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.791288	0.0563769	0.0281885	−	0.999603i	$-0.491026\pi$
$197$	0.791288	0.0563769	0.0281885	+	0.999603i	$0.491026\pi$
$198$	0	0
$199$	15.7477	1.11633	0.558163	−	0.829731i	$-0.311506\pi$
$199$	15.7477	1.11633	0.558163	+	0.829731i	$0.311506\pi$
$200$	0	0
$201$	−19.5390	−1.37818
$202$	0	0
$203$	6.79129	0.476655
$204$	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
$213$	−4.41742	−0.302677
$214$	0	0
$215$	7.41742	0.505864
$216$	0	0
$217$	−1.00000	−0.0678844
$218$	0	0
$219$	39.0780	2.64065
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−8.62614	−0.577649	−0.288824	−	0.957382i	$-0.593264\pi$
$223$	−8.62614	−0.577649	−0.288824	+	0.957382i	$0.593264\pi$
$224$	0	0
$225$	−20.9564	−1.39710
$226$	0	0
$227$	13.7477	0.912469	0.456234	−	0.889860i	$-0.349198\pi$
$227$	13.7477	0.912469	0.456234	+	0.889860i	$0.349198\pi$
$228$	0	0
$229$	26.7477	1.76754	0.883770	−	0.467922i	$-0.154997\pi$
$229$	26.7477	1.76754	0.883770	+	0.467922i	$0.154997\pi$
$230$	0	0
$231$	−2.20871	−0.145323
$232$	0	0
$233$	−27.9564	−1.83149	−0.915744	−	0.401763i	$-0.868398\pi$
$233$	−27.9564	−1.83149	−0.915744	+	0.401763i	$0.868398\pi$
$234$	0	0
$235$	4.87841	0.318232
$236$	0	0
$237$	11.1652	0.725255
$238$	0	0
$239$	24.4955	1.58448	0.792240	−	0.610210i	$-0.208915\pi$
$239$	24.4955	1.58448	0.792240	+	0.610210i	$0.208915\pi$
$240$	0	0
$241$	3.37386	0.217330	0.108665	−	0.994078i	$-0.465342\pi$
$241$	3.37386	0.217330	0.108665	+	0.994078i	$0.465342\pi$
$242$	0	0
$243$	16.1652	1.03699
$244$	0	0
$245$	−0.791288	−0.0505535
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−25.1216	−1.59202
$250$	0	0
$251$	4.41742	0.278825	0.139413	−	0.990234i	$-0.455479\pi$
$251$	4.41742	0.278825	0.139413	+	0.990234i	$0.455479\pi$
$252$	0	0
$253$	−4.74773	−0.298487
$254$	0	0
$255$	6.62614	0.414945
$256$	0	0
$257$	−0.626136	−0.0390573	−0.0195287	−	0.999809i	$-0.506217\pi$
$257$	−0.626136	−0.0390573	−0.0195287	+	0.999809i	$0.506217\pi$
$258$	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
$265$	1.12159	0.0688988
$266$	0	0
$267$	23.3739	1.43046
$268$	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
$271$	−3.74773	−0.227658	−0.113829	−	0.993500i	$-0.536312\pi$
$271$	−3.74773	−0.227658	−0.113829	+	0.993500i	$0.536312\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.46099	0.208705
$276$	0	0
$277$	32.4955	1.95246	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	32.4955	1.95246	0.976231	+	0.216731i	$0.0695395\pi$
$278$	0	0
$279$	4.79129	0.286847
$280$	0	0
$281$	22.7477	1.35702	0.678508	−	0.734593i	$-0.262627\pi$
$281$	22.7477	1.35702	0.678508	+	0.734593i	$0.262627\pi$
$282$	0	0
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	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
$291$	−12.9129	−0.756966
$292$	0	0
$293$	13.5826	0.793503	0.396751	−	0.917926i	$-0.370138\pi$
$293$	13.5826	0.793503	0.396751	+	0.917926i	$0.370138\pi$
$294$	0	0
$295$	−7.12159	−0.414635
$296$	0	0
$297$	3.95644	0.229576
$298$	0	0
$299$	0	0
$300$	0	0
$301$	9.37386	0.540301
$302$	0	0
$303$	−48.9564	−2.81247
$304$	0	0
$305$	−1.58258	−0.0906180
$306$	0	0
$307$	14.1216	0.805962	0.402981	−	0.915208i	$-0.367974\pi$
$307$	14.1216	0.805962	0.402981	+	0.915208i	$0.367974\pi$
$308$	0	0
$309$	2.79129	0.158791
$310$	0	0
$311$	18.9564	1.07492	0.537461	−	0.843289i	$-0.319384\pi$
$311$	18.9564	1.07492	0.537461	+	0.843289i	$0.319384\pi$
$312$	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
$317$	16.5826	0.931370	0.465685	−	0.884950i	$-0.345808\pi$
$317$	16.5826	0.931370	0.465685	+	0.884950i	$0.345808\pi$
$318$	0	0
$319$	5.37386	0.300879
$320$	0	0
$321$	−40.1216	−2.23937
$322$	0	0
$323$	19.1216	1.06395
$324$	0	0
$325$	0	0
$326$	0	0
$327$	27.2087	1.50465
$328$	0	0
$329$	6.16515	0.339896
$330$	0	0
$331$	−27.1216	−1.49074	−0.745369	−	0.666652i	$-0.767727\pi$
$331$	−27.1216	−1.49074	−0.745369	+	0.666652i	$0.767727\pi$
$332$	0	0
$333$	19.1652	1.05024
$334$	0	0
$335$	−5.53901	−0.302629
$336$	0	0
$337$	−6.37386	−0.347206	−0.173603	−	0.984816i	$-0.555541\pi$
$337$	−6.37386	−0.347206	−0.173603	+	0.984816i	$0.555541\pi$
$338$	0	0
$339$	−17.2087	−0.934649
$340$	0	0
$341$	−0.791288	−0.0428506
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	13.2523	0.713478
$346$	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
$349$	29.7477	1.59236	0.796180	−	0.605060i	$-0.206851\pi$
$349$	29.7477	1.59236	0.796180	+	0.605060i	$0.206851\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.7042	0.782624	0.391312	−	0.920258i	$-0.372021\pi$
$353$	14.7042	0.782624	0.391312	+	0.920258i	$0.372021\pi$
$354$	0	0
$355$	−1.25227	−0.0664637
$356$	0	0
$357$	8.37386	0.443192
$358$	0	0
$359$	24.9564	1.31715	0.658575	−	0.752515i	$-0.271159\pi$
$359$	24.9564	1.31715	0.658575	+	0.752515i	$0.271159\pi$
$360$	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
$369$	−36.3303	−1.89128
$370$	0	0
$371$	1.41742	0.0735890
$372$	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
$375$	−20.7042	−1.06916
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−5.62614	−0.288995	−0.144498	−	0.989505i	$-0.546157\pi$
$379$	−5.62614	−0.288995	−0.144498	+	0.989505i	$0.546157\pi$
$380$	0	0
$381$	42.9129	2.19849
$382$	0	0
$383$	38.5390	1.96925	0.984626	−	0.174677i	$-0.0558880\pi$
$383$	38.5390	1.96925	0.984626	+	0.174677i	$0.0558880\pi$
$384$	0	0
$385$	−0.626136	−0.0319109
$386$	0	0
$387$	−44.9129	−2.28305
$388$	0	0
$389$	24.3303	1.23360	0.616798	−	0.787122i	$-0.288430\pi$
$389$	24.3303	1.23360	0.616798	+	0.787122i	$0.288430\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	52.9129	2.66910
$394$	0	0
$395$	3.16515	0.159256
$396$	0	0
$397$	31.4955	1.58071	0.790356	−	0.612648i	$-0.209896\pi$
$397$	31.4955	1.58071	0.790356	+	0.612648i	$0.209896\pi$
$398$	0	0
$399$	17.7913	0.890678
$400$	0	0
$401$	25.1216	1.25451	0.627256	−	0.778813i	$-0.284178\pi$
$401$	25.1216	1.25451	0.627256	+	0.778813i	$0.284178\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.330303	0.0164129
$406$	0	0
$407$	−3.16515	−0.156891
$408$	0	0
$409$	29.1216	1.43997	0.719985	−	0.693990i	$-0.244149\pi$
$409$	29.1216	1.43997	0.719985	+	0.693990i	$0.244149\pi$
$410$	0	0
$411$	−6.62614	−0.326843
$412$	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
$421$	−23.4955	−1.14510	−0.572549	−	0.819870i	$-0.694045\pi$
$421$	−23.4955	−1.14510	−0.572549	+	0.819870i	$0.694045\pi$
$422$	0	0
$423$	−29.5390	−1.43624
$424$	0	0
$425$	−13.1216	−0.636491
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.53901	−0.411310	−0.205655	−	0.978625i	$-0.565932\pi$
$431$	−8.53901	−0.411310	−0.205655	+	0.978625i	$0.565932\pi$
$432$	0	0
$433$	−22.4955	−1.08106	−0.540531	−	0.841324i	$-0.681777\pi$
$433$	−22.4955	−1.08106	−0.540531	+	0.841324i	$0.681777\pi$
$434$	0	0
$435$	−15.0000	−0.719195
$436$	0	0
$437$	38.2432	1.82942
$438$	0	0
$439$	−9.37386	−0.447390	−0.223695	−	0.974659i	$-0.571812\pi$
$439$	−9.37386	−0.447390	−0.223695	+	0.974659i	$0.571812\pi$
$440$	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
$447$	61.2867	2.89876
$448$	0	0
$449$	2.83485	0.133785	0.0668924	−	0.997760i	$-0.478692\pi$
$449$	2.83485	0.133785	0.0668924	+	0.997760i	$0.478692\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	−19.5390	−0.918023
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.4955	−0.818403	−0.409201	−	0.912444i	$-0.634193\pi$
$457$	−17.4955	−0.818403	−0.409201	+	0.912444i	$0.634193\pi$
$458$	0	0
$459$	−15.0000	−0.700140
$460$	0	0
$461$	−24.7913	−1.15465	−0.577323	−	0.816516i	$-0.695903\pi$
$461$	−24.7913	−1.15465	−0.577323	+	0.816516i	$0.695903\pi$
$462$	0	0
$463$	−5.00000	−0.232370	−0.116185	−	0.993228i	$-0.537067\pi$
$463$	−5.00000	−0.232370	−0.116185	+	0.993228i	$0.537067\pi$
$464$	0	0
$465$	2.20871	0.102427
$466$	0	0
$467$	42.1652	1.95117	0.975585	−	0.219621i	$-0.0704821\pi$
$467$	42.1652	1.95117	0.975585	+	0.219621i	$0.0704821\pi$
$468$	0	0
$469$	−7.00000	−0.323230
$470$	0	0
$471$	12.9129	0.594994
$472$	0	0
$473$	7.41742	0.341054
$474$	0	0
$475$	−27.8784	−1.27915
$476$	0	0
$477$	−6.79129	−0.310952
$478$	0	0
$479$	−22.2867	−1.01831	−0.509154	−	0.860676i	$-0.670041\pi$
$479$	−22.2867	−1.01831	−0.509154	+	0.860676i	$0.670041\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	16.7477	0.762048
$484$	0	0
$485$	−3.66061	−0.166220
$486$	0	0
$487$	−18.1216	−0.821168	−0.410584	−	0.911823i	$-0.634675\pi$
$487$	−18.1216	−0.821168	−0.410584	+	0.911823i	$0.634675\pi$
$488$	0	0
$489$	−57.9129	−2.61891
$490$	0	0
$491$	−9.79129	−0.441875	−0.220937	−	0.975288i	$-0.570912\pi$
$491$	−9.79129	−0.441875	−0.220937	+	0.975288i	$0.570912\pi$
$492$	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
$499$	20.1216	0.900766	0.450383	−	0.892835i	$-0.351287\pi$
$499$	20.1216	0.900766	0.450383	+	0.892835i	$0.351287\pi$
$500$	0	0
$501$	−50.7042	−2.26530
$502$	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
$505$	−13.8784	−0.617581
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.74773	−0.0774666	−0.0387333	−	0.999250i	$-0.512332\pi$
$509$	−1.74773	−0.0774666	−0.0387333	+	0.999250i	$0.512332\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	0	0
$513$	−31.8693	−1.40706
$514$	0	0
$515$	0.791288	0.0348683
$516$	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
$525$	−12.2087	−0.532832
$526$	0	0
$527$	3.00000	0.130682
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	43.1216	1.87132
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−11.3739	−0.491735
$536$	0	0
$537$	12.7913	0.551985
$538$	0	0
$539$	−0.791288	−0.0340832
$540$	0	0
$541$	−39.1216	−1.68197	−0.840984	−	0.541060i	$-0.818023\pi$
$541$	−39.1216	−1.68197	−0.840984	+	0.541060i	$0.818023\pi$
$542$	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
$551$	−43.2867	−1.84408
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	8.83485	0.375018
$556$	0	0
$557$	20.2087	0.856271	0.428135	−	0.903715i	$-0.359171\pi$
$557$	20.2087	0.856271	0.428135	+	0.903715i	$0.359171\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	6.62614	0.279756
$562$	0	0
$563$	−39.4955	−1.66453	−0.832267	−	0.554374i	$-0.812958\pi$
$563$	−39.4955	−1.66453	−0.832267	+	0.554374i	$0.812958\pi$
$564$	0	0
$565$	−4.87841	−0.205236
$566$	0	0
$567$	0.417424	0.0175302
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	15.1216	0.632819	0.316409	−	0.948623i	$-0.397523\pi$
$571$	15.1216	0.632819	0.316409	+	0.948623i	$0.397523\pi$
$572$	0	0
$573$	32.2087	1.34554
$574$	0	0
$575$	−26.2432	−1.09442
$576$	0	0
$577$	−26.4955	−1.10302	−0.551510	−	0.834168i	$-0.685948\pi$
$577$	−26.4955	−1.10302	−0.551510	+	0.834168i	$0.685948\pi$
$578$	0	0
$579$	−27.9129	−1.16002
$580$	0	0
$581$	−9.00000	−0.373383
$582$	0	0
$583$	1.12159	0.0464515
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.7042	−0.730729	−0.365365	−	0.930865i	$-0.619056\pi$
$587$	−17.7042	−0.730729	−0.365365	+	0.930865i	$0.619056\pi$
$588$	0	0
$589$	6.37386	0.262630
$590$	0	0
$591$	−2.20871	−0.0908543
$592$	0	0
$593$	6.95644	0.285667	0.142833	−	0.989747i	$-0.454379\pi$
$593$	6.95644	0.285667	0.142833	+	0.989747i	$0.454379\pi$
$594$	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
$603$	33.5390	1.36581
$604$	0	0
$605$	8.20871	0.333732
$606$	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
$609$	−18.9564	−0.768154
$610$	0	0
$611$	0	0
$612$	0	0
$613$	20.1216	0.812703	0.406352	−	0.913717i	$-0.366801\pi$
$613$	20.1216	0.812703	0.406352	+	0.913717i	$0.366801\pi$
$614$	0	0
$615$	−16.7477	−0.675334
$616$	0	0
$617$	−2.66970	−0.107478	−0.0537390	−	0.998555i	$-0.517114\pi$
$617$	−2.66970	−0.107478	−0.0537390	+	0.998555i	$0.517114\pi$
$618$	0	0
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	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
$629$	12.0000	0.478471
$630$	0	0
$631$	29.1216	1.15931	0.579656	−	0.814861i	$-0.303187\pi$
$631$	29.1216	1.15931	0.579656	+	0.814861i	$0.303187\pi$
$632$	0	0
$633$	36.2867	1.44227
$634$	0	0
$635$	12.1652	0.482759
$636$	0	0
$637$	0	0
$638$	0	0
$639$	7.58258	0.299962
$640$	0	0
$641$	40.1216	1.58471	0.792354	−	0.610062i	$-0.208855\pi$
$641$	40.1216	1.58471	0.792354	+	0.610062i	$0.208855\pi$
$642$	0	0
$643$	23.7477	0.936519	0.468259	−	0.883591i	$-0.344881\pi$
$643$	23.7477	0.936519	0.468259	+	0.883591i	$0.344881\pi$
$644$	0	0
$645$	−20.7042	−0.815226
$646$	0	0
$647$	−19.5826	−0.769870	−0.384935	−	0.922944i	$-0.625776\pi$
$647$	−19.5826	−0.769870	−0.384935	+	0.922944i	$0.625776\pi$
$648$	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
$655$	15.0000	0.586098
$656$	0	0
$657$	−67.0780	−2.61696
$658$	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
$661$	−18.7477	−0.729202	−0.364601	−	0.931164i	$-0.618795\pi$
$661$	−18.7477	−0.729202	−0.364601	+	0.931164i	$0.618795\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.04356	0.195581
$666$	0	0
$667$	−40.7477	−1.57776
$668$	0	0
$669$	24.0780	0.930910
$670$	0	0
$671$	−1.58258	−0.0610947
$672$	0	0
$673$	20.4955	0.790042	0.395021	−	0.918672i	$-0.370737\pi$
$673$	20.4955	0.790042	0.395021	+	0.918672i	$0.370737\pi$
$674$	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
$681$	−38.3739	−1.47049
$682$	0	0
$683$	31.7477	1.21479	0.607397	−	0.794399i	$-0.292214\pi$
$683$	31.7477	1.21479	0.607397	+	0.794399i	$0.292214\pi$
$684$	0	0
$685$	−1.87841	−0.0717703
$686$	0	0
$687$	−74.6606	−2.84848
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−5.62614	−0.214028	−0.107014	−	0.994257i	$-0.534129\pi$
$691$	−5.62614	−0.214028	−0.107014	+	0.994257i	$0.534129\pi$
$692$	0	0
$693$	3.79129	0.144019
$694$	0	0
$695$	−10.5826	−0.401420
$696$	0	0
$697$	−22.7477	−0.861632
$698$	0	0
$699$	78.0345	2.95153
$700$	0	0
$701$	−34.7477	−1.31240	−0.656202	−	0.754585i	$-0.727838\pi$
$701$	−34.7477	−1.31240	−0.656202	+	0.754585i	$0.727838\pi$
$702$	0	0
$703$	25.4955	0.961579
$704$	0	0
$705$	−13.6170	−0.512848
$706$	0	0
$707$	−17.5390	−0.659623
$708$	0	0
$709$	25.6261	0.962410	0.481205	−	0.876608i	$-0.340199\pi$
$709$	25.6261	0.962410	0.481205	+	0.876608i	$0.340199\pi$
$710$	0	0
$711$	−19.1652	−0.718749
$712$	0	0
$713$	6.00000	0.224702
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−68.3739	−2.55347
$718$	0	0
$719$	48.8258	1.82089	0.910447	−	0.413626i	$-0.135738\pi$
$719$	48.8258	1.82089	0.910447	+	0.413626i	$0.135738\pi$
$720$	0	0
$721$	1.00000	0.0372419
$722$	0	0
$723$	−9.41742	−0.350238
$724$	0	0
$725$	29.7042	1.10319
$726$	0	0
$727$	−28.4955	−1.05684	−0.528419	−	0.848984i	$-0.677215\pi$
$727$	−28.4955	−1.05684	−0.528419	+	0.848984i	$0.677215\pi$
$728$	0	0
$729$	−43.8693	−1.62479
$730$	0	0
$731$	−28.1216	−1.04011
$732$	0	0
$733$	30.8693	1.14018	0.570092	−	0.821581i	$-0.306907\pi$
$733$	30.8693	1.14018	0.570092	+	0.821581i	$0.306907\pi$
$734$	0	0
$735$	2.20871	0.0814696
$736$	0	0
$737$	−5.53901	−0.204032
$738$	0	0
$739$	−42.7477	−1.57250	−0.786250	−	0.617908i	$-0.787980\pi$
$739$	−42.7477	−1.57250	−0.786250	+	0.617908i	$0.787980\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.7913	1.01956	0.509782	−	0.860304i	$-0.329726\pi$
$743$	27.7913	1.01956	0.509782	+	0.860304i	$0.329726\pi$
$744$	0	0
$745$	17.3739	0.636529
$746$	0	0
$747$	43.1216	1.57774
$748$	0	0
$749$	−14.3739	−0.525210
$750$	0	0
$751$	5.00000	0.182453	0.0912263	−	0.995830i	$-0.470921\pi$
$751$	5.00000	0.182453	0.0912263	+	0.995830i	$0.470921\pi$
$752$	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
$759$	13.2523	0.481027
$760$	0	0
$761$	20.5390	0.744539	0.372269	−	0.928125i	$-0.378580\pi$
$761$	20.5390	0.744539	0.372269	+	0.928125i	$0.378580\pi$
$762$	0	0
$763$	9.74773	0.352891
$764$	0	0
$765$	−11.3739	−0.411223
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−35.0000	−1.26213	−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000	−1.26213	−0.631066	+	0.775729i	$0.717382\pi$
$770$	0	0
$771$	1.74773	0.0629429
$772$	0	0
$773$	42.4955	1.52846	0.764228	−	0.644947i	$-0.223120\pi$
$773$	42.4955	1.52846	0.764228	+	0.644947i	$0.223120\pi$
$774$	0	0
$775$	−4.37386	−0.157114
$776$	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
$785$	3.66061	0.130653
$786$	0	0
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	0	0
$789$	29.5390	1.05162
$790$	0	0
$791$	−6.16515	−0.219208
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−3.13068	−0.111034
$796$	0	0
$797$	49.9129	1.76800	0.884002	−	0.467482i	$-0.154839\pi$
$797$	49.9129	1.76800	0.884002	+	0.467482i	$0.154839\pi$
$798$	0	0
$799$	−18.4955	−0.654322
$800$	0	0
$801$	−40.1216	−1.41763
$802$	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
$811$	−25.8693	−0.908395	−0.454197	−	0.890901i	$-0.650074\pi$
$811$	−25.8693	−0.908395	−0.454197	+	0.890901i	$0.650074\pi$
$812$	0	0
$813$	10.4610	0.366883
$814$	0	0
$815$	−16.4174	−0.575077
$816$	0	0
$817$	−59.7477	−2.09031
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	0	0
$823$	18.2523	0.636234	0.318117	−	0.948051i	$-0.396949\pi$
$823$	18.2523	0.636234	0.318117	+	0.948051i	$0.396949\pi$
$824$	0	0
$825$	−9.66061	−0.336339
$826$	0	0
$827$	−4.12159	−0.143322	−0.0716609	−	0.997429i	$-0.522830\pi$
$827$	−4.12159	−0.143322	−0.0716609	+	0.997429i	$0.522830\pi$
$828$	0	0
$829$	−42.3739	−1.47171	−0.735853	−	0.677142i	$-0.763218\pi$
$829$	−42.3739	−1.47171	−0.735853	+	0.677142i	$0.763218\pi$
$830$	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
$837$	−5.00000	−0.172825
$838$	0	0
$839$	−26.0436	−0.899124	−0.449562	−	0.893249i	$-0.648420\pi$
$839$	−26.0436	−0.899124	−0.449562	+	0.893249i	$0.648420\pi$
$840$	0	0
$841$	17.1216	0.590400
$842$	0	0
$843$	−63.4955	−2.18690
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.3739	0.356450
$848$	0	0
$849$	−72.5735	−2.49072
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	−9.74773	−0.333756	−0.166878	−	0.985978i	$-0.553369\pi$
$853$	−9.74773	−0.333756	−0.166878	+	0.985978i	$0.553369\pi$
$854$	0	0
$855$	−24.1652	−0.826431
$856$	0	0
$857$	20.2087	0.690316	0.345158	−	0.938545i	$-0.387825\pi$
$857$	20.2087	0.690316	0.345158	+	0.938545i	$0.387825\pi$
$858$	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
$863$	2.83485	0.0964994	0.0482497	−	0.998835i	$-0.484636\pi$
$863$	2.83485	0.0964994	0.0482497	+	0.998835i	$0.484636\pi$
$864$	0	0
$865$	−12.1307	−0.412456
$866$	0	0
$867$	22.3303	0.758377
$868$	0	0
$869$	3.16515	0.107370
$870$	0	0
$871$	0	0
$872$	0	0
$873$	22.1652	0.750177
$874$	0	0
$875$	−7.41742	−0.250755
$876$	0	0
$877$	23.7477	0.801904	0.400952	−	0.916099i	$-0.368679\pi$
$877$	23.7477	0.801904	0.400952	+	0.916099i	$0.368679\pi$
$878$	0	0
$879$	−37.9129	−1.27877
$880$	0	0
$881$	−7.41742	−0.249899	−0.124950	−	0.992163i	$-0.539877\pi$
$881$	−7.41742	−0.249899	−0.124950	+	0.992163i	$0.539877\pi$
$882$	0	0
$883$	−29.7477	−1.00109	−0.500545	−	0.865710i	$-0.666867\pi$
$883$	−29.7477	−1.00109	−0.500545	+	0.865710i	$0.666867\pi$
$884$	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
$889$	15.3739	0.515623
$890$	0	0
$891$	0.330303	0.0110656
$892$	0	0
$893$	−39.2958	−1.31498
$894$	0	0
$895$	3.62614	0.121208
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.79129	−0.226502
$900$	0	0
$901$	−4.25227	−0.141664
$902$	0	0
$903$	−26.1652	−0.870722
$904$	0	0
$905$	2.17424	0.0722743
$906$	0	0
$907$	50.9909	1.69313	0.846563	−	0.532289i	$-0.178668\pi$
$907$	50.9909	1.69313	0.846563	+	0.532289i	$0.178668\pi$
$908$	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
$915$	4.41742	0.146036
$916$	0	0
$917$	18.9564	0.625997
$918$	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
$921$	−39.4174	−1.29885
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−17.4955	−0.575247
$926$	0	0
$927$	−4.79129	−0.157367
$928$	0	0
$929$	−46.5826	−1.52832	−0.764162	−	0.645024i	$-0.776847\pi$
$929$	−46.5826	−1.52832	−0.764162	+	0.645024i	$0.776847\pi$
$930$	0	0
$931$	6.37386	0.208895
$932$	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
$941$	−38.0780	−1.24131	−0.620654	−	0.784084i	$-0.713133\pi$
$941$	−38.0780	−1.24131	−0.620654	+	0.784084i	$0.713133\pi$
$942$	0	0
$943$	−45.4955	−1.48154
$944$	0	0
$945$	−3.95644	−0.128703
$946$	0	0
$947$	32.3739	1.05201	0.526005	−	0.850482i	$-0.323689\pi$
$947$	32.3739	1.05201	0.526005	+	0.850482i	$0.323689\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−46.2867	−1.50095
$952$	0	0
$953$	8.66970	0.280839	0.140420	−	0.990092i	$-0.455155\pi$
$953$	8.66970	0.280839	0.140420	+	0.990092i	$0.455155\pi$
$954$	0	0
$955$	9.13068	0.295462
$956$	0	0
$957$	−15.0000	−0.484881
$958$	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
$967$	−3.74773	−0.120519	−0.0602594	−	0.998183i	$-0.519193\pi$
$967$	−3.74773	−0.120519	−0.0602594	+	0.998183i	$0.519193\pi$
$968$	0	0
$969$	−53.3739	−1.71462
$970$	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
$973$	−13.3739	−0.428746
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.8258	1.56207	0.781037	−	0.624485i	$-0.214691\pi$
$977$	48.8258	1.56207	0.781037	+	0.624485i	$0.214691\pi$
$978$	0	0
$979$	6.62614	0.211772
$980$	0	0
$981$	−46.7042	−1.49115
$982$	0	0
$983$	14.5390	0.463723	0.231861	−	0.972749i	$-0.425518\pi$
$983$	14.5390	0.463723	0.231861	+	0.972749i	$0.425518\pi$
$984$	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
$993$	75.7042	2.40240
$994$	0	0
$995$	−12.4610	−0.395040
$996$	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
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.a.h.1.1		2
13.4	even	6		364.2.k.d.29.2	✓	4
13.5	odd	4		4732.2.g.e.337.2		4
13.8	odd	4		4732.2.g.e.337.1		4
13.10	even	6		364.2.k.d.113.2	yes	4
13.12	even	2		4732.2.a.g.1.1		2
39.17	odd	6		3276.2.z.e.757.1		4
39.23	odd	6		3276.2.z.e.3025.1		4
52.23	odd	6		1456.2.s.k.113.1		4
52.43	odd	6		1456.2.s.k.1121.1		4
91.4	even	6		2548.2.i.k.1745.2		4
91.10	odd	6		2548.2.l.l.373.2		4
91.17	odd	6		2548.2.i.i.1745.1		4
91.23	even	6		2548.2.i.k.165.2		4
91.30	even	6		2548.2.l.j.1537.1		4
91.62	odd	6		2548.2.k.e.1569.1		4
91.69	odd	6		2548.2.k.e.393.1		4
91.75	odd	6		2548.2.i.i.165.1		4
91.82	odd	6		2548.2.l.l.1537.2		4
91.88	even	6		2548.2.l.j.373.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.k.d.29.2	✓	4	13.4	even	6
364.2.k.d.113.2	yes	4	13.10	even	6
1456.2.s.k.113.1		4	52.23	odd	6
1456.2.s.k.1121.1		4	52.43	odd	6
2548.2.i.i.165.1		4	91.75	odd	6
2548.2.i.i.1745.1		4	91.17	odd	6
2548.2.i.k.165.2		4	91.23	even	6
2548.2.i.k.1745.2		4	91.4	even	6
2548.2.k.e.393.1		4	91.69	odd	6
2548.2.k.e.1569.1		4	91.62	odd	6
2548.2.l.j.373.1		4	91.88	even	6
2548.2.l.j.1537.1		4	91.30	even	6
2548.2.l.l.373.2		4	91.10	odd	6
2548.2.l.l.1537.2		4	91.82	odd	6
3276.2.z.e.757.1		4	39.17	odd	6
3276.2.z.e.3025.1		4	39.23	odd	6
4732.2.a.g.1.1		2	13.12	even	2
4732.2.a.h.1.1		2	1.1	even	1	trivial
4732.2.g.e.337.1		4	13.8	odd	4
4732.2.g.e.337.2		4	13.5	odd	4

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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.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79129 q^{3} -0.791288 q^{5} -1.00000 q^{7} +4.79129 q^{9} -0.791288 q^{11} +2.20871 q^{15} +3.00000 q^{17} +6.37386 q^{19} +2.79129 q^{21} +6.00000 q^{23} -4.37386 q^{25} -5.00000 q^{27} -6.79129 q^{29} +1.00000 q^{31} +2.20871 q^{33} +0.791288 q^{35} +4.00000 q^{37} -7.58258 q^{41} -9.37386 q^{43} -3.79129 q^{45} -6.16515 q^{47} +1.00000 q^{49} -8.37386 q^{51} -1.41742 q^{53} +0.626136 q^{55} -17.7913 q^{57} +9.00000 q^{59} +2.00000 q^{61} -4.79129 q^{63} +7.00000 q^{67} -16.7477 q^{69} +1.58258 q^{71} -14.0000 q^{73} +12.2087 q^{75} +0.791288 q^{77} -4.00000 q^{79} -0.417424 q^{81} +9.00000 q^{83} -2.37386 q^{85} +18.9564 q^{87} -8.37386 q^{89} -2.79129 q^{93} -5.04356 q^{95} +4.62614 q^{97} -3.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 3 q^{5} - 2 q^{7} + 5 q^{9} + 3 q^{11} + 9 q^{15} + 6 q^{17} - q^{19} + q^{21} + 12 q^{23} + 5 q^{25} - 10 q^{27} - 9 q^{29} + 2 q^{31} + 9 q^{33} - 3 q^{35} + 8 q^{37} - 6 q^{41} - 5 q^{43}+ \cdots - 3 q^{99}+O(q^{100}) \) 2 * q - q^3 + 3 * q^5 - 2 * q^7 + 5 * q^9 + 3 * q^11 + 9 * q^15 + 6 * q^17 - q^19 + q^21 + 12 * q^23 + 5 * q^25 - 10 * q^27 - 9 * q^29 + 2 * q^31 + 9 * q^33 - 3 * q^35 + 8 * q^37 - 6 * q^41 - 5 * q^43 - 3 * q^45 + 6 * q^47 + 2 * q^49 - 3 * q^51 - 12 * q^53 + 15 * q^55 - 31 * q^57 + 18 * q^59 + 4 * q^61 - 5 * q^63 + 14 * q^67 - 6 * q^69 - 6 * q^71 - 28 * q^73 + 29 * q^75 - 3 * q^77 - 8 * q^79 - 10 * q^81 + 18 * q^83 + 9 * q^85 + 15 * q^87 - 3 * q^89 - q^93 - 33 * q^95 + 23 * q^97 - 3 * q^99

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