LMFDB - Embedded newform 5148.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5148,2,Mod(1,5148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5148.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:	$41.1069869606$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	5148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.41421 q^{5} +2.82843 q^{7} +O(q^{10})$	$q+3.41421 q^{5} +2.82843 q^{7} +1.00000 q^{11} -1.00000 q^{13} +6.24264 q^{17} +4.82843 q^{23} +6.65685 q^{25} +0.585786 q^{29} -5.41421 q^{31} +9.65685 q^{35} +0.828427 q^{37} -3.65685 q^{41} -0.242641 q^{43} +8.00000 q^{47} +1.00000 q^{49} -9.31371 q^{53} +3.41421 q^{55} -2.82843 q^{59} -3.17157 q^{61} -3.41421 q^{65} +4.24264 q^{67} +10.8284 q^{71} -5.65685 q^{73} +2.82843 q^{77} -7.07107 q^{79} +17.6569 q^{83} +21.3137 q^{85} +11.8995 q^{89} -2.82843 q^{91} -12.8284 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5}+O(q^{10}) $ 2 * q + 4 * q^5	$ 2 q + 4 q^{5} + 2 q^{11} - 2 q^{13} + 4 q^{17} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 8 q^{31} + 8 q^{35} - 4 q^{37} + 4 q^{41} + 8 q^{43} + 16 q^{47} + 2 q^{49} + 4 q^{53} + 4 q^{55} - 12 q^{61} - 4 q^{65} + 16 q^{71} + 24 q^{83} + 20 q^{85} + 4 q^{89} - 20 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 + 2 * q^11 - 2 * q^13 + 4 * q^17 + 4 * q^23 + 2 * q^25 + 4 * q^29 - 8 * q^31 + 8 * q^35 - 4 * q^37 + 4 * q^41 + 8 * q^43 + 16 * q^47 + 2 * q^49 + 4 * q^53 + 4 * q^55 - 12 * q^61 - 4 * q^65 + 16 * q^71 + 24 * q^83 + 20 * q^85 + 4 * q^89 - 20 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.41421	1.52688	0.763441	−	0.645877i	$-0.223508\pi$
$5$	3.41421	1.52688	0.763441	+	0.645877i	$0.223508\pi$
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.24264	1.51406	0.757031	−	0.653379i	$-0.226649\pi$
$17$	6.24264	1.51406	0.757031	+	0.653379i	$0.226649\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.82843	1.00680	0.503398	−	0.864054i	$-0.332083\pi$
$23$	4.82843	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.585786	0.108778	0.0543889	−	0.998520i	$-0.482679\pi$
$29$	0.585786	0.108778	0.0543889	+	0.998520i	$0.482679\pi$
$30$	0	0
$31$	−5.41421	−0.972421	−0.486211	−	0.873842i	$-0.661621\pi$
$31$	−5.41421	−0.972421	−0.486211	+	0.873842i	$0.661621\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.65685	1.63231
$36$	0	0
$37$	0.828427	0.136193	0.0680963	−	0.997679i	$-0.478307\pi$
$37$	0.828427	0.136193	0.0680963	+	0.997679i	$0.478307\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	0	0
$43$	−0.242641	−0.0370024	−0.0185012	−	0.999829i	$-0.505889\pi$
$43$	−0.242641	−0.0370024	−0.0185012	+	0.999829i	$0.505889\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.31371	−1.27934	−0.639668	−	0.768651i	$-0.720928\pi$
$53$	−9.31371	−1.27934	−0.639668	+	0.768651i	$0.720928\pi$
$54$	0	0
$55$	3.41421	0.460372
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.82843	−0.368230	−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843	−0.368230	−0.184115	+	0.982905i	$0.558942\pi$
$60$	0	0
$61$	−3.17157	−0.406078	−0.203039	−	0.979171i	$-0.565082\pi$
$61$	−3.17157	−0.406078	−0.203039	+	0.979171i	$0.565082\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.41421	−0.423481
$66$	0	0
$67$	4.24264	0.518321	0.259161	−	0.965834i	$-0.416554\pi$
$67$	4.24264	0.518321	0.259161	+	0.965834i	$0.416554\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.8284	1.28510	0.642549	−	0.766245i	$-0.277877\pi$
$71$	10.8284	1.28510	0.642549	+	0.766245i	$0.277877\pi$
$72$	0	0
$73$	−5.65685	−0.662085	−0.331042	−	0.943616i	$-0.607400\pi$
$73$	−5.65685	−0.662085	−0.331042	+	0.943616i	$0.607400\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.82843	0.322329
$78$	0	0
$79$	−7.07107	−0.795557	−0.397779	−	0.917481i	$-0.630219\pi$
$79$	−7.07107	−0.795557	−0.397779	+	0.917481i	$0.630219\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	17.6569	1.93809	0.969046	−	0.246881i	$-0.0794057\pi$
$83$	17.6569	1.93809	0.969046	+	0.246881i	$0.0794057\pi$
$84$	0	0
$85$	21.3137	2.31180
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.8995	1.26134	0.630672	−	0.776050i	$-0.282779\pi$
$89$	11.8995	1.26134	0.630672	+	0.776050i	$0.282779\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.8284	−1.30253	−0.651265	−	0.758851i	$-0.725761\pi$
$97$	−12.8284	−1.30253	−0.651265	+	0.758851i	$0.725761\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.7279	1.86350	0.931749	−	0.363103i	$-0.118283\pi$
$101$	18.7279	1.86350	0.931749	+	0.363103i	$0.118283\pi$
$102$	0	0
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.17157	−0.113260	−0.0566301	−	0.998395i	$-0.518036\pi$
$107$	−1.17157	−0.113260	−0.0566301	+	0.998395i	$0.518036\pi$
$108$	0	0
$109$	−15.3137	−1.46679	−0.733394	−	0.679804i	$-0.762065\pi$
$109$	−15.3137	−1.46679	−0.733394	+	0.679804i	$0.762065\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−18.9706	−1.78460	−0.892300	−	0.451442i	$-0.850910\pi$
$113$	−18.9706	−1.78460	−0.892300	+	0.451442i	$0.850910\pi$
$114$	0	0
$115$	16.4853	1.53726
$116$	0	0
$117$	0	0
$118$	0	0
$119$	17.6569	1.61860
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−7.07107	−0.627456	−0.313728	−	0.949513i	$-0.601578\pi$
$127$	−7.07107	−0.627456	−0.313728	+	0.949513i	$0.601578\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.0711	−1.11674	−0.558368	−	0.829593i	$-0.688572\pi$
$137$	−13.0711	−1.11674	−0.558368	+	0.829593i	$0.688572\pi$
$138$	0	0
$139$	8.24264	0.699132	0.349566	−	0.936912i	$-0.386329\pi$
$139$	8.24264	0.699132	0.349566	+	0.936912i	$0.386329\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.48528	−0.203602	−0.101801	−	0.994805i	$-0.532461\pi$
$149$	−2.48528	−0.203602	−0.101801	+	0.994805i	$0.532461\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−18.4853	−1.48477
$156$	0	0
$157$	6.34315	0.506238	0.253119	−	0.967435i	$-0.418544\pi$
$157$	6.34315	0.506238	0.253119	+	0.967435i	$0.418544\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	13.6569	1.07631
$162$	0	0
$163$	−12.7279	−0.996928	−0.498464	−	0.866910i	$-0.666102\pi$
$163$	−12.7279	−0.996928	−0.498464	+	0.866910i	$0.666102\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.3137	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.8995	−1.51293	−0.756465	−	0.654034i	$-0.773075\pi$
$173$	−19.8995	−1.51293	−0.756465	+	0.654034i	$0.773075\pi$
$174$	0	0
$175$	18.8284	1.42330
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.82843	−0.360894	−0.180447	−	0.983585i	$-0.557754\pi$
$179$	−4.82843	−0.360894	−0.180447	+	0.983585i	$0.557754\pi$
$180$	0	0
$181$	24.9706	1.85605	0.928024	−	0.372521i	$-0.121507\pi$
$181$	24.9706	1.85605	0.928024	+	0.372521i	$0.121507\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.82843	0.207950
$186$	0	0
$187$	6.24264	0.456507
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.34315	0.169544	0.0847720	−	0.996400i	$-0.472984\pi$
$191$	2.34315	0.169544	0.0847720	+	0.996400i	$0.472984\pi$
$192$	0	0
$193$	−5.65685	−0.407189	−0.203595	−	0.979055i	$-0.565262\pi$
$193$	−5.65685	−0.407189	−0.203595	+	0.979055i	$0.565262\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.1421	0.865091	0.432546	−	0.901612i	$-0.357615\pi$
$197$	12.1421	0.865091	0.432546	+	0.901612i	$0.357615\pi$
$198$	0	0
$199$	−4.48528	−0.317953	−0.158977	−	0.987282i	$-0.550819\pi$
$199$	−4.48528	−0.317953	−0.158977	+	0.987282i	$0.550819\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.65685	0.116288
$204$	0	0
$205$	−12.4853	−0.872010
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	10.5858	0.728756	0.364378	−	0.931251i	$-0.381282\pi$
$211$	10.5858	0.728756	0.364378	+	0.931251i	$0.381282\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.828427	−0.0564983
$216$	0	0
$217$	−15.3137	−1.03956
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.24264	−0.419925
$222$	0	0
$223$	−21.8995	−1.46650	−0.733249	−	0.679960i	$-0.761997\pi$
$223$	−21.8995	−1.46650	−0.733249	+	0.679960i	$0.761997\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.1421	−0.805902	−0.402951	−	0.915222i	$-0.632015\pi$
$227$	−12.1421	−0.805902	−0.402951	+	0.915222i	$0.632015\pi$
$228$	0	0
$229$	1.31371	0.0868123	0.0434062	−	0.999058i	$-0.486179\pi$
$229$	1.31371	0.0868123	0.0434062	+	0.999058i	$0.486179\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.58579	−0.562474	−0.281237	−	0.959638i	$-0.590745\pi$
$233$	−8.58579	−0.562474	−0.281237	+	0.959638i	$0.590745\pi$
$234$	0	0
$235$	27.3137	1.78175
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.1421	−1.30289	−0.651443	−	0.758697i	$-0.725836\pi$
$239$	−20.1421	−1.30289	−0.651443	+	0.758697i	$0.725836\pi$
$240$	0	0
$241$	−0.343146	−0.0221040	−0.0110520	−	0.999939i	$-0.503518\pi$
$241$	−0.343146	−0.0221040	−0.0110520	+	0.999939i	$0.503518\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.41421	0.218126
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−25.6569	−1.61945	−0.809723	−	0.586812i	$-0.800383\pi$
$251$	−25.6569	−1.61945	−0.809723	+	0.586812i	$0.800383\pi$
$252$	0	0
$253$	4.82843	0.303561
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.828427	0.0516759	0.0258379	−	0.999666i	$-0.491775\pi$
$257$	0.828427	0.0516759	0.0258379	+	0.999666i	$0.491775\pi$
$258$	0	0
$259$	2.34315	0.145596
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.48528	−0.523225	−0.261612	−	0.965173i	$-0.584254\pi$
$263$	−8.48528	−0.523225	−0.261612	+	0.965173i	$0.584254\pi$
$264$	0	0
$265$	−31.7990	−1.95340
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.82843	0.538279	0.269139	−	0.963101i	$-0.413261\pi$
$269$	8.82843	0.538279	0.269139	+	0.963101i	$0.413261\pi$
$270$	0	0
$271$	1.65685	0.100647	0.0503234	−	0.998733i	$-0.483975\pi$
$271$	1.65685	0.100647	0.0503234	+	0.998733i	$0.483975\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.65685	0.401423
$276$	0	0
$277$	14.9706	0.899494	0.449747	−	0.893156i	$-0.351514\pi$
$277$	14.9706	0.899494	0.449747	+	0.893156i	$0.351514\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	23.6569	1.41125	0.705625	−	0.708586i	$-0.250666\pi$
$281$	23.6569	1.41125	0.705625	+	0.708586i	$0.250666\pi$
$282$	0	0
$283$	27.0711	1.60921	0.804604	−	0.593812i	$-0.202378\pi$
$283$	27.0711	1.60921	0.804604	+	0.593812i	$0.202378\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.3431	−0.610537
$288$	0	0
$289$	21.9706	1.29239
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.51472	0.0884908	0.0442454	−	0.999021i	$-0.485912\pi$
$293$	1.51472	0.0884908	0.0442454	+	0.999021i	$0.485912\pi$
$294$	0	0
$295$	−9.65685	−0.562244
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.82843	−0.279235
$300$	0	0
$301$	−0.686292	−0.0395572
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−10.8284	−0.620034
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.8284	1.40789	0.703945	−	0.710254i	$-0.251420\pi$
$311$	24.8284	1.40789	0.703945	+	0.710254i	$0.251420\pi$
$312$	0	0
$313$	−22.6274	−1.27898	−0.639489	−	0.768801i	$-0.720854\pi$
$313$	−22.6274	−1.27898	−0.639489	+	0.768801i	$0.720854\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.92893	0.389168	0.194584	−	0.980886i	$-0.437664\pi$
$317$	6.92893	0.389168	0.194584	+	0.980886i	$0.437664\pi$
$318$	0	0
$319$	0.585786	0.0327977
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−6.65685	−0.369256
$326$	0	0
$327$	0	0
$328$	0	0
$329$	22.6274	1.24749
$330$	0	0
$331$	31.5563	1.73449	0.867247	−	0.497878i	$-0.165887\pi$
$331$	31.5563	1.73449	0.867247	+	0.497878i	$0.165887\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14.4853	0.791415
$336$	0	0
$337$	9.51472	0.518300	0.259150	−	0.965837i	$-0.416558\pi$
$337$	9.51472	0.518300	0.259150	+	0.965837i	$0.416558\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.41421	−0.293196
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.6569	1.37733	0.688666	−	0.725079i	$-0.258197\pi$
$347$	25.6569	1.37733	0.688666	+	0.725079i	$0.258197\pi$
$348$	0	0
$349$	−35.9411	−1.92388	−0.961942	−	0.273253i	$-0.911900\pi$
$349$	−35.9411	−1.92388	−0.961942	+	0.273253i	$0.911900\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.7279	−0.783888	−0.391944	−	0.919989i	$-0.628197\pi$
$353$	−14.7279	−0.783888	−0.391944	+	0.919989i	$0.628197\pi$
$354$	0	0
$355$	36.9706	1.96219
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.1421	0.851949	0.425975	−	0.904735i	$-0.359931\pi$
$359$	16.1421	0.851949	0.425975	+	0.904735i	$0.359931\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−19.3137	−1.01093
$366$	0	0
$367$	8.68629	0.453421	0.226710	−	0.973962i	$-0.427203\pi$
$367$	8.68629	0.453421	0.226710	+	0.973962i	$0.427203\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−26.3431	−1.36767
$372$	0	0
$373$	19.4558	1.00739	0.503693	−	0.863883i	$-0.331974\pi$
$373$	19.4558	1.00739	0.503693	+	0.863883i	$0.331974\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.585786	−0.0301695
$378$	0	0
$379$	29.6985	1.52551	0.762754	−	0.646688i	$-0.223847\pi$
$379$	29.6985	1.52551	0.762754	+	0.646688i	$0.223847\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.7990	1.01168	0.505841	−	0.862627i	$-0.331182\pi$
$383$	19.7990	1.01168	0.505841	+	0.862627i	$0.331182\pi$
$384$	0	0
$385$	9.65685	0.492159
$386$	0	0
$387$	0	0
$388$	0	0
$389$	17.7990	0.902445	0.451222	−	0.892412i	$-0.350988\pi$
$389$	17.7990	0.902445	0.451222	+	0.892412i	$0.350988\pi$
$390$	0	0
$391$	30.1421	1.52435
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−24.1421	−1.21472
$396$	0	0
$397$	−37.7990	−1.89708	−0.948538	−	0.316662i	$-0.897438\pi$
$397$	−37.7990	−1.89708	−0.948538	+	0.316662i	$0.897438\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	19.2132	0.959462	0.479731	−	0.877416i	$-0.340734\pi$
$401$	19.2132	0.959462	0.479731	+	0.877416i	$0.340734\pi$
$402$	0	0
$403$	5.41421	0.269701
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.828427	0.0410636
$408$	0	0
$409$	2.34315	0.115861	0.0579306	−	0.998321i	$-0.481550\pi$
$409$	2.34315	0.115861	0.0579306	+	0.998321i	$0.481550\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	60.2843	2.95924
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−28.8284	−1.40836	−0.704180	−	0.710021i	$-0.748685\pi$
$419$	−28.8284	−1.40836	−0.704180	+	0.710021i	$0.748685\pi$
$420$	0	0
$421$	−37.3137	−1.81856	−0.909279	−	0.416186i	$-0.863366\pi$
$421$	−37.3137	−1.81856	−0.909279	+	0.416186i	$0.863366\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	41.5563	2.01578
$426$	0	0
$427$	−8.97056	−0.434116
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.6569	0.657828	0.328914	−	0.944360i	$-0.393317\pi$
$431$	13.6569	0.657828	0.328914	+	0.944360i	$0.393317\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	9.41421	0.449316	0.224658	−	0.974438i	$-0.427874\pi$
$439$	9.41421	0.449316	0.224658	+	0.974438i	$0.427874\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.3137	1.10767	0.553834	−	0.832627i	$-0.313164\pi$
$443$	23.3137	1.10767	0.553834	+	0.832627i	$0.313164\pi$
$444$	0	0
$445$	40.6274	1.92592
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8.58579	−0.405188	−0.202594	−	0.979263i	$-0.564937\pi$
$449$	−8.58579	−0.405188	−0.202594	+	0.979263i	$0.564937\pi$
$450$	0	0
$451$	−3.65685	−0.172195
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−9.65685	−0.452720
$456$	0	0
$457$	25.3137	1.18413	0.592063	−	0.805892i	$-0.298314\pi$
$457$	25.3137	1.18413	0.592063	+	0.805892i	$0.298314\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.9706	−0.510950	−0.255475	−	0.966816i	$-0.582232\pi$
$461$	−10.9706	−0.510950	−0.255475	+	0.966816i	$0.582232\pi$
$462$	0	0
$463$	12.9289	0.600858	0.300429	−	0.953804i	$-0.402870\pi$
$463$	12.9289	0.600858	0.300429	+	0.953804i	$0.402870\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.4558	1.08541	0.542704	−	0.839924i	$-0.317401\pi$
$467$	23.4558	1.08541	0.542704	+	0.839924i	$0.317401\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.242641	−0.0111566
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−24.1421	−1.10308	−0.551541	−	0.834148i	$-0.685960\pi$
$479$	−24.1421	−1.10308	−0.551541	+	0.834148i	$0.685960\pi$
$480$	0	0
$481$	−0.828427	−0.0377730
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−43.7990	−1.98881
$486$	0	0
$487$	−11.5563	−0.523668	−0.261834	−	0.965113i	$-0.584327\pi$
$487$	−11.5563	−0.523668	−0.261834	+	0.965113i	$0.584327\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20.9706	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$491$	−20.9706	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$492$	0	0
$493$	3.65685	0.164696
$494$	0	0
$495$	0	0
$496$	0	0
$497$	30.6274	1.37383
$498$	0	0
$499$	9.89949	0.443162	0.221581	−	0.975142i	$-0.428878\pi$
$499$	9.89949	0.443162	0.221581	+	0.975142i	$0.428878\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.4853	1.09174	0.545872	−	0.837868i	$-0.316198\pi$
$503$	24.4853	1.09174	0.545872	+	0.837868i	$0.316198\pi$
$504$	0	0
$505$	63.9411	2.84534
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.55635	−0.423578	−0.211789	−	0.977315i	$-0.567929\pi$
$509$	−9.55635	−0.423578	−0.211789	+	0.977315i	$0.567929\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.65685	−0.425532
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.9706	−0.655872	−0.327936	−	0.944700i	$-0.606353\pi$
$521$	−14.9706	−0.655872	−0.327936	+	0.944700i	$0.606353\pi$
$522$	0	0
$523$	−2.10051	−0.0918487	−0.0459243	−	0.998945i	$-0.514623\pi$
$523$	−2.10051	−0.0918487	−0.0459243	+	0.998945i	$0.514623\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.7990	−1.47231
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.65685	0.158396
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−52.2843	−2.23961
$546$	0	0
$547$	26.1005	1.11598	0.557989	−	0.829849i	$-0.311573\pi$
$547$	26.1005	1.11598	0.557989	+	0.829849i	$0.311573\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−20.0000	−0.850487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−26.4853	−1.12222	−0.561109	−	0.827742i	$-0.689625\pi$
$557$	−26.4853	−1.12222	−0.561109	+	0.827742i	$0.689625\pi$
$558$	0	0
$559$	0.242641	0.0102626
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.65685	0.0698281	0.0349140	−	0.999390i	$-0.488884\pi$
$563$	1.65685	0.0698281	0.0349140	+	0.999390i	$0.488884\pi$
$564$	0	0
$565$	−64.7696	−2.72488
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13.5563	−0.568312	−0.284156	−	0.958778i	$-0.591713\pi$
$569$	−13.5563	−0.568312	−0.284156	+	0.958778i	$0.591713\pi$
$570$	0	0
$571$	−28.0416	−1.17351	−0.586753	−	0.809766i	$-0.699594\pi$
$571$	−28.0416	−1.17351	−0.586753	+	0.809766i	$0.699594\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	32.1421	1.34042
$576$	0	0
$577$	26.9706	1.12280	0.561400	−	0.827545i	$-0.310263\pi$
$577$	26.9706	1.12280	0.561400	+	0.827545i	$0.310263\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	49.9411	2.07191
$582$	0	0
$583$	−9.31371	−0.385734
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.7990	1.31248	0.656242	−	0.754550i	$-0.272145\pi$
$587$	31.7990	1.31248	0.656242	+	0.754550i	$0.272145\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.1716	0.787282	0.393641	−	0.919264i	$-0.371215\pi$
$593$	19.1716	0.787282	0.393641	+	0.919264i	$0.371215\pi$
$594$	0	0
$595$	60.2843	2.47141
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13.6569	0.558004	0.279002	−	0.960291i	$-0.409996\pi$
$599$	13.6569	0.558004	0.279002	+	0.960291i	$0.409996\pi$
$600$	0	0
$601$	43.2548	1.76440	0.882201	−	0.470874i	$-0.156061\pi$
$601$	43.2548	1.76440	0.882201	+	0.470874i	$0.156061\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.41421	0.138808
$606$	0	0
$607$	13.2132	0.536307	0.268154	−	0.963376i	$-0.413586\pi$
$607$	13.2132	0.536307	0.268154	+	0.963376i	$0.413586\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	−37.9411	−1.53243	−0.766214	−	0.642586i	$-0.777862\pi$
$613$	−37.9411	−1.53243	−0.766214	+	0.642586i	$0.777862\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.9289	−1.08412	−0.542059	−	0.840340i	$-0.682355\pi$
$617$	−26.9289	−1.08412	−0.542059	+	0.840340i	$0.682355\pi$
$618$	0	0
$619$	9.41421	0.378389	0.189195	−	0.981940i	$-0.439412\pi$
$619$	9.41421	0.378389	0.189195	+	0.981940i	$0.439412\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	33.6569	1.34843
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.17157	0.206204
$630$	0	0
$631$	−43.0711	−1.71463	−0.857316	−	0.514790i	$-0.827870\pi$
$631$	−43.0711	−1.71463	−0.857316	+	0.514790i	$0.827870\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−24.1421	−0.958051
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−37.5980	−1.48503	−0.742515	−	0.669829i	$-0.766367\pi$
$641$	−37.5980	−1.48503	−0.742515	+	0.669829i	$0.766367\pi$
$642$	0	0
$643$	−7.55635	−0.297993	−0.148997	−	0.988838i	$-0.547604\pi$
$643$	−7.55635	−0.297993	−0.148997	+	0.988838i	$0.547604\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.6569	−1.16593	−0.582966	−	0.812497i	$-0.698108\pi$
$647$	−29.6569	−1.16593	−0.582966	+	0.812497i	$0.698108\pi$
$648$	0	0
$649$	−2.82843	−0.111025
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.8284	1.12814	0.564072	−	0.825726i	$-0.309234\pi$
$653$	28.8284	1.12814	0.564072	+	0.825726i	$0.309234\pi$
$654$	0	0
$655$	−27.3137	−1.06723
$656$	0	0
$657$	0	0
$658$	0	0
$659$	31.1127	1.21198	0.605989	−	0.795473i	$-0.292777\pi$
$659$	31.1127	1.21198	0.605989	+	0.795473i	$0.292777\pi$
$660$	0	0
$661$	8.82843	0.343386	0.171693	−	0.985151i	$-0.445076\pi$
$661$	8.82843	0.343386	0.171693	+	0.985151i	$0.445076\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.82843	0.109517
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.17157	−0.122437
$672$	0	0
$673$	−11.6569	−0.449339	−0.224669	−	0.974435i	$-0.572130\pi$
$673$	−11.6569	−0.449339	−0.224669	+	0.974435i	$0.572130\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.7574	0.528738	0.264369	−	0.964422i	$-0.414836\pi$
$677$	13.7574	0.528738	0.264369	+	0.964422i	$0.414836\pi$
$678$	0	0
$679$	−36.2843	−1.39246
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.82843	−0.261283	−0.130641	−	0.991430i	$-0.541704\pi$
$683$	−6.82843	−0.261283	−0.130641	+	0.991430i	$0.541704\pi$
$684$	0	0
$685$	−44.6274	−1.70513
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.31371	0.354824
$690$	0	0
$691$	−8.44365	−0.321212	−0.160606	−	0.987019i	$-0.551345\pi$
$691$	−8.44365	−0.321212	−0.160606	+	0.987019i	$0.551345\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	28.1421	1.06749
$696$	0	0
$697$	−22.8284	−0.864688
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30.2426	−1.14225	−0.571124	−	0.820864i	$-0.693493\pi$
$701$	−30.2426	−1.14225	−0.571124	+	0.820864i	$0.693493\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	52.9706	1.99216
$708$	0	0
$709$	−48.6274	−1.82624	−0.913120	−	0.407690i	$-0.866334\pi$
$709$	−48.6274	−1.82624	−0.913120	+	0.407690i	$0.866334\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−26.1421	−0.979031
$714$	0	0
$715$	−3.41421	−0.127684
$716$	0	0
$717$	0	0
$718$	0	0
$719$	35.3137	1.31698	0.658490	−	0.752590i	$-0.271196\pi$
$719$	35.3137	1.31698	0.658490	+	0.752590i	$0.271196\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.89949	0.144824
$726$	0	0
$727$	−17.1716	−0.636858	−0.318429	−	0.947947i	$-0.603155\pi$
$727$	−17.1716	−0.636858	−0.318429	+	0.947947i	$0.603155\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.51472	−0.0560239
$732$	0	0
$733$	14.6274	0.540276	0.270138	−	0.962822i	$-0.412931\pi$
$733$	14.6274	0.540276	0.270138	+	0.962822i	$0.412931\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.24264	0.156280
$738$	0	0
$739$	9.17157	0.337382	0.168691	−	0.985669i	$-0.446046\pi$
$739$	9.17157	0.337382	0.168691	+	0.985669i	$0.446046\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−25.9411	−0.951688	−0.475844	−	0.879530i	$-0.657857\pi$
$743$	−25.9411	−0.951688	−0.475844	+	0.879530i	$0.657857\pi$
$744$	0	0
$745$	−8.48528	−0.310877
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.31371	−0.121080
$750$	0	0
$751$	−19.3137	−0.704767	−0.352384	−	0.935856i	$-0.614629\pi$
$751$	−19.3137	−0.704767	−0.352384	+	0.935856i	$0.614629\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	40.9706	1.49107
$756$	0	0
$757$	−24.2843	−0.882627	−0.441313	−	0.897353i	$-0.645487\pi$
$757$	−24.2843	−0.882627	−0.441313	+	0.897353i	$0.645487\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	19.1716	0.694969	0.347484	−	0.937686i	$-0.387036\pi$
$761$	19.1716	0.694969	0.347484	+	0.937686i	$0.387036\pi$
$762$	0	0
$763$	−43.3137	−1.56806
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.82843	0.102129
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4.58579	0.164939	0.0824696	−	0.996594i	$-0.473719\pi$
$773$	4.58579	0.164939	0.0824696	+	0.996594i	$0.473719\pi$
$774$	0	0
$775$	−36.0416	−1.29465
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	10.8284	0.387472
$782$	0	0
$783$	0	0
$784$	0	0
$785$	21.6569	0.772966
$786$	0	0
$787$	−15.7990	−0.563173	−0.281587	−	0.959536i	$-0.590861\pi$
$787$	−15.7990	−0.563173	−0.281587	+	0.959536i	$0.590861\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−53.6569	−1.90782
$792$	0	0
$793$	3.17157	0.112626
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.51472	−0.195341	−0.0976707	−	0.995219i	$-0.531139\pi$
$797$	−5.51472	−0.195341	−0.0976707	+	0.995219i	$0.531139\pi$
$798$	0	0
$799$	49.9411	1.76679
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.65685	−0.199626
$804$	0	0
$805$	46.6274	1.64340
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20.3848	0.716691	0.358345	−	0.933589i	$-0.383341\pi$
$809$	20.3848	0.716691	0.358345	+	0.933589i	$0.383341\pi$
$810$	0	0
$811$	16.2843	0.571818	0.285909	−	0.958257i	$-0.407704\pi$
$811$	16.2843	0.571818	0.285909	+	0.958257i	$0.407704\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−43.4558	−1.52219
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	40.6274	1.41791	0.708953	−	0.705255i	$-0.249168\pi$
$821$	40.6274	1.41791	0.708953	+	0.705255i	$0.249168\pi$
$822$	0	0
$823$	1.45584	0.0507475	0.0253738	−	0.999678i	$-0.491922\pi$
$823$	1.45584	0.0507475	0.0253738	+	0.999678i	$0.491922\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−42.7696	−1.48724	−0.743622	−	0.668601i	$-0.766894\pi$
$827$	−42.7696	−1.48724	−0.743622	+	0.668601i	$0.766894\pi$
$828$	0	0
$829$	−41.3137	−1.43488	−0.717442	−	0.696618i	$-0.754687\pi$
$829$	−41.3137	−1.43488	−0.717442	+	0.696618i	$0.754687\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.24264	0.216295
$834$	0	0
$835$	−38.6274	−1.33676
$836$	0	0
$837$	0	0
$838$	0	0
$839$	41.6569	1.43815	0.719077	−	0.694930i	$-0.244565\pi$
$839$	41.6569	1.43815	0.719077	+	0.694930i	$0.244565\pi$
$840$	0	0
$841$	−28.6569	−0.988167
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.41421	0.117453
$846$	0	0
$847$	2.82843	0.0971859
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	−17.3137	−0.592810	−0.296405	−	0.955062i	$-0.595788\pi$
$853$	−17.3137	−0.592810	−0.296405	+	0.955062i	$0.595788\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.0711	−0.856411	−0.428206	−	0.903681i	$-0.640854\pi$
$857$	−25.0711	−0.856411	−0.428206	+	0.903681i	$0.640854\pi$
$858$	0	0
$859$	−22.8284	−0.778896	−0.389448	−	0.921048i	$-0.627334\pi$
$859$	−22.8284	−0.778896	−0.389448	+	0.921048i	$0.627334\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−20.7696	−0.707004	−0.353502	−	0.935434i	$-0.615009\pi$
$863$	−20.7696	−0.707004	−0.353502	+	0.935434i	$0.615009\pi$
$864$	0	0
$865$	−67.9411	−2.31007
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7.07107	−0.239870
$870$	0	0
$871$	−4.24264	−0.143756
$872$	0	0
$873$	0	0
$874$	0	0
$875$	16.0000	0.540899
$876$	0	0
$877$	27.6569	0.933906	0.466953	−	0.884282i	$-0.345352\pi$
$877$	27.6569	0.933906	0.466953	+	0.884282i	$0.345352\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−13.0294	−0.438973	−0.219486	−	0.975616i	$-0.570438\pi$
$881$	−13.0294	−0.438973	−0.219486	+	0.975616i	$0.570438\pi$
$882$	0	0
$883$	−34.8284	−1.17207	−0.586035	−	0.810286i	$-0.699312\pi$
$883$	−34.8284	−1.17207	−0.586035	+	0.810286i	$0.699312\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.48528	−0.284908	−0.142454	−	0.989801i	$-0.545499\pi$
$887$	−8.48528	−0.284908	−0.142454	+	0.989801i	$0.545499\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−16.4853	−0.551042
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.17157	−0.105778
$900$	0	0
$901$	−58.1421	−1.93700
$902$	0	0
$903$	0	0
$904$	0	0
$905$	85.2548	2.83397
$906$	0	0
$907$	11.3137	0.375666	0.187833	−	0.982201i	$-0.439854\pi$
$907$	11.3137	0.375666	0.187833	+	0.982201i	$0.439854\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44.2843	−1.46720	−0.733602	−	0.679580i	$-0.762162\pi$
$911$	−44.2843	−1.46720	−0.733602	+	0.679580i	$0.762162\pi$
$912$	0	0
$913$	17.6569	0.584357
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−22.6274	−0.747223
$918$	0	0
$919$	−41.4142	−1.36613	−0.683064	−	0.730358i	$-0.739353\pi$
$919$	−41.4142	−1.36613	−0.683064	+	0.730358i	$0.739353\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.8284	−0.356422
$924$	0	0
$925$	5.51472	0.181323
$926$	0	0
$927$	0	0
$928$	0	0
$929$	18.9289	0.621038	0.310519	−	0.950567i	$-0.399497\pi$
$929$	18.9289	0.621038	0.310519	+	0.950567i	$0.399497\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	21.3137	0.697033
$936$	0	0
$937$	22.2843	0.727995	0.363998	−	0.931400i	$-0.381412\pi$
$937$	22.2843	0.727995	0.363998	+	0.931400i	$0.381412\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	−17.6569	−0.574986
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.6274	0.345345	0.172672	−	0.984979i	$-0.444760\pi$
$947$	10.6274	0.345345	0.172672	+	0.984979i	$0.444760\pi$
$948$	0	0
$949$	5.65685	0.183629
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−35.4142	−1.14718	−0.573589	−	0.819143i	$-0.694450\pi$
$953$	−35.4142	−1.14718	−0.573589	+	0.819143i	$0.694450\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36.9706	−1.19384
$960$	0	0
$961$	−1.68629	−0.0543965
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19.3137	−0.621730
$966$	0	0
$967$	46.4264	1.49297	0.746486	−	0.665401i	$-0.231739\pi$
$967$	46.4264	1.49297	0.746486	+	0.665401i	$0.231739\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.1716	1.00034	0.500172	−	0.865926i	$-0.333270\pi$
$971$	31.1716	1.00034	0.500172	+	0.865926i	$0.333270\pi$
$972$	0	0
$973$	23.3137	0.747403
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.5563	−0.433706	−0.216853	−	0.976204i	$-0.569579\pi$
$977$	−13.5563	−0.433706	−0.216853	+	0.976204i	$0.569579\pi$
$978$	0	0
$979$	11.8995	0.380310
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27.5980	−0.880239	−0.440119	−	0.897939i	$-0.645064\pi$
$983$	−27.5980	−0.880239	−0.440119	+	0.897939i	$0.645064\pi$
$984$	0	0
$985$	41.4558	1.32089
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.17157	−0.0372539
$990$	0	0
$991$	−25.4558	−0.808632	−0.404316	−	0.914619i	$-0.632490\pi$
$991$	−25.4558	−0.808632	−0.404316	+	0.914619i	$0.632490\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−15.3137	−0.485477
$996$	0	0
$997$	−3.85786	−0.122180	−0.0610899	−	0.998132i	$-0.519458\pi$
$997$	−3.85786	−0.122180	−0.0610899	+	0.998132i	$0.519458\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.j.1.2		2
3.2	odd	2		1716.2.a.e.1.1	✓	2
12.11	even	2		6864.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.e.1.1	✓	2	3.2	odd	2
5148.2.a.j.1.2		2	1.1	even	1	trivial
6864.2.a.bb.1.1		2	12.11	even	2

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:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q+3.41421 q^{5} +2.82843 q^{7} +O(q^{10})\)	\(q+3.41421 q^{5} +2.82843 q^{7} +1.00000 q^{11} -1.00000 q^{13} +6.24264 q^{17} +4.82843 q^{23} +6.65685 q^{25} +0.585786 q^{29} -5.41421 q^{31} +9.65685 q^{35} +0.828427 q^{37} -3.65685 q^{41} -0.242641 q^{43} +8.00000 q^{47} +1.00000 q^{49} -9.31371 q^{53} +3.41421 q^{55} -2.82843 q^{59} -3.17157 q^{61} -3.41421 q^{65} +4.24264 q^{67} +10.8284 q^{71} -5.65685 q^{73} +2.82843 q^{77} -7.07107 q^{79} +17.6569 q^{83} +21.3137 q^{85} +11.8995 q^{89} -2.82843 q^{91} -12.8284 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5}+O(q^{10}) \) 2 * q + 4 * q^5	\( 2 q + 4 q^{5} + 2 q^{11} - 2 q^{13} + 4 q^{17} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 8 q^{31} + 8 q^{35} - 4 q^{37} + 4 q^{41} + 8 q^{43} + 16 q^{47} + 2 q^{49} + 4 q^{53} + 4 q^{55} - 12 q^{61} - 4 q^{65} + 16 q^{71} + 24 q^{83} + 20 q^{85} + 4 q^{89} - 20 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 + 2 * q^11 - 2 * q^13 + 4 * q^17 + 4 * q^23 + 2 * q^25 + 4 * q^29 - 8 * q^31 + 8 * q^35 - 4 * q^37 + 4 * q^41 + 8 * q^43 + 16 * q^47 + 2 * q^49 + 4 * q^53 + 4 * q^55 - 12 * q^61 - 4 * q^65 + 16 * q^71 + 24 * q^83 + 20 * q^85 + 4 * q^89 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more