LMFDB - Embedded newform 4864.2.a.br.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, 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("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.34309996544.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 8x^{6} + 28x^{5} + 31x^{4} - 36x^{3} - 22x^{2} + 12x + 2 $ x^8 - 4x^7 - 8x^6 + 28x^5 + 31x^4 - 36x^3 - 22x^2 + 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.15093$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.222191 q^{3} -1.55081 q^{5} -3.06334 q^{7} -2.95063 q^{9} +O(q^{10})$	$q-0.222191 q^{3} -1.55081 q^{5} -3.06334 q^{7} -2.95063 q^{9} -0.950631 q^{11} -3.36113 q^{13} +0.344577 q^{15} +1.82843 q^{17} -1.00000 q^{19} +0.680647 q^{21} -3.36113 q^{23} -2.59499 q^{25} +1.32218 q^{27} -4.95873 q^{29} -3.44620 q^{31} +0.211222 q^{33} +4.75066 q^{35} +2.95889 q^{37} +0.746814 q^{39} -4.55687 q^{41} +1.69373 q^{43} +4.57587 q^{45} -3.39941 q^{47} +2.38404 q^{49} -0.406261 q^{51} -4.47142 q^{53} +1.47425 q^{55} +0.222191 q^{57} -0.395014 q^{59} -5.34158 q^{61} +9.03878 q^{63} +5.21247 q^{65} +6.85064 q^{67} +0.746814 q^{69} +5.15207 q^{71} -1.19998 q^{73} +0.576584 q^{75} +2.91210 q^{77} -14.7249 q^{79} +8.55812 q^{81} +3.35564 q^{83} -2.83554 q^{85} +1.10179 q^{87} -9.64561 q^{89} +10.2963 q^{91} +0.765715 q^{93} +1.55081 q^{95} +3.55562 q^{97} +2.80496 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 + 4 * q^9	$ 8 q + 4 q^{3} + 4 q^{9} + 20 q^{11} - 8 q^{17} - 8 q^{19} + 20 q^{25} + 4 q^{27} + 24 q^{33} + 12 q^{35} + 8 q^{41} + 28 q^{43} + 8 q^{49} + 12 q^{51} - 4 q^{57} + 36 q^{59} + 8 q^{65} + 28 q^{67} - 8 q^{73} + 68 q^{75} - 32 q^{81} + 40 q^{83} - 8 q^{89} - 12 q^{91} + 40 q^{97} + 60 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 + 4 * q^9 + 20 * q^11 - 8 * q^17 - 8 * q^19 + 20 * q^25 + 4 * q^27 + 24 * q^33 + 12 * q^35 + 8 * q^41 + 28 * q^43 + 8 * q^49 + 12 * q^51 - 4 * q^57 + 36 * q^59 + 8 * q^65 + 28 * q^67 - 8 * q^73 + 68 * q^75 - 32 * q^81 + 40 * q^83 - 8 * q^89 - 12 * q^91 + 40 * q^97 + 60 * q^99

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
$3$	−0.222191	−0.128282	−0.0641411	−	0.997941i	$-0.520431\pi$
$3$	−0.222191	−0.128282	−0.0641411	+	0.997941i	$0.520431\pi$
$4$	0	0
$5$	−1.55081	−0.693543	−0.346772	−	0.937950i	$-0.612722\pi$
$5$	−1.55081	−0.693543	−0.346772	+	0.937950i	$0.612722\pi$
$6$	0	0
$7$	−3.06334	−1.15783	−0.578917	−	0.815387i	$-0.696524\pi$
$7$	−3.06334	−1.15783	−0.578917	+	0.815387i	$0.696524\pi$
$8$	0	0
$9$	−2.95063	−0.983544
$10$	0	0
$11$	−0.950631	−0.286626	−0.143313	−	0.989677i	$-0.545776\pi$
$11$	−0.950631	−0.286626	−0.143313	+	0.989677i	$0.545776\pi$
$12$	0	0
$13$	−3.36113	−0.932209	−0.466105	−	0.884730i	$-0.654343\pi$
$13$	−3.36113	−0.932209	−0.466105	+	0.884730i	$0.654343\pi$
$14$	0	0
$15$	0.344577	0.0889693
$16$	0	0
$17$	1.82843	0.443459	0.221729	−	0.975108i	$-0.428830\pi$
$17$	1.82843	0.443459	0.221729	+	0.975108i	$0.428830\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.680647	0.148529
$22$	0	0
$23$	−3.36113	−0.700844	−0.350422	−	0.936592i	$-0.613962\pi$
$23$	−3.36113	−0.700844	−0.350422	+	0.936592i	$0.613962\pi$
$24$	0	0
$25$	−2.59499	−0.518998
$26$	0	0
$27$	1.32218	0.254453
$28$	0	0
$29$	−4.95873	−0.920812	−0.460406	−	0.887708i	$-0.652296\pi$
$29$	−4.95873	−0.920812	−0.460406	+	0.887708i	$0.652296\pi$
$30$	0	0
$31$	−3.44620	−0.618955	−0.309478	−	0.950907i	$-0.600154\pi$
$31$	−3.44620	−0.618955	−0.309478	+	0.950907i	$0.600154\pi$
$32$	0	0
$33$	0.211222	0.0367690
$34$	0	0
$35$	4.75066	0.803007
$36$	0	0
$37$	2.95889	0.486439	0.243219	−	0.969971i	$-0.421797\pi$
$37$	2.95889	0.486439	0.243219	+	0.969971i	$0.421797\pi$
$38$	0	0
$39$	0.746814	0.119586
$40$	0	0
$41$	−4.55687	−0.711663	−0.355832	−	0.934550i	$-0.615802\pi$
$41$	−4.55687	−0.711663	−0.355832	+	0.934550i	$0.615802\pi$
$42$	0	0
$43$	1.69373	0.258291	0.129145	−	0.991626i	$-0.458777\pi$
$43$	1.69373	0.258291	0.129145	+	0.991626i	$0.458777\pi$
$44$	0	0
$45$	4.57587	0.682130
$46$	0	0
$47$	−3.39941	−0.495855	−0.247927	−	0.968779i	$-0.579749\pi$
$47$	−3.39941	−0.495855	−0.247927	+	0.968779i	$0.579749\pi$
$48$	0	0
$49$	2.38404	0.340578
$50$	0	0
$51$	−0.406261	−0.0568879
$52$	0	0
$53$	−4.47142	−0.614197	−0.307098	−	0.951678i	$-0.599358\pi$
$53$	−4.47142	−0.614197	−0.307098	+	0.951678i	$0.599358\pi$
$54$	0	0
$55$	1.47425	0.198788
$56$	0	0
$57$	0.222191	0.0294300
$58$	0	0
$59$	−0.395014	−0.0514264	−0.0257132	−	0.999669i	$-0.508186\pi$
$59$	−0.395014	−0.0514264	−0.0257132	+	0.999669i	$0.508186\pi$
$60$	0	0
$61$	−5.34158	−0.683920	−0.341960	−	0.939715i	$-0.611091\pi$
$61$	−5.34158	−0.683920	−0.341960	+	0.939715i	$0.611091\pi$
$62$	0	0
$63$	9.03878	1.13878
$64$	0	0
$65$	5.21247	0.646528
$66$	0	0
$67$	6.85064	0.836939	0.418470	−	0.908231i	$-0.362567\pi$
$67$	6.85064	0.836939	0.418470	+	0.908231i	$0.362567\pi$
$68$	0	0
$69$	0.746814	0.0899058
$70$	0	0
$71$	5.15207	0.611438	0.305719	−	0.952122i	$-0.401103\pi$
$71$	5.15207	0.611438	0.305719	+	0.952122i	$0.401103\pi$
$72$	0	0
$73$	−1.19998	−0.140446	−0.0702232	−	0.997531i	$-0.522371\pi$
$73$	−1.19998	−0.140446	−0.0702232	+	0.997531i	$0.522371\pi$
$74$	0	0
$75$	0.576584	0.0665782
$76$	0	0
$77$	2.91210	0.331865
$78$	0	0
$79$	−14.7249	−1.65669	−0.828343	−	0.560222i	$-0.810716\pi$
$79$	−14.7249	−1.65669	−0.828343	+	0.560222i	$0.810716\pi$
$80$	0	0
$81$	8.55812	0.950902
$82$	0	0
$83$	3.35564	0.368330	0.184165	−	0.982895i	$-0.441042\pi$
$83$	3.35564	0.368330	0.184165	+	0.982895i	$0.441042\pi$
$84$	0	0
$85$	−2.83554	−0.307558
$86$	0	0
$87$	1.10179	0.118124
$88$	0	0
$89$	−9.64561	−1.02243	−0.511216	−	0.859452i	$-0.670805\pi$
$89$	−9.64561	−1.02243	−0.511216	+	0.859452i	$0.670805\pi$
$90$	0	0
$91$	10.2963	1.07934
$92$	0	0
$93$	0.765715	0.0794010
$94$	0	0
$95$	1.55081	0.159110
$96$	0	0
$97$	3.55562	0.361018	0.180509	−	0.983573i	$-0.442225\pi$
$97$	3.55562	0.361018	0.180509	+	0.983573i	$0.442225\pi$
$98$	0	0
$99$	2.80496	0.281909
$100$	0	0
$101$	−13.5381	−1.34709	−0.673545	−	0.739146i	$-0.735229\pi$
$101$	−13.5381	−1.34709	−0.673545	+	0.739146i	$0.735229\pi$
$102$	0	0
$103$	6.64570	0.654820	0.327410	−	0.944882i	$-0.393824\pi$
$103$	6.64570	0.654820	0.327410	+	0.944882i	$0.393824\pi$
$104$	0	0
$105$	−1.05555	−0.103012
$106$	0	0
$107$	15.8519	1.53246	0.766230	−	0.642566i	$-0.222130\pi$
$107$	15.8519	1.53246	0.766230	+	0.642566i	$0.222130\pi$
$108$	0	0
$109$	13.4213	1.28553	0.642764	−	0.766064i	$-0.277788\pi$
$109$	13.4213	1.28553	0.642764	+	0.766064i	$0.277788\pi$
$110$	0	0
$111$	−0.657440	−0.0624015
$112$	0	0
$113$	6.65810	0.626342	0.313171	−	0.949697i	$-0.398609\pi$
$113$	6.65810	0.626342	0.313171	+	0.949697i	$0.398609\pi$
$114$	0	0
$115$	5.21247	0.486065
$116$	0	0
$117$	9.91745	0.916869
$118$	0	0
$119$	−5.60109	−0.513451
$120$	0	0
$121$	−10.0963	−0.917846
$122$	0	0
$123$	1.01250	0.0912937
$124$	0	0
$125$	11.7784	1.05349
$126$	0	0
$127$	6.54782	0.581025	0.290512	−	0.956871i	$-0.406174\pi$
$127$	6.54782	0.581025	0.290512	+	0.956871i	$0.406174\pi$
$128$	0	0
$129$	−0.376331	−0.0331341
$130$	0	0
$131$	14.2838	1.24798	0.623990	−	0.781432i	$-0.285511\pi$
$131$	14.2838	1.24798	0.623990	+	0.781432i	$0.285511\pi$
$132$	0	0
$133$	3.06334	0.265625
$134$	0	0
$135$	−2.05045	−0.176474
$136$	0	0
$137$	−11.9138	−1.01786	−0.508931	−	0.860808i	$-0.669959\pi$
$137$	−11.9138	−1.01786	−0.508931	+	0.860808i	$0.669959\pi$
$138$	0	0
$139$	11.1950	0.949551	0.474775	−	0.880107i	$-0.342529\pi$
$139$	11.1950	0.949551	0.474775	+	0.880107i	$0.342529\pi$
$140$	0	0
$141$	0.755320	0.0636094
$142$	0	0
$143$	3.19519	0.267195
$144$	0	0
$145$	7.69004	0.638623
$146$	0	0
$147$	−0.529714	−0.0436901
$148$	0	0
$149$	7.67749	0.628964	0.314482	−	0.949263i	$-0.398169\pi$
$149$	7.67749	0.628964	0.314482	+	0.949263i	$0.398169\pi$
$150$	0	0
$151$	3.10162	0.252406	0.126203	−	0.992004i	$-0.459721\pi$
$151$	3.10162	0.252406	0.126203	+	0.992004i	$0.459721\pi$
$152$	0	0
$153$	−5.39501	−0.436161
$154$	0	0
$155$	5.34440	0.429272
$156$	0	0
$157$	−14.8994	−1.18910	−0.594550	−	0.804059i	$-0.702670\pi$
$157$	−14.8994	−1.18910	−0.594550	+	0.804059i	$0.702670\pi$
$158$	0	0
$159$	0.993511	0.0787905
$160$	0	0
$161$	10.2963	0.811460
$162$	0	0
$163$	18.5481	1.45280	0.726400	−	0.687272i	$-0.241192\pi$
$163$	18.5481	1.45280	0.726400	+	0.687272i	$0.241192\pi$
$164$	0	0
$165$	−0.327565	−0.0255009
$166$	0	0
$167$	−11.4532	−0.886274	−0.443137	−	0.896454i	$-0.646135\pi$
$167$	−11.4532	−0.886274	−0.443137	+	0.896454i	$0.646135\pi$
$168$	0	0
$169$	−1.70281	−0.130986
$170$	0	0
$171$	2.95063	0.225640
$172$	0	0
$173$	7.20956	0.548133	0.274066	−	0.961711i	$-0.411631\pi$
$173$	7.20956	0.548133	0.274066	+	0.961711i	$0.411631\pi$
$174$	0	0
$175$	7.94933	0.600913
$176$	0	0
$177$	0.0877686	0.00659710
$178$	0	0
$179$	−4.90251	−0.366431	−0.183215	−	0.983073i	$-0.558651\pi$
$179$	−4.90251	−0.366431	−0.183215	+	0.983073i	$0.558651\pi$
$180$	0	0
$181$	19.1775	1.42545	0.712725	−	0.701444i	$-0.247461\pi$
$181$	19.1775	1.42545	0.712725	+	0.701444i	$0.247461\pi$
$182$	0	0
$183$	1.18685	0.0877347
$184$	0	0
$185$	−4.58868	−0.337366
$186$	0	0
$187$	−1.73816	−0.127107
$188$	0	0
$189$	−4.05028	−0.294615
$190$	0	0
$191$	22.8812	1.65563	0.827814	−	0.561003i	$-0.189584\pi$
$191$	22.8812	1.65563	0.827814	+	0.561003i	$0.189584\pi$
$192$	0	0
$193$	−2.11248	−0.152060	−0.0760300	−	0.997106i	$-0.524224\pi$
$193$	−2.11248	−0.152060	−0.0760300	+	0.997106i	$0.524224\pi$
$194$	0	0
$195$	−1.15817	−0.0829380
$196$	0	0
$197$	−6.05012	−0.431053	−0.215526	−	0.976498i	$-0.569147\pi$
$197$	−6.05012	−0.431053	−0.215526	+	0.976498i	$0.569147\pi$
$198$	0	0
$199$	4.03090	0.285743	0.142872	−	0.989741i	$-0.454366\pi$
$199$	4.03090	0.285743	0.142872	+	0.989741i	$0.454366\pi$
$200$	0	0
$201$	−1.52215	−0.107364
$202$	0	0
$203$	15.1903	1.06615
$204$	0	0
$205$	7.06683	0.493569
$206$	0	0
$207$	9.91745	0.689310
$208$	0	0
$209$	0.950631	0.0657565
$210$	0	0
$211$	1.66532	0.114646	0.0573228	−	0.998356i	$-0.481744\pi$
$211$	1.66532	0.114646	0.0573228	+	0.998356i	$0.481744\pi$
$212$	0	0
$213$	−1.14475	−0.0784366
$214$	0	0
$215$	−2.62665	−0.179136
$216$	0	0
$217$	10.5569	0.716647
$218$	0	0
$219$	0.266624	0.0180168
$220$	0	0
$221$	−6.14558	−0.413396
$222$	0	0
$223$	−28.9352	−1.93764	−0.968821	−	0.247761i	$-0.920305\pi$
$223$	−28.9352	−1.93764	−0.968821	+	0.247761i	$0.920305\pi$
$224$	0	0
$225$	7.65685	0.510457
$226$	0	0
$227$	20.0519	1.33089	0.665445	−	0.746447i	$-0.268242\pi$
$227$	20.0519	1.33089	0.665445	+	0.746447i	$0.268242\pi$
$228$	0	0
$229$	14.9953	0.990919	0.495459	−	0.868631i	$-0.335000\pi$
$229$	14.9953	0.990919	0.495459	+	0.868631i	$0.335000\pi$
$230$	0	0
$231$	−0.647045	−0.0425724
$232$	0	0
$233$	11.2738	0.738570	0.369285	−	0.929316i	$-0.379603\pi$
$233$	11.2738	0.738570	0.369285	+	0.929316i	$0.379603\pi$
$234$	0	0
$235$	5.27184	0.343897
$236$	0	0
$237$	3.27176	0.212523
$238$	0	0
$239$	−19.9049	−1.28754	−0.643770	−	0.765219i	$-0.722631\pi$
$239$	−19.9049	−1.28754	−0.643770	+	0.765219i	$0.722631\pi$
$240$	0	0
$241$	−15.8163	−1.01882	−0.509408	−	0.860525i	$-0.670135\pi$
$241$	−15.8163	−1.01882	−0.509408	+	0.860525i	$0.670135\pi$
$242$	0	0
$243$	−5.86808	−0.376437
$244$	0	0
$245$	−3.69720	−0.236205
$246$	0	0
$247$	3.36113	0.213863
$248$	0	0
$249$	−0.745595	−0.0472501
$250$	0	0
$251$	−10.5087	−0.663306	−0.331653	−	0.943401i	$-0.607606\pi$
$251$	−10.5087	−0.663306	−0.331653	+	0.943401i	$0.607606\pi$
$252$	0	0
$253$	3.19519	0.200880
$254$	0	0
$255$	0.630033	0.0394542
$256$	0	0
$257$	−1.70129	−0.106123	−0.0530617	−	0.998591i	$-0.516898\pi$
$257$	−1.70129	−0.106123	−0.0530617	+	0.998591i	$0.516898\pi$
$258$	0	0
$259$	−9.06409	−0.563215
$260$	0	0
$261$	14.6314	0.905659
$262$	0	0
$263$	−13.8359	−0.853157	−0.426578	−	0.904451i	$-0.640281\pi$
$263$	−13.8359	−0.853157	−0.426578	+	0.904451i	$0.640281\pi$
$264$	0	0
$265$	6.93432	0.425972
$266$	0	0
$267$	2.14317	0.131160
$268$	0	0
$269$	−14.9655	−0.912466	−0.456233	−	0.889860i	$-0.650802\pi$
$269$	−14.9655	−0.912466	−0.456233	+	0.889860i	$0.650802\pi$
$270$	0	0
$271$	4.12684	0.250688	0.125344	−	0.992113i	$-0.459997\pi$
$271$	4.12684	0.250688	0.125344	+	0.992113i	$0.459997\pi$
$272$	0	0
$273$	−2.28774	−0.138461
$274$	0	0
$275$	2.46688	0.148758
$276$	0	0
$277$	−25.7721	−1.54849	−0.774247	−	0.632884i	$-0.781871\pi$
$277$	−25.7721	−1.54849	−0.774247	+	0.632884i	$0.781871\pi$
$278$	0	0
$279$	10.1685	0.608769
$280$	0	0
$281$	−10.2694	−0.612621	−0.306311	−	0.951932i	$-0.599095\pi$
$281$	−10.2694	−0.612621	−0.306311	+	0.951932i	$0.599095\pi$
$282$	0	0
$283$	−6.99506	−0.415813	−0.207907	−	0.978149i	$-0.566665\pi$
$283$	−6.99506	−0.415813	−0.207907	+	0.978149i	$0.566665\pi$
$284$	0	0
$285$	−0.344577	−0.0204110
$286$	0	0
$287$	13.9592	0.823987
$288$	0	0
$289$	−13.6569	−0.803344
$290$	0	0
$291$	−0.790027	−0.0463122
$292$	0	0
$293$	−31.8434	−1.86031	−0.930157	−	0.367162i	$-0.880330\pi$
$293$	−31.8434	−1.86031	−0.930157	+	0.367162i	$0.880330\pi$
$294$	0	0
$295$	0.612591	0.0356664
$296$	0	0
$297$	−1.25690	−0.0729330
$298$	0	0
$299$	11.2972	0.653333
$300$	0	0
$301$	−5.18846	−0.299058
$302$	0	0
$303$	3.00805	0.172808
$304$	0	0
$305$	8.28378	0.474328
$306$	0	0
$307$	6.45813	0.368585	0.184292	−	0.982871i	$-0.441001\pi$
$307$	6.45813	0.368585	0.184292	+	0.982871i	$0.441001\pi$
$308$	0	0
$309$	−1.47662	−0.0840018
$310$	0	0
$311$	28.1973	1.59892	0.799462	−	0.600716i	$-0.205118\pi$
$311$	28.1973	1.59892	0.799462	+	0.600716i	$0.205118\pi$
$312$	0	0
$313$	17.1212	0.967746	0.483873	−	0.875138i	$-0.339230\pi$
$313$	17.1212	0.967746	0.483873	+	0.875138i	$0.339230\pi$
$314$	0	0
$315$	−14.0174	−0.789793
$316$	0	0
$317$	−29.5841	−1.66161	−0.830804	−	0.556564i	$-0.812119\pi$
$317$	−29.5841	−1.66161	−0.830804	+	0.556564i	$0.812119\pi$
$318$	0	0
$319$	4.71392	0.263929
$320$	0	0
$321$	−3.52215	−0.196587
$322$	0	0
$323$	−1.82843	−0.101736
$324$	0	0
$325$	8.72209	0.483815
$326$	0	0
$327$	−2.98210	−0.164910
$328$	0	0
$329$	10.4135	0.574117
$330$	0	0
$331$	20.5519	1.12964	0.564818	−	0.825215i	$-0.308946\pi$
$331$	20.5519	1.12964	0.564818	+	0.825215i	$0.308946\pi$
$332$	0	0
$333$	−8.73060	−0.478434
$334$	0	0
$335$	−10.6240	−0.580454
$336$	0	0
$337$	−19.1832	−1.04497	−0.522487	−	0.852647i	$-0.674996\pi$
$337$	−19.1832	−1.04497	−0.522487	+	0.852647i	$0.674996\pi$
$338$	0	0
$339$	−1.47937	−0.0803485
$340$	0	0
$341$	3.27606	0.177409
$342$	0	0
$343$	14.1402	0.763501
$344$	0	0
$345$	−1.15817	−0.0623536
$346$	0	0
$347$	10.5507	0.566390	0.283195	−	0.959062i	$-0.408606\pi$
$347$	10.5507	0.566390	0.283195	+	0.959062i	$0.408606\pi$
$348$	0	0
$349$	29.4693	1.57745	0.788727	−	0.614744i	$-0.210741\pi$
$349$	29.4693	1.57745	0.788727	+	0.614744i	$0.210741\pi$
$350$	0	0
$351$	−4.44401	−0.237204
$352$	0	0
$353$	−3.98903	−0.212315	−0.106157	−	0.994349i	$-0.533855\pi$
$353$	−3.98903	−0.212315	−0.106157	+	0.994349i	$0.533855\pi$
$354$	0	0
$355$	−7.98988	−0.424059
$356$	0	0
$357$	1.24451	0.0658667
$358$	0	0
$359$	14.9377	0.788380	0.394190	−	0.919029i	$-0.371025\pi$
$359$	14.9377	0.788380	0.394190	+	0.919029i	$0.371025\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.24331	0.117743
$364$	0	0
$365$	1.86093	0.0974057
$366$	0	0
$367$	19.1009	0.997059	0.498529	−	0.866873i	$-0.333874\pi$
$367$	19.1009	0.997059	0.498529	+	0.866873i	$0.333874\pi$
$368$	0	0
$369$	13.4456	0.699952
$370$	0	0
$371$	13.6975	0.711137
$372$	0	0
$373$	0.461357	0.0238882	0.0119441	−	0.999929i	$-0.496198\pi$
$373$	0.461357	0.0238882	0.0119441	+	0.999929i	$0.496198\pi$
$374$	0	0
$375$	−2.61706	−0.135144
$376$	0	0
$377$	16.6669	0.858390
$378$	0	0
$379$	−29.9622	−1.53906	−0.769528	−	0.638613i	$-0.779508\pi$
$379$	−29.9622	−1.53906	−0.769528	+	0.638613i	$0.779508\pi$
$380$	0	0
$381$	−1.45487	−0.0745352
$382$	0	0
$383$	−16.9077	−0.863944	−0.431972	−	0.901887i	$-0.642182\pi$
$383$	−16.9077	−0.863944	−0.431972	+	0.901887i	$0.642182\pi$
$384$	0	0
$385$	−4.51612	−0.230163
$386$	0	0
$387$	−4.99756	−0.254040
$388$	0	0
$389$	7.92418	0.401772	0.200886	−	0.979615i	$-0.435618\pi$
$389$	7.92418	0.401772	0.200886	+	0.979615i	$0.435618\pi$
$390$	0	0
$391$	−6.14558	−0.310795
$392$	0	0
$393$	−3.17373	−0.160094
$394$	0	0
$395$	22.8356	1.14898
$396$	0	0
$397$	−13.6718	−0.686170	−0.343085	−	0.939304i	$-0.611472\pi$
$397$	−13.6718	−0.686170	−0.343085	+	0.939304i	$0.611472\pi$
$398$	0	0
$399$	−0.680647	−0.0340750
$400$	0	0
$401$	12.0107	0.599785	0.299893	−	0.953973i	$-0.403049\pi$
$401$	12.0107	0.599785	0.299893	+	0.953973i	$0.403049\pi$
$402$	0	0
$403$	11.5831	0.576996
$404$	0	0
$405$	−13.2720	−0.659492
$406$	0	0
$407$	−2.81281	−0.139426
$408$	0	0
$409$	9.47063	0.468292	0.234146	−	0.972201i	$-0.424771\pi$
$409$	9.47063	0.468292	0.234146	+	0.972201i	$0.424771\pi$
$410$	0	0
$411$	2.64713	0.130574
$412$	0	0
$413$	1.21006	0.0595432
$414$	0	0
$415$	−5.20396	−0.255453
$416$	0	0
$417$	−2.48744	−0.121811
$418$	0	0
$419$	28.0843	1.37201	0.686004	−	0.727598i	$-0.259363\pi$
$419$	28.0843	1.37201	0.686004	+	0.727598i	$0.259363\pi$
$420$	0	0
$421$	−9.83238	−0.479201	−0.239601	−	0.970872i	$-0.577016\pi$
$421$	−9.83238	−0.479201	−0.239601	+	0.970872i	$0.577016\pi$
$422$	0	0
$423$	10.0304	0.487695
$424$	0	0
$425$	−4.74475	−0.230154
$426$	0	0
$427$	16.3631	0.791865
$428$	0	0
$429$	−0.709944	−0.0342764
$430$	0	0
$431$	−8.94284	−0.430761	−0.215381	−	0.976530i	$-0.569099\pi$
$431$	−8.94284	−0.430761	−0.215381	+	0.976530i	$0.569099\pi$
$432$	0	0
$433$	−29.5274	−1.41900	−0.709499	−	0.704707i	$-0.751079\pi$
$433$	−29.5274	−1.41900	−0.709499	+	0.704707i	$0.751079\pi$
$434$	0	0
$435$	−1.70866	−0.0819240
$436$	0	0
$437$	3.36113	0.160785
$438$	0	0
$439$	−20.5113	−0.978953	−0.489477	−	0.872016i	$-0.662812\pi$
$439$	−20.5113	−0.978953	−0.489477	+	0.872016i	$0.662812\pi$
$440$	0	0
$441$	−7.03444	−0.334973
$442$	0	0
$443$	−13.3188	−0.632794	−0.316397	−	0.948627i	$-0.602473\pi$
$443$	−13.3188	−0.632794	−0.316397	+	0.948627i	$0.602473\pi$
$444$	0	0
$445$	14.9585	0.709101
$446$	0	0
$447$	−1.70587	−0.0806850
$448$	0	0
$449$	10.5138	0.496177	0.248089	−	0.968737i	$-0.420198\pi$
$449$	10.5138	0.496177	0.248089	+	0.968737i	$0.420198\pi$
$450$	0	0
$451$	4.33190	0.203981
$452$	0	0
$453$	−0.689153	−0.0323792
$454$	0	0
$455$	−15.9676	−0.748571
$456$	0	0
$457$	26.5831	1.24351	0.621753	−	0.783214i	$-0.286421\pi$
$457$	26.5831	1.24351	0.621753	+	0.783214i	$0.286421\pi$
$458$	0	0
$459$	2.41751	0.112840
$460$	0	0
$461$	−19.2417	−0.896175	−0.448087	−	0.893990i	$-0.647895\pi$
$461$	−19.2417	−0.896175	−0.448087	+	0.893990i	$0.647895\pi$
$462$	0	0
$463$	2.24701	0.104427	0.0522137	−	0.998636i	$-0.483372\pi$
$463$	2.24701	0.104427	0.0522137	+	0.998636i	$0.483372\pi$
$464$	0	0
$465$	−1.18748	−0.0550680
$466$	0	0
$467$	27.5682	1.27570	0.637852	−	0.770159i	$-0.279823\pi$
$467$	27.5682	1.27570	0.637852	+	0.770159i	$0.279823\pi$
$468$	0	0
$469$	−20.9858	−0.969036
$470$	0	0
$471$	3.31051	0.152540
$472$	0	0
$473$	−1.61011	−0.0740329
$474$	0	0
$475$	2.59499	0.119066
$476$	0	0
$477$	13.1935	0.604089
$478$	0	0
$479$	10.5447	0.481802	0.240901	−	0.970550i	$-0.422557\pi$
$479$	10.5447	0.481802	0.240901	+	0.970550i	$0.422557\pi$
$480$	0	0
$481$	−9.94521	−0.453463
$482$	0	0
$483$	−2.28774	−0.104096
$484$	0	0
$485$	−5.51409	−0.250382
$486$	0	0
$487$	−39.7294	−1.80031	−0.900154	−	0.435571i	$-0.856546\pi$
$487$	−39.7294	−1.80031	−0.900154	+	0.435571i	$0.856546\pi$
$488$	0	0
$489$	−4.12123	−0.186369
$490$	0	0
$491$	31.0150	1.39969	0.699844	−	0.714296i	$-0.253253\pi$
$491$	31.0150	1.39969	0.699844	+	0.714296i	$0.253253\pi$
$492$	0	0
$493$	−9.06667	−0.408342
$494$	0	0
$495$	−4.34996	−0.195516
$496$	0	0
$497$	−15.7825	−0.707943
$498$	0	0
$499$	−4.90620	−0.219632	−0.109816	−	0.993952i	$-0.535026\pi$
$499$	−4.90620	−0.219632	−0.109816	+	0.993952i	$0.535026\pi$
$500$	0	0
$501$	2.54480	0.113693
$502$	0	0
$503$	−14.6847	−0.654759	−0.327380	−	0.944893i	$-0.606166\pi$
$503$	−14.6847	−0.654759	−0.327380	+	0.944893i	$0.606166\pi$
$504$	0	0
$505$	20.9950	0.934265
$506$	0	0
$507$	0.378351	0.0168031
$508$	0	0
$509$	20.3988	0.904159	0.452080	−	0.891978i	$-0.350682\pi$
$509$	20.3988	0.904159	0.452080	+	0.891978i	$0.350682\pi$
$510$	0	0
$511$	3.67593	0.162614
$512$	0	0
$513$	−1.32218	−0.0583756
$514$	0	0
$515$	−10.3062	−0.454146
$516$	0	0
$517$	3.23158	0.142125
$518$	0	0
$519$	−1.60190	−0.0703157
$520$	0	0
$521$	22.4356	0.982923	0.491462	−	0.870899i	$-0.336463\pi$
$521$	22.4356	0.982923	0.491462	+	0.870899i	$0.336463\pi$
$522$	0	0
$523$	−9.71531	−0.424821	−0.212410	−	0.977181i	$-0.568131\pi$
$523$	−9.71531	−0.424821	−0.212410	+	0.977181i	$0.568131\pi$
$524$	0	0
$525$	−1.76627	−0.0770865
$526$	0	0
$527$	−6.30112	−0.274481
$528$	0	0
$529$	−11.7028	−0.508818
$530$	0	0
$531$	1.16554	0.0505801
$532$	0	0
$533$	15.3162	0.663419
$534$	0	0
$535$	−24.5833	−1.06283
$536$	0	0
$537$	1.08930	0.0470066
$538$	0	0
$539$	−2.26635	−0.0976185
$540$	0	0
$541$	14.7230	0.632991	0.316496	−	0.948594i	$-0.397494\pi$
$541$	14.7230	0.632991	0.316496	+	0.948594i	$0.397494\pi$
$542$	0	0
$543$	−4.26107	−0.182860
$544$	0	0
$545$	−20.8139	−0.891569
$546$	0	0
$547$	−10.9906	−0.469922	−0.234961	−	0.972005i	$-0.575496\pi$
$547$	−10.9906	−0.469922	−0.234961	+	0.972005i	$0.575496\pi$
$548$	0	0
$549$	15.7610	0.672665
$550$	0	0
$551$	4.95873	0.211249
$552$	0	0
$553$	45.1075	1.91817
$554$	0	0
$555$	1.01956	0.0432781
$556$	0	0
$557$	25.9252	1.09849	0.549243	−	0.835663i	$-0.314916\pi$
$557$	25.9252	1.09849	0.549243	+	0.835663i	$0.314916\pi$
$558$	0	0
$559$	−5.69283	−0.240781
$560$	0	0
$561$	0.386204	0.0163055
$562$	0	0
$563$	−21.0775	−0.888310	−0.444155	−	0.895950i	$-0.646496\pi$
$563$	−21.0775	−0.888310	−0.444155	+	0.895950i	$0.646496\pi$
$564$	0	0
$565$	−10.3255	−0.434395
$566$	0	0
$567$	−26.2164	−1.10099
$568$	0	0
$569$	−11.2369	−0.471076	−0.235538	−	0.971865i	$-0.575685\pi$
$569$	−11.2369	−0.471076	−0.235538	+	0.971865i	$0.575685\pi$
$570$	0	0
$571$	−42.5037	−1.77872	−0.889362	−	0.457204i	$-0.848851\pi$
$571$	−42.5037	−1.77872	−0.889362	+	0.457204i	$0.848851\pi$
$572$	0	0
$573$	−5.08401	−0.212388
$574$	0	0
$575$	8.72209	0.363736
$576$	0	0
$577$	−26.6503	−1.10947	−0.554733	−	0.832029i	$-0.687179\pi$
$577$	−26.6503	−1.10947	−0.554733	+	0.832029i	$0.687179\pi$
$578$	0	0
$579$	0.469376	0.0195066
$580$	0	0
$581$	−10.2795	−0.426464
$582$	0	0
$583$	4.25067	0.176045
$584$	0	0
$585$	−15.3801	−0.635888
$586$	0	0
$587$	33.2195	1.37111	0.685557	−	0.728019i	$-0.259559\pi$
$587$	33.2195	1.37111	0.685557	+	0.728019i	$0.259559\pi$
$588$	0	0
$589$	3.44620	0.141998
$590$	0	0
$591$	1.34428	0.0552964
$592$	0	0
$593$	−25.4149	−1.04367	−0.521833	−	0.853047i	$-0.674752\pi$
$593$	−25.4149	−1.04367	−0.521833	+	0.853047i	$0.674752\pi$
$594$	0	0
$595$	8.68623	0.356101
$596$	0	0
$597$	−0.895632	−0.0366558
$598$	0	0
$599$	46.6431	1.90578	0.952892	−	0.303310i	$-0.0980918\pi$
$599$	46.6431	1.90578	0.952892	+	0.303310i	$0.0980918\pi$
$600$	0	0
$601$	47.4644	1.93611	0.968056	−	0.250734i	$-0.0806718\pi$
$601$	47.4644	1.93611	0.968056	+	0.250734i	$0.0806718\pi$
$602$	0	0
$603$	−20.2137	−0.823166
$604$	0	0
$605$	15.6574	0.636566
$606$	0	0
$607$	−24.4897	−0.994006	−0.497003	−	0.867749i	$-0.665566\pi$
$607$	−24.4897	−0.994006	−0.497003	+	0.867749i	$0.665566\pi$
$608$	0	0
$609$	−3.37514	−0.136768
$610$	0	0
$611$	11.4259	0.462241
$612$	0	0
$613$	18.9950	0.767201	0.383600	−	0.923499i	$-0.374684\pi$
$613$	18.9950	0.767201	0.383600	+	0.923499i	$0.374684\pi$
$614$	0	0
$615$	−1.57019	−0.0633162
$616$	0	0
$617$	−0.219907	−0.00885311	−0.00442656	−	0.999990i	$-0.501409\pi$
$617$	−0.219907	−0.00885311	−0.00442656	+	0.999990i	$0.501409\pi$
$618$	0	0
$619$	−15.9136	−0.639623	−0.319811	−	0.947481i	$-0.603620\pi$
$619$	−15.9136	−0.639623	−0.319811	+	0.947481i	$0.603620\pi$
$620$	0	0
$621$	−4.44401	−0.178332
$622$	0	0
$623$	29.5478	1.18381
$624$	0	0
$625$	−5.29109	−0.211644
$626$	0	0
$627$	−0.211222	−0.00843539
$628$	0	0
$629$	5.41012	0.215716
$630$	0	0
$631$	−27.6207	−1.09956	−0.549781	−	0.835309i	$-0.685289\pi$
$631$	−27.6207	−1.09956	−0.549781	+	0.835309i	$0.685289\pi$
$632$	0	0
$633$	−0.370021	−0.0147070
$634$	0	0
$635$	−10.1544	−0.402966
$636$	0	0
$637$	−8.01308	−0.317490
$638$	0	0
$639$	−15.2019	−0.601376
$640$	0	0
$641$	−4.17066	−0.164731	−0.0823656	−	0.996602i	$-0.526248\pi$
$641$	−4.17066	−0.164731	−0.0823656	+	0.996602i	$0.526248\pi$
$642$	0	0
$643$	−20.8643	−0.822806	−0.411403	−	0.911453i	$-0.634961\pi$
$643$	−20.8643	−0.822806	−0.411403	+	0.911453i	$0.634961\pi$
$644$	0	0
$645$	0.583619	0.0229800
$646$	0	0
$647$	31.7924	1.24989	0.624943	−	0.780670i	$-0.285122\pi$
$647$	31.7924	1.24989	0.624943	+	0.780670i	$0.285122\pi$
$648$	0	0
$649$	0.375512	0.0147401
$650$	0	0
$651$	−2.34564	−0.0919331
$652$	0	0
$653$	−27.2099	−1.06481	−0.532403	−	0.846491i	$-0.678711\pi$
$653$	−27.2099	−1.06481	−0.532403	+	0.846491i	$0.678711\pi$
$654$	0	0
$655$	−22.1514	−0.865528
$656$	0	0
$657$	3.54068	0.138135
$658$	0	0
$659$	−32.8090	−1.27806	−0.639030	−	0.769182i	$-0.720664\pi$
$659$	−32.8090	−1.27806	−0.639030	+	0.769182i	$0.720664\pi$
$660$	0	0
$661$	12.5824	0.489398	0.244699	−	0.969599i	$-0.421311\pi$
$661$	12.5824	0.489398	0.244699	+	0.969599i	$0.421311\pi$
$662$	0	0
$663$	1.36549	0.0530314
$664$	0	0
$665$	−4.75066	−0.184223
$666$	0	0
$667$	16.6669	0.645345
$668$	0	0
$669$	6.42915	0.248565
$670$	0	0
$671$	5.07787	0.196029
$672$	0	0
$673$	−2.58049	−0.0994704	−0.0497352	−	0.998762i	$-0.515838\pi$
$673$	−2.58049	−0.0994704	−0.0497352	+	0.998762i	$0.515838\pi$
$674$	0	0
$675$	−3.43104	−0.132061
$676$	0	0
$677$	5.86443	0.225388	0.112694	−	0.993630i	$-0.464052\pi$
$677$	5.86443	0.225388	0.112694	+	0.993630i	$0.464052\pi$
$678$	0	0
$679$	−10.8921	−0.417999
$680$	0	0
$681$	−4.45535	−0.170729
$682$	0	0
$683$	−13.3162	−0.509531	−0.254765	−	0.967003i	$-0.581998\pi$
$683$	−13.3162	−0.509531	−0.254765	+	0.967003i	$0.581998\pi$
$684$	0	0
$685$	18.4760	0.705931
$686$	0	0
$687$	−3.33183	−0.127117
$688$	0	0
$689$	15.0290	0.572560
$690$	0	0
$691$	−34.9751	−1.33051	−0.665257	−	0.746614i	$-0.731678\pi$
$691$	−34.9751	−1.33051	−0.665257	+	0.746614i	$0.731678\pi$
$692$	0	0
$693$	−8.59255	−0.326404
$694$	0	0
$695$	−17.3614	−0.658555
$696$	0	0
$697$	−8.33190	−0.315593
$698$	0	0
$699$	−2.50494	−0.0947454
$700$	0	0
$701$	4.91851	0.185769	0.0928847	−	0.995677i	$-0.470391\pi$
$701$	4.91851	0.185769	0.0928847	+	0.995677i	$0.470391\pi$
$702$	0	0
$703$	−2.95889	−0.111597
$704$	0	0
$705$	−1.17136	−0.0441159
$706$	0	0
$707$	41.4717	1.55971
$708$	0	0
$709$	27.4930	1.03252	0.516261	−	0.856431i	$-0.327324\pi$
$709$	27.4930	1.03252	0.516261	+	0.856431i	$0.327324\pi$
$710$	0	0
$711$	43.4479	1.62942
$712$	0	0
$713$	11.5831	0.433791
$714$	0	0
$715$	−4.95514	−0.185312
$716$	0	0
$717$	4.42270	0.165169
$718$	0	0
$719$	10.8746	0.405553	0.202777	−	0.979225i	$-0.435004\pi$
$719$	10.8746	0.405553	0.202777	+	0.979225i	$0.435004\pi$
$720$	0	0
$721$	−20.3580	−0.758172
$722$	0	0
$723$	3.51424	0.130696
$724$	0	0
$725$	12.8678	0.477899
$726$	0	0
$727$	−1.13818	−0.0422127	−0.0211063	−	0.999777i	$-0.506719\pi$
$727$	−1.13818	−0.0422127	−0.0211063	+	0.999777i	$0.506719\pi$
$728$	0	0
$729$	−24.3705	−0.902612
$730$	0	0
$731$	3.09686	0.114541
$732$	0	0
$733$	−27.9950	−1.03402	−0.517010	−	0.855980i	$-0.672955\pi$
$733$	−27.9950	−1.03402	−0.517010	+	0.855980i	$0.672955\pi$
$734$	0	0
$735$	0.821486	0.0303010
$736$	0	0
$737$	−6.51243	−0.239889
$738$	0	0
$739$	−44.2375	−1.62730	−0.813652	−	0.581352i	$-0.802524\pi$
$739$	−44.2375	−1.62730	−0.813652	+	0.581352i	$0.802524\pi$
$740$	0	0
$741$	−0.746814	−0.0274349
$742$	0	0
$743$	49.4810	1.81528	0.907640	−	0.419749i	$-0.137882\pi$
$743$	49.4810	1.81528	0.907640	+	0.419749i	$0.137882\pi$
$744$	0	0
$745$	−11.9063	−0.436214
$746$	0	0
$747$	−9.90126	−0.362268
$748$	0	0
$749$	−48.5597	−1.77433
$750$	0	0
$751$	25.9833	0.948145	0.474073	−	0.880486i	$-0.342783\pi$
$751$	25.9833	0.948145	0.474073	+	0.880486i	$0.342783\pi$
$752$	0	0
$753$	2.33495	0.0850904
$754$	0	0
$755$	−4.81002	−0.175055
$756$	0	0
$757$	43.1605	1.56869	0.784347	−	0.620322i	$-0.212998\pi$
$757$	43.1605	1.56869	0.784347	+	0.620322i	$0.212998\pi$
$758$	0	0
$759$	−0.709944	−0.0257693
$760$	0	0
$761$	4.16498	0.150981	0.0754903	−	0.997147i	$-0.475948\pi$
$761$	4.16498	0.150981	0.0754903	+	0.997147i	$0.475948\pi$
$762$	0	0
$763$	−41.1140	−1.48843
$764$	0	0
$765$	8.36664	0.302497
$766$	0	0
$767$	1.32769	0.0479402
$768$	0	0
$769$	5.24441	0.189118	0.0945591	−	0.995519i	$-0.469856\pi$
$769$	5.24441	0.189118	0.0945591	+	0.995519i	$0.469856\pi$
$770$	0	0
$771$	0.378011	0.0136137
$772$	0	0
$773$	3.91186	0.140700	0.0703499	−	0.997522i	$-0.477588\pi$
$773$	3.91186	0.140700	0.0703499	+	0.997522i	$0.477588\pi$
$774$	0	0
$775$	8.94284	0.321236
$776$	0	0
$777$	2.01396	0.0722505
$778$	0	0
$779$	4.55687	0.163267
$780$	0	0
$781$	−4.89772	−0.175254
$782$	0	0
$783$	−6.55632	−0.234304
$784$	0	0
$785$	23.1061	0.824692
$786$	0	0
$787$	−16.8869	−0.601952	−0.300976	−	0.953632i	$-0.597312\pi$
$787$	−16.8869	−0.601952	−0.300976	+	0.953632i	$0.597312\pi$
$788$	0	0
$789$	3.07421	0.109445
$790$	0	0
$791$	−20.3960	−0.725199
$792$	0	0
$793$	17.9537	0.637556
$794$	0	0
$795$	−1.54075	−0.0546447
$796$	0	0
$797$	25.3722	0.898729	0.449365	−	0.893348i	$-0.351650\pi$
$797$	25.3722	0.898729	0.449365	+	0.893348i	$0.351650\pi$
$798$	0	0
$799$	−6.21557	−0.219891
$800$	0	0
$801$	28.4606	1.00561
$802$	0	0
$803$	1.14073	0.0402556
$804$	0	0
$805$	−15.9676	−0.562783
$806$	0	0
$807$	3.32522	0.117053
$808$	0	0
$809$	−31.5640	−1.10973	−0.554866	−	0.831940i	$-0.687230\pi$
$809$	−31.5640	−1.10973	−0.554866	+	0.831940i	$0.687230\pi$
$810$	0	0
$811$	44.1203	1.54927	0.774636	−	0.632407i	$-0.217933\pi$
$811$	44.1203	1.54927	0.774636	+	0.632407i	$0.217933\pi$
$812$	0	0
$813$	−0.916949	−0.0321588
$814$	0	0
$815$	−28.7646	−1.00758
$816$	0	0
$817$	−1.69373	−0.0592560
$818$	0	0
$819$	−30.3805	−1.06158
$820$	0	0
$821$	3.81015	0.132975	0.0664876	−	0.997787i	$-0.478821\pi$
$821$	3.81015	0.132975	0.0664876	+	0.997787i	$0.478821\pi$
$822$	0	0
$823$	34.3301	1.19667	0.598336	−	0.801245i	$-0.295829\pi$
$823$	34.3301	1.19667	0.598336	+	0.801245i	$0.295829\pi$
$824$	0	0
$825$	−0.548119	−0.0190830
$826$	0	0
$827$	−30.7484	−1.06923	−0.534613	−	0.845097i	$-0.679543\pi$
$827$	−30.7484	−1.06923	−0.534613	+	0.845097i	$0.679543\pi$
$828$	0	0
$829$	−10.1917	−0.353971	−0.176986	−	0.984213i	$-0.556635\pi$
$829$	−10.1917	−0.353971	−0.176986	+	0.984213i	$0.556635\pi$
$830$	0	0
$831$	5.72633	0.198644
$832$	0	0
$833$	4.35905	0.151032
$834$	0	0
$835$	17.7617	0.614669
$836$	0	0
$837$	−4.55649	−0.157495
$838$	0	0
$839$	−7.81123	−0.269674	−0.134837	−	0.990868i	$-0.543051\pi$
$839$	−7.81123	−0.269674	−0.134837	+	0.990868i	$0.543051\pi$
$840$	0	0
$841$	−4.41104	−0.152105
$842$	0	0
$843$	2.28177	0.0785884
$844$	0	0
$845$	2.64074	0.0908442
$846$	0	0
$847$	30.9284	1.06271
$848$	0	0
$849$	1.55424	0.0533415
$850$	0	0
$851$	−9.94521	−0.340918
$852$	0	0
$853$	−46.9706	−1.60824	−0.804122	−	0.594465i	$-0.797364\pi$
$853$	−46.9706	−1.60824	−0.804122	+	0.594465i	$0.797364\pi$
$854$	0	0
$855$	−4.57587	−0.156491
$856$	0	0
$857$	2.24191	0.0765821	0.0382911	−	0.999267i	$-0.487809\pi$
$857$	2.24191	0.0765821	0.0382911	+	0.999267i	$0.487809\pi$
$858$	0	0
$859$	−49.9456	−1.70412	−0.852061	−	0.523442i	$-0.824648\pi$
$859$	−49.9456	−1.70412	−0.852061	+	0.523442i	$0.824648\pi$
$860$	0	0
$861$	−3.10162	−0.105703
$862$	0	0
$863$	−10.4322	−0.355115	−0.177557	−	0.984110i	$-0.556820\pi$
$863$	−10.4322	−0.355115	−0.177557	+	0.984110i	$0.556820\pi$
$864$	0	0
$865$	−11.1807	−0.380154
$866$	0	0
$867$	3.03444	0.103055
$868$	0	0
$869$	13.9980	0.474849
$870$	0	0
$871$	−23.0259	−0.780203
$872$	0	0
$873$	−10.4913	−0.355077
$874$	0	0
$875$	−36.0812	−1.21977
$876$	0	0
$877$	23.6385	0.798215	0.399107	−	0.916904i	$-0.369320\pi$
$877$	23.6385	0.798215	0.399107	+	0.916904i	$0.369320\pi$
$878$	0	0
$879$	7.07534	0.238645
$880$	0	0
$881$	−29.2713	−0.986175	−0.493087	−	0.869980i	$-0.664132\pi$
$881$	−29.2713	−0.986175	−0.493087	+	0.869980i	$0.664132\pi$
$882$	0	0
$883$	−42.1875	−1.41972	−0.709862	−	0.704341i	$-0.751243\pi$
$883$	−42.1875	−1.41972	−0.709862	+	0.704341i	$0.751243\pi$
$884$	0	0
$885$	−0.136112	−0.00457537
$886$	0	0
$887$	1.32254	0.0444064	0.0222032	−	0.999753i	$-0.492932\pi$
$887$	1.32254	0.0444064	0.0222032	+	0.999753i	$0.492932\pi$
$888$	0	0
$889$	−20.0582	−0.672730
$890$	0	0
$891$	−8.13561	−0.272553
$892$	0	0
$893$	3.39941	0.113757
$894$	0	0
$895$	7.60286	0.254136
$896$	0	0
$897$	−2.51014	−0.0838110
$898$	0	0
$899$	17.0887	0.569941
$900$	0	0
$901$	−8.17567	−0.272371
$902$	0	0
$903$	1.15283	0.0383638
$904$	0	0
$905$	−29.7406	−0.988611
$906$	0	0
$907$	−3.58771	−0.119128	−0.0595639	−	0.998224i	$-0.518971\pi$
$907$	−3.58771	−0.119128	−0.0595639	+	0.998224i	$0.518971\pi$
$908$	0	0
$909$	39.9459	1.32492
$910$	0	0
$911$	14.6828	0.486464	0.243232	−	0.969968i	$-0.421792\pi$
$911$	14.6828	0.486464	0.243232	+	0.969968i	$0.421792\pi$
$912$	0	0
$913$	−3.18998	−0.105573
$914$	0	0
$915$	−1.84058	−0.0608478
$916$	0	0
$917$	−43.7561	−1.44495
$918$	0	0
$919$	38.2168	1.26066	0.630328	−	0.776329i	$-0.282920\pi$
$919$	38.2168	1.26066	0.630328	+	0.776329i	$0.282920\pi$
$920$	0	0
$921$	−1.43494	−0.0472829
$922$	0	0
$923$	−17.3168	−0.569988
$924$	0	0
$925$	−7.67829	−0.252461
$926$	0	0
$927$	−19.6090	−0.644044
$928$	0	0
$929$	36.8919	1.21038	0.605192	−	0.796080i	$-0.293096\pi$
$929$	36.8919	1.21038	0.605192	+	0.796080i	$0.293096\pi$
$930$	0	0
$931$	−2.38404	−0.0781339
$932$	0	0
$933$	−6.26521	−0.205114
$934$	0	0
$935$	2.69555	0.0881541
$936$	0	0
$937$	33.7347	1.10206	0.551032	−	0.834484i	$-0.314234\pi$
$937$	33.7347	1.10206	0.551032	+	0.834484i	$0.314234\pi$
$938$	0	0
$939$	−3.80418	−0.124145
$940$	0	0
$941$	3.91029	0.127472	0.0637360	−	0.997967i	$-0.479698\pi$
$941$	3.91029	0.127472	0.0637360	+	0.997967i	$0.479698\pi$
$942$	0	0
$943$	15.3162	0.498765
$944$	0	0
$945$	6.28122	0.204328
$946$	0	0
$947$	−46.8518	−1.52248	−0.761240	−	0.648470i	$-0.775409\pi$
$947$	−46.8518	−1.52248	−0.761240	+	0.648470i	$0.775409\pi$
$948$	0	0
$949$	4.03327	0.130925
$950$	0	0
$951$	6.57333	0.213155
$952$	0	0
$953$	26.5470	0.859941	0.429971	−	0.902843i	$-0.358524\pi$
$953$	26.5470	0.859941	0.429971	+	0.902843i	$0.358524\pi$
$954$	0	0
$955$	−35.4844	−1.14825
$956$	0	0
$957$	−1.04739	−0.0338574
$958$	0	0
$959$	36.4959	1.17851
$960$	0	0
$961$	−19.1237	−0.616895
$962$	0	0
$963$	−46.7731	−1.50724
$964$	0	0
$965$	3.27606	0.105460
$966$	0	0
$967$	−6.80878	−0.218956	−0.109478	−	0.993989i	$-0.534918\pi$
$967$	−6.80878	−0.218956	−0.109478	+	0.993989i	$0.534918\pi$
$968$	0	0
$969$	0.406261	0.0130510
$970$	0	0
$971$	48.4549	1.55499	0.777496	−	0.628887i	$-0.216489\pi$
$971$	48.4549	1.55499	0.777496	+	0.628887i	$0.216489\pi$
$972$	0	0
$973$	−34.2942	−1.09942
$974$	0	0
$975$	−1.93797	−0.0620648
$976$	0	0
$977$	46.0137	1.47211	0.736055	−	0.676922i	$-0.236686\pi$
$977$	46.0137	1.47211	0.736055	+	0.676922i	$0.236686\pi$
$978$	0	0
$979$	9.16941	0.293056
$980$	0	0
$981$	−39.6013	−1.26437
$982$	0	0
$983$	−27.8031	−0.886782	−0.443391	−	0.896328i	$-0.646225\pi$
$983$	−27.8031	−0.886782	−0.443391	+	0.896328i	$0.646225\pi$
$984$	0	0
$985$	9.38258	0.298954
$986$	0	0
$987$	−2.31380	−0.0736490
$988$	0	0
$989$	−5.69283	−0.181022
$990$	0	0
$991$	−5.16951	−0.164215	−0.0821074	−	0.996623i	$-0.526165\pi$
$991$	−5.16951	−0.164215	−0.0821074	+	0.996623i	$0.526165\pi$
$992$	0	0
$993$	−4.56646	−0.144912
$994$	0	0
$995$	−6.25116	−0.198175
$996$	0	0
$997$	18.7227	0.592953	0.296476	−	0.955040i	$-0.404188\pi$
$997$	18.7227	0.592953	0.296476	+	0.955040i	$0.404188\pi$
$998$	0	0
$999$	3.91218	0.123776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.br.1.3		8
4.3	odd	2		4864.2.a.bm.1.5		8
8.3	odd	2	inner	4864.2.a.br.1.4		8
8.5	even	2		4864.2.a.bm.1.6		8
16.3	odd	4		2432.2.c.i.1217.10	yes	16
16.5	even	4		2432.2.c.i.1217.9	yes	16
16.11	odd	4		2432.2.c.i.1217.7	✓	16
16.13	even	4		2432.2.c.i.1217.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.i.1217.7	✓	16	16.11	odd	4
2432.2.c.i.1217.8	yes	16	16.13	even	4
2432.2.c.i.1217.9	yes	16	16.5	even	4
2432.2.c.i.1217.10	yes	16	16.3	odd	4
4864.2.a.bm.1.5		8	4.3	odd	2
4864.2.a.bm.1.6		8	8.5	even	2
4864.2.a.br.1.3		8	1.1	even	1	trivial
4864.2.a.br.1.4		8	8.3	odd	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:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q-0.222191 q^{3} -1.55081 q^{5} -3.06334 q^{7} -2.95063 q^{9} +O(q^{10})\)	\(q-0.222191 q^{3} -1.55081 q^{5} -3.06334 q^{7} -2.95063 q^{9} -0.950631 q^{11} -3.36113 q^{13} +0.344577 q^{15} +1.82843 q^{17} -1.00000 q^{19} +0.680647 q^{21} -3.36113 q^{23} -2.59499 q^{25} +1.32218 q^{27} -4.95873 q^{29} -3.44620 q^{31} +0.211222 q^{33} +4.75066 q^{35} +2.95889 q^{37} +0.746814 q^{39} -4.55687 q^{41} +1.69373 q^{43} +4.57587 q^{45} -3.39941 q^{47} +2.38404 q^{49} -0.406261 q^{51} -4.47142 q^{53} +1.47425 q^{55} +0.222191 q^{57} -0.395014 q^{59} -5.34158 q^{61} +9.03878 q^{63} +5.21247 q^{65} +6.85064 q^{67} +0.746814 q^{69} +5.15207 q^{71} -1.19998 q^{73} +0.576584 q^{75} +2.91210 q^{77} -14.7249 q^{79} +8.55812 q^{81} +3.35564 q^{83} -2.83554 q^{85} +1.10179 q^{87} -9.64561 q^{89} +10.2963 q^{91} +0.765715 q^{93} +1.55081 q^{95} +3.55562 q^{97} +2.80496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 + 4 * q^9	\( 8 q + 4 q^{3} + 4 q^{9} + 20 q^{11} - 8 q^{17} - 8 q^{19} + 20 q^{25} + 4 q^{27} + 24 q^{33} + 12 q^{35} + 8 q^{41} + 28 q^{43} + 8 q^{49} + 12 q^{51} - 4 q^{57} + 36 q^{59} + 8 q^{65} + 28 q^{67} - 8 q^{73} + 68 q^{75} - 32 q^{81} + 40 q^{83} - 8 q^{89} - 12 q^{91} + 40 q^{97} + 60 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 + 4 * q^9 + 20 * q^11 - 8 * q^17 - 8 * q^19 + 20 * q^25 + 4 * q^27 + 24 * q^33 + 12 * q^35 + 8 * q^41 + 28 * q^43 + 8 * q^49 + 12 * q^51 - 4 * q^57 + 36 * q^59 + 8 * q^65 + 28 * q^67 - 8 * q^73 + 68 * q^75 - 32 * q^81 + 40 * q^83 - 8 * q^89 - 12 * q^91 + 40 * q^97 + 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more