LMFDB - Embedded newform 5148.2.a.l.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
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			1.1
Root			$2.86620$ 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)

$f(q)$	$=$	$q-4.38350 q^{5} +3.21509 q^{7} +O(q^{10})$	$q-4.38350 q^{5} +3.21509 q^{7} +1.00000 q^{11} +1.00000 q^{13} +1.86620 q^{17} -6.18048 q^{19} -3.48270 q^{23} +14.2151 q^{25} -2.83159 q^{29} -1.08129 q^{31} -14.0934 q^{35} -2.69779 q^{37} +1.21509 q^{41} -6.83159 q^{43} +8.43018 q^{47} +3.33682 q^{49} +12.1626 q^{53} -4.38350 q^{55} +8.76700 q^{59} -11.4648 q^{61} -4.38350 q^{65} +8.81369 q^{67} -4.69779 q^{71} +13.9129 q^{73} +3.21509 q^{77} -13.6678 q^{79} +8.76700 q^{83} -8.18048 q^{85} -18.4769 q^{89} +3.21509 q^{91} +27.0922 q^{95} -12.7670 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{5} - 4 q^{7}+O(q^{10}) $ 3 * q - 4 * q^5 - 4 * q^7	$ 3 q - 4 q^{5} - 4 q^{7} + 3 q^{11} + 3 q^{13} - 2 q^{17} - 8 q^{19} - 12 q^{23} + 29 q^{25} - 4 q^{29} + 18 q^{31} - 6 q^{35} + 4 q^{37} - 10 q^{41} - 16 q^{43} - 2 q^{47} + 19 q^{49} - 6 q^{53} - 4 q^{55} + 8 q^{59} - 4 q^{61} - 4 q^{65} - 10 q^{67} - 2 q^{71} + 16 q^{73} - 4 q^{77} - 12 q^{79} + 8 q^{83} - 14 q^{85} - 10 q^{89} - 4 q^{91} - 22 q^{95} - 20 q^{97}+O(q^{100}) $ 3 * q - 4 * q^5 - 4 * q^7 + 3 * q^11 + 3 * q^13 - 2 * q^17 - 8 * q^19 - 12 * q^23 + 29 * q^25 - 4 * q^29 + 18 * q^31 - 6 * q^35 + 4 * q^37 - 10 * q^41 - 16 * q^43 - 2 * q^47 + 19 * q^49 - 6 * q^53 - 4 * q^55 + 8 * q^59 - 4 * q^61 - 4 * q^65 - 10 * q^67 - 2 * q^71 + 16 * q^73 - 4 * q^77 - 12 * q^79 + 8 * q^83 - 14 * q^85 - 10 * q^89 - 4 * q^91 - 22 * q^95 - 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
$5$	−4.38350	−1.96036	−0.980181	−	0.198104i	$-0.936521\pi$
$5$	−4.38350	−1.96036	−0.980181	+	0.198104i	$0.936521\pi$
$6$	0	0
$7$	3.21509	1.21519	0.607595	−	0.794247i	$-0.292134\pi$
$7$	3.21509	1.21519	0.607595	+	0.794247i	$0.292134\pi$
$8$	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
$17$	1.86620	0.452620	0.226310	−	0.974055i	$-0.427334\pi$
$17$	1.86620	0.452620	0.226310	+	0.974055i	$0.427334\pi$
$18$	0	0
$19$	−6.18048	−1.41790	−0.708950	−	0.705259i	$-0.750831\pi$
$19$	−6.18048	−1.41790	−0.708950	+	0.705259i	$0.750831\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.48270	−0.726192	−0.363096	−	0.931752i	$-0.618280\pi$
$23$	−3.48270	−0.726192	−0.363096	+	0.931752i	$0.618280\pi$
$24$	0	0
$25$	14.2151	2.84302
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.83159	−0.525813	−0.262907	−	0.964821i	$-0.584681\pi$
$29$	−2.83159	−0.525813	−0.262907	+	0.964821i	$0.584681\pi$
$30$	0	0
$31$	−1.08129	−0.194206	−0.0971028	−	0.995274i	$-0.530958\pi$
$31$	−1.08129	−0.194206	−0.0971028	+	0.995274i	$0.530958\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−14.0934	−2.38221
$36$	0	0
$37$	−2.69779	−0.443514	−0.221757	−	0.975102i	$-0.571179\pi$
$37$	−2.69779	−0.443514	−0.221757	+	0.975102i	$0.571179\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.21509	0.189766	0.0948828	−	0.995488i	$-0.469752\pi$
$41$	1.21509	0.189766	0.0948828	+	0.995488i	$0.469752\pi$
$42$	0	0
$43$	−6.83159	−1.04181	−0.520904	−	0.853615i	$-0.674405\pi$
$43$	−6.83159	−1.04181	−0.520904	+	0.853615i	$0.674405\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.43018	1.22967	0.614834	−	0.788656i	$-0.289223\pi$
$47$	8.43018	1.22967	0.614834	+	0.788656i	$0.289223\pi$
$48$	0	0
$49$	3.33682	0.476689
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.1626	1.67066	0.835330	−	0.549750i	$-0.185277\pi$
$53$	12.1626	1.67066	0.835330	+	0.549750i	$0.185277\pi$
$54$	0	0
$55$	−4.38350	−0.591071
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.76700	1.14137	0.570683	−	0.821170i	$-0.306678\pi$
$59$	8.76700	1.14137	0.570683	+	0.821170i	$0.306678\pi$
$60$	0	0
$61$	−11.4648	−1.46792	−0.733958	−	0.679195i	$-0.762329\pi$
$61$	−11.4648	−1.46792	−0.733958	+	0.679195i	$0.762329\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.38350	−0.543707
$66$	0	0
$67$	8.81369	1.07676	0.538382	−	0.842701i	$-0.319036\pi$
$67$	8.81369	1.07676	0.538382	+	0.842701i	$0.319036\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.69779	−0.557525	−0.278762	−	0.960360i	$-0.589924\pi$
$71$	−4.69779	−0.557525	−0.278762	+	0.960360i	$0.589924\pi$
$72$	0	0
$73$	13.9129	1.62838	0.814190	−	0.580599i	$-0.197182\pi$
$73$	13.9129	1.62838	0.814190	+	0.580599i	$0.197182\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.21509	0.366394
$78$	0	0
$79$	−13.6678	−1.53775	−0.768874	−	0.639400i	$-0.779183\pi$
$79$	−13.6678	−1.53775	−0.768874	+	0.639400i	$0.779183\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.76700	0.962304	0.481152	−	0.876637i	$-0.340219\pi$
$83$	8.76700	0.962304	0.481152	+	0.876637i	$0.340219\pi$
$84$	0	0
$85$	−8.18048	−0.887298
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−18.4769	−1.95854	−0.979272	−	0.202550i	$-0.935077\pi$
$89$	−18.4769	−1.95854	−0.979272	+	0.202550i	$0.935077\pi$
$90$	0	0
$91$	3.21509	0.337033
$92$	0	0
$93$	0	0
$94$	0	0
$95$	27.0922	2.77960
$96$	0	0
$97$	−12.7670	−1.29629	−0.648146	−	0.761516i	$-0.724456\pi$
$97$	−12.7670	−1.29629	−0.648146	+	0.761516i	$0.724456\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.49477	0.248239	0.124119	−	0.992267i	$-0.460389\pi$
$101$	2.49477	0.248239	0.124119	+	0.992267i	$0.460389\pi$
$102$	0	0
$103$	19.8258	1.95349	0.976745	−	0.214403i	$-0.0687807\pi$
$103$	19.8258	1.95349	0.976745	+	0.214403i	$0.0687807\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.6978	−1.03419	−0.517097	−	0.855927i	$-0.672987\pi$
$107$	−10.6978	−1.03419	−0.517097	+	0.855927i	$0.672987\pi$
$108$	0	0
$109$	−2.44809	−0.234484	−0.117242	−	0.993103i	$-0.537405\pi$
$109$	−2.44809	−0.234484	−0.117242	+	0.993103i	$0.537405\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.80161	−0.733914	−0.366957	−	0.930238i	$-0.619600\pi$
$113$	−7.80161	−0.733914	−0.366957	+	0.930238i	$0.619600\pi$
$114$	0	0
$115$	15.2664	1.42360
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−40.3944	−3.61298
$126$	0	0
$127$	−10.9008	−0.967290	−0.483645	−	0.875264i	$-0.660687\pi$
$127$	−10.9008	−0.967290	−0.483645	+	0.875264i	$0.660687\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.03461	0.439876	0.219938	−	0.975514i	$-0.429415\pi$
$131$	5.03461	0.439876	0.219938	+	0.975514i	$0.429415\pi$
$132$	0	0
$133$	−19.8708	−1.72302
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.987925	0.0844042	0.0422021	−	0.999109i	$-0.486563\pi$
$137$	0.987925	0.0844042	0.0422021	+	0.999109i	$0.486563\pi$
$138$	0	0
$139$	−7.79698	−0.661331	−0.330666	−	0.943748i	$-0.607273\pi$
$139$	−7.79698	−0.661331	−0.330666	+	0.943748i	$0.607273\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	12.4123	1.03078
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.6107	−0.869260	−0.434630	−	0.900609i	$-0.643121\pi$
$149$	−10.6107	−0.869260	−0.434630	+	0.900609i	$0.643121\pi$
$150$	0	0
$151$	−13.6453	−1.11044	−0.555218	−	0.831705i	$-0.687365\pi$
$151$	−13.6453	−1.11044	−0.555218	+	0.831705i	$0.687365\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.73984	0.380713
$156$	0	0
$157$	4.85412	0.387401	0.193701	−	0.981061i	$-0.437951\pi$
$157$	4.85412	0.387401	0.193701	+	0.981061i	$0.437951\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−11.1972	−0.882462
$162$	0	0
$163$	−10.1851	−0.797760	−0.398880	−	0.917003i	$-0.630601\pi$
$163$	−10.1851	−0.797760	−0.398880	+	0.917003i	$0.630601\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.4302	−1.11664	−0.558321	−	0.829625i	$-0.688554\pi$
$167$	−14.4302	−1.11664	−0.558321	+	0.829625i	$0.688554\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−25.8662	−1.96657	−0.983285	−	0.182071i	$-0.941720\pi$
$173$	−25.8662	−1.96657	−0.983285	+	0.182071i	$0.941720\pi$
$174$	0	0
$175$	45.7028	3.45481
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.3189	−0.771272	−0.385636	−	0.922651i	$-0.626018\pi$
$179$	−10.3189	−0.771272	−0.385636	+	0.922651i	$0.626018\pi$
$180$	0	0
$181$	−5.71449	−0.424755	−0.212377	−	0.977188i	$-0.568121\pi$
$181$	−5.71449	−0.424755	−0.212377	+	0.977188i	$0.568121\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.8258	0.869447
$186$	0	0
$187$	1.86620	0.136470
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.628572	0.0454819	0.0227409	−	0.999741i	$-0.492761\pi$
$191$	0.628572	0.0454819	0.0227409	+	0.999741i	$0.492761\pi$
$192$	0	0
$193$	18.9475	1.36387	0.681935	−	0.731413i	$-0.261139\pi$
$193$	18.9475	1.36387	0.681935	+	0.731413i	$0.261139\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.51730	−0.179350	−0.0896752	−	0.995971i	$-0.528583\pi$
$197$	−2.51730	−0.179350	−0.0896752	+	0.995971i	$0.528583\pi$
$198$	0	0
$199$	−1.56982	−0.111281	−0.0556406	−	0.998451i	$-0.517720\pi$
$199$	−1.56982	−0.111281	−0.0556406	+	0.998451i	$0.517720\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9.10382	−0.638963
$204$	0	0
$205$	−5.32636	−0.372009
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.18048	−0.427513
$210$	0	0
$211$	−19.9596	−1.37407	−0.687037	−	0.726623i	$-0.741089\pi$
$211$	−19.9596	−1.37407	−0.687037	+	0.726623i	$0.741089\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	29.9463	2.04232
$216$	0	0
$217$	−3.47645	−0.235997
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.86620	0.125534
$222$	0	0
$223$	12.5461	0.840148	0.420074	−	0.907490i	$-0.362004\pi$
$223$	12.5461	0.840148	0.420074	+	0.907490i	$0.362004\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.14588	−0.208799	−0.104400	−	0.994535i	$-0.533292\pi$
$227$	−3.14588	−0.208799	−0.104400	+	0.994535i	$0.533292\pi$
$228$	0	0
$229$	−14.5686	−0.962721	−0.481361	−	0.876523i	$-0.659857\pi$
$229$	−14.5686	−0.962721	−0.481361	+	0.876523i	$0.659857\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.52938	−0.100193	−0.0500965	−	0.998744i	$-0.515953\pi$
$233$	−1.52938	−0.100193	−0.0500965	+	0.998744i	$0.515953\pi$
$234$	0	0
$235$	−36.9537	−2.41060
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.5415	0.811239	0.405620	−	0.914042i	$-0.367056\pi$
$239$	12.5415	0.811239	0.405620	+	0.914042i	$0.367056\pi$
$240$	0	0
$241$	−2.40604	−0.154986	−0.0774932	−	0.996993i	$-0.524692\pi$
$241$	−2.40604	−0.154986	−0.0774932	+	0.996993i	$0.524692\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−14.6270	−0.934482
$246$	0	0
$247$	−6.18048	−0.393255
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.0934	−1.26828	−0.634141	−	0.773217i	$-0.718646\pi$
$251$	−20.0934	−1.26828	−0.634141	+	0.773217i	$0.718646\pi$
$252$	0	0
$253$	−3.48270	−0.218955
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−8.67364	−0.538954
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.23300	0.199355	0.0996775	−	0.995020i	$-0.468219\pi$
$263$	3.23300	0.199355	0.0996775	+	0.995020i	$0.468219\pi$
$264$	0	0
$265$	−53.3147	−3.27510
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.19719	−0.316878	−0.158439	−	0.987369i	$-0.550646\pi$
$269$	−5.19719	−0.316878	−0.158439	+	0.987369i	$0.550646\pi$
$270$	0	0
$271$	2.58652	0.157120	0.0785600	−	0.996909i	$-0.474968\pi$
$271$	2.58652	0.157120	0.0785600	+	0.996909i	$0.474968\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	14.2151	0.857202
$276$	0	0
$277$	−29.9642	−1.80037	−0.900187	−	0.435504i	$-0.856570\pi$
$277$	−29.9642	−1.80037	−0.900187	+	0.435504i	$0.856570\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.12173	−0.305537	−0.152768	−	0.988262i	$-0.548819\pi$
$281$	−5.12173	−0.305537	−0.152768	+	0.988262i	$0.548819\pi$
$282$	0	0
$283$	27.2260	1.61842	0.809208	−	0.587522i	$-0.199897\pi$
$283$	27.2260	1.61842	0.809208	+	0.587522i	$0.199897\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.90663	0.230601
$288$	0	0
$289$	−13.5173	−0.795136
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.62113	0.328390	0.164195	−	0.986428i	$-0.447497\pi$
$293$	5.62113	0.328390	0.164195	+	0.986428i	$0.447497\pi$
$294$	0	0
$295$	−38.4302	−2.23749
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.48270	−0.201410
$300$	0	0
$301$	−21.9642	−1.26600
$302$	0	0
$303$	0	0
$304$	0	0
$305$	50.2559	2.87765
$306$	0	0
$307$	−18.6799	−1.06612	−0.533059	−	0.846078i	$-0.678958\pi$
$307$	−18.6799	−1.06612	−0.533059	+	0.846078i	$0.678958\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−22.4481	−1.27291	−0.636457	−	0.771312i	$-0.719601\pi$
$311$	−22.4481	−1.27291	−0.636457	+	0.771312i	$0.719601\pi$
$312$	0	0
$313$	−13.8079	−0.780466	−0.390233	−	0.920716i	$-0.627606\pi$
$313$	−13.8079	−0.780466	−0.390233	+	0.920716i	$0.627606\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.4769	−1.71175	−0.855876	−	0.517182i	$-0.826981\pi$
$317$	−30.4769	−1.71175	−0.855876	+	0.517182i	$0.826981\pi$
$318$	0	0
$319$	−2.83159	−0.158539
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−11.5340	−0.641769
$324$	0	0
$325$	14.2151	0.788511
$326$	0	0
$327$	0	0
$328$	0	0
$329$	27.1038	1.49428
$330$	0	0
$331$	−22.0109	−1.20983	−0.604914	−	0.796291i	$-0.706792\pi$
$331$	−22.0109	−1.20983	−0.604914	+	0.796291i	$0.706792\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−38.6348	−2.11085
$336$	0	0
$337$	25.6966	1.39978	0.699891	−	0.714249i	$-0.253232\pi$
$337$	25.6966	1.39978	0.699891	+	0.714249i	$0.253232\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.08129	−0.0585552
$342$	0	0
$343$	−11.7775	−0.635923
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−34.7312	−1.86447	−0.932234	−	0.361855i	$-0.882143\pi$
$347$	−34.7312	−1.86447	−0.932234	+	0.361855i	$0.882143\pi$
$348$	0	0
$349$	5.05876	0.270789	0.135395	−	0.990792i	$-0.456770\pi$
$349$	5.05876	0.270789	0.135395	+	0.990792i	$0.456770\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.9175	1.16655	0.583276	−	0.812274i	$-0.301771\pi$
$353$	21.9175	1.16655	0.583276	+	0.812274i	$0.301771\pi$
$354$	0	0
$355$	20.5928	1.09295
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.31892	0.227944	0.113972	−	0.993484i	$-0.463643\pi$
$359$	4.31892	0.227944	0.113972	+	0.993484i	$0.463643\pi$
$360$	0	0
$361$	19.1984	1.01044
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−60.9871	−3.19221
$366$	0	0
$367$	−27.6966	−1.44575	−0.722875	−	0.690979i	$-0.757180\pi$
$367$	−27.6966	−1.44575	−0.722875	+	0.690979i	$0.757180\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	39.1038	2.03017
$372$	0	0
$373$	−10.6286	−0.550327	−0.275163	−	0.961398i	$-0.588732\pi$
$373$	−10.6286	−0.550327	−0.275163	+	0.961398i	$0.588732\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.83159	−0.145834
$378$	0	0
$379$	12.2543	0.629463	0.314731	−	0.949181i	$-0.398086\pi$
$379$	12.2543	0.629463	0.314731	+	0.949181i	$0.398086\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	17.6632	0.902546	0.451273	−	0.892386i	$-0.350970\pi$
$383$	17.6632	0.902546	0.451273	+	0.892386i	$0.350970\pi$
$384$	0	0
$385$	−14.0934	−0.718264
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−26.0934	−1.32299	−0.661493	−	0.749951i	$-0.730077\pi$
$389$	−26.0934	−1.32299	−0.661493	+	0.749951i	$0.730077\pi$
$390$	0	0
$391$	−6.49940	−0.328689
$392$	0	0
$393$	0	0
$394$	0	0
$395$	59.9129	3.01454
$396$	0	0
$397$	23.5582	1.18235	0.591175	−	0.806543i	$-0.298664\pi$
$397$	23.5582	1.18235	0.591175	+	0.806543i	$0.298664\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.8829	1.44234	0.721172	−	0.692756i	$-0.243604\pi$
$401$	28.8829	1.44234	0.721172	+	0.692756i	$0.243604\pi$
$402$	0	0
$403$	−1.08129	−0.0538629
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.69779	−0.133724
$408$	0	0
$409$	16.5173	0.816728	0.408364	−	0.912819i	$-0.366099\pi$
$409$	16.5173	0.816728	0.408364	+	0.912819i	$0.366099\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	28.1867	1.38698
$414$	0	0
$415$	−38.4302	−1.88646
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.54145	0.319571	0.159785	−	0.987152i	$-0.448920\pi$
$419$	6.54145	0.319571	0.159785	+	0.987152i	$0.448920\pi$
$420$	0	0
$421$	28.4302	1.38560	0.692801	−	0.721129i	$-0.256376\pi$
$421$	28.4302	1.38560	0.692801	+	0.721129i	$0.256376\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	26.5282	1.28681
$426$	0	0
$427$	−36.8604	−1.78380
$428$	0	0
$429$	0	0
$430$	0	0
$431$	28.3610	1.36610	0.683050	−	0.730372i	$-0.260653\pi$
$431$	28.3610	1.36610	0.683050	+	0.730372i	$0.260653\pi$
$432$	0	0
$433$	18.8604	0.906372	0.453186	−	0.891416i	$-0.350287\pi$
$433$	18.8604	0.906372	0.453186	+	0.891416i	$0.350287\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21.5247	1.02967
$438$	0	0
$439$	−8.76237	−0.418205	−0.209103	−	0.977894i	$-0.567054\pi$
$439$	−8.76237	−0.418205	−0.209103	+	0.977894i	$0.567054\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.9988	0.617592	0.308796	−	0.951128i	$-0.400074\pi$
$443$	12.9988	0.617592	0.308796	+	0.951128i	$0.400074\pi$
$444$	0	0
$445$	80.9934	3.83945
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.58189	0.121847	0.0609235	−	0.998142i	$-0.480595\pi$
$449$	2.58189	0.121847	0.0609235	+	0.998142i	$0.480595\pi$
$450$	0	0
$451$	1.21509	0.0572165
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−14.0934	−0.660707
$456$	0	0
$457$	1.82576	0.0854055	0.0427028	−	0.999088i	$-0.486403\pi$
$457$	1.82576	0.0854055	0.0427028	+	0.999088i	$0.486403\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.0755	0.841858	0.420929	−	0.907094i	$-0.361704\pi$
$461$	18.0755	0.841858	0.420929	+	0.907094i	$0.361704\pi$
$462$	0	0
$463$	19.1747	0.891122	0.445561	−	0.895252i	$-0.353004\pi$
$463$	19.1747	0.891122	0.445561	+	0.895252i	$0.353004\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.51730	0.394134	0.197067	−	0.980390i	$-0.436858\pi$
$467$	8.51730	0.394134	0.197067	+	0.980390i	$0.436858\pi$
$468$	0	0
$469$	28.3368	1.30847
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.83159	−0.314117
$474$	0	0
$475$	−87.8562	−4.03112
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.2855	−0.469957	−0.234978	−	0.972001i	$-0.575502\pi$
$479$	−10.2855	−0.469957	−0.234978	+	0.972001i	$0.575502\pi$
$480$	0	0
$481$	−2.69779	−0.123009
$482$	0	0
$483$	0	0
$484$	0	0
$485$	55.9642	2.54120
$486$	0	0
$487$	−32.2093	−1.45954	−0.729771	−	0.683692i	$-0.760373\pi$
$487$	−32.2093	−1.45954	−0.729771	+	0.683692i	$0.760373\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−9.90663	−0.447080	−0.223540	−	0.974695i	$-0.571761\pi$
$491$	−9.90663	−0.447080	−0.223540	+	0.974695i	$0.571761\pi$
$492$	0	0
$493$	−5.28431	−0.237993
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.1038	−0.677499
$498$	0	0
$499$	−17.6165	−0.788623	−0.394311	−	0.918977i	$-0.629017\pi$
$499$	−17.6165	−0.788623	−0.394311	+	0.918977i	$0.629017\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.2330	0.679206	0.339603	−	0.940569i	$-0.389707\pi$
$503$	15.2330	0.679206	0.339603	+	0.940569i	$0.389707\pi$
$504$	0	0
$505$	−10.9358	−0.486638
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16.8495	−0.746841	−0.373420	−	0.927662i	$-0.621815\pi$
$509$	−16.8495	−0.746841	−0.373420	+	0.927662i	$0.621815\pi$
$510$	0	0
$511$	44.7312	1.97879
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−86.9063	−3.82955
$516$	0	0
$517$	8.43018	0.370759
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−32.2559	−1.41316	−0.706579	−	0.707634i	$-0.749763\pi$
$521$	−32.2559	−1.41316	−0.706579	+	0.707634i	$0.749763\pi$
$522$	0	0
$523$	16.3656	0.715618	0.357809	−	0.933795i	$-0.383524\pi$
$523$	16.3656	0.715618	0.357809	+	0.933795i	$0.383524\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.01790	−0.0879012
$528$	0	0
$529$	−10.8708	−0.472645
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.21509	0.0526315
$534$	0	0
$535$	46.8938	2.02739
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.33682	0.143727
$540$	0	0
$541$	−21.3264	−0.916892	−0.458446	−	0.888722i	$-0.651594\pi$
$541$	−21.3264	−0.916892	−0.458446	+	0.888722i	$0.651594\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.7312	0.459674
$546$	0	0
$547$	26.0529	1.11394	0.556971	−	0.830532i	$-0.311963\pi$
$547$	26.0529	1.11394	0.556971	+	0.830532i	$0.311963\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	17.5006	0.745551
$552$	0	0
$553$	−43.9433	−1.86866
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.74910	−0.285969	−0.142984	−	0.989725i	$-0.545670\pi$
$557$	−6.74910	−0.285969	−0.142984	+	0.989725i	$0.545670\pi$
$558$	0	0
$559$	−6.83159	−0.288945
$560$	0	0
$561$	0	0
$562$	0	0
$563$	43.9642	1.85287	0.926435	−	0.376455i	$-0.122857\pi$
$563$	43.9642	1.85287	0.926435	+	0.376455i	$0.122857\pi$
$564$	0	0
$565$	34.1984	1.43874
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.2722	0.766012	0.383006	−	0.923746i	$-0.374889\pi$
$569$	18.2722	0.766012	0.383006	+	0.923746i	$0.374889\pi$
$570$	0	0
$571$	−14.9250	−0.624590	−0.312295	−	0.949985i	$-0.601098\pi$
$571$	−14.9250	−0.624590	−0.312295	+	0.949985i	$0.601098\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−49.5068	−2.06458
$576$	0	0
$577$	−28.2918	−1.17780	−0.588901	−	0.808206i	$-0.700439\pi$
$577$	−28.2918	−1.17780	−0.588901	+	0.808206i	$0.700439\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	28.1867	1.16938
$582$	0	0
$583$	12.1626	0.503723
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.6966	1.47336	0.736678	−	0.676244i	$-0.236393\pi$
$587$	35.6966	1.47336	0.736678	+	0.676244i	$0.236393\pi$
$588$	0	0
$589$	6.68290	0.275364
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.5173	−0.596154	−0.298077	−	0.954542i	$-0.596345\pi$
$593$	−14.5173	−0.596154	−0.298077	+	0.954542i	$0.596345\pi$
$594$	0	0
$595$	−26.3010	−1.07824
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.09217	0.371496	0.185748	−	0.982597i	$-0.440529\pi$
$599$	9.09217	0.371496	0.185748	+	0.982597i	$0.440529\pi$
$600$	0	0
$601$	22.7219	0.926847	0.463424	−	0.886137i	$-0.346621\pi$
$601$	22.7219	0.926847	0.463424	+	0.886137i	$0.346621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.38350	−0.178215
$606$	0	0
$607$	−35.8996	−1.45712	−0.728560	−	0.684982i	$-0.759810\pi$
$607$	−35.8996	−1.45712	−0.728560	+	0.684982i	$0.759810\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.43018	0.341049
$612$	0	0
$613$	−44.4815	−1.79659	−0.898295	−	0.439392i	$-0.855194\pi$
$613$	−44.4815	−1.79659	−0.898295	+	0.439392i	$0.855194\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.7203	0.914686	0.457343	−	0.889290i	$-0.348801\pi$
$617$	22.7203	0.914686	0.457343	+	0.889290i	$0.348801\pi$
$618$	0	0
$619$	−4.52193	−0.181752	−0.0908759	−	0.995862i	$-0.528967\pi$
$619$	−4.52193	−0.181752	−0.0908759	+	0.995862i	$0.528967\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−59.4048	−2.38000
$624$	0	0
$625$	105.993	4.23974
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.03461	−0.200743
$630$	0	0
$631$	24.7087	0.983636	0.491818	−	0.870698i	$-0.336332\pi$
$631$	24.7087	0.983636	0.491818	+	0.870698i	$0.336332\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	47.7837	1.89624
$636$	0	0
$637$	3.33682	0.132210
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.56862	0.180450	0.0902248	−	0.995921i	$-0.471241\pi$
$641$	4.56862	0.180450	0.0902248	+	0.995921i	$0.471241\pi$
$642$	0	0
$643$	12.7203	0.501641	0.250820	−	0.968034i	$-0.419300\pi$
$643$	12.7203	0.501641	0.250820	+	0.968034i	$0.419300\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−48.0218	−1.88793	−0.943965	−	0.330046i	$-0.892936\pi$
$647$	−48.0218	−1.88793	−0.943965	+	0.330046i	$0.892936\pi$
$648$	0	0
$649$	8.76700	0.344135
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.5352	−0.412275	−0.206137	−	0.978523i	$-0.566089\pi$
$653$	−10.5352	−0.412275	−0.206137	+	0.978523i	$0.566089\pi$
$654$	0	0
$655$	−22.0692	−0.862316
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−23.5340	−0.916755	−0.458377	−	0.888758i	$-0.651569\pi$
$659$	−23.5340	−0.916755	−0.458377	+	0.888758i	$0.651569\pi$
$660$	0	0
$661$	18.4060	0.715912	0.357956	−	0.933738i	$-0.383474\pi$
$661$	18.4060	0.715912	0.357956	+	0.933738i	$0.383474\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	87.1038	3.37774
$666$	0	0
$667$	9.86157	0.381841
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.4648	−0.442593
$672$	0	0
$673$	16.5928	0.639604	0.319802	−	0.947484i	$-0.396384\pi$
$673$	16.5928	0.639604	0.319802	+	0.947484i	$0.396384\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.2272	1.39232	0.696162	−	0.717885i	$-0.254890\pi$
$677$	36.2272	1.39232	0.696162	+	0.717885i	$0.254890\pi$
$678$	0	0
$679$	−41.0471	−1.57524
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.95493	0.227859	0.113930	−	0.993489i	$-0.463656\pi$
$683$	5.95493	0.227859	0.113930	+	0.993489i	$0.463656\pi$
$684$	0	0
$685$	−4.33057	−0.165463
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12.1626	0.463357
$690$	0	0
$691$	7.17466	0.272937	0.136468	−	0.990644i	$-0.456425\pi$
$691$	7.17466	0.272937	0.136468	+	0.990644i	$0.456425\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	34.1781	1.29645
$696$	0	0
$697$	2.26760	0.0858916
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13.3668	−0.504857	−0.252428	−	0.967616i	$-0.581229\pi$
$701$	−13.3668	−0.504857	−0.252428	+	0.967616i	$0.581229\pi$
$702$	0	0
$703$	16.6736	0.628858
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.02092	0.301658
$708$	0	0
$709$	−22.2076	−0.834026	−0.417013	−	0.908901i	$-0.636923\pi$
$709$	−22.2076	−0.834026	−0.417013	+	0.908901i	$0.636923\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.76581	0.141031
$714$	0	0
$715$	−4.38350	−0.163934
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.72194	0.325273	0.162637	−	0.986686i	$-0.448000\pi$
$719$	8.72194	0.325273	0.162637	+	0.986686i	$0.448000\pi$
$720$	0	0
$721$	63.7417	2.37386
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−40.2513	−1.49490
$726$	0	0
$727$	32.1960	1.19408	0.597042	−	0.802210i	$-0.296343\pi$
$727$	32.1960	1.19408	0.597042	+	0.802210i	$0.296343\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12.7491	−0.471543
$732$	0	0
$733$	38.2380	1.41235	0.706177	−	0.708035i	$-0.250418\pi$
$733$	38.2380	1.41235	0.706177	+	0.708035i	$0.250418\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.81369	0.324656
$738$	0	0
$739$	−6.09638	−0.224259	−0.112129	−	0.993694i	$-0.535767\pi$
$739$	−6.09638	−0.224259	−0.112129	+	0.993694i	$0.535767\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.8720	−0.398856	−0.199428	−	0.979912i	$-0.563908\pi$
$743$	−10.8720	−0.398856	−0.199428	+	0.979912i	$0.563908\pi$
$744$	0	0
$745$	46.5119	1.70406
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−34.3944	−1.25674
$750$	0	0
$751$	−47.8350	−1.74552	−0.872762	−	0.488145i	$-0.837674\pi$
$751$	−47.8350	−1.74552	−0.872762	+	0.488145i	$0.837674\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	59.8141	2.17686
$756$	0	0
$757$	23.3201	0.847584	0.423792	−	0.905760i	$-0.360699\pi$
$757$	23.3201	0.847584	0.423792	+	0.905760i	$0.360699\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−52.5990	−1.90671	−0.953356	−	0.301847i	$-0.902397\pi$
$761$	−52.5990	−1.90671	−0.953356	+	0.301847i	$0.902397\pi$
$762$	0	0
$763$	−7.87083	−0.284943
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.76700	0.316558
$768$	0	0
$769$	5.86157	0.211374	0.105687	−	0.994399i	$-0.466296\pi$
$769$	5.86157	0.211374	0.105687	+	0.994399i	$0.466296\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25.3131	0.910449	0.455224	−	0.890377i	$-0.349559\pi$
$773$	25.3131	0.910449	0.455224	+	0.890377i	$0.349559\pi$
$774$	0	0
$775$	−15.3706	−0.552130
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.50986	−0.269069
$780$	0	0
$781$	−4.69779	−0.168100
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−21.2781	−0.759447
$786$	0	0
$787$	−37.4710	−1.33570	−0.667849	−	0.744297i	$-0.732785\pi$
$787$	−37.4710	−1.33570	−0.667849	+	0.744297i	$0.732785\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−25.0829	−0.891845
$792$	0	0
$793$	−11.4648	−0.407127
$794$	0	0
$795$	0	0
$796$	0	0
$797$	43.7565	1.54994	0.774968	−	0.632000i	$-0.217766\pi$
$797$	43.7565	1.54994	0.774968	+	0.632000i	$0.217766\pi$
$798$	0	0
$799$	15.7324	0.556572
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.9129	0.490975
$804$	0	0
$805$	49.0829	1.72995
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−48.9342	−1.72044	−0.860218	−	0.509927i	$-0.829672\pi$
$809$	−48.9342	−1.72044	−0.860218	+	0.509927i	$0.829672\pi$
$810$	0	0
$811$	−34.9568	−1.22750	−0.613749	−	0.789501i	$-0.710339\pi$
$811$	−34.9568	−1.22750	−0.613749	+	0.789501i	$0.710339\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	44.6465	1.56390
$816$	0	0
$817$	42.2225	1.47718
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.3431	0.919379	0.459690	−	0.888080i	$-0.347961\pi$
$821$	26.3431	0.919379	0.459690	+	0.888080i	$0.347961\pi$
$822$	0	0
$823$	52.5928	1.83327	0.916634	−	0.399727i	$-0.130895\pi$
$823$	52.5928	1.83327	0.916634	+	0.399727i	$0.130895\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.42394	−0.223382	−0.111691	−	0.993743i	$-0.535627\pi$
$827$	−6.42394	−0.223382	−0.111691	+	0.993743i	$0.535627\pi$
$828$	0	0
$829$	−25.6032	−0.889237	−0.444618	−	0.895720i	$-0.646661\pi$
$829$	−25.6032	−0.889237	−0.444618	+	0.895720i	$0.646661\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.22717	0.215759
$834$	0	0
$835$	63.2547	2.18902
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36.4994	−1.26010	−0.630050	−	0.776555i	$-0.716965\pi$
$839$	−36.4994	−1.26010	−0.630050	+	0.776555i	$0.716965\pi$
$840$	0	0
$841$	−20.9821	−0.723521
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.38350	−0.150797
$846$	0	0
$847$	3.21509	0.110472
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.39558	0.322076
$852$	0	0
$853$	26.8246	0.918456	0.459228	−	0.888319i	$-0.348126\pi$
$853$	26.8246	0.918456	0.459228	+	0.888319i	$0.348126\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.6320	1.42212	0.711061	−	0.703130i	$-0.248215\pi$
$857$	41.6320	1.42212	0.711061	+	0.703130i	$0.248215\pi$
$858$	0	0
$859$	−3.87083	−0.132071	−0.0660355	−	0.997817i	$-0.521035\pi$
$859$	−3.87083	−0.132071	−0.0660355	+	0.997817i	$0.521035\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−50.7103	−1.72620	−0.863099	−	0.505035i	$-0.831480\pi$
$863$	−50.7103	−1.72620	−0.863099	+	0.505035i	$0.831480\pi$
$864$	0	0
$865$	113.385	3.85519
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13.6678	−0.463649
$870$	0	0
$871$	8.81369	0.298640
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−129.872	−4.39046
$876$	0	0
$877$	1.87083	0.0631734	0.0315867	−	0.999501i	$-0.489944\pi$
$877$	1.87083	0.0631734	0.0315867	+	0.999501i	$0.489944\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	39.1372	1.31857	0.659283	−	0.751894i	$-0.270860\pi$
$881$	39.1372	1.31857	0.659283	+	0.751894i	$0.270860\pi$
$882$	0	0
$883$	14.0334	0.472262	0.236131	−	0.971721i	$-0.424121\pi$
$883$	14.0334	0.472262	0.236131	+	0.971721i	$0.424121\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11.5340	−0.387274	−0.193637	−	0.981073i	$-0.562028\pi$
$887$	−11.5340	−0.387274	−0.193637	+	0.981073i	$0.562028\pi$
$888$	0	0
$889$	−35.0471	−1.17544
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−52.1026	−1.74355
$894$	0	0
$895$	45.2330	1.51197
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.06177	0.102116
$900$	0	0
$901$	22.6978	0.756173
$902$	0	0
$903$	0	0
$904$	0	0
$905$	25.0495	0.832673
$906$	0	0
$907$	16.7670	0.556739	0.278370	−	0.960474i	$-0.410206\pi$
$907$	16.7670	0.556739	0.278370	+	0.960474i	$0.410206\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.6954	−1.61335	−0.806675	−	0.590995i	$-0.798735\pi$
$911$	−48.6954	−1.61335	−0.806675	+	0.590995i	$0.798735\pi$
$912$	0	0
$913$	8.76700	0.290146
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.1867	0.534533
$918$	0	0
$919$	−3.76118	−0.124070	−0.0620349	−	0.998074i	$-0.519759\pi$
$919$	−3.76118	−0.124070	−0.0620349	+	0.998074i	$0.519759\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.69779	−0.154630
$924$	0	0
$925$	−38.3493	−1.26092
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−30.9763	−1.01630	−0.508149	−	0.861269i	$-0.669670\pi$
$929$	−30.9763	−1.01630	−0.508149	+	0.861269i	$0.669670\pi$
$930$	0	0
$931$	−20.6232	−0.675897
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.18048	−0.267530
$936$	0	0
$937$	−40.2801	−1.31589	−0.657947	−	0.753065i	$-0.728575\pi$
$937$	−40.2801	−1.31589	−0.657947	+	0.753065i	$0.728575\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	6.42394	0.209414	0.104707	−	0.994503i	$-0.466609\pi$
$941$	6.42394	0.209414	0.104707	+	0.994503i	$0.466609\pi$
$942$	0	0
$943$	−4.23180	−0.137806
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.6515	0.573597	0.286799	−	0.957991i	$-0.407409\pi$
$947$	17.6515	0.573597	0.286799	+	0.957991i	$0.407409\pi$
$948$	0	0
$949$	13.9129	0.451631
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−23.7278	−0.768618	−0.384309	−	0.923205i	$-0.625560\pi$
$953$	−23.7278	−0.768618	−0.384309	+	0.923205i	$0.625560\pi$
$954$	0	0
$955$	−2.75535	−0.0891610
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.17627	0.102567
$960$	0	0
$961$	−29.8308	−0.962284
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−83.0564	−2.67368
$966$	0	0
$967$	−26.2738	−0.844910	−0.422455	−	0.906384i	$-0.638832\pi$
$967$	−26.2738	−0.844910	−0.422455	+	0.906384i	$0.638832\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.98329	−0.0957385	−0.0478692	−	0.998854i	$-0.515243\pi$
$971$	−2.98329	−0.0957385	−0.0478692	+	0.998854i	$0.515243\pi$
$972$	0	0
$973$	−25.0680	−0.803644
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.7087	−0.342601	−0.171300	−	0.985219i	$-0.554797\pi$
$977$	−10.7087	−0.342601	−0.171300	+	0.985219i	$0.554797\pi$
$978$	0	0
$979$	−18.4769	−0.590523
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.03581	−0.128722	−0.0643611	−	0.997927i	$-0.520501\pi$
$983$	−4.03581	−0.128722	−0.0643611	+	0.997927i	$0.520501\pi$
$984$	0	0
$985$	11.0346	0.351592
$986$	0	0
$987$	0	0
$988$	0	0
$989$	23.7924	0.756553
$990$	0	0
$991$	−6.30101	−0.200158	−0.100079	−	0.994979i	$-0.531910\pi$
$991$	−6.30101	−0.200158	−0.100079	+	0.994979i	$0.531910\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.88129	0.218151
$996$	0	0
$997$	−47.6274	−1.50837	−0.754187	−	0.656660i	$-0.771969\pi$
$997$	−47.6274	−1.50837	−0.754187	+	0.656660i	$0.771969\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.l.1.1		3
3.2	odd	2		1716.2.a.h.1.3	✓	3
12.11	even	2		6864.2.a.bt.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.h.1.3	✓	3	3.2	odd	2
5148.2.a.l.1.1		3	1.1	even	1	trivial
6864.2.a.bt.1.3		3	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)

\(f(q)\)	\(=\)	\(q-4.38350 q^{5} +3.21509 q^{7} +O(q^{10})\)	\(q-4.38350 q^{5} +3.21509 q^{7} +1.00000 q^{11} +1.00000 q^{13} +1.86620 q^{17} -6.18048 q^{19} -3.48270 q^{23} +14.2151 q^{25} -2.83159 q^{29} -1.08129 q^{31} -14.0934 q^{35} -2.69779 q^{37} +1.21509 q^{41} -6.83159 q^{43} +8.43018 q^{47} +3.33682 q^{49} +12.1626 q^{53} -4.38350 q^{55} +8.76700 q^{59} -11.4648 q^{61} -4.38350 q^{65} +8.81369 q^{67} -4.69779 q^{71} +13.9129 q^{73} +3.21509 q^{77} -13.6678 q^{79} +8.76700 q^{83} -8.18048 q^{85} -18.4769 q^{89} +3.21509 q^{91} +27.0922 q^{95} -12.7670 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) 3 * q - 4 * q^5 - 4 * q^7	\( 3 q - 4 q^{5} - 4 q^{7} + 3 q^{11} + 3 q^{13} - 2 q^{17} - 8 q^{19} - 12 q^{23} + 29 q^{25} - 4 q^{29} + 18 q^{31} - 6 q^{35} + 4 q^{37} - 10 q^{41} - 16 q^{43} - 2 q^{47} + 19 q^{49} - 6 q^{53} - 4 q^{55} + 8 q^{59} - 4 q^{61} - 4 q^{65} - 10 q^{67} - 2 q^{71} + 16 q^{73} - 4 q^{77} - 12 q^{79} + 8 q^{83} - 14 q^{85} - 10 q^{89} - 4 q^{91} - 22 q^{95} - 20 q^{97}+O(q^{100}) \) 3 * q - 4 * q^5 - 4 * q^7 + 3 * q^11 + 3 * q^13 - 2 * q^17 - 8 * q^19 - 12 * q^23 + 29 * q^25 - 4 * q^29 + 18 * q^31 - 6 * q^35 + 4 * q^37 - 10 * q^41 - 16 * q^43 - 2 * q^47 + 19 * q^49 - 6 * q^53 - 4 * q^55 + 8 * q^59 - 4 * q^61 - 4 * q^65 - 10 * q^67 - 2 * q^71 + 16 * q^73 - 4 * q^77 - 12 * q^79 + 8 * q^83 - 14 * q^85 - 10 * q^89 - 4 * q^91 - 22 * q^95 - 20 * q^97

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