LMFDB - Embedded newform 132.12.a.b.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [132,12,Mod(1,132)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("132.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(132, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 12, names="a")

Level:	$N$	$=$	$132 = 2^{2} \cdot 3 \cdot 11$
Weight:	$k$	$=$	$12$
Character orbit:	$[\chi]$	$=$	132.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,972] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$101.421299834$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 243453x^{2} + 8521201x + 11492037452$ x^4 - x^3 - 243453x^2 + 8521201x + 11492037452
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{5}\cdot 3\cdot 5$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-222.576$ of defining polynomial
Character	$\chi$	$=$	132.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+243.000 q^{3} +4276.73 q^{5} +49497.0 q^{7} +59049.0 q^{9} +161051. q^{11} -1.68409e6 q^{13} +1.03925e6 q^{15} -7.10007e6 q^{17} -1.65296e7 q^{19} +1.20278e7 q^{21} -1.67687e7 q^{23} -3.05377e7 q^{25} +1.43489e7 q^{27} -5.28256e7 q^{29} -1.20213e8 q^{31} +3.91354e7 q^{33} +2.11685e8 q^{35} +4.84190e7 q^{37} -4.09234e8 q^{39} +2.36748e8 q^{41} -1.82034e8 q^{43} +2.52537e8 q^{45} -2.42104e9 q^{47} +4.72628e8 q^{49} -1.72532e9 q^{51} +3.54466e9 q^{53} +6.88772e8 q^{55} -4.01669e9 q^{57} -2.23157e9 q^{59} +3.76066e9 q^{61} +2.92275e9 q^{63} -7.20240e9 q^{65} -5.99361e8 q^{67} -4.07478e9 q^{69} -1.80963e10 q^{71} -7.08717e9 q^{73} -7.42066e9 q^{75} +7.97154e9 q^{77} +5.37343e10 q^{79} +3.48678e9 q^{81} +5.25766e10 q^{83} -3.03651e10 q^{85} -1.28366e10 q^{87} -6.94888e8 q^{89} -8.33575e10 q^{91} -2.92117e10 q^{93} -7.06926e10 q^{95} -2.56949e9 q^{97} +9.50990e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 972 q^{3} - 7020 q^{5} - 10934 q^{7} + 236196 q^{9} + 644204 q^{11} - 2559568 q^{13} - 1705860 q^{15} - 1606814 q^{17} - 5621902 q^{19} - 2656962 q^{21} + 19030288 q^{23} + 14585700 q^{25} + 57395628 q^{27}+ \cdots + 38039601996 q^{99}+O(q^{100})$ 4 * q + 972 * q^3 - 7020 * q^5 - 10934 * q^7 + 236196 * q^9 + 644204 * q^11 - 2559568 * q^13 - 1705860 * q^15 - 1606814 * q^17 - 5621902 * q^19 - 2656962 * q^21 + 19030288 * q^23 + 14585700 * q^25 + 57395628 * q^27 + 146818034 * q^29 - 158103664 * q^31 + 156541572 * q^33 - 108199300 * q^35 - 770947012 * q^37 - 621975024 * q^39 - 582529706 * q^41 - 2203144746 * q^43 - 414523980 * q^45 - 1693221664 * q^47 - 1348303344 * q^49 - 390455802 * q^51 - 2342022320 * q^53 - 1130578020 * q^55 - 1366122186 * q^57 - 8715508320 * q^59 - 2069677236 * q^61 - 645641766 * q^63 - 15698175680 * q^65 - 5402316304 * q^67 + 4624359984 * q^69 - 13967424104 * q^71 + 3355935748 * q^73 + 3544325100 * q^75 - 1760931634 * q^77 + 32830867762 * q^79 + 13947137604 * q^81 + 10761308844 * q^83 - 7372813140 * q^85 + 35676782262 * q^87 - 51453668352 * q^89 - 15456476028 * q^91 - 38419190352 * q^93 - 56741221900 * q^95 - 118256509380 * q^97 + 38039601996 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	0	0
$2$	0	0
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	0	0
$5$	4276.73	0.612036	0.306018	−	0.952026i	$-0.401003\pi$
$5$	4276.73	0.612036	0.306018	+	0.952026i	$0.401003\pi$
$6$	0	0
$7$	49497.0	1.11311	0.556557	−	0.830809i	$-0.312122\pi$
$7$	49497.0	1.11311	0.556557	+	0.830809i	$0.312122\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	161051.	0.301511
$11$	161051.	0.301511
$12$	0	0
$13$	−1.68409e6	−1.25799	−0.628995	−	0.777409i	$-0.716533\pi$
$13$	−1.68409e6	−1.25799	−0.628995	+	0.777409i	$0.716533\pi$
$14$	0	0
$15$	1.03925e6	0.353359
$16$	0	0
$17$	−7.10007e6	−1.21281	−0.606406	−	0.795155i	$-0.707389\pi$
$17$	−7.10007e6	−1.21281	−0.606406	+	0.795155i	$0.707389\pi$
$18$	0	0
$19$	−1.65296e7	−1.53150	−0.765751	−	0.643137i	$-0.777633\pi$
$19$	−1.65296e7	−1.53150	−0.765751	+	0.643137i	$0.777633\pi$
$20$	0	0
$21$	1.20278e7	0.642657
$22$	0	0
$23$	−1.67687e7	−0.543244	−0.271622	−	0.962404i	$-0.587560\pi$
$23$	−1.67687e7	−0.543244	−0.271622	+	0.962404i	$0.587560\pi$
$24$	0	0
$25$	−3.05377e7	−0.625412
$26$	0	0
$27$	1.43489e7	0.192450
$28$	0	0
$29$	−5.28256e7	−0.478251	−0.239125	−	0.970989i	$-0.576861\pi$
$29$	−5.28256e7	−0.478251	−0.239125	+	0.970989i	$0.576861\pi$
$30$	0	0
$31$	−1.20213e8	−0.754157	−0.377078	−	0.926181i	$-0.623071\pi$
$31$	−1.20213e8	−0.754157	−0.377078	+	0.926181i	$0.623071\pi$
$32$	0	0
$33$	3.91354e7	0.174078
$34$	0	0
$35$	2.11685e8	0.681266
$36$	0	0
$37$	4.84190e7	0.114791	0.0573954	−	0.998352i	$-0.481720\pi$
$37$	4.84190e7	0.114791	0.0573954	+	0.998352i	$0.481720\pi$
$38$	0	0
$39$	−4.09234e8	−0.726301
$40$	0	0
$41$	2.36748e8	0.319135	0.159568	−	0.987187i	$-0.448990\pi$
$41$	2.36748e8	0.319135	0.159568	+	0.987187i	$0.448990\pi$
$42$	0	0
$43$	−1.82034e8	−0.188832	−0.0944160	−	0.995533i	$-0.530098\pi$
$43$	−1.82034e8	−0.188832	−0.0944160	+	0.995533i	$0.530098\pi$
$44$	0	0
$45$	2.52537e8	0.204012
$46$	0	0
$47$	−2.42104e9	−1.53980	−0.769898	−	0.638167i	$-0.779693\pi$
$47$	−2.42104e9	−1.53980	−0.769898	+	0.638167i	$0.779693\pi$
$48$	0	0
$49$	4.72628e8	0.239024
$50$	0	0
$51$	−1.72532e9	−0.700217
$52$	0	0
$53$	3.54466e9	1.16428	0.582139	−	0.813089i	$-0.302216\pi$
$53$	3.54466e9	1.16428	0.582139	+	0.813089i	$0.302216\pi$
$54$	0	0
$55$	6.88772e8	0.184536
$56$	0	0
$57$	−4.01669e9	−0.884213
$58$	0	0
$59$	−2.23157e9	−0.406373	−0.203187	−	0.979140i	$-0.565130\pi$
$59$	−2.23157e9	−0.406373	−0.203187	+	0.979140i	$0.565130\pi$
$60$	0	0
$61$	3.76066e9	0.570099	0.285049	−	0.958513i	$-0.407990\pi$
$61$	3.76066e9	0.570099	0.285049	+	0.958513i	$0.407990\pi$
$62$	0	0
$63$	2.92275e9	0.371038
$64$	0	0
$65$	−7.20240e9	−0.769935
$66$	0	0
$67$	−5.99361e8	−0.0542346	−0.0271173	−	0.999632i	$-0.508633\pi$
$67$	−5.99361e8	−0.0542346	−0.0271173	+	0.999632i	$0.508633\pi$
$68$	0	0
$69$	−4.07478e9	−0.313642
$70$	0	0
$71$	−1.80963e10	−1.19033	−0.595166	−	0.803602i	$-0.702914\pi$
$71$	−1.80963e10	−1.19033	−0.595166	+	0.803602i	$0.702914\pi$
$72$	0	0
$73$	−7.08717e9	−0.400126	−0.200063	−	0.979783i	$-0.564115\pi$
$73$	−7.08717e9	−0.400126	−0.200063	+	0.979783i	$0.564115\pi$
$74$	0	0
$75$	−7.42066e9	−0.361082
$76$	0	0
$77$	7.97154e9	0.335617
$78$	0	0
$79$	5.37343e10	1.96473	0.982364	−	0.186980i	$-0.0598701\pi$
$79$	5.37343e10	1.96473	0.982364	+	0.186980i	$0.0598701\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	5.25766e10	1.46509	0.732543	−	0.680720i	$-0.238333\pi$
$83$	5.25766e10	1.46509	0.732543	+	0.680720i	$0.238333\pi$
$84$	0	0
$85$	−3.03651e10	−0.742284
$86$	0	0
$87$	−1.28366e10	−0.276118
$88$	0	0
$89$	−6.94888e8	−0.0131908	−0.00659538	−	0.999978i	$-0.502099\pi$
$89$	−6.94888e8	−0.0131908	−0.00659538	+	0.999978i	$0.502099\pi$
$90$	0	0
$91$	−8.33575e10	−1.40029
$92$	0	0
$93$	−2.92117e10	−0.435413
$94$	0	0
$95$	−7.06926e10	−0.937334
$96$	0	0
$97$	−2.56949e9	−0.0303811	−0.0151905	−	0.999885i	$-0.504835\pi$
$97$	−2.56949e9	−0.0303811	−0.0151905	+	0.999885i	$0.504835\pi$
$98$	0	0
$99$	9.50990e9	0.100504
$100$	0	0
$101$	5.30371e10	0.502125	0.251063	−	0.967971i	$-0.419220\pi$
$101$	5.30371e10	0.502125	0.251063	+	0.967971i	$0.419220\pi$
$102$	0	0
$103$	−2.11238e11	−1.79542	−0.897712	−	0.440583i	$-0.854772\pi$
$103$	−2.11238e11	−1.79542	−0.897712	+	0.440583i	$0.854772\pi$
$104$	0	0
$105$	5.14395e10	0.393329
$106$	0	0
$107$	1.26755e11	0.873681	0.436841	−	0.899539i	$-0.356097\pi$
$107$	1.26755e11	0.873681	0.436841	+	0.899539i	$0.356097\pi$
$108$	0	0
$109$	−5.87263e10	−0.365584	−0.182792	−	0.983152i	$-0.558513\pi$
$109$	−5.87263e10	−0.365584	−0.182792	+	0.983152i	$0.558513\pi$
$110$	0	0
$111$	1.17658e10	0.0662745
$112$	0	0
$113$	1.46079e11	0.745858	0.372929	−	0.927860i	$-0.378354\pi$
$113$	1.46079e11	0.745858	0.372929	+	0.927860i	$0.378354\pi$
$114$	0	0
$115$	−7.17150e10	−0.332485
$116$	0	0
$117$	−9.94439e10	−0.419330
$118$	0	0
$119$	−3.51432e11	−1.35000
$120$	0	0
$121$	2.59374e10	0.0909091
$122$	0	0
$123$	5.75297e10	0.184253
$124$	0	0
$125$	−3.39426e11	−0.994810
$126$	0	0
$127$	6.00262e10	0.161221	0.0806103	−	0.996746i	$-0.474313\pi$
$127$	6.00262e10	0.161221	0.0806103	+	0.996746i	$0.474313\pi$
$128$	0	0
$129$	−4.42342e10	−0.109022
$130$	0	0
$131$	−4.70497e11	−1.06553	−0.532764	−	0.846264i	$-0.678847\pi$
$131$	−4.70497e11	−1.06553	−0.532764	+	0.846264i	$0.678847\pi$
$132$	0	0
$133$	−8.18166e11	−1.70474
$134$	0	0
$135$	6.13664e10	0.117786
$136$	0	0
$137$	−4.75890e11	−0.842449	−0.421224	−	0.906956i	$-0.638400\pi$
$137$	−4.75890e11	−0.842449	−0.421224	+	0.906956i	$0.638400\pi$
$138$	0	0
$139$	1.10035e10	0.0179866	0.00899332	−	0.999960i	$-0.497137\pi$
$139$	1.10035e10	0.0179866	0.00899332	+	0.999960i	$0.497137\pi$
$140$	0	0
$141$	−5.88312e11	−0.889001
$142$	0	0
$143$	−2.71225e11	−0.379298
$144$	0	0
$145$	−2.25921e11	−0.292706
$146$	0	0
$147$	1.14849e11	0.138001
$148$	0	0
$149$	−7.93521e11	−0.885185	−0.442593	−	0.896723i	$-0.645941\pi$
$149$	−7.93521e11	−0.885185	−0.442593	+	0.896723i	$0.645941\pi$
$150$	0	0
$151$	−8.68743e11	−0.900571	−0.450286	−	0.892885i	$-0.648678\pi$
$151$	−8.68743e11	−0.900571	−0.450286	+	0.892885i	$0.648678\pi$
$152$	0	0
$153$	−4.19252e11	−0.404271
$154$	0	0
$155$	−5.14118e11	−0.461571
$156$	0	0
$157$	5.79479e11	0.484830	0.242415	−	0.970173i	$-0.422060\pi$
$157$	5.79479e11	0.484830	0.242415	+	0.970173i	$0.422060\pi$
$158$	0	0
$159$	8.61351e11	0.672196
$160$	0	0
$161$	−8.29998e11	−0.604693
$162$	0	0
$163$	−1.52481e12	−1.03797	−0.518984	−	0.854784i	$-0.673690\pi$
$163$	−1.52481e12	−1.03797	−0.518984	+	0.854784i	$0.673690\pi$
$164$	0	0
$165$	1.67371e11	0.106542
$166$	0	0
$167$	7.38946e11	0.440222	0.220111	−	0.975475i	$-0.429358\pi$
$167$	7.38946e11	0.440222	0.220111	+	0.975475i	$0.429358\pi$
$168$	0	0
$169$	1.04400e12	0.582538
$170$	0	0
$171$	−9.76057e11	−0.510501
$172$	0	0
$173$	−1.83463e12	−0.900108	−0.450054	−	0.893001i	$-0.648595\pi$
$173$	−1.83463e12	−0.900108	−0.450054	+	0.893001i	$0.648595\pi$
$174$	0	0
$175$	−1.51153e12	−0.696156
$176$	0	0
$177$	−5.42272e11	−0.234620
$178$	0	0
$179$	1.37564e12	0.559516	0.279758	−	0.960071i	$-0.409746\pi$
$179$	1.37564e12	0.559516	0.279758	+	0.960071i	$0.409746\pi$
$180$	0	0
$181$	2.77228e12	1.06073	0.530364	−	0.847770i	$-0.322055\pi$
$181$	2.77228e12	1.06073	0.530364	+	0.847770i	$0.322055\pi$
$182$	0	0
$183$	9.13841e11	0.329147
$184$	0	0
$185$	2.07075e11	0.0702560
$186$	0	0
$187$	−1.14347e12	−0.365676
$188$	0	0
$189$	7.10228e11	0.214219
$190$	0	0
$191$	−1.01129e12	−0.287867	−0.143933	−	0.989587i	$-0.545975\pi$
$191$	−1.01129e12	−0.287867	−0.143933	+	0.989587i	$0.545975\pi$
$192$	0	0
$193$	5.15393e12	1.38539	0.692697	−	0.721229i	$-0.256422\pi$
$193$	5.15393e12	1.38539	0.692697	+	0.721229i	$0.256422\pi$
$194$	0	0
$195$	−1.75018e12	−0.444522
$196$	0	0
$197$	−4.58588e12	−1.10118	−0.550590	−	0.834776i	$-0.685597\pi$
$197$	−4.58588e12	−1.10118	−0.550590	+	0.834776i	$0.685597\pi$
$198$	0	0
$199$	6.52089e12	1.48120	0.740602	−	0.671944i	$-0.234541\pi$
$199$	6.52089e12	1.48120	0.740602	+	0.671944i	$0.234541\pi$
$200$	0	0
$201$	−1.45645e11	−0.0313124
$202$	0	0
$203$	−2.61471e12	−0.532348
$204$	0	0
$205$	1.01251e12	0.195322
$206$	0	0
$207$	−9.90172e11	−0.181081
$208$	0	0
$209$	−2.66211e12	−0.461765
$210$	0	0
$211$	−9.69015e12	−1.59506	−0.797530	−	0.603279i	$-0.793860\pi$
$211$	−9.69015e12	−1.59506	−0.797530	+	0.603279i	$0.793860\pi$
$212$	0	0
$213$	−4.39740e12	−0.687239
$214$	0	0
$215$	−7.78509e11	−0.115572
$216$	0	0
$217$	−5.95018e12	−0.839463
$218$	0	0
$219$	−1.72218e12	−0.231013
$220$	0	0
$221$	1.19572e13	1.52570
$222$	0	0
$223$	−1.37176e13	−1.66572	−0.832861	−	0.553483i	$-0.813299\pi$
$223$	−1.37176e13	−1.66572	−0.832861	+	0.553483i	$0.813299\pi$
$224$	0	0
$225$	−1.80322e12	−0.208471
$226$	0	0
$227$	−1.92745e12	−0.212247	−0.106123	−	0.994353i	$-0.533844\pi$
$227$	−1.92745e12	−0.212247	−0.106123	+	0.994353i	$0.533844\pi$
$228$	0	0
$229$	1.00990e13	1.05970	0.529852	−	0.848090i	$-0.322248\pi$
$229$	1.00990e13	1.05970	0.529852	+	0.848090i	$0.322248\pi$
$230$	0	0
$231$	1.93709e12	0.193768
$232$	0	0
$233$	1.40935e13	1.34450	0.672252	−	0.740322i	$-0.265327\pi$
$233$	1.40935e13	1.34450	0.672252	+	0.740322i	$0.265327\pi$
$234$	0	0
$235$	−1.03541e13	−0.942410
$236$	0	0
$237$	1.30574e13	1.13434
$238$	0	0
$239$	1.20327e13	0.998105	0.499053	−	0.866572i	$-0.333681\pi$
$239$	1.20327e13	0.998105	0.499053	+	0.866572i	$0.333681\pi$
$240$	0	0
$241$	1.69673e12	0.134437	0.0672186	−	0.997738i	$-0.478588\pi$
$241$	1.69673e12	0.134437	0.0672186	+	0.997738i	$0.478588\pi$
$242$	0	0
$243$	8.47289e11	0.0641500
$244$	0	0
$245$	2.02130e12	0.146291
$246$	0	0
$247$	2.78374e13	1.92661
$248$	0	0
$249$	1.27761e13	0.845868
$250$	0	0
$251$	1.22307e13	0.774899	0.387450	−	0.921891i	$-0.373356\pi$
$251$	1.22307e13	0.774899	0.387450	+	0.921891i	$0.373356\pi$
$252$	0	0
$253$	−2.70061e12	−0.163794
$254$	0	0
$255$	−7.37871e12	−0.428558
$256$	0	0
$257$	−6.43742e11	−0.0358162	−0.0179081	−	0.999840i	$-0.505701\pi$
$257$	−6.43742e11	−0.0358162	−0.0179081	+	0.999840i	$0.505701\pi$
$258$	0	0
$259$	2.39660e12	0.127775
$260$	0	0
$261$	−3.11930e12	−0.159417
$262$	0	0
$263$	−1.90576e13	−0.933922	−0.466961	−	0.884278i	$-0.654651\pi$
$263$	−1.90576e13	−0.933922	−0.466961	+	0.884278i	$0.654651\pi$
$264$	0	0
$265$	1.51595e13	0.712580
$266$	0	0
$267$	−1.68858e11	−0.00761569
$268$	0	0
$269$	1.20228e13	0.520436	0.260218	−	0.965550i	$-0.416206\pi$
$269$	1.20228e13	0.520436	0.260218	+	0.965550i	$0.416206\pi$
$270$	0	0
$271$	3.75115e13	1.55895	0.779476	−	0.626432i	$-0.215485\pi$
$271$	3.75115e13	1.55895	0.779476	+	0.626432i	$0.215485\pi$
$272$	0	0
$273$	−2.02559e13	−0.808456
$274$	0	0
$275$	−4.91813e12	−0.188569
$276$	0	0
$277$	1.83109e13	0.674640	0.337320	−	0.941390i	$-0.390480\pi$
$277$	1.83109e13	0.674640	0.337320	+	0.941390i	$0.390480\pi$
$278$	0	0
$279$	−7.09845e12	−0.251386
$280$	0	0
$281$	1.65999e12	0.0565224	0.0282612	−	0.999601i	$-0.491003\pi$
$281$	1.65999e12	0.0565224	0.0282612	+	0.999601i	$0.491003\pi$
$282$	0	0
$283$	5.67061e13	1.85697	0.928484	−	0.371372i	$-0.121113\pi$
$283$	5.67061e13	1.85697	0.928484	+	0.371372i	$0.121113\pi$
$284$	0	0
$285$	−1.71783e13	−0.541170
$286$	0	0
$287$	1.17183e13	0.355234
$288$	0	0
$289$	1.61391e13	0.470912
$290$	0	0
$291$	−6.24387e11	−0.0175405
$292$	0	0
$293$	−4.49516e13	−1.21611	−0.608056	−	0.793894i	$-0.708050\pi$
$293$	−4.49516e13	−1.21611	−0.608056	+	0.793894i	$0.708050\pi$
$294$	0	0
$295$	−9.54383e12	−0.248715
$296$	0	0
$297$	2.31091e12	0.0580259
$298$	0	0
$299$	2.82399e13	0.683396
$300$	0	0
$301$	−9.01013e12	−0.210192
$302$	0	0
$303$	1.28880e13	0.289902
$304$	0	0
$305$	1.60833e13	0.348921
$306$	0	0
$307$	4.60684e13	0.964145	0.482072	−	0.876131i	$-0.339884\pi$
$307$	4.60684e13	0.964145	0.482072	+	0.876131i	$0.339884\pi$
$308$	0	0
$309$	−5.13308e13	−1.03659
$310$	0	0
$311$	3.53257e13	0.688506	0.344253	−	0.938877i	$-0.388132\pi$
$311$	3.53257e13	0.688506	0.344253	+	0.938877i	$0.388132\pi$
$312$	0	0
$313$	3.48768e13	0.656211	0.328105	−	0.944641i	$-0.393590\pi$
$313$	3.48768e13	0.656211	0.328105	+	0.944641i	$0.393590\pi$
$314$	0	0
$315$	1.24998e13	0.227089
$316$	0	0
$317$	−8.21893e13	−1.44208	−0.721040	−	0.692893i	$-0.756336\pi$
$317$	−8.21893e13	−1.44208	−0.721040	+	0.692893i	$0.756336\pi$
$318$	0	0
$319$	−8.50762e12	−0.144198
$320$	0	0
$321$	3.08014e13	0.504420
$322$	0	0
$323$	1.17361e14	1.85742
$324$	0	0
$325$	5.14283e13	0.786762
$326$	0	0
$327$	−1.42705e13	−0.211070
$328$	0	0
$329$	−1.19834e14	−1.71397
$330$	0	0
$331$	3.74524e13	0.518114	0.259057	−	0.965862i	$-0.416588\pi$
$331$	3.74524e13	0.518114	0.259057	+	0.965862i	$0.416588\pi$
$332$	0	0
$333$	2.85910e12	0.0382636
$334$	0	0
$335$	−2.56330e12	−0.0331935
$336$	0	0
$337$	1.00407e14	1.25834	0.629172	−	0.777266i	$-0.283394\pi$
$337$	1.00407e14	1.25834	0.629172	+	0.777266i	$0.283394\pi$
$338$	0	0
$339$	3.54972e13	0.430621
$340$	0	0
$341$	−1.93604e13	−0.227387
$342$	0	0
$343$	−7.44781e13	−0.847054
$344$	0	0
$345$	−1.74267e13	−0.191960
$346$	0	0
$347$	−1.32720e14	−1.41620	−0.708102	−	0.706111i	$-0.750448\pi$
$347$	−1.32720e14	−1.41620	−0.708102	+	0.706111i	$0.750448\pi$
$348$	0	0
$349$	−6.39269e13	−0.660912	−0.330456	−	0.943821i	$-0.607203\pi$
$349$	−6.39269e13	−0.660912	−0.330456	+	0.943821i	$0.607203\pi$
$350$	0	0
$351$	−2.41649e13	−0.242100
$352$	0	0
$353$	−1.90205e14	−1.84698	−0.923488	−	0.383628i	$-0.874675\pi$
$353$	−1.90205e14	−1.84698	−0.923488	+	0.383628i	$0.874675\pi$
$354$	0	0
$355$	−7.73929e13	−0.728526
$356$	0	0
$357$	−8.53980e13	−0.779422
$358$	0	0
$359$	−5.73600e13	−0.507680	−0.253840	−	0.967246i	$-0.581694\pi$
$359$	−5.73600e13	−0.507680	−0.253840	+	0.967246i	$0.581694\pi$
$360$	0	0
$361$	1.56738e14	1.34550
$362$	0	0
$363$	6.30279e12	0.0524864
$364$	0	0
$365$	−3.03099e13	−0.244891
$366$	0	0
$367$	1.01972e14	0.799497	0.399749	−	0.916625i	$-0.369097\pi$
$367$	1.01972e14	0.799497	0.399749	+	0.916625i	$0.369097\pi$
$368$	0	0
$369$	1.39797e13	0.106378
$370$	0	0
$371$	1.75450e14	1.29597
$372$	0	0
$373$	−2.00903e14	−1.44075	−0.720375	−	0.693585i	$-0.756030\pi$
$373$	−2.00903e14	−1.44075	−0.720375	+	0.693585i	$0.756030\pi$
$374$	0	0
$375$	−8.24806e13	−0.574354
$376$	0	0
$377$	8.89631e13	0.601634
$378$	0	0
$379$	−2.71278e14	−1.78196	−0.890982	−	0.454038i	$-0.849983\pi$
$379$	−2.71278e14	−1.78196	−0.890982	+	0.454038i	$0.849983\pi$
$380$	0	0
$381$	1.45864e13	0.0930807
$382$	0	0
$383$	−2.10084e14	−1.30256	−0.651282	−	0.758836i	$-0.725768\pi$
$383$	−2.10084e14	−1.30256	−0.651282	+	0.758836i	$0.725768\pi$
$384$	0	0
$385$	3.40921e13	0.205409
$386$	0	0
$387$	−1.07489e13	−0.0629440
$388$	0	0
$389$	−1.52090e14	−0.865722	−0.432861	−	0.901461i	$-0.642496\pi$
$389$	−1.52090e14	−0.865722	−0.432861	+	0.901461i	$0.642496\pi$
$390$	0	0
$391$	1.19059e14	0.658853
$392$	0	0
$393$	−1.14331e14	−0.615183
$394$	0	0
$395$	2.29807e14	1.20248
$396$	0	0
$397$	−4.81438e13	−0.245015	−0.122508	−	0.992468i	$-0.539094\pi$
$397$	−4.81438e13	−0.245015	−0.122508	+	0.992468i	$0.539094\pi$
$398$	0	0
$399$	−1.98814e14	−0.984231
$400$	0	0
$401$	−3.33450e14	−1.60597	−0.802983	−	0.596002i	$-0.796755\pi$
$401$	−3.33450e14	−1.60597	−0.802983	+	0.596002i	$0.796755\pi$
$402$	0	0
$403$	2.02449e14	0.948722
$404$	0	0
$405$	1.49120e13	0.0680040
$406$	0	0
$407$	7.79794e12	0.0346107
$408$	0	0
$409$	8.74164e13	0.377672	0.188836	−	0.982009i	$-0.439529\pi$
$409$	8.74164e13	0.377672	0.188836	+	0.982009i	$0.439529\pi$
$410$	0	0
$411$	−1.15641e14	−0.486388
$412$	0	0
$413$	−1.10456e14	−0.452340
$414$	0	0
$415$	2.24856e14	0.896685
$416$	0	0
$417$	2.67385e12	0.0103846
$418$	0	0
$419$	2.11975e14	0.801876	0.400938	−	0.916105i	$-0.368684\pi$
$419$	2.11975e14	0.801876	0.400938	+	0.916105i	$0.368684\pi$
$420$	0	0
$421$	−2.17283e14	−0.800707	−0.400354	−	0.916361i	$-0.631113\pi$
$421$	−2.17283e14	−0.800707	−0.400354	+	0.916361i	$0.631113\pi$
$422$	0	0
$423$	−1.42960e14	−0.513265
$424$	0	0
$425$	2.16820e14	0.758507
$426$	0	0
$427$	1.86142e14	0.634585
$428$	0	0
$429$	−6.59076e13	−0.218988
$430$	0	0
$431$	−4.61216e14	−1.49375	−0.746877	−	0.664962i	$-0.768448\pi$
$431$	−4.61216e14	−1.49375	−0.746877	+	0.664962i	$0.768448\pi$
$432$	0	0
$433$	2.26473e14	0.715043	0.357522	−	0.933905i	$-0.383622\pi$
$433$	2.26473e14	0.715043	0.357522	+	0.933905i	$0.383622\pi$
$434$	0	0
$435$	−5.48988e13	−0.168994
$436$	0	0
$437$	2.77179e14	0.831980
$438$	0	0
$439$	5.83617e14	1.70833	0.854167	−	0.519998i	$-0.174067\pi$
$439$	5.83617e14	1.70833	0.854167	+	0.519998i	$0.174067\pi$
$440$	0	0
$441$	2.79082e13	0.0796746
$442$	0	0
$443$	−6.37713e14	−1.77584	−0.887921	−	0.459995i	$-0.847851\pi$
$443$	−6.37713e14	−1.77584	−0.887921	+	0.459995i	$0.847851\pi$
$444$	0	0
$445$	−2.97185e12	−0.00807321
$446$	0	0
$447$	−1.92826e14	−0.511062
$448$	0	0
$449$	−5.12419e14	−1.32517	−0.662583	−	0.748989i	$-0.730540\pi$
$449$	−5.12419e14	−1.32517	−0.662583	+	0.748989i	$0.730540\pi$
$450$	0	0
$451$	3.81285e13	0.0962229
$452$	0	0
$453$	−2.11104e14	−0.519945
$454$	0	0
$455$	−3.56497e14	−0.857025
$456$	0	0
$457$	−1.87583e14	−0.440204	−0.220102	−	0.975477i	$-0.570639\pi$
$457$	−1.87583e14	−0.440204	−0.220102	+	0.975477i	$0.570639\pi$
$458$	0	0
$459$	−1.01878e14	−0.233406
$460$	0	0
$461$	2.48275e14	0.555364	0.277682	−	0.960673i	$-0.410434\pi$
$461$	2.48275e14	0.555364	0.277682	+	0.960673i	$0.410434\pi$
$462$	0	0
$463$	2.80907e14	0.613574	0.306787	−	0.951778i	$-0.400746\pi$
$463$	2.80907e14	0.613574	0.306787	+	0.951778i	$0.400746\pi$
$464$	0	0
$465$	−1.24931e14	−0.266488
$466$	0	0
$467$	5.42322e14	1.12983	0.564916	−	0.825148i	$-0.308908\pi$
$467$	5.42322e14	1.12983	0.564916	+	0.825148i	$0.308908\pi$
$468$	0	0
$469$	−2.96666e13	−0.0603694
$470$	0	0
$471$	1.40813e14	0.279917
$472$	0	0
$473$	−2.93167e13	−0.0569350
$474$	0	0
$475$	5.04776e14	0.957821
$476$	0	0
$477$	2.09308e14	0.388093
$478$	0	0
$479$	−3.32527e14	−0.602534	−0.301267	−	0.953540i	$-0.597410\pi$
$479$	−3.32527e14	−0.602534	−0.301267	+	0.953540i	$0.597410\pi$
$480$	0	0
$481$	−8.15421e13	−0.144406
$482$	0	0
$483$	−2.01690e14	−0.349120
$484$	0	0
$485$	−1.09890e13	−0.0185943
$486$	0	0
$487$	1.10839e15	1.83351	0.916755	−	0.399450i	$-0.130799\pi$
$487$	1.10839e15	1.83351	0.916755	+	0.399450i	$0.130799\pi$
$488$	0	0
$489$	−3.70529e14	−0.599272
$490$	0	0
$491$	8.24501e12	0.0130390	0.00651948	−	0.999979i	$-0.497925\pi$
$491$	8.24501e12	0.0130390	0.00651948	+	0.999979i	$0.497925\pi$
$492$	0	0
$493$	3.75065e14	0.580028
$494$	0	0
$495$	4.06713e13	0.0615119
$496$	0	0
$497$	−8.95712e14	−1.32498
$498$	0	0
$499$	3.32942e14	0.481744	0.240872	−	0.970557i	$-0.422567\pi$
$499$	3.32942e14	0.481744	0.240872	+	0.970557i	$0.422567\pi$
$500$	0	0
$501$	1.79564e14	0.254162
$502$	0	0
$503$	8.44449e14	1.16936	0.584682	−	0.811263i	$-0.301219\pi$
$503$	8.44449e14	1.16936	0.584682	+	0.811263i	$0.301219\pi$
$504$	0	0
$505$	2.26825e14	0.307319
$506$	0	0
$507$	2.53692e14	0.336328
$508$	0	0
$509$	7.37067e14	0.956222	0.478111	−	0.878299i	$-0.341322\pi$
$509$	7.37067e14	0.956222	0.478111	+	0.878299i	$0.341322\pi$
$510$	0	0
$511$	−3.50794e14	−0.445386
$512$	0	0
$513$	−2.37182e14	−0.294738
$514$	0	0
$515$	−9.03407e14	−1.09886
$516$	0	0
$517$	−3.89910e14	−0.464266
$518$	0	0
$519$	−4.45815e14	−0.519678
$520$	0	0
$521$	−5.91556e14	−0.675131	−0.337566	−	0.941302i	$-0.609603\pi$
$521$	−5.91556e14	−0.675131	−0.337566	+	0.941302i	$0.609603\pi$
$522$	0	0
$523$	1.40636e15	1.57159	0.785793	−	0.618490i	$-0.212255\pi$
$523$	1.40636e15	1.57159	0.785793	+	0.618490i	$0.212255\pi$
$524$	0	0
$525$	−3.67301e14	−0.401926
$526$	0	0
$527$	8.53520e14	0.914650
$528$	0	0
$529$	−6.71622e14	−0.704886
$530$	0	0
$531$	−1.31772e14	−0.135458
$532$	0	0
$533$	−3.98705e14	−0.401469
$534$	0	0
$535$	5.42095e14	0.534724
$536$	0	0
$537$	3.34280e14	0.323037
$538$	0	0
$539$	7.61173e13	0.0720684
$540$	0	0
$541$	−7.32827e14	−0.679856	−0.339928	−	0.940452i	$-0.610403\pi$
$541$	−7.32827e14	−0.679856	−0.339928	+	0.940452i	$0.610403\pi$
$542$	0	0
$543$	6.73663e14	0.612412
$544$	0	0
$545$	−2.51157e14	−0.223750
$546$	0	0
$547$	2.06410e15	1.80219	0.901094	−	0.433623i	$-0.142765\pi$
$547$	2.06410e15	1.80219	0.901094	+	0.433623i	$0.142765\pi$
$548$	0	0
$549$	2.22063e14	0.190033
$550$	0	0
$551$	8.73187e14	0.732442
$552$	0	0
$553$	2.65969e15	2.18697
$554$	0	0
$555$	5.03193e13	0.0405623
$556$	0	0
$557$	−4.05216e14	−0.320246	−0.160123	−	0.987097i	$-0.551189\pi$
$557$	−4.05216e14	−0.320246	−0.160123	+	0.987097i	$0.551189\pi$
$558$	0	0
$559$	3.06561e14	0.237549
$560$	0	0
$561$	−2.77864e14	−0.211123
$562$	0	0
$563$	2.36026e15	1.75858	0.879292	−	0.476283i	$-0.158016\pi$
$563$	2.36026e15	1.75858	0.879292	+	0.476283i	$0.158016\pi$
$564$	0	0
$565$	6.24740e14	0.456492
$566$	0	0
$567$	1.72585e14	0.123679
$568$	0	0
$569$	−9.19257e14	−0.646129	−0.323065	−	0.946377i	$-0.604713\pi$
$569$	−9.19257e14	−0.646129	−0.323065	+	0.946377i	$0.604713\pi$
$570$	0	0
$571$	−1.54276e15	−1.06365	−0.531825	−	0.846854i	$-0.678494\pi$
$571$	−1.54276e15	−1.06365	−0.531825	+	0.846854i	$0.678494\pi$
$572$	0	0
$573$	−2.45743e14	−0.166200
$574$	0	0
$575$	5.12076e14	0.339752
$576$	0	0
$577$	−6.19430e14	−0.403205	−0.201602	−	0.979467i	$-0.564615\pi$
$577$	−6.19430e14	−0.403205	−0.201602	+	0.979467i	$0.564615\pi$
$578$	0	0
$579$	1.25240e15	0.799857
$580$	0	0
$581$	2.60239e15	1.63081
$582$	0	0
$583$	5.70870e14	0.351043
$584$	0	0
$585$	−4.25295e14	−0.256645
$586$	0	0
$587$	−2.42732e15	−1.43753	−0.718767	−	0.695251i	$-0.755293\pi$
$587$	−2.42732e15	−1.43753	−0.718767	+	0.695251i	$0.755293\pi$
$588$	0	0
$589$	1.98707e15	1.15499
$590$	0	0
$591$	−1.11437e15	−0.635767
$592$	0	0
$593$	−9.10830e14	−0.510078	−0.255039	−	0.966931i	$-0.582088\pi$
$593$	−9.10830e14	−0.510078	−0.255039	+	0.966931i	$0.582088\pi$
$594$	0	0
$595$	−1.50298e15	−0.826247
$596$	0	0
$597$	1.58458e15	0.855173
$598$	0	0
$599$	−1.18211e15	−0.626338	−0.313169	−	0.949697i	$-0.601391\pi$
$599$	−1.18211e15	−0.626338	−0.313169	+	0.949697i	$0.601391\pi$
$600$	0	0
$601$	−1.28439e14	−0.0668169	−0.0334085	−	0.999442i	$-0.510636\pi$
$601$	−1.28439e14	−0.0668169	−0.0334085	+	0.999442i	$0.510636\pi$
$602$	0	0
$603$	−3.53916e13	−0.0180782
$604$	0	0
$605$	1.10927e14	0.0556396
$606$	0	0
$607$	9.30843e14	0.458499	0.229250	−	0.973368i	$-0.426373\pi$
$607$	9.30843e14	0.458499	0.229250	+	0.973368i	$0.426373\pi$
$608$	0	0
$609$	−6.35375e14	−0.307351
$610$	0	0
$611$	4.07724e15	1.93705
$612$	0	0
$613$	−2.25531e15	−1.05238	−0.526191	−	0.850366i	$-0.676380\pi$
$613$	−2.25531e15	−1.05238	−0.526191	+	0.850366i	$0.676380\pi$
$614$	0	0
$615$	2.46039e14	0.112769
$616$	0	0
$617$	2.22402e15	1.00131	0.500656	−	0.865646i	$-0.333092\pi$
$617$	2.22402e15	1.00131	0.500656	+	0.865646i	$0.333092\pi$
$618$	0	0
$619$	1.30319e15	0.576380	0.288190	−	0.957573i	$-0.406947\pi$
$619$	1.30319e15	0.576380	0.288190	+	0.957573i	$0.406947\pi$
$620$	0	0
$621$	−2.40612e14	−0.104547
$622$	0	0
$623$	−3.43949e13	−0.0146828
$624$	0	0
$625$	3.94653e13	0.0165530
$626$	0	0
$627$	−6.46893e14	−0.266600
$628$	0	0
$629$	−3.43778e14	−0.139220
$630$	0	0
$631$	−3.20240e15	−1.27443	−0.637213	−	0.770688i	$-0.719913\pi$
$631$	−3.20240e15	−1.27443	−0.637213	+	0.770688i	$0.719913\pi$
$632$	0	0
$633$	−2.35471e15	−0.920908
$634$	0	0
$635$	2.56716e14	0.0986727
$636$	0	0
$637$	−7.95949e14	−0.300690
$638$	0	0
$639$	−1.06857e15	−0.396778
$640$	0	0
$641$	−3.39934e15	−1.24072	−0.620362	−	0.784316i	$-0.713014\pi$
$641$	−3.39934e15	−1.24072	−0.620362	+	0.784316i	$0.713014\pi$
$642$	0	0
$643$	3.00031e15	1.07648	0.538240	−	0.842791i	$-0.319089\pi$
$643$	3.00031e15	1.07648	0.538240	+	0.842791i	$0.319089\pi$
$644$	0	0
$645$	−1.89178e14	−0.0667255
$646$	0	0
$647$	−1.48374e15	−0.514497	−0.257249	−	0.966345i	$-0.582816\pi$
$647$	−1.48374e15	−0.514497	−0.257249	+	0.966345i	$0.582816\pi$
$648$	0	0
$649$	−3.59397e14	−0.122526
$650$	0	0
$651$	−1.44589e15	−0.484664
$652$	0	0
$653$	−4.88834e15	−1.61116	−0.805580	−	0.592486i	$-0.798146\pi$
$653$	−4.88834e15	−1.61116	−0.805580	+	0.592486i	$0.798146\pi$
$654$	0	0
$655$	−2.01219e15	−0.652142
$656$	0	0
$657$	−4.18490e14	−0.133375
$658$	0	0
$659$	−2.48991e15	−0.780395	−0.390198	−	0.920731i	$-0.627593\pi$
$659$	−2.48991e15	−0.780395	−0.390198	+	0.920731i	$0.627593\pi$
$660$	0	0
$661$	8.58072e14	0.264494	0.132247	−	0.991217i	$-0.457781\pi$
$661$	8.58072e14	0.264494	0.132247	+	0.991217i	$0.457781\pi$
$662$	0	0
$663$	2.90559e15	0.880866
$664$	0	0
$665$	−3.49908e15	−1.04336
$666$	0	0
$667$	8.85815e14	0.259807
$668$	0	0
$669$	−3.33338e15	−0.961705
$670$	0	0
$671$	6.05658e14	0.171891
$672$	0	0
$673$	−5.80312e15	−1.62024	−0.810119	−	0.586265i	$-0.800598\pi$
$673$	−5.80312e15	−1.62024	−0.810119	+	0.586265i	$0.800598\pi$
$674$	0	0
$675$	−4.38183e14	−0.120361
$676$	0	0
$677$	−1.18029e15	−0.318970	−0.159485	−	0.987200i	$-0.550983\pi$
$677$	−1.18029e15	−0.318970	−0.159485	+	0.987200i	$0.550983\pi$
$678$	0	0
$679$	−1.27182e14	−0.0338176
$680$	0	0
$681$	−4.68371e14	−0.122541
$682$	0	0
$683$	3.90875e15	1.00629	0.503147	−	0.864201i	$-0.332176\pi$
$683$	3.90875e15	1.00629	0.503147	+	0.864201i	$0.332176\pi$
$684$	0	0
$685$	−2.03525e15	−0.515609
$686$	0	0
$687$	2.45406e15	0.611820
$688$	0	0
$689$	−5.96952e15	−1.46465
$690$	0	0
$691$	4.49753e14	0.108604	0.0543018	−	0.998525i	$-0.482707\pi$
$691$	4.49753e14	0.108604	0.0543018	+	0.998525i	$0.482707\pi$
$692$	0	0
$693$	4.70712e14	0.111872
$694$	0	0
$695$	4.70591e13	0.0110085
$696$	0	0
$697$	−1.68092e15	−0.387051
$698$	0	0
$699$	3.42473e15	0.776250
$700$	0	0
$701$	7.75392e15	1.73010	0.865052	−	0.501682i	$-0.167285\pi$
$701$	7.75392e15	1.73010	0.865052	+	0.501682i	$0.167285\pi$
$702$	0	0
$703$	−8.00348e14	−0.175802
$704$	0	0
$705$	−2.51605e15	−0.544100
$706$	0	0
$707$	2.62518e15	0.558923
$708$	0	0
$709$	−1.45922e15	−0.305891	−0.152945	−	0.988235i	$-0.548876\pi$
$709$	−1.45922e15	−0.305891	−0.152945	+	0.988235i	$0.548876\pi$
$710$	0	0
$711$	3.17295e15	0.654909
$712$	0	0
$713$	2.01581e15	0.409691
$714$	0	0
$715$	−1.15995e15	−0.232144
$716$	0	0
$717$	2.92396e15	0.576256
$718$	0	0
$719$	−8.69903e14	−0.168835	−0.0844174	−	0.996430i	$-0.526903\pi$
$719$	−8.69903e14	−0.168835	−0.0844174	+	0.996430i	$0.526903\pi$
$720$	0	0
$721$	−1.04556e16	−1.99851
$722$	0	0
$723$	4.12306e14	0.0776174
$724$	0	0
$725$	1.61317e15	0.299104
$726$	0	0
$727$	−3.21235e15	−0.586657	−0.293328	−	0.956012i	$-0.594763\pi$
$727$	−3.21235e15	−0.586657	−0.293328	+	0.956012i	$0.594763\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	1.29245e15	0.229018
$732$	0	0
$733$	−3.29021e15	−0.574318	−0.287159	−	0.957883i	$-0.592711\pi$
$733$	−3.29021e15	−0.574318	−0.287159	+	0.957883i	$0.592711\pi$
$734$	0	0
$735$	4.91177e14	0.0844612
$736$	0	0
$737$	−9.65276e13	−0.0163524
$738$	0	0
$739$	−7.21135e15	−1.20357	−0.601786	−	0.798657i	$-0.705544\pi$
$739$	−7.21135e15	−1.20357	−0.601786	+	0.798657i	$0.705544\pi$
$740$	0	0
$741$	6.76448e15	1.11233
$742$	0	0
$743$	5.79650e13	0.00939134	0.00469567	−	0.999989i	$-0.498505\pi$
$743$	5.79650e13	0.00939134	0.00469567	+	0.999989i	$0.498505\pi$
$744$	0	0
$745$	−3.39368e15	−0.541765
$746$	0	0
$747$	3.10460e15	0.488362
$748$	0	0
$749$	6.27397e15	0.972507
$750$	0	0
$751$	−1.34738e15	−0.205812	−0.102906	−	0.994691i	$-0.532814\pi$
$751$	−1.34738e15	−0.205812	−0.102906	+	0.994691i	$0.532814\pi$
$752$	0	0
$753$	2.97206e15	0.447388
$754$	0	0
$755$	−3.71538e15	−0.551182
$756$	0	0
$757$	1.35789e15	0.198536	0.0992679	−	0.995061i	$-0.468350\pi$
$757$	1.35789e15	0.198536	0.0992679	+	0.995061i	$0.468350\pi$
$758$	0	0
$759$	−6.56248e14	−0.0945667
$760$	0	0
$761$	4.22137e15	0.599567	0.299783	−	0.954007i	$-0.403086\pi$
$761$	4.22137e15	0.599567	0.299783	+	0.954007i	$0.403086\pi$
$762$	0	0
$763$	−2.90678e15	−0.406937
$764$	0	0
$765$	−1.79303e15	−0.247428
$766$	0	0
$767$	3.75817e15	0.511213
$768$	0	0
$769$	−6.48320e13	−0.00869349	−0.00434675	−	0.999991i	$-0.501384\pi$
$769$	−6.48320e13	−0.00869349	−0.00434675	+	0.999991i	$0.501384\pi$
$770$	0	0
$771$	−1.56429e14	−0.0206785
$772$	0	0
$773$	−1.40880e16	−1.83595	−0.917976	−	0.396635i	$-0.870178\pi$
$773$	−1.40880e16	−1.83595	−0.917976	+	0.396635i	$0.870178\pi$
$774$	0	0
$775$	3.67103e15	0.471659
$776$	0	0
$777$	5.82373e14	0.0737711
$778$	0	0
$779$	−3.91335e15	−0.488756
$780$	0	0
$781$	−2.91442e15	−0.358899
$782$	0	0
$783$	−7.57990e14	−0.0920394
$784$	0	0
$785$	2.47827e15	0.296733
$786$	0	0
$787$	−9.28457e15	−1.09623	−0.548114	−	0.836404i	$-0.684654\pi$
$787$	−9.28457e15	−1.09623	−0.548114	+	0.836404i	$0.684654\pi$
$788$	0	0
$789$	−4.63099e15	−0.539200
$790$	0	0
$791$	7.23047e15	0.830225
$792$	0	0
$793$	−6.33330e15	−0.717179
$794$	0	0
$795$	3.68377e15	0.411408
$796$	0	0
$797$	−1.55817e16	−1.71630	−0.858149	−	0.513400i	$-0.828386\pi$
$797$	−1.55817e16	−1.71630	−0.858149	+	0.513400i	$0.828386\pi$
$798$	0	0
$799$	1.71895e16	1.86748
$800$	0	0
$801$	−4.10324e13	−0.00439692
$802$	0	0
$803$	−1.14140e15	−0.120643
$804$	0	0
$805$	−3.54968e15	−0.370094
$806$	0	0
$807$	2.92154e15	0.300474
$808$	0	0
$809$	1.47984e16	1.50140	0.750701	−	0.660642i	$-0.229716\pi$
$809$	1.47984e16	1.50140	0.750701	+	0.660642i	$0.229716\pi$
$810$	0	0
$811$	−1.83756e14	−0.0183920	−0.00919598	−	0.999958i	$-0.502927\pi$
$811$	−1.83756e14	−0.0183920	−0.00919598	+	0.999958i	$0.502927\pi$
$812$	0	0
$813$	9.11529e15	0.900062
$814$	0	0
$815$	−6.52121e15	−0.635274
$816$	0	0
$817$	3.00895e15	0.289197
$818$	0	0
$819$	−4.92218e15	−0.466762
$820$	0	0
$821$	1.06573e16	0.997146	0.498573	−	0.866848i	$-0.333858\pi$
$821$	1.06573e16	0.997146	0.498573	+	0.866848i	$0.333858\pi$
$822$	0	0
$823$	1.22100e16	1.12724	0.563622	−	0.826033i	$-0.309408\pi$
$823$	1.22100e16	1.12724	0.563622	+	0.826033i	$0.309408\pi$
$824$	0	0
$825$	−1.19511e15	−0.108870
$826$	0	0
$827$	1.51075e16	1.35804	0.679019	−	0.734120i	$-0.262405\pi$
$827$	1.51075e16	1.35804	0.679019	+	0.734120i	$0.262405\pi$
$828$	0	0
$829$	1.65133e16	1.46482	0.732411	−	0.680862i	$-0.238395\pi$
$829$	1.65133e16	1.46482	0.732411	+	0.680862i	$0.238395\pi$
$830$	0	0
$831$	4.44956e15	0.389503
$832$	0	0
$833$	−3.35569e15	−0.289891
$834$	0	0
$835$	3.16027e15	0.269432
$836$	0	0
$837$	−1.72492e15	−0.145138
$838$	0	0
$839$	−1.94218e15	−0.161286	−0.0806432	−	0.996743i	$-0.525697\pi$
$839$	−1.94218e15	−0.161286	−0.0806432	+	0.996743i	$0.525697\pi$
$840$	0	0
$841$	−9.40996e15	−0.771276
$842$	0	0
$843$	4.03378e14	0.0326332
$844$	0	0
$845$	4.46491e15	0.356534
$846$	0	0
$847$	1.28383e15	0.101192
$848$	0	0
$849$	1.37796e16	1.07212
$850$	0	0
$851$	−8.11922e14	−0.0623594
$852$	0	0
$853$	−4.94533e15	−0.374952	−0.187476	−	0.982269i	$-0.560031\pi$
$853$	−4.94533e15	−0.374952	−0.187476	+	0.982269i	$0.560031\pi$
$854$	0	0
$855$	−4.17433e15	−0.312445
$856$	0	0
$857$	6.78557e15	0.501408	0.250704	−	0.968064i	$-0.419338\pi$
$857$	6.78557e15	0.501408	0.250704	+	0.968064i	$0.419338\pi$
$858$	0	0
$859$	−1.35581e16	−0.989089	−0.494545	−	0.869152i	$-0.664665\pi$
$859$	−1.35581e16	−0.989089	−0.494545	+	0.869152i	$0.664665\pi$
$860$	0	0
$861$	2.84755e15	0.205094
$862$	0	0
$863$	2.22653e16	1.58332	0.791660	−	0.610963i	$-0.209217\pi$
$863$	2.22653e16	1.58332	0.791660	+	0.610963i	$0.209217\pi$
$864$	0	0
$865$	−7.84621e15	−0.550898
$866$	0	0
$867$	3.92179e15	0.271881
$868$	0	0
$869$	8.65396e15	0.592388
$870$	0	0
$871$	1.00938e15	0.0682266
$872$	0	0
$873$	−1.51726e14	−0.0101270
$874$	0	0
$875$	−1.68006e16	−1.10734
$876$	0	0
$877$	5.54983e14	0.0361229	0.0180614	−	0.999837i	$-0.494251\pi$
$877$	5.54983e14	0.0361229	0.0180614	+	0.999837i	$0.494251\pi$
$878$	0	0
$879$	−1.09232e16	−0.702122
$880$	0	0
$881$	3.62374e15	0.230032	0.115016	−	0.993364i	$-0.463308\pi$
$881$	3.62374e15	0.230032	0.115016	+	0.993364i	$0.463308\pi$
$882$	0	0
$883$	−2.89325e16	−1.81385	−0.906926	−	0.421291i	$-0.861577\pi$
$883$	−2.89325e16	−1.81385	−0.906926	+	0.421291i	$0.861577\pi$
$884$	0	0
$885$	−2.31915e15	−0.143596
$886$	0	0
$887$	2.22272e16	1.35927	0.679633	−	0.733552i	$-0.262139\pi$
$887$	2.22272e16	1.35927	0.679633	+	0.733552i	$0.262139\pi$
$888$	0	0
$889$	2.97112e15	0.179457
$890$	0	0
$891$	5.61550e14	0.0335013
$892$	0	0
$893$	4.00188e16	2.35820
$894$	0	0
$895$	5.88323e15	0.342444
$896$	0	0
$897$	6.86231e15	0.394559
$898$	0	0
$899$	6.35032e15	0.360676
$900$	0	0
$901$	−2.51673e16	−1.41205
$902$	0	0
$903$	−2.18946e15	−0.121354
$904$	0	0
$905$	1.18563e16	0.649204
$906$	0	0
$907$	1.52787e15	0.0826505	0.0413252	−	0.999146i	$-0.486842\pi$
$907$	1.52787e15	0.0826505	0.0413252	+	0.999146i	$0.486842\pi$
$908$	0	0
$909$	3.13179e15	0.167375
$910$	0	0
$911$	−2.60448e16	−1.37521	−0.687606	−	0.726084i	$-0.741338\pi$
$911$	−2.60448e16	−1.37521	−0.687606	+	0.726084i	$0.741338\pi$
$912$	0	0
$913$	8.46752e15	0.441740
$914$	0	0
$915$	3.90825e15	0.201450
$916$	0	0
$917$	−2.32882e16	−1.18606
$918$	0	0
$919$	9.49506e15	0.477818	0.238909	−	0.971042i	$-0.423210\pi$
$919$	9.49506e15	0.477818	0.238909	+	0.971042i	$0.423210\pi$
$920$	0	0
$921$	1.11946e16	0.556649
$922$	0	0
$923$	3.04758e16	1.49743
$924$	0	0
$925$	−1.47861e15	−0.0717915
$926$	0	0
$927$	−1.24734e16	−0.598475
$928$	0	0
$929$	1.43695e16	0.681326	0.340663	−	0.940186i	$-0.389349\pi$
$929$	1.43695e16	0.681326	0.340663	+	0.940186i	$0.389349\pi$
$930$	0	0
$931$	−7.81236e15	−0.366066
$932$	0	0
$933$	8.58413e15	0.397509
$934$	0	0
$935$	−4.89032e15	−0.223807
$936$	0	0
$937$	2.95515e16	1.33663	0.668315	−	0.743878i	$-0.267016\pi$
$937$	2.95515e16	1.33663	0.668315	+	0.743878i	$0.267016\pi$
$938$	0	0
$939$	8.47507e15	0.378863
$940$	0	0
$941$	−2.65472e16	−1.17294	−0.586471	−	0.809971i	$-0.699483\pi$
$941$	−2.65472e16	−1.17294	−0.586471	+	0.809971i	$0.699483\pi$
$942$	0	0
$943$	−3.96994e15	−0.173368
$944$	0	0
$945$	3.03745e15	0.131110
$946$	0	0
$947$	−2.42393e16	−1.03418	−0.517089	−	0.855932i	$-0.672984\pi$
$947$	−2.42393e16	−1.03418	−0.517089	+	0.855932i	$0.672984\pi$
$948$	0	0
$949$	1.19354e16	0.503354
$950$	0	0
$951$	−1.99720e16	−0.832585
$952$	0	0
$953$	−4.41234e15	−0.181827	−0.0909134	−	0.995859i	$-0.528979\pi$
$953$	−4.41234e15	−0.181827	−0.0909134	+	0.995859i	$0.528979\pi$
$954$	0	0
$955$	−4.32501e15	−0.176185
$956$	0	0
$957$	−2.06735e15	−0.0832528
$958$	0	0
$959$	−2.35551e16	−0.937742
$960$	0	0
$961$	−1.09573e16	−0.431247
$962$	0	0
$963$	7.48473e15	0.291227
$964$	0	0
$965$	2.20420e16	0.847910
$966$	0	0
$967$	1.19219e16	0.453421	0.226710	−	0.973962i	$-0.427203\pi$
$967$	1.19219e16	0.453421	0.226710	+	0.973962i	$0.427203\pi$
$968$	0	0
$969$	2.85188e16	1.07238
$970$	0	0
$971$	6.15632e15	0.228884	0.114442	−	0.993430i	$-0.463492\pi$
$971$	6.15632e15	0.228884	0.114442	+	0.993430i	$0.463492\pi$
$972$	0	0
$973$	5.44641e14	0.0200212
$974$	0	0
$975$	1.24971e16	0.454237
$976$	0	0
$977$	−1.06276e16	−0.381956	−0.190978	−	0.981594i	$-0.561166\pi$
$977$	−1.06276e16	−0.381956	−0.190978	+	0.981594i	$0.561166\pi$
$978$	0	0
$979$	−1.11912e14	−0.00397716
$980$	0	0
$981$	−3.46773e15	−0.121861
$982$	0	0
$983$	3.59591e16	1.24958	0.624790	−	0.780793i	$-0.285185\pi$
$983$	3.59591e16	1.24958	0.624790	+	0.780793i	$0.285185\pi$
$984$	0	0
$985$	−1.96126e16	−0.673962
$986$	0	0
$987$	−2.91197e16	−0.989560
$988$	0	0
$989$	3.05246e15	0.102582
$990$	0	0
$991$	−1.41640e16	−0.470740	−0.235370	−	0.971906i	$-0.575630\pi$
$991$	−1.41640e16	−0.470740	−0.235370	+	0.971906i	$0.575630\pi$
$992$	0	0
$993$	9.10093e15	0.299133
$994$	0	0
$995$	2.78881e16	0.906549
$996$	0	0
$997$	−1.14947e16	−0.369552	−0.184776	−	0.982781i	$-0.559156\pi$
$997$	−1.14947e16	−0.369552	−0.184776	+	0.982781i	$0.559156\pi$
$998$	0	0
$999$	6.94760e14	0.0220915

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	132.12.a.b.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.12.a.b.1.3	✓	4	1.1	even	1	trivial

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	132.12.a.b.1.3

Level	$132$
Weight	$12$
Character	132.1
Self dual	yes
Analytic conductor	$101.421$
Analytic rank	$1$
Dimension	$4$
CM	no
Inner twists	$1$