LMFDB - Embedded newform 147.16.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,16,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	147.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:	$209.759452497$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	147.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-72.0000 q^{2} -2187.00 q^{3} -27584.0 q^{4} +221490. q^{5} +157464. q^{6} +4.34534e6 q^{8} +4.78297e6 q^{9} +O(q^{10})$

\(q-72.0000 q^{2} -2187.00 q^{3} -27584.0 q^{4} +221490. q^{5} +157464. q^{6} +4.34534e6 q^{8} +4.78297e6 q^{9} -1.59473e7 q^{10} +3.71693e7 q^{11} +6.03262e7 q^{12} +2.79974e8 q^{13} -4.84399e8 q^{15} +5.91008e8 q^{16} -2.49291e9 q^{17} -3.44374e8 q^{18} +4.66978e9 q^{19} -6.10958e9 q^{20} -2.67619e9 q^{22} -1.84679e10 q^{23} -9.50327e9 q^{24} +1.85402e10 q^{25} -2.01581e10 q^{26} -1.04604e10 q^{27} -1.15953e11 q^{29} +3.48767e10 q^{30} +5.61870e10 q^{31} -1.84941e11 q^{32} -8.12893e10 q^{33} +1.79490e11 q^{34} -1.31933e11 q^{36} +6.14765e11 q^{37} -3.36224e11 q^{38} -6.12304e11 q^{39} +9.62450e11 q^{40} -5.49860e11 q^{41} -9.82884e11 q^{43} -1.02528e12 q^{44} +1.05938e12 q^{45} +1.32969e12 q^{46} -2.07614e12 q^{47} -1.29253e12 q^{48} -1.33490e12 q^{50} +5.45200e12 q^{51} -7.72281e12 q^{52} -1.20484e13 q^{53} +7.53145e11 q^{54} +8.23263e12 q^{55} -1.02128e13 q^{57} +8.34865e12 q^{58} -2.30879e13 q^{59} +1.33617e13 q^{60} +8.50581e12 q^{61} -4.04547e12 q^{62} -6.05040e12 q^{64} +6.20115e13 q^{65} +5.85283e12 q^{66} -1.23310e13 q^{67} +6.87645e13 q^{68} +4.03894e13 q^{69} +5.89892e13 q^{71} +2.07836e13 q^{72} +5.60983e12 q^{73} -4.42631e13 q^{74} -4.05475e13 q^{75} -1.28811e14 q^{76} +4.40859e13 q^{78} +1.59919e14 q^{79} +1.30902e14 q^{80} +2.28768e13 q^{81} +3.95899e13 q^{82} -5.76759e13 q^{83} -5.52155e14 q^{85} +7.07677e13 q^{86} +2.53590e14 q^{87} +1.61513e14 q^{88} +3.62288e14 q^{89} -7.62753e13 q^{90} +5.09419e14 q^{92} -1.22881e14 q^{93} +1.49482e14 q^{94} +1.03431e15 q^{95} +4.04466e14 q^{96} +5.39787e14 q^{97} +1.77780e14 q^{99} +O(q^{100})\)

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 $

$2$	−72.0000	−0.397748	−0.198874	−	0.980025i	$-0.563728\pi$
$2$	−72.0000	−0.397748	−0.198874	+	0.980025i	$0.563728\pi$
$3$	−2187.00	−0.577350
$3$	−2187.00	−0.577350
$4$	−27584.0	−0.841797
$5$	221490.	1.26788	0.633941	−	0.773381i	$-0.281436\pi$
$5$	221490.	1.26788	0.633941	+	0.773381i	$0.281436\pi$
$6$	157464.	0.229640
$7$	0	0
$7$	0	0
$8$	4.34534e6	0.732570
$9$	4.78297e6	0.333333
$10$	−1.59473e7	−0.504297
$11$	3.71693e7	0.575095	0.287547	−	0.957766i	$-0.407160\pi$
$11$	3.71693e7	0.575095	0.287547	+	0.957766i	$0.407160\pi$
$12$	6.03262e7	0.486012
$13$	2.79974e8	1.23749	0.618747	−	0.785590i	$-0.287641\pi$
$13$	2.79974e8	1.23749	0.618747	+	0.785590i	$0.287641\pi$
$14$	0	0
$15$	−4.84399e8	−0.732012
$16$	5.91008e8	0.550419
$17$	−2.49291e9	−1.47347	−0.736733	−	0.676184i	$-0.763633\pi$
$17$	−2.49291e9	−1.47347	−0.736733	+	0.676184i	$0.763633\pi$
$18$	−3.44374e8	−0.132583
$19$	4.66978e9	1.19852	0.599259	−	0.800555i	$-0.295462\pi$
$19$	4.66978e9	1.19852	0.599259	+	0.800555i	$0.295462\pi$
$20$	−6.10958e9	−1.06730
$21$	0	0
$22$	−2.67619e9	−0.228743
$23$	−1.84679e10	−1.13099	−0.565496	−	0.824751i	$-0.691315\pi$
$23$	−1.84679e10	−1.13099	−0.565496	+	0.824751i	$0.691315\pi$
$24$	−9.50327e9	−0.422950
$25$	1.85402e10	0.607527
$26$	−2.01581e10	−0.492210
$27$	−1.04604e10	−0.192450
$28$	0	0
$29$	−1.15953e11	−1.24824	−0.624121	−	0.781328i	$-0.714543\pi$
$29$	−1.15953e11	−1.24824	−0.624121	+	0.781328i	$0.714543\pi$
$30$	3.48767e10	0.291156
$31$	5.61870e10	0.366795	0.183397	−	0.983039i	$-0.441290\pi$
$31$	5.61870e10	0.366795	0.183397	+	0.983039i	$0.441290\pi$
$32$	−1.84941e11	−0.951498
$33$	−8.12893e10	−0.332031
$34$	1.79490e11	0.586068
$35$	0	0
$36$	−1.31933e11	−0.280599
$37$	6.14765e11	1.06462	0.532312	−	0.846548i	$-0.321323\pi$
$37$	6.14765e11	1.06462	0.532312	+	0.846548i	$0.321323\pi$
$38$	−3.36224e11	−0.476708
$39$	−6.12304e11	−0.714467
$40$	9.62450e11	0.928813
$41$	−5.49860e11	−0.440933	−0.220467	−	0.975394i	$-0.570758\pi$
$41$	−5.49860e11	−0.440933	−0.220467	+	0.975394i	$0.570758\pi$
$42$	0	0
$43$	−9.82884e11	−0.551428	−0.275714	−	0.961240i	$-0.588914\pi$
$43$	−9.82884e11	−0.551428	−0.275714	+	0.961240i	$0.588914\pi$
$44$	−1.02528e12	−0.484113
$45$	1.05938e12	0.422628
$46$	1.32969e12	0.449849
$47$	−2.07614e12	−0.597756	−0.298878	−	0.954291i	$-0.596612\pi$
$47$	−2.07614e12	−0.597756	−0.298878	+	0.954291i	$0.596612\pi$
$48$	−1.29253e12	−0.317784
$49$	0	0
$50$	−1.33490e12	−0.241642
$51$	5.45200e12	0.850706
$52$	−7.72281e12	−1.04172
$53$	−1.20484e13	−1.40883	−0.704417	−	0.709787i	$-0.748791\pi$
$53$	−1.20484e13	−1.40883	−0.704417	+	0.709787i	$0.748791\pi$
$54$	7.53145e11	0.0765466
$55$	8.23263e12	0.729153
$56$	0	0
$57$	−1.02128e13	−0.691965
$58$	8.34865e12	0.496485
$59$	−2.30879e13	−1.20780	−0.603899	−	0.797061i	$-0.706387\pi$
$59$	−2.30879e13	−1.20780	−0.603899	+	0.797061i	$0.706387\pi$
$60$	1.33617e13	0.616206
$61$	8.50581e12	0.346531	0.173265	−	0.984875i	$-0.444568\pi$
$61$	8.50581e12	0.346531	0.173265	+	0.984875i	$0.444568\pi$
$62$	−4.04547e12	−0.145892
$63$	0	0
$64$	−6.05040e12	−0.171963
$65$	6.20115e13	1.56900
$66$	5.85283e12	0.132065
$67$	−1.23310e13	−0.248564	−0.124282	−	0.992247i	$-0.539663\pi$
$67$	−1.23310e13	−0.248564	−0.124282	+	0.992247i	$0.539663\pi$
$68$	6.87645e13	1.24036
$69$	4.03894e13	0.652979
$70$	0	0
$71$	5.89892e13	0.769724	0.384862	−	0.922974i	$-0.374249\pi$
$71$	5.89892e13	0.769724	0.384862	+	0.922974i	$0.374249\pi$
$72$	2.07836e13	0.244190
$73$	5.60983e12	0.0594331	0.0297165	−	0.999558i	$-0.490540\pi$
$73$	5.60983e12	0.0594331	0.0297165	+	0.999558i	$0.490540\pi$
$74$	−4.42631e13	−0.423452
$75$	−4.05475e13	−0.350756
$76$	−1.28811e14	−1.00891
$77$	0	0
$78$	4.40859e13	0.284178
$79$	1.59919e14	0.936906	0.468453	−	0.883488i	$-0.344812\pi$
$79$	1.59919e14	0.936906	0.468453	+	0.883488i	$0.344812\pi$
$80$	1.30902e14	0.697867
$81$	2.28768e13	0.111111
$82$	3.95899e13	0.175380
$83$	−5.76759e13	−0.233297	−0.116648	−	0.993173i	$-0.537215\pi$
$83$	−5.76759e13	−0.233297	−0.116648	+	0.993173i	$0.537215\pi$
$84$	0	0
$85$	−5.52155e14	−1.86818
$86$	7.07677e13	0.219329
$87$	2.53590e14	0.720673
$88$	1.61513e14	0.421297
$89$	3.62288e14	0.868217	0.434109	−	0.900861i	$-0.357063\pi$
$89$	3.62288e14	0.868217	0.434109	+	0.900861i	$0.357063\pi$
$90$	−7.62753e13	−0.168099
$91$	0	0
$92$	5.09419e14	0.952066
$93$	−1.22881e14	−0.211769
$94$	1.49482e14	0.237756
$95$	1.03431e15	1.51958
$96$	4.04466e14	0.549348
$97$	5.39787e14	0.678319	0.339160	−	0.940729i	$-0.389857\pi$
$97$	5.39787e14	0.678319	0.339160	+	0.940729i	$0.389857\pi$
$98$	0	0
$99$	1.77780e14	0.191698
$100$	−5.11414e14	−0.511414
$101$	4.22221e14	0.391859	0.195929	−	0.980618i	$-0.437228\pi$
$101$	4.22221e14	0.391859	0.195929	+	0.980618i	$0.437228\pi$
$102$	−3.92544e14	−0.338366
$103$	1.45689e14	0.116720	0.0583602	−	0.998296i	$-0.481413\pi$
$103$	1.45689e14	0.116720	0.0583602	+	0.998296i	$0.481413\pi$
$104$	1.21658e15	0.906551
$105$	0	0
$106$	8.67483e14	0.560360
$107$	−4.08696e14	−0.246049	−0.123024	−	0.992404i	$-0.539259\pi$
$107$	−4.08696e14	−0.246049	−0.123024	+	0.992404i	$0.539259\pi$
$108$	2.88538e14	0.162004
$109$	1.10134e15	0.577060	0.288530	−	0.957471i	$-0.406833\pi$
$109$	1.10134e15	0.577060	0.288530	+	0.957471i	$0.406833\pi$
$110$	−5.92749e14	−0.290019
$111$	−1.34449e15	−0.614661
$112$	0	0
$113$	−2.13023e15	−0.851802	−0.425901	−	0.904770i	$-0.640043\pi$
$113$	−2.13023e15	−0.851802	−0.425901	+	0.904770i	$0.640043\pi$
$114$	7.35323e14	0.275227
$115$	−4.09046e15	−1.43397
$116$	3.19846e15	1.05077
$117$	1.33911e15	0.412498
$118$	1.66233e15	0.480399
$119$	0	0
$120$	−2.10488e15	−0.536251
$121$	−2.79569e15	−0.669266
$122$	−6.12418e14	−0.137832
$123$	1.20254e15	0.254573
$124$	−1.54986e15	−0.308767
$125$	−2.65286e15	−0.497610
$126$	0	0
$127$	−6.25962e15	−1.04236	−0.521182	−	0.853445i	$-0.674509\pi$
$127$	−6.25962e15	−1.04236	−0.521182	+	0.853445i	$0.674509\pi$
$128$	6.49577e15	1.01990
$129$	2.14957e15	0.318367
$130$	−4.46483e15	−0.624065
$131$	−1.01059e16	−1.33365	−0.666825	−	0.745214i	$-0.732347\pi$
$131$	−1.01059e16	−1.33365	−0.666825	+	0.745214i	$0.732347\pi$
$132$	2.24228e15	0.279503
$133$	0	0
$134$	8.87833e14	0.0988656
$135$	−2.31686e15	−0.244004
$136$	−1.08326e16	−1.07942
$137$	−9.93739e15	−0.937277	−0.468638	−	0.883390i	$-0.655255\pi$
$137$	−9.93739e15	−0.937277	−0.468638	+	0.883390i	$0.655255\pi$
$138$	−2.90803e15	−0.259721
$139$	1.95078e16	1.65043	0.825216	−	0.564817i	$-0.191053\pi$
$139$	1.95078e16	1.65043	0.825216	+	0.564817i	$0.191053\pi$
$140$	0	0
$141$	4.54053e15	0.345114
$142$	−4.24722e15	−0.306156
$143$	1.04065e16	0.711676
$144$	2.82677e15	0.183473
$145$	−2.56825e16	−1.58262
$146$	−4.03908e14	−0.0236394
$147$	0	0
$148$	−1.69577e16	−0.896197
$149$	1.58935e16	0.798589	0.399295	−	0.916823i	$-0.369255\pi$
$149$	1.58935e16	0.798589	0.399295	+	0.916823i	$0.369255\pi$
$150$	2.91942e15	0.139512
$151$	3.10597e16	1.41211	0.706057	−	0.708155i	$-0.250472\pi$
$151$	3.10597e16	1.41211	0.706057	+	0.708155i	$0.250472\pi$
$152$	2.02918e16	0.877999
$153$	−1.19235e16	−0.491155
$154$	0	0
$155$	1.24449e16	0.465053
$156$	1.68898e16	0.601436
$157$	5.41372e16	1.83759	0.918794	−	0.394738i	$-0.129165\pi$
$157$	5.41372e16	1.83759	0.918794	+	0.394738i	$0.129165\pi$
$158$	−1.15141e16	−0.372652
$159$	2.63498e16	0.813390
$160$	−4.09625e16	−1.20639
$161$	0	0
$162$	−1.64713e15	−0.0441942
$163$	−5.54612e16	−1.42096	−0.710481	−	0.703717i	$-0.751522\pi$
$163$	−5.54612e16	−1.42096	−0.710481	+	0.703717i	$0.751522\pi$
$164$	1.51673e16	0.371176
$165$	−1.80048e16	−0.420977
$166$	4.15266e15	0.0927932
$167$	5.83552e15	0.124654	0.0623270	−	0.998056i	$-0.480148\pi$
$167$	5.83552e15	0.124654	0.0623270	+	0.998056i	$0.480148\pi$
$168$	0	0
$169$	2.71997e16	0.531390
$170$	3.97552e16	0.743065
$171$	2.23354e16	0.399506
$172$	2.71119e16	0.464191
$173$	−4.40958e16	−0.722855	−0.361428	−	0.932400i	$-0.617711\pi$
$173$	−4.40958e16	−0.722855	−0.361428	+	0.932400i	$0.617711\pi$
$174$	−1.82585e16	−0.286646
$175$	0	0
$176$	2.19674e16	0.316543
$177$	5.04932e16	0.697323
$178$	−2.60847e16	−0.345331
$179$	−5.10594e16	−0.648153	−0.324077	−	0.946031i	$-0.605054\pi$
$179$	−5.10594e16	−0.648153	−0.324077	+	0.946031i	$0.605054\pi$
$180$	−2.92219e16	−0.355767
$181$	−3.16143e16	−0.369228	−0.184614	−	0.982811i	$-0.559103\pi$
$181$	−3.16143e16	−0.369228	−0.184614	+	0.982811i	$0.559103\pi$
$182$	0	0
$183$	−1.86022e16	−0.200070
$184$	−8.02495e16	−0.828531
$185$	1.36164e17	1.34982
$186$	8.84743e15	0.0842307
$187$	−9.26599e16	−0.847383
$188$	5.72684e16	0.503189
$189$	0	0
$190$	−7.44703e16	−0.604409
$191$	9.44291e16	0.736810	0.368405	−	0.929665i	$-0.379904\pi$
$191$	9.44291e16	0.736810	0.368405	+	0.929665i	$0.379904\pi$
$192$	1.32322e16	0.0992828
$193$	4.17744e16	0.301460	0.150730	−	0.988575i	$-0.451838\pi$
$193$	4.17744e16	0.301460	0.150730	+	0.988575i	$0.451838\pi$
$194$	−3.88646e16	−0.269800
$195$	−1.35619e17	−0.905861
$196$	0	0
$197$	−1.48383e17	−0.918094	−0.459047	−	0.888412i	$-0.651809\pi$
$197$	−1.48383e17	−0.918094	−0.459047	+	0.888412i	$0.651809\pi$
$198$	−1.28001e16	−0.0762475
$199$	−1.97555e17	−1.13316	−0.566579	−	0.824007i	$-0.691733\pi$
$199$	−1.97555e17	−1.13316	−0.566579	+	0.824007i	$0.691733\pi$
$200$	8.05637e16	0.445056
$201$	2.69679e16	0.143508
$202$	−3.03999e16	−0.155861
$203$	0	0
$204$	−1.50388e17	−0.716122
$205$	−1.21788e17	−0.559052
$206$	−1.04896e16	−0.0464253
$207$	−8.83316e16	−0.376997
$208$	1.65467e17	0.681140
$209$	1.73573e17	0.689262
$210$	0	0
$211$	−4.48570e17	−1.65849	−0.829244	−	0.558887i	$-0.811228\pi$
$211$	−4.48570e17	−1.65849	−0.829244	+	0.558887i	$0.811228\pi$
$212$	3.32342e17	1.18595
$213$	−1.29009e17	−0.444400
$214$	2.94261e16	0.0978654
$215$	−2.17699e17	−0.699146
$216$	−4.54538e16	−0.140983
$217$	0	0
$218$	−7.92962e16	−0.229524
$219$	−1.22687e16	−0.0343137
$220$	−2.27089e17	−0.613799
$221$	−6.97951e17	−1.82341
$222$	9.68033e16	0.244480
$223$	−1.59977e17	−0.390634	−0.195317	−	0.980740i	$-0.562574\pi$
$223$	−1.59977e17	−0.390634	−0.195317	+	0.980740i	$0.562574\pi$
$224$	0	0
$225$	8.86774e16	0.202509
$226$	1.53377e17	0.338802
$227$	−1.45803e17	−0.311582	−0.155791	−	0.987790i	$-0.549793\pi$
$227$	−1.45803e17	−0.311582	−0.155791	+	0.987790i	$0.549793\pi$
$228$	2.81710e17	0.582494
$229$	−6.81194e17	−1.36303	−0.681514	−	0.731805i	$-0.738678\pi$
$229$	−6.81194e17	−1.36303	−0.681514	+	0.731805i	$0.738678\pi$
$230$	2.94513e17	0.570356
$231$	0	0
$232$	−5.03858e17	−0.914425
$233$	−3.89035e17	−0.683628	−0.341814	−	0.939768i	$-0.611041\pi$
$233$	−3.89035e17	−0.683628	−0.341814	+	0.939768i	$0.611041\pi$
$234$	−9.64158e16	−0.164070
$235$	−4.59845e17	−0.757884
$236$	6.36857e17	1.01672
$237$	−3.49742e17	−0.540923
$238$	0	0
$239$	6.76227e16	0.0981993	0.0490996	−	0.998794i	$-0.484365\pi$
$239$	6.76227e16	0.0981993	0.0490996	+	0.998794i	$0.484365\pi$
$240$	−2.86283e17	−0.402913
$241$	9.00528e17	1.22848	0.614242	−	0.789118i	$-0.289462\pi$
$241$	9.00528e17	1.22848	0.614242	+	0.789118i	$0.289462\pi$
$242$	2.01290e17	0.266199
$243$	−5.00315e16	−0.0641500
$244$	−2.34624e17	−0.291709
$245$	0	0
$246$	−8.65831e16	−0.101256
$247$	1.30742e18	1.48316
$248$	2.44152e17	0.268703
$249$	1.26137e17	0.134694
$250$	1.91006e17	0.197923
$251$	−1.58929e18	−1.59827	−0.799134	−	0.601153i	$-0.794708\pi$
$251$	−1.58929e18	−1.59827	−0.799134	+	0.601153i	$0.794708\pi$
$252$	0	0
$253$	−6.86440e17	−0.650428
$254$	4.50692e17	0.414598
$255$	1.20756e18	1.07860
$256$	−2.69436e17	−0.233698
$257$	−2.29738e18	−1.93524	−0.967618	−	0.252418i	$-0.918774\pi$
$257$	−2.29738e18	−1.93524	−0.967618	+	0.252418i	$0.918774\pi$
$258$	−1.54769e17	−0.126630
$259$	0	0
$260$	−1.71053e18	−1.32078
$261$	−5.54602e17	−0.416081
$262$	7.27628e17	0.530456
$263$	−5.05657e17	−0.358252	−0.179126	−	0.983826i	$-0.557327\pi$
$263$	−5.05657e17	−0.358252	−0.179126	+	0.983826i	$0.557327\pi$
$264$	−3.53230e17	−0.243236
$265$	−2.66860e18	−1.78624
$266$	0	0
$267$	−7.92323e17	−0.501265
$268$	3.40139e17	0.209240
$269$	2.44731e18	1.46402	0.732009	−	0.681295i	$-0.238583\pi$
$269$	2.44731e18	1.46402	0.732009	+	0.681295i	$0.238583\pi$
$270$	1.66814e17	0.0970521
$271$	1.53807e18	0.870373	0.435187	−	0.900340i	$-0.356682\pi$
$271$	1.53807e18	0.870373	0.435187	+	0.900340i	$0.356682\pi$
$272$	−1.47333e18	−0.811024
$273$	0	0
$274$	7.15492e17	0.372800
$275$	6.89128e17	0.349385
$276$	−1.11410e18	−0.549675
$277$	−7.89959e17	−0.379321	−0.189660	−	0.981850i	$-0.560739\pi$
$277$	−7.89959e17	−0.379321	−0.189660	+	0.981850i	$0.560739\pi$
$278$	−1.40456e18	−0.656456
$279$	2.68741e17	0.122265
$280$	0	0
$281$	−6.15303e17	−0.265333	−0.132667	−	0.991161i	$-0.542354\pi$
$281$	−6.15303e17	−0.265333	−0.132667	+	0.991161i	$0.542354\pi$
$282$	−3.26918e17	−0.137268
$283$	1.26279e18	0.516337	0.258169	−	0.966100i	$-0.416881\pi$
$283$	1.26279e18	0.516337	0.258169	+	0.966100i	$0.416881\pi$
$284$	−1.62716e18	−0.647951
$285$	−2.26204e18	−0.877330
$286$	−7.49265e17	−0.283067
$287$	0	0
$288$	−8.84566e17	−0.317166
$289$	3.35219e18	1.17110
$290$	1.84914e18	0.629485
$291$	−1.18051e18	−0.391628
$292$	−1.54742e17	−0.0500306
$293$	3.49864e18	1.10253	0.551267	−	0.834329i	$-0.314145\pi$
$293$	3.49864e18	1.10253	0.551267	+	0.834329i	$0.314145\pi$
$294$	0	0
$295$	−5.11374e18	−1.53135
$296$	2.67137e18	0.779912
$297$	−3.88804e17	−0.110677
$298$	−1.14433e18	−0.317637
$299$	−5.17055e18	−1.39960
$300$	1.11846e18	0.295265
$301$	0	0
$302$	−2.23630e18	−0.561665
$303$	−9.23398e17	−0.226240
$304$	2.75988e18	0.659687
$305$	1.88395e18	0.439360
$306$	8.58494e17	0.195356
$307$	−1.28861e18	−0.286142	−0.143071	−	0.989712i	$-0.545698\pi$
$307$	−1.28861e18	−0.286142	−0.143071	+	0.989712i	$0.545698\pi$
$308$	0	0
$309$	−3.18621e17	−0.0673886
$310$	−8.96030e17	−0.184974
$311$	−8.43030e18	−1.69879	−0.849396	−	0.527756i	$-0.823033\pi$
$311$	−8.43030e18	−1.69879	−0.849396	+	0.527756i	$0.823033\pi$
$312$	−2.66067e18	−0.523397
$313$	−7.88359e18	−1.51405	−0.757027	−	0.653383i	$-0.773349\pi$
$313$	−7.88359e18	−1.51405	−0.757027	+	0.653383i	$0.773349\pi$
$314$	−3.89788e18	−0.730896
$315$	0	0
$316$	−4.41120e18	−0.788685
$317$	−8.71539e18	−1.52175	−0.760873	−	0.648900i	$-0.775229\pi$
$317$	−8.71539e18	−1.52175	−0.760873	+	0.648900i	$0.775229\pi$
$318$	−1.89719e18	−0.323524
$319$	−4.30991e18	−0.717857
$320$	−1.34010e18	−0.218029
$321$	8.93817e17	0.142056
$322$	0	0
$323$	−1.16414e19	−1.76598
$324$	−6.31033e17	−0.0935330
$325$	5.19079e18	0.751810
$326$	3.99321e18	0.565184
$327$	−2.40862e18	−0.333166
$328$	−2.38933e18	−0.323015
$329$	0	0
$330$	1.29634e18	0.167442
$331$	1.26995e19	1.60353	0.801763	−	0.597643i	$-0.203896\pi$
$331$	1.26995e19	1.60353	0.801763	+	0.597643i	$0.203896\pi$
$332$	1.59093e18	0.196388
$333$	2.94040e18	0.354875
$334$	−4.20157e17	−0.0495808
$335$	−2.73120e18	−0.315150
$336$	0	0
$337$	−6.64250e18	−0.733005	−0.366503	−	0.930417i	$-0.619445\pi$
$337$	−6.64250e18	−0.733005	−0.366503	+	0.930417i	$0.619445\pi$
$338$	−1.95838e18	−0.211359
$339$	4.65882e18	0.491788
$340$	1.52307e19	1.57263
$341$	2.08843e18	0.210942
$342$	−1.60815e18	−0.158903
$343$	0	0
$344$	−4.27097e18	−0.403960
$345$	8.94584e18	0.827900
$346$	3.17490e18	0.287514
$347$	1.87835e18	0.166458	0.0832289	−	0.996530i	$-0.473477\pi$
$347$	1.87835e18	0.166458	0.0832289	+	0.996530i	$0.473477\pi$
$348$	−6.99503e18	−0.606660
$349$	1.48545e19	1.26086	0.630432	−	0.776245i	$-0.282878\pi$
$349$	1.48545e19	1.26086	0.630432	+	0.776245i	$0.282878\pi$
$350$	0	0
$351$	−2.92863e18	−0.238156
$352$	−6.87412e18	−0.547202
$353$	8.23380e18	0.641638	0.320819	−	0.947140i	$-0.396042\pi$
$353$	8.23380e18	0.641638	0.320819	+	0.947140i	$0.396042\pi$
$354$	−3.63551e18	−0.277358
$355$	1.30655e19	0.975919
$356$	−9.99334e18	−0.730863
$357$	0	0
$358$	3.67627e18	0.257801
$359$	1.87417e19	1.28706	0.643532	−	0.765419i	$-0.277468\pi$
$359$	1.87417e19	1.28706	0.643532	+	0.765419i	$0.277468\pi$
$360$	4.60337e18	0.309604
$361$	6.62574e18	0.436446
$362$	2.27623e18	0.146859
$363$	6.11417e18	0.386401
$364$	0	0
$365$	1.24252e18	0.0753542
$366$	1.33936e18	0.0795772
$367$	−8.63117e18	−0.502428	−0.251214	−	0.967932i	$-0.580830\pi$
$367$	−8.63117e18	−0.502428	−0.251214	+	0.967932i	$0.580830\pi$
$368$	−1.09147e19	−0.622519
$369$	−2.62996e18	−0.146978
$370$	−9.80383e18	−0.536887
$371$	0	0
$372$	3.38955e18	0.178267
$373$	−9.95975e18	−0.513372	−0.256686	−	0.966495i	$-0.582631\pi$
$373$	−9.95975e18	−0.513372	−0.256686	+	0.966495i	$0.582631\pi$
$374$	6.67151e18	0.337044
$375$	5.80181e18	0.287295
$376$	−9.02156e18	−0.437898
$377$	−3.24640e19	−1.54469
$378$	0	0
$379$	1.00949e19	0.461645	0.230823	−	0.972996i	$-0.425858\pi$
$379$	1.00949e19	0.461645	0.230823	+	0.972996i	$0.425858\pi$
$380$	−2.85304e19	−1.27918
$381$	1.36898e19	0.601810
$382$	−6.79890e18	−0.293065
$383$	2.71764e19	1.14869	0.574343	−	0.818614i	$-0.305257\pi$
$383$	2.71764e19	1.14869	0.574343	+	0.818614i	$0.305257\pi$
$384$	−1.42062e19	−0.588837
$385$	0	0
$386$	−3.00775e18	−0.119905
$387$	−4.70111e18	−0.183809
$388$	−1.48895e19	−0.571007
$389$	−9.05285e18	−0.340536	−0.170268	−	0.985398i	$-0.554463\pi$
$389$	−9.05285e18	−0.340536	−0.170268	+	0.985398i	$0.554463\pi$
$390$	9.76458e18	0.360304
$391$	4.60389e19	1.66648
$392$	0	0
$393$	2.21017e19	0.769984
$394$	1.06836e19	0.365170
$395$	3.54204e19	1.18789
$396$	−4.90387e18	−0.161371
$397$	1.76495e19	0.569908	0.284954	−	0.958541i	$-0.408022\pi$
$397$	1.76495e19	0.569908	0.284954	+	0.958541i	$0.408022\pi$
$398$	1.42240e19	0.450711
$399$	0	0
$400$	1.09574e19	0.334394
$401$	2.81135e19	0.842040	0.421020	−	0.907051i	$-0.361672\pi$
$401$	2.81135e19	0.842040	0.421020	+	0.907051i	$0.361672\pi$
$402$	−1.94169e18	−0.0570801
$403$	1.57309e19	0.453906
$404$	−1.16465e19	−0.329866
$405$	5.06698e18	0.140876
$406$	0	0
$407$	2.28504e19	0.612260
$408$	2.36908e19	0.623202
$409$	−6.42955e19	−1.66056	−0.830282	−	0.557343i	$-0.811821\pi$
$409$	−6.42955e19	−1.66056	−0.830282	+	0.557343i	$0.811821\pi$
$410$	8.76877e18	0.222362
$411$	2.17331e19	0.541137
$412$	−4.01868e18	−0.0982549
$413$	0	0
$414$	6.35987e18	0.149950
$415$	−1.27746e19	−0.295793
$416$	−5.17787e19	−1.17747
$417$	−4.26636e19	−0.952878
$418$	−1.24972e19	−0.274152
$419$	2.34474e19	0.505230	0.252615	−	0.967567i	$-0.418709\pi$
$419$	2.34474e19	0.505230	0.252615	+	0.967567i	$0.418709\pi$
$420$	0	0
$421$	−7.28227e19	−1.51409	−0.757044	−	0.653364i	$-0.773357\pi$
$421$	−7.28227e19	−1.51409	−0.757044	+	0.653364i	$0.773357\pi$
$422$	3.22971e19	0.659659
$423$	−9.93013e18	−0.199252
$424$	−5.23543e19	−1.03207
$425$	−4.62192e19	−0.895170
$426$	9.28867e18	0.176759
$427$	0	0
$428$	1.12735e19	0.207123
$429$	−2.27589e19	−0.410886
$430$	1.56743e19	0.278084
$431$	−4.71831e19	−0.822635	−0.411317	−	0.911492i	$-0.634931\pi$
$431$	−4.71831e19	−0.822635	−0.411317	+	0.911492i	$0.634931\pi$
$432$	−6.18215e18	−0.105928
$433$	−3.79099e19	−0.638400	−0.319200	−	0.947687i	$-0.603414\pi$
$433$	−3.79099e19	−0.638400	−0.319200	+	0.947687i	$0.603414\pi$
$434$	0	0
$435$	5.61677e19	0.913728
$436$	−3.03793e19	−0.485768
$437$	−8.62412e19	−1.35551
$438$	8.83346e17	0.0136482
$439$	2.26832e19	0.344525	0.172262	−	0.985051i	$-0.444892\pi$
$439$	2.26832e19	0.344525	0.172262	+	0.985051i	$0.444892\pi$
$440$	3.57736e19	0.534156
$441$	0	0
$442$	5.02525e19	0.725255
$443$	−9.94888e19	−1.41171	−0.705856	−	0.708355i	$-0.749438\pi$
$443$	−9.94888e19	−1.41171	−0.705856	+	0.708355i	$0.749438\pi$
$444$	3.70864e19	0.517420
$445$	8.02431e19	1.10080
$446$	1.15183e19	0.155374
$447$	−3.47592e19	−0.461066
$448$	0	0
$449$	1.30159e20	1.66965	0.834825	−	0.550515i	$-0.185568\pi$
$449$	1.30159e20	1.66965	0.834825	+	0.550515i	$0.185568\pi$
$450$	−6.38477e18	−0.0805474
$451$	−2.04379e19	−0.253579
$452$	5.87604e19	0.717045
$453$	−6.79275e19	−0.815285
$454$	1.04978e19	0.123931
$455$	0	0
$456$	−4.43782e19	−0.506913
$457$	−1.01910e20	−1.14510	−0.572550	−	0.819870i	$-0.694046\pi$
$457$	−1.01910e20	−1.14510	−0.572550	+	0.819870i	$0.694046\pi$
$458$	4.90460e19	0.542141
$459$	2.60767e19	0.283569
$460$	1.12831e20	1.20711
$461$	1.08142e20	1.13825	0.569123	−	0.822253i	$-0.307283\pi$
$461$	1.08142e20	1.13825	0.569123	+	0.822253i	$0.307283\pi$
$462$	0	0
$463$	1.69113e20	1.72314	0.861569	−	0.507641i	$-0.169482\pi$
$463$	1.69113e20	1.72314	0.861569	+	0.507641i	$0.169482\pi$
$464$	−6.85294e19	−0.687056
$465$	−2.72169e19	−0.268498
$466$	2.80105e19	0.271911
$467$	4.58418e19	0.437910	0.218955	−	0.975735i	$-0.429735\pi$
$467$	4.58418e19	0.437910	0.218955	+	0.975735i	$0.429735\pi$
$468$	−3.69380e19	−0.347239
$469$	0	0
$470$	3.31089e19	0.301446
$471$	−1.18398e20	−1.06093
$472$	−1.00325e20	−0.884797
$473$	−3.65331e19	−0.317124
$474$	2.51814e19	0.215151
$475$	8.65789e19	0.728132
$476$	0	0
$477$	−5.76270e19	−0.469611
$478$	−4.86884e18	−0.0390585
$479$	−1.31513e20	−1.03861	−0.519303	−	0.854590i	$-0.673808\pi$
$479$	−1.31513e20	−1.03861	−0.519303	+	0.854590i	$0.673808\pi$
$480$	8.95851e19	0.696508
$481$	1.72118e20	1.31747
$482$	−6.48380e19	−0.488626
$483$	0	0
$484$	7.71163e19	0.563386
$485$	1.19557e20	0.860029
$486$	3.60227e18	0.0255155
$487$	6.13671e19	0.428024	0.214012	−	0.976831i	$-0.431347\pi$
$487$	6.13671e19	0.428024	0.214012	+	0.976831i	$0.431347\pi$
$488$	3.69607e19	0.253858
$489$	1.21294e20	0.820392
$490$	0	0
$491$	2.42057e20	1.58784	0.793918	−	0.608024i	$-0.208038\pi$
$491$	2.42057e20	1.58784	0.793918	+	0.608024i	$0.208038\pi$
$492$	−3.31710e19	−0.214299
$493$	2.89062e20	1.83924
$494$	−9.41342e19	−0.589923
$495$	3.93764e19	0.243051
$496$	3.32070e19	0.201891
$497$	0	0
$498$	−9.08188e18	−0.0535742
$499$	1.46049e20	0.848680	0.424340	−	0.905503i	$-0.360506\pi$
$499$	1.46049e20	0.848680	0.424340	+	0.905503i	$0.360506\pi$
$500$	7.31765e19	0.418887
$501$	−1.27623e19	−0.0719690
$502$	1.14429e20	0.635707
$503$	1.68041e20	0.919720	0.459860	−	0.887991i	$-0.347900\pi$
$503$	1.68041e20	0.919720	0.459860	+	0.887991i	$0.347900\pi$
$504$	0	0
$505$	9.35178e19	0.496831
$506$	4.94237e19	0.258706
$507$	−5.94857e19	−0.306798
$508$	1.72665e20	0.877459
$509$	−7.37709e19	−0.369404	−0.184702	−	0.982795i	$-0.559132\pi$
$509$	−7.37709e19	−0.369404	−0.184702	+	0.982795i	$0.559132\pi$
$510$	−8.69446e19	−0.429009
$511$	0	0
$512$	−1.93454e20	−0.926943
$513$	−4.88476e19	−0.230655
$514$	1.65411e20	0.769736
$515$	3.22686e19	0.147988
$516$	−5.92937e19	−0.268001
$517$	−7.71689e19	−0.343766
$518$	0	0
$519$	9.64375e19	0.417341
$520$	2.69461e20	1.14940
$521$	−3.22023e20	−1.35396	−0.676978	−	0.736003i	$-0.736711\pi$
$521$	−3.22023e20	−1.35396	−0.676978	+	0.736003i	$0.736711\pi$
$522$	3.99313e19	0.165495
$523$	1.19064e19	0.0486429	0.0243214	−	0.999704i	$-0.492257\pi$
$523$	1.19064e19	0.0486429	0.0243214	+	0.999704i	$0.492257\pi$
$524$	2.78762e20	1.12266
$525$	0	0
$526$	3.64073e19	0.142494
$527$	−1.40069e20	−0.540460
$528$	−4.80426e19	−0.182756
$529$	7.44293e19	0.279143
$530$	1.92139e20	0.710471
$531$	−1.10429e20	−0.402599
$532$	0	0
$533$	−1.53947e20	−0.545652
$534$	5.70473e19	0.199377
$535$	−9.05220e19	−0.311961
$536$	−5.35825e19	−0.182090
$537$	1.11667e20	0.374211
$538$	−1.76206e20	−0.582309
$539$	0	0
$540$	6.39084e19	0.205402
$541$	1.59033e20	0.504089	0.252045	−	0.967716i	$-0.418897\pi$
$541$	1.59033e20	0.504089	0.252045	+	0.967716i	$0.418897\pi$
$542$	−1.10741e20	−0.346189
$543$	6.91405e19	0.213174
$544$	4.61041e20	1.40200
$545$	2.43935e20	0.731645
$546$	0	0
$547$	−4.25405e20	−1.24136	−0.620680	−	0.784064i	$-0.713143\pi$
$547$	−4.25405e20	−1.24136	−0.620680	+	0.784064i	$0.713143\pi$
$548$	2.74113e20	0.788997
$549$	4.06830e19	0.115510
$550$	−4.96172e19	−0.138967
$551$	−5.41477e20	−1.49604
$552$	1.75506e20	0.478353
$553$	0	0
$554$	5.68771e19	0.150874
$555$	−2.97791e20	−0.779318
$556$	−5.38104e20	−1.38933
$557$	2.71579e20	0.691803	0.345901	−	0.938271i	$-0.387573\pi$
$557$	2.71579e20	0.691803	0.345901	+	0.938271i	$0.387573\pi$
$558$	−1.93493e19	−0.0486306
$559$	−2.75182e20	−0.682389
$560$	0	0
$561$	2.02647e20	0.489237
$562$	4.43018e19	0.105536
$563$	2.68438e20	0.631002	0.315501	−	0.948925i	$-0.397827\pi$
$563$	2.68438e20	0.631002	0.315501	+	0.948925i	$0.397827\pi$
$564$	−1.25246e20	−0.290516
$565$	−4.71825e20	−1.07999
$566$	−9.09210e19	−0.205372
$567$	0	0
$568$	2.56328e20	0.563877
$569$	−1.34893e20	−0.292853	−0.146426	−	0.989222i	$-0.546777\pi$
$569$	−1.34893e20	−0.292853	−0.146426	+	0.989222i	$0.546777\pi$
$570$	1.62867e20	0.348956
$571$	−6.81153e20	−1.44037	−0.720185	−	0.693782i	$-0.755943\pi$
$571$	−6.81153e20	−1.44037	−0.720185	+	0.693782i	$0.755943\pi$
$572$	−2.87052e20	−0.599087
$573$	−2.06517e20	−0.425398
$574$	0	0
$575$	−3.42400e20	−0.687108
$576$	−2.89389e19	−0.0573209
$577$	2.42927e20	0.474961	0.237480	−	0.971392i	$-0.423678\pi$
$577$	2.42927e20	0.474961	0.237480	+	0.971392i	$0.423678\pi$
$578$	−2.41358e20	−0.465803
$579$	−9.13605e19	−0.174048
$580$	7.08427e20	1.33225
$581$	0	0
$582$	8.49970e19	0.155769
$583$	−4.47830e20	−0.810213
$584$	2.43766e19	0.0435389
$585$	2.96599e20	0.522999
$586$	−2.51902e20	−0.438530
$587$	1.06241e20	0.182602	0.0913012	−	0.995823i	$-0.470897\pi$
$587$	1.06241e20	0.182602	0.0913012	+	0.995823i	$0.470897\pi$
$588$	0	0
$589$	2.62381e20	0.439610
$590$	3.68189e20	0.609089
$591$	3.24513e20	0.530062
$592$	3.63331e20	0.585989
$593$	7.70415e20	1.22692	0.613458	−	0.789727i	$-0.289778\pi$
$593$	7.70415e20	1.22692	0.613458	+	0.789727i	$0.289778\pi$
$594$	2.79939e19	0.0440215
$595$	0	0
$596$	−4.38407e20	−0.672250
$597$	4.32053e20	0.654229
$598$	3.72279e20	0.556686
$599$	−7.59796e19	−0.112201	−0.0561004	−	0.998425i	$-0.517867\pi$
$599$	−7.59796e19	−0.112201	−0.0561004	+	0.998425i	$0.517867\pi$
$600$	−1.76193e20	−0.256953
$601$	−7.78003e20	−1.12053	−0.560264	−	0.828314i	$-0.689300\pi$
$601$	−7.78003e20	−1.12053	−0.560264	+	0.828314i	$0.689300\pi$
$602$	0	0
$603$	−5.89788e19	−0.0828546
$604$	−8.56750e20	−1.18871
$605$	−6.19217e20	−0.848551
$606$	6.64846e19	0.0899863
$607$	−3.21867e20	−0.430290	−0.215145	−	0.976582i	$-0.569022\pi$
$607$	−3.21867e20	−0.430290	−0.215145	+	0.976582i	$0.569022\pi$
$608$	−8.63633e20	−1.14039
$609$	0	0
$610$	−1.35645e20	−0.174755
$611$	−5.81267e20	−0.739719
$612$	3.28898e20	0.413453
$613$	6.68867e20	0.830589	0.415294	−	0.909687i	$-0.363679\pi$
$613$	6.68867e20	0.830589	0.415294	+	0.909687i	$0.363679\pi$
$614$	9.27796e19	0.113812
$615$	2.66351e20	0.322769
$616$	0	0
$617$	−8.47904e20	−1.00279	−0.501393	−	0.865220i	$-0.667179\pi$
$617$	−8.47904e20	−1.00279	−0.501393	+	0.865220i	$0.667179\pi$
$618$	2.29407e19	0.0268036
$619$	3.51027e20	0.405192	0.202596	−	0.979262i	$-0.435062\pi$
$619$	3.51027e20	0.405192	0.202596	+	0.979262i	$0.435062\pi$
$620$	−3.43279e20	−0.391480
$621$	1.93181e20	0.217660
$622$	6.06982e20	0.675690
$623$	0	0
$624$	−3.61876e20	−0.393256
$625$	−1.15339e21	−1.23844
$626$	5.67619e20	0.602212
$627$	−3.79603e20	−0.397945
$628$	−1.49332e21	−1.54688
$629$	−1.53256e21	−1.56869
$630$	0	0
$631$	−1.30718e21	−1.30652	−0.653260	−	0.757134i	$-0.726599\pi$
$631$	−1.30718e21	−1.30652	−0.653260	+	0.757134i	$0.726599\pi$
$632$	6.94902e20	0.686349
$633$	9.81023e20	0.957528
$634$	6.27508e20	0.605271
$635$	−1.38644e21	−1.32160
$636$	−7.26833e20	−0.684709
$637$	0	0
$638$	3.10314e20	0.285526
$639$	2.82143e20	0.256575
$640$	1.43875e21	1.29311
$641$	−1.77975e21	−1.58097	−0.790486	−	0.612480i	$-0.790172\pi$
$641$	−1.77975e21	−1.58097	−0.790486	+	0.612480i	$0.790172\pi$
$642$	−6.43548e19	−0.0565026
$643$	5.18269e19	0.0449752	0.0224876	−	0.999747i	$-0.492841\pi$
$643$	5.18269e19	0.0449752	0.0224876	+	0.999747i	$0.492841\pi$
$644$	0	0
$645$	4.76108e20	0.403652
$646$	8.38178e20	0.702413
$647$	−1.80268e21	−1.49327	−0.746634	−	0.665235i	$-0.768331\pi$
$647$	−1.80268e21	−1.49327	−0.746634	+	0.665235i	$0.768331\pi$
$648$	9.94075e19	0.0813967
$649$	−8.58162e20	−0.694599
$650$	−3.73737e20	−0.299031
$651$	0	0
$652$	1.52984e21	1.19616
$653$	6.01117e20	0.464633	0.232317	−	0.972640i	$-0.425369\pi$
$653$	6.01117e20	0.464633	0.232317	+	0.972640i	$0.425369\pi$
$654$	1.73421e20	0.132516
$655$	−2.23837e21	−1.69091
$656$	−3.24971e20	−0.242698
$657$	2.68316e19	0.0198110
$658$	0	0
$659$	−1.34492e21	−0.970635	−0.485318	−	0.874338i	$-0.661296\pi$
$659$	−1.34492e21	−0.970635	−0.485318	+	0.874338i	$0.661296\pi$
$660$	4.96643e20	0.354377
$661$	−3.00301e20	−0.211858	−0.105929	−	0.994374i	$-0.533782\pi$
$661$	−3.00301e20	−0.211858	−0.105929	+	0.994374i	$0.533782\pi$
$662$	−9.14362e20	−0.637798
$663$	1.52642e21	1.05274
$664$	−2.50622e20	−0.170906
$665$	0	0
$666$	−2.11709e20	−0.141151
$667$	2.14142e21	1.41175
$668$	−1.60967e20	−0.104933
$669$	3.49869e20	0.225533
$670$	1.96646e20	0.125350
$671$	3.16155e20	0.199288
$672$	0	0
$673$	−2.48252e21	−1.53031	−0.765156	−	0.643845i	$-0.777338\pi$
$673$	−2.48252e21	−1.53031	−0.765156	+	0.643845i	$0.777338\pi$
$674$	4.78260e20	0.291551
$675$	−1.93937e20	−0.116919
$676$	−7.50276e20	−0.447323
$677$	1.84000e21	1.08493	0.542467	−	0.840077i	$-0.317491\pi$
$677$	1.84000e21	1.08493	0.542467	+	0.840077i	$0.317491\pi$
$678$	−3.35435e20	−0.195608
$679$	0	0
$680$	−2.39930e21	−1.36857
$681$	3.18871e20	0.179892
$682$	−1.50367e20	−0.0839016
$683$	2.19144e21	1.20941	0.604705	−	0.796450i	$-0.293291\pi$
$683$	2.19144e21	1.20941	0.604705	+	0.796450i	$0.293291\pi$
$684$	−6.16100e20	−0.336303
$685$	−2.20103e21	−1.18836
$686$	0	0
$687$	1.48977e21	0.786945
$688$	−5.80892e20	−0.303517
$689$	−3.37324e21	−1.74342
$690$	−6.44101e20	−0.329295
$691$	5.25387e20	0.265701	0.132851	−	0.991136i	$-0.457587\pi$
$691$	5.25387e20	0.265701	0.132851	+	0.991136i	$0.457587\pi$
$692$	1.21634e21	0.608497
$693$	0	0
$694$	−1.35241e20	−0.0662082
$695$	4.32079e21	2.09256
$696$	1.10194e21	0.527943
$697$	1.37075e21	0.649701
$698$	−1.06953e21	−0.501505
$699$	8.50820e20	0.394693
$700$	0	0
$701$	−3.78227e21	−1.71739	−0.858693	−	0.512491i	$-0.828723\pi$
$701$	−3.78227e21	−1.71739	−0.858693	+	0.512491i	$0.828723\pi$
$702$	2.10861e20	0.0947259
$703$	2.87082e21	1.27597
$704$	−2.24889e20	−0.0988949
$705$	1.00568e21	0.437564
$706$	−5.92834e20	−0.255210
$707$	0	0
$708$	−1.39281e21	−0.587004
$709$	−4.24647e21	−1.77085	−0.885424	−	0.464784i	$-0.846132\pi$
$709$	−4.24647e21	−1.77085	−0.885424	+	0.464784i	$0.846132\pi$
$710$	−9.40717e20	−0.388170
$711$	7.64886e20	0.312302
$712$	1.57426e21	0.636030
$713$	−1.03766e21	−0.414842
$714$	0	0
$715$	2.30493e21	0.902322
$716$	1.40842e21	0.545613
$717$	−1.47891e20	−0.0566954
$718$	−1.34940e21	−0.511927
$719$	3.17360e20	0.119148	0.0595738	−	0.998224i	$-0.481026\pi$
$719$	3.17360e20	0.119148	0.0595738	+	0.998224i	$0.481026\pi$
$720$	6.26102e20	0.232622
$721$	0	0
$722$	−4.77053e20	−0.173595
$723$	−1.96946e21	−0.709266
$724$	8.72049e20	0.310815
$725$	−2.14981e21	−0.758340
$726$	−4.40221e20	−0.153690
$727$	−1.25112e21	−0.432307	−0.216154	−	0.976359i	$-0.569351\pi$
$727$	−1.25112e21	−0.432307	−0.216154	+	0.976359i	$0.569351\pi$
$728$	0	0
$729$	1.09419e20	0.0370370
$730$	−8.94615e19	−0.0299719
$731$	2.45025e21	0.812511
$732$	5.13123e20	0.168418
$733$	−4.26993e21	−1.38721	−0.693603	−	0.720357i	$-0.743978\pi$
$733$	−4.26993e21	−1.38721	−0.693603	+	0.720357i	$0.743978\pi$
$734$	6.21444e20	0.199840
$735$	0	0
$736$	3.41547e21	1.07614
$737$	−4.58335e20	−0.142948
$738$	1.89357e20	0.0584601
$739$	1.73302e21	0.529627	0.264813	−	0.964300i	$-0.414690\pi$
$739$	1.73302e21	0.529627	0.264813	+	0.964300i	$0.414690\pi$
$740$	−3.75596e21	−1.13627
$741$	−2.85933e21	−0.856302
$742$	0	0
$743$	−2.63739e21	−0.774032	−0.387016	−	0.922073i	$-0.626494\pi$
$743$	−2.63739e21	−0.774032	−0.387016	+	0.922073i	$0.626494\pi$
$744$	−5.33960e20	−0.155136
$745$	3.52026e21	1.01252
$746$	7.17102e20	0.204192
$747$	−2.75862e20	−0.0777656
$748$	2.55593e21	0.713324
$749$	0	0
$750$	−4.17730e20	−0.114271
$751$	2.27078e21	0.615001	0.307500	−	0.951548i	$-0.400507\pi$
$751$	2.27078e21	0.615001	0.307500	+	0.951548i	$0.400507\pi$
$752$	−1.22702e21	−0.329016
$753$	3.47577e21	0.922761
$754$	2.33741e21	0.614397
$755$	6.87941e21	1.79040
$756$	0	0
$757$	−1.72959e21	−0.441291	−0.220645	−	0.975354i	$-0.570816\pi$
$757$	−1.72959e21	−0.441291	−0.220645	+	0.975354i	$0.570816\pi$
$758$	−7.26834e20	−0.183618
$759$	1.50125e21	0.375525
$760$	4.49443e21	1.11320
$761$	2.06739e20	0.0507035	0.0253518	−	0.999679i	$-0.491929\pi$
$761$	2.06739e20	0.0507035	0.0253518	+	0.999679i	$0.491929\pi$
$762$	−9.85664e20	−0.239368
$763$	0	0
$764$	−2.60473e21	−0.620245
$765$	−2.64094e21	−0.622727
$766$	−1.95670e21	−0.456887
$767$	−6.46402e21	−1.49464
$768$	5.89256e20	0.134926
$769$	−3.01490e21	−0.683636	−0.341818	−	0.939766i	$-0.611043\pi$
$769$	−3.01490e21	−0.683636	−0.341818	+	0.939766i	$0.611043\pi$
$770$	0	0
$771$	5.02437e21	1.11731
$772$	−1.15230e21	−0.253768
$773$	−8.99357e21	−1.96149	−0.980745	−	0.195293i	$-0.937434\pi$
$773$	−8.99357e21	−1.96149	−0.980745	+	0.195293i	$0.937434\pi$
$774$	3.38480e20	0.0731098
$775$	1.04172e21	0.222838
$776$	2.34556e21	0.496916
$777$	0	0
$778$	6.51805e20	0.135447
$779$	−2.56773e21	−0.528467
$780$	3.74092e21	0.762551
$781$	2.19259e21	0.442664
$782$	−3.31480e21	−0.662838
$783$	1.21291e21	0.240224
$784$	0	0
$785$	1.19908e22	2.32985
$786$	−1.59132e21	−0.306259
$787$	5.92807e21	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$787$	5.92807e21	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$788$	4.09299e21	0.772849
$789$	1.10587e21	0.206837
$790$	−2.55027e21	−0.472479
$791$	0	0
$792$	7.72514e20	0.140432
$793$	2.38141e21	0.428830
$794$	−1.27077e21	−0.226680
$795$	5.83622e21	1.03128
$796$	5.44936e21	0.953889
$797$	3.83837e21	0.665593	0.332797	−	0.942999i	$-0.392008\pi$
$797$	3.83837e21	0.665593	0.332797	+	0.942999i	$0.392008\pi$
$798$	0	0
$799$	5.17565e21	0.880773
$800$	−3.42885e21	−0.578060
$801$	1.73281e21	0.289406
$802$	−2.02417e21	−0.334919
$803$	2.08513e20	0.0341796
$804$	−7.43883e20	−0.120805
$805$	0	0
$806$	−1.13263e21	−0.180540
$807$	−5.35226e21	−0.845251
$808$	1.83470e21	0.287064
$809$	−6.17948e21	−0.957940	−0.478970	−	0.877831i	$-0.658990\pi$
$809$	−6.17948e21	−0.957940	−0.478970	+	0.877831i	$0.658990\pi$
$810$	−3.64823e20	−0.0560330
$811$	−1.87862e21	−0.285879	−0.142939	−	0.989731i	$-0.545655\pi$
$811$	−1.87862e21	−0.285879	−0.142939	+	0.989731i	$0.545655\pi$
$812$	0	0
$813$	−3.36375e21	−0.502510
$814$	−1.64523e21	−0.243525
$815$	−1.22841e22	−1.80161
$816$	3.22217e21	0.468245
$817$	−4.58986e21	−0.660897
$818$	4.62927e21	0.660485
$819$	0	0
$820$	3.35941e21	0.470608
$821$	3.93227e21	0.545845	0.272923	−	0.962036i	$-0.412010\pi$
$821$	3.93227e21	0.545845	0.272923	+	0.962036i	$0.412010\pi$
$822$	−1.56478e21	−0.215236
$823$	−7.92927e21	−1.08077	−0.540386	−	0.841417i	$-0.681722\pi$
$823$	−7.92927e21	−1.08077	−0.540386	+	0.841417i	$0.681722\pi$
$824$	6.33068e20	0.0855059
$825$	−1.50712e21	−0.201718
$826$	0	0
$827$	−1.97038e21	−0.258976	−0.129488	−	0.991581i	$-0.541333\pi$
$827$	−1.97038e21	−0.258976	−0.129488	+	0.991581i	$0.541333\pi$
$828$	2.43654e21	0.317355
$829$	−2.43930e21	−0.314851	−0.157426	−	0.987531i	$-0.550319\pi$
$829$	−2.43930e21	−0.314851	−0.157426	+	0.987531i	$0.550319\pi$
$830$	9.19774e20	0.117651
$831$	1.72764e21	0.219001
$832$	−1.69396e21	−0.212803
$833$	0	0
$834$	3.07178e21	0.379005
$835$	1.29251e21	0.158047
$836$	−4.78783e21	−0.580218
$837$	−5.87736e20	−0.0705897
$838$	−1.68821e21	−0.200954
$839$	1.35318e22	1.59639	0.798196	−	0.602398i	$-0.205788\pi$
$839$	1.35318e22	1.59639	0.798196	+	0.602398i	$0.205788\pi$
$840$	0	0
$841$	4.81601e21	0.558107
$842$	5.24323e21	0.602225
$843$	1.34567e21	0.153190
$844$	1.23734e22	1.39611
$845$	6.02446e21	0.673741
$846$	7.14970e20	0.0792519
$847$	0	0
$848$	−7.12068e21	−0.775448
$849$	−2.76173e21	−0.298107
$850$	3.32778e21	0.356052
$851$	−1.13534e22	−1.20408
$852$	3.55859e21	0.374095
$853$	7.30038e21	0.760725	0.380363	−	0.924837i	$-0.375799\pi$
$853$	7.30038e21	0.760725	0.380363	+	0.924837i	$0.375799\pi$
$854$	0	0
$855$	4.94707e21	0.506527
$856$	−1.77592e21	−0.180248
$857$	−1.16475e22	−1.17187	−0.585934	−	0.810359i	$-0.699272\pi$
$857$	−1.16475e22	−1.17187	−0.585934	+	0.810359i	$0.699272\pi$
$858$	1.63864e21	0.163429
$859$	1.05473e22	1.04278	0.521388	−	0.853320i	$-0.325414\pi$
$859$	1.05473e22	1.04278	0.521388	+	0.853320i	$0.325414\pi$
$860$	6.00501e21	0.588539
$861$	0	0
$862$	3.39718e21	0.327201
$863$	1.25514e22	1.19843	0.599215	−	0.800588i	$-0.295480\pi$
$863$	1.25514e22	1.19843	0.599215	+	0.800588i	$0.295480\pi$
$864$	1.93455e21	0.183116
$865$	−9.76678e21	−0.916496
$866$	2.72951e21	0.253922
$867$	−7.33124e21	−0.676136
$868$	0	0
$869$	5.94407e21	0.538810
$870$	−4.04407e21	−0.363433
$871$	−3.45237e21	−0.307596
$872$	4.78569e21	0.422737
$873$	2.58178e21	0.226106
$874$	6.20937e21	0.539153
$875$	0	0
$876$	3.38420e20	0.0288852
$877$	−1.57294e22	−1.33112	−0.665558	−	0.746346i	$-0.731807\pi$
$877$	−1.57294e22	−1.33112	−0.665558	+	0.746346i	$0.731807\pi$
$878$	−1.63319e21	−0.137034
$879$	−7.65152e21	−0.636548
$880$	4.86555e21	0.401339
$881$	−1.80412e22	−1.47553	−0.737764	−	0.675059i	$-0.764118\pi$
$881$	−1.80412e22	−1.47553	−0.737764	+	0.675059i	$0.764118\pi$
$882$	0	0
$883$	2.41866e22	1.94478	0.972388	−	0.233371i	$-0.0749757\pi$
$883$	2.41866e22	1.94478	0.972388	+	0.233371i	$0.0749757\pi$
$884$	1.92523e22	1.53494
$885$	1.11837e22	0.884123
$886$	7.16320e21	0.561505
$887$	−7.05392e21	−0.548282	−0.274141	−	0.961690i	$-0.588393\pi$
$887$	−7.05392e21	−0.548282	−0.274141	+	0.961690i	$0.588393\pi$
$888$	−5.84228e21	−0.450282
$889$	0	0
$890$	−5.77750e21	−0.437840
$891$	8.50315e20	0.0638994
$892$	4.41279e21	0.328834
$893$	−9.69514e21	−0.716421
$894$	2.50266e21	0.183388
$895$	−1.13091e22	−0.821782
$896$	0	0
$897$	1.13080e22	0.808057
$898$	−9.37143e21	−0.664100
$899$	−6.51508e21	−0.457849
$900$	−2.44608e21	−0.170471
$901$	3.00356e22	2.07587
$902$	1.47153e21	0.100860
$903$	0	0
$904$	−9.25660e21	−0.624005
$905$	−7.00226e21	−0.468138
$906$	4.89078e21	0.324277
$907$	2.48498e22	1.63406	0.817030	−	0.576596i	$-0.195619\pi$
$907$	2.48498e22	1.63406	0.817030	+	0.576596i	$0.195619\pi$
$908$	4.02183e21	0.262289
$909$	2.01947e21	0.130620
$910$	0	0
$911$	−2.83848e21	−0.180592	−0.0902958	−	0.995915i	$-0.528781\pi$
$911$	−2.83848e21	−0.180592	−0.0902958	+	0.995915i	$0.528781\pi$
$912$	−6.03585e21	−0.380870
$913$	−2.14377e21	−0.134168
$914$	7.33749e21	0.455461
$915$	−4.12020e21	−0.253665
$916$	1.87901e22	1.14739
$917$	0	0
$918$	−1.87753e21	−0.112789
$919$	−1.87762e22	−1.11877	−0.559387	−	0.828907i	$-0.688963\pi$
$919$	−1.87762e22	−1.11877	−0.559387	+	0.828907i	$0.688963\pi$
$920$	−1.77745e22	−1.05048
$921$	2.81818e21	0.165204
$922$	−7.78620e21	−0.452734
$923$	1.65155e22	0.952528
$924$	0	0
$925$	1.13979e22	0.646787
$926$	−1.21761e22	−0.685374
$927$	6.96825e20	0.0389068
$928$	2.14445e22	1.18770
$929$	2.61729e21	0.143792	0.0718958	−	0.997412i	$-0.477095\pi$
$929$	2.61729e21	0.143792	0.0718958	+	0.997412i	$0.477095\pi$
$930$	1.95962e21	0.106795
$931$	0	0
$932$	1.07311e22	0.575476
$933$	1.84371e22	0.980798
$934$	−3.30061e21	−0.174178
$935$	−2.05232e22	−1.07438
$936$	5.81889e21	0.302184
$937$	−1.24366e22	−0.640701	−0.320350	−	0.947299i	$-0.603801\pi$
$937$	−1.24366e22	−0.640701	−0.320350	+	0.947299i	$0.603801\pi$
$938$	0	0
$939$	1.72414e22	0.874140
$940$	1.26844e22	0.637984
$941$	−1.71266e22	−0.854574	−0.427287	−	0.904116i	$-0.640531\pi$
$941$	−1.71266e22	−0.854574	−0.427287	+	0.904116i	$0.640531\pi$
$942$	8.52465e21	0.421983
$943$	1.01548e22	0.498692
$944$	−1.36451e22	−0.664795
$945$	0	0
$946$	2.63039e21	0.126135
$947$	1.26035e21	0.0599607	0.0299804	−	0.999550i	$-0.490456\pi$
$947$	1.26035e21	0.0599607	0.0299804	+	0.999550i	$0.490456\pi$
$948$	9.64729e21	0.455347
$949$	1.57061e21	0.0735480
$950$	−6.23368e21	−0.289613
$951$	1.90606e22	0.878581
$952$	0	0
$953$	−2.19887e22	−0.997706	−0.498853	−	0.866687i	$-0.666245\pi$
$953$	−2.19887e22	−0.997706	−0.498853	+	0.866687i	$0.666245\pi$
$954$	4.14915e21	0.186787
$955$	2.09151e22	0.934189
$956$	−1.86531e21	−0.0826639
$957$	9.42577e21	0.414455
$958$	9.46890e21	0.413103
$959$	0	0
$960$	2.93081e21	0.125879
$961$	−2.03083e22	−0.865461
$962$	−1.23925e22	−0.524019
$963$	−1.95478e21	−0.0820163
$964$	−2.48402e22	−1.03413
$965$	9.25260e21	0.382216
$966$	0	0
$967$	1.21130e21	0.0492666	0.0246333	−	0.999697i	$-0.492158\pi$
$967$	1.21130e21	0.0492666	0.0246333	+	0.999697i	$0.492158\pi$
$968$	−1.21482e22	−0.490284
$969$	2.54597e22	1.01959
$970$	−8.60813e21	−0.342075
$971$	−1.26825e22	−0.500103	−0.250051	−	0.968233i	$-0.580448\pi$
$971$	−1.26825e22	−0.500103	−0.250051	+	0.968233i	$0.580448\pi$
$972$	1.38007e21	0.0540013
$973$	0	0
$974$	−4.41843e21	−0.170245
$975$	−1.13523e22	−0.434058
$976$	5.02700e21	0.190737
$977$	−3.29367e21	−0.124014	−0.0620070	−	0.998076i	$-0.519750\pi$
$977$	−3.29367e21	−0.124014	−0.0620070	+	0.998076i	$0.519750\pi$
$978$	−8.73314e21	−0.326309
$979$	1.34660e22	0.499307
$980$	0	0
$981$	5.26766e21	0.192353
$982$	−1.74281e22	−0.631558
$983$	−3.74166e22	−1.34559	−0.672796	−	0.739828i	$-0.734907\pi$
$983$	−3.74166e22	−1.34559	−0.672796	+	0.739828i	$0.734907\pi$
$984$	5.22546e21	0.186493
$985$	−3.28653e22	−1.16404
$986$	−2.08125e22	−0.731554
$987$	0	0
$988$	−3.60638e22	−1.24852
$989$	1.81518e22	0.623661
$990$	−2.83510e21	−0.0966729
$991$	−4.40305e22	−1.49005	−0.745026	−	0.667035i	$-0.767563\pi$
$991$	−4.40305e22	−1.49005	−0.745026	+	0.667035i	$0.767563\pi$
$992$	−1.03913e22	−0.349005
$993$	−2.77738e22	−0.925796
$994$	0	0
$995$	−4.37565e22	−1.43671
$996$	−3.47937e21	−0.113385
$997$	−2.62067e22	−0.847614	−0.423807	−	0.905753i	$-0.639306\pi$
$997$	−2.62067e22	−0.847614	−0.423807	+	0.905753i	$0.639306\pi$
$998$	−1.05155e22	−0.337560
$999$	−6.43066e21	−0.204887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.16.a.b.1.1		1
7.6	odd	2		3.16.a.b.1.1	✓	1
21.20	even	2		9.16.a.c.1.1		1
28.27	even	2		48.16.a.a.1.1		1
35.13	even	4		75.16.b.b.49.2		2
35.27	even	4		75.16.b.b.49.1		2
35.34	odd	2		75.16.a.a.1.1		1
84.83	odd	2		144.16.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.16.a.b.1.1	✓	1	7.6	odd	2
9.16.a.c.1.1		1	21.20	even	2
48.16.a.a.1.1		1	28.27	even	2
75.16.a.a.1.1		1	35.34	odd	2
75.16.b.b.49.1		2	35.27	even	4
75.16.b.b.49.2		2	35.13	even	4
144.16.a.l.1.1		1	84.83	odd	2
147.16.a.b.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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