LMFDB - Embedded newform 27.10.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,10,Mod(1,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("27.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	27.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:	$13.9059675764$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.203942560.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 83x^{2} + 1440 $ x^4 - 83*x^2 + 1440
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$7.63546$ of defining polynomial
Character	$\chi$	$=$	27.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+44.4771 q^{2} +1466.22 q^{4} -1050.62 q^{5} +6591.86 q^{7} +42440.8 q^{8} -46728.6 q^{10} +14465.9 q^{11} +5003.49 q^{13} +293187. q^{14} +1.13694e6 q^{16} -444500. q^{17} -487431. q^{19} -1.54044e6 q^{20} +643402. q^{22} -67304.5 q^{23} -849319. q^{25} +222541. q^{26} +9.66510e6 q^{28} -3.41813e6 q^{29} +3.45923e6 q^{31} +2.88383e7 q^{32} -1.97701e7 q^{34} -6.92555e6 q^{35} -3.94510e6 q^{37} -2.16796e7 q^{38} -4.45892e7 q^{40} -1.76061e7 q^{41} +3.30231e7 q^{43} +2.12102e7 q^{44} -2.99351e6 q^{46} -3.71271e7 q^{47} +3.09907e6 q^{49} -3.77753e7 q^{50} +7.33620e6 q^{52} -7.86818e7 q^{53} -1.51982e7 q^{55} +2.79764e8 q^{56} -1.52029e8 q^{58} +5.13054e7 q^{59} -1.92723e7 q^{61} +1.53857e8 q^{62} +7.00529e8 q^{64} -5.25678e6 q^{65} +2.14838e8 q^{67} -6.51733e8 q^{68} -3.08029e8 q^{70} +3.25802e8 q^{71} +1.57548e8 q^{73} -1.75467e8 q^{74} -7.14680e8 q^{76} +9.53573e7 q^{77} +3.58634e7 q^{79} -1.19450e9 q^{80} -7.83069e8 q^{82} +8.07984e6 q^{83} +4.67001e8 q^{85} +1.46877e9 q^{86} +6.13945e8 q^{88} +9.43654e8 q^{89} +3.29823e7 q^{91} -9.86830e7 q^{92} -1.65131e9 q^{94} +5.12106e8 q^{95} +9.67052e8 q^{97} +1.37838e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2236 q^{4} + 11852 q^{7} - 41760 q^{10} + 179684 q^{13} + 2348680 q^{16} + 1011428 q^{19} + 3473568 q^{22} + 2554060 q^{25} + 19793924 q^{28} + 889136 q^{31} - 43111008 q^{34} - 3805156 q^{37} - 133649280 q^{40}+ \cdots + 1935734516 q^{97}+O(q^{100}) $ 4 * q + 2236 * q^4 + 11852 * q^7 - 41760 * q^10 + 179684 * q^13 + 2348680 * q^16 + 1011428 * q^19 + 3473568 * q^22 + 2554060 * q^25 + 19793924 * q^28 + 889136 * q^31 - 43111008 * q^34 - 3805156 * q^37 - 133649280 * q^40 + 11585024 * q^43 + 13428000 * q^46 - 73622328 * q^49 - 44411860 * q^52 + 314722080 * q^55 - 407685888 * q^58 + 219882572 * q^61 + 1519285360 * q^64 + 390561236 * q^67 - 650481120 * q^70 + 1197949508 * q^73 - 2121017572 * q^76 + 49814324 * q^79 - 1767126528 * q^82 + 370444320 * q^85 - 653198976 * q^88 - 47017172 * q^91 - 4146926688 * q^94 + 1935734516 * q^97

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 $

$2$	44.4771	1.96563	0.982815	−	0.184592i	$-0.0590963\pi$
$2$	44.4771	1.96563	0.982815	+	0.184592i	$0.0590963\pi$
$3$	0	0
$3$	0	0
$4$	1466.22	2.86370
$5$	−1050.62	−0.751764	−0.375882	−	0.926668i	$-0.622660\pi$
$5$	−1050.62	−0.751764	−0.375882	+	0.926668i	$0.622660\pi$
$6$	0	0
$7$	6591.86	1.03769	0.518844	−	0.854869i	$-0.326362\pi$
$7$	6591.86	1.03769	0.518844	+	0.854869i	$0.326362\pi$
$8$	42440.8	3.66335
$9$	0	0
$10$	−46728.6	−1.47769
$11$	14465.9	0.297906	0.148953	−	0.988844i	$-0.452410\pi$
$11$	14465.9	0.297906	0.148953	+	0.988844i	$0.452410\pi$
$12$	0	0
$13$	5003.49	0.0485879	0.0242940	−	0.999705i	$-0.492266\pi$
$13$	5003.49	0.0485879	0.0242940	+	0.999705i	$0.492266\pi$
$14$	293187.	2.03971
$15$	0	0
$16$	1.13694e6	4.33709
$17$	−444500.	−1.29078	−0.645389	−	0.763854i	$-0.723305\pi$
$17$	−444500.	−1.29078	−0.645389	+	0.763854i	$0.723305\pi$
$18$	0	0
$19$	−487431.	−0.858069	−0.429035	−	0.903288i	$-0.641146\pi$
$19$	−487431.	−0.858069	−0.429035	+	0.903288i	$0.641146\pi$
$20$	−1.54044e6	−2.15283
$21$	0	0
$22$	643402.	0.585572
$23$	−67304.5	−0.0501498	−0.0250749	−	0.999686i	$-0.507982\pi$
$23$	−67304.5	−0.0501498	−0.0250749	+	0.999686i	$0.507982\pi$
$24$	0	0
$25$	−849319.	−0.434852
$26$	222541.	0.0955059
$27$	0	0
$28$	9.66510e6	2.97163
$29$	−3.41813e6	−0.897425	−0.448712	−	0.893676i	$-0.648117\pi$
$29$	−3.41813e6	−0.897425	−0.448712	+	0.893676i	$0.648117\pi$
$30$	0	0
$31$	3.45923e6	0.672747	0.336374	−	0.941729i	$-0.390800\pi$
$31$	3.45923e6	0.672747	0.336374	+	0.941729i	$0.390800\pi$
$32$	2.88383e7	4.86177
$33$	0	0
$34$	−1.97701e7	−2.53719
$35$	−6.92555e6	−0.780097
$36$	0	0
$37$	−3.94510e6	−0.346059	−0.173030	−	0.984917i	$-0.555356\pi$
$37$	−3.94510e6	−0.346059	−0.173030	+	0.984917i	$0.555356\pi$
$38$	−2.16796e7	−1.68665
$39$	0	0
$40$	−4.45892e7	−2.75397
$41$	−1.76061e7	−0.973052	−0.486526	−	0.873666i	$-0.661736\pi$
$41$	−1.76061e7	−0.973052	−0.486526	+	0.873666i	$0.661736\pi$
$42$	0	0
$43$	3.30231e7	1.47302	0.736512	−	0.676425i	$-0.236472\pi$
$43$	3.30231e7	1.47302	0.736512	+	0.676425i	$0.236472\pi$
$44$	2.12102e7	0.853113
$45$	0	0
$46$	−2.99351e6	−0.0985759
$47$	−3.71271e7	−1.10982	−0.554908	−	0.831912i	$-0.687247\pi$
$47$	−3.71271e7	−1.10982	−0.554908	+	0.831912i	$0.687247\pi$
$48$	0	0
$49$	3.09907e6	0.0767978
$50$	−3.77753e7	−0.854757
$51$	0	0
$52$	7.33620e6	0.139141
$53$	−7.86818e7	−1.36972	−0.684861	−	0.728673i	$-0.740137\pi$
$53$	−7.86818e7	−1.36972	−0.684861	+	0.728673i	$0.740137\pi$
$54$	0	0
$55$	−1.51982e7	−0.223955
$56$	2.79764e8	3.80142
$57$	0	0
$58$	−1.52029e8	−1.76401
$59$	5.13054e7	0.551225	0.275613	−	0.961269i	$-0.411119\pi$
$59$	5.13054e7	0.551225	0.275613	+	0.961269i	$0.411119\pi$
$60$	0	0
$61$	−1.92723e7	−0.178217	−0.0891085	−	0.996022i	$-0.528402\pi$
$61$	−1.92723e7	−0.178217	−0.0891085	+	0.996022i	$0.528402\pi$
$62$	1.53857e8	1.32237
$63$	0	0
$64$	7.00529e8	5.21935
$65$	−5.25678e6	−0.0365266
$66$	0	0
$67$	2.14838e8	1.30249	0.651245	−	0.758867i	$-0.274247\pi$
$67$	2.14838e8	1.30249	0.651245	+	0.758867i	$0.274247\pi$
$68$	−6.51733e8	−3.69640
$69$	0	0
$70$	−3.08029e8	−1.53338
$71$	3.25802e8	1.52157	0.760784	−	0.649006i	$-0.224815\pi$
$71$	3.25802e8	1.52157	0.760784	+	0.649006i	$0.224815\pi$
$72$	0	0
$73$	1.57548e8	0.649322	0.324661	−	0.945830i	$-0.394750\pi$
$73$	1.57548e8	0.649322	0.324661	+	0.945830i	$0.394750\pi$
$74$	−1.75467e8	−0.680225
$75$	0	0
$76$	−7.14680e8	−2.45726
$77$	9.53573e7	0.309133
$78$	0	0
$79$	3.58634e7	0.103593	0.0517964	−	0.998658i	$-0.483505\pi$
$79$	3.58634e7	0.103593	0.0517964	+	0.998658i	$0.483505\pi$
$80$	−1.19450e9	−3.26047
$81$	0	0
$82$	−7.83069e8	−1.91266
$83$	8.07984e6	0.0186875	0.00934375	−	0.999956i	$-0.497026\pi$
$83$	8.07984e6	0.0186875	0.00934375	+	0.999956i	$0.497026\pi$
$84$	0	0
$85$	4.67001e8	0.970359
$86$	1.46877e9	2.89542
$87$	0	0
$88$	6.13945e8	1.09133
$89$	9.43654e8	1.59425	0.797127	−	0.603812i	$-0.206352\pi$
$89$	9.43654e8	1.59425	0.797127	+	0.603812i	$0.206352\pi$
$90$	0	0
$91$	3.29823e7	0.0504191
$92$	−9.86830e7	−0.143614
$93$	0	0
$94$	−1.65131e9	−2.18149
$95$	5.12106e8	0.645065
$96$	0	0
$97$	9.67052e8	1.10912	0.554558	−	0.832145i	$-0.312887\pi$
$97$	9.67052e8	1.10912	0.554558	+	0.832145i	$0.312887\pi$
$98$	1.37838e8	0.150956
$99$	0	0
$100$	−1.24529e9	−1.24529
$101$	−1.68398e9	−1.61024	−0.805121	−	0.593110i	$-0.797900\pi$
$101$	−1.68398e9	−1.61024	−0.805121	+	0.593110i	$0.797900\pi$
$102$	0	0
$103$	1.00611e8	0.0880805	0.0440403	−	0.999030i	$-0.485977\pi$
$103$	1.00611e8	0.0880805	0.0440403	+	0.999030i	$0.485977\pi$
$104$	2.12352e8	0.177995
$105$	0	0
$106$	−3.49954e9	−2.69237
$107$	5.50494e8	0.406000	0.203000	−	0.979179i	$-0.434931\pi$
$107$	5.50494e8	0.406000	0.203000	+	0.979179i	$0.434931\pi$
$108$	0	0
$109$	1.31023e9	0.889055	0.444527	−	0.895765i	$-0.353372\pi$
$109$	1.31023e9	0.889055	0.444527	+	0.895765i	$0.353372\pi$
$110$	−6.75972e8	−0.440212
$111$	0	0
$112$	7.49457e9	4.50055
$113$	−7.75402e7	−0.0447377	−0.0223689	−	0.999750i	$-0.507121\pi$
$113$	−7.75402e7	−0.0447377	−0.0223689	+	0.999750i	$0.507121\pi$
$114$	0	0
$115$	7.07116e7	0.0377008
$116$	−5.01172e9	−2.56996
$117$	0	0
$118$	2.28192e9	1.08351
$119$	−2.93008e9	−1.33943
$120$	0	0
$121$	−2.14869e9	−0.911252
$122$	−8.57176e8	−0.350309
$123$	0	0
$124$	5.07198e9	1.92655
$125$	2.94431e9	1.07867
$126$	0	0
$127$	−1.89287e9	−0.645661	−0.322830	−	0.946457i	$-0.604634\pi$
$127$	−1.89287e9	−0.645661	−0.322830	+	0.946457i	$0.604634\pi$
$128$	1.63923e10	5.39755
$129$	0	0
$130$	−2.33806e8	−0.0717978
$131$	−4.62187e9	−1.37119	−0.685593	−	0.727985i	$-0.740457\pi$
$131$	−4.62187e9	−1.37119	−0.685593	+	0.727985i	$0.740457\pi$
$132$	0	0
$133$	−3.21308e9	−0.890409
$134$	9.55538e9	2.56022
$135$	0	0
$136$	−1.88649e10	−4.72857
$137$	−6.82694e9	−1.65571	−0.827854	−	0.560944i	$-0.810438\pi$
$137$	−6.82694e9	−1.65571	−0.827854	+	0.560944i	$0.810438\pi$
$138$	0	0
$139$	4.87668e9	1.10805	0.554023	−	0.832501i	$-0.313092\pi$
$139$	4.87668e9	1.10805	0.554023	+	0.832501i	$0.313092\pi$
$140$	−1.01544e10	−2.23397
$141$	0	0
$142$	1.44907e10	2.99084
$143$	7.23801e7	0.0144746
$144$	0	0
$145$	3.59117e9	0.674651
$146$	7.00728e9	1.27633
$147$	0	0
$148$	−5.78437e9	−0.991011
$149$	7.64937e9	1.27142	0.635708	−	0.771930i	$-0.280708\pi$
$149$	7.64937e9	1.27142	0.635708	+	0.771930i	$0.280708\pi$
$150$	0	0
$151$	−4.19589e9	−0.656791	−0.328396	−	0.944540i	$-0.606508\pi$
$151$	−4.19589e9	−0.656791	−0.328396	+	0.944540i	$0.606508\pi$
$152$	−2.06870e10	−3.14341
$153$	0	0
$154$	4.24122e9	0.607642
$155$	−3.63434e9	−0.505747
$156$	0	0
$157$	−7.60479e8	−0.0998938	−0.0499469	−	0.998752i	$-0.515905\pi$
$157$	−7.60479e8	−0.0998938	−0.0499469	+	0.998752i	$0.515905\pi$
$158$	1.59510e9	0.203625
$159$	0	0
$160$	−3.02981e10	−3.65490
$161$	−4.43662e8	−0.0520398
$162$	0	0
$163$	−7.55113e9	−0.837853	−0.418926	−	0.908020i	$-0.637593\pi$
$163$	−7.55113e9	−0.837853	−0.418926	+	0.908020i	$0.637593\pi$
$164$	−2.58143e10	−2.78653
$165$	0	0
$166$	3.59368e8	0.0367327
$167$	8.05764e9	0.801648	0.400824	−	0.916155i	$-0.368724\pi$
$167$	8.05764e9	0.801648	0.400824	+	0.916155i	$0.368724\pi$
$168$	0	0
$169$	−1.05795e10	−0.997639
$170$	2.07709e10	1.90737
$171$	0	0
$172$	4.84190e10	4.21830
$173$	4.12318e9	0.349965	0.174982	−	0.984572i	$-0.444013\pi$
$173$	4.12318e9	0.349965	0.174982	+	0.984572i	$0.444013\pi$
$174$	0	0
$175$	−5.59860e9	−0.451240
$176$	1.64469e10	1.29204
$177$	0	0
$178$	4.19710e10	3.13371
$179$	2.46170e10	1.79224	0.896120	−	0.443811i	$-0.146374\pi$
$179$	2.46170e10	1.79224	0.896120	+	0.443811i	$0.146374\pi$
$180$	0	0
$181$	−4.17661e9	−0.289248	−0.144624	−	0.989487i	$-0.546197\pi$
$181$	−4.17661e9	−0.289248	−0.144624	+	0.989487i	$0.546197\pi$
$182$	1.46696e9	0.0991054
$183$	0	0
$184$	−2.85646e9	−0.183716
$185$	4.14481e9	0.260155
$186$	0	0
$187$	−6.43009e9	−0.384530
$188$	−5.44364e10	−3.17818
$189$	0	0
$190$	2.27770e10	1.26796
$191$	1.46430e10	0.796122	0.398061	−	0.917359i	$-0.369683\pi$
$191$	1.46430e10	0.796122	0.398061	+	0.917359i	$0.369683\pi$
$192$	0	0
$193$	1.58666e10	0.823144	0.411572	−	0.911377i	$-0.364980\pi$
$193$	1.58666e10	0.823144	0.411572	+	0.911377i	$0.364980\pi$
$194$	4.30117e10	2.18011
$195$	0	0
$196$	4.54390e9	0.219926
$197$	−4.01582e9	−0.189966	−0.0949832	−	0.995479i	$-0.530280\pi$
$197$	−4.01582e9	−0.189966	−0.0949832	+	0.995479i	$0.530280\pi$
$198$	0	0
$199$	−3.90309e10	−1.76429	−0.882144	−	0.470980i	$-0.843901\pi$
$199$	−3.90309e10	−1.76429	−0.882144	+	0.470980i	$0.843901\pi$
$200$	−3.60458e10	−1.59301
$201$	0	0
$202$	−7.48987e10	−3.16514
$203$	−2.25319e10	−0.931248
$204$	0	0
$205$	1.84973e10	0.731505
$206$	4.47491e9	0.173134
$207$	0	0
$208$	5.68869e9	0.210730
$209$	−7.05114e9	−0.255624
$210$	0	0
$211$	2.70099e10	0.938108	0.469054	−	0.883170i	$-0.344595\pi$
$211$	2.70099e10	0.938108	0.469054	+	0.883170i	$0.344595\pi$
$212$	−1.15364e11	−3.92248
$213$	0	0
$214$	2.44844e10	0.798046
$215$	−3.46948e10	−1.10737
$216$	0	0
$217$	2.28028e10	0.698102
$218$	5.82753e10	1.74755
$219$	0	0
$220$	−2.22838e10	−0.641340
$221$	−2.22405e9	−0.0627162
$222$	0	0
$223$	−5.73036e10	−1.55171	−0.775854	−	0.630912i	$-0.782681\pi$
$223$	−5.73036e10	−1.55171	−0.775854	+	0.630912i	$0.782681\pi$
$224$	1.90098e11	5.04500
$225$	0	0
$226$	−3.44877e9	−0.0879379
$227$	3.50057e10	0.875029	0.437514	−	0.899211i	$-0.355859\pi$
$227$	3.50057e10	0.875029	0.437514	+	0.899211i	$0.355859\pi$
$228$	0	0
$229$	6.47530e10	1.55597	0.777984	−	0.628284i	$-0.216243\pi$
$229$	6.47530e10	1.55597	0.777984	+	0.628284i	$0.216243\pi$
$230$	3.14505e9	0.0741058
$231$	0	0
$232$	−1.45068e11	−3.28758
$233$	−5.91061e10	−1.31380	−0.656902	−	0.753976i	$-0.728134\pi$
$233$	−5.91061e10	−1.31380	−0.656902	+	0.753976i	$0.728134\pi$
$234$	0	0
$235$	3.90066e10	0.834319
$236$	7.52248e10	1.57855
$237$	0	0
$238$	−1.30322e11	−2.63281
$239$	−4.04702e10	−0.802315	−0.401158	−	0.916009i	$-0.631392\pi$
$239$	−4.04702e10	−0.802315	−0.401158	+	0.916009i	$0.631392\pi$
$240$	0	0
$241$	3.37089e9	0.0643677	0.0321839	−	0.999482i	$-0.489754\pi$
$241$	3.37089e9	0.0643677	0.0321839	+	0.999482i	$0.489754\pi$
$242$	−9.55674e10	−1.79119
$243$	0	0
$244$	−2.82573e10	−0.510361
$245$	−3.25595e9	−0.0577338
$246$	0	0
$247$	−2.43886e9	−0.0416918
$248$	1.46813e11	2.46451
$249$	0	0
$250$	1.30954e11	2.12026
$251$	−4.69295e10	−0.746301	−0.373151	−	0.927771i	$-0.621723\pi$
$251$	−4.69295e10	−0.746301	−0.373151	+	0.927771i	$0.621723\pi$
$252$	0	0
$253$	−9.73621e8	−0.0149399
$254$	−8.41895e10	−1.26913
$255$	0	0
$256$	3.70414e11	5.39023
$257$	−1.74946e10	−0.250152	−0.125076	−	0.992147i	$-0.539917\pi$
$257$	−1.74946e10	−0.250152	−0.125076	+	0.992147i	$0.539917\pi$
$258$	0	0
$259$	−2.60056e10	−0.359102
$260$	−7.70757e9	−0.104601
$261$	0	0
$262$	−2.05567e11	−2.69525
$263$	1.00629e11	1.29695	0.648476	−	0.761235i	$-0.275407\pi$
$263$	1.00629e11	1.29695	0.648476	+	0.761235i	$0.275407\pi$
$264$	0	0
$265$	8.26648e10	1.02971
$266$	−1.42909e11	−1.75021
$267$	0	0
$268$	3.14999e11	3.72995
$269$	2.18445e10	0.254365	0.127182	−	0.991879i	$-0.459407\pi$
$269$	2.18445e10	0.254365	0.127182	+	0.991879i	$0.459407\pi$
$270$	0	0
$271$	5.19462e10	0.585049	0.292524	−	0.956258i	$-0.405505\pi$
$271$	5.19462e10	0.585049	0.292524	+	0.956258i	$0.405505\pi$
$272$	−5.05371e11	−5.59822
$273$	0	0
$274$	−3.03643e11	−3.25451
$275$	−1.22862e10	−0.129545
$276$	0	0
$277$	−8.37704e9	−0.0854932	−0.0427466	−	0.999086i	$-0.513611\pi$
$277$	−8.37704e9	−0.0854932	−0.0427466	+	0.999086i	$0.513611\pi$
$278$	2.16901e11	2.17801
$279$	0	0
$280$	−2.93926e11	−2.85777
$281$	8.16325e10	0.781061	0.390530	−	0.920590i	$-0.372292\pi$
$281$	8.16325e10	0.781061	0.390530	+	0.920590i	$0.372292\pi$
$282$	0	0
$283$	−6.11475e10	−0.566683	−0.283341	−	0.959019i	$-0.591443\pi$
$283$	−6.11475e10	−0.566683	−0.283341	+	0.959019i	$0.591443\pi$
$284$	4.77696e11	4.35732
$285$	0	0
$286$	3.21926e9	0.0284517
$287$	−1.16057e11	−1.00972
$288$	0	0
$289$	7.89921e10	0.666106
$290$	1.59725e11	1.32612
$291$	0	0
$292$	2.30999e11	1.85946
$293$	1.73459e11	1.37497	0.687486	−	0.726198i	$-0.258714\pi$
$293$	1.73459e11	1.37497	0.687486	+	0.726198i	$0.258714\pi$
$294$	0	0
$295$	−5.39026e10	−0.414391
$296$	−1.67433e11	−1.26774
$297$	0	0
$298$	3.40222e11	2.49913
$299$	−3.36758e8	−0.00243667
$300$	0	0
$301$	2.17684e11	1.52854
$302$	−1.86621e11	−1.29101
$303$	0	0
$304$	−5.54182e11	−3.72153
$305$	2.02479e10	0.133977
$306$	0	0
$307$	−1.89397e11	−1.21689	−0.608443	−	0.793598i	$-0.708205\pi$
$307$	−1.89397e11	−1.21689	−0.608443	+	0.793598i	$0.708205\pi$
$308$	1.39814e11	0.885266
$309$	0	0
$310$	−1.61645e11	−0.994112
$311$	1.47676e11	0.895137	0.447569	−	0.894250i	$-0.352290\pi$
$311$	1.47676e11	0.895137	0.447569	+	0.894250i	$0.352290\pi$
$312$	0	0
$313$	4.64248e9	0.0273402	0.0136701	−	0.999907i	$-0.495649\pi$
$313$	4.64248e9	0.0273402	0.0136701	+	0.999907i	$0.495649\pi$
$314$	−3.38239e10	−0.196354
$315$	0	0
$316$	5.25835e10	0.296659
$317$	1.76364e11	0.980943	0.490471	−	0.871457i	$-0.336825\pi$
$317$	1.76364e11	0.980943	0.490471	+	0.871457i	$0.336825\pi$
$318$	0	0
$319$	−4.94464e10	−0.267348
$320$	−7.35991e11	−3.92372
$321$	0	0
$322$	−1.97328e10	−0.102291
$323$	2.16663e11	1.10758
$324$	0	0
$325$	−4.24956e9	−0.0211285
$326$	−3.35853e11	−1.64691
$327$	0	0
$328$	−7.47217e11	−3.56463
$329$	−2.44737e11	−1.15164
$330$	0	0
$331$	−3.09857e11	−1.41885	−0.709423	−	0.704783i	$-0.751044\pi$
$331$	−3.09857e11	−1.41885	−0.709423	+	0.704783i	$0.751044\pi$
$332$	1.18468e10	0.0535155
$333$	0	0
$334$	3.58381e11	1.57574
$335$	−2.25714e11	−0.979165
$336$	0	0
$337$	3.13658e11	1.32471	0.662356	−	0.749189i	$-0.269557\pi$
$337$	3.13658e11	1.32471	0.662356	+	0.749189i	$0.269557\pi$
$338$	−4.70544e11	−1.96099
$339$	0	0
$340$	6.84724e11	2.77882
$341$	5.00409e10	0.200415
$342$	0	0
$343$	−2.45577e11	−0.957996
$344$	1.40153e12	5.39620
$345$	0	0
$346$	1.83387e11	0.687902
$347$	−8.10841e10	−0.300229	−0.150115	−	0.988669i	$-0.547964\pi$
$347$	−8.10841e10	−0.300229	−0.150115	+	0.988669i	$0.547964\pi$
$348$	0	0
$349$	2.68777e11	0.969791	0.484896	−	0.874572i	$-0.338858\pi$
$349$	2.68777e11	0.969791	0.484896	+	0.874572i	$0.338858\pi$
$350$	−2.49010e11	−0.886972
$351$	0	0
$352$	4.17172e11	1.44835
$353$	−3.92960e11	−1.34698	−0.673491	−	0.739195i	$-0.735206\pi$
$353$	−3.92960e11	−1.34698	−0.673491	+	0.739195i	$0.735206\pi$
$354$	0	0
$355$	−3.42295e11	−1.14386
$356$	1.38360e12	4.56547
$357$	0	0
$358$	1.09489e12	3.52288
$359$	2.48781e11	0.790484	0.395242	−	0.918577i	$-0.370661\pi$
$359$	2.48781e11	0.790484	0.395242	+	0.918577i	$0.370661\pi$
$360$	0	0
$361$	−8.50984e10	−0.263718
$362$	−1.85764e11	−0.568555
$363$	0	0
$364$	4.83592e10	0.144385
$365$	−1.65523e11	−0.488136
$366$	0	0
$367$	−3.25248e11	−0.935875	−0.467937	−	0.883762i	$-0.655003\pi$
$367$	−3.25248e11	−0.935875	−0.467937	+	0.883762i	$0.655003\pi$
$368$	−7.65214e10	−0.217504
$369$	0	0
$370$	1.84349e11	0.511368
$371$	−5.18660e11	−1.42135
$372$	0	0
$373$	−3.55232e11	−0.950217	−0.475108	−	0.879927i	$-0.657591\pi$
$373$	−3.55232e11	−0.950217	−0.475108	+	0.879927i	$0.657591\pi$
$374$	−2.85992e11	−0.755844
$375$	0	0
$376$	−1.57571e12	−4.06565
$377$	−1.71026e10	−0.0436040
$378$	0	0
$379$	−4.17590e11	−1.03962	−0.519809	−	0.854283i	$-0.673997\pi$
$379$	−4.17590e11	−1.03962	−0.519809	+	0.854283i	$0.673997\pi$
$380$	7.50858e11	1.84727
$381$	0	0
$382$	6.51279e11	1.56488
$383$	−8.39723e9	−0.0199408	−0.00997038	−	0.999950i	$-0.503174\pi$
$383$	−8.39723e9	−0.0199408	−0.00997038	+	0.999950i	$0.503174\pi$
$384$	0	0
$385$	−1.00184e11	−0.232395
$386$	7.05701e11	1.61800
$387$	0	0
$388$	1.41791e12	3.17618
$389$	−4.47803e11	−0.991549	−0.495774	−	0.868451i	$-0.665116\pi$
$389$	−4.47803e11	−0.991549	−0.495774	+	0.868451i	$0.665116\pi$
$390$	0	0
$391$	2.99168e10	0.0647322
$392$	1.31527e11	0.281337
$393$	0	0
$394$	−1.78612e11	−0.373404
$395$	−3.76788e10	−0.0778773
$396$	0	0
$397$	8.93449e11	1.80515	0.902573	−	0.430537i	$-0.141676\pi$
$397$	8.93449e11	1.80515	0.902573	+	0.430537i	$0.141676\pi$
$398$	−1.73598e12	−3.46794
$399$	0	0
$400$	−9.65628e11	−1.88599
$401$	2.21707e10	0.0428183	0.0214092	−	0.999771i	$-0.493185\pi$
$401$	2.21707e10	0.0428183	0.0214092	+	0.999771i	$0.493185\pi$
$402$	0	0
$403$	1.73082e10	0.0326874
$404$	−2.46908e12	−4.61126
$405$	0	0
$406$	−1.00215e12	−1.83049
$407$	−5.70695e10	−0.103093
$408$	0	0
$409$	4.79724e11	0.847688	0.423844	−	0.905735i	$-0.360680\pi$
$409$	4.79724e11	0.847688	0.423844	+	0.905735i	$0.360680\pi$
$410$	8.22709e11	1.43787
$411$	0	0
$412$	1.47518e11	0.252236
$413$	3.38198e11	0.572000
$414$	0	0
$415$	−8.48885e9	−0.0140486
$416$	1.44292e11	0.236223
$417$	0	0
$418$	−3.13614e11	−0.502462
$419$	8.35323e11	1.32401	0.662005	−	0.749500i	$-0.269706\pi$
$419$	8.35323e11	1.32401	0.662005	+	0.749500i	$0.269706\pi$
$420$	0	0
$421$	2.06281e11	0.320029	0.160014	−	0.987115i	$-0.448846\pi$
$421$	2.06281e11	0.320029	0.160014	+	0.987115i	$0.448846\pi$
$422$	1.20133e12	1.84397
$423$	0	0
$424$	−3.33932e12	−5.01778
$425$	3.77522e11	0.561297
$426$	0	0
$427$	−1.27040e11	−0.184934
$428$	8.07144e11	1.16266
$429$	0	0
$430$	−1.54312e12	−2.17667
$431$	3.46311e11	0.483414	0.241707	−	0.970349i	$-0.422293\pi$
$431$	3.46311e11	0.483414	0.241707	+	0.970349i	$0.422293\pi$
$432$	0	0
$433$	−7.41282e9	−0.0101342	−0.00506708	−	0.999987i	$-0.501613\pi$
$433$	−7.41282e9	−0.0101342	−0.00506708	+	0.999987i	$0.501613\pi$
$434$	1.01420e12	1.37221
$435$	0	0
$436$	1.92108e12	2.54599
$437$	3.28063e10	0.0430320
$438$	0	0
$439$	9.59814e11	1.23338	0.616690	−	0.787206i	$-0.288473\pi$
$439$	9.59814e11	1.23338	0.616690	+	0.787206i	$0.288473\pi$
$440$	−6.45024e11	−0.820425
$441$	0	0
$442$	−9.89194e10	−0.123277
$443$	2.88512e11	0.355915	0.177958	−	0.984038i	$-0.443051\pi$
$443$	2.88512e11	0.355915	0.177958	+	0.984038i	$0.443051\pi$
$444$	0	0
$445$	−9.91423e11	−1.19850
$446$	−2.54870e12	−3.05008
$447$	0	0
$448$	4.61780e12	5.41606
$449$	1.00029e12	1.16150	0.580749	−	0.814083i	$-0.302760\pi$
$449$	1.00029e12	1.16150	0.580749	+	0.814083i	$0.302760\pi$
$450$	0	0
$451$	−2.54688e11	−0.289878
$452$	−1.13691e11	−0.128116
$453$	0	0
$454$	1.55695e12	1.71998
$455$	−3.46520e10	−0.0379033
$456$	0	0
$457$	2.83855e11	0.304420	0.152210	−	0.988348i	$-0.451361\pi$
$457$	2.83855e11	0.304420	0.152210	+	0.988348i	$0.451361\pi$
$458$	2.88003e12	3.05846
$459$	0	0
$460$	1.03678e11	0.107964
$461$	−6.45819e8	−0.000665973 0	−0.000332987	−	1.00000i	$-0.500106\pi$
$461$	−6.45819e8	−0.000665973 0	−0.000332987		1.00000i	$0.500106\pi$
$462$	0	0
$463$	−4.83071e11	−0.488536	−0.244268	−	0.969708i	$-0.578548\pi$
$463$	−4.83071e11	−0.488536	−0.244268	+	0.969708i	$0.578548\pi$
$464$	−3.88622e12	−3.89222
$465$	0	0
$466$	−2.62887e12	−2.58245
$467$	−7.10128e11	−0.690893	−0.345447	−	0.938438i	$-0.612273\pi$
$467$	−7.10128e11	−0.690893	−0.345447	+	0.938438i	$0.612273\pi$
$468$	0	0
$469$	1.41618e12	1.35158
$470$	1.73490e12	1.63996
$471$	0	0
$472$	2.17744e12	2.01933
$473$	4.77709e11	0.438822
$474$	0	0
$475$	4.13985e11	0.373133
$476$	−4.29613e12	−3.83572
$477$	0	0
$478$	−1.80000e12	−1.57706
$479$	1.54619e10	0.0134200	0.00671002	−	0.999977i	$-0.497864\pi$
$479$	1.54619e10	0.0134200	0.00671002	+	0.999977i	$0.497864\pi$
$480$	0	0
$481$	−1.97393e10	−0.0168143
$482$	1.49928e11	0.126523
$483$	0	0
$484$	−3.15044e12	−2.60956
$485$	−1.01601e12	−0.833793
$486$	0	0
$487$	−5.65405e11	−0.455490	−0.227745	−	0.973721i	$-0.573135\pi$
$487$	−5.65405e11	−0.455490	−0.227745	+	0.973721i	$0.573135\pi$
$488$	−8.17932e11	−0.652872
$489$	0	0
$490$	−1.44815e11	−0.113483
$491$	−3.72733e11	−0.289421	−0.144711	−	0.989474i	$-0.546225\pi$
$491$	−3.72733e11	−0.289421	−0.144711	+	0.989474i	$0.546225\pi$
$492$	0	0
$493$	1.51936e12	1.15838
$494$	−1.08473e11	−0.0819506
$495$	0	0
$496$	3.93295e12	2.91777
$497$	2.14764e12	1.57891
$498$	0	0
$499$	−3.57615e11	−0.258204	−0.129102	−	0.991631i	$-0.541209\pi$
$499$	−3.57615e11	−0.258204	−0.129102	+	0.991631i	$0.541209\pi$
$500$	4.31699e12	3.08899
$501$	0	0
$502$	−2.08729e12	−1.46695
$503$	−1.08145e12	−0.753267	−0.376633	−	0.926362i	$-0.622918\pi$
$503$	−1.08145e12	−0.753267	−0.376633	+	0.926362i	$0.622918\pi$
$504$	0	0
$505$	1.76923e12	1.21052
$506$	−4.33039e10	−0.0293663
$507$	0	0
$508$	−2.77536e12	−1.84898
$509$	−6.97300e11	−0.460458	−0.230229	−	0.973137i	$-0.573947\pi$
$509$	−6.97300e11	−0.460458	−0.230229	+	0.973137i	$0.573947\pi$
$510$	0	0
$511$	1.03853e12	0.673794
$512$	8.08206e12	5.19765
$513$	0	0
$514$	−7.78108e11	−0.491706
$515$	−1.05705e11	−0.0662157
$516$	0	0
$517$	−5.37078e11	−0.330621
$518$	−1.15665e12	−0.705861
$519$	0	0
$520$	−2.23102e11	−0.133810
$521$	−1.96135e12	−1.16623	−0.583116	−	0.812389i	$-0.698167\pi$
$521$	−1.96135e12	−1.16623	−0.583116	+	0.812389i	$0.698167\pi$
$522$	0	0
$523$	−6.03647e10	−0.0352797	−0.0176399	−	0.999844i	$-0.505615\pi$
$523$	−6.03647e10	−0.0352797	−0.0176399	+	0.999844i	$0.505615\pi$
$524$	−6.77665e12	−3.92667
$525$	0	0
$526$	4.47570e12	2.54933
$527$	−1.53763e12	−0.868367
$528$	0	0
$529$	−1.79662e12	−0.997485
$530$	3.67669e12	2.02402
$531$	0	0
$532$	−4.71107e12	−2.54987
$533$	−8.80920e10	−0.0472785
$534$	0	0
$535$	−5.78361e11	−0.305216
$536$	9.11790e12	4.77148
$537$	0	0
$538$	9.71580e11	0.499987
$539$	4.48308e10	0.0228785
$540$	0	0
$541$	−1.58437e12	−0.795185	−0.397592	−	0.917562i	$-0.630154\pi$
$541$	−1.58437e12	−0.795185	−0.397592	+	0.917562i	$0.630154\pi$
$542$	2.31042e12	1.14999
$543$	0	0
$544$	−1.28186e13	−6.27546
$545$	−1.37656e12	−0.668359
$546$	0	0
$547$	1.47236e12	0.703189	0.351595	−	0.936152i	$-0.385640\pi$
$547$	1.47236e12	0.703189	0.351595	+	0.936152i	$0.385640\pi$
$548$	−1.00098e13	−4.74146
$549$	0	0
$550$	−5.46454e11	−0.254637
$551$	1.66611e12	0.770052
$552$	0	0
$553$	2.36407e11	0.107497
$554$	−3.72587e11	−0.168048
$555$	0	0
$556$	7.15027e12	3.17311
$557$	−1.23459e12	−0.543470	−0.271735	−	0.962372i	$-0.587597\pi$
$557$	−1.23459e12	−0.543470	−0.271735	+	0.962372i	$0.587597\pi$
$558$	0	0
$559$	1.65231e11	0.0715711
$560$	−7.87396e12	−3.38335
$561$	0	0
$562$	3.63078e12	1.53528
$563$	3.22798e11	0.135407	0.0677037	−	0.997705i	$-0.478433\pi$
$563$	3.22798e11	0.135407	0.0677037	+	0.997705i	$0.478433\pi$
$564$	0	0
$565$	8.14654e10	0.0336322
$566$	−2.71967e12	−1.11389
$567$	0	0
$568$	1.38273e13	5.57404
$569$	−1.51796e12	−0.607091	−0.303546	−	0.952817i	$-0.598170\pi$
$569$	−1.51796e12	−0.607091	−0.303546	+	0.952817i	$0.598170\pi$
$570$	0	0
$571$	6.04069e11	0.237807	0.118903	−	0.992906i	$-0.462062\pi$
$571$	6.04069e11	0.237807	0.118903	+	0.992906i	$0.462062\pi$
$572$	1.06125e11	0.0414510
$573$	0	0
$574$	−5.16188e12	−1.98475
$575$	5.71630e10	0.0218077
$576$	0	0
$577$	−4.16132e12	−1.56293	−0.781466	−	0.623948i	$-0.785528\pi$
$577$	−4.16132e12	−1.56293	−0.781466	+	0.623948i	$0.785528\pi$
$578$	3.51334e12	1.30932
$579$	0	0
$580$	5.26542e12	1.93200
$581$	5.32612e10	0.0193918
$582$	0	0
$583$	−1.13820e12	−0.408048
$584$	6.68646e12	2.37869
$585$	0	0
$586$	7.71498e12	2.70269
$587$	1.29792e12	0.451207	0.225604	−	0.974219i	$-0.427565\pi$
$587$	1.29792e12	0.451207	0.225604	+	0.974219i	$0.427565\pi$
$588$	0	0
$589$	−1.68614e12	−0.577264
$590$	−2.39743e12	−0.814540
$591$	0	0
$592$	−4.48536e12	−1.50089
$593$	6.89642e11	0.229022	0.114511	−	0.993422i	$-0.463470\pi$
$593$	6.89642e11	0.229022	0.114511	+	0.993422i	$0.463470\pi$
$594$	0	0
$595$	3.07841e12	1.00693
$596$	1.12156e13	3.64096
$597$	0	0
$598$	−1.49780e10	−0.00478960
$599$	−6.81664e11	−0.216346	−0.108173	−	0.994132i	$-0.534500\pi$
$599$	−6.81664e11	−0.216346	−0.108173	+	0.994132i	$0.534500\pi$
$600$	0	0
$601$	−4.73397e12	−1.48010	−0.740049	−	0.672553i	$-0.765198\pi$
$601$	−4.73397e12	−1.48010	−0.740049	+	0.672553i	$0.765198\pi$
$602$	9.68195e12	3.00454
$603$	0	0
$604$	−6.15208e12	−1.88086
$605$	2.25745e12	0.685046
$606$	0	0
$607$	−3.38422e12	−1.01183	−0.505917	−	0.862582i	$-0.668846\pi$
$607$	−3.38422e12	−1.01183	−0.505917	+	0.862582i	$0.668846\pi$
$608$	−1.40567e13	−4.17173
$609$	0	0
$610$	9.00568e11	0.263349
$611$	−1.85765e11	−0.0539237
$612$	0	0
$613$	−2.77900e12	−0.794906	−0.397453	−	0.917622i	$-0.630106\pi$
$613$	−2.77900e12	−0.794906	−0.397453	+	0.917622i	$0.630106\pi$
$614$	−8.42382e12	−2.39195
$615$	0	0
$616$	4.04704e12	1.13246
$617$	5.70613e12	1.58511	0.792554	−	0.609802i	$-0.208751\pi$
$617$	5.70613e12	1.58511	0.792554	+	0.609802i	$0.208751\pi$
$618$	0	0
$619$	5.70022e12	1.56057	0.780285	−	0.625424i	$-0.215074\pi$
$619$	5.70022e12	1.56057	0.780285	+	0.625424i	$0.215074\pi$
$620$	−5.32873e12	−1.44831
$621$	0	0
$622$	6.56823e12	1.75951
$623$	6.22044e12	1.65434
$624$	0	0
$625$	−1.43453e12	−0.376053
$626$	2.06484e11	0.0537406
$627$	0	0
$628$	−1.11503e12	−0.286066
$629$	1.75360e12	0.446685
$630$	0	0
$631$	−7.58053e11	−0.190356	−0.0951782	−	0.995460i	$-0.530342\pi$
$631$	−7.58053e11	−0.190356	−0.0951782	+	0.995460i	$0.530342\pi$
$632$	1.52207e12	0.379497
$633$	0	0
$634$	7.84417e12	1.92817
$635$	1.98869e12	0.485384
$636$	0	0
$637$	1.55062e10	0.00373144
$638$	−2.19924e12	−0.525507
$639$	0	0
$640$	−1.72222e13	−4.05768
$641$	−3.07750e12	−0.720007	−0.360003	−	0.932951i	$-0.617224\pi$
$641$	−3.07750e12	−0.720007	−0.360003	+	0.932951i	$0.617224\pi$
$642$	0	0
$643$	2.10425e12	0.485455	0.242727	−	0.970095i	$-0.421958\pi$
$643$	2.10425e12	0.485455	0.242727	+	0.970095i	$0.421958\pi$
$644$	−6.50505e11	−0.149027
$645$	0	0
$646$	9.63655e12	2.17709
$647$	−1.41884e12	−0.318319	−0.159160	−	0.987253i	$-0.550878\pi$
$647$	−1.41884e12	−0.318319	−0.159160	+	0.987253i	$0.550878\pi$
$648$	0	0
$649$	7.42180e11	0.164213
$650$	−1.89008e11	−0.0415309
$651$	0	0
$652$	−1.10716e13	−2.39936
$653$	3.40288e11	0.0732382	0.0366191	−	0.999329i	$-0.488341\pi$
$653$	3.40288e11	0.0732382	0.0366191	+	0.999329i	$0.488341\pi$
$654$	0	0
$655$	4.85583e12	1.03081
$656$	−2.00171e13	−4.22022
$657$	0	0
$658$	−1.08852e13	−2.26371
$659$	−4.62024e12	−0.954290	−0.477145	−	0.878825i	$-0.658328\pi$
$659$	−4.62024e12	−0.954290	−0.477145	+	0.878825i	$0.658328\pi$
$660$	0	0
$661$	−4.63497e12	−0.944365	−0.472183	−	0.881501i	$-0.656534\pi$
$661$	−4.63497e12	−0.944365	−0.472183	+	0.881501i	$0.656534\pi$
$662$	−1.37815e13	−2.78893
$663$	0	0
$664$	3.42915e11	0.0684589
$665$	3.37573e12	0.669377
$666$	0	0
$667$	2.30056e11	0.0450057
$668$	1.18142e13	2.29568
$669$	0	0
$670$	−1.00391e13	−1.92468
$671$	−2.78791e11	−0.0530918
$672$	0	0
$673$	−1.00517e13	−1.88874	−0.944370	−	0.328886i	$-0.893327\pi$
$673$	−1.00517e13	−1.88874	−0.944370	+	0.328886i	$0.893327\pi$
$674$	1.39506e13	2.60390
$675$	0	0
$676$	−1.55118e13	−2.85694
$677$	−8.83179e12	−1.61584	−0.807922	−	0.589289i	$-0.799408\pi$
$677$	−8.83179e12	−1.61584	−0.807922	+	0.589289i	$0.799408\pi$
$678$	0	0
$679$	6.37467e12	1.15092
$680$	1.98199e13	3.55477
$681$	0	0
$682$	2.22568e12	0.393942
$683$	1.10788e13	1.94805	0.974027	−	0.226430i	$-0.0727055\pi$
$683$	1.10788e13	1.94805	0.974027	+	0.226430i	$0.0727055\pi$
$684$	0	0
$685$	7.17253e12	1.24470
$686$	−1.09226e13	−1.88307
$687$	0	0
$688$	3.75454e13	6.38864
$689$	−3.93684e11	−0.0665520
$690$	0	0
$691$	−6.32931e11	−0.105610	−0.0528050	−	0.998605i	$-0.516816\pi$
$691$	−6.32931e11	−0.105610	−0.0528050	+	0.998605i	$0.516816\pi$
$692$	6.04547e12	1.00220
$693$	0	0
$694$	−3.60639e12	−0.590140
$695$	−5.12355e12	−0.832988
$696$	0	0
$697$	7.82591e12	1.25599
$698$	1.19545e13	1.90625
$699$	0	0
$700$	−8.20875e12	−1.29222
$701$	−3.84382e12	−0.601219	−0.300609	−	0.953747i	$-0.597190\pi$
$701$	−3.84382e12	−0.601219	−0.300609	+	0.953747i	$0.597190\pi$
$702$	0	0
$703$	1.92297e12	0.296943
$704$	1.01338e13	1.55487
$705$	0	0
$706$	−1.74777e13	−2.64767
$707$	−1.11006e13	−1.67093
$708$	0	0
$709$	−6.15935e12	−0.915433	−0.457716	−	0.889098i	$-0.651333\pi$
$709$	−6.15935e12	−0.915433	−0.457716	+	0.889098i	$0.651333\pi$
$710$	−1.52243e13	−2.24840
$711$	0	0
$712$	4.00494e13	5.84031
$713$	−2.32822e11	−0.0337381
$714$	0	0
$715$	−7.60441e10	−0.0108815
$716$	3.60938e13	5.13245
$717$	0	0
$718$	1.10651e13	1.55380
$719$	−6.51194e12	−0.908721	−0.454361	−	0.890818i	$-0.650132\pi$
$719$	−6.51194e12	−0.908721	−0.454361	+	0.890818i	$0.650132\pi$
$720$	0	0
$721$	6.63217e11	0.0914001
$722$	−3.78493e12	−0.518371
$723$	0	0
$724$	−6.12381e12	−0.828321
$725$	2.90309e12	0.390247
$726$	0	0
$727$	2.05785e12	0.273218	0.136609	−	0.990625i	$-0.456380\pi$
$727$	2.05785e12	0.273218	0.136609	+	0.990625i	$0.456380\pi$
$728$	1.39980e12	0.184703
$729$	0	0
$730$	−7.36200e12	−0.959496
$731$	−1.46788e13	−1.90135
$732$	0	0
$733$	7.96245e12	1.01878	0.509388	−	0.860537i	$-0.329872\pi$
$733$	7.96245e12	1.01878	0.509388	+	0.860537i	$0.329872\pi$
$734$	−1.44661e13	−1.83958
$735$	0	0
$736$	−1.94095e12	−0.243817
$737$	3.10783e12	0.388019
$738$	0	0
$739$	−4.20258e12	−0.518341	−0.259171	−	0.965832i	$-0.583449\pi$
$739$	−4.20258e12	−0.518341	−0.259171	+	0.965832i	$0.583449\pi$
$740$	6.07719e12	0.745006
$741$	0	0
$742$	−2.30685e13	−2.79384
$743$	5.46810e12	0.658244	0.329122	−	0.944287i	$-0.393247\pi$
$743$	5.46810e12	0.658244	0.329122	+	0.944287i	$0.393247\pi$
$744$	0	0
$745$	−8.03659e12	−0.955804
$746$	−1.57997e13	−1.86777
$747$	0	0
$748$	−9.42791e12	−1.10118
$749$	3.62878e12	0.421302
$750$	0	0
$751$	7.47523e12	0.857521	0.428760	−	0.903418i	$-0.358950\pi$
$751$	7.47523e12	0.857521	0.428760	+	0.903418i	$0.358950\pi$
$752$	−4.22114e13	−4.81338
$753$	0	0
$754$	−7.60675e11	−0.0857093
$755$	4.40829e12	0.493752
$756$	0	0
$757$	1.40395e12	0.155389	0.0776947	−	0.996977i	$-0.475244\pi$
$757$	1.40395e12	0.155389	0.0776947	+	0.996977i	$0.475244\pi$
$758$	−1.85732e13	−2.04350
$759$	0	0
$760$	2.17342e13	2.36310
$761$	−7.95047e12	−0.859334	−0.429667	−	0.902987i	$-0.641369\pi$
$761$	−7.95047e12	−0.859334	−0.429667	+	0.902987i	$0.641369\pi$
$762$	0	0
$763$	8.63686e12	0.922562
$764$	2.14698e13	2.27986
$765$	0	0
$766$	−3.73485e11	−0.0391961
$767$	2.56706e11	0.0267829
$768$	0	0
$769$	−6.95899e12	−0.717592	−0.358796	−	0.933416i	$-0.616813\pi$
$769$	−6.95899e12	−0.717592	−0.358796	+	0.933416i	$0.616813\pi$
$770$	−4.45592e12	−0.456803
$771$	0	0
$772$	2.32639e13	2.35724
$773$	1.22597e13	1.23501	0.617506	−	0.786566i	$-0.288143\pi$
$773$	1.22597e13	1.23501	0.617506	+	0.786566i	$0.288143\pi$
$774$	0	0
$775$	−2.93799e12	−0.292545
$776$	4.10424e13	4.06308
$777$	0	0
$778$	−1.99170e13	−1.94902
$779$	8.58177e12	0.834945
$780$	0	0
$781$	4.71302e12	0.453283
$782$	1.33062e12	0.127240
$783$	0	0
$784$	3.52346e12	0.333079
$785$	7.98975e11	0.0750965
$786$	0	0
$787$	1.19814e13	1.11332	0.556662	−	0.830739i	$-0.312082\pi$
$787$	1.19814e13	1.11332	0.556662	+	0.830739i	$0.312082\pi$
$788$	−5.88807e12	−0.544007
$789$	0	0
$790$	−1.67585e12	−0.153078
$791$	−5.11135e11	−0.0464238
$792$	0	0
$793$	−9.64288e10	−0.00865919
$794$	3.97380e13	3.54825
$795$	0	0
$796$	−5.72277e13	−5.05240
$797$	−2.90159e12	−0.254726	−0.127363	−	0.991856i	$-0.540651\pi$
$797$	−2.90159e12	−0.254726	−0.127363	+	0.991856i	$0.540651\pi$
$798$	0	0
$799$	1.65030e13	1.43253
$800$	−2.44929e13	−2.11415
$801$	0	0
$802$	9.86089e11	0.0841650
$803$	2.27908e12	0.193437
$804$	0	0
$805$	4.66121e11	0.0391217
$806$	7.69821e11	0.0642513
$807$	0	0
$808$	−7.14695e13	−5.89888
$809$	−1.25435e13	−1.02956	−0.514779	−	0.857323i	$-0.672126\pi$
$809$	−1.25435e13	−1.02956	−0.514779	+	0.857323i	$0.672126\pi$
$810$	0	0
$811$	−1.31788e13	−1.06975	−0.534874	−	0.844932i	$-0.679641\pi$
$811$	−1.31788e13	−1.06975	−0.534874	+	0.844932i	$0.679641\pi$
$812$	−3.30366e13	−2.66682
$813$	0	0
$814$	−2.53829e12	−0.202643
$815$	7.93338e12	0.629867
$816$	0	0
$817$	−1.60965e13	−1.26396
$818$	2.13367e13	1.66624
$819$	0	0
$820$	2.71211e13	2.09481
$821$	1.39450e13	1.07121	0.535606	−	0.844468i	$-0.320083\pi$
$821$	1.39450e13	1.07121	0.535606	+	0.844468i	$0.320083\pi$
$822$	0	0
$823$	1.81146e13	1.37635	0.688177	−	0.725543i	$-0.258411\pi$
$823$	1.81146e13	1.37635	0.688177	+	0.725543i	$0.258411\pi$
$824$	4.27003e12	0.322670
$825$	0	0
$826$	1.50421e13	1.12434
$827$	9.15727e12	0.680756	0.340378	−	0.940289i	$-0.389445\pi$
$827$	9.15727e12	0.680756	0.340378	+	0.940289i	$0.389445\pi$
$828$	0	0
$829$	1.41975e13	1.04404	0.522020	−	0.852933i	$-0.325179\pi$
$829$	1.41975e13	1.04404	0.522020	+	0.852933i	$0.325179\pi$
$830$	−3.77560e11	−0.0276143
$831$	0	0
$832$	3.50509e12	0.253597
$833$	−1.37753e12	−0.0991289
$834$	0	0
$835$	−8.46553e12	−0.602650
$836$	−1.03385e13	−0.732030
$837$	0	0
$838$	3.71528e13	2.60251
$839$	2.26357e13	1.57712	0.788561	−	0.614956i	$-0.210826\pi$
$839$	2.26357e13	1.57712	0.788561	+	0.614956i	$0.210826\pi$
$840$	0	0
$841$	−2.82351e12	−0.194629
$842$	9.17478e12	0.629059
$843$	0	0
$844$	3.96024e13	2.68646
$845$	1.11150e13	0.749989
$846$	0	0
$847$	−1.41638e13	−0.945596
$848$	−8.94567e13	−5.94062
$849$	0	0
$850$	1.67911e13	1.10330
$851$	2.65523e11	0.0173548
$852$	0	0
$853$	−2.75478e13	−1.78163	−0.890813	−	0.454370i	$-0.849864\pi$
$853$	−2.75478e13	−1.78163	−0.890813	+	0.454370i	$0.849864\pi$
$854$	−5.65039e12	−0.363511
$855$	0	0
$856$	2.33634e13	1.48732
$857$	−1.19508e13	−0.756801	−0.378401	−	0.925642i	$-0.623526\pi$
$857$	−1.19508e13	−0.756801	−0.378401	+	0.925642i	$0.623526\pi$
$858$	0	0
$859$	−1.68633e13	−1.05675	−0.528377	−	0.849010i	$-0.677199\pi$
$859$	−1.68633e13	−1.05675	−0.528377	+	0.849010i	$0.677199\pi$
$860$	−5.08700e13	−3.17117
$861$	0	0
$862$	1.54029e13	0.950213
$863$	−2.70257e13	−1.65855	−0.829275	−	0.558840i	$-0.811247\pi$
$863$	−2.70257e13	−1.65855	−0.829275	+	0.558840i	$0.811247\pi$
$864$	0	0
$865$	−4.33190e12	−0.263091
$866$	−3.29701e11	−0.0199200
$867$	0	0
$868$	3.34338e13	1.99916
$869$	5.18797e11	0.0308609
$870$	0	0
$871$	1.07494e12	0.0632853
$872$	5.56072e13	3.25692
$873$	0	0
$874$	1.45913e12	0.0845849
$875$	1.94085e13	1.11932
$876$	0	0
$877$	1.47392e13	0.841350	0.420675	−	0.907211i	$-0.361793\pi$
$877$	1.47392e13	0.841350	0.420675	+	0.907211i	$0.361793\pi$
$878$	4.26898e13	2.42437
$879$	0	0
$880$	−1.72795e13	−0.971312
$881$	−2.25153e13	−1.25918	−0.629589	−	0.776928i	$-0.716777\pi$
$881$	−2.25153e13	−1.25918	−0.629589	+	0.776928i	$0.716777\pi$
$882$	0	0
$883$	−2.11103e13	−1.16861	−0.584307	−	0.811533i	$-0.698634\pi$
$883$	−2.11103e13	−1.16861	−0.584307	+	0.811533i	$0.698634\pi$
$884$	−3.26094e12	−0.179601
$885$	0	0
$886$	1.28322e13	0.699598
$887$	4.72702e12	0.256408	0.128204	−	0.991748i	$-0.459079\pi$
$887$	4.72702e12	0.256408	0.128204	+	0.991748i	$0.459079\pi$
$888$	0	0
$889$	−1.24775e13	−0.669995
$890$	−4.40956e13	−2.35581
$891$	0	0
$892$	−8.40195e13	−4.44363
$893$	1.80969e13	0.952299
$894$	0	0
$895$	−2.58631e13	−1.34734
$896$	1.08056e14	5.60097
$897$	0	0
$898$	4.44902e13	2.28308
$899$	−1.18241e13	−0.603740
$900$	0	0
$901$	3.49740e13	1.76801
$902$	−1.13278e13	−0.569792
$903$	0	0
$904$	−3.29087e12	−0.163890
$905$	4.38804e12	0.217446
$906$	0	0
$907$	1.32865e13	0.651895	0.325948	−	0.945388i	$-0.394317\pi$
$907$	1.32865e13	0.651895	0.325948	+	0.945388i	$0.394317\pi$
$908$	5.13259e13	2.50582
$909$	0	0
$910$	−1.54122e12	−0.0745038
$911$	−2.09108e13	−1.00586	−0.502930	−	0.864327i	$-0.667745\pi$
$911$	−2.09108e13	−1.00586	−0.502930	+	0.864327i	$0.667745\pi$
$912$	0	0
$913$	1.16882e11	0.00556711
$914$	1.26251e13	0.598378
$915$	0	0
$916$	9.49419e13	4.45583
$917$	−3.04667e13	−1.42286
$918$	0	0
$919$	−1.12670e13	−0.521060	−0.260530	−	0.965466i	$-0.583897\pi$
$919$	−1.12670e13	−0.521060	−0.260530	+	0.965466i	$0.583897\pi$
$920$	3.00106e12	0.138111
$921$	0	0
$922$	−2.87242e10	−0.00130906
$923$	1.63015e12	0.0739298
$924$	0	0
$925$	3.35065e12	0.150484
$926$	−2.14856e13	−0.960281
$927$	0	0
$928$	−9.85731e13	−4.36307
$929$	2.48251e13	1.09350	0.546752	−	0.837294i	$-0.315864\pi$
$929$	2.48251e13	1.09350	0.546752	+	0.837294i	$0.315864\pi$
$930$	0	0
$931$	−1.51058e12	−0.0658978
$932$	−8.66623e13	−3.76235
$933$	0	0
$934$	−3.15845e13	−1.35804
$935$	6.75560e12	0.289076
$936$	0	0
$937$	2.05299e13	0.870081	0.435040	−	0.900411i	$-0.356734\pi$
$937$	2.05299e13	0.870081	0.435040	+	0.900411i	$0.356734\pi$
$938$	6.29878e13	2.65671
$939$	0	0
$940$	5.71921e13	2.38924
$941$	−2.89837e13	−1.20504	−0.602519	−	0.798104i	$-0.705836\pi$
$941$	−2.89837e13	−1.20504	−0.602519	+	0.798104i	$0.705836\pi$
$942$	0	0
$943$	1.18497e12	0.0487983
$944$	5.83313e13	2.39072
$945$	0	0
$946$	2.12471e13	0.862562
$947$	−2.78208e13	−1.12407	−0.562036	−	0.827113i	$-0.689982\pi$
$947$	−2.78208e13	−1.12407	−0.562036	+	0.827113i	$0.689982\pi$
$948$	0	0
$949$	7.88290e11	0.0315492
$950$	1.84129e13	0.733441
$951$	0	0
$952$	−1.24355e14	−4.90679
$953$	2.85882e13	1.12271	0.561357	−	0.827573i	$-0.310279\pi$
$953$	2.85882e13	1.12271	0.561357	+	0.827573i	$0.310279\pi$
$954$	0	0
$955$	−1.53842e13	−0.598496
$956$	−5.93381e13	−2.29759
$957$	0	0
$958$	6.87702e11	0.0263788
$959$	−4.50023e13	−1.71811
$960$	0	0
$961$	−1.44733e13	−0.547411
$962$	−8.77947e11	−0.0330507
$963$	0	0
$964$	4.94246e12	0.184330
$965$	−1.66698e13	−0.618810
$966$	0	0
$967$	−5.14935e13	−1.89380	−0.946898	−	0.321534i	$-0.895802\pi$
$967$	−5.14935e13	−1.89380	−0.946898	+	0.321534i	$0.895802\pi$
$968$	−9.11919e13	−3.33824
$969$	0	0
$970$	−4.51890e13	−1.63893
$971$	−3.75474e13	−1.35548	−0.677740	−	0.735301i	$-0.737041\pi$
$971$	−3.75474e13	−1.35548	−0.677740	+	0.735301i	$0.737041\pi$
$972$	0	0
$973$	3.21464e13	1.14981
$974$	−2.51476e13	−0.895325
$975$	0	0
$976$	−2.19115e13	−0.772944
$977$	−3.25385e13	−1.14254	−0.571271	−	0.820762i	$-0.693549\pi$
$977$	−3.25385e13	−1.14254	−0.571271	+	0.820762i	$0.693549\pi$
$978$	0	0
$979$	1.36508e13	0.474937
$980$	−4.77392e12	−0.165332
$981$	0	0
$982$	−1.65781e13	−0.568895
$983$	5.63366e13	1.92442	0.962209	−	0.272310i	$-0.0877877\pi$
$983$	5.63366e13	1.92442	0.962209	+	0.272310i	$0.0877877\pi$
$984$	0	0
$985$	4.21911e12	0.142810
$986$	6.75768e13	2.27694
$987$	0	0
$988$	−3.57589e12	−0.119393
$989$	−2.22260e12	−0.0738718
$990$	0	0
$991$	−2.09674e13	−0.690578	−0.345289	−	0.938496i	$-0.612219\pi$
$991$	−2.09674e13	−0.690578	−0.345289	+	0.938496i	$0.612219\pi$
$992$	9.97583e13	3.27074
$993$	0	0
$994$	9.55210e13	3.10356
$995$	4.10067e13	1.32633
$996$	0	0
$997$	4.85780e13	1.55708	0.778541	−	0.627593i	$-0.215960\pi$
$997$	4.85780e13	1.55708	0.778541	+	0.627593i	$0.215960\pi$
$998$	−1.59057e13	−0.507533
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.a.d.1.4	yes	4
3.2	odd	2	inner	27.10.a.d.1.1	✓	4
9.2	odd	6		81.10.c.j.28.4		8
9.4	even	3		81.10.c.j.55.1		8
9.5	odd	6		81.10.c.j.55.4		8
9.7	even	3		81.10.c.j.28.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.a.d.1.1	✓	4	3.2	odd	2	inner
27.10.a.d.1.4	yes	4	1.1	even	1	trivial
81.10.c.j.28.1		8	9.7	even	3
81.10.c.j.28.4		8	9.2	odd	6
81.10.c.j.55.1		8	9.4	even	3
81.10.c.j.55.4		8	9.5	odd	6

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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q+44.4771 q^{2} +1466.22 q^{4} -1050.62 q^{5} +6591.86 q^{7} +42440.8 q^{8} -46728.6 q^{10} +14465.9 q^{11} +5003.49 q^{13} +293187. q^{14} +1.13694e6 q^{16} -444500. q^{17} -487431. q^{19} -1.54044e6 q^{20} +643402. q^{22} -67304.5 q^{23} -849319. q^{25} +222541. q^{26} +9.66510e6 q^{28} -3.41813e6 q^{29} +3.45923e6 q^{31} +2.88383e7 q^{32} -1.97701e7 q^{34} -6.92555e6 q^{35} -3.94510e6 q^{37} -2.16796e7 q^{38} -4.45892e7 q^{40} -1.76061e7 q^{41} +3.30231e7 q^{43} +2.12102e7 q^{44} -2.99351e6 q^{46} -3.71271e7 q^{47} +3.09907e6 q^{49} -3.77753e7 q^{50} +7.33620e6 q^{52} -7.86818e7 q^{53} -1.51982e7 q^{55} +2.79764e8 q^{56} -1.52029e8 q^{58} +5.13054e7 q^{59} -1.92723e7 q^{61} +1.53857e8 q^{62} +7.00529e8 q^{64} -5.25678e6 q^{65} +2.14838e8 q^{67} -6.51733e8 q^{68} -3.08029e8 q^{70} +3.25802e8 q^{71} +1.57548e8 q^{73} -1.75467e8 q^{74} -7.14680e8 q^{76} +9.53573e7 q^{77} +3.58634e7 q^{79} -1.19450e9 q^{80} -7.83069e8 q^{82} +8.07984e6 q^{83} +4.67001e8 q^{85} +1.46877e9 q^{86} +6.13945e8 q^{88} +9.43654e8 q^{89} +3.29823e7 q^{91} -9.86830e7 q^{92} -1.65131e9 q^{94} +5.12106e8 q^{95} +9.67052e8 q^{97} +1.37838e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2236 q^{4} + 11852 q^{7} - 41760 q^{10} + 179684 q^{13} + 2348680 q^{16} + 1011428 q^{19} + 3473568 q^{22} + 2554060 q^{25} + 19793924 q^{28} + 889136 q^{31} - 43111008 q^{34} - 3805156 q^{37} - 133649280 q^{40}+ \cdots + 1935734516 q^{97}+O(q^{100}) \) 4 * q + 2236 * q^4 + 11852 * q^7 - 41760 * q^10 + 179684 * q^13 + 2348680 * q^16 + 1011428 * q^19 + 3473568 * q^22 + 2554060 * q^25 + 19793924 * q^28 + 889136 * q^31 - 43111008 * q^34 - 3805156 * q^37 - 133649280 * q^40 + 11585024 * q^43 + 13428000 * q^46 - 73622328 * q^49 - 44411860 * q^52 + 314722080 * q^55 - 407685888 * q^58 + 219882572 * q^61 + 1519285360 * q^64 + 390561236 * q^67 - 650481120 * q^70 + 1197949508 * q^73 - 2121017572 * q^76 + 49814324 * q^79 - 1767126528 * q^82 + 370444320 * q^85 - 653198976 * q^88 - 47017172 * q^91 - 4146926688 * q^94 + 1935734516 * q^97

Introduction

L-functions

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