LMFDB - Embedded newform 81.8.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,8,Mod(1,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("81.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	81.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:	$25.3031870642$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 401x^{4} - 1212x^{3} + 17752x^{2} + 15108x - 22632 $ x^6 - 401x^4 - 1212x^3 + 17752x^2 + 15108x - 22632
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{9} $
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-16.1993$ of defining polynomial
Character	$\chi$	$=$	81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-14.2319 q^{2} +74.5469 q^{4} +290.607 q^{5} +1111.88 q^{7} +760.739 q^{8} -4135.89 q^{10} +4490.71 q^{11} +2436.58 q^{13} -15824.2 q^{14} -20368.8 q^{16} +15905.4 q^{17} -49949.6 q^{19} +21663.9 q^{20} -63911.4 q^{22} +69385.1 q^{23} +6327.45 q^{25} -34677.2 q^{26} +82887.2 q^{28} -94071.6 q^{29} -19927.2 q^{31} +192512. q^{32} -226364. q^{34} +323120. q^{35} +331750. q^{37} +710878. q^{38} +221076. q^{40} +242266. q^{41} +831426. q^{43} +334769. q^{44} -987481. q^{46} -160011. q^{47} +412732. q^{49} -90051.7 q^{50} +181639. q^{52} +311589. q^{53} +1.30503e6 q^{55} +845850. q^{56} +1.33882e6 q^{58} -312353. q^{59} -57447.8 q^{61} +283601. q^{62} -132603. q^{64} +708087. q^{65} -4.10199e6 q^{67} +1.18570e6 q^{68} -4.59861e6 q^{70} -403110. q^{71} -823496. q^{73} -4.72144e6 q^{74} -3.72359e6 q^{76} +4.99313e6 q^{77} +978827. q^{79} -5.91931e6 q^{80} -3.44791e6 q^{82} -3.70408e6 q^{83} +4.62222e6 q^{85} -1.18328e7 q^{86} +3.41626e6 q^{88} +2.09023e6 q^{89} +2.70918e6 q^{91} +5.17244e6 q^{92} +2.27726e6 q^{94} -1.45157e7 q^{95} +3.50249e6 q^{97} -5.87396e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 9 q^{2} + 321 q^{4} + 180 q^{5} + 84 q^{7} + 2961 q^{8} + 126 q^{10} + 8460 q^{11} + 1848 q^{13} + 16272 q^{14} + 12417 q^{16} + 15282 q^{17} + 12216 q^{19} + 40788 q^{20} + 35001 q^{22} + 51588 q^{23}+ \cdots - 47916657 q^{98}+O(q^{100}) $ 6 * q + 9 * q^2 + 321 * q^4 + 180 * q^5 + 84 * q^7 + 2961 * q^8 + 126 * q^10 + 8460 * q^11 + 1848 * q^13 + 16272 * q^14 + 12417 * q^16 + 15282 * q^17 + 12216 * q^19 + 40788 * q^20 + 35001 * q^22 + 51588 * q^23 - 4746 * q^25 + 268236 * q^26 + 37758 * q^28 + 414648 * q^29 - 8196 * q^31 + 1048977 * q^32 + 106623 * q^34 + 1105308 * q^35 + 69672 * q^37 + 1952685 * q^38 - 305496 * q^40 + 1731582 * q^41 - 408372 * q^43 + 2584557 * q^44 - 842004 * q^46 + 1631484 * q^47 + 179010 * q^49 + 1654461 * q^50 - 681594 * q^52 + 1417824 * q^53 - 8028 * q^55 + 1784466 * q^56 + 948384 * q^58 + 2055636 * q^59 + 2723196 * q^61 - 513414 * q^62 + 3589089 * q^64 + 1387620 * q^65 - 3806556 * q^67 - 2142639 * q^68 - 953442 * q^70 + 1204200 * q^71 - 5335026 * q^73 - 9846504 * q^74 + 6727827 * q^76 - 3478824 * q^77 - 6020916 * q^79 - 19036224 * q^80 + 4701501 * q^82 - 9605052 * q^83 + 1698624 * q^85 - 34278561 * q^86 + 16459029 * q^88 - 12315132 * q^89 + 6785052 * q^91 - 39143394 * q^92 - 12602808 * q^94 - 10422072 * q^95 - 9977226 * q^97 - 47916657 * q^98

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 $

$2$	−14.2319	−1.25793	−0.628967	−	0.777432i	$-0.716522\pi$
$2$	−14.2319	−1.25793	−0.628967	+	0.777432i	$0.716522\pi$
$3$	0	0
$3$	0	0
$4$	74.5469	0.582398
$5$	290.607	1.03971	0.519854	−	0.854255i	$-0.325986\pi$
$5$	290.607	1.03971	0.519854	+	0.854255i	$0.325986\pi$
$6$	0	0
$7$	1111.88	1.22522	0.612611	−	0.790385i	$-0.290119\pi$
$7$	1111.88	1.22522	0.612611	+	0.790385i	$0.290119\pi$
$8$	760.739	0.525316
$9$	0	0
$10$	−4135.89	−1.30788
$11$	4490.71	1.01728	0.508640	−	0.860979i	$-0.330148\pi$
$11$	4490.71	1.01728	0.508640	+	0.860979i	$0.330148\pi$
$12$	0	0
$13$	2436.58	0.307595	0.153797	−	0.988102i	$-0.450850\pi$
$13$	2436.58	0.307595	0.153797	+	0.988102i	$0.450850\pi$
$14$	−15824.2	−1.54125
$15$	0	0
$16$	−20368.8	−1.24321
$17$	15905.4	0.785187	0.392593	−	0.919712i	$-0.371578\pi$
$17$	15905.4	0.785187	0.392593	+	0.919712i	$0.371578\pi$
$18$	0	0
$19$	−49949.6	−1.67069	−0.835343	−	0.549730i	$-0.814731\pi$
$19$	−49949.6	−1.67069	−0.835343	+	0.549730i	$0.814731\pi$
$20$	21663.9	0.605523
$21$	0	0
$22$	−63911.4	−1.27967
$23$	69385.1	1.18910	0.594550	−	0.804058i	$-0.297330\pi$
$23$	69385.1	1.18910	0.594550	+	0.804058i	$0.297330\pi$
$24$	0	0
$25$	6327.45	0.0809914
$26$	−34677.2	−0.386934
$27$	0	0
$28$	82887.2	0.713566
$29$	−94071.6	−0.716251	−0.358126	−	0.933673i	$-0.616584\pi$
$29$	−94071.6	−0.716251	−0.358126	+	0.933673i	$0.616584\pi$
$30$	0	0
$31$	−19927.2	−0.120138	−0.0600689	−	0.998194i	$-0.519132\pi$
$31$	−19927.2	−0.120138	−0.0600689	+	0.998194i	$0.519132\pi$
$32$	192512.	1.03856
$33$	0	0
$34$	−226364.	−0.987713
$35$	323120.	1.27387
$36$	0	0
$37$	331750.	1.07673	0.538363	−	0.842713i	$-0.319043\pi$
$37$	331750.	1.07673	0.538363	+	0.842713i	$0.319043\pi$
$38$	710878.	2.10161
$39$	0	0
$40$	221076.	0.546175
$41$	242266.	0.548971	0.274486	−	0.961591i	$-0.411492\pi$
$41$	242266.	0.548971	0.274486	+	0.961591i	$0.411492\pi$
$42$	0	0
$43$	831426.	1.59472	0.797359	−	0.603505i	$-0.206230\pi$
$43$	831426.	1.59472	0.797359	+	0.603505i	$0.206230\pi$
$44$	334769.	0.592462
$45$	0	0
$46$	−987481.	−1.49581
$47$	−160011.	−0.224805	−0.112403	−	0.993663i	$-0.535855\pi$
$47$	−160011.	−0.224805	−0.112403	+	0.993663i	$0.535855\pi$
$48$	0	0
$49$	412732.	0.501167
$50$	−90051.7	−0.101882
$51$	0	0
$52$	181639.	0.179142
$53$	311589.	0.287486	0.143743	−	0.989615i	$-0.454086\pi$
$53$	311589.	0.287486	0.143743	+	0.989615i	$0.454086\pi$
$54$	0	0
$55$	1.30503e6	1.05767
$56$	845850.	0.643628
$57$	0	0
$58$	1.33882e6	0.900997
$59$	−312353.	−0.197999	−0.0989997	−	0.995087i	$-0.531564\pi$
$59$	−312353.	−0.197999	−0.0989997	+	0.995087i	$0.531564\pi$
$60$	0	0
$61$	−57447.8	−0.0324055	−0.0162028	−	0.999869i	$-0.505158\pi$
$61$	−57447.8	−0.0324055	−0.0162028	+	0.999869i	$0.505158\pi$
$62$	283601.	0.151125
$63$	0	0
$64$	−132603.	−0.0632302
$65$	708087.	0.319808
$66$	0	0
$67$	−4.10199e6	−1.66622	−0.833111	−	0.553105i	$-0.813443\pi$
$67$	−4.10199e6	−1.66622	−0.833111	+	0.553105i	$0.813443\pi$
$68$	1.18570e6	0.457291
$69$	0	0
$70$	−4.59861e6	−1.60245
$71$	−403110.	−0.133666	−0.0668328	−	0.997764i	$-0.521289\pi$
$71$	−403110.	−0.133666	−0.0668328	+	0.997764i	$0.521289\pi$
$72$	0	0
$73$	−823496.	−0.247760	−0.123880	−	0.992297i	$-0.539534\pi$
$73$	−823496.	−0.247760	−0.123880	+	0.992297i	$0.539534\pi$
$74$	−4.72144e6	−1.35445
$75$	0	0
$76$	−3.72359e6	−0.973003
$77$	4.99313e6	1.24639
$78$	0	0
$79$	978827.	0.223363	0.111682	−	0.993744i	$-0.464376\pi$
$79$	978827.	0.223363	0.111682	+	0.993744i	$0.464376\pi$
$80$	−5.91931e6	−1.29258
$81$	0	0
$82$	−3.44791e6	−0.690569
$83$	−3.70408e6	−0.711060	−0.355530	−	0.934665i	$-0.615700\pi$
$83$	−3.70408e6	−0.711060	−0.355530	+	0.934665i	$0.615700\pi$
$84$	0	0
$85$	4.62222e6	0.816365
$86$	−1.18328e7	−2.00605
$87$	0	0
$88$	3.41626e6	0.534394
$89$	2.09023e6	0.314289	0.157145	−	0.987576i	$-0.449771\pi$
$89$	2.09023e6	0.314289	0.157145	+	0.987576i	$0.449771\pi$
$90$	0	0
$91$	2.70918e6	0.376872
$92$	5.17244e6	0.692530
$93$	0	0
$94$	2.27726e6	0.282790
$95$	−1.45157e7	−1.73702
$96$	0	0
$97$	3.50249e6	0.389651	0.194826	−	0.980838i	$-0.437586\pi$
$97$	3.50249e6	0.389651	0.194826	+	0.980838i	$0.437586\pi$
$98$	−5.87396e6	−0.630435
$99$	0	0
$100$	471692.	0.0471692
$101$	554721.	0.0535735	0.0267868	−	0.999641i	$-0.491472\pi$
$101$	554721.	0.0535735	0.0267868	+	0.999641i	$0.491472\pi$
$102$	0	0
$103$	2.03492e7	1.83492	0.917459	−	0.397830i	$-0.130236\pi$
$103$	2.03492e7	1.83492	0.917459	+	0.397830i	$0.130236\pi$
$104$	1.85360e6	0.161584
$105$	0	0
$106$	−4.43450e6	−0.361638
$107$	6.49941e6	0.512897	0.256449	−	0.966558i	$-0.417448\pi$
$107$	6.49941e6	0.512897	0.256449	+	0.966558i	$0.417448\pi$
$108$	0	0
$109$	2.79312e6	0.206584	0.103292	−	0.994651i	$-0.467062\pi$
$109$	2.79312e6	0.206584	0.103292	+	0.994651i	$0.467062\pi$
$110$	−1.85731e7	−1.33048
$111$	0	0
$112$	−2.26476e7	−1.52321
$113$	2.08344e7	1.35833	0.679167	−	0.733984i	$-0.262341\pi$
$113$	2.08344e7	1.35833	0.679167	+	0.733984i	$0.262341\pi$
$114$	0	0
$115$	2.01638e7	1.23632
$116$	−7.01274e6	−0.417143
$117$	0	0
$118$	4.44538e6	0.249070
$119$	1.76849e7	0.962028
$120$	0	0
$121$	679329.	0.0348603
$122$	817591.	0.0407640
$123$	0	0
$124$	−1.48551e6	−0.0699679
$125$	−2.08649e7	−0.955500
$126$	0	0
$127$	8.89911e6	0.385508	0.192754	−	0.981247i	$-0.438258\pi$
$127$	8.89911e6	0.385508	0.192754	+	0.981247i	$0.438258\pi$
$128$	−2.27543e7	−0.959021
$129$	0	0
$130$	−1.00774e7	−0.402298
$131$	1.27174e7	0.494251	0.247126	−	0.968983i	$-0.420514\pi$
$131$	1.27174e7	0.494251	0.247126	+	0.968983i	$0.420514\pi$
$132$	0	0
$133$	−5.55379e7	−2.04696
$134$	5.83791e7	2.09600
$135$	0	0
$136$	1.20998e7	0.412471
$137$	4.90104e7	1.62842	0.814209	−	0.580572i	$-0.197171\pi$
$137$	4.90104e7	1.62842	0.814209	+	0.580572i	$0.197171\pi$
$138$	0	0
$139$	−1.57907e6	−0.0498713	−0.0249357	−	0.999689i	$-0.507938\pi$
$139$	−1.57907e6	−0.0498713	−0.0249357	+	0.999689i	$0.507938\pi$
$140$	2.40876e7	0.741900
$141$	0	0
$142$	5.73703e6	0.168143
$143$	1.09420e7	0.312910
$144$	0	0
$145$	−2.73379e7	−0.744692
$146$	1.17199e7	0.311666
$147$	0	0
$148$	2.47310e7	0.627083
$149$	−3.91468e7	−0.969492	−0.484746	−	0.874655i	$-0.661088\pi$
$149$	−3.91468e7	−0.969492	−0.484746	+	0.874655i	$0.661088\pi$
$150$	0	0
$151$	1.83819e7	0.434481	0.217240	−	0.976118i	$-0.430294\pi$
$151$	1.83819e7	0.434481	0.217240	+	0.976118i	$0.430294\pi$
$152$	−3.79986e7	−0.877638
$153$	0	0
$154$	−7.10617e7	−1.56788
$155$	−5.79097e6	−0.124908
$156$	0	0
$157$	−6.64336e7	−1.37006	−0.685029	−	0.728516i	$-0.740210\pi$
$157$	−6.64336e7	−1.37006	−0.685029	+	0.728516i	$0.740210\pi$
$158$	−1.39306e7	−0.280976
$159$	0	0
$160$	5.59452e7	1.07980
$161$	7.71478e7	1.45691
$162$	0	0
$163$	−3.94682e7	−0.713823	−0.356912	−	0.934138i	$-0.616170\pi$
$163$	−3.94682e7	−0.713823	−0.356912	+	0.934138i	$0.616170\pi$
$164$	1.80602e7	0.319720
$165$	0	0
$166$	5.27160e7	0.894467
$167$	9.72355e7	1.61554	0.807769	−	0.589499i	$-0.200675\pi$
$167$	9.72355e7	1.61554	0.807769	+	0.589499i	$0.200675\pi$
$168$	0	0
$169$	−5.68116e7	−0.905385
$170$	−6.57829e7	−1.02693
$171$	0	0
$172$	6.19802e7	0.928760
$173$	5.54029e7	0.813526	0.406763	−	0.913534i	$-0.366657\pi$
$173$	5.54029e7	0.813526	0.406763	+	0.913534i	$0.366657\pi$
$174$	0	0
$175$	7.03537e6	0.0992324
$176$	−9.14703e7	−1.26469
$177$	0	0
$178$	−2.97480e7	−0.395355
$179$	9.87173e7	1.28649	0.643247	−	0.765659i	$-0.277587\pi$
$179$	9.87173e7	1.28649	0.643247	+	0.765659i	$0.277587\pi$
$180$	0	0
$181$	9.03305e7	1.13230	0.566148	−	0.824304i	$-0.308433\pi$
$181$	9.03305e7	1.13230	0.566148	+	0.824304i	$0.308433\pi$
$182$	−3.85568e7	−0.474079
$183$	0	0
$184$	5.27839e7	0.624654
$185$	9.64090e7	1.11948
$186$	0	0
$187$	7.14265e7	0.798756
$188$	−1.19283e7	−0.130926
$189$	0	0
$190$	2.06586e8	2.18506
$191$	−1.04182e8	−1.08188	−0.540938	−	0.841062i	$-0.681931\pi$
$191$	−1.04182e8	−1.08188	−0.540938	+	0.841062i	$0.681931\pi$
$192$	0	0
$193$	1.33029e7	0.133198	0.0665989	−	0.997780i	$-0.478785\pi$
$193$	1.33029e7	0.133198	0.0665989	+	0.997780i	$0.478785\pi$
$194$	−4.98471e7	−0.490156
$195$	0	0
$196$	3.07679e7	0.291878
$197$	−1.71308e8	−1.59641	−0.798206	−	0.602384i	$-0.794217\pi$
$197$	−1.71308e8	−1.59641	−0.798206	+	0.602384i	$0.794217\pi$
$198$	0	0
$199$	−1.24021e8	−1.11561	−0.557803	−	0.829974i	$-0.688355\pi$
$199$	−1.24021e8	−1.11561	−0.557803	+	0.829974i	$0.688355\pi$
$200$	4.81354e6	0.0425461
$201$	0	0
$202$	−7.89474e6	−0.0673919
$203$	−1.04596e8	−0.877566
$204$	0	0
$205$	7.04043e7	0.570769
$206$	−2.89608e8	−2.30821
$207$	0	0
$208$	−4.96301e7	−0.382405
$209$	−2.24309e8	−1.69956
$210$	0	0
$211$	−1.86497e8	−1.36673	−0.683367	−	0.730075i	$-0.739485\pi$
$211$	−1.86497e8	−1.36673	−0.683367	+	0.730075i	$0.739485\pi$
$212$	2.32280e7	0.167431
$213$	0	0
$214$	−9.24989e7	−0.645191
$215$	2.41618e8	1.65804
$216$	0	0
$217$	−2.21566e7	−0.147195
$218$	−3.97514e7	−0.259869
$219$	0	0
$220$	9.72862e7	0.615987
$221$	3.87547e7	0.241519
$222$	0	0
$223$	−4.20500e7	−0.253921	−0.126961	−	0.991908i	$-0.540522\pi$
$223$	−4.20500e7	−0.253921	−0.126961	+	0.991908i	$0.540522\pi$
$224$	2.14050e8	1.27247
$225$	0	0
$226$	−2.96513e8	−1.70869
$227$	3.13034e8	1.77623	0.888117	−	0.459617i	$-0.152013\pi$
$227$	3.13034e8	1.77623	0.888117	+	0.459617i	$0.152013\pi$
$228$	0	0
$229$	1.67115e8	0.919585	0.459792	−	0.888026i	$-0.347924\pi$
$229$	1.67115e8	0.919585	0.459792	+	0.888026i	$0.347924\pi$
$230$	−2.86969e8	−1.55520
$231$	0	0
$232$	−7.15639e7	−0.376258
$233$	−2.55430e8	−1.32290	−0.661449	−	0.749990i	$-0.730058\pi$
$233$	−2.55430e8	−1.32290	−0.661449	+	0.749990i	$0.730058\pi$
$234$	0	0
$235$	−4.65002e7	−0.233732
$236$	−2.32850e7	−0.115314
$237$	0	0
$238$	−2.51689e8	−1.21017
$239$	−1.76894e8	−0.838148	−0.419074	−	0.907952i	$-0.637645\pi$
$239$	−1.76894e8	−0.838148	−0.419074	+	0.907952i	$0.637645\pi$
$240$	0	0
$241$	−3.13989e8	−1.44496	−0.722478	−	0.691394i	$-0.756997\pi$
$241$	−3.13989e8	−1.44496	−0.722478	+	0.691394i	$0.756997\pi$
$242$	−9.66814e6	−0.0438520
$243$	0	0
$244$	−4.28256e6	−0.0188729
$245$	1.19943e8	0.521067
$246$	0	0
$247$	−1.21706e8	−0.513894
$248$	−1.51594e7	−0.0631103
$249$	0	0
$250$	2.96947e8	1.20196
$251$	1.34891e8	0.538425	0.269212	−	0.963081i	$-0.413237\pi$
$251$	1.34891e8	0.538425	0.269212	+	0.963081i	$0.413237\pi$
$252$	0	0
$253$	3.11588e8	1.20965
$254$	−1.26651e8	−0.484944
$255$	0	0
$256$	3.40810e8	1.26962
$257$	2.35523e8	0.865500	0.432750	−	0.901514i	$-0.357543\pi$
$257$	2.35523e8	0.865500	0.432750	+	0.901514i	$0.357543\pi$
$258$	0	0
$259$	3.68866e8	1.31923
$260$	5.27857e7	0.186256
$261$	0	0
$262$	−1.80992e8	−0.621736
$263$	−1.66665e8	−0.564936	−0.282468	−	0.959277i	$-0.591153\pi$
$263$	−1.66665e8	−0.564936	−0.282468	+	0.959277i	$0.591153\pi$
$264$	0	0
$265$	9.05500e7	0.298901
$266$	7.90410e8	2.57494
$267$	0	0
$268$	−3.05791e8	−0.970404
$269$	−1.49508e8	−0.468308	−0.234154	−	0.972200i	$-0.575232\pi$
$269$	−1.49508e8	−0.468308	−0.234154	+	0.972200i	$0.575232\pi$
$270$	0	0
$271$	3.02665e8	0.923783	0.461892	−	0.886936i	$-0.347171\pi$
$271$	3.02665e8	0.923783	0.461892	+	0.886936i	$0.347171\pi$
$272$	−3.23973e8	−0.976153
$273$	0	0
$274$	−6.97510e8	−2.04844
$275$	2.84148e7	0.0823910
$276$	0	0
$277$	3.26957e7	0.0924297	0.0462148	−	0.998932i	$-0.485284\pi$
$277$	3.26957e7	0.0924297	0.0462148	+	0.998932i	$0.485284\pi$
$278$	2.24732e7	0.0627348
$279$	0	0
$280$	2.45810e8	0.669185
$281$	5.01936e8	1.34951	0.674755	−	0.738042i	$-0.264249\pi$
$281$	5.01936e8	1.34951	0.674755	+	0.738042i	$0.264249\pi$
$282$	0	0
$283$	3.25269e8	0.853081	0.426541	−	0.904468i	$-0.359732\pi$
$283$	3.25269e8	0.853081	0.426541	+	0.904468i	$0.359732\pi$
$284$	−3.00506e7	−0.0778466
$285$	0	0
$286$	−1.55725e8	−0.393620
$287$	2.69371e8	0.672611
$288$	0	0
$289$	−1.57357e8	−0.383481
$290$	3.89070e8	0.936773
$291$	0	0
$292$	−6.13891e7	−0.144295
$293$	−1.43514e8	−0.333318	−0.166659	−	0.986015i	$-0.553298\pi$
$293$	−1.43514e8	−0.333318	−0.166659	+	0.986015i	$0.553298\pi$
$294$	0	0
$295$	−9.07721e7	−0.205861
$296$	2.52375e8	0.565622
$297$	0	0
$298$	5.57133e8	1.21956
$299$	1.69062e8	0.365761
$300$	0	0
$301$	9.24445e8	1.95388
$302$	−2.61609e8	−0.546548
$303$	0	0
$304$	1.01741e9	2.07701
$305$	−1.66947e7	−0.0336923
$306$	0	0
$307$	−6.60558e8	−1.30295	−0.651474	−	0.758671i	$-0.725849\pi$
$307$	−6.60558e8	−1.30295	−0.651474	+	0.758671i	$0.725849\pi$
$308$	3.72222e8	0.725897
$309$	0	0
$310$	8.24165e7	0.157126
$311$	1.05741e8	0.199333	0.0996667	−	0.995021i	$-0.468222\pi$
$311$	1.05741e8	0.199333	0.0996667	+	0.995021i	$0.468222\pi$
$312$	0	0
$313$	−1.18721e8	−0.218838	−0.109419	−	0.993996i	$-0.534899\pi$
$313$	−1.18721e8	−0.218838	−0.109419	+	0.993996i	$0.534899\pi$
$314$	9.45476e8	1.72344
$315$	0	0
$316$	7.29686e7	0.130086
$317$	−4.34669e8	−0.766392	−0.383196	−	0.923667i	$-0.625177\pi$
$317$	−4.34669e8	−0.766392	−0.383196	+	0.923667i	$0.625177\pi$
$318$	0	0
$319$	−4.22448e8	−0.728629
$320$	−3.85355e7	−0.0657409
$321$	0	0
$322$	−1.09796e9	−1.83270
$323$	−7.94468e8	−1.31180
$324$	0	0
$325$	1.54173e7	0.0249125
$326$	5.61707e8	0.897943
$327$	0	0
$328$	1.84301e8	0.288383
$329$	−1.77913e8	−0.275436
$330$	0	0
$331$	7.34259e8	1.11289	0.556444	−	0.830885i	$-0.312165\pi$
$331$	7.34259e8	1.11289	0.556444	+	0.830885i	$0.312165\pi$
$332$	−2.76127e8	−0.414120
$333$	0	0
$334$	−1.38385e9	−2.03224
$335$	−1.19207e9	−1.73238
$336$	0	0
$337$	−4.71545e8	−0.671148	−0.335574	−	0.942014i	$-0.608930\pi$
$337$	−4.71545e8	−0.671148	−0.335574	+	0.942014i	$0.608930\pi$
$338$	8.08537e8	1.13892
$339$	0	0
$340$	3.44572e8	0.475449
$341$	−8.94871e7	−0.122214
$342$	0	0
$343$	−4.56772e8	−0.611181
$344$	6.32498e8	0.837731
$345$	0	0
$346$	−7.88489e8	−1.02336
$347$	−1.10671e9	−1.42194	−0.710970	−	0.703222i	$-0.751744\pi$
$347$	−1.10671e9	−1.42194	−0.710970	+	0.703222i	$0.751744\pi$
$348$	0	0
$349$	−7.96176e7	−0.100258	−0.0501291	−	0.998743i	$-0.515963\pi$
$349$	−7.96176e7	−0.100258	−0.0501291	+	0.998743i	$0.515963\pi$
$350$	−1.00127e8	−0.124828
$351$	0	0
$352$	8.64514e8	1.05651
$353$	−1.01821e8	−0.123205	−0.0616024	−	0.998101i	$-0.519621\pi$
$353$	−1.01821e8	−0.123205	−0.0616024	+	0.998101i	$0.519621\pi$
$354$	0	0
$355$	−1.17147e8	−0.138973
$356$	1.55820e8	0.183041
$357$	0	0
$358$	−1.40494e9	−1.61832
$359$	9.69307e7	0.110568	0.0552842	−	0.998471i	$-0.482394\pi$
$359$	9.69307e7	0.110568	0.0552842	+	0.998471i	$0.482394\pi$
$360$	0	0
$361$	1.60109e9	1.79119
$362$	−1.28558e9	−1.42435
$363$	0	0
$364$	2.01961e8	0.219489
$365$	−2.39314e8	−0.257598
$366$	0	0
$367$	−1.54401e9	−1.63050	−0.815249	−	0.579111i	$-0.803400\pi$
$367$	−1.54401e9	−1.63050	−0.815249	+	0.579111i	$0.803400\pi$
$368$	−1.41329e9	−1.47830
$369$	0	0
$370$	−1.37208e9	−1.40823
$371$	3.46449e8	0.352234
$372$	0	0
$373$	−1.34954e9	−1.34650	−0.673250	−	0.739415i	$-0.735102\pi$
$373$	−1.34954e9	−1.34650	−0.673250	+	0.739415i	$0.735102\pi$
$374$	−1.01654e9	−1.00478
$375$	0	0
$376$	−1.21726e8	−0.118094
$377$	−2.29213e8	−0.220315
$378$	0	0
$379$	1.61145e9	1.52048	0.760238	−	0.649645i	$-0.225082\pi$
$379$	1.61145e9	1.52048	0.760238	+	0.649645i	$0.225082\pi$
$380$	−1.08210e9	−1.01164
$381$	0	0
$382$	1.48271e9	1.36093
$383$	1.26424e9	1.14983	0.574915	−	0.818213i	$-0.305035\pi$
$383$	1.26424e9	1.14983	0.574915	+	0.818213i	$0.305035\pi$
$384$	0	0
$385$	1.45104e9	1.29588
$386$	−1.89326e8	−0.167554
$387$	0	0
$388$	2.61100e8	0.226932
$389$	−1.83733e9	−1.58257	−0.791286	−	0.611446i	$-0.790588\pi$
$389$	−1.83733e9	−1.58257	−0.791286	+	0.611446i	$0.790588\pi$
$390$	0	0
$391$	1.10360e9	0.933666
$392$	3.13982e8	0.263271
$393$	0	0
$394$	2.43803e9	2.00818
$395$	2.84454e8	0.232232
$396$	0	0
$397$	−1.85478e9	−1.48773	−0.743867	−	0.668327i	$-0.767011\pi$
$397$	−1.85478e9	−1.48773	−0.743867	+	0.668327i	$0.767011\pi$
$398$	1.76506e9	1.40336
$399$	0	0
$400$	−1.28882e8	−0.100689
$401$	−1.20857e9	−0.935978	−0.467989	−	0.883734i	$-0.655021\pi$
$401$	−1.20857e9	−0.935978	−0.467989	+	0.883734i	$0.655021\pi$
$402$	0	0
$403$	−4.85541e7	−0.0369537
$404$	4.13528e7	0.0312011
$405$	0	0
$406$	1.48860e9	1.10392
$407$	1.48980e9	1.09533
$408$	0	0
$409$	−1.22143e9	−0.882750	−0.441375	−	0.897323i	$-0.645509\pi$
$409$	−1.22143e9	−0.882750	−0.441375	+	0.897323i	$0.645509\pi$
$410$	−1.00199e9	−0.717990
$411$	0	0
$412$	1.51697e9	1.06865
$413$	−3.47299e8	−0.242593
$414$	0	0
$415$	−1.07643e9	−0.739295
$416$	4.69070e8	0.319456
$417$	0	0
$418$	3.19235e9	2.13793
$419$	1.52070e9	1.00994	0.504969	−	0.863137i	$-0.331504\pi$
$419$	1.52070e9	1.00994	0.504969	+	0.863137i	$0.331504\pi$
$420$	0	0
$421$	1.08736e9	0.710210	0.355105	−	0.934826i	$-0.384445\pi$
$421$	1.08736e9	0.710210	0.355105	+	0.934826i	$0.384445\pi$
$422$	2.65421e9	1.71926
$423$	0	0
$424$	2.37038e8	0.151021
$425$	1.00641e8	0.0635934
$426$	0	0
$427$	−6.38750e7	−0.0397039
$428$	4.84511e8	0.298710
$429$	0	0
$430$	−3.43869e9	−2.08571
$431$	−1.33391e9	−0.802521	−0.401261	−	0.915964i	$-0.631428\pi$
$431$	−1.33391e9	−0.802521	−0.401261	+	0.915964i	$0.631428\pi$
$432$	0	0
$433$	−3.41286e8	−0.202027	−0.101014	−	0.994885i	$-0.532209\pi$
$433$	−3.41286e8	−0.202027	−0.101014	+	0.994885i	$0.532209\pi$
$434$	3.15330e8	0.185162
$435$	0	0
$436$	2.08219e8	0.120314
$437$	−3.46576e9	−1.98661
$438$	0	0
$439$	−1.35623e9	−0.765083	−0.382541	−	0.923938i	$-0.624951\pi$
$439$	−1.35623e9	−0.765083	−0.382541	+	0.923938i	$0.624951\pi$
$440$	9.92789e8	0.555613
$441$	0	0
$442$	−5.51554e8	−0.303815
$443$	−1.15939e9	−0.633604	−0.316802	−	0.948492i	$-0.602609\pi$
$443$	−1.15939e9	−0.633604	−0.316802	+	0.948492i	$0.602609\pi$
$444$	0	0
$445$	6.07436e8	0.326769
$446$	5.98451e8	0.319416
$447$	0	0
$448$	−1.47439e8	−0.0774710
$449$	9.76057e8	0.508877	0.254439	−	0.967089i	$-0.418109\pi$
$449$	9.76057e8	0.508877	0.254439	+	0.967089i	$0.418109\pi$
$450$	0	0
$451$	1.08795e9	0.558458
$452$	1.55314e9	0.791091
$453$	0	0
$454$	−4.45506e9	−2.23439
$455$	7.87307e8	0.391836
$456$	0	0
$457$	−1.88838e9	−0.925513	−0.462757	−	0.886485i	$-0.653140\pi$
$457$	−1.88838e9	−0.925513	−0.462757	+	0.886485i	$0.653140\pi$
$458$	−2.37837e9	−1.15678
$459$	0	0
$460$	1.50315e9	0.720028
$461$	−1.41417e9	−0.672278	−0.336139	−	0.941812i	$-0.609121\pi$
$461$	−1.41417e9	−0.672278	−0.336139	+	0.941812i	$0.609121\pi$
$462$	0	0
$463$	−1.02607e9	−0.480445	−0.240222	−	0.970718i	$-0.577220\pi$
$463$	−1.02607e9	−0.480445	−0.240222	+	0.970718i	$0.577220\pi$
$464$	1.91612e9	0.890451
$465$	0	0
$466$	3.63525e9	1.66412
$467$	4.39875e8	0.199858	0.0999288	−	0.994995i	$-0.468139\pi$
$467$	4.39875e8	0.199858	0.0999288	+	0.994995i	$0.468139\pi$
$468$	0	0
$469$	−4.56092e9	−2.04149
$470$	6.61786e8	0.294019
$471$	0	0
$472$	−2.37619e8	−0.104012
$473$	3.73370e9	1.62228
$474$	0	0
$475$	−3.16054e8	−0.135311
$476$	1.31835e9	0.560283
$477$	0	0
$478$	2.51754e9	1.05434
$479$	2.46738e9	1.02580	0.512899	−	0.858449i	$-0.328572\pi$
$479$	2.46738e9	1.02580	0.512899	+	0.858449i	$0.328572\pi$
$480$	0	0
$481$	8.08336e8	0.331195
$482$	4.46866e9	1.81766
$483$	0	0
$484$	5.06419e7	0.0203026
$485$	1.01785e9	0.405123
$486$	0	0
$487$	9.54719e8	0.374563	0.187281	−	0.982306i	$-0.440032\pi$
$487$	9.54719e8	0.374563	0.187281	+	0.982306i	$0.440032\pi$
$488$	−4.37028e7	−0.0170231
$489$	0	0
$490$	−1.70702e9	−0.655468
$491$	1.96760e9	0.750157	0.375079	−	0.926993i	$-0.377616\pi$
$491$	1.96760e9	0.750157	0.375079	+	0.926993i	$0.377616\pi$
$492$	0	0
$493$	−1.49624e9	−0.562391
$494$	1.73211e9	0.646445
$495$	0	0
$496$	4.05891e8	0.149356
$497$	−4.48210e8	−0.163770
$498$	0	0
$499$	1.04545e9	0.376662	0.188331	−	0.982106i	$-0.439692\pi$
$499$	1.04545e9	0.376662	0.188331	+	0.982106i	$0.439692\pi$
$500$	−1.55541e9	−0.556481
$501$	0	0
$502$	−1.91975e9	−0.677303
$503$	1.15052e9	0.403093	0.201546	−	0.979479i	$-0.435403\pi$
$503$	1.15052e9	0.403093	0.201546	+	0.979479i	$0.435403\pi$
$504$	0	0
$505$	1.61206e8	0.0557008
$506$	−4.43450e9	−1.52166
$507$	0	0
$508$	6.63401e8	0.224519
$509$	−5.00188e9	−1.68121	−0.840603	−	0.541651i	$-0.817799\pi$
$509$	−5.00188e9	−1.68121	−0.840603	+	0.541651i	$0.817799\pi$
$510$	0	0
$511$	−9.15628e8	−0.303561
$512$	−1.93782e9	−0.638071
$513$	0	0
$514$	−3.35194e9	−1.08874
$515$	5.91362e9	1.90778
$516$	0	0
$517$	−7.18562e8	−0.228690
$518$	−5.24967e9	−1.65950
$519$	0	0
$520$	5.38669e8	0.168001
$521$	−1.31851e9	−0.408460	−0.204230	−	0.978923i	$-0.565469\pi$
$521$	−1.31851e9	−0.408460	−0.204230	+	0.978923i	$0.565469\pi$
$522$	0	0
$523$	1.46804e9	0.448727	0.224363	−	0.974506i	$-0.427970\pi$
$523$	1.46804e9	0.448727	0.224363	+	0.974506i	$0.427970\pi$
$524$	9.48041e8	0.287851
$525$	0	0
$526$	2.37196e9	0.710653
$527$	−3.16949e8	−0.0943306
$528$	0	0
$529$	1.40946e9	0.413961
$530$	−1.28870e9	−0.375998
$531$	0	0
$532$	−4.14018e9	−1.19214
$533$	5.90301e8	0.168861
$534$	0	0
$535$	1.88877e9	0.533263
$536$	−3.12054e9	−0.875293
$537$	0	0
$538$	2.12778e9	0.589100
$539$	1.85346e9	0.509827
$540$	0	0
$541$	3.28515e9	0.892000	0.446000	−	0.895033i	$-0.352848\pi$
$541$	3.28515e9	0.892000	0.446000	+	0.895033i	$0.352848\pi$
$542$	−4.30750e9	−1.16206
$543$	0	0
$544$	3.06197e9	0.815464
$545$	8.11701e8	0.214787
$546$	0	0
$547$	3.43780e9	0.898100	0.449050	−	0.893507i	$-0.351762\pi$
$547$	3.43780e9	0.898100	0.449050	+	0.893507i	$0.351762\pi$
$548$	3.65357e9	0.948387
$549$	0	0
$550$	−4.04396e8	−0.103642
$551$	4.69884e9	1.19663
$552$	0	0
$553$	1.08834e9	0.273669
$554$	−4.65322e8	−0.116270
$555$	0	0
$556$	−1.17715e8	−0.0290449
$557$	−7.14256e9	−1.75130	−0.875651	−	0.482945i	$-0.839567\pi$
$557$	−7.14256e9	−1.75130	−0.875651	+	0.482945i	$0.839567\pi$
$558$	0	0
$559$	2.02584e9	0.490527
$560$	−6.58155e9	−1.58369
$561$	0	0
$562$	−7.14350e9	−1.69759
$563$	4.62695e9	1.09274	0.546369	−	0.837545i	$-0.316010\pi$
$563$	4.62695e9	1.09274	0.546369	+	0.837545i	$0.316010\pi$
$564$	0	0
$565$	6.05463e9	1.41227
$566$	−4.62920e9	−1.07312
$567$	0	0
$568$	−3.06662e8	−0.0702167
$569$	−5.65485e9	−1.28685	−0.643425	−	0.765509i	$-0.722487\pi$
$569$	−5.65485e9	−1.28685	−0.643425	+	0.765509i	$0.722487\pi$
$570$	0	0
$571$	7.68026e9	1.72643	0.863216	−	0.504835i	$-0.168447\pi$
$571$	7.68026e9	1.72643	0.863216	+	0.504835i	$0.168447\pi$
$572$	8.15691e8	0.182238
$573$	0	0
$574$	−3.83366e9	−0.846100
$575$	4.39031e8	0.0963070
$576$	0	0
$577$	2.51579e9	0.545204	0.272602	−	0.962127i	$-0.412116\pi$
$577$	2.51579e9	0.545204	0.272602	+	0.962127i	$0.412116\pi$
$578$	2.23949e9	0.482394
$579$	0	0
$580$	−2.03795e9	−0.433707
$581$	−4.11848e9	−0.871206
$582$	0	0
$583$	1.39926e9	0.292454
$584$	−6.26465e8	−0.130152
$585$	0	0
$586$	2.04248e9	0.419292
$587$	−4.50051e9	−0.918392	−0.459196	−	0.888335i	$-0.651862\pi$
$587$	−4.50051e9	−0.918392	−0.459196	+	0.888335i	$0.651862\pi$
$588$	0	0
$589$	9.95354e8	0.200712
$590$	1.29186e9	0.258960
$591$	0	0
$592$	−6.75735e9	−1.33860
$593$	8.41758e9	1.65766	0.828831	−	0.559499i	$-0.189007\pi$
$593$	8.41758e9	1.65766	0.828831	+	0.559499i	$0.189007\pi$
$594$	0	0
$595$	5.13935e9	1.00023
$596$	−2.91827e9	−0.564630
$597$	0	0
$598$	−2.40608e9	−0.460103
$599$	3.82499e9	0.727171	0.363586	−	0.931561i	$-0.381552\pi$
$599$	3.82499e9	0.727171	0.363586	+	0.931561i	$0.381552\pi$
$600$	0	0
$601$	−7.25024e8	−0.136236	−0.0681179	−	0.997677i	$-0.521699\pi$
$601$	−7.25024e8	−0.136236	−0.0681179	+	0.997677i	$0.521699\pi$
$602$	−1.31566e10	−2.45786
$603$	0	0
$604$	1.37031e9	0.253041
$605$	1.97418e8	0.0362445
$606$	0	0
$607$	−1.08492e10	−1.96896	−0.984481	−	0.175489i	$-0.943849\pi$
$607$	−1.08492e10	−1.96896	−0.984481	+	0.175489i	$0.943849\pi$
$608$	−9.61588e9	−1.73511
$609$	0	0
$610$	2.37598e8	0.0423826
$611$	−3.89879e8	−0.0691489
$612$	0	0
$613$	−7.59484e9	−1.33170	−0.665851	−	0.746085i	$-0.731931\pi$
$613$	−7.59484e9	−1.33170	−0.665851	+	0.746085i	$0.731931\pi$
$614$	9.40100e9	1.63902
$615$	0	0
$616$	3.79847e9	0.654751
$617$	1.18234e9	0.202648	0.101324	−	0.994853i	$-0.467692\pi$
$617$	1.18234e9	0.202648	0.101324	+	0.994853i	$0.467692\pi$
$618$	0	0
$619$	−9.62007e9	−1.63028	−0.815138	−	0.579267i	$-0.803339\pi$
$619$	−9.62007e9	−1.63028	−0.815138	+	0.579267i	$0.803339\pi$
$620$	−4.31699e8	−0.0727462
$621$	0	0
$622$	−1.50489e9	−0.250748
$623$	2.32409e9	0.385074
$624$	0	0
$625$	−6.55781e9	−1.07443
$626$	1.68963e9	0.275284
$627$	0	0
$628$	−4.95242e9	−0.797918
$629$	5.27662e9	0.845432
$630$	0	0
$631$	−4.21153e9	−0.667325	−0.333662	−	0.942693i	$-0.608285\pi$
$631$	−4.21153e9	−0.667325	−0.333662	+	0.942693i	$0.608285\pi$
$632$	7.44632e8	0.117336
$633$	0	0
$634$	6.18616e9	0.964070
$635$	2.58614e9	0.400816
$636$	0	0
$637$	1.00566e9	0.154156
$638$	6.01224e9	0.916567
$639$	0	0
$640$	−6.61256e9	−0.997102
$641$	−7.92285e9	−1.18817	−0.594084	−	0.804403i	$-0.702485\pi$
$641$	−7.92285e9	−1.18817	−0.594084	+	0.804403i	$0.702485\pi$
$642$	0	0
$643$	−2.92338e9	−0.433657	−0.216829	−	0.976210i	$-0.569571\pi$
$643$	−2.92338e9	−0.433657	−0.216829	+	0.976210i	$0.569571\pi$
$644$	5.75113e9	0.848502
$645$	0	0
$646$	1.13068e10	1.65016
$647$	1.00413e10	1.45755	0.728774	−	0.684754i	$-0.240090\pi$
$647$	1.00413e10	1.45755	0.728774	+	0.684754i	$0.240090\pi$
$648$	0	0
$649$	−1.40269e9	−0.201421
$650$	−2.19418e8	−0.0313383
$651$	0	0
$652$	−2.94223e9	−0.415729
$653$	1.48330e9	0.208465	0.104233	−	0.994553i	$-0.466761\pi$
$653$	1.48330e9	0.208465	0.104233	+	0.994553i	$0.466761\pi$
$654$	0	0
$655$	3.69576e9	0.513877
$656$	−4.93466e9	−0.682487
$657$	0	0
$658$	2.53203e9	0.346480
$659$	−3.35085e9	−0.456096	−0.228048	−	0.973650i	$-0.573234\pi$
$659$	−3.35085e9	−0.456096	−0.228048	+	0.973650i	$0.573234\pi$
$660$	0	0
$661$	3.40036e9	0.457952	0.228976	−	0.973432i	$-0.426462\pi$
$661$	3.40036e9	0.457952	0.228976	+	0.973432i	$0.426462\pi$
$662$	−1.04499e10	−1.39994
$663$	0	0
$664$	−2.81783e9	−0.373531
$665$	−1.61397e10	−2.12824
$666$	0	0
$667$	−6.52716e9	−0.851695
$668$	7.24860e9	0.940886
$669$	0	0
$670$	1.69654e10	2.17922
$671$	−2.57982e8	−0.0329655
$672$	0	0
$673$	8.96600e9	1.13382	0.566912	−	0.823778i	$-0.308138\pi$
$673$	8.96600e9	1.13382	0.566912	+	0.823778i	$0.308138\pi$
$674$	6.71098e9	0.844260
$675$	0	0
$676$	−4.23513e9	−0.527294
$677$	1.48883e10	1.84410	0.922052	−	0.387067i	$-0.126512\pi$
$677$	1.48883e10	1.84410	0.922052	+	0.387067i	$0.126512\pi$
$678$	0	0
$679$	3.89435e9	0.477409
$680$	3.51630e9	0.428849
$681$	0	0
$682$	1.27357e9	0.153737
$683$	7.71503e9	0.926542	0.463271	−	0.886217i	$-0.346676\pi$
$683$	7.71503e9	0.926542	0.463271	+	0.886217i	$0.346676\pi$
$684$	0	0
$685$	1.42428e10	1.69308
$686$	6.50073e9	0.768825
$687$	0	0
$688$	−1.69351e10	−1.98257
$689$	7.59211e8	0.0884291
$690$	0	0
$691$	−2.52996e9	−0.291703	−0.145851	−	0.989307i	$-0.546592\pi$
$691$	−2.52996e9	−0.291703	−0.145851	+	0.989307i	$0.546592\pi$
$692$	4.13012e9	0.473796
$693$	0	0
$694$	1.57506e10	1.78871
$695$	−4.58890e8	−0.0518516
$696$	0	0
$697$	3.85334e9	0.431045
$698$	1.13311e9	0.126118
$699$	0	0
$700$	5.24465e8	0.0577927
$701$	1.41167e10	1.54782	0.773908	−	0.633298i	$-0.218299\pi$
$701$	1.41167e10	1.54782	0.773908	+	0.633298i	$0.218299\pi$
$702$	0	0
$703$	−1.65708e10	−1.79887
$704$	−5.95484e8	−0.0643229
$705$	0	0
$706$	1.44911e9	0.154983
$707$	6.16783e8	0.0656394
$708$	0	0
$709$	−1.29347e10	−1.36299	−0.681496	−	0.731822i	$-0.738670\pi$
$709$	−1.29347e10	−1.36299	−0.681496	+	0.731822i	$0.738670\pi$
$710$	1.66722e9	0.174819
$711$	0	0
$712$	1.59012e9	0.165101
$713$	−1.38265e9	−0.142856
$714$	0	0
$715$	3.17982e9	0.325335
$716$	7.35907e9	0.749251
$717$	0	0
$718$	−1.37951e9	−0.139088
$719$	−1.22962e10	−1.23373	−0.616864	−	0.787070i	$-0.711597\pi$
$719$	−1.22962e10	−1.23373	−0.616864	+	0.787070i	$0.711597\pi$
$720$	0	0
$721$	2.26258e10	2.24818
$722$	−2.27866e10	−2.25320
$723$	0	0
$724$	6.73386e9	0.659446
$725$	−5.95234e8	−0.0580102
$726$	0	0
$727$	6.74804e9	0.651339	0.325670	−	0.945484i	$-0.394410\pi$
$727$	6.74804e9	0.651339	0.325670	+	0.945484i	$0.394410\pi$
$728$	2.06098e9	0.197977
$729$	0	0
$730$	3.40589e9	0.324041
$731$	1.32242e10	1.25215
$732$	0	0
$733$	3.74245e9	0.350988	0.175494	−	0.984480i	$-0.443848\pi$
$733$	3.74245e9	0.350988	0.175494	+	0.984480i	$0.443848\pi$
$734$	2.19743e10	2.05106
$735$	0	0
$736$	1.33574e10	1.23495
$737$	−1.84209e10	−1.69502
$738$	0	0
$739$	1.27399e9	0.116121	0.0580606	−	0.998313i	$-0.481508\pi$
$739$	1.27399e9	0.116121	0.0580606	+	0.998313i	$0.481508\pi$
$740$	7.18700e9	0.651983
$741$	0	0
$742$	−4.93063e9	−0.443087
$743$	7.25033e9	0.648481	0.324240	−	0.945975i	$-0.394891\pi$
$743$	7.25033e9	0.648481	0.324240	+	0.945975i	$0.394891\pi$
$744$	0	0
$745$	−1.13763e10	−1.00799
$746$	1.92066e10	1.69381
$747$	0	0
$748$	5.32463e9	0.465194
$749$	7.22656e9	0.628413
$750$	0	0
$751$	−1.96482e10	−1.69271	−0.846357	−	0.532616i	$-0.821209\pi$
$751$	−1.96482e10	−1.69271	−0.846357	+	0.532616i	$0.821209\pi$
$752$	3.25922e9	0.279480
$753$	0	0
$754$	3.26213e9	0.277142
$755$	5.34190e9	0.451733
$756$	0	0
$757$	−1.81025e10	−1.51671	−0.758356	−	0.651841i	$-0.773997\pi$
$757$	−1.81025e10	−1.51671	−0.758356	+	0.651841i	$0.773997\pi$
$758$	−2.29340e10	−1.91266
$759$	0	0
$760$	−1.10427e10	−0.912486
$761$	−4.86560e9	−0.400212	−0.200106	−	0.979774i	$-0.564129\pi$
$761$	−4.86560e9	−0.400212	−0.200106	+	0.979774i	$0.564129\pi$
$762$	0	0
$763$	3.10561e9	0.253111
$764$	−7.76648e9	−0.630083
$765$	0	0
$766$	−1.79925e10	−1.44641
$767$	−7.61073e8	−0.0609036
$768$	0	0
$769$	−6.64704e9	−0.527092	−0.263546	−	0.964647i	$-0.584892\pi$
$769$	−6.64704e9	−0.527092	−0.263546	+	0.964647i	$0.584892\pi$
$770$	−2.06510e10	−1.63014
$771$	0	0
$772$	9.91692e8	0.0775740
$773$	−1.79263e10	−1.39593	−0.697963	−	0.716134i	$-0.745910\pi$
$773$	−1.79263e10	−1.39593	−0.697963	+	0.716134i	$0.745910\pi$
$774$	0	0
$775$	−1.26088e8	−0.00973012
$776$	2.66448e9	0.204690
$777$	0	0
$778$	2.61487e10	1.99077
$779$	−1.21011e10	−0.917158
$780$	0	0
$781$	−1.81025e9	−0.135976
$782$	−1.57063e10	−1.17449
$783$	0	0
$784$	−8.40685e9	−0.623056
$785$	−1.93061e10	−1.42446
$786$	0	0
$787$	9.90537e9	0.724368	0.362184	−	0.932107i	$-0.382031\pi$
$787$	9.90537e9	0.724368	0.362184	+	0.932107i	$0.382031\pi$
$788$	−1.27705e10	−0.929747
$789$	0	0
$790$	−4.04832e9	−0.292133
$791$	2.31653e10	1.66426
$792$	0	0
$793$	−1.39976e8	−0.00996777
$794$	2.63970e10	1.87147
$795$	0	0
$796$	−9.24540e9	−0.649726
$797$	−2.37203e10	−1.65965	−0.829824	−	0.558025i	$-0.811559\pi$
$797$	−2.37203e10	−1.65965	−0.829824	+	0.558025i	$0.811559\pi$
$798$	0	0
$799$	−2.54503e9	−0.176514
$800$	1.21811e9	0.0841145
$801$	0	0
$802$	1.72002e10	1.17740
$803$	−3.69808e9	−0.252042
$804$	0	0
$805$	2.24197e10	1.51476
$806$	6.91017e8	0.0464853
$807$	0	0
$808$	4.21998e8	0.0281430
$809$	−2.49700e10	−1.65805	−0.829027	−	0.559208i	$-0.811105\pi$
$809$	−2.49700e10	−1.65805	−0.829027	+	0.559208i	$0.811105\pi$
$810$	0	0
$811$	−1.12321e9	−0.0739412	−0.0369706	−	0.999316i	$-0.511771\pi$
$811$	−1.12321e9	−0.0739412	−0.0369706	+	0.999316i	$0.511771\pi$
$812$	−7.79732e9	−0.511092
$813$	0	0
$814$	−2.12026e10	−1.37786
$815$	−1.14697e10	−0.742167
$816$	0	0
$817$	−4.15294e10	−2.66427
$818$	1.73833e10	1.11044
$819$	0	0
$820$	5.24842e9	0.332415
$821$	−1.49128e10	−0.940497	−0.470249	−	0.882534i	$-0.655836\pi$
$821$	−1.49128e10	−0.940497	−0.470249	+	0.882534i	$0.655836\pi$
$822$	0	0
$823$	−2.81003e9	−0.175716	−0.0878580	−	0.996133i	$-0.528002\pi$
$823$	−2.81003e9	−0.175716	−0.0878580	+	0.996133i	$0.528002\pi$
$824$	1.54804e10	0.963912
$825$	0	0
$826$	4.94272e9	0.305166
$827$	1.92634e10	1.18430	0.592151	−	0.805827i	$-0.298279\pi$
$827$	1.92634e10	1.18430	0.592151	+	0.805827i	$0.298279\pi$
$828$	0	0
$829$	−3.12798e10	−1.90688	−0.953440	−	0.301582i	$-0.902485\pi$
$829$	−3.12798e10	−1.90688	−0.953440	+	0.301582i	$0.902485\pi$
$830$	1.53196e10	0.929984
$831$	0	0
$832$	−3.23099e8	−0.0194493
$833$	6.56467e9	0.393510
$834$	0	0
$835$	2.82573e10	1.67969
$836$	−1.67216e10	−0.989818
$837$	0	0
$838$	−2.16425e10	−1.27044
$839$	1.53071e10	0.894800	0.447400	−	0.894334i	$-0.352350\pi$
$839$	1.53071e10	0.894800	0.447400	+	0.894334i	$0.352350\pi$
$840$	0	0
$841$	−8.40042e9	−0.486984
$842$	−1.54752e10	−0.893398
$843$	0	0
$844$	−1.39028e10	−0.795983
$845$	−1.65099e10	−0.941336
$846$	0	0
$847$	7.55332e8	0.0427116
$848$	−6.34668e9	−0.357406
$849$	0	0
$850$	−1.43231e9	−0.0799963
$851$	2.30185e10	1.28034
$852$	0	0
$853$	3.19206e10	1.76096	0.880480	−	0.474083i	$-0.157220\pi$
$853$	3.19206e10	1.76096	0.880480	+	0.474083i	$0.157220\pi$
$854$	9.09063e8	0.0499449
$855$	0	0
$856$	4.94435e9	0.269433
$857$	−2.96637e10	−1.60988	−0.804938	−	0.593359i	$-0.797802\pi$
$857$	−2.96637e10	−1.60988	−0.804938	+	0.593359i	$0.797802\pi$
$858$	0	0
$859$	−1.70319e10	−0.916829	−0.458414	−	0.888739i	$-0.651582\pi$
$859$	−1.70319e10	−0.916829	−0.458414	+	0.888739i	$0.651582\pi$
$860$	1.80119e10	0.965639
$861$	0	0
$862$	1.89841e10	1.00952
$863$	−3.16249e9	−0.167491	−0.0837454	−	0.996487i	$-0.526688\pi$
$863$	−3.16249e9	−0.167491	−0.0837454	+	0.996487i	$0.526688\pi$
$864$	0	0
$865$	1.61005e10	0.845829
$866$	4.85714e9	0.254137
$867$	0	0
$868$	−1.65171e9	−0.0857262
$869$	4.39563e9	0.227223
$870$	0	0
$871$	−9.99483e9	−0.512521
$872$	2.12484e9	0.108522
$873$	0	0
$874$	4.93243e10	2.49903
$875$	−2.31992e10	−1.17070
$876$	0	0
$877$	6.39795e9	0.320289	0.160145	−	0.987094i	$-0.448804\pi$
$877$	6.39795e9	0.320289	0.160145	+	0.987094i	$0.448804\pi$
$878$	1.93018e10	0.962424
$879$	0	0
$880$	−2.65819e10	−1.31491
$881$	−1.45065e10	−0.714739	−0.357369	−	0.933963i	$-0.616326\pi$
$881$	−1.45065e10	−0.714739	−0.357369	+	0.933963i	$0.616326\pi$
$882$	0	0
$883$	2.39105e10	1.16876	0.584382	−	0.811479i	$-0.301337\pi$
$883$	2.39105e10	1.16876	0.584382	+	0.811479i	$0.301337\pi$
$884$	2.88905e9	0.140660
$885$	0	0
$886$	1.65004e10	0.797032
$887$	−3.08024e10	−1.48201	−0.741007	−	0.671497i	$-0.765652\pi$
$887$	−3.08024e10	−1.48201	−0.741007	+	0.671497i	$0.765652\pi$
$888$	0	0
$889$	9.89473e9	0.472333
$890$	−8.64497e9	−0.411054
$891$	0	0
$892$	−3.13470e9	−0.147883
$893$	7.99247e9	0.375579
$894$	0	0
$895$	2.86880e10	1.33758
$896$	−2.53000e10	−1.17501
$897$	0	0
$898$	−1.38911e10	−0.640134
$899$	1.87458e9	0.0860488
$900$	0	0
$901$	4.95594e9	0.225730
$902$	−1.54836e10	−0.702503
$903$	0	0
$904$	1.58495e10	0.713555
$905$	2.62507e10	1.17726
$906$	0	0
$907$	1.21312e10	0.539858	0.269929	−	0.962880i	$-0.413000\pi$
$907$	1.21312e10	0.539858	0.269929	+	0.962880i	$0.413000\pi$
$908$	2.33357e10	1.03448
$909$	0	0
$910$	−1.12049e10	−0.492904
$911$	1.94714e10	0.853261	0.426630	−	0.904426i	$-0.359701\pi$
$911$	1.94714e10	0.853261	0.426630	+	0.904426i	$0.359701\pi$
$912$	0	0
$913$	−1.66339e10	−0.723348
$914$	2.68752e10	1.16423
$915$	0	0
$916$	1.24579e10	0.535564
$917$	1.41402e10	0.605567
$918$	0	0
$919$	2.10755e10	0.895721	0.447861	−	0.894103i	$-0.352186\pi$
$919$	2.10755e10	0.895721	0.447861	+	0.894103i	$0.352186\pi$
$920$	1.53394e10	0.649457
$921$	0	0
$922$	2.01264e10	0.845682
$923$	−9.82211e8	−0.0411149
$924$	0	0
$925$	2.09914e9	0.0872056
$926$	1.46029e10	0.604368
$927$	0	0
$928$	−1.81099e10	−0.743870
$929$	6.96282e9	0.284925	0.142462	−	0.989800i	$-0.454498\pi$
$929$	6.96282e9	0.284925	0.142462	+	0.989800i	$0.454498\pi$
$930$	0	0
$931$	−2.06158e10	−0.837292
$932$	−1.90415e10	−0.770453
$933$	0	0
$934$	−6.26026e9	−0.251408
$935$	2.07571e10	0.830472
$936$	0	0
$937$	1.58623e10	0.629908	0.314954	−	0.949107i	$-0.398011\pi$
$937$	1.58623e10	0.629908	0.314954	+	0.949107i	$0.398011\pi$
$938$	6.49105e10	2.56806
$939$	0	0
$940$	−3.46645e9	−0.136125
$941$	−3.24983e10	−1.27144	−0.635722	−	0.771918i	$-0.719297\pi$
$941$	−3.24983e10	−1.27144	−0.635722	+	0.771918i	$0.719297\pi$
$942$	0	0
$943$	1.68097e10	0.652782
$944$	6.36225e9	0.246155
$945$	0	0
$946$	−5.31376e10	−2.04072
$947$	5.41742e9	0.207285	0.103642	−	0.994615i	$-0.466950\pi$
$947$	5.41742e9	0.207285	0.103642	+	0.994615i	$0.466950\pi$
$948$	0	0
$949$	−2.00651e9	−0.0762097
$950$	4.49805e9	0.170213
$951$	0	0
$952$	1.34536e10	0.505369
$953$	2.62658e9	0.0983028	0.0491514	−	0.998791i	$-0.484348\pi$
$953$	2.62658e9	0.0983028	0.0491514	+	0.998791i	$0.484348\pi$
$954$	0	0
$955$	−3.02762e10	−1.12484
$956$	−1.31869e10	−0.488136
$957$	0	0
$958$	−3.51155e10	−1.29039
$959$	5.44936e10	1.99517
$960$	0	0
$961$	−2.71155e10	−0.985567
$962$	−1.15042e10	−0.416622
$963$	0	0
$964$	−2.34069e10	−0.841539
$965$	3.86593e9	0.138487
$966$	0	0
$967$	3.91506e10	1.39234	0.696171	−	0.717876i	$-0.254886\pi$
$967$	3.91506e10	1.39234	0.696171	+	0.717876i	$0.254886\pi$
$968$	5.16792e8	0.0183127
$969$	0	0
$970$	−1.44859e10	−0.509618
$971$	−2.49003e10	−0.872846	−0.436423	−	0.899742i	$-0.643755\pi$
$971$	−2.49003e10	−0.872846	−0.436423	+	0.899742i	$0.643755\pi$
$972$	0	0
$973$	−1.75574e9	−0.0611034
$974$	−1.35875e10	−0.471175
$975$	0	0
$976$	1.17014e9	0.0402869
$977$	4.45804e9	0.152937	0.0764686	−	0.997072i	$-0.475636\pi$
$977$	4.45804e9	0.152937	0.0764686	+	0.997072i	$0.475636\pi$
$978$	0	0
$979$	9.38663e9	0.319720
$980$	8.94137e9	0.303468
$981$	0	0
$982$	−2.80027e10	−0.943648
$983$	−2.65557e10	−0.891702	−0.445851	−	0.895107i	$-0.647099\pi$
$983$	−2.65557e10	−0.891702	−0.445851	+	0.895107i	$0.647099\pi$
$984$	0	0
$985$	−4.97832e10	−1.65980
$986$	2.12944e10	0.707451
$987$	0	0
$988$	−9.07282e9	−0.299291
$989$	5.76886e10	1.89628
$990$	0	0
$991$	6.89473e9	0.225040	0.112520	−	0.993649i	$-0.464108\pi$
$991$	6.89473e9	0.225040	0.112520	+	0.993649i	$0.464108\pi$
$992$	−3.83621e9	−0.124770
$993$	0	0
$994$	6.37888e9	0.206012
$995$	−3.60415e10	−1.15990
$996$	0	0
$997$	−5.13227e8	−0.0164012	−0.00820062	−	0.999966i	$-0.502610\pi$
$997$	−5.13227e8	−0.0164012	−0.00820062	+	0.999966i	$0.502610\pi$
$998$	−1.48787e10	−0.473816
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.8.a.e.1.2		6
3.2	odd	2		81.8.a.c.1.5		6
9.2	odd	6		27.8.c.a.10.2		12
9.4	even	3		9.8.c.a.7.5	yes	12
9.5	odd	6		27.8.c.a.19.2		12
9.7	even	3		9.8.c.a.4.5	✓	12
36.7	odd	6		144.8.i.c.49.1		12
36.11	even	6		432.8.i.c.145.5		12
36.23	even	6		432.8.i.c.289.5		12
36.31	odd	6		144.8.i.c.97.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.8.c.a.4.5	✓	12	9.7	even	3
9.8.c.a.7.5	yes	12	9.4	even	3
27.8.c.a.10.2		12	9.2	odd	6
27.8.c.a.19.2		12	9.5	odd	6
81.8.a.c.1.5		6	3.2	odd	2
81.8.a.e.1.2		6	1.1	even	1	trivial
144.8.i.c.49.1		12	36.7	odd	6
144.8.i.c.97.1		12	36.31	odd	6
432.8.i.c.145.5		12	36.11	even	6
432.8.i.c.289.5		12	36.23	even	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:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q-14.2319 q^{2} +74.5469 q^{4} +290.607 q^{5} +1111.88 q^{7} +760.739 q^{8} -4135.89 q^{10} +4490.71 q^{11} +2436.58 q^{13} -15824.2 q^{14} -20368.8 q^{16} +15905.4 q^{17} -49949.6 q^{19} +21663.9 q^{20} -63911.4 q^{22} +69385.1 q^{23} +6327.45 q^{25} -34677.2 q^{26} +82887.2 q^{28} -94071.6 q^{29} -19927.2 q^{31} +192512. q^{32} -226364. q^{34} +323120. q^{35} +331750. q^{37} +710878. q^{38} +221076. q^{40} +242266. q^{41} +831426. q^{43} +334769. q^{44} -987481. q^{46} -160011. q^{47} +412732. q^{49} -90051.7 q^{50} +181639. q^{52} +311589. q^{53} +1.30503e6 q^{55} +845850. q^{56} +1.33882e6 q^{58} -312353. q^{59} -57447.8 q^{61} +283601. q^{62} -132603. q^{64} +708087. q^{65} -4.10199e6 q^{67} +1.18570e6 q^{68} -4.59861e6 q^{70} -403110. q^{71} -823496. q^{73} -4.72144e6 q^{74} -3.72359e6 q^{76} +4.99313e6 q^{77} +978827. q^{79} -5.91931e6 q^{80} -3.44791e6 q^{82} -3.70408e6 q^{83} +4.62222e6 q^{85} -1.18328e7 q^{86} +3.41626e6 q^{88} +2.09023e6 q^{89} +2.70918e6 q^{91} +5.17244e6 q^{92} +2.27726e6 q^{94} -1.45157e7 q^{95} +3.50249e6 q^{97} -5.87396e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 9 q^{2} + 321 q^{4} + 180 q^{5} + 84 q^{7} + 2961 q^{8} + 126 q^{10} + 8460 q^{11} + 1848 q^{13} + 16272 q^{14} + 12417 q^{16} + 15282 q^{17} + 12216 q^{19} + 40788 q^{20} + 35001 q^{22} + 51588 q^{23}+ \cdots - 47916657 q^{98}+O(q^{100}) \) 6 * q + 9 * q^2 + 321 * q^4 + 180 * q^5 + 84 * q^7 + 2961 * q^8 + 126 * q^10 + 8460 * q^11 + 1848 * q^13 + 16272 * q^14 + 12417 * q^16 + 15282 * q^17 + 12216 * q^19 + 40788 * q^20 + 35001 * q^22 + 51588 * q^23 - 4746 * q^25 + 268236 * q^26 + 37758 * q^28 + 414648 * q^29 - 8196 * q^31 + 1048977 * q^32 + 106623 * q^34 + 1105308 * q^35 + 69672 * q^37 + 1952685 * q^38 - 305496 * q^40 + 1731582 * q^41 - 408372 * q^43 + 2584557 * q^44 - 842004 * q^46 + 1631484 * q^47 + 179010 * q^49 + 1654461 * q^50 - 681594 * q^52 + 1417824 * q^53 - 8028 * q^55 + 1784466 * q^56 + 948384 * q^58 + 2055636 * q^59 + 2723196 * q^61 - 513414 * q^62 + 3589089 * q^64 + 1387620 * q^65 - 3806556 * q^67 - 2142639 * q^68 - 953442 * q^70 + 1204200 * q^71 - 5335026 * q^73 - 9846504 * q^74 + 6727827 * q^76 - 3478824 * q^77 - 6020916 * q^79 - 19036224 * q^80 + 4701501 * q^82 - 9605052 * q^83 + 1698624 * q^85 - 34278561 * q^86 + 16459029 * q^88 - 12315132 * q^89 + 6785052 * q^91 - 39143394 * q^92 - 12602808 * q^94 - 10422072 * q^95 - 9977226 * q^97 - 47916657 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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