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Properties

Label 81.8.a.e.1.4
Level $81$
Weight $8$
Character 81.1
Self dual yes
Analytic conductor $25.303$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,8,Mod(1,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 81.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.3031870642\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 401x^{4} - 1212x^{3} + 17752x^{2} + 15108x - 22632 \) Copy content Toggle raw display
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 1.4
Root \(20.3986\) of defining polynomial
Character \(\chi\) \(=\) 81.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.18840 q^{2} -89.7037 q^{4} -335.903 q^{5} -884.050 q^{7} -1347.24 q^{8} -2078.70 q^{10} +4213.90 q^{11} +12515.3 q^{13} -5470.85 q^{14} +3144.84 q^{16} -742.627 q^{17} +9111.12 q^{19} +30131.8 q^{20} +26077.3 q^{22} +45302.5 q^{23} +34705.9 q^{25} +77449.4 q^{26} +79302.6 q^{28} -34583.7 q^{29} -277546. q^{31} +191908. q^{32} -4595.67 q^{34} +296955. q^{35} -209817. q^{37} +56383.2 q^{38} +452541. q^{40} +106932. q^{41} +17025.8 q^{43} -378003. q^{44} +280350. q^{46} +1.35148e6 q^{47} -41998.3 q^{49} +214774. q^{50} -1.12267e6 q^{52} +1.83419e6 q^{53} -1.41546e6 q^{55} +1.19103e6 q^{56} -214018. q^{58} +871149. q^{59} +974577. q^{61} -1.71756e6 q^{62} +785063. q^{64} -4.20392e6 q^{65} -286453. q^{67} +66616.4 q^{68} +1.83768e6 q^{70} +967923. q^{71} +4.50531e6 q^{73} -1.29843e6 q^{74} -817301. q^{76} -3.72530e6 q^{77} +2.45529e6 q^{79} -1.05636e6 q^{80} +661738. q^{82} +1.39188e6 q^{83} +249451. q^{85} +105362. q^{86} -5.67713e6 q^{88} -7.88308e6 q^{89} -1.10641e7 q^{91} -4.06380e6 q^{92} +8.36347e6 q^{94} -3.06045e6 q^{95} -6.87447e6 q^{97} -259902. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{2} + 321 q^{4} + 180 q^{5} + 84 q^{7} + 2961 q^{8} + 126 q^{10} + 8460 q^{11} + 1848 q^{13} + 16272 q^{14} + 12417 q^{16} + 15282 q^{17} + 12216 q^{19} + 40788 q^{20} + 35001 q^{22} + 51588 q^{23}+ \cdots - 47916657 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.18840 0.546982 0.273491 − 0.961875i \(-0.411822\pi\)
0.273491 + 0.961875i \(0.411822\pi\)
\(3\) 0 0
\(4\) −89.7037 −0.700810
\(5\) −335.903 −1.20176 −0.600882 − 0.799338i \(-0.705184\pi\)
−0.600882 + 0.799338i \(0.705184\pi\)
\(6\) 0 0
\(7\) −884.050 −0.974168 −0.487084 − 0.873355i \(-0.661939\pi\)
−0.487084 + 0.873355i \(0.661939\pi\)
\(8\) −1347.24 −0.930313
\(9\) 0 0
\(10\) −2078.70 −0.657343
\(11\) 4213.90 0.954575 0.477288 − 0.878747i \(-0.341620\pi\)
0.477288 + 0.878747i \(0.341620\pi\)
\(12\) 0 0
\(13\) 12515.3 1.57993 0.789966 − 0.613151i \(-0.210098\pi\)
0.789966 + 0.613151i \(0.210098\pi\)
\(14\) −5470.85 −0.532853
\(15\) 0 0
\(16\) 3144.84 0.191946
\(17\) −742.627 −0.0366606 −0.0183303 − 0.999832i \(-0.505835\pi\)
−0.0183303 + 0.999832i \(0.505835\pi\)
\(18\) 0 0
\(19\) 9111.12 0.304743 0.152372 − 0.988323i \(-0.451309\pi\)
0.152372 + 0.988323i \(0.451309\pi\)
\(20\) 30131.8 0.842208
\(21\) 0 0
\(22\) 26077.3 0.522136
\(23\) 45302.5 0.776380 0.388190 − 0.921579i \(-0.373100\pi\)
0.388190 + 0.921579i \(0.373100\pi\)
\(24\) 0 0
\(25\) 34705.9 0.444235
\(26\) 77449.4 0.864195
\(27\) 0 0
\(28\) 79302.6 0.682707
\(29\) −34583.7 −0.263317 −0.131658 − 0.991295i \(-0.542030\pi\)
−0.131658 + 0.991295i \(0.542030\pi\)
\(30\) 0 0
\(31\) −277546. −1.67328 −0.836639 − 0.547754i \(-0.815483\pi\)
−0.836639 + 0.547754i \(0.815483\pi\)
\(32\) 191908. 1.03530
\(33\) 0 0
\(34\) −4595.67 −0.0200527
\(35\) 296955. 1.17072
\(36\) 0 0
\(37\) −209817. −0.680981 −0.340491 − 0.940248i \(-0.610593\pi\)
−0.340491 + 0.940248i \(0.610593\pi\)
\(38\) 56383.2 0.166689
\(39\) 0 0
\(40\) 452541. 1.11802
\(41\) 106932. 0.242306 0.121153 − 0.992634i \(-0.461341\pi\)
0.121153 + 0.992634i \(0.461341\pi\)
\(42\) 0 0
\(43\) 17025.8 0.0326564 0.0163282 − 0.999867i \(-0.494802\pi\)
0.0163282 + 0.999867i \(0.494802\pi\)
\(44\) −378003. −0.668976
\(45\) 0 0
\(46\) 280350. 0.424666
\(47\) 1.35148e6 1.89874 0.949371 − 0.314158i \(-0.101722\pi\)
0.949371 + 0.314158i \(0.101722\pi\)
\(48\) 0 0
\(49\) −41998.3 −0.0509971
\(50\) 214774. 0.242989
\(51\) 0 0
\(52\) −1.12267e6 −1.10723
\(53\) 1.83419e6 1.69230 0.846152 − 0.532942i \(-0.178914\pi\)
0.846152 + 0.532942i \(0.178914\pi\)
\(54\) 0 0
\(55\) −1.41546e6 −1.14717
\(56\) 1.19103e6 0.906281
\(57\) 0 0
\(58\) −214018. −0.144029
\(59\) 871149. 0.552218 0.276109 − 0.961126i \(-0.410955\pi\)
0.276109 + 0.961126i \(0.410955\pi\)
\(60\) 0 0
\(61\) 974577. 0.549746 0.274873 − 0.961481i \(-0.411364\pi\)
0.274873 + 0.961481i \(0.411364\pi\)
\(62\) −1.71756e6 −0.915254
\(63\) 0 0
\(64\) 785063. 0.374347
\(65\) −4.20392e6 −1.89870
\(66\) 0 0
\(67\) −286453. −0.116357 −0.0581785 − 0.998306i \(-0.518529\pi\)
−0.0581785 + 0.998306i \(0.518529\pi\)
\(68\) 66616.4 0.0256921
\(69\) 0 0
\(70\) 1.83768e6 0.640363
\(71\) 967923. 0.320950 0.160475 − 0.987040i \(-0.448697\pi\)
0.160475 + 0.987040i \(0.448697\pi\)
\(72\) 0 0
\(73\) 4.50531e6 1.35548 0.677742 − 0.735299i \(-0.262958\pi\)
0.677742 + 0.735299i \(0.262958\pi\)
\(74\) −1.29843e6 −0.372485
\(75\) 0 0
\(76\) −817301. −0.213567
\(77\) −3.72530e6 −0.929917
\(78\) 0 0
\(79\) 2.45529e6 0.560284 0.280142 − 0.959959i \(-0.409618\pi\)
0.280142 + 0.959959i \(0.409618\pi\)
\(80\) −1.05636e6 −0.230673
\(81\) 0 0
\(82\) 661738. 0.132537
\(83\) 1.39188e6 0.267195 0.133598 − 0.991036i \(-0.457347\pi\)
0.133598 + 0.991036i \(0.457347\pi\)
\(84\) 0 0
\(85\) 249451. 0.0440574
\(86\) 105362. 0.0178625
\(87\) 0 0
\(88\) −5.67713e6 −0.888054
\(89\) −7.88308e6 −1.18531 −0.592653 − 0.805458i \(-0.701920\pi\)
−0.592653 + 0.805458i \(0.701920\pi\)
\(90\) 0 0
\(91\) −1.10641e7 −1.53912
\(92\) −4.06380e6 −0.544096
\(93\) 0 0
\(94\) 8.36347e6 1.03858
\(95\) −3.06045e6 −0.366229
\(96\) 0 0
\(97\) −6.87447e6 −0.764783 −0.382391 − 0.924000i \(-0.624899\pi\)
−0.382391 + 0.924000i \(0.624899\pi\)
\(98\) −259902. −0.0278945
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 81.8.a.e.1.4 6
3.2 odd 2 81.8.a.c.1.3 6
9.2 odd 6 27.8.c.a.10.4 12
9.4 even 3 9.8.c.a.7.3 yes 12
9.5 odd 6 27.8.c.a.19.4 12
9.7 even 3 9.8.c.a.4.3 ✓ 12
36.7 odd 6 144.8.i.c.49.2 12
36.11 even 6 432.8.i.c.145.1 12
36.23 even 6 432.8.i.c.289.1 12
36.31 odd 6 144.8.i.c.97.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.3 ✓ 12 9.7 even 3
9.8.c.a.7.3 yes 12 9.4 even 3
27.8.c.a.10.4 12 9.2 odd 6
27.8.c.a.19.4 12 9.5 odd 6
81.8.a.c.1.3 6 3.2 odd 2
81.8.a.e.1.4 6 1.1 even 1 trivial
144.8.i.c.49.2 12 36.7 odd 6
144.8.i.c.97.2 12 36.31 odd 6
432.8.i.c.145.1 12 36.11 even 6
432.8.i.c.289.1 12 36.23 even 6