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Properties

Label 81.10.e.a
Level $81$
Weight $10$
Character orbit 81.e
Analytic conductor $41.718$
Analytic rank $0$
Dimension $156$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,10,Mod(10,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.10");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 81.e (of order \(9\), degree \(6\), not minimal)

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: no
Analytic conductor: \(41.7179027293\)
Analytic rank: \(0\)
Dimension: \(156\)
Relative dimension: \(26\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 156 q + 6 q^{2} - 6 q^{4} - 2382 q^{5} - 6 q^{7} - 36861 q^{8} - 3 q^{10} + 121767 q^{11} - 6 q^{13} + 720417 q^{14} + 1530 q^{16} - 1002249 q^{17} - 3 q^{19} + 8448573 q^{20} - 1585014 q^{22} - 9316482 q^{23}+ \cdots - 6901039926 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −7.14775 − 40.5369i 0 −1111.03 + 404.381i 610.019 + 511.867i 0 4530.75 + 1649.06i 13796.1 + 23895.6i 0 16389.2 − 28387.0i
10.2 −6.66051 − 37.7736i 0 −901.361 + 328.069i −1267.19 − 1063.30i 0 −6241.55 − 2271.74i 8576.64 + 14855.2i 0 −31724.5 + 54948.4i
10.3 −6.61278 − 37.5029i 0 −881.618 + 320.883i −1241.60 − 1041.82i 0 9160.78 + 3334.25i 8115.13 + 14055.8i 0 −30861.0 + 53452.8i
10.4 −6.40153 − 36.3049i 0 −795.940 + 289.699i 170.489 + 143.057i 0 −5644.86 − 2054.56i 6175.28 + 10695.9i 0 4102.29 − 7105.37i
10.5 −6.21254 − 35.2331i 0 −721.652 + 262.660i 1770.10 + 1485.29i 0 1506.43 + 548.294i 4578.80 + 7930.71i 0 41334.5 − 71593.4i
10.6 −3.96930 − 22.5110i 0 −9.86871 + 3.59192i −1862.39 − 1562.73i 0 7319.38 + 2664.03i −5731.70 − 9927.59i 0 −27786.2 + 48127.2i
10.7 −3.81300 − 21.6246i 0 28.0377 − 10.2049i −47.8760 − 40.1728i 0 −9290.87 − 3381.60i −5948.89 − 10303.8i 0 −686.169 + 1188.48i
10.8 −3.56307 − 20.2072i 0 85.4874 − 31.1149i 748.976 + 628.466i 0 −3373.46 − 1227.84i −6186.19 − 10714.8i 0 10030.9 − 17374.0i
10.9 −3.47036 − 19.6814i 0 105.808 − 38.5110i 245.404 + 205.919i 0 4352.22 + 1584.08i −6241.31 − 10810.3i 0 3201.13 − 5544.51i
10.10 −2.61607 − 14.8364i 0 267.846 − 97.4881i −1174.80 − 985.775i 0 −3599.21 − 1310.01i −6003.80 − 10398.9i 0 −11552.0 + 20008.7i
10.11 −2.60890 − 14.7958i 0 269.014 − 97.9130i 1952.85 + 1638.63i 0 7550.25 + 2748.07i −5996.68 − 10386.6i 0 19150.1 − 33168.9i
10.12 −0.0974639 − 0.552746i 0 480.827 − 175.007i −567.016 − 475.783i 0 1109.98 + 404.000i −287.283 − 497.589i 0 −207.723 + 359.787i
10.13 0.622466 + 3.53018i 0 469.048 − 170.719i −22.7101 − 19.0560i 0 10919.6 + 3974.40i 1812.30 + 3139.00i 0 53.1349 − 92.0324i
10.14 0.725207 + 4.11285i 0 464.733 − 169.149i 1430.66 + 1200.46i 0 −10647.4 − 3875.35i 2101.85 + 3640.50i 0 −3899.81 + 6754.67i
10.15 0.907148 + 5.14469i 0 455.478 − 165.780i 1094.11 + 918.064i 0 184.507 + 67.1550i 2603.43 + 4509.28i 0 −3730.64 + 6461.66i
10.16 1.50531 + 8.53705i 0 410.507 − 149.412i −1797.12 − 1507.96i 0 1.39853 + 0.509023i 4112.68 + 7123.37i 0 10168.3 − 17612.1i
10.17 1.82927 + 10.3743i 0 376.843 − 137.160i −469.551 − 394.000i 0 −6352.63 − 2312.17i 4809.06 + 8329.54i 0 3228.53 − 5591.99i
10.18 3.18108 + 18.0408i 0 165.772 − 60.3360i 1375.78 + 1154.42i 0 2117.11 + 770.566i 6305.53 + 10921.5i 0 −16450.1 + 28492.4i
10.19 4.45544 + 25.2681i 0 −137.502 + 50.0467i 84.0511 + 70.5273i 0 7618.26 + 2772.82i 4691.20 + 8125.40i 0 −1407.60 + 2438.04i
10.20 4.55647 + 25.8410i 0 −165.874 + 60.3733i −1409.37 − 1182.60i 0 −8682.75 − 3160.26i 4401.44 + 7623.52i 0 24137.8 − 41807.9i
See next 80 embeddings (of 156 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.10.e.a 156
3.b odd 2 1 27.10.e.a ✓ 156
27.e even 9 1 inner 81.10.e.a 156
27.f odd 18 1 27.10.e.a ✓ 156
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.10.e.a ✓ 156 3.b odd 2 1
27.10.e.a ✓ 156 27.f odd 18 1
81.10.e.a 156 1.a even 1 1 trivial
81.10.e.a 156 27.e even 9 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(81, [\chi])\).