Properties

Label 81.9.d.g
Level $81$
Weight $9$
Character orbit 81.d
Analytic conductor $32.998$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,9,Mod(26,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.26");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.d (of order \(6\), degree \(2\), 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: \(32.9976674150\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2048 q^{4} - 3692 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2048 q^{4} - 3692 q^{7} + 21504 q^{10} - 63860 q^{13} - 95116 q^{16} + 370216 q^{19} + 691980 q^{22} + 541712 q^{25} - 1994264 q^{28} - 571136 q^{31} - 1027656 q^{34} + 8708536 q^{37} + 2973768 q^{40} - 5453084 q^{43} - 8468088 q^{46} - 14602560 q^{49} + 25384960 q^{52} - 28905144 q^{55} - 1213068 q^{58} - 31037996 q^{61} + 48506000 q^{64} + 62356828 q^{67} - 11075124 q^{70} - 42547328 q^{73} - 69593876 q^{76} + 151045036 q^{79} - 80402280 q^{82} - 160995564 q^{85} - 366976068 q^{88} + 747361592 q^{91} + 544741584 q^{94} + 133878688 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −25.8080 14.9003i 0 316.037 + 547.392i −153.960 + 88.8891i 0 −1695.41 + 2936.53i 11207.2i 0 5297.89
26.2 −22.2403 12.8404i 0 201.753 + 349.447i 176.989 102.185i 0 1851.21 3206.39i 3788.08i 0 −5248.38
26.3 −20.8866 12.0589i 0 162.833 + 282.035i 378.029 218.255i 0 −275.376 + 476.966i 1680.18i 0 −10527.6
26.4 −19.4167 11.2102i 0 123.338 + 213.628i −935.412 + 540.060i 0 503.318 871.773i 209.045i 0 24216.8
26.5 −12.9399 7.47088i 0 −16.3720 28.3571i 943.515 544.739i 0 −654.245 + 1133.19i 4314.34i 0 −16278.7
26.6 −8.67350 5.00765i 0 −77.8470 134.835i −514.754 + 297.193i 0 −2204.76 + 3818.75i 4123.23i 0 5952.96
26.7 −7.89343 4.55727i 0 −86.4626 149.758i 157.475 90.9184i 0 436.536 756.103i 3909.46i 0 −1657.36
26.8 −5.00808 2.89142i 0 −111.279 192.742i −542.191 + 313.034i 0 1115.72 1932.48i 2767.43i 0 3620.45
26.9 5.00808 + 2.89142i 0 −111.279 192.742i 542.191 313.034i 0 1115.72 1932.48i 2767.43i 0 3620.45
26.10 7.89343 + 4.55727i 0 −86.4626 149.758i −157.475 + 90.9184i 0 436.536 756.103i 3909.46i 0 −1657.36
26.11 8.67350 + 5.00765i 0 −77.8470 134.835i 514.754 297.193i 0 −2204.76 + 3818.75i 4123.23i 0 5952.96
26.12 12.9399 + 7.47088i 0 −16.3720 28.3571i −943.515 + 544.739i 0 −654.245 + 1133.19i 4314.34i 0 −16278.7
26.13 19.4167 + 11.2102i 0 123.338 + 213.628i 935.412 540.060i 0 503.318 871.773i 209.045i 0 24216.8
26.14 20.8866 + 12.0589i 0 162.833 + 282.035i −378.029 + 218.255i 0 −275.376 + 476.966i 1680.18i 0 −10527.6
26.15 22.2403 + 12.8404i 0 201.753 + 349.447i −176.989 + 102.185i 0 1851.21 3206.39i 3788.08i 0 −5248.38
26.16 25.8080 + 14.9003i 0 316.037 + 547.392i 153.960 88.8891i 0 −1695.41 + 2936.53i 11207.2i 0 5297.89
53.1 −25.8080 + 14.9003i 0 316.037 547.392i −153.960 88.8891i 0 −1695.41 2936.53i 11207.2i 0 5297.89
53.2 −22.2403 + 12.8404i 0 201.753 349.447i 176.989 + 102.185i 0 1851.21 + 3206.39i 3788.08i 0 −5248.38
53.3 −20.8866 + 12.0589i 0 162.833 282.035i 378.029 + 218.255i 0 −275.376 476.966i 1680.18i 0 −10527.6
53.4 −19.4167 + 11.2102i 0 123.338 213.628i −935.412 540.060i 0 503.318 + 871.773i 209.045i 0 24216.8
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.9.d.g 32
3.b odd 2 1 inner 81.9.d.g 32
9.c even 3 1 81.9.b.b 16
9.c even 3 1 inner 81.9.d.g 32
9.d odd 6 1 81.9.b.b 16
9.d odd 6 1 inner 81.9.d.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.9.b.b 16 9.c even 3 1
81.9.b.b 16 9.d odd 6 1
81.9.d.g 32 1.a even 1 1 trivial
81.9.d.g 32 3.b odd 2 1 inner
81.9.d.g 32 9.c even 3 1 inner
81.9.d.g 32 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 3072 T_{2}^{30} + 5659875 T_{2}^{28} - 6840915480 T_{2}^{26} + 6132249070245 T_{2}^{24} + \cdots + 11\!\cdots\!96 \) acting on \(S_{9}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display