Properties

Label 99.8.d.a
Level $99$
Weight $8$
Character orbit 99.d
Analytic conductor $30.926$
Analytic rank $0$
Dimension $28$
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] = [99,8,Mod(98,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.98");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 99.d (of order \(2\), degree \(1\), 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: \(30.9261175229\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 1792 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 1792 q^{4} + 136936 q^{16} + 69696 q^{22} - 877308 q^{25} - 21616 q^{31} + 161304 q^{34} + 408464 q^{37} - 5023892 q^{49} + 439472 q^{55} + 2851344 q^{58} - 15376208 q^{64} - 17729872 q^{67} + 3438408 q^{70} + 54779784 q^{82} + 26540976 q^{88} - 19094256 q^{91} + 10561400 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
98.1 −20.6546 0 298.612 288.556i 0 754.006i −3523.91 0 5960.01i
98.2 −20.6546 0 298.612 288.556i 0 754.006i −3523.91 0 5960.01i
98.3 −19.4265 0 249.387 529.951i 0 235.028i −2358.12 0 10295.1i
98.4 −19.4265 0 249.387 529.951i 0 235.028i −2358.12 0 10295.1i
98.5 −16.3395 0 138.981 19.0280i 0 1638.46i −179.422 0 310.908i
98.6 −16.3395 0 138.981 19.0280i 0 1638.46i −179.422 0 310.908i
98.7 −13.8089 0 62.6859 205.633i 0 449.641i 901.916 0 2839.57i
98.8 −13.8089 0 62.6859 205.633i 0 449.641i 901.916 0 2839.57i
98.9 −7.31902 0 −74.4319 81.1133i 0 179.404i 1481.60 0 593.670i
98.10 −7.31902 0 −74.4319 81.1133i 0 179.404i 1481.60 0 593.670i
98.11 −5.16121 0 −101.362 347.376i 0 1160.84i 1183.79 0 1792.88i
98.12 −5.16121 0 −101.362 347.376i 0 1160.84i 1183.79 0 1792.88i
98.13 −1.45890 0 −125.872 481.864i 0 1459.64i 370.374 0 702.992i
98.14 −1.45890 0 −125.872 481.864i 0 1459.64i 370.374 0 702.992i
98.15 1.45890 0 −125.872 481.864i 0 1459.64i −370.374 0 702.992i
98.16 1.45890 0 −125.872 481.864i 0 1459.64i −370.374 0 702.992i
98.17 5.16121 0 −101.362 347.376i 0 1160.84i −1183.79 0 1792.88i
98.18 5.16121 0 −101.362 347.376i 0 1160.84i −1183.79 0 1792.88i
98.19 7.31902 0 −74.4319 81.1133i 0 179.404i −1481.60 0 593.670i
98.20 7.31902 0 −74.4319 81.1133i 0 179.404i −1481.60 0 593.670i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 98.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.8.d.a 28
3.b odd 2 1 inner 99.8.d.a 28
11.b odd 2 1 inner 99.8.d.a 28
33.d even 2 1 inner 99.8.d.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.8.d.a 28 1.a even 1 1 trivial
99.8.d.a 28 3.b odd 2 1 inner
99.8.d.a 28 11.b odd 2 1 inner
99.8.d.a 28 33.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(99, [\chi])\).