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Properties

Label 288.8.f.a
Level $288$
Weight $8$
Character orbit 288.f
Analytic conductor $89.967$
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] = [288,8,Mod(143,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("288.143");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 288.f (of order \(2\), degree \(1\), 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: \(89.9668873394\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 72)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 121168 q^{19} + 437500 q^{25} - 1505696 q^{43} - 2272076 q^{49} + 776272 q^{67} - 2534128 q^{73} + 3406992 q^{91} - 26311456 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1 0 0 0 −527.660 0 − 1312.48i 0 0 0
143.2 0 0 0 −527.660 0 1312.48i 0 0 0
143.3 0 0 0 −412.177 0 359.164i 0 0 0
143.4 0 0 0 −412.177 0 − 359.164i 0 0 0
143.5 0 0 0 −294.266 0 − 328.633i 0 0 0
143.6 0 0 0 −294.266 0 328.633i 0 0 0
143.7 0 0 0 −238.250 0 − 737.398i 0 0 0
143.8 0 0 0 −238.250 0 737.398i 0 0 0
143.9 0 0 0 −168.010 0 − 1488.58i 0 0 0
143.10 0 0 0 −168.010 0 1488.58i 0 0 0
143.11 0 0 0 −166.398 0 750.222i 0 0 0
143.12 0 0 0 −166.398 0 − 750.222i 0 0 0
143.13 0 0 0 −93.0830 0 − 1025.06i 0 0 0
143.14 0 0 0 −93.0830 0 1025.06i 0 0 0
143.15 0 0 0 93.0830 0 − 1025.06i 0 0 0
143.16 0 0 0 93.0830 0 1025.06i 0 0 0
143.17 0 0 0 166.398 0 750.222i 0 0 0
143.18 0 0 0 166.398 0 − 750.222i 0 0 0
143.19 0 0 0 168.010 0 − 1488.58i 0 0 0
143.20 0 0 0 168.010 0 1488.58i 0 0 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.8.f.a 28
3.b odd 2 1 inner 288.8.f.a 28
4.b odd 2 1 72.8.f.a ✓ 28
8.b even 2 1 72.8.f.a ✓ 28
8.d odd 2 1 inner 288.8.f.a 28
12.b even 2 1 72.8.f.a ✓ 28
24.f even 2 1 inner 288.8.f.a 28
24.h odd 2 1 72.8.f.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.8.f.a ✓ 28 4.b odd 2 1
72.8.f.a ✓ 28 8.b even 2 1
72.8.f.a ✓ 28 12.b even 2 1
72.8.f.a ✓ 28 24.h odd 2 1
288.8.f.a 28 1.a even 1 1 trivial
288.8.f.a 28 3.b odd 2 1 inner
288.8.f.a 28 8.d odd 2 1 inner
288.8.f.a 28 24.f even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(288, [\chi])\).