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Properties

Label 1288.3.k.a
Level $1288$
Weight $3$
Character orbit 1288.k
Analytic conductor $35.095$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1288,3,Mod(505,1288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1288.505");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1288.k (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: \(35.0954580496\)
Analytic rank: \(0\)
Dimension: \(72\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 216 q^{9} + 32 q^{23} - 392 q^{25} - 192 q^{27} + 16 q^{31} - 16 q^{39} - 32 q^{41} + 288 q^{47} - 504 q^{49} + 400 q^{55} + 192 q^{59} + 280 q^{69} + 128 q^{71} + 224 q^{73} + 464 q^{75} + 1144 q^{81} + 176 q^{85} + 656 q^{87} - 672 q^{93} - 304 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1 0 −5.75276 0 − 5.27290i 0 − 2.64575i 0 24.0942 0
505.2 0 −5.75276 0 5.27290i 0 2.64575i 0 24.0942 0
505.3 0 −5.65025 0 9.71356i 0 − 2.64575i 0 22.9253 0
505.4 0 −5.65025 0 − 9.71356i 0 2.64575i 0 22.9253 0
505.5 0 −5.64165 0 − 0.728697i 0 2.64575i 0 22.8283 0
505.6 0 −5.64165 0 0.728697i 0 − 2.64575i 0 22.8283 0
505.7 0 −5.02575 0 0.0734019i 0 − 2.64575i 0 16.2581 0
505.8 0 −5.02575 0 − 0.0734019i 0 2.64575i 0 16.2581 0
505.9 0 −4.26924 0 2.65037i 0 2.64575i 0 9.22643 0
505.10 0 −4.26924 0 − 2.65037i 0 − 2.64575i 0 9.22643 0
505.11 0 −4.16857 0 − 8.93510i 0 2.64575i 0 8.37701 0
505.12 0 −4.16857 0 8.93510i 0 − 2.64575i 0 8.37701 0
505.13 0 −3.78357 0 − 4.93176i 0 − 2.64575i 0 5.31542 0
505.14 0 −3.78357 0 4.93176i 0 2.64575i 0 5.31542 0
505.15 0 −3.76058 0 − 8.18528i 0 − 2.64575i 0 5.14196 0
505.16 0 −3.76058 0 8.18528i 0 2.64575i 0 5.14196 0
505.17 0 −3.28116 0 4.07711i 0 − 2.64575i 0 1.76599 0
505.18 0 −3.28116 0 − 4.07711i 0 2.64575i 0 1.76599 0
505.19 0 −2.68180 0 − 2.05390i 0 2.64575i 0 −1.80797 0
505.20 0 −2.68180 0 2.05390i 0 − 2.64575i 0 −1.80797 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 505.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1288.3.k.a ✓ 72
23.b odd 2 1 inner 1288.3.k.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.3.k.a ✓ 72 1.a even 1 1 trivial
1288.3.k.a ✓ 72 23.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1288, [\chi])\).