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Properties

Label 945.2.g.b
Level $945$
Weight $2$
Character orbit 945.g
Analytic conductor $7.546$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(944,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.944");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.g (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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{4} + 32 q^{16} + 20 q^{25} - 24 q^{46} + 8 q^{49} + 56 q^{64} + 40 q^{79} + 76 q^{85} + 40 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
944.1 −2.58501 0 4.68227 −2.03254 − 0.932083i 0 −0.737868 − 2.54078i −6.93370 0 5.25414 + 2.40944i
944.2 −2.58501 0 4.68227 −2.03254 + 0.932083i 0 −0.737868 + 2.54078i −6.93370 0 5.25414 − 2.40944i
944.3 −2.58501 0 4.68227 2.03254 − 0.932083i 0 0.737868 + 2.54078i −6.93370 0 −5.25414 + 2.40944i
944.4 −2.58501 0 4.68227 2.03254 + 0.932083i 0 0.737868 − 2.54078i −6.93370 0 −5.25414 − 2.40944i
944.5 −1.92418 0 1.70246 −1.22993 − 1.86742i 0 2.62359 + 0.341697i 0.572523 0 2.36661 + 3.59325i
944.6 −1.92418 0 1.70246 −1.22993 + 1.86742i 0 2.62359 − 0.341697i 0.572523 0 2.36661 − 3.59325i
944.7 −1.92418 0 1.70246 1.22993 − 1.86742i 0 −2.62359 − 0.341697i 0.572523 0 −2.36661 + 3.59325i
944.8 −1.92418 0 1.70246 1.22993 + 1.86742i 0 −2.62359 + 0.341697i 0.572523 0 −2.36661 − 3.59325i
944.9 −1.10299 0 −0.783408 −0.826134 − 2.07786i 0 1.13699 + 2.38899i 3.07008 0 0.911219 + 2.29186i
944.10 −1.10299 0 −0.783408 −0.826134 + 2.07786i 0 1.13699 − 2.38899i 3.07008 0 0.911219 − 2.29186i
944.11 −1.10299 0 −0.783408 0.826134 − 2.07786i 0 −1.13699 − 2.38899i 3.07008 0 −0.911219 + 2.29186i
944.12 −1.10299 0 −0.783408 0.826134 + 2.07786i 0 −1.13699 + 2.38899i 3.07008 0 −0.911219 − 2.29186i
944.13 −0.631409 0 −1.60132 −2.21891 − 0.276494i 0 −2.40407 + 1.10473i 2.27391 0 1.40104 + 0.174581i
944.14 −0.631409 0 −1.60132 −2.21891 + 0.276494i 0 −2.40407 − 1.10473i 2.27391 0 1.40104 − 0.174581i
944.15 −0.631409 0 −1.60132 2.21891 − 0.276494i 0 2.40407 − 1.10473i 2.27391 0 −1.40104 + 0.174581i
944.16 −0.631409 0 −1.60132 2.21891 + 0.276494i 0 2.40407 + 1.10473i 2.27391 0 −1.40104 − 0.174581i
944.17 0.631409 0 −1.60132 −2.21891 − 0.276494i 0 2.40407 + 1.10473i −2.27391 0 −1.40104 − 0.174581i
944.18 0.631409 0 −1.60132 −2.21891 + 0.276494i 0 2.40407 − 1.10473i −2.27391 0 −1.40104 + 0.174581i
944.19 0.631409 0 −1.60132 2.21891 − 0.276494i 0 −2.40407 − 1.10473i −2.27391 0 1.40104 − 0.174581i
944.20 0.631409 0 −1.60132 2.21891 + 0.276494i 0 −2.40407 + 1.10473i −2.27391 0 1.40104 + 0.174581i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 944.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.g.b ✓ 32
3.b odd 2 1 inner 945.2.g.b ✓ 32
5.b even 2 1 inner 945.2.g.b ✓ 32
7.b odd 2 1 inner 945.2.g.b ✓ 32
15.d odd 2 1 inner 945.2.g.b ✓ 32
21.c even 2 1 inner 945.2.g.b ✓ 32
35.c odd 2 1 inner 945.2.g.b ✓ 32
105.g even 2 1 inner 945.2.g.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.g.b ✓ 32 1.a even 1 1 trivial
945.2.g.b ✓ 32 3.b odd 2 1 inner
945.2.g.b ✓ 32 5.b even 2 1 inner
945.2.g.b ✓ 32 7.b odd 2 1 inner
945.2.g.b ✓ 32 15.d odd 2 1 inner
945.2.g.b ✓ 32 21.c even 2 1 inner
945.2.g.b ✓ 32 35.c odd 2 1 inner
945.2.g.b ✓ 32 105.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 12T_{2}^{6} + 42T_{2}^{4} - 45T_{2}^{2} + 12 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display