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Properties

Label 936.2.g.f
Level $936$
Weight $2$
Character orbit 936.g
Analytic conductor $7.474$
Analytic rank $0$
Dimension $24$
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] = [936,2,Mod(469,936)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(936, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("936.469");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.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: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} - 8 q^{7} + 12 q^{10} - 4 q^{16} + 4 q^{22} - 24 q^{25} + 8 q^{28} + 40 q^{31} - 16 q^{34} - 36 q^{40} - 24 q^{46} + 24 q^{49} - 4 q^{52} - 16 q^{55} - 24 q^{58} + 8 q^{64} - 16 q^{70} - 16 q^{76} + 68 q^{82} + 28 q^{88} + 12 q^{94} - 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
469.1 −1.40752 − 0.137446i 0 1.96222 + 0.386915i − 4.18566i 0 2.70923 −2.70868 − 0.814289i 0 −0.575302 + 5.89140i
469.2 −1.40752 + 0.137446i 0 1.96222 − 0.386915i 4.18566i 0 2.70923 −2.70868 + 0.814289i 0 −0.575302 − 5.89140i
469.3 −1.36264 − 0.378437i 0 1.71357 + 1.03135i 0.839709i 0 −0.482293 −1.94468 − 2.05383i 0 0.317777 − 1.14422i
469.4 −1.36264 + 0.378437i 0 1.71357 − 1.03135i − 0.839709i 0 −0.482293 −1.94468 + 2.05383i 0 0.317777 + 1.14422i
469.5 −1.21070 − 0.730889i 0 0.931601 + 1.76978i − 2.32782i 0 −4.85152 0.165621 − 2.82357i 0 −1.70138 + 2.81830i
469.6 −1.21070 + 0.730889i 0 0.931601 − 1.76978i 2.32782i 0 −4.85152 0.165621 + 2.82357i 0 −1.70138 − 2.81830i
469.7 −0.913088 − 1.07994i 0 −0.332541 + 1.97216i − 0.592273i 0 3.54968 2.43345 − 1.44163i 0 −0.639619 + 0.540797i
469.8 −0.913088 + 1.07994i 0 −0.332541 − 1.97216i 0.592273i 0 3.54968 2.43345 + 1.44163i 0 −0.639619 − 0.540797i
469.9 −0.900006 − 1.09087i 0 −0.379977 + 1.96357i 3.06504i 0 −1.53955 2.48398 − 1.35272i 0 3.34355 − 2.75856i
469.10 −0.900006 + 1.09087i 0 −0.379977 − 1.96357i − 3.06504i 0 −1.53955 2.48398 + 1.35272i 0 3.34355 + 2.75856i
469.11 −0.229270 − 1.39551i 0 −1.89487 + 0.639895i 1.61588i 0 −1.38556 1.32741 + 2.49759i 0 2.25497 − 0.370474i
469.12 −0.229270 + 1.39551i 0 −1.89487 − 0.639895i − 1.61588i 0 −1.38556 1.32741 − 2.49759i 0 2.25497 + 0.370474i
469.13 0.229270 − 1.39551i 0 −1.89487 − 0.639895i 1.61588i 0 −1.38556 −1.32741 + 2.49759i 0 2.25497 + 0.370474i
469.14 0.229270 + 1.39551i 0 −1.89487 + 0.639895i − 1.61588i 0 −1.38556 −1.32741 − 2.49759i 0 2.25497 − 0.370474i
469.15 0.900006 − 1.09087i 0 −0.379977 − 1.96357i 3.06504i 0 −1.53955 −2.48398 − 1.35272i 0 3.34355 + 2.75856i
469.16 0.900006 + 1.09087i 0 −0.379977 + 1.96357i − 3.06504i 0 −1.53955 −2.48398 + 1.35272i 0 3.34355 − 2.75856i
469.17 0.913088 − 1.07994i 0 −0.332541 − 1.97216i − 0.592273i 0 3.54968 −2.43345 − 1.44163i 0 −0.639619 − 0.540797i
469.18 0.913088 + 1.07994i 0 −0.332541 + 1.97216i 0.592273i 0 3.54968 −2.43345 + 1.44163i 0 −0.639619 + 0.540797i
469.19 1.21070 − 0.730889i 0 0.931601 − 1.76978i − 2.32782i 0 −4.85152 −0.165621 − 2.82357i 0 −1.70138 − 2.81830i
469.20 1.21070 + 0.730889i 0 0.931601 + 1.76978i 2.32782i 0 −4.85152 −0.165621 + 2.82357i 0 −1.70138 + 2.81830i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 469.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.g.f ✓ 24
3.b odd 2 1 inner 936.2.g.f ✓ 24
4.b odd 2 1 3744.2.g.f 24
8.b even 2 1 inner 936.2.g.f ✓ 24
8.d odd 2 1 3744.2.g.f 24
12.b even 2 1 3744.2.g.f 24
24.f even 2 1 3744.2.g.f 24
24.h odd 2 1 inner 936.2.g.f ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.g.f ✓ 24 1.a even 1 1 trivial
936.2.g.f ✓ 24 3.b odd 2 1 inner
936.2.g.f ✓ 24 8.b even 2 1 inner
936.2.g.f ✓ 24 24.h odd 2 1 inner
3744.2.g.f 24 4.b odd 2 1
3744.2.g.f 24 8.d odd 2 1
3744.2.g.f 24 12.b even 2 1
3744.2.g.f 24 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 36T_{5}^{10} + 432T_{5}^{8} + 2128T_{5}^{6} + 4224T_{5}^{4} + 2880T_{5}^{2} + 576 \) acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\). Copy content Toggle raw display