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Properties

Label 936.2.j.a
Level $936$
Weight $2$
Character orbit 936.j
Analytic conductor $7.474$
Analytic rank $0$
Dimension $48$
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(755,936)]
 
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([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("936.755");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.j (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: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{4} + 16 q^{10} + 8 q^{16} + 32 q^{19} + 48 q^{25} - 24 q^{28} + 32 q^{34} - 32 q^{40} - 32 q^{43} + 24 q^{46} - 48 q^{49} + 8 q^{52} - 40 q^{58} + 40 q^{64} + 32 q^{67} - 40 q^{70} + 40 q^{76} + 16 q^{82} + 48 q^{88} + 32 q^{94} - 32 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
755.1 −1.40511 − 0.160221i 0 1.94866 + 0.450256i 3.43430 0 − 1.48385i −2.66594 − 0.944875i 0 −4.82556 − 0.550247i
755.2 −1.40511 + 0.160221i 0 1.94866 − 0.450256i 3.43430 0 1.48385i −2.66594 + 0.944875i 0 −4.82556 + 0.550247i
755.3 −1.40416 − 0.168324i 0 1.94333 + 0.472708i −2.74521 0 − 0.0889956i −2.64918 − 0.990868i 0 3.85471 + 0.462085i
755.4 −1.40416 + 0.168324i 0 1.94333 − 0.472708i −2.74521 0 0.0889956i −2.64918 + 0.990868i 0 3.85471 − 0.462085i
755.5 −1.32582 − 0.492152i 0 1.51557 + 1.30500i −0.678099 0 4.56073i −1.36711 − 2.47609i 0 0.899035 + 0.333728i
755.6 −1.32582 + 0.492152i 0 1.51557 − 1.30500i −0.678099 0 − 4.56073i −1.36711 + 2.47609i 0 0.899035 − 0.333728i
755.7 −1.18697 − 0.768828i 0 0.817808 + 1.82515i −0.327243 0 2.84373i 0.432514 − 2.79516i 0 0.388429 + 0.251594i
755.8 −1.18697 + 0.768828i 0 0.817808 − 1.82515i −0.327243 0 − 2.84373i 0.432514 + 2.79516i 0 0.388429 − 0.251594i
755.9 −1.15766 − 0.812299i 0 0.680340 + 1.88073i −3.23777 0 − 4.33596i 0.740114 − 2.72988i 0 3.74823 + 2.63004i
755.10 −1.15766 + 0.812299i 0 0.680340 − 1.88073i −3.23777 0 4.33596i 0.740114 + 2.72988i 0 3.74823 − 2.63004i
755.11 −1.01985 − 0.979751i 0 0.0801759 + 1.99839i 2.22260 0 − 0.534598i 1.87616 − 2.11661i 0 −2.26672 − 2.17760i
755.12 −1.01985 + 0.979751i 0 0.0801759 − 1.99839i 2.22260 0 0.534598i 1.87616 + 2.11661i 0 −2.26672 + 2.17760i
755.13 −0.668898 − 1.24602i 0 −1.10515 + 1.66693i −3.68104 0 5.03597i 2.81626 + 0.262038i 0 2.46224 + 4.58666i
755.14 −0.668898 + 1.24602i 0 −1.10515 − 1.66693i −3.68104 0 − 5.03597i 2.81626 − 0.262038i 0 2.46224 − 4.58666i
755.15 −0.634171 − 1.26405i 0 −1.19565 + 1.60325i 0.743520 0 − 1.72000i 2.78484 + 0.494635i 0 −0.471519 − 0.939848i
755.16 −0.634171 + 1.26405i 0 −1.19565 − 1.60325i 0.743520 0 1.72000i 2.78484 − 0.494635i 0 −0.471519 + 0.939848i
755.17 −0.513202 − 1.31781i 0 −1.47325 + 1.35261i −0.398187 0 2.67298i 2.53855 + 1.24730i 0 0.204351 + 0.524735i
755.18 −0.513202 + 1.31781i 0 −1.47325 − 1.35261i −0.398187 0 − 2.67298i 2.53855 − 1.24730i 0 0.204351 − 0.524735i
755.19 −0.508510 − 1.31963i 0 −1.48284 + 1.34209i −2.44507 0 − 2.55268i 2.52509 + 1.27433i 0 1.24334 + 3.22658i
755.20 −0.508510 + 1.31963i 0 −1.48284 − 1.34209i −2.44507 0 2.55268i 2.52509 − 1.27433i 0 1.24334 − 3.22658i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 755.48
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 936.2.j.a ✓ 48
3.b odd 2 1 inner 936.2.j.a ✓ 48
4.b odd 2 1 3744.2.j.a 48
8.b even 2 1 3744.2.j.a 48
8.d odd 2 1 inner 936.2.j.a ✓ 48
12.b even 2 1 3744.2.j.a 48
24.f even 2 1 inner 936.2.j.a ✓ 48
24.h odd 2 1 3744.2.j.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.j.a ✓ 48 1.a even 1 1 trivial
936.2.j.a ✓ 48 3.b odd 2 1 inner
936.2.j.a ✓ 48 8.d odd 2 1 inner
936.2.j.a ✓ 48 24.f even 2 1 inner
3744.2.j.a 48 4.b odd 2 1
3744.2.j.a 48 8.b even 2 1
3744.2.j.a 48 12.b even 2 1
3744.2.j.a 48 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(936, [\chi])\).