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Properties

Label 735.2.j.f
Level $735$
Weight $2$
Character orbit 735.j
Analytic conductor $5.869$
Analytic rank $0$
Dimension $24$
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] = [735,2,Mod(197,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("735.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.j (of order \(4\), degree \(2\), 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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 16 q^{15} - 48 q^{16} + 36 q^{18} + 8 q^{22} + 32 q^{25} + 20 q^{30} + 16 q^{36} + 80 q^{37} + 40 q^{43} - 48 q^{46} + 76 q^{51} + 4 q^{57} + 40 q^{58} - 88 q^{60} + 8 q^{67} - 120 q^{72} - 124 q^{78} + 20 q^{81} - 136 q^{85} - 128 q^{88} - 36 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −1.85196 + 1.85196i −0.170698 + 1.72362i − 4.85952i −2.22031 − 0.265025i −2.87595 − 3.50820i 0 5.29573 + 5.29573i −2.94172 − 0.588437i 4.60274 − 3.62111i
197.2 −1.85196 + 1.85196i 0.170698 − 1.72362i − 4.85952i 2.22031 + 0.265025i 2.87595 + 3.50820i 0 5.29573 + 5.29573i −2.94172 − 0.588437i −4.60274 + 3.62111i
197.3 −1.13449 + 1.13449i −1.53019 − 0.811487i − 0.574141i −1.66945 + 1.48759i 2.65662 − 0.815364i 0 −1.61762 − 1.61762i 1.68298 + 2.48346i 0.206319 − 3.58164i
197.4 −1.13449 + 1.13449i 1.53019 + 0.811487i − 0.574141i 1.66945 − 1.48759i −2.65662 + 0.815364i 0 −1.61762 − 1.61762i 1.68298 + 2.48346i −0.206319 + 3.58164i
197.5 −0.532135 + 0.532135i −0.772460 + 1.55026i 1.43366i 1.33535 + 1.79355i −0.413895 − 1.23600i 0 −1.82717 − 1.82717i −1.80661 − 2.39503i −1.66500 − 0.243824i
197.6 −0.532135 + 0.532135i 0.772460 − 1.55026i 1.43366i −1.33535 − 1.79355i 0.413895 + 1.23600i 0 −1.82717 − 1.82717i −1.80661 − 2.39503i 1.66500 + 0.243824i
197.7 0.532135 − 0.532135i −1.55026 + 0.772460i 1.43366i −1.33535 − 1.79355i −0.413895 + 1.23600i 0 1.82717 + 1.82717i 1.80661 − 2.39503i −1.66500 − 0.243824i
197.8 0.532135 − 0.532135i 1.55026 − 0.772460i 1.43366i 1.33535 + 1.79355i 0.413895 − 1.23600i 0 1.82717 + 1.82717i 1.80661 − 2.39503i 1.66500 + 0.243824i
197.9 1.13449 − 1.13449i −0.811487 − 1.53019i − 0.574141i −1.66945 + 1.48759i −2.65662 − 0.815364i 0 1.61762 + 1.61762i −1.68298 + 2.48346i −0.206319 + 3.58164i
197.10 1.13449 − 1.13449i 0.811487 + 1.53019i − 0.574141i 1.66945 − 1.48759i 2.65662 + 0.815364i 0 1.61762 + 1.61762i −1.68298 + 2.48346i 0.206319 − 3.58164i
197.11 1.85196 − 1.85196i −1.72362 + 0.170698i − 4.85952i 2.22031 + 0.265025i −2.87595 + 3.50820i 0 −5.29573 − 5.29573i 2.94172 − 0.588437i 4.60274 − 3.62111i
197.12 1.85196 − 1.85196i 1.72362 − 0.170698i − 4.85952i −2.22031 − 0.265025i 2.87595 − 3.50820i 0 −5.29573 − 5.29573i 2.94172 − 0.588437i −4.60274 + 3.62111i
638.1 −1.85196 − 1.85196i −0.170698 − 1.72362i 4.85952i −2.22031 + 0.265025i −2.87595 + 3.50820i 0 5.29573 − 5.29573i −2.94172 + 0.588437i 4.60274 + 3.62111i
638.2 −1.85196 − 1.85196i 0.170698 + 1.72362i 4.85952i 2.22031 − 0.265025i 2.87595 − 3.50820i 0 5.29573 − 5.29573i −2.94172 + 0.588437i −4.60274 − 3.62111i
638.3 −1.13449 − 1.13449i −1.53019 + 0.811487i 0.574141i −1.66945 − 1.48759i 2.65662 + 0.815364i 0 −1.61762 + 1.61762i 1.68298 − 2.48346i 0.206319 + 3.58164i
638.4 −1.13449 − 1.13449i 1.53019 − 0.811487i 0.574141i 1.66945 + 1.48759i −2.65662 − 0.815364i 0 −1.61762 + 1.61762i 1.68298 − 2.48346i −0.206319 − 3.58164i
638.5 −0.532135 − 0.532135i −0.772460 − 1.55026i − 1.43366i 1.33535 − 1.79355i −0.413895 + 1.23600i 0 −1.82717 + 1.82717i −1.80661 + 2.39503i −1.66500 + 0.243824i
638.6 −0.532135 − 0.532135i 0.772460 + 1.55026i − 1.43366i −1.33535 + 1.79355i 0.413895 − 1.23600i 0 −1.82717 + 1.82717i −1.80661 + 2.39503i 1.66500 − 0.243824i
638.7 0.532135 + 0.532135i −1.55026 − 0.772460i − 1.43366i −1.33535 + 1.79355i −0.413895 − 1.23600i 0 1.82717 − 1.82717i 1.80661 + 2.39503i −1.66500 + 0.243824i
638.8 0.532135 + 0.532135i 1.55026 + 0.772460i − 1.43366i 1.33535 − 1.79355i 0.413895 + 1.23600i 0 1.82717 − 1.82717i 1.80661 + 2.39503i 1.66500 − 0.243824i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.b odd 2 1 inner
15.e even 4 1 inner
21.c even 2 1 inner
35.f even 4 1 inner
105.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.j.f ✓ 24
3.b odd 2 1 inner 735.2.j.f ✓ 24
5.c odd 4 1 inner 735.2.j.f ✓ 24
7.b odd 2 1 inner 735.2.j.f ✓ 24
7.c even 3 2 735.2.y.h 48
7.d odd 6 2 735.2.y.h 48
15.e even 4 1 inner 735.2.j.f ✓ 24
21.c even 2 1 inner 735.2.j.f ✓ 24
21.g even 6 2 735.2.y.h 48
21.h odd 6 2 735.2.y.h 48
35.f even 4 1 inner 735.2.j.f ✓ 24
35.k even 12 2 735.2.y.h 48
35.l odd 12 2 735.2.y.h 48
105.k odd 4 1 inner 735.2.j.f ✓ 24
105.w odd 12 2 735.2.y.h 48
105.x even 12 2 735.2.y.h 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.j.f ✓ 24 1.a even 1 1 trivial
735.2.j.f ✓ 24 3.b odd 2 1 inner
735.2.j.f ✓ 24 5.c odd 4 1 inner
735.2.j.f ✓ 24 7.b odd 2 1 inner
735.2.j.f ✓ 24 15.e even 4 1 inner
735.2.j.f ✓ 24 21.c even 2 1 inner
735.2.j.f ✓ 24 35.f even 4 1 inner
735.2.j.f ✓ 24 105.k odd 4 1 inner
735.2.y.h 48 7.c even 3 2
735.2.y.h 48 7.d odd 6 2
735.2.y.h 48 21.g even 6 2
735.2.y.h 48 21.h odd 6 2
735.2.y.h 48 35.k even 12 2
735.2.y.h 48 35.l odd 12 2
735.2.y.h 48 105.w odd 12 2
735.2.y.h 48 105.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{12} + 54T_{2}^{8} + 329T_{2}^{4} + 100 \) Copy content Toggle raw display
\( T_{13}^{12} + 473T_{13}^{8} + 54576T_{13}^{4} + 6400 \) Copy content Toggle raw display
\( T_{17}^{12} + 1273T_{17}^{8} + 233240T_{17}^{4} + 11316496 \) Copy content Toggle raw display