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Properties

Label 475.2.j.d
Level $475$
Weight $2$
Character orbit 475.j
Analytic conductor $3.793$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(49,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.j (of order \(6\), degree \(2\), not 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{4} + 2 q^{6} + 14 q^{9} - 4 q^{11} - 12 q^{14} + 12 q^{16} + 12 q^{19} - 6 q^{21} + 22 q^{24} + 76 q^{26} + 6 q^{29} - 12 q^{31} - 2 q^{34} - 26 q^{36} - 32 q^{39} - 22 q^{41} + 42 q^{44} - 48 q^{46}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −1.87935 − 1.08504i −1.22342 − 0.706345i 1.35464 + 2.34630i 0 1.53283 + 2.65494i − 1.76171i − 1.53919i −0.502155 − 0.869757i 0
49.2 −1.87935 − 1.08504i 2.55640 + 1.47594i 1.35464 + 2.34630i 0 −3.20292 − 5.54761i 0.591620i − 1.53919i 2.85679 + 4.94811i 0
49.3 −1.28275 − 0.740597i −2.47401 − 1.42837i 0.0969683 + 0.167954i 0 2.11569 + 3.66449i − 3.78541i 2.67513i 2.58048 + 4.46952i 0
49.4 −1.28275 − 0.740597i 0.157277 + 0.0908038i 0.0969683 + 0.167954i 0 −0.134498 − 0.232958i 1.30422i 2.67513i −1.48351 − 2.56951i 0
49.5 −0.269427 − 0.155554i −1.94349 − 1.12208i −0.951606 − 1.64823i 0 0.349087 + 0.604636i 3.96928i 1.21432i 1.01811 + 1.76343i 0
49.6 −0.269427 − 0.155554i 0.891863 + 0.514917i −0.951606 − 1.64823i 0 −0.160195 − 0.277466i − 3.28038i 1.21432i −0.969720 − 1.67960i 0
49.7 0.269427 + 0.155554i −0.891863 − 0.514917i −0.951606 − 1.64823i 0 −0.160195 − 0.277466i 3.28038i − 1.21432i −0.969720 − 1.67960i 0
49.8 0.269427 + 0.155554i 1.94349 + 1.12208i −0.951606 − 1.64823i 0 0.349087 + 0.604636i − 3.96928i − 1.21432i 1.01811 + 1.76343i 0
49.9 1.28275 + 0.740597i −0.157277 − 0.0908038i 0.0969683 + 0.167954i 0 −0.134498 − 0.232958i − 1.30422i − 2.67513i −1.48351 − 2.56951i 0
49.10 1.28275 + 0.740597i 2.47401 + 1.42837i 0.0969683 + 0.167954i 0 2.11569 + 3.66449i 3.78541i − 2.67513i 2.58048 + 4.46952i 0
49.11 1.87935 + 1.08504i −2.55640 − 1.47594i 1.35464 + 2.34630i 0 −3.20292 − 5.54761i − 0.591620i 1.53919i 2.85679 + 4.94811i 0
49.12 1.87935 + 1.08504i 1.22342 + 0.706345i 1.35464 + 2.34630i 0 1.53283 + 2.65494i 1.76171i 1.53919i −0.502155 − 0.869757i 0
349.1 −1.87935 + 1.08504i −1.22342 + 0.706345i 1.35464 − 2.34630i 0 1.53283 − 2.65494i 1.76171i 1.53919i −0.502155 + 0.869757i 0
349.2 −1.87935 + 1.08504i 2.55640 − 1.47594i 1.35464 − 2.34630i 0 −3.20292 + 5.54761i − 0.591620i 1.53919i 2.85679 − 4.94811i 0
349.3 −1.28275 + 0.740597i −2.47401 + 1.42837i 0.0969683 − 0.167954i 0 2.11569 − 3.66449i 3.78541i − 2.67513i 2.58048 − 4.46952i 0
349.4 −1.28275 + 0.740597i 0.157277 − 0.0908038i 0.0969683 − 0.167954i 0 −0.134498 + 0.232958i − 1.30422i − 2.67513i −1.48351 + 2.56951i 0
349.5 −0.269427 + 0.155554i −1.94349 + 1.12208i −0.951606 + 1.64823i 0 0.349087 − 0.604636i − 3.96928i − 1.21432i 1.01811 − 1.76343i 0
349.6 −0.269427 + 0.155554i 0.891863 − 0.514917i −0.951606 + 1.64823i 0 −0.160195 + 0.277466i 3.28038i − 1.21432i −0.969720 + 1.67960i 0
349.7 0.269427 − 0.155554i −0.891863 + 0.514917i −0.951606 + 1.64823i 0 −0.160195 + 0.277466i − 3.28038i 1.21432i −0.969720 + 1.67960i 0
349.8 0.269427 − 0.155554i 1.94349 − 1.12208i −0.951606 + 1.64823i 0 0.349087 − 0.604636i 3.96928i 1.21432i 1.01811 − 1.76343i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.j.d 24
5.b even 2 1 inner 475.2.j.d 24
5.c odd 4 1 475.2.e.f ✓ 12
5.c odd 4 1 475.2.e.h yes 12
19.c even 3 1 inner 475.2.j.d 24
95.i even 6 1 inner 475.2.j.d 24
95.l even 12 1 9025.2.a.bs 6
95.l even 12 1 9025.2.a.by 6
95.m odd 12 1 475.2.e.f ✓ 12
95.m odd 12 1 475.2.e.h yes 12
95.m odd 12 1 9025.2.a.br 6
95.m odd 12 1 9025.2.a.bz 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.e.f ✓ 12 5.c odd 4 1
475.2.e.f ✓ 12 95.m odd 12 1
475.2.e.h yes 12 5.c odd 4 1
475.2.e.h yes 12 95.m odd 12 1
475.2.j.d 24 1.a even 1 1 trivial
475.2.j.d 24 5.b even 2 1 inner
475.2.j.d 24 19.c even 3 1 inner
475.2.j.d 24 95.i even 6 1 inner
9025.2.a.br 6 95.m odd 12 1
9025.2.a.bs 6 95.l even 12 1
9025.2.a.by 6 95.l even 12 1
9025.2.a.bz 6 95.m odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 7T_{2}^{10} + 38T_{2}^{8} - 75T_{2}^{6} + 114T_{2}^{4} - 11T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display