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Properties

Label 675.2.n.b
Level $675$
Weight $2$
Character orbit 675.n
Analytic conductor $5.390$
Analytic rank $0$
Dimension $80$
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] = [675,2,Mod(109,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.n (of order \(10\), degree \(4\), 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 20 q^{4} + 10 q^{10} + 4 q^{16} + 6 q^{19} - 40 q^{22} + 10 q^{25} + 30 q^{28} - 6 q^{31} + 44 q^{34} - 60 q^{37} + 40 q^{40} + 44 q^{46} - 108 q^{49} - 50 q^{52} - 60 q^{58} + 48 q^{61} - 52 q^{64} - 40 q^{67} - 120 q^{70} - 80 q^{73} + 64 q^{76} - 40 q^{79} - 40 q^{85} - 50 q^{88} - 50 q^{91} + 42 q^{94} + 70 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.58416 + 2.18041i 0 −1.62659 − 5.00613i 1.49307 − 1.66455i 0 − 0.722182i 8.36574 + 2.71819i 0 1.26414 + 5.89243i
109.2 −1.34519 + 1.85150i 0 −1.00047 − 3.07913i −2.23015 + 0.162622i 0 0.334831i 2.69370 + 0.875235i 0 2.69888 − 4.34787i
109.3 −1.30549 + 1.79686i 0 −0.906351 − 2.78946i −0.443474 + 2.19165i 0 − 2.64404i 1.97083 + 0.640363i 0 −3.35913 − 3.65805i
109.4 −1.26170 + 1.73658i 0 −0.805787 − 2.47996i 1.14706 − 1.91944i 0 2.82491i 1.24036 + 0.403017i 0 1.88601 + 4.41371i
109.5 −1.00014 + 1.37658i 0 −0.276650 − 0.851441i 1.99748 + 1.00502i 0 5.07116i −1.78777 − 0.580880i 0 −3.38125 + 1.74454i
109.6 −0.880531 + 1.21195i 0 −0.0754459 − 0.232199i −1.65643 − 1.50208i 0 − 2.26877i −2.50161 − 0.812823i 0 3.27898 − 0.684873i
109.7 −0.833723 + 1.14752i 0 −0.00367749 − 0.0113181i 2.22834 + 0.185702i 0 − 3.83208i −2.68193 − 0.871412i 0 −2.07092 + 2.40225i
109.8 −0.385700 + 0.530871i 0 0.484975 + 1.49260i −0.892535 + 2.05021i 0 0.997946i −2.22758 − 0.723786i 0 −0.744148 − 1.26459i
109.9 −0.342882 + 0.471937i 0 0.512878 + 1.57848i −2.23461 + 0.0806925i 0 4.06534i −2.03039 − 0.659713i 0 0.728127 − 1.08226i
109.10 −0.111328 + 0.153229i 0 0.606949 + 1.86800i −0.508893 − 2.17739i 0 − 1.92500i −0.714065 − 0.232014i 0 0.390294 + 0.164426i
109.11 0.111328 − 0.153229i 0 0.606949 + 1.86800i 0.508893 + 2.17739i 0 − 1.92500i 0.714065 + 0.232014i 0 0.390294 + 0.164426i
109.12 0.342882 − 0.471937i 0 0.512878 + 1.57848i 2.23461 − 0.0806925i 0 4.06534i 2.03039 + 0.659713i 0 0.728127 − 1.08226i
109.13 0.385700 − 0.530871i 0 0.484975 + 1.49260i 0.892535 − 2.05021i 0 0.997946i 2.22758 + 0.723786i 0 −0.744148 − 1.26459i
109.14 0.833723 − 1.14752i 0 −0.00367749 − 0.0113181i −2.22834 − 0.185702i 0 − 3.83208i 2.68193 + 0.871412i 0 −2.07092 + 2.40225i
109.15 0.880531 − 1.21195i 0 −0.0754459 − 0.232199i 1.65643 + 1.50208i 0 − 2.26877i 2.50161 + 0.812823i 0 3.27898 − 0.684873i
109.16 1.00014 − 1.37658i 0 −0.276650 − 0.851441i −1.99748 − 1.00502i 0 5.07116i 1.78777 + 0.580880i 0 −3.38125 + 1.74454i
109.17 1.26170 − 1.73658i 0 −0.805787 − 2.47996i −1.14706 + 1.91944i 0 2.82491i −1.24036 − 0.403017i 0 1.88601 + 4.41371i
109.18 1.30549 − 1.79686i 0 −0.906351 − 2.78946i 0.443474 − 2.19165i 0 − 2.64404i −1.97083 − 0.640363i 0 −3.35913 − 3.65805i
109.19 1.34519 − 1.85150i 0 −1.00047 − 3.07913i 2.23015 − 0.162622i 0 0.334831i −2.69370 − 0.875235i 0 2.69888 − 4.34787i
109.20 1.58416 − 2.18041i 0 −1.62659 − 5.00613i −1.49307 + 1.66455i 0 − 0.722182i −8.36574 − 2.71819i 0 1.26414 + 5.89243i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.n.b ✓ 80
3.b odd 2 1 inner 675.2.n.b ✓ 80
25.e even 10 1 inner 675.2.n.b ✓ 80
75.h odd 10 1 inner 675.2.n.b ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.2.n.b ✓ 80 1.a even 1 1 trivial
675.2.n.b ✓ 80 3.b odd 2 1 inner
675.2.n.b ✓ 80 25.e even 10 1 inner
675.2.n.b ✓ 80 75.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} - 30 T_{2}^{78} + 519 T_{2}^{76} - 6808 T_{2}^{74} + 75989 T_{2}^{72} - 730006 T_{2}^{70} + \cdots + 3748096 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display