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Properties

Label 1610.2.g.a
Level $1610$
Weight $2$
Character orbit 1610.g
Analytic conductor $12.856$
Analytic rank $0$
Dimension $96$
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] = [1610,2,Mod(1609,1610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1610.1609");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1610 = 2 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1610.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: \(12.8559147254\)
Analytic rank: \(0\)
Dimension: \(96\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 96 q^{4} + 88 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 96 q^{4} + 88 q^{9} + 96 q^{16} + 8 q^{25} + 8 q^{35} - 88 q^{36} - 16 q^{39} - 16 q^{46} + 36 q^{49} + 24 q^{50} - 96 q^{64} - 12 q^{70} + 8 q^{71} + 128 q^{81} + 16 q^{85} - 32 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1609.1 − 1.00000i 0.209068 −1.00000 1.92511 + 1.13752i − 0.209068i −0.855049 − 2.50378i 1.00000i −2.95629 1.13752 − 1.92511i
1609.2 1.00000i 0.209068 −1.00000 1.92511 − 1.13752i 0.209068i −0.855049 + 2.50378i − 1.00000i −2.95629 1.13752 + 1.92511i
1609.3 − 1.00000i 3.32094 −1.00000 −0.0164715 + 2.23601i − 3.32094i 2.20809 − 1.45752i 1.00000i 8.02867 2.23601 + 0.0164715i
1609.4 1.00000i 3.32094 −1.00000 −0.0164715 − 2.23601i 3.32094i 2.20809 + 1.45752i − 1.00000i 8.02867 2.23601 − 0.0164715i
1609.5 − 1.00000i 0.209068 −1.00000 −1.92511 − 1.13752i − 0.209068i 0.855049 + 2.50378i 1.00000i −2.95629 −1.13752 + 1.92511i
1609.6 1.00000i 0.209068 −1.00000 −1.92511 + 1.13752i 0.209068i 0.855049 − 2.50378i − 1.00000i −2.95629 −1.13752 − 1.92511i
1609.7 − 1.00000i 2.59834 −1.00000 −1.15059 − 1.91733i − 2.59834i 2.54535 − 0.721949i 1.00000i 3.75138 −1.91733 + 1.15059i
1609.8 1.00000i 2.59834 −1.00000 −1.15059 + 1.91733i 2.59834i 2.54535 + 0.721949i − 1.00000i 3.75138 −1.91733 − 1.15059i
1609.9 − 1.00000i −0.842190 −1.00000 1.91245 − 1.15867i 0.842190i 1.90341 + 1.83767i 1.00000i −2.29072 −1.15867 − 1.91245i
1609.10 1.00000i −0.842190 −1.00000 1.91245 + 1.15867i − 0.842190i 1.90341 − 1.83767i − 1.00000i −2.29072 −1.15867 + 1.91245i
1609.11 − 1.00000i 0.842190 −1.00000 −1.91245 + 1.15867i − 0.842190i 1.90341 − 1.83767i 1.00000i −2.29072 1.15867 + 1.91245i
1609.12 1.00000i 0.842190 −1.00000 −1.91245 − 1.15867i 0.842190i 1.90341 + 1.83767i − 1.00000i −2.29072 1.15867 − 1.91245i
1609.13 − 1.00000i −0.574023 −1.00000 −0.755769 − 2.10447i 0.574023i 2.38007 − 1.15553i 1.00000i −2.67050 −2.10447 + 0.755769i
1609.14 1.00000i −0.574023 −1.00000 −0.755769 + 2.10447i − 0.574023i 2.38007 + 1.15553i − 1.00000i −2.67050 −2.10447 − 0.755769i
1609.15 − 1.00000i 2.20972 −1.00000 −2.21000 − 0.340423i − 2.20972i −2.40927 − 1.09336i 1.00000i 1.88286 −0.340423 + 2.21000i
1609.16 1.00000i 2.20972 −1.00000 −2.21000 + 0.340423i 2.20972i −2.40927 + 1.09336i − 1.00000i 1.88286 −0.340423 − 2.21000i
1609.17 − 1.00000i −0.421830 −1.00000 −1.54042 + 1.62083i 0.421830i −2.63295 − 0.259941i 1.00000i −2.82206 1.62083 + 1.54042i
1609.18 1.00000i −0.421830 −1.00000 −1.54042 − 1.62083i − 0.421830i −2.63295 + 0.259941i − 1.00000i −2.82206 1.62083 − 1.54042i
1609.19 − 1.00000i −1.23117 −1.00000 0.749972 − 2.10655i 1.23117i 0.117671 − 2.64313i 1.00000i −1.48421 −2.10655 − 0.749972i
1609.20 1.00000i −1.23117 −1.00000 0.749972 + 2.10655i − 1.23117i 0.117671 + 2.64313i − 1.00000i −1.48421 −2.10655 + 0.749972i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1609.96
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
23.b odd 2 1 inner
35.c odd 2 1 inner
115.c odd 2 1 inner
161.c even 2 1 inner
805.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1610.2.g.a ✓ 96
5.b even 2 1 inner 1610.2.g.a ✓ 96
7.b odd 2 1 inner 1610.2.g.a ✓ 96
23.b odd 2 1 inner 1610.2.g.a ✓ 96
35.c odd 2 1 inner 1610.2.g.a ✓ 96
115.c odd 2 1 inner 1610.2.g.a ✓ 96
161.c even 2 1 inner 1610.2.g.a ✓ 96
805.d even 2 1 inner 1610.2.g.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1610.2.g.a ✓ 96 1.a even 1 1 trivial
1610.2.g.a ✓ 96 5.b even 2 1 inner
1610.2.g.a ✓ 96 7.b odd 2 1 inner
1610.2.g.a ✓ 96 23.b odd 2 1 inner
1610.2.g.a ✓ 96 35.c odd 2 1 inner
1610.2.g.a ✓ 96 115.c odd 2 1 inner
1610.2.g.a ✓ 96 161.c even 2 1 inner
1610.2.g.a ✓ 96 805.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1610, [\chi])\).