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Properties

Label 690.2.j.a
Level $690$
Weight $2$
Character orbit 690.j
Analytic conductor $5.510$
Analytic rank $0$
Dimension $24$
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] = [690,2,Mod(367,690)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(690, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("690.367");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.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: \(5.50967773947\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{6} - 24 q^{16} - 8 q^{23} - 16 q^{25} - 16 q^{26} + 16 q^{31} - 16 q^{35} + 24 q^{36} - 8 q^{46} - 8 q^{47} + 24 q^{50} + 24 q^{55} + 16 q^{58} - 56 q^{62} - 32 q^{70} - 16 q^{71} - 48 q^{73} - 24 q^{81} + 24 q^{82} + 16 q^{87} - 8 q^{92} + 56 q^{93} + 24 q^{95} + 24 q^{96} - 32 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
367.1 −0.707107 − 0.707107i 0.707107 − 0.707107i 1.00000i −2.13084 + 0.677875i −1.00000 −1.05927 + 1.05927i 0.707107 − 0.707107i − 1.00000i 1.98606 + 1.02740i
367.2 −0.707107 − 0.707107i 0.707107 − 0.707107i 1.00000i −0.899538 − 2.04715i −1.00000 −1.80902 + 1.80902i 0.707107 − 0.707107i − 1.00000i −0.811485 + 2.08362i
367.3 −0.707107 − 0.707107i 0.707107 − 0.707107i 1.00000i −0.299475 − 2.21592i −1.00000 3.52616 − 3.52616i 0.707107 − 0.707107i − 1.00000i −1.35513 + 1.77865i
367.4 −0.707107 − 0.707107i 0.707107 − 0.707107i 1.00000i 0.299475 + 2.21592i −1.00000 −3.52616 + 3.52616i 0.707107 − 0.707107i − 1.00000i 1.35513 − 1.77865i
367.5 −0.707107 − 0.707107i 0.707107 − 0.707107i 1.00000i 0.899538 + 2.04715i −1.00000 1.80902 − 1.80902i 0.707107 − 0.707107i − 1.00000i 0.811485 − 2.08362i
367.6 −0.707107 − 0.707107i 0.707107 − 0.707107i 1.00000i 2.13084 − 0.677875i −1.00000 1.05927 − 1.05927i 0.707107 − 0.707107i − 1.00000i −1.98606 − 1.02740i
367.7 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i −2.22154 − 0.254460i −1.00000 2.54671 − 2.54671i −0.707107 + 0.707107i − 1.00000i −1.39094 − 1.75080i
367.8 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i −1.57090 + 1.59131i −1.00000 −1.37926 + 1.37926i −0.707107 + 0.707107i − 1.00000i −2.23602 + 0.0144369i
367.9 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i −0.397104 − 2.20052i −1.00000 −1.66838 + 1.66838i −0.707107 + 0.707107i − 1.00000i 1.27521 − 1.83680i
367.10 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0.397104 + 2.20052i −1.00000 1.66838 − 1.66838i −0.707107 + 0.707107i − 1.00000i −1.27521 + 1.83680i
367.11 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 1.57090 − 1.59131i −1.00000 1.37926 − 1.37926i −0.707107 + 0.707107i − 1.00000i 2.23602 − 0.0144369i
367.12 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 2.22154 + 0.254460i −1.00000 −2.54671 + 2.54671i −0.707107 + 0.707107i − 1.00000i 1.39094 + 1.75080i
643.1 −0.707107 + 0.707107i 0.707107 + 0.707107i − 1.00000i −2.13084 − 0.677875i −1.00000 −1.05927 − 1.05927i 0.707107 + 0.707107i 1.00000i 1.98606 − 1.02740i
643.2 −0.707107 + 0.707107i 0.707107 + 0.707107i − 1.00000i −0.899538 + 2.04715i −1.00000 −1.80902 − 1.80902i 0.707107 + 0.707107i 1.00000i −0.811485 − 2.08362i
643.3 −0.707107 + 0.707107i 0.707107 + 0.707107i − 1.00000i −0.299475 + 2.21592i −1.00000 3.52616 + 3.52616i 0.707107 + 0.707107i 1.00000i −1.35513 − 1.77865i
643.4 −0.707107 + 0.707107i 0.707107 + 0.707107i − 1.00000i 0.299475 − 2.21592i −1.00000 −3.52616 − 3.52616i 0.707107 + 0.707107i 1.00000i 1.35513 + 1.77865i
643.5 −0.707107 + 0.707107i 0.707107 + 0.707107i − 1.00000i 0.899538 − 2.04715i −1.00000 1.80902 + 1.80902i 0.707107 + 0.707107i 1.00000i 0.811485 + 2.08362i
643.6 −0.707107 + 0.707107i 0.707107 + 0.707107i − 1.00000i 2.13084 + 0.677875i −1.00000 1.05927 + 1.05927i 0.707107 + 0.707107i 1.00000i −1.98606 + 1.02740i
643.7 0.707107 − 0.707107i −0.707107 − 0.707107i − 1.00000i −2.22154 + 0.254460i −1.00000 2.54671 + 2.54671i −0.707107 − 0.707107i 1.00000i −1.39094 + 1.75080i
643.8 0.707107 − 0.707107i −0.707107 − 0.707107i − 1.00000i −1.57090 − 1.59131i −1.00000 −1.37926 − 1.37926i −0.707107 − 0.707107i 1.00000i −2.23602 − 0.0144369i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 367.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.j.a ✓ 24
5.c odd 4 1 inner 690.2.j.a ✓ 24
23.b odd 2 1 inner 690.2.j.a ✓ 24
115.e even 4 1 inner 690.2.j.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.j.a ✓ 24 1.a even 1 1 trivial
690.2.j.a ✓ 24 5.c odd 4 1 inner
690.2.j.a ✓ 24 23.b odd 2 1 inner
690.2.j.a ✓ 24 115.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 880T_{7}^{20} + 180320T_{7}^{16} + 11978496T_{7}^{12} + 320323840T_{7}^{8} + 3331522560T_{7}^{4} + 10070523904 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\). Copy content Toggle raw display