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Properties

Label 8619.2.a.br
Level $8619$
Weight $2$
Character orbit 8619.a
Self dual yes
Analytic conductor $68.823$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8619,2,Mod(1,8619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8619, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8619.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8619.a (trivial)

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: yes
Analytic conductor: \(68.8230615021\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 663)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 22 q^{3} + 32 q^{4} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 22 q^{3} + 32 q^{4} + 22 q^{9} + 26 q^{10} - 32 q^{12} + 18 q^{14} + 36 q^{16} - 22 q^{17} + 22 q^{22} - 20 q^{23} + 46 q^{25} - 22 q^{27} + 6 q^{29} - 26 q^{30} - 14 q^{35} + 32 q^{36} + 38 q^{38} + 108 q^{40} - 18 q^{42} - 30 q^{43} - 36 q^{48} + 68 q^{49} + 22 q^{51} - 10 q^{53} - 18 q^{55} - 30 q^{56} + 92 q^{61} + 52 q^{62} + 96 q^{64} - 22 q^{66} - 32 q^{68} + 20 q^{69} + 42 q^{74} - 46 q^{75} + 28 q^{77} + 68 q^{79} + 22 q^{81} + 6 q^{82} - 6 q^{87} + 134 q^{88} + 26 q^{90} + 40 q^{92} - 94 q^{94} - 72 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.72119 −1.00000 5.40489 −4.00369 2.72119 0.629071 −9.26535 1.00000 10.8948
1.2 −2.70494 −1.00000 5.31671 −1.50456 2.70494 4.66371 −8.97152 1.00000 4.06976
1.3 −2.41403 −1.00000 3.82754 3.65533 2.41403 −4.23232 −4.41173 1.00000 −8.82407
1.4 −2.35757 −1.00000 3.55811 −3.59766 2.35757 −4.00925 −3.67335 1.00000 8.48171
1.5 −1.80553 −1.00000 1.25995 −0.537668 1.80553 −3.31395 1.33619 1.00000 0.970777
1.6 −1.61354 −1.00000 0.603526 −1.43426 1.61354 −1.65360 2.25327 1.00000 2.31424
1.7 −1.57992 −1.00000 0.496139 1.82584 1.57992 1.19505 2.37598 1.00000 −2.88467
1.8 −1.40063 −1.00000 −0.0382282 −0.691348 1.40063 3.92751 2.85481 1.00000 0.968325
1.9 −1.03403 −1.00000 −0.930781 3.65775 1.03403 −3.14151 3.03052 1.00000 −3.78223
1.10 −0.604879 −1.00000 −1.63412 0.897196 0.604879 2.90433 2.19821 1.00000 −0.542695
1.11 −0.369141 −1.00000 −1.86373 −3.61390 0.369141 −2.44806 1.42626 1.00000 1.33404
1.12 0.369141 −1.00000 −1.86373 3.61390 −0.369141 2.44806 −1.42626 1.00000 1.33404
1.13 0.604879 −1.00000 −1.63412 −0.897196 −0.604879 −2.90433 −2.19821 1.00000 −0.542695
1.14 1.03403 −1.00000 −0.930781 −3.65775 −1.03403 3.14151 −3.03052 1.00000 −3.78223
1.15 1.40063 −1.00000 −0.0382282 0.691348 −1.40063 −3.92751 −2.85481 1.00000 0.968325
1.16 1.57992 −1.00000 0.496139 −1.82584 −1.57992 −1.19505 −2.37598 1.00000 −2.88467
1.17 1.61354 −1.00000 0.603526 1.43426 −1.61354 1.65360 −2.25327 1.00000 2.31424
1.18 1.80553 −1.00000 1.25995 0.537668 −1.80553 3.31395 −1.33619 1.00000 0.970777
1.19 2.35757 −1.00000 3.55811 3.59766 −2.35757 4.00925 3.67335 1.00000 8.48171
1.20 2.41403 −1.00000 3.82754 −3.65533 −2.41403 4.23232 4.41173 1.00000 −8.82407
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(13\) \(-1\)
\(17\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8619.2.a.br 22
13.b even 2 1 inner 8619.2.a.br 22
13.f odd 12 2 663.2.z.f ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.z.f ✓ 22 13.f odd 12 2
8619.2.a.br 22 1.a even 1 1 trivial
8619.2.a.br 22 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{22} - 38 T_{2}^{20} + 621 T_{2}^{18} - 5722 T_{2}^{16} + 32768 T_{2}^{14} - 121273 T_{2}^{12} + \cdots - 3888 \) Copy content Toggle raw display
\( T_{5}^{22} - 78 T_{5}^{20} + 2555 T_{5}^{18} - 45557 T_{5}^{16} + 479787 T_{5}^{14} - 3043583 T_{5}^{12} + \cdots - 836352 \) Copy content Toggle raw display
\( T_{7}^{22} - 111 T_{7}^{20} + 5348 T_{7}^{18} - 146746 T_{7}^{16} + 2528268 T_{7}^{14} + \cdots - 817938432 \) Copy content Toggle raw display