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Properties

Label 8281.2.a.cz
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $36$
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] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 14 q^{2} + 38 q^{4} + 42 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 14 q^{2} + 38 q^{4} + 42 q^{8} + 36 q^{9} + 52 q^{11} + 44 q^{15} + 42 q^{16} + 46 q^{18} + 68 q^{22} - 8 q^{23} + 32 q^{25} - 8 q^{29} + 118 q^{32} + 66 q^{36} + 28 q^{37} - 16 q^{43} + 130 q^{44} + 48 q^{46} + 90 q^{50} + 4 q^{51} + 52 q^{57} - 34 q^{58} + 100 q^{60} + 62 q^{64} + 40 q^{67} + 188 q^{71} - 42 q^{72} + 96 q^{74} + 8 q^{79} - 4 q^{81} + 64 q^{85} + 58 q^{86} + 156 q^{88} - 76 q^{92} - 16 q^{93} - 64 q^{95} + 156 q^{99}+O(q^{100}) \) 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 −2.32760 −2.53008 3.41771 −3.60017 5.88901 0 −3.29985 3.40131 8.37976
1.2 −2.32760 2.53008 3.41771 3.60017 −5.88901 0 −3.29985 3.40131 −8.37976
1.3 −2.16753 −1.28225 2.69819 −0.501045 2.77933 0 −1.51336 −1.35582 1.08603
1.4 −2.16753 1.28225 2.69819 0.501045 −2.77933 0 −1.51336 −1.35582 −1.08603
1.5 −1.75367 −1.73084 1.07535 −2.34349 3.03532 0 1.62153 −0.00419585 4.10970
1.6 −1.75367 1.73084 1.07535 2.34349 −3.03532 0 1.62153 −0.00419585 −4.10970
1.7 −1.63103 −1.44692 0.660273 0.621753 2.35997 0 2.18514 −0.906425 −1.01410
1.8 −1.63103 1.44692 0.660273 −0.621753 −2.35997 0 2.18514 −0.906425 1.01410
1.9 −1.43547 −2.56073 0.0605620 1.41193 3.67584 0 2.78400 3.55734 −2.02678
1.10 −1.43547 2.56073 0.0605620 −1.41193 −3.67584 0 2.78400 3.55734 2.02678
1.11 −0.739620 −0.0660476 −1.45296 3.87852 0.0488502 0 2.55388 −2.99564 −2.86864
1.12 −0.739620 0.0660476 −1.45296 −3.87852 −0.0488502 0 2.55388 −2.99564 2.86864
1.13 −0.101893 −0.462556 −1.98962 1.45067 0.0471315 0 0.406516 −2.78604 −0.147814
1.14 −0.101893 0.462556 −1.98962 −1.45067 −0.0471315 0 0.406516 −2.78604 0.147814
1.15 0.0222049 −2.31666 −1.99951 0.574345 −0.0514412 0 −0.0888086 2.36693 0.0127533
1.16 0.0222049 2.31666 −1.99951 −0.574345 0.0514412 0 −0.0888086 2.36693 −0.0127533
1.17 0.286533 −1.75963 −1.91790 −2.61493 −0.504191 0 −1.12261 0.0962853 −0.749265
1.18 0.286533 1.75963 −1.91790 2.61493 0.504191 0 −1.12261 0.0962853 0.749265
1.19 0.488498 −1.17176 −1.76137 1.82556 −0.572403 0 −1.83742 −1.62698 0.891784
1.20 0.488498 1.17176 −1.76137 −1.82556 0.572403 0 −1.83742 −1.62698 −0.891784
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(13\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8281.2.a.cz yes 36
7.b odd 2 1 inner 8281.2.a.cz yes 36
13.b even 2 1 8281.2.a.cy ✓ 36
91.b odd 2 1 8281.2.a.cy ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8281.2.a.cy ✓ 36 13.b even 2 1
8281.2.a.cy ✓ 36 91.b odd 2 1
8281.2.a.cz yes 36 1.a even 1 1 trivial
8281.2.a.cz yes 36 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{18} - 7 T_{2}^{17} - 3 T_{2}^{16} + 119 T_{2}^{15} - 151 T_{2}^{14} - 751 T_{2}^{13} + 1582 T_{2}^{12} + \cdots - 1 \) Copy content Toggle raw display
\( T_{3}^{36} - 72 T_{3}^{34} + 2350 T_{3}^{32} - 46038 T_{3}^{30} + 604211 T_{3}^{28} - 5615288 T_{3}^{26} + \cdots + 64 \) Copy content Toggle raw display
\( T_{5}^{36} - 106 T_{5}^{34} + 5000 T_{5}^{32} - 138680 T_{5}^{30} + 2519796 T_{5}^{28} - 31661560 T_{5}^{26} + \cdots + 440896 \) Copy content Toggle raw display
\( T_{11}^{18} - 26 T_{11}^{17} + 235 T_{11}^{16} - 518 T_{11}^{15} - 4738 T_{11}^{14} + 28538 T_{11}^{13} + \cdots - 1257397 \) Copy content Toggle raw display
\( T_{17}^{36} - 306 T_{17}^{34} + 42253 T_{17}^{32} - 3483504 T_{17}^{30} + 191250552 T_{17}^{28} + \cdots + 109435199032384 \) Copy content Toggle raw display