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Properties

Label 8032.2.a.g
Level $8032$
Weight $2$
Character orbit 8032.a
Self dual yes
Analytic conductor $64.136$
Analytic rank $1$
Dimension $30$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8032,2,Mod(1,8032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8032, 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("8032.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8032 = 2^{5} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8032.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: \(64.1358429035\)
Analytic rank: \(1\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9} - 31 q^{11} + 5 q^{13} - 8 q^{15} - q^{17} - 29 q^{19} + 6 q^{21} - 27 q^{23} + 34 q^{25} - 36 q^{27} + q^{29} + 9 q^{31} - 8 q^{33} - 51 q^{35} + 5 q^{37} + 8 q^{39} - 3 q^{41} - 43 q^{43} - 42 q^{47} + 41 q^{49} - 54 q^{51} - 11 q^{53} + 10 q^{55} + 2 q^{57} - 49 q^{59} + 21 q^{61} - 15 q^{63} - 14 q^{65} - 68 q^{67} - 10 q^{69} - 44 q^{71} + 25 q^{73} - 51 q^{75} - 20 q^{77} + 49 q^{79} + 6 q^{81} - 88 q^{83} - 5 q^{85} - 16 q^{87} - 16 q^{89} - 46 q^{91} - 4 q^{93} - 28 q^{95} - 4 q^{97} - 71 q^{99}+O(q^{100}) \) 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 0 −3.43537 0 2.52824 0 −2.67250 0 8.80174 0
1.2 0 −3.13996 0 −2.26457 0 2.38671 0 6.85933 0
1.3 0 −3.00046 0 2.74997 0 3.93356 0 6.00279 0
1.4 0 −2.99531 0 −2.48297 0 −3.43053 0 5.97187 0
1.5 0 −2.71488 0 −2.26191 0 1.04506 0 4.37055 0
1.6 0 −2.62556 0 1.77921 0 −2.01732 0 3.89356 0
1.7 0 −2.20649 0 2.47534 0 −0.802937 0 1.86861 0
1.8 0 −2.04450 0 0.222963 0 −5.15673 0 1.17996 0
1.9 0 −1.89707 0 −3.37686 0 3.27287 0 0.598876 0
1.10 0 −1.77983 0 3.93815 0 −3.71151 0 0.167788 0
1.11 0 −1.73507 0 −3.61768 0 3.69185 0 0.0104735 0
1.12 0 −1.21978 0 2.91757 0 −2.62203 0 −1.51215 0
1.13 0 −0.933880 0 0.0426876 0 2.12724 0 −2.12787 0
1.14 0 −0.824754 0 −2.00155 0 −1.90493 0 −2.31978 0
1.15 0 −0.692263 0 3.18782 0 3.79509 0 −2.52077 0
1.16 0 −0.369540 0 −3.94367 0 −3.05773 0 −2.86344 0
1.17 0 −0.193782 0 1.98943 0 −0.152201 0 −2.96245 0
1.18 0 0.222021 0 0.0345121 0 5.16899 0 −2.95071 0
1.19 0 0.685335 0 −1.94665 0 −1.34986 0 −2.53032 0
1.20 0 1.13597 0 −1.24613 0 −4.57789 0 −1.70957 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8032.2.a.g ✓ 30
4.b odd 2 1 8032.2.a.j yes 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8032.2.a.g ✓ 30 1.a even 1 1 trivial
8032.2.a.j yes 30 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8032))\):

\( T_{3}^{30} + 9 T_{3}^{29} - 22 T_{3}^{28} - 411 T_{3}^{27} - 250 T_{3}^{26} + 8070 T_{3}^{25} + \cdots - 138600 \) Copy content Toggle raw display
\( T_{7}^{30} + 5 T_{7}^{29} - 113 T_{7}^{28} - 589 T_{7}^{27} + 5493 T_{7}^{26} + 30333 T_{7}^{25} + \cdots + 3343679661 \) Copy content Toggle raw display
\( T_{11}^{30} + 31 T_{11}^{29} + 307 T_{11}^{28} - 164 T_{11}^{27} - 23879 T_{11}^{26} + \cdots + 829293330432 \) Copy content Toggle raw display