Properties

Label 3808.2.a.i
Level $3808$
Weight $2$
Character orbit 3808.a
Self dual yes
Analytic conductor $30.407$
Analytic rank $1$
Dimension $6$
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] = [3808,2,Mod(1,3808)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3808, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3808.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3808.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: \(30.4070330897\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.80686992.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 10x^{4} + 15x^{3} + 8x^{2} - 15x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{4} - 1) q^{5} + q^{7} + ( - \beta_{2} - \beta_1) q^{9} + ( - \beta_{5} + \beta_{3} - \beta_1 - 1) q^{11} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{13} + (\beta_{5} - \beta_{4} - \beta_1) q^{15}+ \cdots + ( - 2 \beta_{5} + 4 \beta_{4} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} + 2 q^{15} - 6 q^{17} - 10 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} - 8 q^{27} - 14 q^{29} + 8 q^{31} - 8 q^{33} - 6 q^{35} - 4 q^{37} + 14 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 10x^{4} + 15x^{3} + 8x^{2} - 15x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} - 9\nu^{3} + 6\nu^{2} + 7\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 9\nu^{2} + 6\nu + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 9\nu^{3} + 8\nu^{2} + 9\nu - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu^{4} + 11\nu^{3} - 13\nu^{2} - 15\nu + 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + 10\nu^{3} - 5\nu^{2} - 12\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} - 3\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{5} - 3\beta_{4} - \beta_{3} + 3\beta_{2} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -15\beta_{5} + 15\beta_{4} + 18\beta_{3} - 11\beta_{2} - 33\beta _1 + 29 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 71\beta_{5} - 53\beta_{4} - 30\beta_{3} + 53\beta_{2} + 123\beta _1 - 11 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.76333
−3.19746
0.243788
0.856749
2.39657
−1.06298
0 −2.64896 0 −3.47027 0 1.00000 0 4.01699 0
1.2 0 −2.52160 0 0.600745 0 1.00000 0 3.35847 0
1.3 0 −1.03321 0 1.36630 0 1.00000 0 −1.93247 0
1.4 0 0.601565 0 −4.19936 0 1.00000 0 −2.63812 0
1.5 0 1.20648 0 0.863883 0 1.00000 0 −1.54442 0
1.6 0 2.39573 0 −1.16130 0 1.00000 0 2.73954 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(17\) \( +1 \)

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 3808.2.a.i 6
4.b odd 2 1 3808.2.a.m yes 6
8.b even 2 1 7616.2.a.cb 6
8.d odd 2 1 7616.2.a.bx 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3808.2.a.i 6 1.a even 1 1 trivial
3808.2.a.m yes 6 4.b odd 2 1
7616.2.a.bx 6 8.d odd 2 1
7616.2.a.cb 6 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3808))\):

\( T_{3}^{6} + 2T_{3}^{5} - 9T_{3}^{4} - 14T_{3}^{3} + 21T_{3}^{2} + 14T_{3} - 12 \) Copy content Toggle raw display
\( T_{11}^{6} + 2T_{11}^{5} - 34T_{11}^{4} - 92T_{11}^{3} + 196T_{11}^{2} + 576T_{11} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} + \cdots - 12 \) Copy content Toggle raw display
$5$ \( T^{6} + 6 T^{5} + \cdots - 12 \) Copy content Toggle raw display
$7$ \( (T - 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 2 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( (T + 1)^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 10 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{6} - 4 T^{5} + \cdots - 5184 \) Copy content Toggle raw display
$29$ \( T^{6} + 14 T^{5} + \cdots + 768 \) Copy content Toggle raw display
$31$ \( T^{6} - 8 T^{5} + \cdots - 8128 \) Copy content Toggle raw display
$37$ \( T^{6} + 4 T^{5} + \cdots - 10176 \) Copy content Toggle raw display
$41$ \( T^{6} + 2 T^{5} + \cdots + 1084 \) Copy content Toggle raw display
$43$ \( T^{6} + 8 T^{5} + \cdots + 5392 \) Copy content Toggle raw display
$47$ \( T^{6} + 2 T^{5} + \cdots - 3072 \) Copy content Toggle raw display
$53$ \( T^{6} + 10 T^{5} + \cdots + 1764 \) Copy content Toggle raw display
$59$ \( T^{6} + 14 T^{5} + \cdots - 68864 \) Copy content Toggle raw display
$61$ \( T^{6} + 20 T^{5} + \cdots - 29572 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + \cdots - 576 \) Copy content Toggle raw display
$71$ \( T^{6} - 20 T^{5} + \cdots - 15552 \) Copy content Toggle raw display
$73$ \( T^{6} + 6 T^{5} + \cdots - 2916 \) Copy content Toggle raw display
$79$ \( T^{6} - 38 T^{5} + \cdots + 665024 \) Copy content Toggle raw display
$83$ \( T^{6} + 16 T^{5} + \cdots + 35184 \) Copy content Toggle raw display
$89$ \( T^{6} + 2 T^{5} + \cdots + 760432 \) Copy content Toggle raw display
$97$ \( T^{6} + 12 T^{5} + \cdots - 18324 \) Copy content Toggle raw display
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