Properties

Label 7616.2.a.bu
Level $7616$
Weight $2$
Character orbit 7616.a
Self dual yes
Analytic conductor $60.814$
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] = [7616,2,Mod(1,7616)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7616, 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("7616.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7616.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: \(60.8140661794\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.109859312.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 7x^{4} + 15x^{3} + 13x^{2} - 9x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3808)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 1) q^{3} + (\beta_1 - 1) q^{5} + q^{7} + (2 \beta_{5} + \beta_{4} - \beta_{2} + 2) q^{9} + (\beta_{2} - 1) q^{11} + (\beta_{3} - \beta_1 - 1) q^{13} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 1) q^{15}+ \cdots + (2 \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} - 8 q^{13} - 6 q^{17} - 6 q^{19} - 4 q^{21} + 18 q^{23} + 12 q^{25} - 22 q^{27} + 8 q^{29} - 4 q^{31} - 6 q^{33} - 4 q^{35} + 8 q^{37} + 14 q^{39}+ \cdots - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 7x^{4} + 15x^{3} + 13x^{2} - 9x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} + 11\nu^{3} - 8\nu^{2} - 8\nu - 19 ) / 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} + 22\nu^{3} - 30\nu^{2} - 63\nu + 3 ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 11\nu^{3} - 37\nu^{2} + 73\nu + 18 ) / 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{5} + 13\nu^{4} + 11\nu^{3} - 67\nu^{2} - 12\nu + 32 ) / 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{5} + 10\nu^{4} + 33\nu^{3} - 49\nu^{2} - 60\nu + 17 ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - \beta_{4} + \beta_{3} + 3\beta_{2} + 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{5} - 7\beta_{4} + 7\beta_{3} + 13\beta_{2} + 2\beta _1 + 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -24\beta_{5} - 13\beta_{4} + 17\beta_{3} + 43\beta_{2} + 16\beta _1 + 63 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -74\beta_{5} - 63\beta_{4} + 73\beta_{3} + 163\beta_{2} + 32\beta _1 + 189 ) / 2 \) 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
−0.184385
2.13809
−1.10086
3.55174
−2.04046
0.635879
0 −3.38207 0 −2.62360 0 1.00000 0 8.43840 0
1.2 0 −2.59220 0 3.54237 0 1.00000 0 3.71950 0
1.3 0 −1.07243 0 −3.18060 0 1.00000 0 −1.84989 0
1.4 0 −0.531974 0 −0.110568 0 1.00000 0 −2.71700 0
1.5 0 1.73670 0 1.54368 0 1.00000 0 0.0161353 0
1.6 0 1.84197 0 −3.17127 0 1.00000 0 0.392864 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 7616.2.a.bu 6
4.b odd 2 1 7616.2.a.cc 6
8.b even 2 1 3808.2.a.p yes 6
8.d odd 2 1 3808.2.a.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3808.2.a.h 6 8.d odd 2 1
3808.2.a.p yes 6 8.b even 2 1
7616.2.a.bu 6 1.a even 1 1 trivial
7616.2.a.cc 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7616))\):

\( T_{3}^{6} + 4T_{3}^{5} - 5T_{3}^{4} - 26T_{3}^{3} + 3T_{3}^{2} + 38T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{6} + 4T_{5}^{5} - 13T_{5}^{4} - 62T_{5}^{3} + 5T_{5}^{2} + 146T_{5} + 16 \) Copy content Toggle raw display
\( T_{11}^{6} + 8T_{11}^{5} - 2T_{11}^{4} - 124T_{11}^{3} - 156T_{11}^{2} + 184T_{11} - 32 \) Copy content Toggle raw display
\( T_{19}^{6} + 6T_{19}^{5} - 8T_{19}^{4} - 60T_{19}^{3} - 4T_{19}^{2} + 64T_{19} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 4 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{6} + 4 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T - 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 8 T^{5} + \cdots - 32 \) Copy content Toggle raw display
$13$ \( T^{6} + 8 T^{5} + \cdots - 256 \) Copy content Toggle raw display
$17$ \( (T + 1)^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 6 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$23$ \( T^{6} - 18 T^{5} + \cdots - 256 \) Copy content Toggle raw display
$29$ \( T^{6} - 8 T^{5} + \cdots - 40864 \) Copy content Toggle raw display
$31$ \( T^{6} + 4 T^{5} + \cdots - 11584 \) Copy content Toggle raw display
$37$ \( T^{6} - 8 T^{5} + \cdots + 1088 \) Copy content Toggle raw display
$41$ \( T^{6} - 16 T^{5} + \cdots - 85252 \) Copy content Toggle raw display
$43$ \( T^{6} + 12 T^{5} + \cdots - 7936 \) Copy content Toggle raw display
$47$ \( T^{6} - 18 T^{5} + \cdots + 20096 \) Copy content Toggle raw display
$53$ \( T^{6} - 2 T^{5} + \cdots - 25652 \) Copy content Toggle raw display
$59$ \( T^{6} + 16 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} + \cdots + 41248 \) Copy content Toggle raw display
$67$ \( T^{6} + 12 T^{5} + \cdots - 1888 \) Copy content Toggle raw display
$71$ \( T^{6} + 2 T^{5} + \cdots + 161024 \) Copy content Toggle raw display
$73$ \( T^{6} - 8 T^{5} + \cdots - 120964 \) Copy content Toggle raw display
$79$ \( T^{6} + 18 T^{5} + \cdots + 741376 \) Copy content Toggle raw display
$83$ \( T^{6} + 28 T^{5} + \cdots + 46768 \) Copy content Toggle raw display
$89$ \( T^{6} + 2 T^{5} + \cdots - 66544 \) Copy content Toggle raw display
$97$ \( T^{6} + 18 T^{5} + \cdots - 4516 \) Copy content Toggle raw display
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