Properties

Label 7616.2.a.br
Level $7616$
Weight $2$
Character orbit 7616.a
Self dual yes
Analytic conductor $60.814$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.804272.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 9x^{3} - 6x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} - q^{7} + (\beta_{4} + \beta_{3} + \beta_{2} + \cdots + 1) q^{9} + ( - \beta_{3} + \beta_1 + 1) q^{11} + ( - \beta_{4} + 1) q^{13} + (\beta_{4} + \beta_{3} + \beta_{2} + 2) q^{15}+ \cdots + (\beta_{4} + 5 \beta_{3} + \cdots + 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{5} - 5 q^{7} + 3 q^{9} + 4 q^{11} + 6 q^{13} + 8 q^{15} - 5 q^{17} + 6 q^{19} + 18 q^{23} + 5 q^{25} - 18 q^{27} + 14 q^{29} + 16 q^{31} - 26 q^{33} - 2 q^{35} - 2 q^{37} + 6 q^{39} - 10 q^{41}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - \nu^{3} - 8\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{4} - \nu^{3} - 17\nu^{2} - 4\nu + 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\nu^{4} + 2\nu^{3} + 26\nu^{2} + \nu - 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 2\beta_{3} - \beta_{2} + 9\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{4} + 10\beta_{3} + 8\beta_{2} + 15\beta _1 + 33 \) 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
3.24142
0.582493
−0.540819
−0.783617
−2.49947
0 −3.24142 0 −1.62440 0 −1.00000 0 7.50679 0
1.2 0 −0.582493 0 3.85102 0 −1.00000 0 −2.66070 0
1.3 0 0.540819 0 −2.15728 0 −1.00000 0 −2.70751 0
1.4 0 0.783617 0 −0.768652 0 −1.00000 0 −2.38595 0
1.5 0 2.49947 0 2.69931 0 −1.00000 0 3.24737 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.br 5
4.b odd 2 1 7616.2.a.bs 5
8.b even 2 1 3808.2.a.e 5
8.d odd 2 1 3808.2.a.f yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3808.2.a.e 5 8.b even 2 1
3808.2.a.f yes 5 8.d odd 2 1
7616.2.a.br 5 1.a even 1 1 trivial
7616.2.a.bs 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7616))\):

\( T_{3}^{5} - 9T_{3}^{3} + 6T_{3}^{2} + 3T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{5} - 2T_{5}^{4} - 13T_{5}^{3} + 8T_{5}^{2} + 49T_{5} + 28 \) Copy content Toggle raw display
\( T_{11}^{5} - 4T_{11}^{4} - 22T_{11}^{3} + 76T_{11}^{2} + 100T_{11} - 200 \) Copy content Toggle raw display
\( T_{19}^{5} - 6T_{19}^{4} - 36T_{19}^{3} + 204T_{19}^{2} + 156T_{19} - 848 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 9 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( T^{5} - 2 T^{4} + \cdots + 28 \) Copy content Toggle raw display
$7$ \( (T + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 4 T^{4} + \cdots - 200 \) Copy content Toggle raw display
$13$ \( T^{5} - 6 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$17$ \( (T + 1)^{5} \) Copy content Toggle raw display
$19$ \( T^{5} - 6 T^{4} + \cdots - 848 \) Copy content Toggle raw display
$23$ \( T^{5} - 18 T^{4} + \cdots - 464 \) Copy content Toggle raw display
$29$ \( T^{5} - 14 T^{4} + \cdots - 3232 \) Copy content Toggle raw display
$31$ \( T^{5} - 16 T^{4} + \cdots - 112 \) Copy content Toggle raw display
$37$ \( T^{5} + 2 T^{4} + \cdots + 1616 \) Copy content Toggle raw display
$41$ \( T^{5} + 10 T^{4} + \cdots - 134 \) Copy content Toggle raw display
$43$ \( T^{5} - 20 T^{4} + \cdots - 33500 \) Copy content Toggle raw display
$47$ \( T^{5} - 26 T^{4} + \cdots + 15552 \) Copy content Toggle raw display
$53$ \( T^{5} - 145 T^{3} + \cdots - 12702 \) Copy content Toggle raw display
$59$ \( T^{5} - 200 T^{3} + \cdots - 3584 \) Copy content Toggle raw display
$61$ \( T^{5} + 8 T^{4} + \cdots + 488 \) Copy content Toggle raw display
$67$ \( T^{5} + 12 T^{4} + \cdots + 6932 \) Copy content Toggle raw display
$71$ \( T^{5} - 10 T^{4} + \cdots - 2512 \) Copy content Toggle raw display
$73$ \( T^{5} + 14 T^{4} + \cdots + 27606 \) Copy content Toggle raw display
$79$ \( T^{5} - 34 T^{4} + \cdots - 10704 \) Copy content Toggle raw display
$83$ \( T^{5} + 20 T^{4} + \cdots - 512 \) Copy content Toggle raw display
$89$ \( T^{5} + 4 T^{4} + \cdots - 48168 \) Copy content Toggle raw display
$97$ \( T^{5} + 12 T^{4} + \cdots + 114074 \) Copy content Toggle raw display
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