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Properties

Label 7616.2.a.bn
Level $7616$
Weight $2$
Character orbit 7616.a
Self dual yes
Analytic conductor $60.814$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.9301.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 119)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} + \beta_{2} q^{5} - q^{7} + ( - \beta_{3} + 2) q^{9} + \beta_1 q^{11} + ( - \beta_{3} + \beta_{2} - 1) q^{13} + ( - \beta_{3} - \beta_{2} + 4) q^{15} - q^{17} + ( - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{19}+ \cdots + (\beta_{3} - 3 \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 6 q^{9} + 2 q^{11} - 8 q^{13} + 16 q^{15} - 4 q^{17} + 10 q^{19} - 2 q^{21} + 6 q^{23} - 2 q^{25} + 2 q^{27} - 2 q^{29} - 12 q^{31} - 2 q^{33} + 2 q^{35} - 6 q^{37} + 18 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 3\nu^{2} + 2\nu - 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 3\beta_{2} + 5\beta _1 + 5 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
0.869986
−1.68863
2.60312
−0.784476
0 −2.57835 0 −3.57835 0 −1.00000 0 3.64787 0
1.2 0 0.0879544 0 −0.912046 0 −1.00000 0 −2.99226 0
1.3 0 1.45066 0 0.450660 0 −1.00000 0 −0.895586 0
1.4 0 3.03973 0 2.03973 0 −1.00000 0 6.23998 0
\(n\): e.g. 2-40 or 990-1000
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.bn 4
4.b odd 2 1 7616.2.a.bk 4
8.b even 2 1 1904.2.a.s 4
8.d odd 2 1 119.2.a.a ✓ 4
24.f even 2 1 1071.2.a.k 4
40.e odd 2 1 2975.2.a.k 4
56.e even 2 1 833.2.a.e 4
56.k odd 6 2 833.2.e.e 8
56.m even 6 2 833.2.e.f 8
136.e odd 2 1 2023.2.a.e 4
168.e odd 2 1 7497.2.a.bl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.2.a.a ✓ 4 8.d odd 2 1
833.2.a.e 4 56.e even 2 1
833.2.e.e 8 56.k odd 6 2
833.2.e.f 8 56.m even 6 2
1071.2.a.k 4 24.f even 2 1
1904.2.a.s 4 8.b even 2 1
2023.2.a.e 4 136.e odd 2 1
2975.2.a.k 4 40.e odd 2 1
7497.2.a.bl 4 168.e odd 2 1
7616.2.a.bk 4 4.b odd 2 1
7616.2.a.bn 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7616))\):

\( T_{3}^{4} - 2T_{3}^{3} - 7T_{3}^{2} + 12T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{4} + 2T_{5}^{3} - 7T_{5}^{2} - 4T_{5} + 3 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} - 20T_{11}^{2} + 8T_{11} + 48 \) Copy content Toggle raw display
\( T_{19}^{4} - 10T_{19}^{3} - 20T_{19}^{2} + 392T_{19} - 784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 48 \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} + \cdots - 368 \) Copy content Toggle raw display
$17$ \( (T + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + \cdots - 784 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + \cdots - 240 \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} + \cdots + 48 \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + \cdots - 917 \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} + \cdots + 80 \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + \cdots - 237 \) Copy content Toggle raw display
$43$ \( T^{4} + 12 T^{3} + \cdots - 115 \) Copy content Toggle raw display
$47$ \( T^{4} + 2 T^{3} + \cdots + 1776 \) Copy content Toggle raw display
$53$ \( T^{4} - 26 T^{3} + \cdots + 801 \) Copy content Toggle raw display
$59$ \( T^{4} + 4 T^{3} + \cdots - 768 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + \cdots + 6451 \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} + \cdots + 1949 \) Copy content Toggle raw display
$71$ \( T^{4} - 14 T^{3} + \cdots - 3312 \) Copy content Toggle raw display
$73$ \( T^{4} - 20 T^{3} + \cdots + 131 \) Copy content Toggle raw display
$79$ \( T^{4} - 14 T^{3} + \cdots - 400 \) Copy content Toggle raw display
$83$ \( T^{4} + 28 T^{3} + \cdots + 1200 \) Copy content Toggle raw display
$89$ \( T^{4} + 10 T^{3} + \cdots + 720 \) Copy content Toggle raw display
$97$ \( T^{4} - 26 T^{3} + \cdots - 1901 \) Copy content Toggle raw display
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