Properties

Label 7488.2.a.cg
Level $7488$
Weight $2$
Character orbit 7488.a
Self dual yes
Analytic conductor $59.792$
Analytic rank $0$
Dimension $2$
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] = [7488,2,Mod(1,7488)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7488, 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("7488.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7488.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: \(59.7919810335\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1248)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{5} + (\beta - 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{5} + (\beta - 3) q^{7} + 2 \beta q^{11} + q^{13} + 2 \beta q^{17} + (\beta + 1) q^{19} + (2 \beta + 2) q^{23} + (2 \beta + 1) q^{25} + (2 \beta - 4) q^{29} + (\beta - 7) q^{31} + (2 \beta - 2) q^{35} - 2 \beta q^{37} + ( - \beta + 7) q^{41} + ( - 2 \beta + 2) q^{43} + (2 \beta + 4) q^{47} + ( - 6 \beta + 7) q^{49} + ( - 2 \beta - 4) q^{53} + ( - 2 \beta - 10) q^{55} + ( - 4 \beta - 2) q^{59} + ( - 2 \beta - 8) q^{61} + ( - \beta - 1) q^{65} + (3 \beta - 1) q^{67} + 10 q^{71} + 2 \beta q^{73} + ( - 6 \beta + 10) q^{77} + 4 \beta q^{79} + ( - 4 \beta - 6) q^{83} + ( - 2 \beta - 10) q^{85} + ( - 3 \beta + 9) q^{89} + (\beta - 3) q^{91} + ( - 2 \beta - 6) q^{95} + ( - 2 \beta + 12) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 6 q^{7} + 2 q^{13} + 2 q^{19} + 4 q^{23} + 2 q^{25} - 8 q^{29} - 14 q^{31} - 4 q^{35} + 14 q^{41} + 4 q^{43} + 8 q^{47} + 14 q^{49} - 8 q^{53} - 20 q^{55} - 4 q^{59} - 16 q^{61} - 2 q^{65} - 2 q^{67} + 20 q^{71} + 20 q^{77} - 12 q^{83} - 20 q^{85} + 18 q^{89} - 6 q^{91} - 12 q^{95} + 24 q^{97}+O(q^{100}) \) 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
1.61803
−0.618034
0 0 0 −3.23607 0 −0.763932 0 0 0
1.2 0 0 0 1.23607 0 −5.23607 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(13\) \( -1 \)

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 7488.2.a.cg 2
3.b odd 2 1 2496.2.a.bg 2
4.b odd 2 1 7488.2.a.ch 2
8.b even 2 1 3744.2.a.v 2
8.d odd 2 1 3744.2.a.w 2
12.b even 2 1 2496.2.a.bj 2
24.f even 2 1 1248.2.a.k 2
24.h odd 2 1 1248.2.a.m yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.a.k 2 24.f even 2 1
1248.2.a.m yes 2 24.h odd 2 1
2496.2.a.bg 2 3.b odd 2 1
2496.2.a.bj 2 12.b even 2 1
3744.2.a.v 2 8.b even 2 1
3744.2.a.w 2 8.d odd 2 1
7488.2.a.cg 2 1.a even 1 1 trivial
7488.2.a.ch 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7488))\):

\( T_{5}^{2} + 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 20 \) Copy content Toggle raw display
\( T_{17}^{2} - 20 \) Copy content Toggle raw display
\( T_{19}^{2} - 2T_{19} - 4 \) Copy content Toggle raw display
\( T_{29}^{2} + 8T_{29} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 20 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$31$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$37$ \( T^{2} - 20 \) Copy content Toggle raw display
$41$ \( T^{2} - 14T + 44 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$71$ \( (T - 10)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 20 \) Copy content Toggle raw display
$79$ \( T^{2} - 80 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 24T + 124 \) Copy content Toggle raw display
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