Properties

Label 4680.2.a.bi
Level $4680$
Weight $2$
Character orbit 4680.a
Self dual yes
Analytic conductor $37.370$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4680,2,Mod(1,4680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.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: \(37.3699881460\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + (\beta_{2} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + (\beta_{2} + 1) q^{7} + (\beta_{2} + \beta_1 - 1) q^{11} + q^{13} + (\beta_{2} - 1) q^{17} + (\beta_1 + 2) q^{19} + (\beta_{2} - 1) q^{23} + q^{25} + \beta_1 q^{29} + 2 q^{31} + ( - \beta_{2} - 1) q^{35} + (\beta_{2} - \beta_1 + 1) q^{37} + ( - 3 \beta_{2} - \beta_1 + 1) q^{41} + (2 \beta_{2} + 2) q^{43} + (2 \beta_{2} - 2) q^{47} + (\beta_{2} - \beta_1) q^{49} + ( - \beta_{2} - \beta_1 + 1) q^{53} + ( - \beta_{2} - \beta_1 + 1) q^{55} - 2 \beta_1 q^{59} + ( - \beta_{2} + \beta_1 + 3) q^{61} - q^{65} + ( - 2 \beta_{2} + 4) q^{67} + ( - 3 \beta_{2} + \beta_1 - 1) q^{71} + (2 \beta_{2} - \beta_1) q^{73} + ( - 3 \beta_{2} + \beta_1 + 3) q^{77} + (3 \beta_{2} + \beta_1 + 5) q^{79} + ( - \beta_{2} + 1) q^{85} + (3 \beta_{2} + \beta_1 + 3) q^{89} + (\beta_{2} + 1) q^{91} + ( - \beta_1 - 2) q^{95} + ( - 3 \beta_{2} - 2 \beta_1 + 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + 3 q^{7} - 3 q^{11} + 3 q^{13} - 3 q^{17} + 6 q^{19} - 3 q^{23} + 3 q^{25} + 6 q^{31} - 3 q^{35} + 3 q^{37} + 3 q^{41} + 6 q^{43} - 6 q^{47} + 3 q^{53} + 3 q^{55} + 9 q^{61} - 3 q^{65} + 12 q^{67} - 3 q^{71} + 9 q^{77} + 15 q^{79} + 3 q^{85} + 9 q^{89} + 3 q^{91} - 6 q^{95} + 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + \beta _1 + 8 ) / 2 \) 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
−0.523976
2.66908
−2.14510
0 0 0 −1.00000 0 −2.20147 0 0 0
1.2 0 0 0 −1.00000 0 1.45490 0 0 0
1.3 0 0 0 −1.00000 0 3.74657 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(1\)
\(13\) \(-1\)

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 4680.2.a.bi 3
3.b odd 2 1 4680.2.a.bk yes 3
4.b odd 2 1 9360.2.a.cx 3
12.b even 2 1 9360.2.a.dc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4680.2.a.bi 3 1.a even 1 1 trivial
4680.2.a.bk yes 3 3.b odd 2 1
9360.2.a.cx 3 4.b odd 2 1
9360.2.a.dc 3 12.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4680))\):

\( T_{7}^{3} - 3T_{7}^{2} - 6T_{7} + 12 \) Copy content Toggle raw display
\( T_{11}^{3} + 3T_{11}^{2} - 24T_{11} - 64 \) Copy content Toggle raw display
\( T_{17}^{3} + 3T_{17}^{2} - 6T_{17} - 4 \) Copy content Toggle raw display
\( T_{19}^{3} - 6T_{19}^{2} - 12T_{19} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 3 T^{2} + \cdots + 12 \) Copy content Toggle raw display
$11$ \( T^{3} + 3 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 3 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$19$ \( T^{3} - 6 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{3} + 3 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$29$ \( T^{3} - 24T - 24 \) Copy content Toggle raw display
$31$ \( (T - 2)^{3} \) Copy content Toggle raw display
$37$ \( T^{3} - 3 T^{2} + \cdots - 36 \) Copy content Toggle raw display
$41$ \( T^{3} - 3 T^{2} + \cdots - 196 \) Copy content Toggle raw display
$43$ \( T^{3} - 6 T^{2} + \cdots + 96 \) Copy content Toggle raw display
$47$ \( T^{3} + 6 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$53$ \( T^{3} - 3 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( T^{3} - 96T + 192 \) Copy content Toggle raw display
$61$ \( T^{3} - 9 T^{2} + \cdots + 164 \) Copy content Toggle raw display
$67$ \( T^{3} - 12 T^{2} + \cdots + 48 \) Copy content Toggle raw display
$71$ \( T^{3} + 3 T^{2} + \cdots + 304 \) Copy content Toggle raw display
$73$ \( T^{3} - 72T - 232 \) Copy content Toggle raw display
$79$ \( T^{3} - 15 T^{2} + \cdots + 592 \) Copy content Toggle raw display
$83$ \( T^{3} \) Copy content Toggle raw display
$89$ \( T^{3} - 9 T^{2} + \cdots + 516 \) Copy content Toggle raw display
$97$ \( T^{3} - 15 T^{2} + \cdots + 628 \) Copy content Toggle raw display
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