Properties

Label 4304.2.a.e
Level $4304$
Weight $2$
Character orbit 4304.a
Self dual yes
Analytic conductor $34.368$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4304,2,Mod(1,4304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4304.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4304 = 2^{4} \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4304.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: \(34.3676130300\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4913.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 538)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{3} + ( - \beta_{3} + 1) q^{5} + \beta_1 q^{7} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{9} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{11} + ( - \beta_{3} + \beta_{2} + 2 \beta_1) q^{13}+ \cdots + ( - \beta_{2} + \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} + 5 q^{5} + q^{7} + 3 q^{9} - 7 q^{11} + 4 q^{13} - 8 q^{15} + 4 q^{17} - 2 q^{19} - 5 q^{21} - 16 q^{23} - q^{25} - 6 q^{27} + q^{31} + q^{33} - 3 q^{35} - 2 q^{37} - 3 q^{39} - q^{41}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} + 4\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 6\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 6\beta _1 + 3 \) 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
−0.344151
0.487928
2.90570
−2.04948
0 −3.04948 0 1.48793 0 −0.344151 0 6.29934 0
1.2 0 −1.34415 0 3.90570 0 0.487928 0 −1.19326 0
1.3 0 −0.512072 0 −1.04948 0 2.90570 0 −2.73778 0
1.4 0 1.90570 0 0.655849 0 −2.04948 0 0.631706 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(269\) \( -1 \)

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 4304.2.a.e 4
4.b odd 2 1 538.2.a.c 4
12.b even 2 1 4842.2.a.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.2.a.c 4 4.b odd 2 1
4304.2.a.e 4 1.a even 1 1 trivial
4842.2.a.j 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3T_{3}^{3} - 3T_{3}^{2} - 10T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4304))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{4} - 5 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} - 6 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 7 T^{3} + \cdots - 13 \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + \cdots + 52 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 16 T^{3} + \cdots + 52 \) Copy content Toggle raw display
$29$ \( T^{4} - 51 T^{2} + \cdots + 17 \) Copy content Toggle raw display
$31$ \( T^{4} - T^{3} + \cdots - 67 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + \cdots - 188 \) Copy content Toggle raw display
$41$ \( T^{4} + T^{3} + \cdots + 52 \) Copy content Toggle raw display
$43$ \( T^{4} + 9 T^{3} + \cdots + 1937 \) Copy content Toggle raw display
$47$ \( T^{4} + 13 T^{3} + \cdots - 1172 \) Copy content Toggle raw display
$53$ \( (T^{2} - 14 T + 32)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 19 T^{3} + \cdots - 188 \) Copy content Toggle raw display
$61$ \( T^{4} + 20 T^{3} + \cdots + 268 \) Copy content Toggle raw display
$67$ \( T^{4} + 15 T^{3} + \cdots + 101 \) Copy content Toggle raw display
$71$ \( T^{4} + 15 T^{3} + \cdots - 2228 \) Copy content Toggle raw display
$73$ \( T^{4} + 7 T^{3} + \cdots + 6481 \) Copy content Toggle raw display
$79$ \( T^{4} - 17 T^{3} + \cdots + 68 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( T^{4} - 20 T^{3} + \cdots - 191 \) Copy content Toggle raw display
$97$ \( T^{4} - 3 T^{3} + \cdots + 27196 \) Copy content Toggle raw display
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