Properties

Label 4020.2.a.d
Level $4020$
Weight $2$
Character orbit 4020.a
Self dual yes
Analytic conductor $32.100$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4020,2,Mod(1,4020)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4020, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4020.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.0998616126\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + (\beta_{3} - \beta_{2}) q^{7} + q^{9} + ( - 3 \beta_{3} + \beta_1 - 3) q^{11} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{13} + q^{15} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1 - 4) q^{17}+ \cdots + ( - 3 \beta_{3} + \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 5 q^{11} - 9 q^{13} + 4 q^{15} - 8 q^{17} - 7 q^{19} - 3 q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 9 q^{29} - 9 q^{31} - 5 q^{33} - 3 q^{35} - 20 q^{37} - 9 q^{39}+ \cdots - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{15} + \zeta_{15}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−1.95630
1.33826
1.82709
−0.209057
0 1.00000 0 1.00000 0 −3.44512 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 −1.40898 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 −0.720227 0 1.00000 0
1.4 0 1.00000 0 1.00000 0 2.57433 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)
\(67\) \( -1 \)

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 4020.2.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.d 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 3T_{7}^{3} - 6T_{7}^{2} - 18T_{7} - 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 3 T^{3} + \cdots - 9 \) Copy content Toggle raw display
$11$ \( T^{4} + 5 T^{3} + \cdots - 5 \) Copy content Toggle raw display
$13$ \( T^{4} + 9 T^{3} + \cdots - 29 \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + \cdots - 269 \) Copy content Toggle raw display
$19$ \( T^{4} + 7 T^{3} + \cdots - 59 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + \cdots - 719 \) Copy content Toggle raw display
$29$ \( T^{4} + 9 T^{3} + \cdots + 181 \) Copy content Toggle raw display
$31$ \( T^{4} + 9 T^{3} + \cdots + 691 \) Copy content Toggle raw display
$37$ \( T^{4} + 20 T^{3} + \cdots - 1055 \) Copy content Toggle raw display
$41$ \( T^{4} + 11 T^{3} + \cdots - 29 \) Copy content Toggle raw display
$43$ \( T^{4} - T^{3} + \cdots - 29 \) Copy content Toggle raw display
$47$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$53$ \( T^{4} + 15 T^{3} + \cdots + 45 \) Copy content Toggle raw display
$59$ \( T^{4} - 7 T^{3} + \cdots - 269 \) Copy content Toggle raw display
$61$ \( T^{4} + 7 T^{3} + \cdots + 61 \) Copy content Toggle raw display
$67$ \( (T - 1)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 190 T^{2} + \cdots + 4495 \) Copy content Toggle raw display
$73$ \( T^{4} + 26 T^{3} + \cdots - 3719 \) Copy content Toggle raw display
$79$ \( T^{4} + 9 T^{3} + \cdots + 261 \) Copy content Toggle raw display
$83$ \( T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{4} - 10 T^{3} + \cdots + 3775 \) Copy content Toggle raw display
$97$ \( T^{4} + 11 T^{3} + \cdots - 149 \) Copy content Toggle raw display
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