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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4536,2,Mod(1,4536)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4536.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4536, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4536 = 2^{3} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4536.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-4,0,4,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(36.2201423569\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{15 +4 \sqrt{3}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 9x^{2} - 6x + 6 \) 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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1 - 1) q^{5} + q^{7} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{3} + \beta_1 + 1) q^{13} + ( - \beta_{3} - \beta_1 - 1) q^{17} + (\beta_{3} - \beta_{2}) q^{19}+ \cdots + (2 \beta_{3} + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 4 q^{7} - 6 q^{11} + 6 q^{13} - 2 q^{17} - 2 q^{19} - 10 q^{23} + 2 q^{25} - 10 q^{29} - 2 q^{31} - 4 q^{35} + 6 q^{37} - 8 q^{41} + 2 q^{43} - 10 q^{47} + 4 q^{49} - 4 q^{53} - 6 q^{55}+ \cdots + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 9x^{2} - 6x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 6\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 7\beta _1 + 5 \) 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.

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
0.554520
3.20740
−2.28657
−1.47535
0 0 0 −3.28657 0 1.00000 0 0 0
1.2 0 0 0 −2.47535 0 1.00000 0 0 0
1.3 0 0 0 −0.445480 0 1.00000 0 0 0
1.4 0 0 0 2.20740 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)

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 4536.2.a.w ✓ 4
3.b odd 2 1 4536.2.a.bb yes 4
4.b odd 2 1 9072.2.a.cf 4
12.b even 2 1 9072.2.a.ck 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4536.2.a.w ✓ 4 1.a even 1 1 trivial
4536.2.a.bb yes 4 3.b odd 2 1
9072.2.a.cf 4 4.b odd 2 1
9072.2.a.ck 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4536))\):

\( T_{5}^{4} + 4T_{5}^{3} - 3T_{5}^{2} - 20T_{5} - 8 \) Copy content Toggle raw display
\( T_{11}^{4} + 6T_{11}^{3} - 9T_{11}^{2} - 54T_{11} + 54 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + \cdots + 54 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + \cdots - 48 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 52 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 10 T^{3} + \cdots - 104 \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} + \cdots - 128 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} + \cdots - 3 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + \cdots - 104 \) Copy content Toggle raw display
$47$ \( T^{4} + 10 T^{3} + \cdots - 392 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + \cdots + 568 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + \cdots + 3768 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + \cdots + 5812 \) Copy content Toggle raw display
$67$ \( T^{4} - 129 T^{2} + \cdots + 3246 \) Copy content Toggle raw display
$71$ \( T^{4} - 159 T^{2} + \cdots + 1752 \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + \cdots - 752 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} + \cdots + 118 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + \cdots - 48 \) Copy content Toggle raw display
$89$ \( T^{4} + 26 T^{3} + \cdots - 5024 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
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