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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2268,2,Mod(1,2268)] 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("2268.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-4,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.1100711784\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{19})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 11x^{2} + 16 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - q^{7} + ( - \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} - 2) q^{13} + (2 \beta_{2} - \beta_1) q^{17} + 2 \beta_{3} q^{19} + (2 \beta_{2} - 4 \beta_1) q^{23} + (\beta_{3} + 1) q^{25}+ \cdots + (2 \beta_{3} + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7} - 6 q^{13} - 4 q^{19} + 2 q^{25} - 24 q^{31} - 16 q^{37} - 10 q^{43} + 4 q^{49} - 28 q^{55} - 30 q^{61} - 14 q^{67} - 26 q^{73} - 30 q^{79} - 10 q^{85} + 6 q^{91} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 11x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 7\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} + 7\beta_1 \) 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
−3.04547
−1.31342
1.31342
3.04547
0 0 0 −3.04547 0 −1.00000 0 0 0
1.2 0 0 0 −1.31342 0 −1.00000 0 0 0
1.3 0 0 0 1.31342 0 −1.00000 0 0 0
1.4 0 0 0 3.04547 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.a.k ✓ 4
3.b odd 2 1 inner 2268.2.a.k ✓ 4
4.b odd 2 1 9072.2.a.ch 4
9.c even 3 2 2268.2.j.r 8
9.d odd 6 2 2268.2.j.r 8
12.b even 2 1 9072.2.a.ch 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.a.k ✓ 4 1.a even 1 1 trivial
2268.2.a.k ✓ 4 3.b odd 2 1 inner
2268.2.j.r 8 9.c even 3 2
2268.2.j.r 8 9.d odd 6 2
9072.2.a.ch 4 4.b odd 2 1
9072.2.a.ch 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2268))\):

\( T_{5}^{4} - 11T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} - 23T_{11}^{2} + 4 \) 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
$5$ \( T^{4} - 11T^{2} + 16 \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 23T^{2} + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} + 3 T - 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 23T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 56)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 76)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 83T^{2} + 1024 \) Copy content Toggle raw display
$31$ \( (T + 6)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T - 41)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 5 T - 8)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 123T^{2} + 576 \) Copy content Toggle raw display
$59$ \( (T^{2} - 76)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 15 T + 42)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 7 T - 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 83T^{2} + 1024 \) Copy content Toggle raw display
$73$ \( (T^{2} + 13 T + 28)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 15 T + 42)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 92T^{2} + 64 \) Copy content Toggle raw display
$89$ \( T^{4} - 267T^{2} + 2304 \) Copy content Toggle raw display
$97$ \( (T^{2} - 14 T - 8)^{2} \) Copy content Toggle raw display
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