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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4608,2,Mod(1,4608)] 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("4608.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4608, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,-8,0,0,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{10 +4 \sqrt{2}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1536)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - \beta_{3} q^{7} + ( - \beta_{2} - 2) q^{11} + (\beta_{3} - \beta_1) q^{13} + (\beta_{2} - 2) q^{17} - \beta_{2} q^{19} - 2 \beta_1 q^{23} + (2 \beta_{2} + 5) q^{25} + (2 \beta_{3} - \beta_1) q^{29}+ \cdots + (2 \beta_{2} + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{11} - 8 q^{17} + 20 q^{25} - 8 q^{35} - 24 q^{41} - 16 q^{43} + 28 q^{49} - 16 q^{59} - 32 q^{65} - 32 q^{67} + 16 q^{73} - 24 q^{83} - 40 q^{89} - 48 q^{91} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} - 2\nu^{2} - 8\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 3\nu^{2} + 2\nu - 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 4\beta_{2} - 5\beta _1 + 6 ) / 2 \) 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
2.68554
0.334904
−1.74912
−1.27133
0 0 0 −3.95687 0 −1.63899 0 0 0
1.2 0 0 0 −2.08402 0 5.03127 0 0 0
1.3 0 0 0 2.08402 0 −5.03127 0 0 0
1.4 0 0 0 3.95687 0 1.63899 0 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4608.2.a.t 4
3.b odd 2 1 1536.2.a.n yes 4
4.b odd 2 1 4608.2.a.ba 4
8.b even 2 1 4608.2.a.ba 4
8.d odd 2 1 inner 4608.2.a.t 4
12.b even 2 1 1536.2.a.m ✓ 4
16.e even 4 2 4608.2.d.p 8
16.f odd 4 2 4608.2.d.p 8
24.f even 2 1 1536.2.a.n yes 4
24.h odd 2 1 1536.2.a.m ✓ 4
48.i odd 4 2 1536.2.d.g 8
48.k even 4 2 1536.2.d.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.m ✓ 4 12.b even 2 1
1536.2.a.m ✓ 4 24.h odd 2 1
1536.2.a.n yes 4 3.b odd 2 1
1536.2.a.n yes 4 24.f even 2 1
1536.2.d.g 8 48.i odd 4 2
1536.2.d.g 8 48.k even 4 2
4608.2.a.t 4 1.a even 1 1 trivial
4608.2.a.t 4 8.d odd 2 1 inner
4608.2.a.ba 4 4.b odd 2 1
4608.2.a.ba 4 8.b even 2 1
4608.2.d.p 8 16.e even 4 2
4608.2.d.p 8 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4608))\):

\( T_{5}^{4} - 20T_{5}^{2} + 68 \) Copy content Toggle raw display
\( T_{7}^{4} - 28T_{7}^{2} + 68 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 4 \) Copy content Toggle raw display
\( T_{23}^{4} - 80T_{23}^{2} + 1088 \) 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
$5$ \( T^{4} - 20T^{2} + 68 \) Copy content Toggle raw display
$7$ \( T^{4} - 28T^{2} + 68 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 40T^{2} + 272 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 80T^{2} + 1088 \) Copy content Toggle raw display
$29$ \( T^{4} - 116T^{2} + 3332 \) Copy content Toggle raw display
$31$ \( T^{4} - 28T^{2} + 68 \) Copy content Toggle raw display
$37$ \( T^{4} - 56T^{2} + 272 \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 56)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 80T^{2} + 1088 \) Copy content Toggle raw display
$53$ \( T^{4} - 148T^{2} + 68 \) Copy content Toggle raw display
$59$ \( (T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 56T^{2} + 272 \) Copy content Toggle raw display
$67$ \( (T^{2} + 16 T + 32)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 112T^{2} + 1088 \) Copy content Toggle raw display
$73$ \( (T - 4)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - 28T^{2} + 68 \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
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