Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,8,Mod(1,432)] 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("432.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,324,0,980] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(134.950331009\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1289}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 322 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 108)
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 \(\beta = 9\sqrt{1289}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 162) q^{5} + ( - 3 \beta + 490) q^{7} + (4 \beta - 405) q^{11} + (24 \beta - 3940) q^{13} + (40 \beta - 10368) q^{17} + (30 \beta - 5228) q^{19} + ( - 40 \beta + 30294) q^{23} + ( - 324 \beta + 52528) q^{25}+ \cdots + ( - 18552 \beta + 1494455) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 324 q^{5} + 980 q^{7} - 810 q^{11} - 7880 q^{13} - 20736 q^{17} - 10456 q^{19} + 60588 q^{23} + 105056 q^{25} + 32400 q^{29} + 145988 q^{31} + 785214 q^{35} - 10460 q^{37} - 1085400 q^{41} + 606440 q^{43}+ \cdots + 2988910 q^{97}+O(q^{100}) \) 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
18.4513
−17.4513
0 0 0 −161.124 0 −479.371 0 0 0
1.2 0 0 0 485.124 0 1459.37 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

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 432.8.a.r 2
3.b odd 2 1 432.8.a.i 2
4.b odd 2 1 108.8.a.e yes 2
12.b even 2 1 108.8.a.b ✓ 2
36.f odd 6 2 324.8.e.g 4
36.h even 6 2 324.8.e.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.8.a.b ✓ 2 12.b even 2 1
108.8.a.e yes 2 4.b odd 2 1
324.8.e.g 4 36.f odd 6 2
324.8.e.j 4 36.h even 6 2
432.8.a.i 2 3.b odd 2 1
432.8.a.r 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(432))\):

\( T_{5}^{2} - 324T_{5} - 78165 \) Copy content Toggle raw display
\( T_{7}^{2} - 980T_{7} - 699581 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 324T - 78165 \) Copy content Toggle raw display
$7$ \( T^{2} - 980T - 699581 \) Copy content Toggle raw display
$11$ \( T^{2} + 810 T - 1506519 \) Copy content Toggle raw display
$13$ \( T^{2} + 7880 T - 44615984 \) Copy content Toggle raw display
$17$ \( T^{2} + 20736 T - 59558976 \) Copy content Toggle raw display
$19$ \( T^{2} + 10456 T - 66636116 \) Copy content Toggle raw display
$23$ \( T^{2} - 60588 T + 750672036 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 40131731556 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 11797562189 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 3821580476 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 276972137100 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 239610916916 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 485764195716 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 883947989259 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 1945901264400 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 572702600000 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 894770581324 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 473739274944 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 4430654478791 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 211199920000 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 1188922519881 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 3197429581356 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 33701749740911 \) Copy content Toggle raw display
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