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Properties

Label 432.5.e.c
Level $432$
Weight $5$
Character orbit 432.e
Analytic conductor $44.656$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,5,Mod(161,432)] 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("432.161"); S:= CuspForms(chi, 5); 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, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 432.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-34] 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: no
Analytic conductor: \(44.6558240522\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 6\sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - 17 q^{7} - 11 \beta q^{11} + 95 q^{13} + 21 \beta q^{17} - 209 q^{19} + 59 \beta q^{23} + 409 q^{25} - 22 \beta q^{29} - 950 q^{31} - 17 \beta q^{35} - 1177 q^{37} - 146 \beta q^{41} + \cdots - 2809 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 34 q^{7} + 190 q^{13} - 418 q^{19} + 818 q^{25} - 1900 q^{31} - 2354 q^{37} - 2860 q^{43} - 4224 q^{49} + 4752 q^{55} - 2882 q^{61} - 6994 q^{67} - 18050 q^{73} - 10546 q^{79} - 9072 q^{85} - 3230 q^{91}+ \cdots - 5618 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
− 2.44949i
2.44949i
0 0 0 − 14.6969i 0 −17.0000 0 0 0
161.2 0 0 0 14.6969i 0 −17.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 432.5.e.c 2
3.b odd 2 1 inner 432.5.e.c 2
4.b odd 2 1 27.5.b.b ✓ 2
12.b even 2 1 27.5.b.b ✓ 2
20.d odd 2 1 675.5.c.d 2
20.e even 4 2 675.5.d.g 4
36.f odd 6 2 81.5.d.d 4
36.h even 6 2 81.5.d.d 4
60.h even 2 1 675.5.c.d 2
60.l odd 4 2 675.5.d.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.5.b.b ✓ 2 4.b odd 2 1
27.5.b.b ✓ 2 12.b even 2 1
81.5.d.d 4 36.f odd 6 2
81.5.d.d 4 36.h even 6 2
432.5.e.c 2 1.a even 1 1 trivial
432.5.e.c 2 3.b odd 2 1 inner
675.5.c.d 2 20.d odd 2 1
675.5.c.d 2 60.h even 2 1
675.5.d.g 4 20.e even 4 2
675.5.d.g 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{2} + 216 \) Copy content Toggle raw display
\( T_{7} + 17 \) 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
$5$ \( T^{2} + 216 \) Copy content Toggle raw display
$7$ \( (T + 17)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 26136 \) Copy content Toggle raw display
$13$ \( (T - 95)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 95256 \) Copy content Toggle raw display
$19$ \( (T + 209)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 751896 \) Copy content Toggle raw display
$29$ \( T^{2} + 104544 \) Copy content Toggle raw display
$31$ \( (T + 950)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1177)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4604256 \) Copy content Toggle raw display
$43$ \( (T + 1430)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2472984 \) Copy content Toggle raw display
$53$ \( T^{2} + 8468064 \) Copy content Toggle raw display
$59$ \( T^{2} + 4541400 \) Copy content Toggle raw display
$61$ \( (T + 1441)^{2} \) Copy content Toggle raw display
$67$ \( (T + 3497)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 3763584 \) Copy content Toggle raw display
$73$ \( (T + 9025)^{2} \) Copy content Toggle raw display
$79$ \( (T + 5273)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 37740384 \) Copy content Toggle raw display
$89$ \( T^{2} + 124433496 \) Copy content Toggle raw display
$97$ \( (T + 2809)^{2} \) Copy content Toggle raw display
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