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Properties

Label 432.9.e.j
Level $432$
Weight $9$
Character orbit 432.e
Analytic conductor $175.988$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,9,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, 9); 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, 9, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 432.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,-1470] 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: \(175.987559546\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 285x^{4} + 19144x^{2} + 260100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{24} \)
Twist minimal: no (minimal twist has level 108)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + ( - \beta_{3} - 245) q^{7} + ( - \beta_{4} + 6 \beta_1) q^{11} + ( - \beta_{5} - \beta_{3} + 11102) q^{13} + ( - 5 \beta_{4} - \beta_{2} - 7 \beta_1) q^{17} + ( - \beta_{5} - 53 \beta_{3} - 1520) q^{19}+ \cdots + (884 \beta_{5} - 24316 \beta_{3} + 27026507) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 1470 q^{7} + 66612 q^{13} - 9120 q^{19} - 269472 q^{25} + 1507254 q^{31} + 1950744 q^{37} - 2092476 q^{43} + 8065008 q^{49} + 14493006 q^{55} + 5993688 q^{61} + 44150316 q^{67} + 5298582 q^{73} - 100415604 q^{79}+ \cdots + 162159042 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 285x^{4} + 19144x^{2} + 260100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 735\nu^{5} + 185301\nu^{3} + 8219814\nu ) / 22168 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4419\nu^{5} - 520425\nu^{3} + 39034314\nu ) / 55420 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 81\nu^{4} + 29187\nu^{2} + 1613466 ) / 326 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14133\nu^{5} - 3255255\nu^{3} - 327663282\nu ) / 110840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4131\nu^{4} - 854793\nu^{2} - 22081086 ) / 326 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -27\beta_{4} + 7\beta_{2} - 87\beta_1 ) / 52488 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 51\beta_{3} - 184680 ) / 1944 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4023\beta_{4} + 3857\beta_{2} + 24747\beta_1 ) / 52488 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -1081\beta_{5} - 31659\beta_{3} + 83469528 ) / 5832 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -712287\beta_{4} - 1050673\beta_{2} - 3682947\beta_1 ) / 52488 \) Copy content Toggle raw display

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

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
4.26679i
− 8.61216i
− 13.8789i
13.8789i
8.61216i
− 4.26679i
0 0 0 − 979.684i 0 −3646.68 0 0 0
161.2 0 0 0 − 575.178i 0 79.3070 0 0 0
161.3 0 0 0 − 126.493i 0 2832.37 0 0 0
161.4 0 0 0 126.493i 0 2832.37 0 0 0
161.5 0 0 0 575.178i 0 79.3070 0 0 0
161.6 0 0 0 979.684i 0 −3646.68 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 161.6
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 432.9.e.j 6
3.b odd 2 1 inner 432.9.e.j 6
4.b odd 2 1 108.9.c.d ✓ 6
12.b even 2 1 108.9.c.d ✓ 6
36.f odd 6 2 324.9.g.g 12
36.h even 6 2 324.9.g.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.9.c.d ✓ 6 4.b odd 2 1
108.9.c.d ✓ 6 12.b even 2 1
324.9.g.g 12 36.f odd 6 2
324.9.g.g 12 36.h even 6 2
432.9.e.j 6 1.a even 1 1 trivial
432.9.e.j 6 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{6} + 1306611T_{5}^{4} + 338174115075T_{5}^{2} + 5080549720100625 \) Copy content Toggle raw display
\( T_{7}^{3} + 735T_{7}^{2} - 10393341T_{7} + 819143045 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{3} + 735 T^{2} + \cdots + 819143045)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 17\!\cdots\!89 \) Copy content Toggle raw display
$13$ \( (T^{3} + \cdots + 68399921049544)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots - 12\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 48\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 19\!\cdots\!75)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots - 23\!\cdots\!12)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 63\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{3} + \cdots - 49\!\cdots\!08)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 76\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 41\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + \cdots + 58\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots - 46\!\cdots\!01)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 94\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots + 47\!\cdots\!49)^{2} \) Copy content Toggle raw display
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