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Properties

Label 432.9.e.k
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 = [6,0,0,0,0,0,1698] 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: 6.0.6171673600.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 40x^{3} + 225x^{2} + 150x + 50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{21} \)
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 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.

\(f(q)\) \(=\) \( q + (\beta_{4} + 2 \beta_1) q^{5} + (\beta_{2} + 283) q^{7} + (5 \beta_{5} - 14 \beta_{4} + 42 \beta_1) q^{11} + (5 \beta_{3} - 2 \beta_{2} + 6974) q^{13} + ( - 42 \beta_{5} + 28 \beta_{4} + 55 \beta_1) q^{17}+ \cdots + (3580 \beta_{3} - 4008 \beta_{2} + 8955851) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 1698 q^{7} + 41844 q^{13} + 36384 q^{19} - 1646688 q^{25} - 2058474 q^{31} - 9395880 q^{37} + 2737284 q^{43} + 28900656 q^{49} + 26674542 q^{55} - 40180776 q^{61} - 111355284 q^{67} + 12821718 q^{73}+ \cdots + 53735106 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 40x^{3} + 225x^{2} + 150x + 50 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2360\nu^{5} - 5352\nu^{4} + 3664\nu^{3} + 109696\nu^{2} + 466280\nu + 165400 ) / 1085 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 126\nu^{5} - 1188\nu^{4} + 2142\nu^{3} + 2520\nu^{2} + 1260\nu - 127728 ) / 31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6336\nu^{5} - 33912\nu^{4} + 107712\nu^{3} + 126720\nu^{2} + 63360\nu - 1136880 ) / 1085 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7085\nu^{5} + 17259\nu^{4} - 31258\nu^{3} - 256927\nu^{2} - 1656035\nu - 585925 ) / 1085 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -46063\nu^{5} + 107679\nu^{4} - 126212\nu^{3} - 1708319\nu^{2} - 9792715\nu - 3469625 ) / 1085 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 9\beta_{4} - 2\beta_{3} + \beta_{2} - 4\beta _1 + 972 ) / 2916 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 80\beta_{5} - 216\beta_{4} + 913\beta_1 ) / 17496 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 70\beta_{5} - 342\beta_{4} + 65\beta_{3} - 46\beta_{2} + 251\beta _1 - 120528 ) / 5832 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 245\beta_{3} - 352\beta_{2} - 1193616 ) / 5832 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5230\beta_{5} + 22302\beta_{4} + 3735\beta_{3} - 3366\beta_{2} - 30821\beta _1 - 9937728 ) / 17496 \) 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
−2.05620 + 2.05620i
3.41249 + 3.41249i
−0.356289 + 0.356289i
−0.356289 − 0.356289i
3.41249 − 3.41249i
−2.05620 − 2.05620i
0 0 0 − 1141.55i 0 618.310 0 0 0
161.2 0 0 0 − 799.282i 0 4073.62 0 0 0
161.3 0 0 0 − 230.735i 0 −3842.93 0 0 0
161.4 0 0 0 230.735i 0 −3842.93 0 0 0
161.5 0 0 0 799.282i 0 4073.62 0 0 0
161.6 0 0 0 1141.55i 0 618.310 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.k 6
3.b odd 2 1 inner 432.9.e.k 6
4.b odd 2 1 27.9.b.d ✓ 6
12.b even 2 1 27.9.b.d ✓ 6
36.f odd 6 2 81.9.d.f 12
36.h even 6 2 81.9.d.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.9.b.d ✓ 6 4.b odd 2 1
27.9.b.d ✓ 6 12.b even 2 1
81.9.d.f 12 36.f odd 6 2
81.9.d.f 12 36.h even 6 2
432.9.e.k 6 1.a even 1 1 trivial
432.9.e.k 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 \(S_{9}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{6} + 1995219T_{5}^{4} + 935893851075T_{5}^{2} + 44321375623280625 \) Copy content Toggle raw display
\( T_{7}^{3} - 849T_{7}^{2} - 15511965T_{7} + 9679397525 \) 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
$5$ \( T^{6} + \cdots + 44\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{3} - 849 T^{2} + \cdots + 9679397525)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 69\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{3} + \cdots + 3314333225800)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots - 54072465923584)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots - 83\!\cdots\!01)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{3} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 28\!\cdots\!89 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 30\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + \cdots - 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots + 43\!\cdots\!75)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots - 45\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 14\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots + 69\!\cdots\!25)^{2} \) Copy content Toggle raw display
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