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Properties

Label 432.9.q.a
Level $432$
Weight $9$
Character orbit 432.q
Analytic conductor $175.988$
Analytic rank $0$
Dimension $14$
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(17,432)] 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("432.17"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 432.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,0,0,-438] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(175.987559546\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5 x^{13} - 930122 x^{12} + 122593669 x^{11} + 316468329343 x^{10} - 78164131766942 x^{9} + \cdots + 19\!\cdots\!59 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{30} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{9} + 21 \beta_1 - 21) q^{5} + (\beta_{10} + 2 \beta_{9} + \cdots + 133 \beta_1) q^{7} + (\beta_{13} + 5 \beta_{11} + \cdots - 2733) q^{11} + (8 \beta_{13} - \beta_{12} + \cdots + 219) q^{13}+ \cdots + (16514 \beta_{13} - 26750 \beta_{12} + \cdots - 16514) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 438 q^{5} - 922 q^{7} - 28677 q^{11} + 1684 q^{13} + 269630 q^{19} - 1000452 q^{23} + 65177 q^{25} - 3797682 q^{29} + 164132 q^{31} - 1671668 q^{37} + 10239447 q^{41} - 791815 q^{43} + 31148628 q^{47}+ \cdots + 127049161 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 5 x^{13} - 930122 x^{12} + 122593669 x^{11} + 316468329343 x^{10} - 78164131766942 x^{9} + \cdots + 19\!\cdots\!59 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 28\!\cdots\!70 \nu^{13} + \cdots - 31\!\cdots\!67 ) / 24\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 98\!\cdots\!20 \nu^{13} + \cdots + 31\!\cdots\!37 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16\!\cdots\!80 \nu^{13} + \cdots - 19\!\cdots\!13 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 41\!\cdots\!09 \nu^{13} + \cdots - 71\!\cdots\!46 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43\!\cdots\!29 \nu^{13} + \cdots - 71\!\cdots\!53 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38\!\cdots\!99 \nu^{13} + \cdots + 18\!\cdots\!85 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 99\!\cdots\!89 \nu^{13} + \cdots - 11\!\cdots\!60 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 48\!\cdots\!21 \nu^{13} + \cdots - 54\!\cdots\!44 ) / 24\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 49\!\cdots\!31 \nu^{13} + \cdots - 55\!\cdots\!01 ) / 24\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 38\!\cdots\!67 \nu^{13} + \cdots - 23\!\cdots\!52 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15\!\cdots\!91 \nu^{13} + \cdots - 12\!\cdots\!13 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 18\!\cdots\!01 \nu^{13} + \cdots - 15\!\cdots\!34 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!90 \nu^{13} + \cdots - 19\!\cdots\!73 ) / 56\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} + 3\beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4 \beta_{12} + 8 \beta_{11} + 189 \beta_{9} - 179 \beta_{8} - 4 \beta_{7} - 1028 \beta_{5} + \cdots + 398126 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 36 \beta_{13} + 37816 \beta_{12} + 85282 \beta_{11} + 522 \beta_{10} - 771036 \beta_{9} + \cdots - 225152616 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 189360 \beta_{13} - 30337356 \beta_{12} - 24473492 \beta_{11} - 239388 \beta_{10} + \cdots + 296075539075 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 79721700 \beta_{13} + 78798275392 \beta_{12} + 114619292734 \beta_{11} + 1003607430 \beta_{10} + \cdots - 316497036045849 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 74273642184 \beta_{13} - 17859498882776 \beta_{12} - 17277346273692 \beta_{11} + \cdots + 89\!\cdots\!47 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 534292162208 \beta_{13} + \cdots - 42\!\cdots\!87 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 76\!\cdots\!52 \beta_{13} + \cdots + 91\!\cdots\!76 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 40\!\cdots\!32 \beta_{13} + \cdots - 16\!\cdots\!10 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 90\!\cdots\!32 \beta_{13} + \cdots + 10\!\cdots\!79 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 88\!\cdots\!88 \beta_{13} + \cdots - 19\!\cdots\!67 ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 34\!\cdots\!40 \beta_{13} + \cdots + 40\!\cdots\!77 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 15\!\cdots\!36 \beta_{13} + \cdots - 24\!\cdots\!83 ) / 9 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
430.310 − 0.866025i
359.735 − 0.866025i
200.431 − 0.866025i
105.587 − 0.866025i
−51.8259 − 0.866025i
−423.533 − 0.866025i
−618.204 − 0.866025i
430.310 + 0.866025i
359.735 + 0.866025i
200.431 + 0.866025i
105.587 + 0.866025i
−51.8259 + 0.866025i
−423.533 + 0.866025i
−618.204 + 0.866025i
0 0 0 −676.216 − 390.413i 0 −2168.61 − 3756.14i 0 0 0
17.2 0 0 0 −570.352 − 329.293i 0 1512.41 + 2619.58i 0 0 0
17.3 0 0 0 −331.396 − 191.331i 0 −467.516 − 809.762i 0 0 0
17.4 0 0 0 −189.131 − 109.195i 0 −1404.67 − 2432.96i 0 0 0
17.5 0 0 0 46.9888 + 27.1290i 0 921.012 + 1595.24i 0 0 0
17.6 0 0 0 604.549 + 349.037i 0 1124.45 + 1947.61i 0 0 0
17.7 0 0 0 896.557 + 517.627i 0 21.9132 + 37.9547i 0 0 0
305.1 0 0 0 −676.216 + 390.413i 0 −2168.61 + 3756.14i 0 0 0
305.2 0 0 0 −570.352 + 329.293i 0 1512.41 − 2619.58i 0 0 0
305.3 0 0 0 −331.396 + 191.331i 0 −467.516 + 809.762i 0 0 0
305.4 0 0 0 −189.131 + 109.195i 0 −1404.67 + 2432.96i 0 0 0
305.5 0 0 0 46.9888 − 27.1290i 0 921.012 − 1595.24i 0 0 0
305.6 0 0 0 604.549 − 349.037i 0 1124.45 − 1947.61i 0 0 0
305.7 0 0 0 896.557 − 517.627i 0 21.9132 − 37.9547i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.9.q.a 14
3.b odd 2 1 144.9.q.a 14
4.b odd 2 1 27.9.d.a 14
9.c even 3 1 144.9.q.a 14
9.d odd 6 1 inner 432.9.q.a 14
12.b even 2 1 9.9.d.a ✓ 14
36.f odd 6 1 9.9.d.a ✓ 14
36.f odd 6 1 81.9.b.a 14
36.h even 6 1 27.9.d.a 14
36.h even 6 1 81.9.b.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.9.d.a ✓ 14 12.b even 2 1
9.9.d.a ✓ 14 36.f odd 6 1
27.9.d.a 14 4.b odd 2 1
27.9.d.a 14 36.h even 6 1
81.9.b.a 14 36.f odd 6 1
81.9.b.a 14 36.h even 6 1
144.9.q.a 14 3.b odd 2 1
144.9.q.a 14 9.c even 3 1
432.9.q.a 14 1.a even 1 1 trivial
432.9.q.a 14 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 438 T_{5}^{13} - 1303854 T_{5}^{12} - 599097276 T_{5}^{11} + 1339373715651 T_{5}^{10} + \cdots + 28\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(432, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 15\!\cdots\!43 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots - 18\!\cdots\!88)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 83\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 26\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 20\!\cdots\!23 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 11\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 35\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 10\!\cdots\!67 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 26\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 48\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 49\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 55\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
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