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Properties

Label 432.9.g.h
Level $432$
Weight $9$
Character orbit 432.g
Analytic conductor $175.988$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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] = [432,9,Mod(271,432)] 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("432.271"); 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([1, 0, 0])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 432.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,57036] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(175.987559546\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3658 x^{10} + 10323672 x^{8} + 10806771976 x^{6} + 8657863222624 x^{4} + 575997114675360 x^{2} + 35\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{25} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{5} + (\beta_{3} - 79 \beta_1) q^{7} + (\beta_{7} + \beta_{6}) q^{11} + ( - \beta_{4} + 4753) q^{13} + (\beta_{11} - 2 \beta_{9} - 19 \beta_{8}) q^{17} + (\beta_{5} - 22 \beta_{3} + 6591 \beta_1) q^{19}+ \cdots + (1395 \beta_{4} + 2863 \beta_{2} + 7508881) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 57036 q^{13} + 306996 q^{25} - 2032140 q^{37} - 6810648 q^{49} + 28116276 q^{61} + 33515148 q^{73} + 95803200 q^{85} + 90106572 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3658 x^{10} + 10323672 x^{8} + 10806771976 x^{6} + 8657863222624 x^{4} + 575997114675360 x^{2} + 35\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 46967975917 \nu^{10} + 168057385563551 \nu^{8} + \cdots + 14\!\cdots\!60 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10035723 \nu^{10} - 28322994132 \nu^{8} - 79933652672688 \nu^{6} + \cdots + 75\!\cdots\!56 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 332363300786267 \nu^{10} + \cdots + 99\!\cdots\!60 ) / 48\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2870580749 \nu^{10} + 8101403527116 \nu^{8} + \cdots - 56\!\cdots\!92 ) / 20\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!37 \nu^{10} + \cdots + 30\!\cdots\!60 ) / 97\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 155854636448696 \nu^{11} + \cdots - 27\!\cdots\!40 \nu ) / 87\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 66\!\cdots\!21 \nu^{11} + \cdots - 79\!\cdots\!20 \nu ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!51 \nu^{11} + \cdots + 69\!\cdots\!00 \nu ) / 73\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 92\!\cdots\!71 \nu^{11} + \cdots - 39\!\cdots\!00 \nu ) / 73\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 36\!\cdots\!24 \nu^{11} + \cdots - 43\!\cdots\!80 \nu ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 90\!\cdots\!59 \nu^{11} + \cdots + 38\!\cdots\!00 \nu ) / 30\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - 3\beta_{9} - 195\beta_{8} - 192\beta_{7} + 9\beta_{6} ) / 10368 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{3} + \beta_{2} + 5487\beta _1 - 5487 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -1303\beta_{11} + 645\beta_{9} + 177093\beta_{8} ) / 2592 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -84\beta_{5} + 84\beta_{4} + 7572\beta_{3} + 2552\beta_{2} - 10899570\beta _1 - 10899570 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1554961 \beta_{11} - 1554961 \beta_{10} + 1064397 \beta_{9} - 200169459 \beta_{8} + \cdots + 3193191 \beta_{6} ) / 2592 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -614544\beta_{4} - 12555848\beta_{2} + 47886141432 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1871660935 \beta_{11} + 1871660935 \beta_{10} + 2369120043 \beta_{9} - 235345963797 \beta_{8} + \cdots - 7107360129 \beta_{6} ) / 1296 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 867188448 \beta_{5} + 867188448 \beta_{4} - 45185325216 \beta_{3} + 15350837888 \beta_{2} + \cdots - 55294341240648 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -2265461219617\beta_{11} - 3504387613341\beta_{9} + 281624210431011\beta_{8} ) / 324 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2248580806080 \beta_{5} + 2248580806080 \beta_{4} + 110073572379888 \beta_{3} + 37440717728656 \beta_{2} + \cdots - 13\!\cdots\!12 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 27\!\cdots\!23 \beta_{11} + \cdots + 13\!\cdots\!85 \beta_{6} ) / 324 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
−4.08892 + 7.08221i
−4.08892 − 7.08221i
−24.6650 − 42.7211i
−24.6650 + 42.7211i
−17.0122 + 29.4661i
−17.0122 − 29.4661i
17.0122 − 29.4661i
17.0122 + 29.4661i
24.6650 + 42.7211i
24.6650 − 42.7211i
4.08892 − 7.08221i
4.08892 + 7.08221i
0 0 0 −891.067 0 − 2857.34i 0 0 0
271.2 0 0 0 −891.067 0 2857.34i 0 0 0
271.3 0 0 0 −612.710 0 − 3291.22i 0 0 0
271.4 0 0 0 −612.710 0 3291.22i 0 0 0
271.5 0 0 0 −281.444 0 − 23.3884i 0 0 0
271.6 0 0 0 −281.444 0 23.3884i 0 0 0
271.7 0 0 0 281.444 0 − 23.3884i 0 0 0
271.8 0 0 0 281.444 0 23.3884i 0 0 0
271.9 0 0 0 612.710 0 − 3291.22i 0 0 0
271.10 0 0 0 612.710 0 3291.22i 0 0 0
271.11 0 0 0 891.067 0 − 2857.34i 0 0 0
271.12 0 0 0 891.067 0 2857.34i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.9.g.h ✓ 12
3.b odd 2 1 inner 432.9.g.h ✓ 12
4.b odd 2 1 inner 432.9.g.h ✓ 12
12.b even 2 1 inner 432.9.g.h ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.9.g.h ✓ 12 1.a even 1 1 trivial
432.9.g.h ✓ 12 3.b odd 2 1 inner
432.9.g.h ✓ 12 4.b odd 2 1 inner
432.9.g.h ✓ 12 12.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{6} - 1248624T_{5}^{4} + 390708048960T_{5}^{2} - 23610942139392000 \) Copy content Toggle raw display
\( T_{7}^{6} + 18997065T_{7}^{4} + 88448074023771T_{7}^{2} + 48376854501867675 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 48\!\cdots\!75)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + \cdots + 38534560881119)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 70\!\cdots\!75)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots + 22\!\cdots\!85)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 10\!\cdots\!01)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 47\!\cdots\!87)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots - 36\!\cdots\!73)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 11\!\cdots\!47)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots - 49\!\cdots\!05)^{4} \) Copy content Toggle raw display
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