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Properties

Label 432.9.e.m
Level $432$
Weight $9$
Character orbit 432.e
Analytic conductor $175.988$
Analytic rank $0$
Dimension $8$
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] = [432,9,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 = [8,0,0,0,0,0,1800] 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 496x^{6} - 160x^{5} + 86460x^{4} + 40960x^{3} - 6057696x^{2} - 1832640x + 155713284 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{51}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 216)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{2} + 225) q^{7} + (2 \beta_{5} + \beta_{3} - \beta_1) q^{11} + (\beta_{6} + 6223) q^{13} + ( - \beta_{7} + 12 \beta_{5} + \cdots + 3 \beta_1) q^{17} + ( - 4 \beta_{6} - \beta_{4} + \cdots - 18959) q^{19}+ \cdots + ( - 210 \beta_{6} + 213 \beta_{4} + \cdots - 6043681) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1800 q^{7} + 49784 q^{13} - 151672 q^{19} - 1541624 q^{25} + 380144 q^{31} + 4801656 q^{37} - 2403344 q^{43} - 10499072 q^{49} + 3453184 q^{55} + 8213624 q^{61} - 23495032 q^{67} - 13673224 q^{73}+ \cdots - 48349448 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 496x^{6} - 160x^{5} + 86460x^{4} + 40960x^{3} - 6057696x^{2} - 1832640x + 155713284 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 26690626 \nu^{7} + 491878066 \nu^{6} - 13809795843 \nu^{5} - 199410212638 \nu^{4} + \cdots - 10\!\cdots\!28 ) / 249260744070 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 509128 \nu^{7} + 12293518 \nu^{6} + 152475054 \nu^{5} - 4452736704 \nu^{4} + \cdots - 7355946509664 ) / 4615939705 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 70832224 \nu^{7} - 1613193836 \nu^{6} - 34822174782 \nu^{5} + 202516505828 \nu^{4} + \cdots - 17\!\cdots\!32 ) / 124630372035 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4604128 \nu^{7} - 293405122 \nu^{6} + 696921294 \nu^{5} + 108617662816 \nu^{4} + \cdots + 90041494460256 ) / 4615939705 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8536064 \nu^{7} - 14318080 \nu^{6} - 3444387072 \nu^{5} + 4074004480 \nu^{4} + \cdots + 18101109841920 ) / 7331198355 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13664688 \nu^{7} - 103857242 \nu^{6} - 4855218474 \nu^{5} + 36306408736 \nu^{4} + \cdots + 80650347229536 ) / 4615939705 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2133171626 \nu^{7} - 30858498862 \nu^{6} - 1017985018803 \nu^{5} + 13623202977826 \nu^{4} + \cdots + 79\!\cdots\!36 ) / 124630372035 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -18\beta_{6} + 27\beta_{5} - 2\beta_{4} - 180\beta_{2} ) / 55296 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -8\beta_{7} + 12\beta_{6} - 59\beta_{5} + 44\beta_{4} - 80\beta_{3} - 1032\beta_{2} + 1360\beta _1 + 6856704 ) / 55296 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 24 \beta_{7} - 897 \beta_{6} + 2532 \beta_{5} + 31 \beta_{4} - 96 \beta_{3} - 5154 \beta_{2} + \cdots + 829440 ) / 13824 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 320 \beta_{7} + 360 \beta_{6} - 3455 \beta_{5} + 2600 \beta_{4} - 8192 \beta_{3} + \cdots + 252619776 ) / 13824 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 20640 \beta_{7} - 320706 \beta_{6} + 1290459 \beta_{5} + 26702 \beta_{4} - 87360 \beta_{3} + \cdots + 663552000 ) / 27648 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 105496 \beta_{7} - 43476 \beta_{6} - 712753 \beta_{5} + 955468 \beta_{4} - 4682992 \beta_{3} + \cdots + 80061308928 ) / 27648 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1463448 \beta_{7} - 13286589 \beta_{6} + 76413744 \beta_{5} + 1405379 \beta_{4} + \cdots + 42610821120 ) / 6912 \) 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
7.66643 − 1.41421i
−13.3882 + 1.41421i
14.0326 + 1.41421i
−8.31078 + 1.41421i
−8.31078 − 1.41421i
14.0326 − 1.41421i
−13.3882 − 1.41421i
7.66643 + 1.41421i
0 0 0 − 1231.05i 0 1020.63 0 0 0
161.2 0 0 0 − 894.505i 0 702.397 0 0 0
161.3 0 0 0 − 119.285i 0 2411.25 0 0 0
161.4 0 0 0 − 58.8653i 0 −3234.27 0 0 0
161.5 0 0 0 58.8653i 0 −3234.27 0 0 0
161.6 0 0 0 119.285i 0 2411.25 0 0 0
161.7 0 0 0 894.505i 0 702.397 0 0 0
161.8 0 0 0 1231.05i 0 1020.63 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 161.8
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.m 8
3.b odd 2 1 inner 432.9.e.m 8
4.b odd 2 1 216.9.e.a ✓ 8
12.b even 2 1 216.9.e.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.9.e.a ✓ 8 4.b odd 2 1
216.9.e.a ✓ 8 12.b even 2 1
432.9.e.m 8 1.a even 1 1 trivial
432.9.e.m 8 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 \(S_{9}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{8} + 2333312T_{5}^{6} + 1253616359424T_{5}^{4} + 21569845320089600T_{5}^{2} + 59786892096962560000 \) Copy content Toggle raw display
\( T_{7}^{4} - 900T_{7}^{3} - 8499834T_{7}^{2} + 14027255676T_{7} - 5590734666111 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots - 5590734666111)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 78\!\cdots\!17)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 10\!\cdots\!67)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 84\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 95\!\cdots\!40)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 11\!\cdots\!63)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 89\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 36\!\cdots\!80)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 50\!\cdots\!41)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 60\!\cdots\!79)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 45\!\cdots\!57)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 14\!\cdots\!77)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 26\!\cdots\!61)^{2} \) Copy content Toggle raw display
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