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Properties

Label 432.6.i.c
Level $432$
Weight $6$
Character orbit 432.i
Analytic conductor $69.286$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,6,Mod(145,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.145");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(69.2858101592\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 40x^{6} + 568x^{4} + 3363x^{2} + 7056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{6} + \beta_{2} + \cdots - 1) q^{5} + ( - 3 \beta_{6} + 3 \beta_{4} + \cdots - 7) q^{7} + ( - 2 \beta_{6} - 6 \beta_{4} + \cdots + 107) q^{11} + ( - 3 \beta_{7} + 3 \beta_{6} + \cdots + 3) q^{13}+ \cdots + ( - 2142 \beta_{6} - 5748 \beta_{4} + \cdots + 4549) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 78 q^{5} - 28 q^{7} + 444 q^{11} - 182 q^{13} + 4356 q^{17} - 952 q^{19} + 8844 q^{23} - 1654 q^{25} - 12018 q^{29} - 1132 q^{31} - 16224 q^{35} - 15176 q^{37} - 1248 q^{41} + 6092 q^{43} - 60 q^{47}+ \cdots + 33976 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 40x^{6} + 568x^{4} + 3363x^{2} + 7056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 40\nu^{5} + 484\nu^{3} + 1683\nu + 84 ) / 168 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{4} + 60\nu^{2} + 9\nu + 252 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{4} + 60\nu^{2} - 9\nu + 252 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{7} - 28\nu^{6} - 276\nu^{5} - 868\nu^{4} - 2592\nu^{3} - 8428\nu^{2} - 7167\nu - 25116 ) / 84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{7} + 28\nu^{6} - 276\nu^{5} + 868\nu^{4} - 2592\nu^{3} + 8428\nu^{2} - 7167\nu + 25116 ) / 84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -71\nu^{7} + 56\nu^{6} - 2252\nu^{5} + 1988\nu^{4} - 22772\nu^{3} + 22652\nu^{2} - 72705\nu + 78876 ) / 168 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{6} + 71\nu^{4} + 809\nu^{2} + 2820 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 2\beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - 180 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 4\beta_{6} + 7\beta_{5} + 7\beta_{4} + 12\beta_{3} - 12\beta_{2} - 32\beta _1 + 14 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -40\beta_{7} + 40\beta_{5} - 40\beta_{4} + 23\beta_{3} + 23\beta_{2} + 2088 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -42\beta_{7} + 84\beta_{6} - 138\beta_{5} - 138\beta_{4} - 157\beta_{3} + 157\beta_{2} + 996\beta _1 - 456 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 638\beta_{7} - 611\beta_{5} + 611\beta_{4} - 412\beta_{3} - 412\beta_{2} - 26694 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 712 \beta_{7} - 1424 \beta_{6} + 2132 \beta_{5} + 2132 \beta_{4} + 2155 \beta_{3} - 2155 \beta_{2} + \cdots + 9952 ) / 18 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
3.62198i
2.56934i
− 3.84183i
− 2.34949i
− 3.62198i
− 2.56934i
3.84183i
2.34949i
0 0 0 −43.7155 + 75.7175i 0 −20.6033 − 35.6859i 0 0 0
145.2 0 0 0 −23.6560 + 40.9735i 0 −1.01546 − 1.75882i 0 0 0
145.3 0 0 0 −4.05388 + 7.02152i 0 87.7139 + 151.925i 0 0 0
145.4 0 0 0 32.4255 − 56.1625i 0 −80.0952 − 138.729i 0 0 0
289.1 0 0 0 −43.7155 − 75.7175i 0 −20.6033 + 35.6859i 0 0 0
289.2 0 0 0 −23.6560 − 40.9735i 0 −1.01546 + 1.75882i 0 0 0
289.3 0 0 0 −4.05388 − 7.02152i 0 87.7139 − 151.925i 0 0 0
289.4 0 0 0 32.4255 + 56.1625i 0 −80.0952 + 138.729i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.6.i.c 8
3.b odd 2 1 144.6.i.c 8
4.b odd 2 1 27.6.c.a 8
9.c even 3 1 inner 432.6.i.c 8
9.d odd 6 1 144.6.i.c 8
12.b even 2 1 9.6.c.a ✓ 8
36.f odd 6 1 27.6.c.a 8
36.f odd 6 1 81.6.a.d 4
36.h even 6 1 9.6.c.a ✓ 8
36.h even 6 1 81.6.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.6.c.a ✓ 8 12.b even 2 1
9.6.c.a ✓ 8 36.h even 6 1
27.6.c.a 8 4.b odd 2 1
27.6.c.a 8 36.f odd 6 1
81.6.a.c 4 36.h even 6 1
81.6.a.d 4 36.f odd 6 1
144.6.i.c 8 3.b odd 2 1
144.6.i.c 8 9.d odd 6 1
432.6.i.c 8 1.a even 1 1 trivial
432.6.i.c 8 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 78 T_{5}^{7} + 10119 T_{5}^{6} + 296406 T_{5}^{5} + 42290505 T_{5}^{4} + \cdots + 4730520600576 \) acting on \(S_{6}^{\mathrm{new}}(432, [\chi])\). 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
$5$ \( T^{8} + \cdots + 4730520600576 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 5530756283536 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 14\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( (T^{4} - 2178 T^{3} + \cdots - 917747509932)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 1740240514672)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 64\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 87\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 24412884987584)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 72\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 99\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 92\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 83\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 53\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 82\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 27\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 30\!\cdots\!60)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 26\!\cdots\!09 \) Copy content Toggle raw display
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