Properties

Label 432.5.q.a
Level $432$
Weight $5$
Character orbit 432.q
Analytic conductor $44.656$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,5,Mod(17,432)]
 
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, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.17");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 432.q (of order \(6\), 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: \(44.6558240522\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.39400128.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + 2 \beta_{4} + \beta_{3} + 2) q^{5} + ( - 2 \beta_{5} + \beta_{4} + 12 \beta_{2} + 5 \beta_1 + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + 2 \beta_{4} + \beta_{3} + 2) q^{5} + ( - 2 \beta_{5} + \beta_{4} + 12 \beta_{2} + 5 \beta_1 + 3) q^{7} + (\beta_{5} - 52 \beta_{4} - 2 \beta_{3} + \beta_{2} + 5 \beta_1 + 53) q^{11} + ( - 5 \beta_{5} + 10 \beta_{4} + 5 \beta_{3} + 8 \beta_{2} + 16 \beta_1) q^{13} + ( - 14 \beta_{5} - 190 \beta_{4} + 7 \beta_{3} - 17 \beta_{2} - 24 \beta_1 - 88) q^{17} + ( - 9 \beta_{3} + 9 \beta_{2} + 52) q^{19} + ( - 14 \beta_{5} - 55 \beta_{4} - 14 \beta_{3} - 43 \beta_{2} - 82) q^{23} + (35 \beta_{5} - 57 \beta_{4} + 33 \beta_{2} + 34 \beta_1 - 92) q^{25} + ( - 21 \beta_{5} - 96 \beta_{4} + 42 \beta_{3} - 21 \beta_{2} + 64 \beta_1 + 75) q^{29} + ( - 46 \beta_{5} + 329 \beta_{4} + 46 \beta_{3} - 101 \beta_{2} - 202 \beta_1) q^{31} + ( - 24 \beta_{5} - 750 \beta_{4} + 12 \beta_{3} - 179 \beta_{2} + \cdots - 363) q^{35}+ \cdots + (122 \beta_{5} - 9139 \beta_{4} + 906 \beta_{2} + 514 \beta_1 - 9261) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{5} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 12 q^{5} - 12 q^{7} + 483 q^{11} - 6 q^{13} + 258 q^{19} - 282 q^{23} - 273 q^{25} + 1056 q^{29} - 1290 q^{31} + 12 q^{37} - 7629 q^{41} + 285 q^{43} - 9642 q^{47} - 1863 q^{49} - 2016 q^{55} + 6225 q^{59} + 3630 q^{61} + 7158 q^{65} + 5055 q^{67} - 14622 q^{73} - 2580 q^{77} - 4764 q^{79} - 1866 q^{83} + 12366 q^{85} - 34836 q^{91} - 13362 q^{95} - 28959 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 11\nu^{4} - 121\nu^{3} + 98\nu^{2} + 1118\nu - 220 ) / 1098 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 11\nu^{4} + 121\nu^{3} - 98\nu^{2} + 529\nu + 220 ) / 549 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -17\nu^{5} + 187\nu^{4} - 410\nu^{3} + 1666\nu^{2} + 889\nu + 10534 ) / 549 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 55\nu^{5} - 56\nu^{4} + 616\nu^{3} + 649\nu^{2} + 5488\nu + 22 ) / 1098 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 347\nu^{5} - 340\nu^{4} + 3740\nu^{3} + 5339\nu^{2} + 32954\nu + 7166 ) / 366 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 19\beta_{4} + 3\beta_{2} + 2\beta _1 - 20 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 11\beta_{2} - 12\beta _1 - 26 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{5} + 215\beta_{4} + 11\beta_{3} - 34\beta_{2} - 68\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -23\beta_{5} + 503\beta_{4} - 293\beta_{2} - 158\beta _1 + 526 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−1.28901 + 2.23263i
1.89154 3.27625i
−0.102534 + 0.177594i
−1.28901 2.23263i
1.89154 + 3.27625i
−0.102534 0.177594i
0 0 0 −13.8760 8.01130i 0 36.2418 + 62.7727i 0 0 0
17.2 0 0 0 −10.2044 5.89150i 0 −26.6364 46.1356i 0 0 0
17.3 0 0 0 30.0804 + 17.3669i 0 −15.6054 27.0294i 0 0 0
305.1 0 0 0 −13.8760 + 8.01130i 0 36.2418 62.7727i 0 0 0
305.2 0 0 0 −10.2044 + 5.89150i 0 −26.6364 + 46.1356i 0 0 0
305.3 0 0 0 30.0804 17.3669i 0 −15.6054 + 27.0294i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.3
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 432.5.q.a 6
3.b odd 2 1 144.5.q.a 6
4.b odd 2 1 27.5.d.a 6
9.c even 3 1 144.5.q.a 6
9.c even 3 1 1296.5.e.c 6
9.d odd 6 1 inner 432.5.q.a 6
9.d odd 6 1 1296.5.e.c 6
12.b even 2 1 9.5.d.a 6
36.f odd 6 1 9.5.d.a 6
36.f odd 6 1 81.5.b.a 6
36.h even 6 1 27.5.d.a 6
36.h even 6 1 81.5.b.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.5.d.a 6 12.b even 2 1
9.5.d.a 6 36.f odd 6 1
27.5.d.a 6 4.b odd 2 1
27.5.d.a 6 36.h even 6 1
81.5.b.a 6 36.f odd 6 1
81.5.b.a 6 36.h even 6 1
144.5.q.a 6 3.b odd 2 1
144.5.q.a 6 9.c even 3 1
432.5.q.a 6 1.a even 1 1 trivial
432.5.q.a 6 9.d odd 6 1 inner
1296.5.e.c 6 9.c even 3 1
1296.5.e.c 6 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 12T_{5}^{5} - 729T_{5}^{4} + 9324T_{5}^{3} + 649161T_{5}^{2} + 8825166T_{5} + 43001388 \) acting on \(S_{5}^{\mathrm{new}}(432, [\chi])\). 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
$5$ \( T^{6} - 12 T^{5} - 729 T^{4} + \cdots + 43001388 \) Copy content Toggle raw display
$7$ \( T^{6} + 12 T^{5} + \cdots + 14524588324 \) Copy content Toggle raw display
$11$ \( T^{6} - 483 T^{5} + \cdots + 481294471563 \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} + \cdots + 708378089104 \) Copy content Toggle raw display
$17$ \( T^{6} + 155115 T^{4} + \cdots + 47166451632 \) Copy content Toggle raw display
$19$ \( (T^{3} - 129 T^{2} - 18024 T + 1195028)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 282 T^{5} + \cdots + 10049071819968 \) Copy content Toggle raw display
$29$ \( T^{6} - 1056 T^{5} + \cdots + 92\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{6} + 1290 T^{5} + \cdots + 63\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T^{3} - 6 T^{2} - 2222952 T + 1276743376)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 7629 T^{5} + \cdots + 32\!\cdots\!63 \) Copy content Toggle raw display
$43$ \( T^{6} - 285 T^{5} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{6} + 9642 T^{5} + \cdots + 65\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{6} + 26693064 T^{4} + \cdots + 34\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{6} - 6225 T^{5} + \cdots + 74\!\cdots\!87 \) Copy content Toggle raw display
$61$ \( T^{6} - 3630 T^{5} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{6} - 5055 T^{5} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( T^{6} + 64502244 T^{4} + \cdots + 26\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( (T^{3} + 7311 T^{2} + \cdots - 15741832472)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 4764 T^{5} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{6} + 1866 T^{5} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{6} + 283928328 T^{4} + \cdots + 53\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{6} + 28959 T^{5} + \cdots + 52\!\cdots\!89 \) Copy content Toggle raw display
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