Properties

Label 432.5.g.c
Level $432$
Weight $5$
Character orbit 432.g
Analytic conductor $44.656$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,5,Mod(271,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.271");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 432.g (of order \(2\), degree \(1\), 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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 10x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} - 7 \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} - 7 \beta_{2} q^{7} - \beta_{3} q^{11} - 203 q^{13} + \beta_1 q^{17} + 83 \beta_{2} q^{19} - 5 \beta_{3} q^{23} + 455 q^{25} + 28 \beta_1 q^{29} - 32 \beta_{2} q^{31} + 7 \beta_{3} q^{35} + 11 q^{37} - 96 \beta_1 q^{41} + 392 \beta_{2} q^{43} + 7 \beta_{3} q^{47} + 1078 q^{49} + 42 \beta_1 q^{53} + 1080 \beta_{2} q^{55} + 35 \beta_{3} q^{59} - 1981 q^{61} + 203 \beta_1 q^{65} + 1155 \beta_{2} q^{67} - 38 \beta_{3} q^{71} - 4151 q^{73} - 189 \beta_1 q^{77} + 1127 \beta_{2} q^{79} - 14 \beta_{3} q^{83} - 1080 q^{85} - 55 \beta_1 q^{89} + 1421 \beta_{2} q^{91} - 83 \beta_{3} q^{95} + 16849 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 812 q^{13} + 1820 q^{25} + 44 q^{37} + 4312 q^{49} - 7924 q^{61} - 16604 q^{73} - 4320 q^{85} + 67396 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 10x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{3} + 60\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{2} - 15 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27\nu^{3} ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 9\beta_1 ) / 108 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{2} + 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{3} ) / 27 \) Copy content Toggle raw display

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
2.73861 + 1.58114i
2.73861 1.58114i
−2.73861 1.58114i
−2.73861 + 1.58114i
0 0 0 −32.8634 0 36.3731i 0 0 0
271.2 0 0 0 −32.8634 0 36.3731i 0 0 0
271.3 0 0 0 32.8634 0 36.3731i 0 0 0
271.4 0 0 0 32.8634 0 36.3731i 0 0 0
\(n\): e.g. 2-40 or 990-1000
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 432.5.g.c 4
3.b odd 2 1 inner 432.5.g.c 4
4.b odd 2 1 inner 432.5.g.c 4
12.b even 2 1 inner 432.5.g.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.5.g.c 4 1.a even 1 1 trivial
432.5.g.c 4 3.b odd 2 1 inner
432.5.g.c 4 4.b odd 2 1 inner
432.5.g.c 4 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 \(S_{5}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{2} - 1080 \) Copy content Toggle raw display
\( T_{7}^{2} + 1323 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 1080)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1323)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 29160)^{2} \) Copy content Toggle raw display
$13$ \( (T + 203)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 1080)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 186003)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 729000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 846720)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 27648)^{2} \) Copy content Toggle raw display
$37$ \( (T - 11)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 9953280)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4148928)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 1428840)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 1905120)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 35721000)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1981)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 36018675)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 42107040)^{2} \) Copy content Toggle raw display
$73$ \( (T + 4151)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 34293483)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 5715360)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3267000)^{2} \) Copy content Toggle raw display
$97$ \( (T - 16849)^{4} \) Copy content Toggle raw display
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