Properties

Label 432.3.e.c
Level $432$
Weight $3$
Character orbit 432.e
Analytic conductor $11.771$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 432.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.7711474204\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{5} -5 q^{7} +O(q^{10})\) \( q + 3 i q^{5} -5 q^{7} -15 i q^{11} -10 q^{13} -18 i q^{17} + 16 q^{19} -12 i q^{23} + 16 q^{25} -30 i q^{29} + q^{31} -15 i q^{35} + 20 q^{37} -60 i q^{41} -50 q^{43} -6 i q^{47} -24 q^{49} + 27 i q^{53} + 45 q^{55} -30 i q^{59} -76 q^{61} -30 i q^{65} + 10 q^{67} -90 i q^{71} + 65 q^{73} + 75 i q^{77} -14 q^{79} + 3 i q^{83} + 54 q^{85} -90 i q^{89} + 50 q^{91} + 48 i q^{95} -85 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 10q^{7} + O(q^{10}) \) \( 2q - 10q^{7} - 20q^{13} + 32q^{19} + 32q^{25} + 2q^{31} + 40q^{37} - 100q^{43} - 48q^{49} + 90q^{55} - 152q^{61} + 20q^{67} + 130q^{73} - 28q^{79} + 108q^{85} + 100q^{91} - 170q^{97} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.00000i
1.00000i
0 0 0 3.00000i 0 −5.00000 0 0 0
161.2 0 0 0 3.00000i 0 −5.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.3.e.c 2
3.b odd 2 1 inner 432.3.e.c 2
4.b odd 2 1 27.3.b.b 2
8.b even 2 1 1728.3.e.g 2
8.d odd 2 1 1728.3.e.m 2
9.c even 3 2 1296.3.q.j 4
9.d odd 6 2 1296.3.q.j 4
12.b even 2 1 27.3.b.b 2
20.d odd 2 1 675.3.c.h 2
20.e even 4 1 675.3.d.a 2
20.e even 4 1 675.3.d.d 2
24.f even 2 1 1728.3.e.m 2
24.h odd 2 1 1728.3.e.g 2
36.f odd 6 2 81.3.d.b 4
36.h even 6 2 81.3.d.b 4
60.h even 2 1 675.3.c.h 2
60.l odd 4 1 675.3.d.a 2
60.l odd 4 1 675.3.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.b.b 2 4.b odd 2 1
27.3.b.b 2 12.b even 2 1
81.3.d.b 4 36.f odd 6 2
81.3.d.b 4 36.h even 6 2
432.3.e.c 2 1.a even 1 1 trivial
432.3.e.c 2 3.b odd 2 1 inner
675.3.c.h 2 20.d odd 2 1
675.3.c.h 2 60.h even 2 1
675.3.d.a 2 20.e even 4 1
675.3.d.a 2 60.l odd 4 1
675.3.d.d 2 20.e even 4 1
675.3.d.d 2 60.l odd 4 1
1296.3.q.j 4 9.c even 3 2
1296.3.q.j 4 9.d odd 6 2
1728.3.e.g 2 8.b even 2 1
1728.3.e.g 2 24.h odd 2 1
1728.3.e.m 2 8.d odd 2 1
1728.3.e.m 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{2} + 9 \)
\( T_{7} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 9 + T^{2} \)
$7$ \( ( 5 + T )^{2} \)
$11$ \( 225 + T^{2} \)
$13$ \( ( 10 + T )^{2} \)
$17$ \( 324 + T^{2} \)
$19$ \( ( -16 + T )^{2} \)
$23$ \( 144 + T^{2} \)
$29$ \( 900 + T^{2} \)
$31$ \( ( -1 + T )^{2} \)
$37$ \( ( -20 + T )^{2} \)
$41$ \( 3600 + T^{2} \)
$43$ \( ( 50 + T )^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( 729 + T^{2} \)
$59$ \( 900 + T^{2} \)
$61$ \( ( 76 + T )^{2} \)
$67$ \( ( -10 + T )^{2} \)
$71$ \( 8100 + T^{2} \)
$73$ \( ( -65 + T )^{2} \)
$79$ \( ( 14 + T )^{2} \)
$83$ \( 9 + T^{2} \)
$89$ \( 8100 + T^{2} \)
$97$ \( ( 85 + T )^{2} \)
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