Properties

Label 432.2.c.c
Level 432
Weight 2
Character orbit 432.c
Analytic conductor 3.450
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 432.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.44953736732\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \zeta_{12}^{3} q^{5} + ( -1 + 2 \zeta_{12}^{2} ) q^{7} +O(q^{10})\) \( q + 3 \zeta_{12}^{3} q^{5} + ( -1 + 2 \zeta_{12}^{2} ) q^{7} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{11} -2 q^{13} + 6 \zeta_{12}^{3} q^{17} + ( 4 - 8 \zeta_{12}^{2} ) q^{19} -4 q^{25} + 6 \zeta_{12}^{3} q^{29} + ( -3 + 6 \zeta_{12}^{2} ) q^{31} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{35} + 8 q^{37} + ( -6 + 12 \zeta_{12}^{2} ) q^{43} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{47} + 4 q^{49} -9 \zeta_{12}^{3} q^{53} + ( 9 - 18 \zeta_{12}^{2} ) q^{55} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{59} -4 q^{61} -6 \zeta_{12}^{3} q^{65} + ( -2 + 4 \zeta_{12}^{2} ) q^{67} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{71} + q^{73} -9 \zeta_{12}^{3} q^{77} + ( -2 + 4 \zeta_{12}^{2} ) q^{79} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{83} -18 q^{85} -6 \zeta_{12}^{3} q^{89} + ( 2 - 4 \zeta_{12}^{2} ) q^{91} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{95} -5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 8q^{13} - 16q^{25} + 32q^{37} + 16q^{49} - 16q^{61} + 4q^{73} - 72q^{85} - 20q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0 0 0 3.00000i 0 1.73205i 0 0 0
431.2 0 0 0 3.00000i 0 1.73205i 0 0 0
431.3 0 0 0 3.00000i 0 1.73205i 0 0 0
431.4 0 0 0 3.00000i 0 1.73205i 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 432.2.c.c 4
3.b odd 2 1 inner 432.2.c.c 4
4.b odd 2 1 inner 432.2.c.c 4
8.b even 2 1 1728.2.c.e 4
8.d odd 2 1 1728.2.c.e 4
9.c even 3 1 1296.2.s.g 4
9.c even 3 1 1296.2.s.i 4
9.d odd 6 1 1296.2.s.g 4
9.d odd 6 1 1296.2.s.i 4
12.b even 2 1 inner 432.2.c.c 4
24.f even 2 1 1728.2.c.e 4
24.h odd 2 1 1728.2.c.e 4
36.f odd 6 1 1296.2.s.g 4
36.f odd 6 1 1296.2.s.i 4
36.h even 6 1 1296.2.s.g 4
36.h even 6 1 1296.2.s.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.c.c 4 1.a even 1 1 trivial
432.2.c.c 4 3.b odd 2 1 inner
432.2.c.c 4 4.b odd 2 1 inner
432.2.c.c 4 12.b even 2 1 inner
1296.2.s.g 4 9.c even 3 1
1296.2.s.g 4 9.d odd 6 1
1296.2.s.g 4 36.f odd 6 1
1296.2.s.g 4 36.h even 6 1
1296.2.s.i 4 9.c even 3 1
1296.2.s.i 4 9.d odd 6 1
1296.2.s.i 4 36.f odd 6 1
1296.2.s.i 4 36.h even 6 1
1728.2.c.e 4 8.b even 2 1
1728.2.c.e 4 8.d odd 2 1
1728.2.c.e 4 24.f even 2 1
1728.2.c.e 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{2} + 9 \)
\( T_{7}^{2} + 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( ( 1 - T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 5 T + 7 T^{2} )^{2}( 1 + 5 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 5 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{4} \)
$17$ \( ( 1 + 2 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 23 T^{2} )^{4} \)
$29$ \( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 35 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 8 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 41 T^{2} )^{4} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2}( 1 + 8 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 - 14 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 25 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 10 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 4 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 34 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - T + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 146 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 139 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 142 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 5 T + 97 T^{2} )^{4} \)
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