Properties

Label 432.2.c.a
Level 432
Weight 2
Character orbit 432.c
Analytic conductor 3.450
Analytic rank 0
Dimension 2
CM discriminant -3
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - 6 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( 3 - 6 \zeta_{6} ) q^{7} -5 q^{13} + ( 3 - 6 \zeta_{6} ) q^{19} + 5 q^{25} + ( 6 - 12 \zeta_{6} ) q^{31} + 11 q^{37} + ( -6 + 12 \zeta_{6} ) q^{43} -20 q^{49} - q^{61} + ( -9 + 18 \zeta_{6} ) q^{67} + 7 q^{73} + ( 3 - 6 \zeta_{6} ) q^{79} + ( -15 + 30 \zeta_{6} ) q^{91} + 19 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 10q^{13} + 10q^{25} + 22q^{37} - 40q^{49} - 2q^{61} + 14q^{73} + 38q^{97} + O(q^{100}) \)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 5.19615i 0 0 0
431.2 0 0 0 0 0 5.19615i 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 CM by \(\Q(\sqrt{-3}) \)
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.a 2
3.b odd 2 1 CM 432.2.c.a 2
4.b odd 2 1 inner 432.2.c.a 2
8.b even 2 1 1728.2.c.b 2
8.d odd 2 1 1728.2.c.b 2
9.c even 3 1 1296.2.s.a 2
9.c even 3 1 1296.2.s.f 2
9.d odd 6 1 1296.2.s.a 2
9.d odd 6 1 1296.2.s.f 2
12.b even 2 1 inner 432.2.c.a 2
24.f even 2 1 1728.2.c.b 2
24.h odd 2 1 1728.2.c.b 2
36.f odd 6 1 1296.2.s.a 2
36.f odd 6 1 1296.2.s.f 2
36.h even 6 1 1296.2.s.a 2
36.h even 6 1 1296.2.s.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.c.a 2 1.a even 1 1 trivial
432.2.c.a 2 3.b odd 2 1 CM
432.2.c.a 2 4.b odd 2 1 inner
432.2.c.a 2 12.b even 2 1 inner
1296.2.s.a 2 9.c even 3 1
1296.2.s.a 2 9.d odd 6 1
1296.2.s.a 2 36.f odd 6 1
1296.2.s.a 2 36.h even 6 1
1296.2.s.f 2 9.c even 3 1
1296.2.s.f 2 9.d odd 6 1
1296.2.s.f 2 36.f odd 6 1
1296.2.s.f 2 36.h even 6 1
1728.2.c.b 2 8.b even 2 1
1728.2.c.b 2 8.d odd 2 1
1728.2.c.b 2 24.f even 2 1
1728.2.c.b 2 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 \(S_{2}^{\mathrm{new}}(432, [\chi])\):

\( T_{5} \)
\( T_{7}^{2} + 27 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( ( 1 - 5 T^{2} )^{2} \)
$7$ \( ( 1 - T + 7 T^{2} )( 1 + T + 7 T^{2} ) \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( ( 1 + 5 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 17 T^{2} )^{2} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \)
$37$ \( ( 1 - 11 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 41 T^{2} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 + T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 5 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 7 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 17 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( ( 1 - 89 T^{2} )^{2} \)
$97$ \( ( 1 - 19 T + 97 T^{2} )^{2} \)
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