Properties

Label 432.2.s.b
Level $432$
Weight $2$
Character orbit 432.s
Analytic conductor $3.450$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 432.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.44953736732\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + \zeta_{6} ) q^{5} + ( 1 + \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -2 + \zeta_{6} ) q^{5} + ( 1 + \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{11} + 5 \zeta_{6} q^{13} + ( -4 + 8 \zeta_{6} ) q^{17} + ( 2 - 4 \zeta_{6} ) q^{19} + 9 \zeta_{6} q^{23} + ( -2 + 2 \zeta_{6} ) q^{25} + ( -1 - \zeta_{6} ) q^{29} + ( 6 - 3 \zeta_{6} ) q^{31} -3 q^{35} + 2 q^{37} + ( 6 - 3 \zeta_{6} ) q^{41} + ( -3 - 3 \zeta_{6} ) q^{43} + ( 3 - 3 \zeta_{6} ) q^{47} -4 \zeta_{6} q^{49} + ( -3 + 6 \zeta_{6} ) q^{55} -3 \zeta_{6} q^{59} + ( 1 - \zeta_{6} ) q^{61} + ( -5 - 5 \zeta_{6} ) q^{65} + ( -10 + 5 \zeta_{6} ) q^{67} -12 q^{71} -2 q^{73} + ( 6 - 3 \zeta_{6} ) q^{77} + ( 5 + 5 \zeta_{6} ) q^{79} + ( 15 - 15 \zeta_{6} ) q^{83} -12 \zeta_{6} q^{85} + ( 4 - 8 \zeta_{6} ) q^{89} + ( -5 + 10 \zeta_{6} ) q^{91} + 6 \zeta_{6} q^{95} + ( 5 - 5 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} + 3q^{7} + O(q^{10}) \) \( 2q - 3q^{5} + 3q^{7} + 3q^{11} + 5q^{13} + 9q^{23} - 2q^{25} - 3q^{29} + 9q^{31} - 6q^{35} + 4q^{37} + 9q^{41} - 9q^{43} + 3q^{47} - 4q^{49} - 3q^{59} + q^{61} - 15q^{65} - 15q^{67} - 24q^{71} - 4q^{73} + 9q^{77} + 15q^{79} + 15q^{83} - 12q^{85} + 6q^{95} + 5q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −1.50000 + 0.866025i 0 1.50000 + 0.866025i 0 0 0
287.1 0 0 0 −1.50000 0.866025i 0 1.50000 0.866025i 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
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.s.b 2
3.b odd 2 1 144.2.s.c yes 2
4.b odd 2 1 432.2.s.a 2
8.b even 2 1 1728.2.s.d 2
8.d odd 2 1 1728.2.s.c 2
9.c even 3 1 144.2.s.b 2
9.c even 3 1 1296.2.c.a 2
9.d odd 6 1 432.2.s.a 2
9.d odd 6 1 1296.2.c.c 2
12.b even 2 1 144.2.s.b 2
24.f even 2 1 576.2.s.b 2
24.h odd 2 1 576.2.s.c 2
36.f odd 6 1 144.2.s.c yes 2
36.f odd 6 1 1296.2.c.c 2
36.h even 6 1 inner 432.2.s.b 2
36.h even 6 1 1296.2.c.a 2
72.j odd 6 1 1728.2.s.c 2
72.j odd 6 1 5184.2.c.b 2
72.l even 6 1 1728.2.s.d 2
72.l even 6 1 5184.2.c.d 2
72.n even 6 1 576.2.s.b 2
72.n even 6 1 5184.2.c.d 2
72.p odd 6 1 576.2.s.c 2
72.p odd 6 1 5184.2.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.b 2 9.c even 3 1
144.2.s.b 2 12.b even 2 1
144.2.s.c yes 2 3.b odd 2 1
144.2.s.c yes 2 36.f odd 6 1
432.2.s.a 2 4.b odd 2 1
432.2.s.a 2 9.d odd 6 1
432.2.s.b 2 1.a even 1 1 trivial
432.2.s.b 2 36.h even 6 1 inner
576.2.s.b 2 24.f even 2 1
576.2.s.b 2 72.n even 6 1
576.2.s.c 2 24.h odd 2 1
576.2.s.c 2 72.p odd 6 1
1296.2.c.a 2 9.c even 3 1
1296.2.c.a 2 36.h even 6 1
1296.2.c.c 2 9.d odd 6 1
1296.2.c.c 2 36.f odd 6 1
1728.2.s.c 2 8.d odd 2 1
1728.2.s.c 2 72.j odd 6 1
1728.2.s.d 2 8.b even 2 1
1728.2.s.d 2 72.l even 6 1
5184.2.c.b 2 72.j odd 6 1
5184.2.c.b 2 72.p odd 6 1
5184.2.c.d 2 72.l even 6 1
5184.2.c.d 2 72.n even 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - 4 T + 7 T^{2} )( 1 + T + 7 T^{2} ) \)
$11$ \( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \)
$17$ \( 1 + 14 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \)
$29$ \( 1 + 3 T + 32 T^{2} + 87 T^{3} + 841 T^{4} \)
$31$ \( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 9 T + 68 T^{2} - 369 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 9 T + 70 T^{2} + 387 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \)
$67$ \( 1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 2 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 15 T + 154 T^{2} - 1185 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 130 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \)
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