Properties

Label 4032.2.a.x
Level $4032$
Weight $2$
Character orbit 4032.a
Self dual yes
Analytic conductor $32.196$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1956820950\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{7} + O(q^{10}) \) \( q + q^{7} + 2q^{11} + 2q^{13} - 4q^{17} - 4q^{19} - 6q^{23} - 5q^{25} - 2q^{29} + 6q^{37} - 8q^{41} - 8q^{43} - 4q^{47} + q^{49} - 6q^{53} + 14q^{61} + 4q^{67} - 2q^{71} - 2q^{73} + 2q^{77} - 4q^{79} - 12q^{83} + 2q^{91} + 6q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 0 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.a.x 1
3.b odd 2 1 1344.2.a.e 1
4.b odd 2 1 4032.2.a.q 1
8.b even 2 1 2016.2.a.i 1
8.d odd 2 1 2016.2.a.d 1
12.b even 2 1 1344.2.a.p 1
21.c even 2 1 9408.2.a.ch 1
24.f even 2 1 672.2.a.c 1
24.h odd 2 1 672.2.a.g yes 1
48.i odd 4 2 5376.2.c.j 2
48.k even 4 2 5376.2.c.y 2
84.h odd 2 1 9408.2.a.w 1
168.e odd 2 1 4704.2.a.z 1
168.i even 2 1 4704.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.c 1 24.f even 2 1
672.2.a.g yes 1 24.h odd 2 1
1344.2.a.e 1 3.b odd 2 1
1344.2.a.p 1 12.b even 2 1
2016.2.a.d 1 8.d odd 2 1
2016.2.a.i 1 8.b even 2 1
4032.2.a.q 1 4.b odd 2 1
4032.2.a.x 1 1.a even 1 1 trivial
4704.2.a.k 1 168.i even 2 1
4704.2.a.z 1 168.e odd 2 1
5376.2.c.j 2 48.i odd 4 2
5376.2.c.y 2 48.k even 4 2
9408.2.a.w 1 84.h odd 2 1
9408.2.a.ch 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4032))\):

\( T_{5} \)
\( T_{11} - 2 \)
\( T_{13} - 2 \)
\( T_{17} + 4 \)
\( T_{19} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 5 T^{2} \)
$7$ \( 1 - T \)
$11$ \( 1 - 2 T + 11 T^{2} \)
$13$ \( 1 - 2 T + 13 T^{2} \)
$17$ \( 1 + 4 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 6 T + 23 T^{2} \)
$29$ \( 1 + 2 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 + 8 T + 41 T^{2} \)
$43$ \( 1 + 8 T + 43 T^{2} \)
$47$ \( 1 + 4 T + 47 T^{2} \)
$53$ \( 1 + 6 T + 53 T^{2} \)
$59$ \( 1 + 59 T^{2} \)
$61$ \( 1 - 14 T + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 + 2 T + 71 T^{2} \)
$73$ \( 1 + 2 T + 73 T^{2} \)
$79$ \( 1 + 4 T + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 + 89 T^{2} \)
$97$ \( 1 - 6 T + 97 T^{2} \)
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