Properties

Label 4032.2.a.e
Level $4032$
Weight $2$
Character orbit 4032.a
Self dual yes
Analytic conductor $32.196$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{5} - q^{7} + O(q^{10}) \) \( q - 2 q^{5} - q^{7} - 4 q^{11} - 6 q^{13} - 2 q^{17} + 4 q^{19} - 8 q^{23} - q^{25} - 2 q^{29} + 2 q^{35} + 10 q^{37} + 6 q^{41} + 4 q^{43} + q^{49} + 6 q^{53} + 8 q^{55} + 4 q^{59} - 6 q^{61} + 12 q^{65} - 4 q^{67} - 8 q^{71} + 10 q^{73} + 4 q^{77} - 4 q^{83} + 4 q^{85} + 6 q^{89} + 6 q^{91} - 8 q^{95} - 14 q^{97} + O(q^{100}) \)

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 −2.00000 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.e 1
3.b odd 2 1 1344.2.a.q 1
4.b odd 2 1 4032.2.a.m 1
8.b even 2 1 126.2.a.a 1
8.d odd 2 1 1008.2.a.j 1
12.b even 2 1 1344.2.a.i 1
21.c even 2 1 9408.2.a.n 1
24.f even 2 1 336.2.a.d 1
24.h odd 2 1 42.2.a.a 1
40.f even 2 1 3150.2.a.bo 1
40.i odd 4 2 3150.2.g.r 2
48.i odd 4 2 5376.2.c.bc 2
48.k even 4 2 5376.2.c.e 2
56.e even 2 1 7056.2.a.k 1
56.h odd 2 1 882.2.a.b 1
56.j odd 6 2 882.2.g.j 2
56.p even 6 2 882.2.g.h 2
72.j odd 6 2 1134.2.f.g 2
72.n even 6 2 1134.2.f.j 2
84.h odd 2 1 9408.2.a.bw 1
120.i odd 2 1 1050.2.a.i 1
120.m even 2 1 8400.2.a.k 1
120.w even 4 2 1050.2.g.a 2
168.e odd 2 1 2352.2.a.l 1
168.i even 2 1 294.2.a.g 1
168.s odd 6 2 294.2.e.c 2
168.v even 6 2 2352.2.q.i 2
168.ba even 6 2 294.2.e.a 2
168.be odd 6 2 2352.2.q.n 2
264.m even 2 1 5082.2.a.d 1
312.b odd 2 1 7098.2.a.f 1
840.u even 2 1 7350.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 24.h odd 2 1
126.2.a.a 1 8.b even 2 1
294.2.a.g 1 168.i even 2 1
294.2.e.a 2 168.ba even 6 2
294.2.e.c 2 168.s odd 6 2
336.2.a.d 1 24.f even 2 1
882.2.a.b 1 56.h odd 2 1
882.2.g.h 2 56.p even 6 2
882.2.g.j 2 56.j odd 6 2
1008.2.a.j 1 8.d odd 2 1
1050.2.a.i 1 120.i odd 2 1
1050.2.g.a 2 120.w even 4 2
1134.2.f.g 2 72.j odd 6 2
1134.2.f.j 2 72.n even 6 2
1344.2.a.i 1 12.b even 2 1
1344.2.a.q 1 3.b odd 2 1
2352.2.a.l 1 168.e odd 2 1
2352.2.q.i 2 168.v even 6 2
2352.2.q.n 2 168.be odd 6 2
3150.2.a.bo 1 40.f even 2 1
3150.2.g.r 2 40.i odd 4 2
4032.2.a.e 1 1.a even 1 1 trivial
4032.2.a.m 1 4.b odd 2 1
5082.2.a.d 1 264.m even 2 1
5376.2.c.e 2 48.k even 4 2
5376.2.c.bc 2 48.i odd 4 2
7056.2.a.k 1 56.e even 2 1
7098.2.a.f 1 312.b odd 2 1
7350.2.a.f 1 840.u even 2 1
8400.2.a.k 1 120.m even 2 1
9408.2.a.n 1 21.c even 2 1
9408.2.a.bw 1 84.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4032))\):

\( T_{5} + 2 \)
\( T_{11} + 4 \)
\( T_{13} + 6 \)
\( T_{17} + 2 \)
\( T_{19} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 2 + T \)
$7$ \( 1 + T \)
$11$ \( 4 + T \)
$13$ \( 6 + T \)
$17$ \( 2 + T \)
$19$ \( -4 + T \)
$23$ \( 8 + T \)
$29$ \( 2 + T \)
$31$ \( T \)
$37$ \( -10 + T \)
$41$ \( -6 + T \)
$43$ \( -4 + T \)
$47$ \( T \)
$53$ \( -6 + T \)
$59$ \( -4 + T \)
$61$ \( 6 + T \)
$67$ \( 4 + T \)
$71$ \( 8 + T \)
$73$ \( -10 + T \)
$79$ \( T \)
$83$ \( 4 + T \)
$89$ \( -6 + T \)
$97$ \( 14 + T \)
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