Properties

Label 1152.2.a.h
Level $1152$
Weight $2$
Character orbit 1152.a
Self dual yes
Analytic conductor $9.199$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{5} + 4q^{7} + O(q^{10}) \) \( q - 2q^{5} + 4q^{7} - 2q^{11} + 2q^{13} + 2q^{17} - 2q^{19} + 4q^{23} - q^{25} + 6q^{29} - 8q^{35} + 10q^{37} + 6q^{41} - 6q^{43} - 8q^{47} + 9q^{49} + 6q^{53} + 4q^{55} + 14q^{59} + 2q^{61} - 4q^{65} - 10q^{67} + 12q^{71} + 14q^{73} - 8q^{77} + 8q^{79} - 6q^{83} - 4q^{85} + 2q^{89} + 8q^{91} + 4q^{95} - 2q^{97} + O(q^{100}) \)

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 4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1152.2.a.h 1
3.b odd 2 1 128.2.a.b yes 1
4.b odd 2 1 1152.2.a.c 1
8.b even 2 1 1152.2.a.r 1
8.d odd 2 1 1152.2.a.m 1
12.b even 2 1 128.2.a.d yes 1
15.d odd 2 1 3200.2.a.u 1
15.e even 4 2 3200.2.c.k 2
16.e even 4 2 2304.2.d.b 2
16.f odd 4 2 2304.2.d.r 2
21.c even 2 1 6272.2.a.g 1
24.f even 2 1 128.2.a.a 1
24.h odd 2 1 128.2.a.c yes 1
48.i odd 4 2 256.2.b.a 2
48.k even 4 2 256.2.b.c 2
60.h even 2 1 3200.2.a.h 1
60.l odd 4 2 3200.2.c.e 2
84.h odd 2 1 6272.2.a.a 1
96.o even 8 4 1024.2.e.i 4
96.p odd 8 4 1024.2.e.m 4
120.i odd 2 1 3200.2.a.e 1
120.m even 2 1 3200.2.a.x 1
120.q odd 4 2 3200.2.c.l 2
120.w even 4 2 3200.2.c.f 2
168.e odd 2 1 6272.2.a.h 1
168.i even 2 1 6272.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 24.f even 2 1
128.2.a.b yes 1 3.b odd 2 1
128.2.a.c yes 1 24.h odd 2 1
128.2.a.d yes 1 12.b even 2 1
256.2.b.a 2 48.i odd 4 2
256.2.b.c 2 48.k even 4 2
1024.2.e.i 4 96.o even 8 4
1024.2.e.m 4 96.p odd 8 4
1152.2.a.c 1 4.b odd 2 1
1152.2.a.h 1 1.a even 1 1 trivial
1152.2.a.m 1 8.d odd 2 1
1152.2.a.r 1 8.b even 2 1
2304.2.d.b 2 16.e even 4 2
2304.2.d.r 2 16.f odd 4 2
3200.2.a.e 1 120.i odd 2 1
3200.2.a.h 1 60.h even 2 1
3200.2.a.u 1 15.d odd 2 1
3200.2.a.x 1 120.m even 2 1
3200.2.c.e 2 60.l odd 4 2
3200.2.c.f 2 120.w even 4 2
3200.2.c.k 2 15.e even 4 2
3200.2.c.l 2 120.q odd 4 2
6272.2.a.a 1 84.h odd 2 1
6272.2.a.b 1 168.i even 2 1
6272.2.a.g 1 21.c even 2 1
6272.2.a.h 1 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1152))\):

\( T_{5} + 2 \)
\( T_{7} - 4 \)
\( T_{11} + 2 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 - 4 T + 7 T^{2} \)
$11$ \( 1 + 2 T + 11 T^{2} \)
$13$ \( 1 - 2 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 2 T + 19 T^{2} \)
$23$ \( 1 - 4 T + 23 T^{2} \)
$29$ \( 1 - 6 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 - 10 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 + 6 T + 43 T^{2} \)
$47$ \( 1 + 8 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 - 14 T + 59 T^{2} \)
$61$ \( 1 - 2 T + 61 T^{2} \)
$67$ \( 1 + 10 T + 67 T^{2} \)
$71$ \( 1 - 12 T + 71 T^{2} \)
$73$ \( 1 - 14 T + 73 T^{2} \)
$79$ \( 1 - 8 T + 79 T^{2} \)
$83$ \( 1 + 6 T + 83 T^{2} \)
$89$ \( 1 - 2 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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