Properties

Label 1152.4.a.l
Level $1152$
Weight $4$
Character orbit 1152.a
Self dual yes
Analytic conductor $67.970$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.9702003266\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 8 q^{5} + 10 q^{7} + O(q^{10}) \) \( q + 8 q^{5} + 10 q^{7} + 68 q^{11} + 46 q^{13} + 74 q^{17} - 16 q^{19} - 20 q^{23} - 61 q^{25} + 228 q^{29} + 162 q^{31} + 80 q^{35} - 262 q^{37} - 30 q^{41} - 264 q^{43} + 124 q^{47} - 243 q^{49} - 204 q^{53} + 544 q^{55} + 340 q^{59} - 950 q^{61} + 368 q^{65} + 436 q^{67} - 780 q^{71} + 518 q^{73} + 680 q^{77} + 1010 q^{79} + 852 q^{83} + 592 q^{85} + 686 q^{89} + 460 q^{91} - 128 q^{95} - 806 q^{97} + O(q^{100}) \)

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 8.00000 0 10.0000 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 1152.4.a.l 1
3.b odd 2 1 384.4.a.e yes 1
4.b odd 2 1 1152.4.a.k 1
8.b even 2 1 1152.4.a.b 1
8.d odd 2 1 1152.4.a.a 1
12.b even 2 1 384.4.a.a 1
24.f even 2 1 384.4.a.h yes 1
24.h odd 2 1 384.4.a.d yes 1
48.i odd 4 2 768.4.d.e 2
48.k even 4 2 768.4.d.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.a.a 1 12.b even 2 1
384.4.a.d yes 1 24.h odd 2 1
384.4.a.e yes 1 3.b odd 2 1
384.4.a.h yes 1 24.f even 2 1
768.4.d.e 2 48.i odd 4 2
768.4.d.l 2 48.k even 4 2
1152.4.a.a 1 8.d odd 2 1
1152.4.a.b 1 8.b even 2 1
1152.4.a.k 1 4.b odd 2 1
1152.4.a.l 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1152))\):

\( T_{5} - 8 \)
\( T_{7} - 10 \)
\( T_{13} - 46 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -8 + T \)
$7$ \( -10 + T \)
$11$ \( -68 + T \)
$13$ \( -46 + T \)
$17$ \( -74 + T \)
$19$ \( 16 + T \)
$23$ \( 20 + T \)
$29$ \( -228 + T \)
$31$ \( -162 + T \)
$37$ \( 262 + T \)
$41$ \( 30 + T \)
$43$ \( 264 + T \)
$47$ \( -124 + T \)
$53$ \( 204 + T \)
$59$ \( -340 + T \)
$61$ \( 950 + T \)
$67$ \( -436 + T \)
$71$ \( 780 + T \)
$73$ \( -518 + T \)
$79$ \( -1010 + T \)
$83$ \( -852 + T \)
$89$ \( -686 + T \)
$97$ \( 806 + T \)
show more
show less