Properties

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

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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: \(1\)
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

\(f(q)\) \(=\) \( q - 4 q^{5} - 10 q^{7} + O(q^{10}) \) \( q - 4 q^{5} - 10 q^{7} + 4 q^{11} + 26 q^{13} - 14 q^{17} + 8 q^{19} + 148 q^{23} - 109 q^{25} - 72 q^{29} - 18 q^{31} + 40 q^{35} + 262 q^{37} + 378 q^{41} - 432 q^{43} + 148 q^{47} - 243 q^{49} - 360 q^{53} - 16 q^{55} + 428 q^{59} - 442 q^{61} - 104 q^{65} - 692 q^{67} + 540 q^{71} - 1018 q^{73} - 40 q^{77} - 386 q^{79} - 108 q^{83} + 56 q^{85} + 382 q^{89} - 260 q^{91} - 32 q^{95} + 298 q^{97} + O(q^{100}) \)

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 −4.00000 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.e 1
3.b odd 2 1 384.4.a.c yes 1
4.b odd 2 1 1152.4.a.f 1
8.b even 2 1 1152.4.a.g 1
8.d odd 2 1 1152.4.a.h 1
12.b even 2 1 384.4.a.g yes 1
24.f even 2 1 384.4.a.b 1
24.h odd 2 1 384.4.a.f yes 1
48.i odd 4 2 768.4.d.k 2
48.k even 4 2 768.4.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.a.b 1 24.f even 2 1
384.4.a.c yes 1 3.b odd 2 1
384.4.a.f yes 1 24.h odd 2 1
384.4.a.g yes 1 12.b even 2 1
768.4.d.f 2 48.k even 4 2
768.4.d.k 2 48.i odd 4 2
1152.4.a.e 1 1.a even 1 1 trivial
1152.4.a.f 1 4.b odd 2 1
1152.4.a.g 1 8.b even 2 1
1152.4.a.h 1 8.d odd 2 1

Hecke kernels

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

\( T_{5} + 4 \)
\( T_{7} + 10 \)
\( T_{13} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 4 + T \)
$7$ \( 10 + T \)
$11$ \( -4 + T \)
$13$ \( -26 + T \)
$17$ \( 14 + T \)
$19$ \( -8 + T \)
$23$ \( -148 + T \)
$29$ \( 72 + T \)
$31$ \( 18 + T \)
$37$ \( -262 + T \)
$41$ \( -378 + T \)
$43$ \( 432 + T \)
$47$ \( -148 + T \)
$53$ \( 360 + T \)
$59$ \( -428 + T \)
$61$ \( 442 + T \)
$67$ \( 692 + T \)
$71$ \( -540 + T \)
$73$ \( 1018 + T \)
$79$ \( 386 + T \)
$83$ \( 108 + T \)
$89$ \( -382 + T \)
$97$ \( -298 + T \)
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