Properties

Label 384.2.a.a
Level $384$
Weight $2$
Character orbit 384.a
Self dual yes
Analytic conductor $3.066$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 4q^{5} + 2q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} - 4q^{5} + 2q^{7} + q^{9} + 4q^{11} + 2q^{13} + 4q^{15} - 2q^{17} + 8q^{19} - 2q^{21} + 4q^{23} + 11q^{25} - q^{27} - 6q^{31} - 4q^{33} - 8q^{35} - 2q^{37} - 2q^{39} + 6q^{41} - 4q^{45} + 4q^{47} - 3q^{49} + 2q^{51} - 16q^{55} - 8q^{57} - 4q^{59} + 14q^{61} + 2q^{63} - 8q^{65} + 4q^{67} - 4q^{69} + 12q^{71} - 10q^{73} - 11q^{75} + 8q^{77} + 10q^{79} + q^{81} - 12q^{83} + 8q^{85} - 14q^{89} + 4q^{91} + 6q^{93} - 32q^{95} + 10q^{97} + 4q^{99} + O(q^{100}) \)

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 −1.00000 0 −4.00000 0 2.00000 0 1.00000 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 384.2.a.a 1
3.b odd 2 1 1152.2.a.t 1
4.b odd 2 1 384.2.a.e yes 1
5.b even 2 1 9600.2.a.bk 1
8.b even 2 1 384.2.a.h yes 1
8.d odd 2 1 384.2.a.d yes 1
12.b even 2 1 1152.2.a.s 1
16.e even 4 2 768.2.d.c 2
16.f odd 4 2 768.2.d.f 2
20.d odd 2 1 9600.2.a.t 1
24.f even 2 1 1152.2.a.a 1
24.h odd 2 1 1152.2.a.b 1
40.e odd 2 1 9600.2.a.bz 1
40.f even 2 1 9600.2.a.e 1
48.i odd 4 2 2304.2.d.f 2
48.k even 4 2 2304.2.d.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.a 1 1.a even 1 1 trivial
384.2.a.d yes 1 8.d odd 2 1
384.2.a.e yes 1 4.b odd 2 1
384.2.a.h yes 1 8.b even 2 1
768.2.d.c 2 16.e even 4 2
768.2.d.f 2 16.f odd 4 2
1152.2.a.a 1 24.f even 2 1
1152.2.a.b 1 24.h odd 2 1
1152.2.a.s 1 12.b even 2 1
1152.2.a.t 1 3.b odd 2 1
2304.2.d.f 2 48.i odd 4 2
2304.2.d.o 2 48.k even 4 2
9600.2.a.e 1 40.f even 2 1
9600.2.a.t 1 20.d odd 2 1
9600.2.a.bk 1 5.b even 2 1
9600.2.a.bz 1 40.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(384))\):

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

Hecke characteristic polynomials

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