Properties

Label 384.2.a.f
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

\(f(q)\) \(=\) \( q + q^{3} - 2q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} - 2q^{7} + q^{9} + 4q^{11} + 6q^{13} + 6q^{17} - 2q^{21} - 4q^{23} - 5q^{25} + q^{27} + 4q^{29} - 10q^{31} + 4q^{33} + 2q^{37} + 6q^{39} - 2q^{41} - 8q^{43} + 12q^{47} - 3q^{49} + 6q^{51} - 12q^{53} + 4q^{59} + 2q^{61} - 2q^{63} - 4q^{67} - 4q^{69} + 4q^{71} - 10q^{73} - 5q^{75} - 8q^{77} + 6q^{79} + q^{81} - 12q^{83} + 4q^{87} + 2q^{89} - 12q^{91} - 10q^{93} - 6q^{97} + 4q^{99} + O(q^{100}) \)

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 0 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.f yes 1
3.b odd 2 1 1152.2.a.i 1
4.b odd 2 1 384.2.a.c yes 1
5.b even 2 1 9600.2.a.w 1
8.b even 2 1 384.2.a.b 1
8.d odd 2 1 384.2.a.g yes 1
12.b even 2 1 1152.2.a.l 1
16.e even 4 2 768.2.d.g 2
16.f odd 4 2 768.2.d.b 2
20.d odd 2 1 9600.2.a.bh 1
24.f even 2 1 1152.2.a.k 1
24.h odd 2 1 1152.2.a.j 1
40.e odd 2 1 9600.2.a.h 1
40.f even 2 1 9600.2.a.bw 1
48.i odd 4 2 2304.2.d.m 2
48.k even 4 2 2304.2.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.b 1 8.b even 2 1
384.2.a.c yes 1 4.b odd 2 1
384.2.a.f yes 1 1.a even 1 1 trivial
384.2.a.g yes 1 8.d odd 2 1
768.2.d.b 2 16.f odd 4 2
768.2.d.g 2 16.e even 4 2
1152.2.a.i 1 3.b odd 2 1
1152.2.a.j 1 24.h odd 2 1
1152.2.a.k 1 24.f even 2 1
1152.2.a.l 1 12.b even 2 1
2304.2.d.d 2 48.k even 4 2
2304.2.d.m 2 48.i odd 4 2
9600.2.a.h 1 40.e odd 2 1
9600.2.a.w 1 5.b even 2 1
9600.2.a.bh 1 20.d odd 2 1
9600.2.a.bw 1 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(384))\):

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

Hecke characteristic polynomials

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