Properties

Label 384.2.a.c
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

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

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.c yes 1
3.b odd 2 1 1152.2.a.l 1
4.b odd 2 1 384.2.a.f yes 1
5.b even 2 1 9600.2.a.bh 1
8.b even 2 1 384.2.a.g yes 1
8.d odd 2 1 384.2.a.b 1
12.b even 2 1 1152.2.a.i 1
16.e even 4 2 768.2.d.b 2
16.f odd 4 2 768.2.d.g 2
20.d odd 2 1 9600.2.a.w 1
24.f even 2 1 1152.2.a.j 1
24.h odd 2 1 1152.2.a.k 1
40.e odd 2 1 9600.2.a.bw 1
40.f even 2 1 9600.2.a.h 1
48.i odd 4 2 2304.2.d.d 2
48.k even 4 2 2304.2.d.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.b 1 8.d odd 2 1
384.2.a.c yes 1 1.a even 1 1 trivial
384.2.a.f yes 1 4.b odd 2 1
384.2.a.g yes 1 8.b even 2 1
768.2.d.b 2 16.e even 4 2
768.2.d.g 2 16.f odd 4 2
1152.2.a.i 1 12.b even 2 1
1152.2.a.j 1 24.f even 2 1
1152.2.a.k 1 24.h odd 2 1
1152.2.a.l 1 3.b odd 2 1
2304.2.d.d 2 48.i odd 4 2
2304.2.d.m 2 48.k even 4 2
9600.2.a.h 1 40.f even 2 1
9600.2.a.w 1 20.d odd 2 1
9600.2.a.bh 1 5.b even 2 1
9600.2.a.bw 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(384))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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