Properties

Label 384.4.a.f
Level $384$
Weight $4$
Character orbit 384.a
Self dual yes
Analytic conductor $22.657$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.6567334422\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + 3 q^{3} - 4 q^{5} - 10 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 4 q^{5} - 10 q^{7} + 9 q^{9} + 4 q^{11} - 26 q^{13} - 12 q^{15} + 14 q^{17} - 8 q^{19} - 30 q^{21} - 148 q^{23} - 109 q^{25} + 27 q^{27} - 72 q^{29} - 18 q^{31} + 12 q^{33} + 40 q^{35} - 262 q^{37} - 78 q^{39} - 378 q^{41} + 432 q^{43} - 36 q^{45} - 148 q^{47} - 243 q^{49} + 42 q^{51} - 360 q^{53} - 16 q^{55} - 24 q^{57} + 428 q^{59} + 442 q^{61} - 90 q^{63} + 104 q^{65} + 692 q^{67} - 444 q^{69} - 540 q^{71} - 1018 q^{73} - 327 q^{75} - 40 q^{77} - 386 q^{79} + 81 q^{81} - 108 q^{83} - 56 q^{85} - 216 q^{87} - 382 q^{89} + 260 q^{91} - 54 q^{93} + 32 q^{95} + 298 q^{97} + 36 q^{99}+O(q^{100}) \) 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 3.00000 0 −4.00000 0 −10.0000 0 9.00000 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 384.4.a.f yes 1
3.b odd 2 1 1152.4.a.g 1
4.b odd 2 1 384.4.a.b 1
8.b even 2 1 384.4.a.c yes 1
8.d odd 2 1 384.4.a.g yes 1
12.b even 2 1 1152.4.a.h 1
16.e even 4 2 768.4.d.k 2
16.f odd 4 2 768.4.d.f 2
24.f even 2 1 1152.4.a.f 1
24.h odd 2 1 1152.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.a.b 1 4.b odd 2 1
384.4.a.c yes 1 8.b even 2 1
384.4.a.f yes 1 1.a even 1 1 trivial
384.4.a.g yes 1 8.d odd 2 1
768.4.d.f 2 16.f odd 4 2
768.4.d.k 2 16.e even 4 2
1152.4.a.e 1 24.h odd 2 1
1152.4.a.f 1 24.f even 2 1
1152.4.a.g 1 3.b odd 2 1
1152.4.a.h 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(384))\):

\( T_{5} + 4 \) Copy content Toggle raw display
\( T_{7} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

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