Properties

Label 384.4.a.e
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

\(f(q)\) \(=\) \( q + 3 q^{3} - 8 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 8 q^{5} + 10 q^{7} + 9 q^{9} - 68 q^{11} + 46 q^{13} - 24 q^{15} - 74 q^{17} - 16 q^{19} + 30 q^{21} + 20 q^{23} - 61 q^{25} + 27 q^{27} - 228 q^{29} + 162 q^{31} - 204 q^{33} - 80 q^{35} - 262 q^{37} + 138 q^{39} + 30 q^{41} - 264 q^{43} - 72 q^{45} - 124 q^{47} - 243 q^{49} - 222 q^{51} + 204 q^{53} + 544 q^{55} - 48 q^{57} - 340 q^{59} - 950 q^{61} + 90 q^{63} - 368 q^{65} + 436 q^{67} + 60 q^{69} + 780 q^{71} + 518 q^{73} - 183 q^{75} - 680 q^{77} + 1010 q^{79} + 81 q^{81} - 852 q^{83} + 592 q^{85} - 684 q^{87} - 686 q^{89} + 460 q^{91} + 486 q^{93} + 128 q^{95} - 806 q^{97} - 612 q^{99}+O(q^{100}) \) 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 −8.00000 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.e yes 1
3.b odd 2 1 1152.4.a.l 1
4.b odd 2 1 384.4.a.a 1
8.b even 2 1 384.4.a.d yes 1
8.d odd 2 1 384.4.a.h yes 1
12.b even 2 1 1152.4.a.k 1
16.e even 4 2 768.4.d.e 2
16.f odd 4 2 768.4.d.l 2
24.f even 2 1 1152.4.a.a 1
24.h odd 2 1 1152.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.a.a 1 4.b odd 2 1
384.4.a.d yes 1 8.b even 2 1
384.4.a.e yes 1 1.a even 1 1 trivial
384.4.a.h yes 1 8.d odd 2 1
768.4.d.e 2 16.e even 4 2
768.4.d.l 2 16.f odd 4 2
1152.4.a.a 1 24.f even 2 1
1152.4.a.b 1 24.h odd 2 1
1152.4.a.k 1 12.b even 2 1
1152.4.a.l 1 3.b odd 2 1

Hecke kernels

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

\( T_{5} + 8 \) 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
$5$ \( T + 8 \) Copy content Toggle raw display
$7$ \( T - 10 \) Copy content Toggle raw display
$11$ \( T + 68 \) Copy content Toggle raw display
$13$ \( T - 46 \) Copy content Toggle raw display
$17$ \( T + 74 \) Copy content Toggle raw display
$19$ \( T + 16 \) Copy content Toggle raw display
$23$ \( T - 20 \) Copy content Toggle raw display
$29$ \( T + 228 \) Copy content Toggle raw display
$31$ \( T - 162 \) Copy content Toggle raw display
$37$ \( T + 262 \) Copy content Toggle raw display
$41$ \( T - 30 \) Copy content Toggle raw display
$43$ \( T + 264 \) Copy content Toggle raw display
$47$ \( T + 124 \) Copy content Toggle raw display
$53$ \( T - 204 \) Copy content Toggle raw display
$59$ \( T + 340 \) Copy content Toggle raw display
$61$ \( T + 950 \) Copy content Toggle raw display
$67$ \( T - 436 \) Copy content Toggle raw display
$71$ \( T - 780 \) Copy content Toggle raw display
$73$ \( T - 518 \) Copy content Toggle raw display
$79$ \( T - 1010 \) Copy content Toggle raw display
$83$ \( T + 852 \) Copy content Toggle raw display
$89$ \( T + 686 \) Copy content Toggle raw display
$97$ \( T + 806 \) Copy content Toggle raw display
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