Properties

Label 384.3.h.a
Level $384$
Weight $3$
Character orbit 384.h
Self dual yes
Analytic conductor $10.463$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 q^{3} + \beta q^{5} + \beta q^{7} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + \beta q^{5} + \beta q^{7} + 9 q^{9} -10 q^{11} -3 \beta q^{15} -3 \beta q^{21} + 71 q^{25} -27 q^{27} -3 \beta q^{29} + 5 \beta q^{31} + 30 q^{33} + 96 q^{35} + 9 \beta q^{45} + 47 q^{49} + 5 \beta q^{53} -10 \beta q^{55} + 10 q^{59} + 9 \beta q^{63} -50 q^{73} -213 q^{75} -10 \beta q^{77} -15 \beta q^{79} + 81 q^{81} + 134 q^{83} + 9 \beta q^{87} -15 \beta q^{93} -190 q^{97} -90 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} + 18q^{9} - 20q^{11} + 142q^{25} - 54q^{27} + 60q^{33} + 192q^{35} + 94q^{49} + 20q^{59} - 100q^{73} - 426q^{75} + 162q^{81} + 268q^{83} - 380q^{97} - 180q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−2.44949
2.44949
0 −3.00000 0 −9.79796 0 −9.79796 0 9.00000 0
65.2 0 −3.00000 0 9.79796 0 9.79796 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
8.d odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.3.h.a 2
3.b odd 2 1 384.3.h.d yes 2
4.b odd 2 1 384.3.h.d yes 2
8.b even 2 1 384.3.h.d yes 2
8.d odd 2 1 inner 384.3.h.a 2
12.b even 2 1 inner 384.3.h.a 2
16.e even 4 2 768.3.e.j 4
16.f odd 4 2 768.3.e.j 4
24.f even 2 1 384.3.h.d yes 2
24.h odd 2 1 CM 384.3.h.a 2
48.i odd 4 2 768.3.e.j 4
48.k even 4 2 768.3.e.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.h.a 2 1.a even 1 1 trivial
384.3.h.a 2 8.d odd 2 1 inner
384.3.h.a 2 12.b even 2 1 inner
384.3.h.a 2 24.h odd 2 1 CM
384.3.h.d yes 2 3.b odd 2 1
384.3.h.d yes 2 4.b odd 2 1
384.3.h.d yes 2 8.b even 2 1
384.3.h.d yes 2 24.f even 2 1
768.3.e.j 4 16.e even 4 2
768.3.e.j 4 16.f odd 4 2
768.3.e.j 4 48.i odd 4 2
768.3.e.j 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} - 96 \)
\( T_{11} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 3 + T )^{2} \)
$5$ \( -96 + T^{2} \)
$7$ \( -96 + T^{2} \)
$11$ \( ( 10 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( -864 + T^{2} \)
$31$ \( -2400 + T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( -2400 + T^{2} \)
$59$ \( ( -10 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 50 + T )^{2} \)
$79$ \( -21600 + T^{2} \)
$83$ \( ( -134 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 190 + T )^{2} \)
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