Properties

Label 384.1.h.b
Level $384$
Weight $1$
Character orbit 384.h
Self dual yes
Analytic conductor $0.192$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -8, -24, 12
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.191640964851\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-2}, \sqrt{3})\)
Artin image $D_4$
Artin field Galois closure of 4.0.3072.2

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + q^{9} + O(q^{10}) \) \( q + q^{3} + q^{9} - 2q^{11} - q^{25} + q^{27} - 2q^{33} - q^{49} + 2q^{59} - 2q^{73} - q^{75} + q^{81} - 2q^{83} + 2q^{97} - 2q^{99} + O(q^{100}) \)

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
0
0 1.00000 0 0 0 0 0 1.00000 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
12.b even 2 1 RM by \(\Q(\sqrt{3}) \)
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.1.h.b yes 1
3.b odd 2 1 384.1.h.a 1
4.b odd 2 1 384.1.h.a 1
8.b even 2 1 384.1.h.a 1
8.d odd 2 1 CM 384.1.h.b yes 1
12.b even 2 1 RM 384.1.h.b yes 1
16.e even 4 2 768.1.e.c 2
16.f odd 4 2 768.1.e.c 2
24.f even 2 1 384.1.h.a 1
24.h odd 2 1 CM 384.1.h.b yes 1
32.g even 8 4 3072.1.i.f 4
32.h odd 8 4 3072.1.i.f 4
48.i odd 4 2 768.1.e.c 2
48.k even 4 2 768.1.e.c 2
96.o even 8 4 3072.1.i.f 4
96.p odd 8 4 3072.1.i.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.1.h.a 1 3.b odd 2 1
384.1.h.a 1 4.b odd 2 1
384.1.h.a 1 8.b even 2 1
384.1.h.a 1 24.f even 2 1
384.1.h.b yes 1 1.a even 1 1 trivial
384.1.h.b yes 1 8.d odd 2 1 CM
384.1.h.b yes 1 12.b even 2 1 RM
384.1.h.b yes 1 24.h odd 2 1 CM
768.1.e.c 2 16.e even 4 2
768.1.e.c 2 16.f odd 4 2
768.1.e.c 2 48.i odd 4 2
768.1.e.c 2 48.k even 4 2
3072.1.i.f 4 32.g even 8 4
3072.1.i.f 4 32.h odd 8 4
3072.1.i.f 4 96.o even 8 4
3072.1.i.f 4 96.p odd 8 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} + 2 \) acting on \(S_{1}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

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