Properties

Label 384.3.h.b
Level $384$
Weight $3$
Character orbit 384.h
Analytic conductor $10.463$
Analytic rank $0$
Dimension $2$
CM discriminant -8
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: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{3} + ( -7 - 2 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{3} + ( -7 - 2 \beta ) q^{9} -14 q^{11} -12 \beta q^{17} + 6 \beta q^{19} -25 q^{25} + ( 23 - 5 \beta ) q^{27} + ( 14 - 14 \beta ) q^{33} -24 \beta q^{41} -30 \beta q^{43} -49 q^{49} + ( 96 + 12 \beta ) q^{51} + ( -48 - 6 \beta ) q^{57} -82 q^{59} + 42 \beta q^{67} + 142 q^{73} + ( 25 - 25 \beta ) q^{75} + ( 17 + 28 \beta ) q^{81} -158 q^{83} + 36 \beta q^{89} -94 q^{97} + ( 98 + 28 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 14q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 14q^{9} - 28q^{11} - 50q^{25} + 46q^{27} + 28q^{33} - 98q^{49} + 192q^{51} - 96q^{57} - 164q^{59} + 284q^{73} + 50q^{75} + 34q^{81} - 316q^{83} - 188q^{97} + 196q^{99} + O(q^{100}) \)

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
1.41421i
1.41421i
0 −1.00000 2.82843i 0 0 0 0 0 −7.00000 + 5.65685i 0
65.2 0 −1.00000 + 2.82843i 0 0 0 0 0 −7.00000 5.65685i 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 inner
24.h odd 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.b 2
3.b odd 2 1 384.3.h.c yes 2
4.b odd 2 1 384.3.h.c yes 2
8.b even 2 1 384.3.h.c yes 2
8.d odd 2 1 CM 384.3.h.b 2
12.b even 2 1 inner 384.3.h.b 2
16.e even 4 2 768.3.e.k 4
16.f odd 4 2 768.3.e.k 4
24.f even 2 1 384.3.h.c yes 2
24.h odd 2 1 inner 384.3.h.b 2
48.i odd 4 2 768.3.e.k 4
48.k even 4 2 768.3.e.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.h.b 2 1.a even 1 1 trivial
384.3.h.b 2 8.d odd 2 1 CM
384.3.h.b 2 12.b even 2 1 inner
384.3.h.b 2 24.h odd 2 1 inner
384.3.h.c yes 2 3.b odd 2 1
384.3.h.c yes 2 4.b odd 2 1
384.3.h.c yes 2 8.b even 2 1
384.3.h.c yes 2 24.f even 2 1
768.3.e.k 4 16.e even 4 2
768.3.e.k 4 16.f odd 4 2
768.3.e.k 4 48.i odd 4 2
768.3.e.k 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 \(S_{3}^{\mathrm{new}}(384, [\chi])\):

\( T_{5} \)
\( T_{11} + 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 14 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( 1152 + T^{2} \)
$19$ \( 288 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( 4608 + T^{2} \)
$43$ \( 7200 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( ( 82 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( 14112 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -142 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( ( 158 + T )^{2} \)
$89$ \( 10368 + T^{2} \)
$97$ \( ( 94 + T )^{2} \)
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