Properties

Label 384.4.f.a
Level $384$
Weight $4$
Character orbit 384.f
Analytic conductor $22.657$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} -7 \beta_{1} q^{5} -\beta_{3} q^{7} -27 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} -7 \beta_{1} q^{5} -\beta_{3} q^{7} -27 q^{9} + 14 \beta_{2} q^{11} + 7 \beta_{3} q^{15} -27 \beta_{1} q^{21} + 267 q^{25} -27 \beta_{2} q^{27} -79 \beta_{1} q^{29} + 23 \beta_{3} q^{31} -378 q^{33} -56 \beta_{2} q^{35} + 189 \beta_{1} q^{45} + 127 q^{49} + 205 \beta_{1} q^{53} + 98 \beta_{3} q^{55} + 138 \beta_{2} q^{59} + 27 \beta_{3} q^{63} -322 q^{73} + 267 \beta_{2} q^{75} -378 \beta_{1} q^{77} -21 \beta_{3} q^{79} + 729 q^{81} -170 \beta_{2} q^{83} + 79 \beta_{3} q^{87} + 621 \beta_{1} q^{93} -574 q^{97} -378 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 108q^{9} + O(q^{10}) \) \( 4q - 108q^{9} + 1068q^{25} - 1512q^{33} + 508q^{49} - 1288q^{73} + 2916q^{81} - 2296q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)
\(\beta_{2}\)\(=\)\( 3 \nu^{2} + 3 \)
\(\beta_{3}\)\(=\)\( 3 \nu^{3} + 12 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - 3 \beta_{1}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} - 3\)\()/3\)
\(\nu^{3}\)\(=\)\(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 5.19615i 0 −19.7990 0 14.6969i 0 −27.0000 0
191.2 0 5.19615i 0 19.7990 0 14.6969i 0 −27.0000 0
191.3 0 5.19615i 0 −19.7990 0 14.6969i 0 −27.0000 0
191.4 0 5.19615i 0 19.7990 0 14.6969i 0 −27.0000 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 \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.4.f.a 4
3.b odd 2 1 inner 384.4.f.a 4
4.b odd 2 1 inner 384.4.f.a 4
8.b even 2 1 inner 384.4.f.a 4
8.d odd 2 1 inner 384.4.f.a 4
12.b even 2 1 inner 384.4.f.a 4
16.e even 4 2 768.4.c.q 4
16.f odd 4 2 768.4.c.q 4
24.f even 2 1 inner 384.4.f.a 4
24.h odd 2 1 CM 384.4.f.a 4
48.i odd 4 2 768.4.c.q 4
48.k even 4 2 768.4.c.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.f.a 4 1.a even 1 1 trivial
384.4.f.a 4 3.b odd 2 1 inner
384.4.f.a 4 4.b odd 2 1 inner
384.4.f.a 4 8.b even 2 1 inner
384.4.f.a 4 8.d odd 2 1 inner
384.4.f.a 4 12.b even 2 1 inner
384.4.f.a 4 24.f even 2 1 inner
384.4.f.a 4 24.h odd 2 1 CM
768.4.c.q 4 16.e even 4 2
768.4.c.q 4 16.f odd 4 2
768.4.c.q 4 48.i odd 4 2
768.4.c.q 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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} - 392 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 27 + T^{2} )^{2} \)
$5$ \( ( -392 + T^{2} )^{2} \)
$7$ \( ( 216 + T^{2} )^{2} \)
$11$ \( ( 5292 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( -49928 + T^{2} )^{2} \)
$31$ \( ( 114264 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( -336200 + T^{2} )^{2} \)
$59$ \( ( 514188 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 322 + T )^{4} \)
$79$ \( ( 95256 + T^{2} )^{2} \)
$83$ \( ( 780300 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( 574 + T )^{4} \)
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