Properties

Label 384.5.b.a
Level $384$
Weight $5$
Character orbit 384.b
Analytic conductor $39.694$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - 2 x^{3} + 5 x^{2} - 4 x + 61\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \beta_{1} q^{3} + \beta_{2} q^{5} + \beta_{3} q^{7} + 27 q^{9} +O(q^{10})\) \( q + 3 \beta_{1} q^{3} + \beta_{2} q^{5} + \beta_{3} q^{7} + 27 q^{9} -52 \beta_{1} q^{11} + 2 \beta_{2} q^{13} + 3 \beta_{3} q^{15} -338 q^{17} -4 \beta_{1} q^{19} + 9 \beta_{2} q^{21} + 14 \beta_{3} q^{23} -287 q^{25} + 81 \beta_{1} q^{27} -43 \beta_{2} q^{29} -25 \beta_{3} q^{31} -468 q^{33} -912 \beta_{1} q^{35} + 8 \beta_{2} q^{37} + 6 \beta_{3} q^{39} + 578 q^{41} -1172 \beta_{1} q^{43} + 27 \beta_{2} q^{45} -42 \beta_{3} q^{47} -335 q^{49} -1014 \beta_{1} q^{51} -81 \beta_{2} q^{53} -52 \beta_{3} q^{55} -36 q^{57} -692 \beta_{1} q^{59} + 212 \beta_{2} q^{61} + 27 \beta_{3} q^{63} -1824 q^{65} -4772 \beta_{1} q^{67} + 126 \beta_{2} q^{69} + 82 \beta_{3} q^{71} + 8734 q^{73} -861 \beta_{1} q^{75} -156 \beta_{2} q^{77} + 215 \beta_{3} q^{79} + 729 q^{81} -7620 \beta_{1} q^{83} -338 \beta_{2} q^{85} -129 \beta_{3} q^{87} + 910 q^{89} -1824 \beta_{1} q^{91} -225 \beta_{2} q^{93} -4 \beta_{3} q^{95} + 5422 q^{97} -1404 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 108q^{9} + O(q^{10}) \) \( 4q + 108q^{9} - 1352q^{17} - 1148q^{25} - 1872q^{33} + 2312q^{41} - 1340q^{49} - 144q^{57} - 7296q^{65} + 34936q^{73} + 2916q^{81} + 3640q^{89} + 21688q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} + 5 x^{2} - 4 x + 61\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} + 3 \nu^{2} + 7 \nu - 4 \)\()/31\)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} - 4 \nu + 8 \)
\(\beta_{3}\)\(=\)\((\)\( 48 \nu^{3} - 72 \nu^{2} + 576 \nu - 276 \)\()/31\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 24 \beta_{1} + 12\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 6 \beta_{2} + 24 \beta_{1} - 36\)\()/24\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{3} + 9 \beta_{2} - 252 \beta_{1} - 60\)\()/24\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
−1.23205 + 2.17945i
−1.23205 2.17945i
2.23205 2.17945i
2.23205 + 2.17945i
0 −5.19615 0 30.1993i 0 52.3068i 0 27.0000 0
319.2 0 −5.19615 0 30.1993i 0 52.3068i 0 27.0000 0
319.3 0 5.19615 0 30.1993i 0 52.3068i 0 27.0000 0
319.4 0 5.19615 0 30.1993i 0 52.3068i 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.5.b.a 4
3.b odd 2 1 1152.5.b.j 4
4.b odd 2 1 inner 384.5.b.a 4
8.b even 2 1 inner 384.5.b.a 4
8.d odd 2 1 inner 384.5.b.a 4
12.b even 2 1 1152.5.b.j 4
16.e even 4 2 768.5.g.d 4
16.f odd 4 2 768.5.g.d 4
24.f even 2 1 1152.5.b.j 4
24.h odd 2 1 1152.5.b.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.b.a 4 1.a even 1 1 trivial
384.5.b.a 4 4.b odd 2 1 inner
384.5.b.a 4 8.b even 2 1 inner
384.5.b.a 4 8.d odd 2 1 inner
768.5.g.d 4 16.e even 4 2
768.5.g.d 4 16.f odd 4 2
1152.5.b.j 4 3.b odd 2 1
1152.5.b.j 4 12.b even 2 1
1152.5.b.j 4 24.f even 2 1
1152.5.b.j 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 912 \) acting on \(S_{5}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -27 + T^{2} )^{2} \)
$5$ \( ( 912 + T^{2} )^{2} \)
$7$ \( ( 2736 + T^{2} )^{2} \)
$11$ \( ( -8112 + T^{2} )^{2} \)
$13$ \( ( 3648 + T^{2} )^{2} \)
$17$ \( ( 338 + T )^{4} \)
$19$ \( ( -48 + T^{2} )^{2} \)
$23$ \( ( 536256 + T^{2} )^{2} \)
$29$ \( ( 1686288 + T^{2} )^{2} \)
$31$ \( ( 1710000 + T^{2} )^{2} \)
$37$ \( ( 58368 + T^{2} )^{2} \)
$41$ \( ( -578 + T )^{4} \)
$43$ \( ( -4120752 + T^{2} )^{2} \)
$47$ \( ( 4826304 + T^{2} )^{2} \)
$53$ \( ( 5983632 + T^{2} )^{2} \)
$59$ \( ( -1436592 + T^{2} )^{2} \)
$61$ \( ( 40988928 + T^{2} )^{2} \)
$67$ \( ( -68315952 + T^{2} )^{2} \)
$71$ \( ( 18396864 + T^{2} )^{2} \)
$73$ \( ( -8734 + T )^{4} \)
$79$ \( ( 126471600 + T^{2} )^{2} \)
$83$ \( ( -174193200 + T^{2} )^{2} \)
$89$ \( ( -910 + T )^{4} \)
$97$ \( ( -5422 + T )^{4} \)
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