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: \( x^{4} - 2x^{3} + 5x^{2} - 4x + 61 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} + \beta_{2} q^{5} + \beta_{3} q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} + \beta_{2} q^{5} + \beta_{3} q^{7} + 27 q^{9} - 52 \beta_1 q^{11} + 2 \beta_{2} q^{13} + 3 \beta_{3} q^{15} - 338 q^{17} - 4 \beta_1 q^{19} + 9 \beta_{2} q^{21} + 14 \beta_{3} q^{23} - 287 q^{25} + 81 \beta_1 q^{27} - 43 \beta_{2} q^{29} - 25 \beta_{3} q^{31} - 468 q^{33} - 912 \beta_1 q^{35} + 8 \beta_{2} q^{37} + 6 \beta_{3} q^{39} + 578 q^{41} - 1172 \beta_1 q^{43} + 27 \beta_{2} q^{45} - 42 \beta_{3} q^{47} - 335 q^{49} - 1014 \beta_1 q^{51} - 81 \beta_{2} q^{53} - 52 \beta_{3} q^{55} - 36 q^{57} - 692 \beta_1 q^{59} + 212 \beta_{2} q^{61} + 27 \beta_{3} q^{63} - 1824 q^{65} - 4772 \beta_1 q^{67} + 126 \beta_{2} q^{69} + 82 \beta_{3} q^{71} + 8734 q^{73} - 861 \beta_1 q^{75} - 156 \beta_{2} q^{77} + 215 \beta_{3} q^{79} + 729 q^{81} - 7620 \beta_1 q^{83} - 338 \beta_{2} q^{85} - 129 \beta_{3} q^{87} + 910 q^{89} - 1824 \beta_1 q^{91} - 225 \beta_{2} q^{93} - 4 \beta_{3} q^{95} + 5422 q^{97} - 1404 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 108 q^{9} - 1352 q^{17} - 1148 q^{25} - 1872 q^{33} + 2312 q^{41} - 1340 q^{49} - 144 q^{57} - 7296 q^{65} + 34936 q^{73} + 2916 q^{81} + 3640 q^{89} + 21688 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} + 7\nu - 4 ) / 31 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 4\nu + 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 48\nu^{3} - 72\nu^{2} + 576\nu - 276 ) / 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 24\beta _1 + 12 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 6\beta_{2} + 24\beta _1 - 36 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{3} + 9\beta_{2} - 252\beta _1 - 60 ) / 24 \) Copy content Toggle raw display

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 \(S_{5}^{\mathrm{new}}(384, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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