Properties

Label 384.5.b.c
Level $384$
Weight $5$
Character orbit 384.b
Analytic conductor $39.694$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} -\beta_{5} q^{7} + 27 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} -\beta_{5} q^{7} + 27 q^{9} + ( 7 \beta_{2} + \beta_{6} ) q^{11} + ( -2 \beta_{1} + 15 \beta_{3} ) q^{13} + ( 5 \beta_{4} + 4 \beta_{5} ) q^{15} + ( 174 + \beta_{7} ) q^{17} + ( 45 \beta_{2} + 3 \beta_{6} ) q^{19} + ( -3 \beta_{1} + 18 \beta_{3} ) q^{21} + ( -12 \beta_{4} - 18 \beta_{5} ) q^{23} + ( -447 - 2 \beta_{7} ) q^{25} -27 \beta_{2} q^{27} + ( -3 \beta_{1} + 89 \beta_{3} ) q^{29} + ( 36 \beta_{4} + 5 \beta_{5} ) q^{31} + ( -180 - 3 \beta_{7} ) q^{33} + ( -157 \beta_{2} - 7 \beta_{6} ) q^{35} + ( 20 \beta_{1} + 21 \beta_{3} ) q^{37} + ( 3 \beta_{4} - 21 \beta_{5} ) q^{39} + ( -126 + 9 \beta_{7} ) q^{41} + ( 153 \beta_{2} - 9 \beta_{6} ) q^{43} + ( 27 \beta_{1} - 27 \beta_{3} ) q^{45} + ( 96 \beta_{4} - 54 \beta_{5} ) q^{47} + ( 1297 - 4 \beta_{7} ) q^{49} + ( -177 \beta_{2} - 9 \beta_{6} ) q^{51} + ( -25 \beta_{1} - 5 \beta_{3} ) q^{53} + ( 36 \beta_{4} - 160 \beta_{5} ) q^{55} + ( -1188 - 9 \beta_{7} ) q^{57} + 252 \beta_{2} q^{59} + ( -88 \beta_{1} - 249 \beta_{3} ) q^{61} -27 \beta_{5} q^{63} + ( 2976 + 17 \beta_{7} ) q^{65} + ( -432 \beta_{2} + 12 \beta_{6} ) q^{67} + ( -90 \beta_{1} + 216 \beta_{3} ) q^{69} + ( 300 \beta_{4} + 18 \beta_{5} ) q^{71} + ( -1282 - 4 \beta_{7} ) q^{73} + ( 453 \beta_{2} + 18 \beta_{6} ) q^{75} + ( 148 \beta_{1} - 456 \beta_{3} ) q^{77} + ( 108 \beta_{4} - 243 \beta_{5} ) q^{79} + 729 q^{81} + ( -403 \beta_{2} + 35 \beta_{6} ) q^{83} + ( 238 \beta_{1} - 1182 \beta_{3} ) q^{85} + ( 71 \beta_{4} - 98 \beta_{5} ) q^{87} + ( -3954 - 10 \beta_{7} ) q^{89} + ( 873 \beta_{2} + 27 \beta_{6} ) q^{91} + ( 123 \beta_{1} + 234 \beta_{3} ) q^{93} + ( -12 \beta_{4} - 576 \beta_{5} ) q^{95} + ( -5650 - 10 \beta_{7} ) q^{97} + ( 189 \beta_{2} + 27 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 216q^{9} + O(q^{10}) \) \( 8q + 216q^{9} + 1392q^{17} - 3576q^{25} - 1440q^{33} - 1008q^{41} + 10376q^{49} - 9504q^{57} + 23808q^{65} - 10256q^{73} + 5832q^{81} - 31632q^{89} - 45200q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 24 \nu^{6} + 108 \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{7} - 3 \nu^{5} + 3 \nu^{3} + 24 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{7} - 24 \nu^{6} + 25 \nu^{5} - 40 \nu^{4} + 55 \nu^{3} - 120 \nu^{2} + 184 \nu - 208 \)\()/10\)
\(\beta_{5}\)\(=\)\((\)\( 7 \nu^{7} + 48 \nu^{6} + 25 \nu^{5} + 80 \nu^{4} + 55 \nu^{3} + 240 \nu^{2} + 184 \nu + 416 \)\()/10\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} - 128 \nu^{6} + \nu^{5} - 384 \nu^{4} - \nu^{3} - 128 \nu^{2} - 8 \nu - 768 \)\()/8\)
\(\beta_{7}\)\(=\)\( -36 \nu^{7} - 60 \nu^{5} - 132 \nu^{3} - 192 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + 8 \beta_{5} + 16 \beta_{4} - 12 \beta_{3} + 32 \beta_{2}\)\()/384\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{6} + 12 \beta_{5} - 12 \beta_{4} + \beta_{2} - 8 \beta_{1} - 288\)\()/384\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + 60 \beta_{3}\)\()/192\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} - 3 \beta_{2} - 24 \beta_{1} - 96\)\()/384\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{7} + 8 \beta_{5} + 16 \beta_{4} - 132 \beta_{3} - 352 \beta_{2}\)\()/384\)
\(\nu^{6}\)\(=\)\((\)\(5 \beta_{1} - 108\)\()/24\)
\(\nu^{7}\)\(=\)\((\)\(-7 \beta_{7} - 56 \beta_{5} - 112 \beta_{4} - 156 \beta_{3} + 416 \beta_{2}\)\()/384\)

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
1.09445 + 0.895644i
−0.228425 + 1.39564i
−0.228425 1.39564i
1.09445 0.895644i
0.228425 1.39564i
−1.09445 0.895644i
−1.09445 + 0.895644i
0.228425 + 1.39564i
0 −5.19615 0 39.7490i 0 46.0431i 0 27.0000 0
319.2 0 −5.19615 0 23.7490i 0 9.38251i 0 27.0000 0
319.3 0 −5.19615 0 23.7490i 0 9.38251i 0 27.0000 0
319.4 0 −5.19615 0 39.7490i 0 46.0431i 0 27.0000 0
319.5 0 5.19615 0 39.7490i 0 46.0431i 0 27.0000 0
319.6 0 5.19615 0 23.7490i 0 9.38251i 0 27.0000 0
319.7 0 5.19615 0 23.7490i 0 9.38251i 0 27.0000 0
319.8 0 5.19615 0 39.7490i 0 46.0431i 0 27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.8
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.c 8
3.b odd 2 1 1152.5.b.k 8
4.b odd 2 1 inner 384.5.b.c 8
8.b even 2 1 inner 384.5.b.c 8
8.d odd 2 1 inner 384.5.b.c 8
12.b even 2 1 1152.5.b.k 8
16.e even 4 1 768.5.g.c 4
16.e even 4 1 768.5.g.g 4
16.f odd 4 1 768.5.g.c 4
16.f odd 4 1 768.5.g.g 4
24.f even 2 1 1152.5.b.k 8
24.h odd 2 1 1152.5.b.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.b.c 8 1.a even 1 1 trivial
384.5.b.c 8 4.b odd 2 1 inner
384.5.b.c 8 8.b even 2 1 inner
384.5.b.c 8 8.d odd 2 1 inner
768.5.g.c 4 16.e even 4 1
768.5.g.c 4 16.f odd 4 1
768.5.g.g 4 16.e even 4 1
768.5.g.g 4 16.f odd 4 1
1152.5.b.k 8 3.b odd 2 1
1152.5.b.k 8 12.b even 2 1
1152.5.b.k 8 24.f even 2 1
1152.5.b.k 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2144 T_{5}^{2} + 891136 \) acting on \(S_{5}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -27 + T^{2} )^{4} \)
$5$ \( ( 891136 + 2144 T^{2} + T^{4} )^{2} \)
$7$ \( ( 186624 + 2208 T^{2} + T^{4} )^{2} \)
$11$ \( ( 412252416 - 45408 T^{2} + T^{4} )^{2} \)
$13$ \( ( 107495424 + 36864 T^{2} + T^{4} )^{2} \)
$17$ \( ( -34236 - 348 T + T^{2} )^{4} \)
$19$ \( ( 19955517696 - 491616 T^{2} + T^{4} )^{2} \)
$23$ \( ( 36790308864 + 825984 T^{2} + T^{4} )^{2} \)
$29$ \( ( 247876528384 + 1032032 T^{2} + T^{4} )^{2} \)
$31$ \( ( 189245880576 + 1389216 T^{2} + T^{4} )^{2} \)
$37$ \( ( 140607000576 + 862848 T^{2} + T^{4} )^{2} \)
$41$ \( ( -5209596 + 252 T + T^{2} )^{4} \)
$43$ \( ( 1176686901504 - 4797792 T^{2} + T^{4} )^{2} \)
$47$ \( ( 54724012314624 + 17165952 T^{2} + T^{4} )^{2} \)
$53$ \( ( 394886560000 + 1263200 T^{2} + T^{4} )^{2} \)
$59$ \( ( -1714608 + T^{2} )^{4} \)
$61$ \( ( 14729384300544 + 23548032 T^{2} + T^{4} )^{2} \)
$67$ \( ( 4145361152256 - 16458336 T^{2} + T^{4} )^{2} \)
$71$ \( ( 424196534145024 + 94718592 T^{2} + T^{4} )^{2} \)
$73$ \( ( 611332 + 2564 T + T^{2} )^{4} \)
$79$ \( ( 3797136795711744 + 147736224 T^{2} + T^{4} )^{2} \)
$83$ \( ( 470880972843264 - 61970016 T^{2} + T^{4} )^{2} \)
$89$ \( ( 9182916 + 7908 T + T^{2} )^{4} \)
$97$ \( ( 25471300 + 11300 T + T^{2} )^{4} \)
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