Properties

Label 384.3.e.d
Level $384$
Weight $3$
Character orbit 384.e
Analytic conductor $10.463$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 18 x^{6} + 99 x^{4} + 170 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{3} q^{5} + ( 1 + \beta_{2} ) q^{7} + ( \beta_{2} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{3} q^{5} + ( 1 + \beta_{2} ) q^{7} + ( \beta_{2} + \beta_{7} ) q^{9} + ( \beta_{4} + \beta_{5} - \beta_{7} ) q^{11} + ( 1 - \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{13} + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{15} + ( 1 - 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( -3 + \beta_{1} - 2 \beta_{2} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{19} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{21} + ( 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{23} + ( -2 - 5 \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{25} + ( -6 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{27} + ( -2 + 6 \beta_{1} + 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{29} + ( -9 + 4 \beta_{1} - \beta_{2} - 2 \beta_{5} - 2 \beta_{7} ) q^{31} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{33} + ( -1 + 3 \beta_{1} + 8 \beta_{3} + 5 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{35} + ( -1 + 9 \beta_{1} - 4 \beta_{2} - 3 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{37} + ( -13 + \beta_{1} - 2 \beta_{3} + 5 \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{39} + ( -2 + 6 \beta_{1} - 6 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{41} + ( 17 + \beta_{1} + 2 \beta_{2} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{43} + ( -10 + 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{45} + ( -2 + 6 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{47} + ( 2 + 9 \beta_{1} + 2 \beta_{2} + 3 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} ) q^{49} + ( 23 - \beta_{1} - 4 \beta_{3} - 5 \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{51} + ( 4 - 12 \beta_{1} + \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{53} + ( 22 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{7} ) q^{55} + ( -1 - 4 \beta_{1} - \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{57} + ( -1 + 3 \beta_{1} + 8 \beta_{4} + \beta_{6} ) q^{59} + ( -13 - 11 \beta_{1} + 4 \beta_{2} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{61} + ( 31 + 3 \beta_{2} + 4 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} ) q^{63} + ( 4 - 12 \beta_{1} + 2 \beta_{3} - 6 \beta_{4} - 4 \beta_{6} ) q^{65} + ( -37 + \beta_{1} + 8 \beta_{2} + 4 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{67} + ( 8 + 2 \beta_{1} - 6 \beta_{2} + 6 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} ) q^{69} + ( -2 + 6 \beta_{1} + 4 \beta_{3} - 10 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{71} + ( -2 - 12 \beta_{1} + 4 \beta_{2} + 4 \beta_{6} ) q^{73} + ( -43 - 10 \beta_{2} - 9 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{75} + ( -4 + 12 \beta_{1} - 2 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{77} + ( -41 - 16 \beta_{1} + 3 \beta_{2} + 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{79} + ( 10 - 3 \beta_{1} + 4 \beta_{3} + 8 \beta_{4} - 5 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{81} + ( 2 - 6 \beta_{1} - 8 \beta_{3} + 5 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{83} + ( 16 + 8 \beta_{1} + 16 \beta_{2} + 8 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} ) q^{85} + ( -48 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} + 7 \beta_{7} ) q^{87} + ( -5 + 15 \beta_{1} + 16 \beta_{3} - 8 \beta_{4} + \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{89} + ( 40 + 2 \beta_{1} - 10 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{91} + ( 27 - 11 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} + 8 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{93} + ( 8 - 24 \beta_{1} - 4 \beta_{3} - 16 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} ) q^{95} + ( 5 + 13 \beta_{1} - 2 \beta_{2} - 5 \beta_{5} - \beta_{6} - 5 \beta_{7} ) q^{97} + ( 55 - 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{3} + 8q^{7} + O(q^{10}) \) \( 8q + 4q^{3} + 8q^{7} + 16q^{15} - 24q^{19} + 16q^{21} - 40q^{25} - 44q^{27} - 56q^{31} + 8q^{33} + 32q^{37} - 104q^{39} + 136q^{43} - 80q^{45} + 72q^{49} + 176q^{51} + 192q^{55} - 40q^{57} - 160q^{61} + 264q^{63} - 280q^{67} + 80q^{69} - 80q^{73} - 348q^{75} - 408q^{79} + 72q^{81} + 192q^{85} - 368q^{87} + 336q^{91} + 160q^{93} + 96q^{97} + 432q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 18 x^{6} + 99 x^{4} + 170 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 16 \nu^{4} + 70 \nu^{2} + 6 \nu + 63 \)\()/6\)
\(\beta_{2}\)\(=\)\( \nu^{4} + 11 \nu^{2} + 18 \)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{7} + 33 \nu^{5} + 159 \nu^{3} + 193 \nu \)\()/9\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} + 36 \nu^{5} + 180 \nu^{3} + 178 \nu \)\()/9\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - 16 \nu^{5} - 3 \nu^{4} - 70 \nu^{3} - 27 \nu^{2} - 57 \nu - 27 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 16 \nu^{4} - 70 \nu^{2} + 6 \nu - 61 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 16 \nu^{5} - 3 \nu^{4} + 70 \nu^{3} - 27 \nu^{2} + 63 \nu - 27 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + 3 \beta_{1} - 1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{7} - \beta_{6} + 3 \beta_{5} + 6 \beta_{2} - 3 \beta_{1} - 53\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{7} - 13 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 12 \beta_{3} - 39 \beta_{1} + 13\)\()/12\)
\(\nu^{4}\)\(=\)\((\)\(-33 \beta_{7} + 11 \beta_{6} - 33 \beta_{5} - 54 \beta_{2} + 33 \beta_{1} + 367\)\()/12\)
\(\nu^{5}\)\(=\)\((\)\(21 \beta_{7} + 101 \beta_{6} - 21 \beta_{5} + 57 \beta_{4} - 120 \beta_{3} + 303 \beta_{1} - 101\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(159 \beta_{7} - 59 \beta_{6} + 159 \beta_{5} + 222 \beta_{2} - 141 \beta_{1} - 1453\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(-54 \beta_{7} - 413 \beta_{6} + 54 \beta_{5} - 351 \beta_{4} + 540 \beta_{3} - 1239 \beta_{1} + 413\)\()/6\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
1.32750i
1.32750i
2.98985i
2.98985i
2.55118i
2.55118i
0.888828i
0.888828i
0 −2.69031 1.32750i 0 0.640013i 0 2.72077 0 5.47550 + 7.14275i 0
257.2 0 −2.69031 + 1.32750i 0 0.640013i 0 2.72077 0 5.47550 7.14275i 0
257.3 0 0.246559 2.98985i 0 6.63641i 0 0.578158 0 −8.87842 1.47435i 0
257.4 0 0.246559 + 2.98985i 0 6.63641i 0 0.578158 0 −8.87842 + 1.47435i 0
257.5 0 1.57844 2.55118i 0 1.31534i 0 −10.2329 0 −4.01705 8.05378i 0
257.6 0 1.57844 + 2.55118i 0 1.31534i 0 −10.2329 0 −4.01705 + 8.05378i 0
257.7 0 2.86531 0.888828i 0 8.59176i 0 10.9340 0 7.41997 5.09353i 0
257.8 0 2.86531 + 0.888828i 0 8.59176i 0 10.9340 0 7.41997 + 5.09353i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.8
Significant digits:
Format:

Inner twists

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

\( T_{7}^{4} - 4 T_{7}^{3} - 108 T_{7}^{2} + 368 T_{7} - 176 \)
\( T_{13}^{4} - 360 T_{13}^{2} + 256 T_{13} + 7824 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 6561 - 2916 T + 648 T^{2} + 36 T^{3} - 66 T^{4} + 4 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$5$ \( 2304 + 7040 T^{2} + 3504 T^{4} + 120 T^{6} + T^{8} \)
$7$ \( ( -176 + 368 T - 108 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$11$ \( 20214016 + 3167360 T^{2} + 69552 T^{4} + 488 T^{6} + T^{8} \)
$13$ \( ( 7824 + 256 T - 360 T^{2} + T^{4} )^{2} \)
$17$ \( 991494144 + 60416000 T^{2} + 480000 T^{4} + 1248 T^{6} + T^{8} \)
$19$ \( ( -98352 - 17584 T - 780 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$23$ \( 48132849664 + 478846976 T^{2} + 1671936 T^{4} + 2336 T^{6} + T^{8} \)
$29$ \( 78767790336 + 896411520 T^{2} + 3074736 T^{4} + 3704 T^{6} + T^{8} \)
$31$ \( ( 50256 - 17520 T - 828 T^{2} + 28 T^{3} + T^{4} )^{2} \)
$37$ \( ( -488432 + 90944 T - 3528 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$41$ \( 3916979306496 + 31405105152 T^{2} + 34597632 T^{4} + 10976 T^{6} + T^{8} \)
$43$ \( ( -917424 + 92496 T - 588 T^{2} - 68 T^{3} + T^{4} )^{2} \)
$47$ \( 424144797696 + 3025141760 T^{2} + 6905856 T^{4} + 5376 T^{6} + T^{8} \)
$53$ \( 870911869798656 + 673606710144 T^{2} + 188831664 T^{4} + 22776 T^{6} + T^{8} \)
$59$ \( 12745356964096 + 31548679808 T^{2} + 26455344 T^{4} + 8840 T^{6} + T^{8} \)
$61$ \( ( -4584688 - 262208 T - 2184 T^{2} + 80 T^{3} + T^{4} )^{2} \)
$67$ \( ( -6784816 - 335312 T + 900 T^{2} + 140 T^{3} + T^{4} )^{2} \)
$71$ \( 1706597351424 + 36124434432 T^{2} + 77642496 T^{4} + 22688 T^{6} + T^{8} \)
$73$ \( ( -899312 - 192608 T - 5544 T^{2} + 40 T^{3} + T^{4} )^{2} \)
$79$ \( ( -38762928 - 676400 T + 7524 T^{2} + 204 T^{3} + T^{4} )^{2} \)
$83$ \( 56638749360384 + 102766133376 T^{2} + 59473584 T^{4} + 13416 T^{6} + T^{8} \)
$89$ \( 9213001971859456 + 4688908992512 T^{2} + 730308096 T^{4} + 45632 T^{6} + T^{8} \)
$97$ \( ( 8416272 + 274112 T - 8136 T^{2} - 48 T^{3} + T^{4} )^{2} \)
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