Properties

Label 384.8.a.l
Level $384$
Weight $8$
Character orbit 384.a
Self dual yes
Analytic conductor $119.956$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 384.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(119.955849786\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - 286x - 1680 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 27 q^{3} + (\beta_1 + 103) q^{5} + (\beta_{2} + 2) q^{7} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 27 q^{3} + (\beta_1 + 103) q^{5} + (\beta_{2} + 2) q^{7} + 729 q^{9} + (2 \beta_{2} + 8 \beta_1 + 492) q^{11} + (3 \beta_{2} + 5 \beta_1 + 1025) q^{13} + (27 \beta_1 + 2781) q^{15} + (16 \beta_{2} + 50 \beta_1 - 1692) q^{17} + (8 \beta_{2} - 2 \beta_1 + 4722) q^{19} + (27 \beta_{2} + 54) q^{21} + ( - 34 \beta_{2} - 4 \beta_1 + 19992) q^{23} + (30 \beta_{2} + 58 \beta_1 + 10529) q^{25} + 19683 q^{27} + (2 \beta_{2} - 383 \beta_1 + 23867) q^{29} + ( - 103 \beta_{2} + 448 \beta_1 + 59306) q^{31} + (54 \beta_{2} + 216 \beta_1 + 13284) q^{33} + (250 \beta_{2} + 580 \beta_1 - 28060) q^{35} + (173 \beta_{2} - 567 \beta_1 - 61527) q^{37} + (81 \beta_{2} + 135 \beta_1 + 27675) q^{39} + (36 \beta_{2} + 146 \beta_1 + 219132) q^{41} + ( - 284 \beta_{2} + 1342 \beta_1 - 39734) q^{43} + (729 \beta_1 + 75087) q^{45} + ( - 258 \beta_{2} + 448 \beta_1 + 221860) q^{47} + ( - 574 \beta_{2} + 1890 \beta_1 + 402179) q^{49} + (432 \beta_{2} + 1350 \beta_1 - 45684) q^{51} + ( - 814 \beta_{2} - 3603 \beta_1 - 38497) q^{53} + (740 \beta_{2} + 1288 \beta_1 + 618504) q^{55} + (216 \beta_{2} - 54 \beta_1 + 127494) q^{57} + ( - 1364 \beta_{2} - 3636 \beta_1 - 345648) q^{59} + ( - 115 \beta_{2} + 2629 \beta_1 + 1195645) q^{61} + (729 \beta_{2} + 1458) q^{63} + (900 \beta_{2} + 2534 \beta_1 + 411002) q^{65} + ( - 1380 \beta_{2} - 1988 \beta_1 - 782296) q^{67} + ( - 918 \beta_{2} - 108 \beta_1 + 539784) q^{69} + (2054 \beta_{2} + 1460 \beta_1 + 2104328) q^{71} + ( - 2938 \beta_{2} - 9690 \beta_1 - 707288) q^{73} + (810 \beta_{2} + 1566 \beta_1 + 284283) q^{75} + (516 \beta_{2} + 8420 \beta_1 + 2226292) q^{77} + ( - 4627 \beta_{2} - 11692 \beta_1 + 669046) q^{79} + 531441 q^{81} + (2838 \beta_{2} - 13452 \beta_1 - 2481312) q^{83} + (5500 \beta_{2} + 5306 \beta_1 + 3275718) q^{85} + (54 \beta_{2} - 10341 \beta_1 + 644409) q^{87} + (2720 \beta_{2} - 27228 \beta_1 - 4442546) q^{89} + (32 \beta_{2} + 8570 \beta_1 + 3537874) q^{91} + ( - 2781 \beta_{2} + 12096 \beta_1 + 1601262) q^{93} + (1940 \beta_{2} + 9436 \beta_1 + 104148) q^{95} + ( - 204 \beta_{2} + 22864 \beta_1 + 994426) q^{97} + (1458 \beta_{2} + 5832 \beta_1 + 358668) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 81 q^{3} + 308 q^{5} + 6 q^{7} + 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 81 q^{3} + 308 q^{5} + 6 q^{7} + 2187 q^{9} + 1468 q^{11} + 3070 q^{13} + 8316 q^{15} - 5126 q^{17} + 14168 q^{19} + 162 q^{21} + 59980 q^{23} + 31529 q^{25} + 59049 q^{27} + 71984 q^{29} + 177470 q^{31} + 39636 q^{33} - 84760 q^{35} - 184014 q^{37} + 82890 q^{39} + 657250 q^{41} - 120544 q^{43} + 224532 q^{45} + 665132 q^{47} + 1204647 q^{49} - 138402 q^{51} - 111888 q^{53} + 1854224 q^{55} + 382536 q^{57} - 1033308 q^{59} + 3584306 q^{61} + 4374 q^{63} + 1230472 q^{65} - 2344900 q^{67} + 1619460 q^{69} + 6311524 q^{71} - 2112174 q^{73} + 851283 q^{75} + 6670456 q^{77} + 2018830 q^{79} + 1594323 q^{81} - 7430484 q^{83} + 9821848 q^{85} + 1943568 q^{87} - 13300410 q^{89} + 10605052 q^{91} + 4791690 q^{93} + 303008 q^{95} + 2960414 q^{97} + 1070172 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 286x - 1680 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu^{2} - 24\nu - 763 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -12\nu^{2} + 168\nu + 2288 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 3\beta _1 + 1 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 7\beta _1 + 3053 ) / 16 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−7.15472
−12.1582
19.3129
0 27.0000 0 −283.526 0 473.726 0 729.000 0
1.2 0 27.0000 0 223.082 0 −1526.43 0 729.000 0
1.3 0 27.0000 0 368.444 0 1058.71 0 729.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.8.a.l yes 3
4.b odd 2 1 384.8.a.j yes 3
8.b even 2 1 384.8.a.i 3
8.d odd 2 1 384.8.a.k yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.8.a.i 3 8.b even 2 1
384.8.a.j yes 3 4.b odd 2 1
384.8.a.k yes 3 8.d odd 2 1
384.8.a.l yes 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(384))\):

\( T_{5}^{3} - 308T_{5}^{2} - 85520T_{5} + 23304000 \) Copy content Toggle raw display
\( T_{7}^{3} - 6T_{7}^{2} - 1837620T_{7} + 765563000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 27)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 308 T^{2} + \cdots + 23304000 \) Copy content Toggle raw display
$7$ \( T^{3} - 6 T^{2} - 1837620 T + 765563000 \) Copy content Toggle raw display
$11$ \( T^{3} - 1468 T^{2} + \cdots - 12568576832 \) Copy content Toggle raw display
$13$ \( T^{3} - 3070 T^{2} + \cdots + 8296093848 \) Copy content Toggle raw display
$17$ \( T^{3} + 5126 T^{2} + \cdots - 7729742082552 \) Copy content Toggle raw display
$19$ \( T^{3} - 14168 T^{2} + \cdots + 907808232960 \) Copy content Toggle raw display
$23$ \( T^{3} - 59980 T^{2} + \cdots + 6681673416640 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots - 329426101684224 \) Copy content Toggle raw display
$31$ \( T^{3} - 177470 T^{2} + \cdots + 30\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{3} + 184014 T^{2} + \cdots - 27\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{3} - 657250 T^{2} + \cdots - 96\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{3} + 120544 T^{2} + \cdots + 63\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{3} - 665132 T^{2} + \cdots + 33\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{3} + 111888 T^{2} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{3} + 1033308 T^{2} + \cdots + 14\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{3} - 3584306 T^{2} + \cdots - 37\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{3} + 2344900 T^{2} + \cdots - 20\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{3} - 6311524 T^{2} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{3} + 2112174 T^{2} + \cdots + 27\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{3} - 2018830 T^{2} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + 7430484 T^{2} + \cdots - 10\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{3} + 13300410 T^{2} + \cdots - 76\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{3} - 2960414 T^{2} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
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