Properties

Label 384.8.d.c
Level $384$
Weight $8$
Character orbit 384.d
Analytic conductor $119.956$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(119.955849786\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 449x^{6} + 50632x^{4} + 69129x^{2} + 18225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q - 27 \beta_1 q^{3} + (\beta_{5} + 28 \beta_1) q^{5} + ( - \beta_{3} - 360) q^{7} - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 27 \beta_1 q^{3} + (\beta_{5} + 28 \beta_1) q^{5} + ( - \beta_{3} - 360) q^{7} - 729 q^{9} + ( - \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 36 \beta_1) q^{11} + ( - \beta_{7} - 4 \beta_{6} - 17 \beta_{5} + 540 \beta_1) q^{13} + ( - 27 \beta_{2} + 756) q^{15} + ( - 11 \beta_{4} - 13 \beta_{3} - 33 \beta_{2} + 2862) q^{17} + (5 \beta_{7} - 3 \beta_{6} - 39 \beta_{5} + 4068 \beta_1) q^{19} + ( - 27 \beta_{7} + 9720 \beta_1) q^{21} + (24 \beta_{4} + 22 \beta_{3} + 156 \beta_{2} - 25920) q^{23} + ( - 10 \beta_{4} - 74 \beta_{3} + 198 \beta_{2} - 25587) q^{25} + 19683 \beta_1 q^{27} + ( - 58 \beta_{7} - 52 \beta_{6} + 59 \beta_{5} + 61972 \beta_1) q^{29} + ( - 60 \beta_{4} - 59 \beta_{3} + 528 \beta_{2} + 2232) q^{31} + (81 \beta_{4} + 27 \beta_{3} - 81 \beta_{2} + 972) q^{33} + ( - 21 \beta_{7} - 105 \beta_{6} - 1329 \beta_{5} - 35280 \beta_1) q^{35} + ( - 177 \beta_{7} + 228 \beta_{6} + 397 \beta_{5} + \cdots + 115844 \beta_1) q^{37}+ \cdots + (729 \beta_{7} - 2187 \beta_{6} - 2187 \beta_{5} - 26244 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2880 q^{7} - 5832 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2880 q^{7} - 5832 q^{9} + 6048 q^{15} + 22896 q^{17} - 207360 q^{23} - 204696 q^{25} + 17856 q^{31} + 7776 q^{33} + 116640 q^{39} + 687056 q^{41} + 1987200 q^{47} + 4815560 q^{49} - 2077056 q^{55} + 878688 q^{57} + 2099520 q^{63} + 12871808 q^{65} + 6336000 q^{71} + 8920752 q^{73} - 1251648 q^{79} + 4251528 q^{81} + 13385952 q^{87} + 4447408 q^{89} + 31607424 q^{95} - 14157584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 449x^{6} + 50632x^{4} + 69129x^{2} + 18225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 131\nu^{7} + 58684\nu^{5} + 6584732\nu^{3} + 5063679\nu ) / 1626480 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 65\nu^{6} + 27156\nu^{4} + 2829796\nu^{2} + 2486853 ) / 9036 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 57\nu^{6} + 24308\nu^{4} + 2496836\nu^{2} - 8497827 ) / 9036 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 553\nu^{6} + 249076\nu^{4} + 28104132\nu^{2} + 20876013 ) / 4518 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -713\nu^{7} - 320092\nu^{5} - 36039416\nu^{3} - 40729437\nu ) / 101655 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7231\nu^{7} + 3248924\nu^{5} + 366814612\nu^{3} + 540861579\nu ) / 101655 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9022\nu^{7} + 4049168\nu^{5} + 455154004\nu^{3} + 338539158\nu ) / 101655 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} - 25\beta_{5} - 192\beta_1 ) / 768 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - 73\beta_{3} + 47\beta_{2} - 86208 ) / 768 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 71\beta_{7} + 143\beta_{6} + 2783\beta_{5} + 37824\beta_1 ) / 384 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -31\beta_{4} + 16303\beta_{3} - 13769\beta_{2} + 19264704 ) / 768 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -22159\beta_{7} - 63991\beta_{6} - 1252807\beta_{5} - 28166592\beta_1 ) / 768 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3823\beta_{4} - 454133\beta_{3} + 476635\beta_{2} - 540598368 ) / 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2827561\beta_{7} + 14328841\beta_{6} + 282409921\beta_{5} + 8832273600\beta_1 ) / 768 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
14.6646i
0.596953i
1.01122i
15.2503i
15.2503i
1.01122i
0.596953i
14.6646i
0 27.0000i 0 344.306i 0 −1669.75 0 −729.000 0
193.2 0 27.0000i 0 135.999i 0 678.568 0 −729.000 0
193.3 0 27.0000i 0 69.8856i 0 860.192 0 −729.000 0
193.4 0 27.0000i 0 522.419i 0 −1309.01 0 −729.000 0
193.5 0 27.0000i 0 522.419i 0 −1309.01 0 −729.000 0
193.6 0 27.0000i 0 69.8856i 0 860.192 0 −729.000 0
193.7 0 27.0000i 0 135.999i 0 678.568 0 −729.000 0
193.8 0 27.0000i 0 344.306i 0 −1669.75 0 −729.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.8.d.c 8
4.b odd 2 1 384.8.d.d yes 8
8.b even 2 1 inner 384.8.d.c 8
8.d odd 2 1 384.8.d.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.8.d.c 8 1.a even 1 1 trivial
384.8.d.c 8 8.b even 2 1 inner
384.8.d.d yes 8 4.b odd 2 1
384.8.d.d yes 8 8.d odd 2 1

Hecke kernels

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

\( T_{5}^{8} + 414848T_{5}^{6} + 41596678144T_{5}^{4} + 791786461593600T_{5}^{2} + 2922622746624000000 \) Copy content Toggle raw display
\( T_{7}^{4} + 1440T_{7}^{3} - 1814176T_{7}^{2} - 1624601088T_{7} + 1275801663744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 414848 T^{6} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1440 T^{3} + \cdots + 1275801663744)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 83236928 T^{6} + \cdots + 49\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{8} + 269496640 T^{6} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{4} - 11448 T^{3} + \cdots + 58\!\cdots\!48)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 878694464 T^{6} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( (T^{4} + 103680 T^{3} + \cdots + 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 69547739776 T^{6} + \cdots + 58\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( (T^{4} - 8928 T^{3} + \cdots - 45\!\cdots\!92)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 646806181952 T^{6} + \cdots + 78\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} - 343528 T^{3} + \cdots - 32\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 703896484928 T^{6} + \cdots + 68\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( (T^{4} - 993600 T^{3} + \cdots - 24\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 4748734157440 T^{6} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{8} + 10826005492800 T^{6} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{8} + 12824477047872 T^{6} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{8} + 35890312491072 T^{6} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} - 3168000 T^{3} + \cdots + 15\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 4460376 T^{3} + \cdots - 79\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 625824 T^{3} + \cdots - 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 93693006217280 T^{6} + \cdots + 36\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{4} - 2223704 T^{3} + \cdots + 69\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 7078792 T^{3} + \cdots - 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
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