Properties

Label 384.9.b.b
Level $384$
Weight $9$
Character orbit 384.b
Analytic conductor $156.433$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(156.433386263\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 4826 x^{6} + 8748877 x^{4} + 7060845096 x^{2} + 2140819627716\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{11} \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{2} q^{5} -\beta_{4} q^{7} + 2187 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{2} q^{5} -\beta_{4} q^{7} + 2187 q^{9} + ( -44 \beta_{1} + \beta_{5} ) q^{11} + ( 22 \beta_{2} + \beta_{3} ) q^{13} + ( 2 \beta_{4} + \beta_{7} ) q^{15} + ( 34606 - \beta_{6} ) q^{17} + ( 2420 \beta_{1} - 5 \beta_{5} ) q^{19} + ( -27 \beta_{2} + 27 \beta_{3} ) q^{21} + ( 6 \beta_{4} + 4 \beta_{7} ) q^{23} + ( -355247 - 22 \beta_{6} ) q^{25} -2187 \beta_{1} q^{27} + ( -99 \beta_{2} + 110 \beta_{3} ) q^{29} + ( 77 \beta_{4} + 20 \beta_{7} ) q^{31} + ( 96876 + 27 \beta_{6} ) q^{33} + ( -6584 \beta_{1} + 57 \beta_{5} ) q^{35} + ( -1876 \beta_{2} + 477 \beta_{3} ) q^{37} + ( -35 \beta_{4} + 23 \beta_{7} ) q^{39} + ( 728354 + 23 \beta_{6} ) q^{41} + ( 49476 \beta_{1} - 145 \beta_{5} ) q^{43} + 2187 \beta_{2} q^{45} + ( 10 \beta_{4} + 128 \beta_{7} ) q^{47} + ( -2074991 + 204 \beta_{6} ) q^{49} + ( -34630 \beta_{1} - 81 \beta_{5} ) q^{51} + ( 1495 \beta_{2} + 842 \beta_{3} ) q^{53} + ( 2648 \beta_{4} + 404 \beta_{7} ) q^{55} + ( -5295780 - 135 \beta_{6} ) q^{57} + ( -98180 \beta_{1} + 848 \beta_{5} ) q^{59} + ( 400 \beta_{2} - 3413 \beta_{3} ) q^{61} -2187 \beta_{4} q^{63} + ( -16620384 - 449 \beta_{6} ) q^{65} + ( -150228 \beta_{1} - 580 \beta_{5} ) q^{67} + ( 8694 \beta_{2} + 54 \beta_{3} ) q^{69} + ( -3622 \beta_{4} + 636 \beta_{7} ) q^{71} + ( -7219970 - 340 \beta_{6} ) q^{73} + ( 354719 \beta_{1} - 1782 \beta_{5} ) q^{75} + ( -24748 \beta_{2} - 8516 \beta_{3} ) q^{77} + ( 9797 \beta_{4} - 1348 \beta_{7} ) q^{79} + 4782969 q^{81} + ( 714228 \beta_{1} + 1347 \beta_{5} ) q^{83} + ( 3630 \beta_{2} + 1840 \beta_{3} ) q^{85} + ( -8888 \beta_{4} + 11 \beta_{7} ) q^{87} + ( -32347346 + 1546 \beta_{6} ) q^{89} + ( 139112 \beta_{1} + 1923 \beta_{5} ) q^{91} + ( 44739 \beta_{2} - 999 \beta_{3} ) q^{93} + ( -17640 \beta_{4} - 4220 \beta_{7} ) q^{95} + ( -96436178 - 1190 \beta_{6} ) q^{97} + ( -96228 \beta_{1} + 2187 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 17496q^{9} + O(q^{10}) \) \( 8q + 17496q^{9} + 276848q^{17} - 2841976q^{25} + 775008q^{33} + 5826832q^{41} - 16599928q^{49} - 42366240q^{57} - 132963072q^{65} - 57759760q^{73} + 38263752q^{81} - 258778768q^{89} - 771489424q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 4826 x^{6} + 8748877 x^{4} + 7060845096 x^{2} + 2140819627716\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -22041 \nu^{7} - 80033094 \nu^{5} - 97666072335 \nu^{3} - 40034401078230 \nu \)\()/ 84091362124 \)
\(\beta_{2}\)\(=\)\((\)\( -135 \nu^{6} - 530927 \nu^{4} - 696017072 \nu^{2} - 304279869852 \)\()/1206926\)
\(\beta_{3}\)\(=\)\((\)\( 402 \nu^{6} + 1366418 \nu^{4} + 1557414512 \nu^{2} + 595243216008 \)\()/603463\)
\(\beta_{4}\)\(=\)\((\)\( 443391 \nu^{7} + 1598960541 \nu^{5} + 1920327097770 \nu^{3} + 770822874559080 \nu \)\()/ 42045681062 \)
\(\beta_{5}\)\(=\)\((\)\( 461414 \nu^{7} + 2224833092 \nu^{5} + 3357073959114 \nu^{3} + 1626110629448292 \nu \)\()/ 21022840531 \)
\(\beta_{6}\)\(=\)\((\)\( -768 \nu^{6} - 2775168 \nu^{4} - 3337694208 \nu^{2} - 1335834521664 \)\()/86209\)
\(\beta_{7}\)\(=\)\((\)\( -1072761 \nu^{7} - 3866206107 \nu^{5} - 4636501305750 \nu^{3} - 1847566336831416 \nu \)\()/ 3822334642 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + 29 \beta_{4} + 96 \beta_{1}\)\()/2592\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + 8 \beta_{3} - 32 \beta_{2} - 463296\)\()/384\)
\(\nu^{3}\)\(=\)\((\)\(-1384 \beta_{7} - 81 \beta_{5} - 48200 \beta_{4} - 464088 \beta_{1}\)\()/3456\)
\(\nu^{4}\)\(=\)\((\)\(-2401 \beta_{6} - 20288 \beta_{3} + 70400 \beta_{2} + 556082112\)\()/384\)
\(\nu^{5}\)\(=\)\((\)\(154880 \beta_{7} + 36195 \beta_{5} + 6632064 \beta_{4} + 104020488 \beta_{1}\)\()/384\)
\(\nu^{6}\)\(=\)\((\)\(4286953 \beta_{6} + 38543040 \beta_{3} - 115319808 \beta_{2} - 663855941568\)\()/384\)
\(\nu^{7}\)\(=\)\((\)\(-450207760 \beta_{7} - 274643145 \beta_{5} - 24462797648 \beta_{4} - 529545004824 \beta_{1}\)\()/1152\)

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
1.73205 + 33.5532i
1.73205 35.9608i
1.73205 + 35.9608i
1.73205 33.5532i
−1.73205 33.5532i
−1.73205 + 35.9608i
−1.73205 35.9608i
−1.73205 + 33.5532i
0 −46.7654 0 1205.32i 0 1133.44i 0 2187.00 0
319.2 0 −46.7654 0 197.353i 0 3794.06i 0 2187.00 0
319.3 0 −46.7654 0 197.353i 0 3794.06i 0 2187.00 0
319.4 0 −46.7654 0 1205.32i 0 1133.44i 0 2187.00 0
319.5 0 46.7654 0 1205.32i 0 1133.44i 0 2187.00 0
319.6 0 46.7654 0 197.353i 0 3794.06i 0 2187.00 0
319.7 0 46.7654 0 197.353i 0 3794.06i 0 2187.00 0
319.8 0 46.7654 0 1205.32i 0 1133.44i 0 2187.00 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 384.9.b.b 8
4.b odd 2 1 inner 384.9.b.b 8
8.b even 2 1 inner 384.9.b.b 8
8.d odd 2 1 inner 384.9.b.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.9.b.b 8 1.a even 1 1 trivial
384.9.b.b 8 4.b odd 2 1 inner
384.9.b.b 8 8.b even 2 1 inner
384.9.b.b 8 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 1491744 T_{5}^{2} + 56583532800 \) acting on \(S_{9}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -2187 + T^{2} )^{4} \)
$5$ \( ( 56583532800 + 1491744 T^{2} + T^{4} )^{2} \)
$7$ \( ( 18492829141248 + 15679584 T^{2} + T^{4} )^{2} \)
$11$ \( ( 115520687528622336 - 696931680 T^{2} + T^{4} )^{2} \)
$13$ \( ( 81910692156837888 + 786981504 T^{2} + T^{4} )^{2} \)
$17$ \( ( 165051460 - 69212 T + T^{2} )^{4} \)
$19$ \( ( 17802225831238560000 - 42855996000 T^{2} + T^{4} )^{2} \)
$23$ \( ( 98677366453592690688 + 51799867776 T^{2} + T^{4} )^{2} \)
$29$ \( ( \)\(21\!\cdots\!00\)\( + 566761587744 T^{2} + T^{4} )^{2} \)
$31$ \( ( \)\(26\!\cdots\!88\)\( + 1369173987936 T^{2} + T^{4} )^{2} \)
$37$ \( ( \)\(56\!\cdots\!28\)\( + 15049518744576 T^{2} + T^{4} )^{2} \)
$41$ \( ( -15705528188 - 1456708 T + T^{2} )^{4} \)
$43$ \( ( \)\(35\!\cdots\!24\)\( - 25198148122464 T^{2} + T^{4} )^{2} \)
$47$ \( ( \)\(21\!\cdots\!12\)\( + 52582489089408 T^{2} + T^{4} )^{2} \)
$53$ \( ( \)\(48\!\cdots\!00\)\( + 37287632692512 T^{2} + T^{4} )^{2} \)
$59$ \( ( \)\(51\!\cdots\!00\)\( - 537373081395552 T^{2} + T^{4} )^{2} \)
$61$ \( ( \)\(49\!\cdots\!00\)\( + 539483534602752 T^{2} + T^{4} )^{2} \)
$67$ \( ( \)\(44\!\cdots\!24\)\( - 330049356887136 T^{2} + T^{4} )^{2} \)
$71$ \( ( \)\(54\!\cdots\!92\)\( + 1515456173420928 T^{2} + T^{4} )^{2} \)
$73$ \( ( -67231781704700 + 14439940 T + T^{2} )^{4} \)
$79$ \( ( \)\(13\!\cdots\!00\)\( + 7402940888084064 T^{2} + T^{4} )^{2} \)
$83$ \( ( \)\(24\!\cdots\!84\)\( - 3477726014338656 T^{2} + T^{4} )^{2} \)
$89$ \( ( -1421500804154300 + 64694692 T + T^{2} )^{4} \)
$97$ \( ( 7837779508054084 + 192872356 T + T^{2} )^{4} \)
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