Properties

Label 288.7.b.b
Level 288
Weight 7
Character orbit 288.b
Analytic conductor 66.256
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 288.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(66.2555760825\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.3803625.2
Defining polynomial: \(x^{4} - x^{3} + 6 x^{2} - 16 x + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + \beta_{2} q^{5} + ( \beta_{2} - \beta_{3} ) q^{7} + ( 244 + 57 \beta_{1} ) q^{11} + ( \beta_{2} - 8 \beta_{3} ) q^{13} + ( -1042 + 200 \beta_{1} ) q^{17} + ( 364 - 65 \beta_{1} ) q^{19} + ( -91 \beta_{2} - 5 \beta_{3} ) q^{23} + ( -5975 + 880 \beta_{1} ) q^{25} + ( -13 \beta_{2} - 64 \beta_{3} ) q^{29} + ( -28 \beta_{2} + 60 \beta_{3} ) q^{31} + ( -12480 - 800 \beta_{1} ) q^{35} + ( 207 \beta_{2} - 56 \beta_{3} ) q^{37} + ( 29486 - 1104 \beta_{1} ) q^{41} + ( -49364 - 2121 \beta_{1} ) q^{43} + ( -638 \beta_{2} + 286 \beta_{3} ) q^{47} + ( 529 - 5696 \beta_{1} ) q^{49} + ( 1053 \beta_{2} - 56 \beta_{3} ) q^{53} + ( -611 \beta_{2} - 285 \beta_{3} ) q^{55} + ( 135508 - 327 \beta_{1} ) q^{59} + ( 845 \beta_{2} - 328 \beta_{3} ) q^{61} + ( 51360 - 12560 \beta_{1} ) q^{65} + ( 197548 + 5967 \beta_{1} ) q^{67} + ( -1977 \beta_{2} - 743 \beta_{3} ) q^{71} + ( 110978 - 8536 \beta_{1} ) q^{73} + ( 1612 \beta_{2} - 1384 \beta_{3} ) q^{77} + ( -1118 \beta_{2} - 546 \beta_{3} ) q^{79} + ( -866252 + 3089 \beta_{1} ) q^{83} + ( -4042 \beta_{2} - 1000 \beta_{3} ) q^{85} + ( -190306 + 11928 \beta_{1} ) q^{89} + ( -849600 - 39968 \beta_{1} ) q^{91} + ( 1339 \beta_{2} + 325 \beta_{3} ) q^{95} + ( -231694 - 42696 \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 976q^{11} - 4168q^{17} + 1456q^{19} - 23900q^{25} - 49920q^{35} + 117944q^{41} - 197456q^{43} + 2116q^{49} + 542032q^{59} + 205440q^{65} + 790192q^{67} + 443912q^{73} - 3465008q^{83} - 761224q^{89} - 3398400q^{91} - 926776q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 6 x^{2} - 16 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 10 \nu + 8 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 17 \nu^{2} + 38 \nu - 64 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{3} + 9 \nu^{2} + 138 \nu - 64 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 16 \beta_{1} + 32\)\()/128\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{3} - 13 \beta_{2} + 16 \beta_{1} - 352\)\()/128\)
\(\nu^{3}\)\(=\)\((\)\(13 \beta_{3} - 3 \beta_{2} - 336 \beta_{1} + 992\)\()/128\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
−2.31174 3.26433i
2.81174 + 2.84502i
2.81174 2.84502i
−2.31174 + 3.26433i
0 0 0 199.084i 0 19.6656i 0 0 0
271.2 0 0 0 59.7107i 0 483.584i 0 0 0
271.3 0 0 0 59.7107i 0 483.584i 0 0 0
271.4 0 0 0 199.084i 0 19.6656i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 288.7.b.b 4
3.b odd 2 1 32.7.d.b 4
4.b odd 2 1 72.7.b.b 4
8.b even 2 1 72.7.b.b 4
8.d odd 2 1 inner 288.7.b.b 4
12.b even 2 1 8.7.d.b 4
24.f even 2 1 32.7.d.b 4
24.h odd 2 1 8.7.d.b 4
48.i odd 4 2 256.7.c.l 8
48.k even 4 2 256.7.c.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.b 4 12.b even 2 1
8.7.d.b 4 24.h odd 2 1
32.7.d.b 4 3.b odd 2 1
32.7.d.b 4 24.f even 2 1
72.7.b.b 4 4.b odd 2 1
72.7.b.b 4 8.b even 2 1
256.7.c.l 8 48.i odd 4 2
256.7.c.l 8 48.k even 4 2
288.7.b.b 4 1.a even 1 1 trivial
288.7.b.b 4 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 43200 T_{5}^{2} + 141312000 \) acting on \(S_{7}^{\mathrm{new}}(288, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 19300 T^{2} + 256155750 T^{4} - 4711914062500 T^{6} + 59604644775390625 T^{8} \)
$7$ \( 1 - 236356 T^{2} + 28021959366 T^{4} - 3271471277679556 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 - 488 T + 2238078 T^{2} - 864521768 T^{3} + 3138428376721 T^{4} )^{2} \)
$13$ \( 1 - 4994596 T^{2} + 30260873415846 T^{4} - \)\(11\!\cdots\!76\)\( T^{6} + \)\(54\!\cdots\!61\)\( T^{8} \)
$17$ \( ( 1 + 2084 T + 32560902 T^{2} + 50302693796 T^{3} + 582622237229761 T^{4} )^{2} \)
$19$ \( ( 1 - 728 T + 92449758 T^{2} - 34249401368 T^{3} + 2213314919066161 T^{4} )^{2} \)
$23$ \( 1 - 212117956 T^{2} + 23026671552237126 T^{4} - \)\(46\!\cdots\!76\)\( T^{6} + \)\(48\!\cdots\!41\)\( T^{8} \)
$29$ \( 1 - 1409719204 T^{2} + 1160515330165289766 T^{4} - \)\(49\!\cdots\!64\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \)
$31$ \( 1 - 2758360324 T^{2} + 3362667870952277766 T^{4} - \)\(21\!\cdots\!64\)\( T^{6} + \)\(62\!\cdots\!21\)\( T^{8} \)
$37$ \( 1 - 8121202276 T^{2} + 29600495645847907686 T^{4} - \)\(53\!\cdots\!56\)\( T^{6} + \)\(43\!\cdots\!61\)\( T^{8} \)
$41$ \( ( 1 - 58972 T + 9857729958 T^{2} - 280123147300252 T^{3} + 22563490300366186081 T^{4} )^{2} \)
$43$ \( ( 1 + 98728 T + 13190101374 T^{2} + 624095531101672 T^{3} + 39959630797262576401 T^{4} )^{2} \)
$47$ \( 1 - 13578767236 T^{2} + \)\(16\!\cdots\!86\)\( T^{4} - \)\(15\!\cdots\!76\)\( T^{6} + \)\(13\!\cdots\!81\)\( T^{8} \)
$53$ \( 1 - 42194545636 T^{2} + \)\(11\!\cdots\!86\)\( T^{4} - \)\(20\!\cdots\!76\)\( T^{6} + \)\(24\!\cdots\!81\)\( T^{8} \)
$59$ \( ( 1 - 271016 T + 102678575166 T^{2} - 11431599505249256 T^{3} + \)\(17\!\cdots\!81\)\( T^{4} )^{2} \)
$61$ \( 1 - 160868902564 T^{2} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(42\!\cdots\!44\)\( T^{6} + \)\(70\!\cdots\!41\)\( T^{8} \)
$67$ \( ( 1 - 395096 T + 204987839262 T^{2} - 35739744961443224 T^{3} + \)\(81\!\cdots\!61\)\( T^{4} )^{2} \)
$71$ \( 1 - 164364621124 T^{2} + \)\(21\!\cdots\!06\)\( T^{4} - \)\(26\!\cdots\!84\)\( T^{6} + \)\(26\!\cdots\!81\)\( T^{8} \)
$73$ \( ( 1 - 221956 T + 284381984742 T^{2} - 33589539530201284 T^{3} + \)\(22\!\cdots\!21\)\( T^{4} )^{2} \)
$79$ \( 1 - 828257339524 T^{2} + \)\(28\!\cdots\!06\)\( T^{4} - \)\(48\!\cdots\!84\)\( T^{6} + \)\(34\!\cdots\!81\)\( T^{8} \)
$83$ \( ( 1 + 1732504 T + 1400265667422 T^{2} + 566425504623285976 T^{3} + \)\(10\!\cdots\!61\)\( T^{4} )^{2} \)
$89$ \( ( 1 + 380612 T + 970422538278 T^{2} + 189157043115248132 T^{3} + \)\(24\!\cdots\!21\)\( T^{4} )^{2} \)
$97$ \( ( 1 + 463388 T + 953987784774 T^{2} + 385989231420039452 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \)
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