Properties

Label 288.3.h.a
Level $288$
Weight $3$
Character orbit 288.h
Analytic conductor $7.847$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(7.84743161358\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.33808912384.2
Defining polynomial: \(x^{8} - 2 x^{7} + 11 x^{6} - 18 x^{5} + 47 x^{4} - 28 x^{3} - 44 x^{2} + 48 x + 36\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 72)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + \beta_{4} q^{7} +O(q^{10})\) \( q + \beta_{2} q^{5} + \beta_{4} q^{7} + ( -\beta_{2} - \beta_{3} ) q^{11} + \beta_{7} q^{13} + ( \beta_{5} + 2 \beta_{6} ) q^{17} + ( \beta_{1} + \beta_{7} ) q^{19} + ( 3 \beta_{5} + \beta_{6} ) q^{23} + ( 5 + 4 \beta_{4} ) q^{25} + ( \beta_{2} - 2 \beta_{3} ) q^{29} + ( 16 + 3 \beta_{4} ) q^{31} + ( 7 \beta_{2} - \beta_{3} ) q^{35} + ( \beta_{1} + \beta_{7} ) q^{37} + ( \beta_{5} - 2 \beta_{6} ) q^{41} + ( -2 \beta_{1} + 2 \beta_{7} ) q^{43} + ( -7 \beta_{5} + 3 \beta_{6} ) q^{47} + 3 q^{49} + ( -5 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -32 - 2 \beta_{4} ) q^{55} + ( -4 \beta_{2} + 4 \beta_{3} ) q^{59} + ( \beta_{1} - 3 \beta_{7} ) q^{61} -2 \beta_{6} q^{65} + ( \beta_{1} - 3 \beta_{7} ) q^{67} + ( 15 \beta_{5} - 3 \beta_{6} ) q^{71} + ( -20 - 8 \beta_{4} ) q^{73} + ( -4 \beta_{2} + 8 \beta_{3} ) q^{77} + ( 48 - 11 \beta_{4} ) q^{79} + ( -7 \beta_{2} + \beta_{3} ) q^{83} + ( -5 \beta_{1} - 2 \beta_{7} ) q^{85} + ( -7 \beta_{5} + 4 \beta_{6} ) q^{89} + ( \beta_{1} - 7 \beta_{7} ) q^{91} + ( -34 \beta_{5} - 14 \beta_{6} ) q^{95} + ( -24 - 4 \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 40q^{25} + 128q^{31} + 24q^{49} - 256q^{55} - 160q^{73} + 384q^{79} - 192q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 11 x^{6} - 18 x^{5} + 47 x^{4} - 28 x^{3} - 44 x^{2} + 48 x + 36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -449 \nu^{7} - 3617 \nu^{6} - 2407 \nu^{5} - 40233 \nu^{4} + 8129 \nu^{3} - 164911 \nu^{2} + 13066 \nu + 80772 \)\()/9753\)
\(\beta_{2}\)\(=\)\((\)\( 432 \nu^{7} - 495 \nu^{6} + 4126 \nu^{5} - 4929 \nu^{4} + 13046 \nu^{3} - 4339 \nu^{2} - 30774 \nu - 7582 \)\()/6502\)
\(\beta_{3}\)\(=\)\((\)\( 574 \nu^{7} - 1335 \nu^{6} + 6596 \nu^{5} - 10929 \nu^{4} + 24348 \nu^{3} - 4215 \nu^{2} - 53442 \nu + 73338 \)\()/6502\)
\(\beta_{4}\)\(=\)\((\)\( -321 \nu^{7} + 571 \nu^{6} - 3111 \nu^{5} + 4543 \nu^{4} - 12087 \nu^{3} + 3337 \nu^{2} + 27066 \nu - 10576 \)\()/3251\)
\(\beta_{5}\)\(=\)\((\)\( -3067 \nu^{7} + 6833 \nu^{6} - 36773 \nu^{5} + 65991 \nu^{4} - 176981 \nu^{3} + 149203 \nu^{2} - 2632 \nu - 110994 \)\()/19506\)
\(\beta_{6}\)\(=\)\((\)\( 5357 \nu^{7} - 17449 \nu^{6} + 72943 \nu^{5} - 171543 \nu^{4} + 379759 \nu^{3} - 515711 \nu^{2} + 31220 \nu + 336090 \)\()/19506\)
\(\beta_{7}\)\(=\)\((\)\( -6103 \nu^{7} + 16001 \nu^{6} - 78593 \nu^{5} + 162129 \nu^{4} - 402137 \nu^{3} + 432823 \nu^{2} - 17722 \nu - 298464 \)\()/9753\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{7} - 16 \beta_{5} + 6 \beta_{4} + \beta_{1} + 12\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + \beta_{1} - 36\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(-9 \beta_{7} - 9 \beta_{6} + 19 \beta_{5} - 24 \beta_{4} - 9 \beta_{3} - 27 \beta_{2} - \beta_{1} - 12\)\()/24\)
\(\nu^{4}\)\(=\)\((\)\(15 \beta_{7} + 32 \beta_{6} - 26 \beta_{4} - 16 \beta_{3} - 16 \beta_{2} - 11 \beta_{1} + 76\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(51 \beta_{7} + 150 \beta_{6} + 86 \beta_{5} + 354 \beta_{4} + 210 \beta_{3} + 270 \beta_{2} - 59 \beta_{1} - 948\)\()/48\)
\(\nu^{6}\)\(=\)\((\)\(-38 \beta_{7} - 49 \beta_{6} + 67 \beta_{5} + 41 \beta_{4} + 3 \beta_{3} + 65 \beta_{2} + 7 \beta_{1} + 180\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(231 \beta_{7} - 588 \beta_{6} - 2308 \beta_{5} - 1770 \beta_{4} - 1848 \beta_{3} + 445 \beta_{1} + 15468\)\()/48\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.15139 + 0.593052i
1.15139 0.593052i
−0.651388 2.66948i
−0.651388 + 2.66948i
−0.651388 + 0.158947i
−0.651388 0.158947i
1.15139 2.23537i
1.15139 + 2.23537i
0 0 0 −7.67101 0 7.21110 0 0 0
17.2 0 0 0 −7.67101 0 7.21110 0 0 0
17.3 0 0 0 −1.07498 0 −7.21110 0 0 0
17.4 0 0 0 −1.07498 0 −7.21110 0 0 0
17.5 0 0 0 1.07498 0 −7.21110 0 0 0
17.6 0 0 0 1.07498 0 −7.21110 0 0 0
17.7 0 0 0 7.67101 0 7.21110 0 0 0
17.8 0 0 0 7.67101 0 7.21110 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.3.h.a 8
3.b odd 2 1 inner 288.3.h.a 8
4.b odd 2 1 72.3.h.a 8
8.b even 2 1 inner 288.3.h.a 8
8.d odd 2 1 72.3.h.a 8
12.b even 2 1 72.3.h.a 8
16.e even 4 2 2304.3.e.o 8
16.f odd 4 2 2304.3.e.n 8
24.f even 2 1 72.3.h.a 8
24.h odd 2 1 inner 288.3.h.a 8
48.i odd 4 2 2304.3.e.o 8
48.k even 4 2 2304.3.e.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.h.a 8 4.b odd 2 1
72.3.h.a 8 8.d odd 2 1
72.3.h.a 8 12.b even 2 1
72.3.h.a 8 24.f even 2 1
288.3.h.a 8 1.a even 1 1 trivial
288.3.h.a 8 3.b odd 2 1 inner
288.3.h.a 8 8.b even 2 1 inner
288.3.h.a 8 24.h odd 2 1 inner
2304.3.e.n 8 16.f odd 4 2
2304.3.e.n 8 48.k even 4 2
2304.3.e.o 8 16.e even 4 2
2304.3.e.o 8 48.i odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(288, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 68 - 60 T^{2} + T^{4} )^{2} \)
$7$ \( ( -52 + T^{2} )^{4} \)
$11$ \( ( 9792 - 304 T^{2} + T^{4} )^{2} \)
$13$ \( ( 2448 + 472 T^{2} + T^{4} )^{2} \)
$17$ \( ( 171396 + 836 T^{2} + T^{4} )^{2} \)
$19$ \( ( 352512 + 1504 T^{2} + T^{4} )^{2} \)
$23$ \( ( 576 + 464 T^{2} + T^{4} )^{2} \)
$29$ \( ( 103428 - 988 T^{2} + T^{4} )^{2} \)
$31$ \( ( -212 - 32 T + T^{2} )^{4} \)
$37$ \( ( 352512 + 1504 T^{2} + T^{4} )^{2} \)
$41$ \( ( 133956 + 932 T^{2} + T^{4} )^{2} \)
$43$ \( ( 10027008 + 6400 T^{2} + T^{4} )^{2} \)
$47$ \( ( 46656 + 4176 T^{2} + T^{4} )^{2} \)
$53$ \( ( 1591812 - 2524 T^{2} + T^{4} )^{2} \)
$59$ \( ( 4456448 - 4608 T^{2} + T^{4} )^{2} \)
$61$ \( ( 3172608 + 5472 T^{2} + T^{4} )^{2} \)
$67$ \( ( 3172608 + 5472 T^{2} + T^{4} )^{2} \)
$71$ \( ( 13483584 + 11088 T^{2} + T^{4} )^{2} \)
$73$ \( ( -2928 + 40 T + T^{2} )^{4} \)
$79$ \( ( -3988 - 96 T + T^{2} )^{4} \)
$83$ \( ( 183872 - 3120 T^{2} + T^{4} )^{2} \)
$89$ \( ( 171396 + 5828 T^{2} + T^{4} )^{2} \)
$97$ \( ( -256 + 48 T + T^{2} )^{4} \)
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