Properties

Label 2736.2.s.bb
Level $2736$
Weight $2$
Character orbit 2736.s
Analytic conductor $21.847$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.764411904.5
Defining polynomial: \(x^{8} - 6 x^{6} + 21 x^{4} - 54 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} + ( 1 + \beta_{5} ) q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} + ( 1 + \beta_{5} ) q^{7} + ( \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} ) q^{11} + ( 1 + \beta_{3} ) q^{13} + 2 \beta_{1} q^{17} + ( -1 + 2 \beta_{4} - \beta_{5} ) q^{19} + ( -\beta_{2} + \beta_{7} ) q^{23} + ( -1 - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{25} + ( -2 \beta_{2} + 2 \beta_{7} ) q^{29} + ( 7 + \beta_{5} ) q^{31} + ( \beta_{1} - \beta_{6} ) q^{35} + ( -1 + 2 \beta_{5} ) q^{37} -2 \beta_{6} q^{41} + ( \beta_{3} + 3 \beta_{4} ) q^{43} + 2 \beta_{7} q^{47} + 2 \beta_{5} q^{49} -\beta_{2} q^{53} + 6 \beta_{3} q^{55} + ( -3 \beta_{1} - \beta_{6} ) q^{59} + ( -5 - 5 \beta_{3} ) q^{61} + ( -\beta_{1} + \beta_{2} ) q^{65} + ( -7 - 7 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{6} ) q^{71} + 5 \beta_{3} q^{73} + ( \beta_{1} - \beta_{2} + 4 \beta_{6} - 4 \beta_{7} ) q^{77} + ( 7 \beta_{3} + 3 \beta_{4} ) q^{79} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{83} + ( 12 + 12 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{85} + 5 \beta_{2} q^{89} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{91} + ( -\beta_{1} + 4 \beta_{2} + \beta_{6} - 2 \beta_{7} ) q^{95} + ( 8 \beta_{3} + 2 \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{7} + O(q^{10}) \) \( 8q + 8q^{7} + 4q^{13} - 8q^{19} - 4q^{25} + 56q^{31} - 8q^{37} - 4q^{43} - 24q^{55} - 20q^{61} - 28q^{67} - 20q^{73} - 28q^{79} + 48q^{85} + 4q^{91} - 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 6 x^{6} + 21 x^{4} - 54 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} - 21 \nu^{5} + 87 \nu^{3} - 162 \nu \)\()/135\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 24 \nu^{5} + 48 \nu^{3} + 27 \nu \)\()/135\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{6} + 3 \nu^{4} - 6 \nu^{2} - 9 \)\()/45\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{6} - 3 \nu^{4} + 51 \nu^{2} - 81 \)\()/45\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 6 \nu^{4} - 12 \nu^{2} + 27 \)\()/9\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{7} + 12 \nu^{5} - 114 \nu^{3} + 54 \nu \)\()/135\)
\(\beta_{7}\)\(=\)\((\)\( -22 \nu^{7} + 78 \nu^{5} - 246 \nu^{3} + 486 \nu \)\()/135\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} + 4 \beta_{2} - 2 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - 3 \beta_{6} + 2 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 3\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{7} - 7 \beta_{6} - 10 \beta_{2} + 2 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(3 \beta_{5} - 30 \beta_{3} - 15\)
\(\nu^{7}\)\(=\)\((\)\(-9 \beta_{7} - 6 \beta_{6} - 6 \beta_{2} - 30 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1 - \beta_{3}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−1.27970 + 1.16721i
−1.69185 0.370982i
1.69185 + 0.370982i
1.27970 1.16721i
−1.27970 1.16721i
−1.69185 + 0.370982i
1.69185 0.370982i
1.27970 + 1.16721i
0 0 0 −1.65068 2.85906i 0 −1.44949 0 0 0
577.2 0 0 0 −0.524648 0.908716i 0 3.44949 0 0 0
577.3 0 0 0 0.524648 + 0.908716i 0 3.44949 0 0 0
577.4 0 0 0 1.65068 + 2.85906i 0 −1.44949 0 0 0
1873.1 0 0 0 −1.65068 + 2.85906i 0 −1.44949 0 0 0
1873.2 0 0 0 −0.524648 + 0.908716i 0 3.44949 0 0 0
1873.3 0 0 0 0.524648 0.908716i 0 3.44949 0 0 0
1873.4 0 0 0 1.65068 2.85906i 0 −1.44949 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1873.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.s.bb 8
3.b odd 2 1 inner 2736.2.s.bb 8
4.b odd 2 1 171.2.f.c 8
12.b even 2 1 171.2.f.c 8
19.c even 3 1 inner 2736.2.s.bb 8
57.h odd 6 1 inner 2736.2.s.bb 8
76.f even 6 1 3249.2.a.be 4
76.g odd 6 1 171.2.f.c 8
76.g odd 6 1 3249.2.a.bd 4
228.m even 6 1 171.2.f.c 8
228.m even 6 1 3249.2.a.bd 4
228.n odd 6 1 3249.2.a.be 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.f.c 8 4.b odd 2 1
171.2.f.c 8 12.b even 2 1
171.2.f.c 8 76.g odd 6 1
171.2.f.c 8 228.m even 6 1
2736.2.s.bb 8 1.a even 1 1 trivial
2736.2.s.bb 8 3.b odd 2 1 inner
2736.2.s.bb 8 19.c even 3 1 inner
2736.2.s.bb 8 57.h odd 6 1 inner
3249.2.a.bd 4 76.g odd 6 1
3249.2.a.bd 4 228.m even 6 1
3249.2.a.be 4 76.f even 6 1
3249.2.a.be 4 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{8} + 12 T_{5}^{6} + 132 T_{5}^{4} + 144 T_{5}^{2} + 144 \)
\( T_{7}^{2} - 2 T_{7} - 5 \)
\( T_{11}^{4} - 36 T_{11}^{2} + 108 \)
\( T_{13}^{2} - T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 144 + 144 T^{2} + 132 T^{4} + 12 T^{6} + T^{8} \)
$7$ \( ( -5 - 2 T + T^{2} )^{4} \)
$11$ \( ( 108 - 36 T^{2} + T^{4} )^{2} \)
$13$ \( ( 1 - T + T^{2} )^{4} \)
$17$ \( 36864 + 9216 T^{2} + 2112 T^{4} + 48 T^{6} + T^{8} \)
$19$ \( ( 19 + 2 T + T^{2} )^{4} \)
$23$ \( 90000 + 10800 T^{2} + 996 T^{4} + 36 T^{6} + T^{8} \)
$29$ \( 23040000 + 691200 T^{2} + 15936 T^{4} + 144 T^{6} + T^{8} \)
$31$ \( ( 43 - 14 T + T^{2} )^{4} \)
$37$ \( ( -23 + 2 T + T^{2} )^{4} \)
$41$ \( 589824 + 73728 T^{2} + 8448 T^{4} + 96 T^{6} + T^{8} \)
$43$ \( ( 2809 - 106 T + 57 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$47$ \( 589824 + 73728 T^{2} + 8448 T^{4} + 96 T^{6} + T^{8} \)
$53$ \( 144 + 144 T^{2} + 132 T^{4} + 12 T^{6} + T^{8} \)
$59$ \( 18766224 + 571824 T^{2} + 13092 T^{4} + 132 T^{6} + T^{8} \)
$61$ \( ( 25 + 5 T + T^{2} )^{4} \)
$67$ \( ( 25 - 70 T + 201 T^{2} + 14 T^{3} + T^{4} )^{2} \)
$71$ \( 23040000 + 691200 T^{2} + 15936 T^{4} + 144 T^{6} + T^{8} \)
$73$ \( ( 25 + 5 T + T^{2} )^{4} \)
$79$ \( ( 25 - 70 T + 201 T^{2} + 14 T^{3} + T^{4} )^{2} \)
$83$ \( ( 1728 - 144 T^{2} + T^{4} )^{2} \)
$89$ \( 56250000 + 2250000 T^{2} + 82500 T^{4} + 300 T^{6} + T^{8} \)
$97$ \( ( 1600 + 640 T + 216 T^{2} + 16 T^{3} + T^{4} )^{2} \)
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