Properties

Label 2736.2.bm.p
Level $2736$
Weight $2$
Character orbit 2736.bm
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.bm (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.897122304.10
Defining polynomial: \(x^{8} - 8 x^{6} + 51 x^{4} - 104 x^{2} + 169\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 8 x^{6} + 51 x^{4} - 104 x^{2} + 169\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{6} + 68 \nu^{4} - 323 \nu^{2} + 1092 \)\()/663\)
\(\beta_{2}\)\(=\)\((\)\( 6 \nu^{7} + 17 \nu^{5} + 85 \nu^{3} + 923 \nu \)\()/663\)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{7} + 17 \nu^{5} + 85 \nu^{3} - 1040 \nu \)\()/663\)
\(\beta_{4}\)\(=\)\((\)\( 8 \nu^{6} - 51 \nu^{4} + 408 \nu^{2} - 832 \)\()/663\)
\(\beta_{5}\)\(=\)\((\)\( 15 \nu^{6} - 68 \nu^{4} + 323 \nu^{2} + 208 \)\()/663\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 8 \nu^{5} - 38 \nu^{3} + 13 \nu \)\()/39\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + 8 \nu^{5} - 38 \nu^{3} + 91 \nu \)\()/39\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{6}\)\()/2\)
\(\nu^{2}\)\(=\)\(-2 \beta_{5} + 4 \beta_{4} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} - 3 \beta_{6} + 5 \beta_{3} + 3 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\(-8 \beta_{5} + 19 \beta_{4} + 16 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{7} + 16 \beta_{6} + 11 \beta_{3} + 27 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(51 \beta_{5} + 51 \beta_{1} - 100\)
\(\nu^{7}\)\(=\)\((\)\(-151 \beta_{7} + 151 \beta_{6} - 102 \beta_{3} + 102 \beta_{2}\)\()/2\)

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1 + \beta_{4}\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
1.30421 + 0.752986i
2.07341 + 1.19709i
−2.07341 1.19709i
−1.30421 0.752986i
1.30421 0.752986i
2.07341 1.19709i
−2.07341 + 1.19709i
−1.30421 + 0.752986i
0 0 0 −2.05719 + 3.56317i 0 1.00000i 0 0 0
559.2 0 0 0 −0.876327 + 1.51784i 0 1.00000i 0 0 0
559.3 0 0 0 0.876327 1.51784i 0 1.00000i 0 0 0
559.4 0 0 0 2.05719 3.56317i 0 1.00000i 0 0 0
1855.1 0 0 0 −2.05719 3.56317i 0 1.00000i 0 0 0
1855.2 0 0 0 −0.876327 1.51784i 0 1.00000i 0 0 0
1855.3 0 0 0 0.876327 + 1.51784i 0 1.00000i 0 0 0
1855.4 0 0 0 2.05719 + 3.56317i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1855.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
76.f even 6 1 inner
228.n 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.bm.p 8
3.b odd 2 1 inner 2736.2.bm.p 8
4.b odd 2 1 2736.2.bm.q yes 8
12.b even 2 1 2736.2.bm.q yes 8
19.d odd 6 1 2736.2.bm.q yes 8
57.f even 6 1 2736.2.bm.q yes 8
76.f even 6 1 inner 2736.2.bm.p 8
228.n odd 6 1 inner 2736.2.bm.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.2.bm.p 8 1.a even 1 1 trivial
2736.2.bm.p 8 3.b odd 2 1 inner
2736.2.bm.p 8 76.f even 6 1 inner
2736.2.bm.p 8 228.n odd 6 1 inner
2736.2.bm.q yes 8 4.b odd 2 1
2736.2.bm.q yes 8 12.b even 2 1
2736.2.bm.q yes 8 19.d odd 6 1
2736.2.bm.q yes 8 57.f even 6 1

Hecke kernels

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

\( T_{5}^{8} + 20 T_{5}^{6} + 348 T_{5}^{4} + 1040 T_{5}^{2} + 2704 \)
\( T_{7}^{2} + 1 \)
\( T_{11}^{4} + 20 T_{11}^{2} + 52 \)
\( T_{23}^{8} - 44 T_{23}^{6} + 1884 T_{23}^{4} - 2288 T_{23}^{2} + 2704 \)
\( T_{31}^{2} + 12 T_{31} + 33 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 2704 + 1040 T^{2} + 348 T^{4} + 20 T^{6} + T^{8} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( ( 52 + 20 T^{2} + T^{4} )^{2} \)
$13$ \( ( 3 + 3 T + T^{2} )^{4} \)
$17$ \( 43264 + 6656 T^{2} + 816 T^{4} + 32 T^{6} + T^{8} \)
$19$ \( ( 361 + 26 T^{2} + T^{4} )^{2} \)
$23$ \( 2704 - 2288 T^{2} + 1884 T^{4} - 44 T^{6} + T^{8} \)
$29$ \( 3504384 - 179712 T^{2} + 7344 T^{4} - 96 T^{6} + T^{8} \)
$31$ \( ( 33 + 12 T + T^{2} )^{4} \)
$37$ \( ( 1089 + 78 T^{2} + T^{4} )^{2} \)
$41$ \( T^{8} \)
$43$ \( ( 9 - 36 T + 51 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$47$ \( 11075584 - 425984 T^{2} + 13056 T^{4} - 128 T^{6} + T^{8} \)
$53$ \( 219024 - 61776 T^{2} + 16956 T^{4} - 132 T^{6} + T^{8} \)
$59$ \( 219024 + 61776 T^{2} + 16956 T^{4} + 132 T^{6} + T^{8} \)
$61$ \( ( 1521 + 234 T + 75 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$67$ \( ( 81 + 108 T + 135 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$71$ \( 283855104 + 4852224 T^{2} + 66096 T^{4} + 288 T^{6} + T^{8} \)
$73$ \( ( 6889 - 830 T + 183 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$79$ \( ( 1521 + 468 T + 183 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$83$ \( ( 208 + 128 T^{2} + T^{4} )^{2} \)
$89$ \( 17740944 - 758160 T^{2} + 28188 T^{4} - 180 T^{6} + T^{8} \)
$97$ \( ( 1296 - 36 T^{2} + T^{4} )^{2} \)
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