Properties

Label 2736.2.bm.n
Level $2736$
Weight $2$
Character orbit 2736.bm
Analytic conductor $21.847$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.954288.1
Defining polynomial: \(x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{5} + ( -1 - \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{7} +O(q^{10})\) \( q + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{5} + ( -1 - \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{7} + ( \beta_{1} - \beta_{5} ) q^{11} + ( -2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{13} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{17} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{19} + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{23} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{25} + ( -2 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{29} + \beta_{2} q^{31} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{35} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{37} + ( 2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{41} + ( -4 + \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{43} + ( 4 + 2 \beta_{3} - 4 \beta_{5} ) q^{47} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} ) q^{49} + ( -8 - 4 \beta_{3} - \beta_{5} ) q^{53} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{55} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{61} + ( 5 + 4 \beta_{1} - \beta_{2} + 10 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{65} + ( -2 \beta_{1} - 7 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{67} + ( 5 + \beta_{1} + \beta_{2} + 5 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{71} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{73} + ( 5 + \beta_{2} ) q^{77} + ( 4 - 3 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{79} + ( 1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{83} + ( 4 \beta_{1} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{85} + ( -4 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{89} + ( -2 \beta_{3} - 5 \beta_{4} ) q^{91} + ( 6 - 3 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} ) q^{95} + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{5} + O(q^{10}) \) \( 6q + 2q^{5} - 3q^{13} + 2q^{17} - 4q^{19} - 12q^{23} - 5q^{25} - 6q^{29} + 2q^{31} - 6q^{35} + 12q^{41} - 33q^{43} + 18q^{47} - 8q^{49} - 36q^{53} + 12q^{55} - 2q^{59} + 3q^{61} + 19q^{67} + 16q^{71} + 11q^{73} + 32q^{77} + 9q^{79} - 8q^{85} - 6q^{89} + q^{91} + 26q^{95} - 24q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 5 \nu^{4} + \nu^{3} + 9 \nu^{2} + 21 \nu - 45 \)\()/27\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 4 \nu^{4} + \nu^{3} + 18 \nu^{2} - 33 \nu + 9 \)\()/27\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{5} - \nu^{4} - 2 \nu^{3} + 12 \nu + 9 \)\()/27\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{5} - 8 \nu^{4} + 2 \nu^{3} + 9 \nu^{2} - 12 \nu + 18 \)\()/27\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{5} - \nu^{4} + 4 \nu^{3} - 3 \nu^{2} + 12 \nu + 27 \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{5} - 8 \beta_{3} + \beta_{2} + \beta_{1} - 5\)\()/2\)
\(\nu^{4}\)\(=\)\(-3 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{5}\)\(=\)\((\)\(-3 \beta_{5} + 9 \beta_{4} - 11 \beta_{3} - 8 \beta_{2} + 4 \beta_{1} + 16\)\()/2\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−1.62241 + 0.606458i
1.71903 + 0.211943i
0.403374 1.68443i
−1.62241 0.606458i
1.71903 0.211943i
0.403374 + 1.68443i
0 0 0 −1.33641 + 2.31473i 0 3.93569i 0 0 0
559.2 0 0 0 0.675970 1.17081i 0 1.45735i 0 0 0
559.3 0 0 0 1.66044 2.87597i 0 2.71781i 0 0 0
1855.1 0 0 0 −1.33641 2.31473i 0 3.93569i 0 0 0
1855.2 0 0 0 0.675970 + 1.17081i 0 1.45735i 0 0 0
1855.3 0 0 0 1.66044 + 2.87597i 0 2.71781i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1855.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.f even 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.n 6
3.b odd 2 1 912.2.bb.e 6
4.b odd 2 1 2736.2.bm.o 6
12.b even 2 1 912.2.bb.f yes 6
19.d odd 6 1 2736.2.bm.o 6
57.f even 6 1 912.2.bb.f yes 6
76.f even 6 1 inner 2736.2.bm.n 6
228.n odd 6 1 912.2.bb.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bb.e 6 3.b odd 2 1
912.2.bb.e 6 228.n odd 6 1
912.2.bb.f yes 6 12.b even 2 1
912.2.bb.f yes 6 57.f even 6 1
2736.2.bm.n 6 1.a even 1 1 trivial
2736.2.bm.n 6 76.f even 6 1 inner
2736.2.bm.o 6 4.b odd 2 1
2736.2.bm.o 6 19.d odd 6 1

Hecke kernels

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

\( T_{5}^{6} - 2 T_{5}^{5} + 12 T_{5}^{4} - 8 T_{5}^{3} + 88 T_{5}^{2} - 96 T_{5} + 144 \)
\( T_{7}^{6} + 25 T_{7}^{4} + 163 T_{7}^{2} + 243 \)
\( T_{11}^{6} + 16 T_{11}^{4} + 64 T_{11}^{2} + 48 \)
\( T_{23}^{6} + 12 T_{23}^{5} + 28 T_{23}^{4} - 240 T_{23}^{3} - 416 T_{23}^{2} + 4080 T_{23} + 13872 \)
\( T_{31}^{3} - T_{31}^{2} - 9 T_{31} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 144 - 96 T + 88 T^{2} - 8 T^{3} + 12 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( 243 + 163 T^{2} + 25 T^{4} + T^{6} \)
$11$ \( 48 + 64 T^{2} + 16 T^{4} + T^{6} \)
$13$ \( 2883 + 2139 T + 436 T^{2} - 69 T^{3} - 20 T^{4} + 3 T^{5} + T^{6} \)
$17$ \( 576 - 480 T + 448 T^{2} - 8 T^{3} + 24 T^{4} - 2 T^{5} + T^{6} \)
$19$ \( 6859 + 1444 T + 323 T^{2} + 136 T^{3} + 17 T^{4} + 4 T^{5} + T^{6} \)
$23$ \( 13872 + 4080 T - 416 T^{2} - 240 T^{3} + 28 T^{4} + 12 T^{5} + T^{6} \)
$29$ \( 192 - 480 T + 448 T^{2} - 120 T^{3} - 8 T^{4} + 6 T^{5} + T^{6} \)
$31$ \( ( -3 - 9 T - T^{2} + T^{3} )^{2} \)
$37$ \( 328683 + 14971 T^{2} + 217 T^{4} + T^{6} \)
$41$ \( 248832 - 82944 T + 5760 T^{2} + 1152 T^{3} - 48 T^{4} - 12 T^{5} + T^{6} \)
$43$ \( 2187 + 7533 T + 9540 T^{2} + 3069 T^{3} + 456 T^{4} + 33 T^{5} + T^{6} \)
$47$ \( 714432 - 134688 T - 320 T^{2} + 1656 T^{3} + 16 T^{4} - 18 T^{5} + T^{6} \)
$53$ \( 80688 + 66912 T + 24400 T^{2} + 4896 T^{3} + 568 T^{4} + 36 T^{5} + T^{6} \)
$59$ \( 144 + 96 T + 88 T^{2} + 8 T^{3} + 12 T^{4} + 2 T^{5} + T^{6} \)
$61$ \( 638401 - 112659 T + 22278 T^{2} - 1175 T^{3} + 150 T^{4} - 3 T^{5} + T^{6} \)
$67$ \( 10201 - 8383 T + 4970 T^{2} - 1375 T^{3} + 278 T^{4} - 19 T^{5} + T^{6} \)
$71$ \( 2304 + 1920 T + 2368 T^{2} - 736 T^{3} + 216 T^{4} - 16 T^{5} + T^{6} \)
$73$ \( 33489 - 8235 T + 4038 T^{2} + 129 T^{3} + 166 T^{4} - 11 T^{5} + T^{6} \)
$79$ \( 151321 - 22173 T + 6750 T^{2} - 265 T^{3} + 138 T^{4} - 9 T^{5} + T^{6} \)
$83$ \( 15552 + 4464 T^{2} + 132 T^{4} + T^{6} \)
$89$ \( 215472 - 112560 T + 21208 T^{2} - 840 T^{3} - 128 T^{4} + 6 T^{5} + T^{6} \)
$97$ \( 62208 - 6912 T - 3200 T^{2} + 384 T^{3} + 208 T^{4} + 24 T^{5} + T^{6} \)
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