Properties

Label 2736.2.s.z
Level $2736$
Weight $2$
Character orbit 2736.s
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.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
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: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{2} ) q^{5} + ( -\beta_{1} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{2} ) q^{5} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( \beta_{1} + \beta_{3} + \beta_{4} ) q^{11} + \beta_{5} q^{13} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{19} + ( -4 + 4 \beta_{2} + \beta_{3} - \beta_{5} ) q^{23} + ( -2 + 2 \beta_{2} + \beta_{5} ) q^{25} + ( 2 - 2 \beta_{2} + 2 \beta_{3} ) q^{29} + ( -5 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{31} + ( \beta_{1} - 6 \beta_{2} + \beta_{4} - \beta_{5} ) q^{35} - q^{37} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{43} + ( 6 - 6 \beta_{2} ) q^{47} + ( -1 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{49} + ( -3 + 3 \beta_{2} - 3 \beta_{3} ) q^{53} + ( -2 \beta_{1} + 3 \beta_{2} + \beta_{4} - \beta_{5} ) q^{55} + ( \beta_{1} + \beta_{4} - \beta_{5} ) q^{59} + ( -4 + 4 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{61} + ( -3 - \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{65} + ( 2 - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{67} + ( -2 \beta_{1} + 6 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{71} + ( -2 \beta_{1} - 7 \beta_{2} ) q^{73} + ( -3 + 3 \beta_{1} + 3 \beta_{3} ) q^{77} + ( -\beta_{1} + 3 \beta_{2} + \beta_{4} - \beta_{5} ) q^{79} + ( -2 + 2 \beta_{4} ) q^{83} + ( -5 + 5 \beta_{2} + \beta_{3} ) q^{89} + ( 3 - 3 \beta_{2} + \beta_{3} - \beta_{5} ) q^{91} + ( -4 - 3 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{95} + ( -2 \beta_{1} - \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{5} + 2q^{7} + O(q^{10}) \) \( 6q + 2q^{5} + 2q^{7} + q^{13} - 4q^{19} - 14q^{23} - 5q^{25} + 4q^{29} - 30q^{31} - 18q^{35} - 6q^{37} - 4q^{41} - 3q^{43} + 18q^{47} - 4q^{49} - 6q^{53} + 12q^{55} - 13q^{61} - 12q^{65} + 9q^{67} + 18q^{71} - 19q^{73} - 24q^{77} + 11q^{79} - 8q^{83} - 16q^{89} + 7q^{91} + 2q^{95} + 2q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} - 4 \nu^{4} + \nu^{3} - 9 \nu^{2} + 21 \nu + 9 \)\()/27\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{5} - \nu^{4} - 2 \nu^{3} + 12 \nu + 36 \)\()/27\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{5} + 8 \nu^{4} - 2 \nu^{3} - 9 \nu^{2} + 12 \nu - 18 \)\()/27\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{5} - 2 \nu^{4} + 14 \nu^{3} + 18 \nu^{2} + 24 \nu + 45 \)\()/27\)
\(\beta_{5}\)\(=\)\((\)\( 10 \nu^{5} + 5 \nu^{4} - 8 \nu^{3} + 36 \nu^{2} - 6 \nu - 153 \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 4 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} - 2 \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{5} + 7 \beta_{4} + 2 \beta_{3} - 24 \beta_{2} + 4 \beta_{1} + 9\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{5} + \beta_{4} + 8 \beta_{3} - 6 \beta_{2} - 2 \beta_{1} + 18\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(7 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 33 \beta_{2} + 22 \beta_{1} + 81\)\()/6\)

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\) \(1\) \(1\)

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
−1.62241 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 3.67282 0 0 0
577.2 0 0 0 0.675970 + 1.17081i 0 −0.351939 0 0 0
577.3 0 0 0 1.66044 + 2.87597i 0 −2.32088 0 0 0
1873.1 0 0 0 −1.33641 + 2.31473i 0 3.67282 0 0 0
1873.2 0 0 0 0.675970 1.17081i 0 −0.351939 0 0 0
1873.3 0 0 0 1.66044 2.87597i 0 −2.32088 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1873.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.s.z 6
3.b odd 2 1 912.2.q.l 6
4.b odd 2 1 171.2.f.b 6
12.b even 2 1 57.2.e.b 6
19.c even 3 1 inner 2736.2.s.z 6
57.h odd 6 1 912.2.q.l 6
76.f even 6 1 3249.2.a.t 3
76.g odd 6 1 171.2.f.b 6
76.g odd 6 1 3249.2.a.y 3
228.m even 6 1 57.2.e.b 6
228.m even 6 1 1083.2.a.l 3
228.n odd 6 1 1083.2.a.o 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.b 6 12.b even 2 1
57.2.e.b 6 228.m even 6 1
171.2.f.b 6 4.b odd 2 1
171.2.f.b 6 76.g odd 6 1
912.2.q.l 6 3.b odd 2 1
912.2.q.l 6 57.h odd 6 1
1083.2.a.l 3 228.m even 6 1
1083.2.a.o 3 228.n odd 6 1
2736.2.s.z 6 1.a even 1 1 trivial
2736.2.s.z 6 19.c even 3 1 inner
3249.2.a.t 3 76.f even 6 1
3249.2.a.y 3 76.g odd 6 1

Hecke kernels

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

\( T_{5}^{6} - 2 T_{5}^{5} + 12 T_{5}^{4} - 8 T_{5}^{3} + 88 T_{5}^{2} - 96 T_{5} + 144 \)
\( T_{7}^{3} - T_{7}^{2} - 9 T_{7} - 3 \)
\( T_{11}^{3} - 24 T_{11} - 36 \)
\( T_{13}^{6} - T_{13}^{5} + 22 T_{13}^{4} + 27 T_{13}^{3} + 438 T_{13}^{2} + 63 T_{13} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 144 - 96 T + 88 T^{2} - 8 T^{3} + 12 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( ( -3 - 9 T - T^{2} + T^{3} )^{2} \)
$11$ \( ( -36 - 24 T + T^{3} )^{2} \)
$13$ \( 9 + 63 T + 438 T^{2} + 27 T^{3} + 22 T^{4} - T^{5} + T^{6} \)
$17$ \( T^{6} \)
$19$ \( 6859 + 1444 T + 323 T^{2} + 136 T^{3} + 17 T^{4} + 4 T^{5} + T^{6} \)
$23$ \( 24336 - 4368 T + 2968 T^{2} + 704 T^{3} + 168 T^{4} + 14 T^{5} + T^{6} \)
$29$ \( 9216 - 3072 T + 1408 T^{2} - 64 T^{3} + 48 T^{4} - 4 T^{5} + T^{6} \)
$31$ \( ( -31 + 51 T + 15 T^{2} + T^{3} )^{2} \)
$37$ \( ( 1 + T )^{6} \)
$41$ \( 9216 + 3072 T + 1408 T^{2} + 64 T^{3} + 48 T^{4} + 4 T^{5} + T^{6} \)
$43$ \( 169 - 273 T + 402 T^{2} - 89 T^{3} + 30 T^{4} + 3 T^{5} + T^{6} \)
$47$ \( ( 36 - 6 T + T^{2} )^{3} \)
$53$ \( 104976 + 23328 T + 7128 T^{2} + 216 T^{3} + 108 T^{4} + 6 T^{5} + T^{6} \)
$59$ \( 1296 + 864 T + 576 T^{2} + 72 T^{3} + 24 T^{4} + T^{6} \)
$61$ \( 5329 - 803 T + 1070 T^{2} + 289 T^{3} + 158 T^{4} + 13 T^{5} + T^{6} \)
$67$ \( 292681 - 43821 T + 11430 T^{2} - 353 T^{3} + 162 T^{4} - 9 T^{5} + T^{6} \)
$71$ \( 419904 + 7776 T + 11808 T^{2} - 1512 T^{3} + 312 T^{4} - 18 T^{5} + T^{6} \)
$73$ \( 961 - 2573 T + 7478 T^{2} + 1639 T^{3} + 278 T^{4} + 19 T^{5} + T^{6} \)
$79$ \( 29241 + 513 T + 1890 T^{2} - 375 T^{3} + 118 T^{4} - 11 T^{5} + T^{6} \)
$83$ \( ( -192 - 80 T + 4 T^{2} + T^{3} )^{2} \)
$89$ \( 11664 + 8208 T + 4048 T^{2} + 1000 T^{3} + 180 T^{4} + 16 T^{5} + T^{6} \)
$97$ \( 2096704 - 388064 T + 74720 T^{2} - 2360 T^{3} + 272 T^{4} - 2 T^{5} + T^{6} \)
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