Properties

Label 2736.2.s.y
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: 6.0.2696112.1
Defining polynomial: \(x^{6} - x^{5} + 5 x^{4} + 18 x^{2} - 8 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} + ( 1 - \beta_{2} ) q^{11} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{13} + ( 3 + 2 \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{19} + ( \beta_{1} + \beta_{2} ) q^{23} + ( -\beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{25} + ( -3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{29} + ( 3 + 3 \beta_{2} + 3 \beta_{3} ) q^{31} + ( -4 + 4 \beta_{4} - 2 \beta_{5} ) q^{35} + ( -5 - 3 \beta_{2} - 3 \beta_{3} ) q^{37} + ( -7 + 2 \beta_{1} + 7 \beta_{4} - 2 \beta_{5} ) q^{41} + ( 3 - 4 \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{43} + ( 5 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{47} + ( 3 - 2 \beta_{2} ) q^{49} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{53} + ( 3 + \beta_{1} - 3 \beta_{4} + \beta_{5} ) q^{55} + ( -4 - \beta_{1} + 4 \beta_{4} + \beta_{5} ) q^{59} + ( -3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} ) q^{61} + ( -7 - 3 \beta_{3} ) q^{65} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{67} + ( 1 - \beta_{4} - \beta_{5} ) q^{71} + ( 3 + 2 \beta_{1} - 3 \beta_{4} ) q^{73} + ( -3 + \beta_{2} - 3 \beta_{3} ) q^{77} + ( -7 + 2 \beta_{1} + 7 \beta_{4} + \beta_{5} ) q^{79} + ( 5 + 3 \beta_{2} - 2 \beta_{3} ) q^{83} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 5 \beta_{4} + \beta_{5} ) q^{85} + ( 6 \beta_{1} + 6 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{89} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 14 \beta_{4} - 2 \beta_{5} ) q^{91} + ( 4 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 5 \beta_{4} + \beta_{5} ) q^{95} + ( -1 + 2 \beta_{1} + \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + q^{5} + 4q^{7} + O(q^{10}) \) \( 6q + q^{5} + 4q^{7} + 8q^{11} + q^{13} + 11q^{17} - q^{23} + 6q^{25} - 3q^{29} + 12q^{31} - 12q^{35} - 24q^{37} - 19q^{41} + 5q^{43} - 17q^{47} + 22q^{49} - 5q^{53} + 10q^{55} - 13q^{59} + 3q^{61} - 42q^{65} - 9q^{67} + 3q^{71} + 11q^{73} - 20q^{77} - 19q^{79} + 24q^{83} - 17q^{85} + 3q^{89} + 44q^{91} + 13q^{95} - q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 5 x^{4} + 18 x^{2} - 8 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 5 \nu^{4} + 25 \nu^{3} - 18 \nu^{2} + 8 \nu - 40 \)\()/82\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{5} - 10 \nu^{4} + 9 \nu^{3} - 36 \nu^{2} + 16 \nu - 121 \)\()/41\)
\(\beta_{4}\)\(=\)\((\)\( -10 \nu^{5} + 9 \nu^{4} - 45 \nu^{3} - 25 \nu^{2} - 162 \nu + 72 \)\()/82\)
\(\beta_{5}\)\(=\)\((\)\( -26 \nu^{5} + 7 \nu^{4} - 117 \nu^{3} - 65 \nu^{2} - 454 \nu - 26 \)\()/82\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 3 \beta_{4} - \beta_{3}\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + 4 \beta_{2} - 1\)
\(\nu^{4}\)\(=\)\(-5 \beta_{5} + 13 \beta_{4} - 2 \beta_{1} - 13\)
\(\nu^{5}\)\(=\)\(-7 \beta_{5} + 11 \beta_{4} + 7 \beta_{3} - 18 \beta_{2} - 18 \beta_{1}\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−0.906803 1.57063i
0.235342 + 0.407624i
1.17146 + 2.02903i
−0.906803 + 1.57063i
0.235342 0.407624i
1.17146 2.02903i
0 0 0 −0.906803 1.57063i 0 2.52444 0 0 0
577.2 0 0 0 0.235342 + 0.407624i 0 3.30777 0 0 0
577.3 0 0 0 1.17146 + 2.02903i 0 −3.83221 0 0 0
1873.1 0 0 0 −0.906803 + 1.57063i 0 2.52444 0 0 0
1873.2 0 0 0 0.235342 0.407624i 0 3.30777 0 0 0
1873.3 0 0 0 1.17146 2.02903i 0 −3.83221 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.y 6
3.b odd 2 1 304.2.i.f 6
4.b odd 2 1 1368.2.s.k 6
12.b even 2 1 152.2.i.c 6
19.c even 3 1 inner 2736.2.s.y 6
24.f even 2 1 1216.2.i.n 6
24.h odd 2 1 1216.2.i.m 6
57.f even 6 1 5776.2.a.bq 3
57.h odd 6 1 304.2.i.f 6
57.h odd 6 1 5776.2.a.bk 3
76.g odd 6 1 1368.2.s.k 6
228.m even 6 1 152.2.i.c 6
228.m even 6 1 2888.2.a.r 3
228.n odd 6 1 2888.2.a.n 3
456.u even 6 1 1216.2.i.n 6
456.x odd 6 1 1216.2.i.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.i.c 6 12.b even 2 1
152.2.i.c 6 228.m even 6 1
304.2.i.f 6 3.b odd 2 1
304.2.i.f 6 57.h odd 6 1
1216.2.i.m 6 24.h odd 2 1
1216.2.i.m 6 456.x odd 6 1
1216.2.i.n 6 24.f even 2 1
1216.2.i.n 6 456.u even 6 1
1368.2.s.k 6 4.b odd 2 1
1368.2.s.k 6 76.g odd 6 1
2736.2.s.y 6 1.a even 1 1 trivial
2736.2.s.y 6 19.c even 3 1 inner
2888.2.a.n 3 228.n odd 6 1
2888.2.a.r 3 228.m even 6 1
5776.2.a.bk 3 57.h odd 6 1
5776.2.a.bq 3 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 \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{6} - T_{5}^{5} + 5 T_{5}^{4} + 18 T_{5}^{2} - 8 T_{5} + 4 \)
\( T_{7}^{3} - 2 T_{7}^{2} - 14 T_{7} + 32 \)
\( T_{11}^{3} - 4 T_{11}^{2} + T_{11} + 4 \)
\( T_{13}^{6} - T_{13}^{5} + 33 T_{13}^{4} - 120 T_{13}^{3} + 1100 T_{13}^{2} - 2432 T_{13} + 5776 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 4 - 8 T + 18 T^{2} + 5 T^{4} - T^{5} + T^{6} \)
$7$ \( ( 32 - 14 T - 2 T^{2} + T^{3} )^{2} \)
$11$ \( ( 4 + T - 4 T^{2} + T^{3} )^{2} \)
$13$ \( 5776 - 2432 T + 1100 T^{2} - 120 T^{3} + 33 T^{4} - T^{5} + T^{6} \)
$17$ \( 1024 + 768 T + 928 T^{2} - 328 T^{3} + 97 T^{4} - 11 T^{5} + T^{6} \)
$19$ \( 6859 + 342 T^{2} + 16 T^{3} + 18 T^{4} + T^{6} \)
$23$ \( 4 + 8 T + 18 T^{2} + 5 T^{4} + T^{5} + T^{6} \)
$29$ \( 4 - 80 T + 1594 T^{2} - 124 T^{3} + 49 T^{4} + 3 T^{5} + T^{6} \)
$31$ \( ( 216 - 54 T - 6 T^{2} + T^{3} )^{2} \)
$37$ \( ( -292 - 18 T + 12 T^{2} + T^{3} )^{2} \)
$41$ \( 1681 + 3731 T + 7502 T^{2} + 1647 T^{3} + 270 T^{4} + 19 T^{5} + T^{6} \)
$43$ \( 118336 - 28896 T + 8776 T^{2} - 268 T^{3} + 109 T^{4} - 5 T^{5} + T^{6} \)
$47$ \( 521284 + 12274 T^{2} + 1444 T^{3} + 289 T^{4} + 17 T^{5} + T^{6} \)
$53$ \( 1936 + 352 T + 284 T^{2} + 48 T^{3} + 33 T^{4} + 5 T^{5} + T^{6} \)
$59$ \( 2809 + 2597 T + 1712 T^{2} + 531 T^{3} + 120 T^{4} + 13 T^{5} + T^{6} \)
$61$ \( 58564 + 24200 T + 9274 T^{2} + 784 T^{3} + 109 T^{4} - 3 T^{5} + T^{6} \)
$67$ \( 529 + 713 T + 1168 T^{2} - 233 T^{3} + 112 T^{4} + 9 T^{5} + T^{6} \)
$71$ \( 16 - 16 T + 28 T^{2} + 4 T^{3} + 13 T^{4} - 3 T^{5} + T^{6} \)
$73$ \( 361 + 437 T + 738 T^{2} - 291 T^{3} + 98 T^{4} - 11 T^{5} + T^{6} \)
$79$ \( 256 - 1408 T + 8048 T^{2} + 1704 T^{3} + 273 T^{4} + 19 T^{5} + T^{6} \)
$83$ \( ( 632 - 43 T - 12 T^{2} + T^{3} )^{2} \)
$89$ \( 295936 - 100096 T + 35488 T^{2} - 536 T^{3} + 193 T^{4} - 3 T^{5} + T^{6} \)
$97$ \( 1 + 17 T + 290 T^{2} - 15 T^{3} + 18 T^{4} + T^{5} + T^{6} \)
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