Properties

Label 2736.2.a.bf
Level $2736$
Weight $2$
Character orbit 2736.a
Self dual yes
Analytic conductor $21.847$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13068.1
Defining polynomial: \(x^{4} - x^{3} - 6 x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 171)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + ( -1 - \beta_{1} ) q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( -1 - \beta_{1} ) q^{7} + \beta_{3} q^{11} + 2 q^{13} -\beta_{2} q^{17} - q^{19} + ( -\beta_{2} + \beta_{3} ) q^{23} + ( 2 - \beta_{1} ) q^{25} + ( \beta_{2} - \beta_{3} ) q^{29} + ( 2 + 2 \beta_{1} ) q^{31} + ( -\beta_{2} - 2 \beta_{3} ) q^{35} + ( 4 - 2 \beta_{1} ) q^{37} + ( -\beta_{2} - \beta_{3} ) q^{41} + ( 1 - 3 \beta_{1} ) q^{43} + ( 2 \beta_{2} - \beta_{3} ) q^{47} + ( 2 + \beta_{1} ) q^{49} + ( \beta_{2} + 3 \beta_{3} ) q^{53} + ( 3 + 3 \beta_{1} ) q^{55} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 5 + 3 \beta_{1} ) q^{61} -2 \beta_{2} q^{65} + 4 q^{67} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 5 + 3 \beta_{1} ) q^{73} + ( 3 \beta_{2} - 2 \beta_{3} ) q^{77} + 4 q^{79} + ( 3 \beta_{2} + \beta_{3} ) q^{83} + ( 7 - \beta_{1} ) q^{85} + ( \beta_{2} + 3 \beta_{3} ) q^{89} + ( -2 - 2 \beta_{1} ) q^{91} + \beta_{2} q^{95} + ( -6 - 4 \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{7} + O(q^{10}) \) \( 4q - 2q^{7} + 8q^{13} - 4q^{19} + 10q^{25} + 4q^{31} + 20q^{37} + 10q^{43} + 6q^{49} + 6q^{55} + 14q^{61} + 16q^{67} + 14q^{73} + 16q^{79} + 30q^{85} - 4q^{91} - 16q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6 x^{2} - x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu^{2} - 7 \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( 2 \nu^{3} - 3 \nu^{2} - 9 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 5 \beta_{2} - 2 \beta_{1} + 12\)\()/4\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} + 3 \beta_{2} - 3 \beta_{1} + 4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.04374
−1.82405
−0.548230
0.328543
0 0 0 −3.22060 0 2.37228 0 0 0
1.2 0 0 0 −2.15121 0 −3.37228 0 0 0
1.3 0 0 0 2.15121 0 −3.37228 0 0 0
1.4 0 0 0 3.22060 0 2.37228 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.a.bf 4
3.b odd 2 1 inner 2736.2.a.bf 4
4.b odd 2 1 171.2.a.e 4
12.b even 2 1 171.2.a.e 4
20.d odd 2 1 4275.2.a.bp 4
28.d even 2 1 8379.2.a.bw 4
60.h even 2 1 4275.2.a.bp 4
76.d even 2 1 3249.2.a.bf 4
84.h odd 2 1 8379.2.a.bw 4
228.b odd 2 1 3249.2.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.a.e 4 4.b odd 2 1
171.2.a.e 4 12.b even 2 1
2736.2.a.bf 4 1.a even 1 1 trivial
2736.2.a.bf 4 3.b odd 2 1 inner
3249.2.a.bf 4 76.d even 2 1
3249.2.a.bf 4 228.b odd 2 1
4275.2.a.bp 4 20.d odd 2 1
4275.2.a.bp 4 60.h even 2 1
8379.2.a.bw 4 28.d even 2 1
8379.2.a.bw 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2736))\):

\( T_{5}^{4} - 15 T_{5}^{2} + 48 \)
\( T_{7}^{2} + T_{7} - 8 \)
\( T_{11}^{4} - 27 T_{11}^{2} + 108 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 48 - 15 T^{2} + T^{4} \)
$7$ \( ( -8 + T + T^{2} )^{2} \)
$11$ \( 108 - 27 T^{2} + T^{4} \)
$13$ \( ( -2 + T )^{4} \)
$17$ \( 48 - 15 T^{2} + T^{4} \)
$19$ \( ( 1 + T )^{4} \)
$23$ \( 48 - 48 T^{2} + T^{4} \)
$29$ \( 48 - 48 T^{2} + T^{4} \)
$31$ \( ( -32 - 2 T + T^{2} )^{2} \)
$37$ \( ( -8 - 10 T + T^{2} )^{2} \)
$41$ \( 192 - 36 T^{2} + T^{4} \)
$43$ \( ( -68 - 5 T + T^{2} )^{2} \)
$47$ \( 1452 - 99 T^{2} + T^{4} \)
$53$ \( 13872 - 240 T^{2} + T^{4} \)
$59$ \( 3072 - 144 T^{2} + T^{4} \)
$61$ \( ( -62 - 7 T + T^{2} )^{2} \)
$67$ \( ( -4 + T )^{4} \)
$71$ \( 768 - 192 T^{2} + T^{4} \)
$73$ \( ( -62 - 7 T + T^{2} )^{2} \)
$79$ \( ( -4 + T )^{4} \)
$83$ \( 432 - 144 T^{2} + T^{4} \)
$89$ \( 13872 - 240 T^{2} + T^{4} \)
$97$ \( ( -116 + 8 T + T^{2} )^{2} \)
show more
show less