Properties

Label 2736.2.a.bc
Level $2736$
Weight $2$
Character orbit 2736.a
Self dual yes
Analytic conductor $21.847$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \(x^{3} - x^{2} - 8 x + 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1368)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( -3 - \beta_{1} ) q^{11} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{17} + q^{19} + ( -2 + 2 \beta_{1} ) q^{23} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{25} + ( 2 - 2 \beta_{2} ) q^{29} + 6 q^{31} + ( -9 + 3 \beta_{1} - 2 \beta_{2} ) q^{35} + 4 q^{37} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 3 + \beta_{1} + \beta_{2} ) q^{43} + ( -3 + \beta_{1} + 2 \beta_{2} ) q^{47} + ( 10 - \beta_{1} + 3 \beta_{2} ) q^{49} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -3 - \beta_{1} - \beta_{2} ) q^{55} + ( 4 - 4 \beta_{1} + 4 \beta_{2} ) q^{59} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{61} + ( 10 - 6 \beta_{1} ) q^{65} + 4 \beta_{2} q^{67} -4 \beta_{2} q^{71} + ( 5 + \beta_{1} - 3 \beta_{2} ) q^{73} + ( 5 + \beta_{1} + 6 \beta_{2} ) q^{77} + ( 8 - 2 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -6 + 2 \beta_{2} ) q^{83} + ( -5 + 5 \beta_{1} + \beta_{2} ) q^{85} + ( 2 + 2 \beta_{2} ) q^{89} + ( -2 + 10 \beta_{1} - 2 \beta_{2} ) q^{91} + ( -1 + \beta_{1} ) q^{95} + ( 6 + 4 \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{5} + 2q^{7} + O(q^{10}) \) \( 3q - 2q^{5} + 2q^{7} - 10q^{11} - 4q^{13} + 8q^{17} + 3q^{19} - 4q^{23} + 3q^{25} + 6q^{29} + 18q^{31} - 24q^{35} + 12q^{37} + 14q^{41} + 10q^{43} - 8q^{47} + 29q^{49} + 10q^{53} - 10q^{55} + 8q^{59} + 24q^{65} + 16q^{73} + 16q^{77} + 22q^{79} - 18q^{83} - 10q^{85} + 6q^{89} + 4q^{91} - 2q^{95} + 22q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 8 x + 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 6\)

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
−2.91729
1.31955
2.59774
0 0 0 −3.91729 0 4.32401 0 0 0
1.2 0 0 0 0.319551 0 2.61968 0 0 0
1.3 0 0 0 1.59774 0 −4.94370 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

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.a.bc 3
3.b odd 2 1 2736.2.a.be 3
4.b odd 2 1 1368.2.a.m 3
12.b even 2 1 1368.2.a.o yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.a.m 3 4.b odd 2 1
1368.2.a.o yes 3 12.b even 2 1
2736.2.a.bc 3 1.a even 1 1 trivial
2736.2.a.be 3 3.b odd 2 1

Hecke kernels

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

\( T_{5}^{3} + 2 T_{5}^{2} - 7 T_{5} + 2 \)
\( T_{7}^{3} - 2 T_{7}^{2} - 23 T_{7} + 56 \)
\( T_{11}^{3} + 10 T_{11}^{2} + 25 T_{11} + 2 \)
\( T_{13}^{3} + 4 T_{13}^{2} - 44 T_{13} - 160 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( 2 - 7 T + 2 T^{2} + T^{3} \)
$7$ \( 56 - 23 T - 2 T^{2} + T^{3} \)
$11$ \( 2 + 25 T + 10 T^{2} + T^{3} \)
$13$ \( -160 - 44 T + 4 T^{2} + T^{3} \)
$17$ \( 152 - 15 T - 8 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( 16 - 28 T + 4 T^{2} + T^{3} \)
$29$ \( 104 - 28 T - 6 T^{2} + T^{3} \)
$31$ \( ( -6 + T )^{3} \)
$37$ \( ( -4 + T )^{3} \)
$41$ \( 32 + 16 T - 14 T^{2} + T^{3} \)
$43$ \( 4 + 9 T - 10 T^{2} + T^{3} \)
$47$ \( -320 - 39 T + 8 T^{2} + T^{3} \)
$53$ \( 664 - 92 T - 10 T^{2} + T^{3} \)
$59$ \( 1280 - 176 T - 8 T^{2} + T^{3} \)
$61$ \( -190 - 67 T + T^{3} \)
$67$ \( -256 - 160 T + T^{3} \)
$71$ \( 256 - 160 T + T^{3} \)
$73$ \( 122 + 5 T - 16 T^{2} + T^{3} \)
$79$ \( 64 + 112 T - 22 T^{2} + T^{3} \)
$83$ \( -56 + 68 T + 18 T^{2} + T^{3} \)
$89$ \( 40 - 28 T - 6 T^{2} + T^{3} \)
$97$ \( 1048 + 28 T - 22 T^{2} + T^{3} \)
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