Properties

Label 2736.2.a.y
Level $2736$
Weight $2$
Character orbit 2736.a
Self dual yes
Analytic conductor $21.847$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 228)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{5} + ( -1 + \beta ) q^{7} +O(q^{10})\) \( q + ( -1 - \beta ) q^{5} + ( -1 + \beta ) q^{7} + ( -1 - \beta ) q^{11} + 2 q^{13} + ( 1 + \beta ) q^{17} - q^{19} + ( -4 + 2 \beta ) q^{23} + ( 4 + 3 \beta ) q^{25} + ( 2 + 2 \beta ) q^{29} + ( 2 - 2 \beta ) q^{31} + ( -7 - \beta ) q^{35} -2 \beta q^{37} + ( -1 + \beta ) q^{43} + ( -11 + \beta ) q^{47} + ( 2 - \beta ) q^{49} + ( -2 - 2 \beta ) q^{53} + ( 9 + 3 \beta ) q^{55} + ( -5 - \beta ) q^{61} + ( -2 - 2 \beta ) q^{65} -4 \beta q^{67} -12 q^{71} + ( -3 + \beta ) q^{73} + ( -7 - \beta ) q^{77} -8 q^{79} + ( 4 - 2 \beta ) q^{83} + ( -9 - 3 \beta ) q^{85} + ( -10 + 2 \beta ) q^{89} + ( -2 + 2 \beta ) q^{91} + ( 1 + \beta ) q^{95} + 14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} - q^{7} + O(q^{10}) \) \( 2q - 3q^{5} - q^{7} - 3q^{11} + 4q^{13} + 3q^{17} - 2q^{19} - 6q^{23} + 11q^{25} + 6q^{29} + 2q^{31} - 15q^{35} - 2q^{37} - q^{43} - 21q^{47} + 3q^{49} - 6q^{53} + 21q^{55} - 11q^{61} - 6q^{65} - 4q^{67} - 24q^{71} - 5q^{73} - 15q^{77} - 16q^{79} + 6q^{83} - 21q^{85} - 18q^{89} - 2q^{91} + 3q^{95} + 28q^{97} + O(q^{100}) \)

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
3.37228
−2.37228
0 0 0 −4.37228 0 2.37228 0 0 0
1.2 0 0 0 1.37228 0 −3.37228 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

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.y 2
3.b odd 2 1 912.2.a.n 2
4.b odd 2 1 684.2.a.d 2
12.b even 2 1 228.2.a.c 2
24.f even 2 1 3648.2.a.bk 2
24.h odd 2 1 3648.2.a.bq 2
60.h even 2 1 5700.2.a.t 2
60.l odd 4 2 5700.2.f.m 4
228.b odd 2 1 4332.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.a.c 2 12.b even 2 1
684.2.a.d 2 4.b odd 2 1
912.2.a.n 2 3.b odd 2 1
2736.2.a.y 2 1.a even 1 1 trivial
3648.2.a.bk 2 24.f even 2 1
3648.2.a.bq 2 24.h odd 2 1
4332.2.a.i 2 228.b odd 2 1
5700.2.a.t 2 60.h even 2 1
5700.2.f.m 4 60.l odd 4 2

Hecke kernels

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

\( T_{5}^{2} + 3 T_{5} - 6 \)
\( T_{7}^{2} + T_{7} - 8 \)
\( T_{11}^{2} + 3 T_{11} - 6 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -6 + 3 T + T^{2} \)
$7$ \( -8 + T + T^{2} \)
$11$ \( -6 + 3 T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -6 - 3 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -24 + 6 T + T^{2} \)
$29$ \( -24 - 6 T + T^{2} \)
$31$ \( -32 - 2 T + T^{2} \)
$37$ \( -32 + 2 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( -8 + T + T^{2} \)
$47$ \( 102 + 21 T + T^{2} \)
$53$ \( -24 + 6 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 22 + 11 T + T^{2} \)
$67$ \( -128 + 4 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( -2 + 5 T + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( -24 - 6 T + T^{2} \)
$89$ \( 48 + 18 T + T^{2} \)
$97$ \( ( -14 + T )^{2} \)
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