Properties

Label 2736.2.a.ba
Level $2736$
Weight $2$
Character orbit 2736.a
Self dual yes
Analytic conductor $21.847$
Analytic rank $1$
Dimension $2$
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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.8470699930\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 684)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} -3 q^{7} +O(q^{10})\) \( q + \beta q^{5} -3 q^{7} -\beta q^{11} + 2 q^{13} -3 \beta q^{17} + q^{19} + 2 \beta q^{23} + 2 q^{25} + 2 \beta q^{29} -10 q^{31} -3 \beta q^{35} + 4 q^{37} -4 \beta q^{41} - q^{43} -\beta q^{47} + 2 q^{49} -2 \beta q^{53} -7 q^{55} -7 q^{61} + 2 \beta q^{65} -12 q^{67} -4 \beta q^{71} -3 q^{73} + 3 \beta q^{77} + 4 q^{79} + 6 \beta q^{83} -21 q^{85} + 6 \beta q^{89} -6 q^{91} + \beta q^{95} -14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{7} + O(q^{10}) \) \( 2q - 6q^{7} + 4q^{13} + 2q^{19} + 4q^{25} - 20q^{31} + 8q^{37} - 2q^{43} + 4q^{49} - 14q^{55} - 14q^{61} - 24q^{67} - 6q^{73} + 8q^{79} - 42q^{85} - 12q^{91} - 28q^{97} + O(q^{100}) \)

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
−2.64575
2.64575
0 0 0 −2.64575 0 −3.00000 0 0 0
1.2 0 0 0 2.64575 0 −3.00000 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.ba 2
3.b odd 2 1 inner 2736.2.a.ba 2
4.b odd 2 1 684.2.a.e 2
12.b even 2 1 684.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.a.e 2 4.b odd 2 1
684.2.a.e 2 12.b even 2 1
2736.2.a.ba 2 1.a even 1 1 trivial
2736.2.a.ba 2 3.b odd 2 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -7 + T^{2} \)
$7$ \( ( 3 + T )^{2} \)
$11$ \( -7 + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -63 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -28 + T^{2} \)
$29$ \( -28 + T^{2} \)
$31$ \( ( 10 + T )^{2} \)
$37$ \( ( -4 + T )^{2} \)
$41$ \( -112 + T^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( -7 + T^{2} \)
$53$ \( -28 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 7 + T )^{2} \)
$67$ \( ( 12 + T )^{2} \)
$71$ \( -112 + T^{2} \)
$73$ \( ( 3 + T )^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( -252 + T^{2} \)
$89$ \( -252 + T^{2} \)
$97$ \( ( 14 + T )^{2} \)
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