Properties

Label 2736.2.a.a
Level $2736$
Weight $2$
Character orbit 2736.a
Self dual yes
Analytic conductor $21.847$
Analytic rank $2$
Dimension $1$
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: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 456)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
0 0 0 −4.00000 0 −4.00000 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

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.a 1
3.b odd 2 1 912.2.a.l 1
4.b odd 2 1 1368.2.a.b 1
12.b even 2 1 456.2.a.b 1
24.f even 2 1 3648.2.a.s 1
24.h odd 2 1 3648.2.a.a 1
228.b odd 2 1 8664.2.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.a.b 1 12.b even 2 1
912.2.a.l 1 3.b odd 2 1
1368.2.a.b 1 4.b odd 2 1
2736.2.a.a 1 1.a even 1 1 trivial
3648.2.a.a 1 24.h odd 2 1
3648.2.a.s 1 24.f even 2 1
8664.2.a.o 1 228.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2736))\):

\( T_{5} + 4 \)
\( T_{7} + 4 \)
\( T_{11} + 4 \)
\( T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 4 + T \)
$7$ \( 4 + T \)
$11$ \( 4 + T \)
$13$ \( 4 + T \)
$17$ \( 6 + T \)
$19$ \( 1 + T \)
$23$ \( 6 + T \)
$29$ \( 2 + T \)
$31$ \( 2 + T \)
$37$ \( -4 + T \)
$41$ \( -6 + T \)
$43$ \( 4 + T \)
$47$ \( 2 + T \)
$53$ \( -6 + T \)
$59$ \( 4 + T \)
$61$ \( 10 + T \)
$67$ \( 8 + T \)
$71$ \( T \)
$73$ \( 2 + T \)
$79$ \( 14 + T \)
$83$ \( 16 + T \)
$89$ \( -18 + T \)
$97$ \( -14 + T \)
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