Properties

Label 756.2.a.g
Level 756
Weight 2
Character orbit 756.a
Self dual yes
Analytic conductor 6.037
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{5} - q^{7} +O(q^{10})\) \( q -\beta q^{5} - q^{7} + \beta q^{11} + 6 q^{13} -2 \beta q^{17} - q^{19} + \beta q^{23} + 8 q^{25} + 2 \beta q^{29} + 9 q^{31} + \beta q^{35} - q^{37} + 3 \beta q^{41} + 8 q^{43} + q^{49} -13 q^{55} -4 \beta q^{59} -6 \beta q^{65} -2 q^{67} + \beta q^{71} + 4 q^{73} -\beta q^{77} -2 \beta q^{83} + 26 q^{85} + 3 \beta q^{89} -6 q^{91} + \beta q^{95} -8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} + O(q^{10}) \) \( 2q - 2q^{7} + 12q^{13} - 2q^{19} + 16q^{25} + 18q^{31} - 2q^{37} + 16q^{43} + 2q^{49} - 26q^{55} - 4q^{67} + 8q^{73} + 52q^{85} - 12q^{91} - 16q^{97} + O(q^{100}) \)

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
2.30278
−1.30278
0 0 0 −3.60555 0 −1.00000 0 0 0
1.2 0 0 0 3.60555 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 756.2.a.g 2
3.b odd 2 1 inner 756.2.a.g 2
4.b odd 2 1 3024.2.a.bj 2
7.b odd 2 1 5292.2.a.o 2
9.c even 3 2 2268.2.j.q 4
9.d odd 6 2 2268.2.j.q 4
12.b even 2 1 3024.2.a.bj 2
21.c even 2 1 5292.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.a.g 2 1.a even 1 1 trivial
756.2.a.g 2 3.b odd 2 1 inner
2268.2.j.q 4 9.c even 3 2
2268.2.j.q 4 9.d odd 6 2
3024.2.a.bj 2 4.b odd 2 1
3024.2.a.bj 2 12.b even 2 1
5292.2.a.o 2 7.b odd 2 1
5292.2.a.o 2 21.c even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(756))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 3 T^{2} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 + 9 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 18 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + T + 19 T^{2} )^{2} \)
$23$ \( 1 + 33 T^{2} + 529 T^{4} \)
$29$ \( 1 + 6 T^{2} + 841 T^{4} \)
$31$ \( ( 1 - 9 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + T + 37 T^{2} )^{2} \)
$41$ \( 1 - 35 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{2} \)
$59$ \( 1 - 90 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 + 61 T^{2} )^{2} \)
$67$ \( ( 1 + 2 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 129 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 - 4 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( 1 + 114 T^{2} + 6889 T^{4} \)
$89$ \( 1 + 61 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + 8 T + 97 T^{2} )^{2} \)
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