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: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
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.

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

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:

Atkin-Lehner signs

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

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 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

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

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -13 + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -13 + T^{2} \)
$13$ \( ( -6 + T )^{2} \)
$17$ \( -52 + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -13 + T^{2} \)
$29$ \( -52 + T^{2} \)
$31$ \( ( -9 + T )^{2} \)
$37$ \( ( 1 + T )^{2} \)
$41$ \( -117 + T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( -208 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( ( 2 + T )^{2} \)
$71$ \( -13 + T^{2} \)
$73$ \( ( -4 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( -52 + T^{2} \)
$89$ \( -117 + T^{2} \)
$97$ \( ( 8 + T )^{2} \)
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