Properties

Label 784.4.a.x
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.2574974445\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 392)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} -\beta q^{5} + 5 q^{9} +O(q^{10})\) \( q + \beta q^{3} -\beta q^{5} + 5 q^{9} + 4 q^{11} + \beta q^{13} -32 q^{15} -22 \beta q^{17} + 23 \beta q^{19} -120 q^{23} -93 q^{25} -22 \beta q^{27} + 218 q^{29} -26 \beta q^{31} + 4 \beta q^{33} + 130 q^{37} + 32 q^{39} -26 \beta q^{41} -332 q^{43} -5 \beta q^{45} -22 \beta q^{47} -704 q^{51} -498 q^{53} -4 \beta q^{55} + 736 q^{57} -97 \beta q^{59} + 115 \beta q^{61} -32 q^{65} -156 q^{67} -120 \beta q^{69} -240 q^{71} -93 \beta q^{75} -1112 q^{79} -839 q^{81} -5 \beta q^{83} + 704 q^{85} + 218 \beta q^{87} -240 \beta q^{89} -832 q^{93} -736 q^{95} + 262 \beta q^{97} + 20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10q^{9} + O(q^{10}) \) \( 2q + 10q^{9} + 8q^{11} - 64q^{15} - 240q^{23} - 186q^{25} + 436q^{29} + 260q^{37} + 64q^{39} - 664q^{43} - 1408q^{51} - 996q^{53} + 1472q^{57} - 64q^{65} - 312q^{67} - 480q^{71} - 2224q^{79} - 1678q^{81} + 1408q^{85} - 1664q^{93} - 1472q^{95} + 40q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −5.65685 0 5.65685 0 0 0 5.00000 0
1.2 0 5.65685 0 −5.65685 0 0 0 5.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.a.x 2
4.b odd 2 1 392.4.a.g 2
7.b odd 2 1 inner 784.4.a.x 2
28.d even 2 1 392.4.a.g 2
28.f even 6 2 392.4.i.j 4
28.g odd 6 2 392.4.i.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.4.a.g 2 4.b odd 2 1
392.4.a.g 2 28.d even 2 1
392.4.i.j 4 28.f even 6 2
392.4.i.j 4 28.g odd 6 2
784.4.a.x 2 1.a even 1 1 trivial
784.4.a.x 2 7.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3}^{2} - 32 \)
\( T_{5}^{2} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -32 + T^{2} \)
$5$ \( -32 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( -15488 + T^{2} \)
$19$ \( -16928 + T^{2} \)
$23$ \( ( 120 + T )^{2} \)
$29$ \( ( -218 + T )^{2} \)
$31$ \( -21632 + T^{2} \)
$37$ \( ( -130 + T )^{2} \)
$41$ \( -21632 + T^{2} \)
$43$ \( ( 332 + T )^{2} \)
$47$ \( -15488 + T^{2} \)
$53$ \( ( 498 + T )^{2} \)
$59$ \( -301088 + T^{2} \)
$61$ \( -423200 + T^{2} \)
$67$ \( ( 156 + T )^{2} \)
$71$ \( ( 240 + T )^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 1112 + T )^{2} \)
$83$ \( -800 + T^{2} \)
$89$ \( -1843200 + T^{2} \)
$97$ \( -2196608 + T^{2} \)
show more
show less