Properties

Label 784.2.a.k
Level $784$
Weight $2$
Character orbit 784.a
Self dual yes
Analytic conductor $6.260$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(6.26027151847\)
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: \( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} -2 \beta q^{5} - q^{9} +O(q^{10})\) \( q + \beta q^{3} -2 \beta q^{5} - q^{9} -6 q^{11} + 4 \beta q^{13} -4 q^{15} -\beta q^{17} -3 \beta q^{19} -4 q^{23} + 3 q^{25} -4 \beta q^{27} -6 q^{29} + 2 \beta q^{31} -6 \beta q^{33} + 2 q^{37} + 8 q^{39} + \beta q^{41} -10 q^{43} + 2 \beta q^{45} -2 \beta q^{47} -2 q^{51} -2 q^{53} + 12 \beta q^{55} -6 q^{57} + \beta q^{59} + 6 \beta q^{61} -16 q^{65} -4 q^{67} -4 \beta q^{69} + 12 q^{71} + 7 \beta q^{73} + 3 \beta q^{75} + 4 q^{79} -5 q^{81} + \beta q^{83} + 4 q^{85} -6 \beta q^{87} + 3 \beta q^{89} + 4 q^{93} + 12 q^{95} -9 \beta q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 12q^{11} - 8q^{15} - 8q^{23} + 6q^{25} - 12q^{29} + 4q^{37} + 16q^{39} - 20q^{43} - 4q^{51} - 4q^{53} - 12q^{57} - 32q^{65} - 8q^{67} + 24q^{71} + 8q^{79} - 10q^{81} + 8q^{85} + 8q^{93} + 24q^{95} + 12q^{99} + O(q^{100}) \)

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 −1.41421 0 2.82843 0 0 0 −1.00000 0
1.2 0 1.41421 0 −2.82843 0 0 0 −1.00000 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 784.2.a.k 2
3.b odd 2 1 7056.2.a.ct 2
4.b odd 2 1 392.2.a.g 2
7.b odd 2 1 inner 784.2.a.k 2
7.c even 3 2 784.2.i.n 4
7.d odd 6 2 784.2.i.n 4
8.b even 2 1 3136.2.a.bp 2
8.d odd 2 1 3136.2.a.bk 2
12.b even 2 1 3528.2.a.be 2
20.d odd 2 1 9800.2.a.bv 2
21.c even 2 1 7056.2.a.ct 2
28.d even 2 1 392.2.a.g 2
28.f even 6 2 392.2.i.h 4
28.g odd 6 2 392.2.i.h 4
56.e even 2 1 3136.2.a.bk 2
56.h odd 2 1 3136.2.a.bp 2
84.h odd 2 1 3528.2.a.be 2
84.j odd 6 2 3528.2.s.bj 4
84.n even 6 2 3528.2.s.bj 4
140.c even 2 1 9800.2.a.bv 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.g 2 4.b odd 2 1
392.2.a.g 2 28.d even 2 1
392.2.i.h 4 28.f even 6 2
392.2.i.h 4 28.g odd 6 2
784.2.a.k 2 1.a even 1 1 trivial
784.2.a.k 2 7.b odd 2 1 inner
784.2.i.n 4 7.c even 3 2
784.2.i.n 4 7.d odd 6 2
3136.2.a.bk 2 8.d odd 2 1
3136.2.a.bk 2 56.e even 2 1
3136.2.a.bp 2 8.b even 2 1
3136.2.a.bp 2 56.h odd 2 1
3528.2.a.be 2 12.b even 2 1
3528.2.a.be 2 84.h odd 2 1
3528.2.s.bj 4 84.j odd 6 2
3528.2.s.bj 4 84.n even 6 2
7056.2.a.ct 2 3.b odd 2 1
7056.2.a.ct 2 21.c even 2 1
9800.2.a.bv 2 20.d odd 2 1
9800.2.a.bv 2 140.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3}^{2} - 2 \)
\( T_{5}^{2} - 8 \)
\( T_{11} + 6 \)