Properties

Label 784.2.a.d
Level 784
Weight 2
Character orbit 784.a
Self dual yes
Analytic conductor 6.260
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
0 −1.00000 0 3.00000 0 0 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 784.2.a.d 1
3.b odd 2 1 7056.2.a.f 1
4.b odd 2 1 196.2.a.b 1
7.b odd 2 1 784.2.a.g 1
7.c even 3 2 112.2.i.b 2
7.d odd 6 2 784.2.i.d 2
8.b even 2 1 3136.2.a.s 1
8.d odd 2 1 3136.2.a.h 1
12.b even 2 1 1764.2.a.a 1
20.d odd 2 1 4900.2.a.g 1
20.e even 4 2 4900.2.e.i 2
21.c even 2 1 7056.2.a.bw 1
21.h odd 6 2 1008.2.s.p 2
28.d even 2 1 196.2.a.a 1
28.f even 6 2 196.2.e.a 2
28.g odd 6 2 28.2.e.a 2
56.e even 2 1 3136.2.a.v 1
56.h odd 2 1 3136.2.a.k 1
56.k odd 6 2 448.2.i.e 2
56.p even 6 2 448.2.i.c 2
84.h odd 2 1 1764.2.a.j 1
84.j odd 6 2 1764.2.k.b 2
84.n even 6 2 252.2.k.c 2
140.c even 2 1 4900.2.a.n 1
140.j odd 4 2 4900.2.e.h 2
140.p odd 6 2 700.2.i.c 2
140.w even 12 4 700.2.r.b 4
252.o even 6 2 2268.2.l.a 2
252.u odd 6 2 2268.2.i.a 2
252.bb even 6 2 2268.2.i.h 2
252.bl odd 6 2 2268.2.l.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 28.g odd 6 2
112.2.i.b 2 7.c even 3 2
196.2.a.a 1 28.d even 2 1
196.2.a.b 1 4.b odd 2 1
196.2.e.a 2 28.f even 6 2
252.2.k.c 2 84.n even 6 2
448.2.i.c 2 56.p even 6 2
448.2.i.e 2 56.k odd 6 2
700.2.i.c 2 140.p odd 6 2
700.2.r.b 4 140.w even 12 4
784.2.a.d 1 1.a even 1 1 trivial
784.2.a.g 1 7.b odd 2 1
784.2.i.d 2 7.d odd 6 2
1008.2.s.p 2 21.h odd 6 2
1764.2.a.a 1 12.b even 2 1
1764.2.a.j 1 84.h odd 2 1
1764.2.k.b 2 84.j odd 6 2
2268.2.i.a 2 252.u odd 6 2
2268.2.i.h 2 252.bb even 6 2
2268.2.l.a 2 252.o even 6 2
2268.2.l.h 2 252.bl odd 6 2
3136.2.a.h 1 8.d odd 2 1
3136.2.a.k 1 56.h odd 2 1
3136.2.a.s 1 8.b even 2 1
3136.2.a.v 1 56.e even 2 1
4900.2.a.g 1 20.d odd 2 1
4900.2.a.n 1 140.c even 2 1
4900.2.e.h 2 140.j odd 4 2
4900.2.e.i 2 20.e even 4 2
7056.2.a.f 1 3.b odd 2 1
7056.2.a.bw 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3} + 1 \)
\( T_{5} - 3 \)
\( T_{11} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T + 3 T^{2} \)
$5$ \( 1 - 3 T + 5 T^{2} \)
$7$ 1
$11$ \( 1 - 3 T + 11 T^{2} \)
$13$ \( 1 - 2 T + 13 T^{2} \)
$17$ \( 1 - 3 T + 17 T^{2} \)
$19$ \( 1 - T + 19 T^{2} \)
$23$ \( 1 + 3 T + 23 T^{2} \)
$29$ \( 1 + 6 T + 29 T^{2} \)
$31$ \( 1 - 7 T + 31 T^{2} \)
$37$ \( 1 + T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 - 9 T + 47 T^{2} \)
$53$ \( 1 - 3 T + 53 T^{2} \)
$59$ \( 1 + 9 T + 59 T^{2} \)
$61$ \( 1 + T + 61 T^{2} \)
$67$ \( 1 - 7 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + T + 73 T^{2} \)
$79$ \( 1 - 13 T + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 - 15 T + 89 T^{2} \)
$97$ \( 1 + 10 T + 97 T^{2} \)
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