Properties

Label 77.2.a.a
Level 77
Weight 2
Character orbit 77.a
Self dual yes
Analytic conductor 0.615
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 77 = 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 77.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.614848095564\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} - 2q^{4} - q^{5} - q^{7} + 6q^{9} + O(q^{10}) \) \( q - 3q^{3} - 2q^{4} - q^{5} - q^{7} + 6q^{9} - q^{11} + 6q^{12} - 4q^{13} + 3q^{15} + 4q^{16} + 2q^{17} - 6q^{19} + 2q^{20} + 3q^{21} - 5q^{23} - 4q^{25} - 9q^{27} + 2q^{28} + 10q^{29} + q^{31} + 3q^{33} + q^{35} - 12q^{36} - 5q^{37} + 12q^{39} - 2q^{41} - 8q^{43} + 2q^{44} - 6q^{45} + 8q^{47} - 12q^{48} + q^{49} - 6q^{51} + 8q^{52} - 6q^{53} + q^{55} + 18q^{57} + 3q^{59} - 6q^{60} - 2q^{61} - 6q^{63} - 8q^{64} + 4q^{65} - 3q^{67} - 4q^{68} + 15q^{69} + q^{71} + 10q^{73} + 12q^{75} + 12q^{76} + q^{77} + 6q^{79} - 4q^{80} + 9q^{81} + 12q^{83} - 6q^{84} - 2q^{85} - 30q^{87} - 15q^{89} + 4q^{91} + 10q^{92} - 3q^{93} + 6q^{95} - 5q^{97} - 6q^{99} + O(q^{100}) \)

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 −3.00000 −2.00000 −1.00000 0 −1.00000 0 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 77.2.a.a 1
3.b odd 2 1 693.2.a.c 1
4.b odd 2 1 1232.2.a.l 1
5.b even 2 1 1925.2.a.h 1
5.c odd 4 2 1925.2.b.e 2
7.b odd 2 1 539.2.a.c 1
7.c even 3 2 539.2.e.f 2
7.d odd 6 2 539.2.e.c 2
8.b even 2 1 4928.2.a.bj 1
8.d odd 2 1 4928.2.a.a 1
11.b odd 2 1 847.2.a.b 1
11.c even 5 4 847.2.f.i 4
11.d odd 10 4 847.2.f.h 4
21.c even 2 1 4851.2.a.j 1
28.d even 2 1 8624.2.a.a 1
33.d even 2 1 7623.2.a.j 1
77.b even 2 1 5929.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.a 1 1.a even 1 1 trivial
539.2.a.c 1 7.b odd 2 1
539.2.e.c 2 7.d odd 6 2
539.2.e.f 2 7.c even 3 2
693.2.a.c 1 3.b odd 2 1
847.2.a.b 1 11.b odd 2 1
847.2.f.h 4 11.d odd 10 4
847.2.f.i 4 11.c even 5 4
1232.2.a.l 1 4.b odd 2 1
1925.2.a.h 1 5.b even 2 1
1925.2.b.e 2 5.c odd 4 2
4851.2.a.j 1 21.c even 2 1
4928.2.a.a 1 8.d odd 2 1
4928.2.a.bj 1 8.b even 2 1
5929.2.a.f 1 77.b even 2 1
7623.2.a.j 1 33.d even 2 1
8624.2.a.a 1 28.d even 2 1

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(11\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(77))\):

\( T_{2} \)
\( T_{3} + 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} \)
$3$ \( 1 + 3 T + 3 T^{2} \)
$5$ \( 1 + T + 5 T^{2} \)
$7$ \( 1 + T \)
$11$ \( 1 + T \)
$13$ \( 1 + 4 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 6 T + 19 T^{2} \)
$23$ \( 1 + 5 T + 23 T^{2} \)
$29$ \( 1 - 10 T + 29 T^{2} \)
$31$ \( 1 - T + 31 T^{2} \)
$37$ \( 1 + 5 T + 37 T^{2} \)
$41$ \( 1 + 2 T + 41 T^{2} \)
$43$ \( 1 + 8 T + 43 T^{2} \)
$47$ \( 1 - 8 T + 47 T^{2} \)
$53$ \( 1 + 6 T + 53 T^{2} \)
$59$ \( 1 - 3 T + 59 T^{2} \)
$61$ \( 1 + 2 T + 61 T^{2} \)
$67$ \( 1 + 3 T + 67 T^{2} \)
$71$ \( 1 - T + 71 T^{2} \)
$73$ \( 1 - 10 T + 73 T^{2} \)
$79$ \( 1 - 6 T + 79 T^{2} \)
$83$ \( 1 - 12 T + 83 T^{2} \)
$89$ \( 1 + 15 T + 89 T^{2} \)
$97$ \( 1 + 5 T + 97 T^{2} \)
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