Properties

Label 77.2.a.d
Level $77$
Weight $2$
Character orbit 77.a
Self dual yes
Analytic conductor $0.615$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( 1 + \beta ) q^{3} + 3 q^{4} -2 q^{5} + ( -5 - \beta ) q^{6} + q^{7} -\beta q^{8} + ( 3 + 2 \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{2} + ( 1 + \beta ) q^{3} + 3 q^{4} -2 q^{5} + ( -5 - \beta ) q^{6} + q^{7} -\beta q^{8} + ( 3 + 2 \beta ) q^{9} + 2 \beta q^{10} - q^{11} + ( 3 + 3 \beta ) q^{12} + ( 1 - \beta ) q^{13} -\beta q^{14} + ( -2 - 2 \beta ) q^{15} - q^{16} + ( -1 + \beta ) q^{17} + ( -10 - 3 \beta ) q^{18} + ( 2 - 2 \beta ) q^{19} -6 q^{20} + ( 1 + \beta ) q^{21} + \beta q^{22} + ( -2 - 2 \beta ) q^{23} + ( -5 - \beta ) q^{24} - q^{25} + ( 5 - \beta ) q^{26} + ( 10 + 2 \beta ) q^{27} + 3 q^{28} + ( 4 - 2 \beta ) q^{29} + ( 10 + 2 \beta ) q^{30} + ( -5 - \beta ) q^{31} + 3 \beta q^{32} + ( -1 - \beta ) q^{33} + ( -5 + \beta ) q^{34} -2 q^{35} + ( 9 + 6 \beta ) q^{36} + ( -4 + 2 \beta ) q^{37} + ( 10 - 2 \beta ) q^{38} -4 q^{39} + 2 \beta q^{40} + ( -9 + \beta ) q^{41} + ( -5 - \beta ) q^{42} + 8 q^{43} -3 q^{44} + ( -6 - 4 \beta ) q^{45} + ( 10 + 2 \beta ) q^{46} + ( 5 + \beta ) q^{47} + ( -1 - \beta ) q^{48} + q^{49} + \beta q^{50} + 4 q^{51} + ( 3 - 3 \beta ) q^{52} + ( 4 + 2 \beta ) q^{53} + ( -10 - 10 \beta ) q^{54} + 2 q^{55} -\beta q^{56} -8 q^{57} + ( 10 - 4 \beta ) q^{58} + ( 1 + \beta ) q^{59} + ( -6 - 6 \beta ) q^{60} + ( -5 + \beta ) q^{61} + ( 5 + 5 \beta ) q^{62} + ( 3 + 2 \beta ) q^{63} -13 q^{64} + ( -2 + 2 \beta ) q^{65} + ( 5 + \beta ) q^{66} + ( 10 - 2 \beta ) q^{67} + ( -3 + 3 \beta ) q^{68} + ( -12 - 4 \beta ) q^{69} + 2 \beta q^{70} + ( -6 + 2 \beta ) q^{71} + ( -10 - 3 \beta ) q^{72} + ( -3 - \beta ) q^{73} + ( -10 + 4 \beta ) q^{74} + ( -1 - \beta ) q^{75} + ( 6 - 6 \beta ) q^{76} - q^{77} + 4 \beta q^{78} + 4 \beta q^{79} + 2 q^{80} + ( 11 + 6 \beta ) q^{81} + ( -5 + 9 \beta ) q^{82} + ( 2 + 6 \beta ) q^{83} + ( 3 + 3 \beta ) q^{84} + ( 2 - 2 \beta ) q^{85} -8 \beta q^{86} + ( -6 + 2 \beta ) q^{87} + \beta q^{88} + 2 q^{89} + ( 20 + 6 \beta ) q^{90} + ( 1 - \beta ) q^{91} + ( -6 - 6 \beta ) q^{92} + ( -10 - 6 \beta ) q^{93} + ( -5 - 5 \beta ) q^{94} + ( -4 + 4 \beta ) q^{95} + ( 15 + 3 \beta ) q^{96} + ( 4 - 6 \beta ) q^{97} -\beta q^{98} + ( -3 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 6q^{4} - 4q^{5} - 10q^{6} + 2q^{7} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 6q^{4} - 4q^{5} - 10q^{6} + 2q^{7} + 6q^{9} - 2q^{11} + 6q^{12} + 2q^{13} - 4q^{15} - 2q^{16} - 2q^{17} - 20q^{18} + 4q^{19} - 12q^{20} + 2q^{21} - 4q^{23} - 10q^{24} - 2q^{25} + 10q^{26} + 20q^{27} + 6q^{28} + 8q^{29} + 20q^{30} - 10q^{31} - 2q^{33} - 10q^{34} - 4q^{35} + 18q^{36} - 8q^{37} + 20q^{38} - 8q^{39} - 18q^{41} - 10q^{42} + 16q^{43} - 6q^{44} - 12q^{45} + 20q^{46} + 10q^{47} - 2q^{48} + 2q^{49} + 8q^{51} + 6q^{52} + 8q^{53} - 20q^{54} + 4q^{55} - 16q^{57} + 20q^{58} + 2q^{59} - 12q^{60} - 10q^{61} + 10q^{62} + 6q^{63} - 26q^{64} - 4q^{65} + 10q^{66} + 20q^{67} - 6q^{68} - 24q^{69} - 12q^{71} - 20q^{72} - 6q^{73} - 20q^{74} - 2q^{75} + 12q^{76} - 2q^{77} + 4q^{80} + 22q^{81} - 10q^{82} + 4q^{83} + 6q^{84} + 4q^{85} - 12q^{87} + 4q^{89} + 40q^{90} + 2q^{91} - 12q^{92} - 20q^{93} - 10q^{94} - 8q^{95} + 30q^{96} + 8q^{97} - 6q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
−2.23607 3.23607 3.00000 −2.00000 −7.23607 1.00000 −2.23607 7.47214 4.47214
1.2 2.23607 −1.23607 3.00000 −2.00000 −2.76393 1.00000 2.23607 −1.47214 −4.47214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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

Hecke kernels

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

\( T_{2}^{2} - 5 \)
\( T_{3}^{2} - 2 T_{3} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 + T^{2} \)
$3$ \( -4 - 2 T + T^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -4 - 2 T + T^{2} \)
$17$ \( -4 + 2 T + T^{2} \)
$19$ \( -16 - 4 T + T^{2} \)
$23$ \( -16 + 4 T + T^{2} \)
$29$ \( -4 - 8 T + T^{2} \)
$31$ \( 20 + 10 T + T^{2} \)
$37$ \( -4 + 8 T + T^{2} \)
$41$ \( 76 + 18 T + T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( 20 - 10 T + T^{2} \)
$53$ \( -4 - 8 T + T^{2} \)
$59$ \( -4 - 2 T + T^{2} \)
$61$ \( 20 + 10 T + T^{2} \)
$67$ \( 80 - 20 T + T^{2} \)
$71$ \( 16 + 12 T + T^{2} \)
$73$ \( 4 + 6 T + T^{2} \)
$79$ \( -80 + T^{2} \)
$83$ \( -176 - 4 T + T^{2} \)
$89$ \( ( -2 + T )^{2} \)
$97$ \( -164 - 8 T + T^{2} \)
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