Properties

Label 87.2.a.a
Level $87$
Weight $2$
Character orbit 87.a
Self dual yes
Analytic conductor $0.695$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 87.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.694698497585\)
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: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
−0.618034 1.00000 −1.61803 3.23607 −0.618034 0.236068 2.23607 1.00000 −2.00000
1.2 1.61803 1.00000 0.618034 −1.23607 1.61803 −4.23607 −2.23607 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(29\) \(1\)

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 87.2.a.a 2
3.b odd 2 1 261.2.a.b 2
4.b odd 2 1 1392.2.a.q 2
5.b even 2 1 2175.2.a.l 2
5.c odd 4 2 2175.2.c.k 4
7.b odd 2 1 4263.2.a.j 2
8.b even 2 1 5568.2.a.bl 2
8.d odd 2 1 5568.2.a.bs 2
12.b even 2 1 4176.2.a.bn 2
15.d odd 2 1 6525.2.a.ba 2
29.b even 2 1 2523.2.a.c 2
87.d odd 2 1 7569.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.2.a.a 2 1.a even 1 1 trivial
261.2.a.b 2 3.b odd 2 1
1392.2.a.q 2 4.b odd 2 1
2175.2.a.l 2 5.b even 2 1
2175.2.c.k 4 5.c odd 4 2
2523.2.a.c 2 29.b even 2 1
4176.2.a.bn 2 12.b even 2 1
4263.2.a.j 2 7.b odd 2 1
5568.2.a.bl 2 8.b even 2 1
5568.2.a.bs 2 8.d odd 2 1
6525.2.a.ba 2 15.d odd 2 1
7569.2.a.k 2 87.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(87))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - T + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -4 - 2 T + T^{2} \)
$7$ \( -1 + 4 T + T^{2} \)
$11$ \( -1 - 4 T + T^{2} \)
$13$ \( -19 + 2 T + T^{2} \)
$17$ \( ( -3 + T )^{2} \)
$19$ \( 20 + 10 T + T^{2} \)
$23$ \( -44 + 2 T + T^{2} \)
$29$ \( ( 1 + T )^{2} \)
$31$ \( -36 + 6 T + T^{2} \)
$37$ \( 4 - 6 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -41 + 4 T + T^{2} \)
$53$ \( 76 - 18 T + T^{2} \)
$59$ \( -20 + T^{2} \)
$61$ \( 4 + 6 T + T^{2} \)
$67$ \( -121 + 4 T + T^{2} \)
$71$ \( 4 + 6 T + T^{2} \)
$73$ \( 76 - 18 T + T^{2} \)
$79$ \( 220 + 30 T + T^{2} \)
$83$ \( -44 + 12 T + T^{2} \)
$89$ \( ( -5 + T )^{2} \)
$97$ \( -236 - 6 T + T^{2} \)
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