Properties

Label 89.2.a.a
Level 89
Weight 2
Character orbit 89.a
Self dual yes
Analytic conductor 0.711
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 89.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.710668577989\)
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

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

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
−1.00000 −1.00000 −1.00000 −1.00000 1.00000 −4.00000 3.00000 −2.00000 1.00000
\(n\): 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 89.2.a.a 1
3.b odd 2 1 801.2.a.d 1
4.b odd 2 1 1424.2.a.e 1
5.b even 2 1 2225.2.a.b 1
7.b odd 2 1 4361.2.a.a 1
8.b even 2 1 5696.2.a.k 1
8.d odd 2 1 5696.2.a.g 1
89.b even 2 1 7921.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.2.a.a 1 1.a even 1 1 trivial
801.2.a.d 1 3.b odd 2 1
1424.2.a.e 1 4.b odd 2 1
2225.2.a.b 1 5.b even 2 1
4361.2.a.a 1 7.b odd 2 1
5696.2.a.g 1 8.d odd 2 1
5696.2.a.k 1 8.b even 2 1
7921.2.a.a 1 89.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(89\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(89))\).

Hecke Characteristic Polynomials

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