Properties

Label 69.2.a.a
Level $69$
Weight $2$
Character orbit 69.a
Self dual yes
Analytic conductor $0.551$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.550967773947\)
Analytic rank: \(0\)
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

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

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(1\)

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 69.2.a.a 1
3.b odd 2 1 207.2.a.a 1
4.b odd 2 1 1104.2.a.c 1
5.b even 2 1 1725.2.a.e 1
5.c odd 4 2 1725.2.b.g 2
7.b odd 2 1 3381.2.a.k 1
8.b even 2 1 4416.2.a.f 1
8.d odd 2 1 4416.2.a.x 1
11.b odd 2 1 8349.2.a.a 1
12.b even 2 1 3312.2.a.k 1
15.d odd 2 1 5175.2.a.v 1
23.b odd 2 1 1587.2.a.e 1
69.c even 2 1 4761.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.a.a 1 1.a even 1 1 trivial
207.2.a.a 1 3.b odd 2 1
1104.2.a.c 1 4.b odd 2 1
1587.2.a.e 1 23.b odd 2 1
1725.2.a.e 1 5.b even 2 1
1725.2.b.g 2 5.c odd 4 2
3312.2.a.k 1 12.b even 2 1
3381.2.a.k 1 7.b odd 2 1
4416.2.a.f 1 8.b even 2 1
4416.2.a.x 1 8.d odd 2 1
4761.2.a.b 1 69.c even 2 1
5175.2.a.v 1 15.d odd 2 1
8349.2.a.a 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(69))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( 2 + T \)
$11$ \( -4 + T \)
$13$ \( 6 + T \)
$17$ \( -4 + T \)
$19$ \( -2 + T \)
$23$ \( 1 + T \)
$29$ \( -2 + T \)
$31$ \( -4 + T \)
$37$ \( -2 + T \)
$41$ \( -2 + T \)
$43$ \( -10 + T \)
$47$ \( T \)
$53$ \( 12 + T \)
$59$ \( 12 + T \)
$61$ \( 6 + T \)
$67$ \( 10 + T \)
$71$ \( -8 + T \)
$73$ \( 14 + T \)
$79$ \( -10 + T \)
$83$ \( -12 + T \)
$89$ \( 16 + T \)
$97$ \( 10 + T \)
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