Properties

Label 690.2.a.k
Level $690$
Weight $2$
Character orbit 690.a
Self dual yes
Analytic conductor $5.510$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.50967773947\)
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

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

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

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 690.2.a.k 1
3.b odd 2 1 2070.2.a.b 1
4.b odd 2 1 5520.2.a.i 1
5.b even 2 1 3450.2.a.d 1
5.c odd 4 2 3450.2.d.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.k 1 1.a even 1 1 trivial
2070.2.a.b 1 3.b odd 2 1
3450.2.a.d 1 5.b even 2 1
3450.2.d.t 2 5.c odd 4 2
5520.2.a.i 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(690))\):

\( T_{7} \)
\( T_{11} - 4 \)
\( T_{17} + 6 \)

Hecke characteristic polynomials

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