Properties

Label 690.2.a.g
Level $690$
Weight $2$
Character orbit 690.a
Self dual yes
Analytic conductor $5.510$
Analytic rank $1$
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: \(1\)
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

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

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 −2.00000 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.g 1
3.b odd 2 1 2070.2.a.e 1
4.b odd 2 1 5520.2.a.z 1
5.b even 2 1 3450.2.a.l 1
5.c odd 4 2 3450.2.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.g 1 1.a even 1 1 trivial
2070.2.a.e 1 3.b odd 2 1
3450.2.a.l 1 5.b even 2 1
3450.2.d.d 2 5.c odd 4 2
5520.2.a.z 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(690))\):

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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