Properties

Label 6690.2.a.j
Level 6690
Weight 2
Character orbit 6690.a
Self dual yes
Analytic conductor 53.420
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6690 = 2 \cdot 3 \cdot 5 \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.4199189522\)
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

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

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 1.00000 1.00000 1.00000 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 6690.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6690.2.a.j 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(223\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6690))\):

\( T_{7} - 1 \)
\( T_{11} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 - T \)
$5$ \( 1 - T \)
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( 1 - 5 T + 11 T^{2} \)
$13$ \( 1 + 13 T^{2} \)
$17$ \( 1 + 3 T + 17 T^{2} \)
$19$ \( 1 + 8 T + 19 T^{2} \)
$23$ \( 1 - 3 T + 23 T^{2} \)
$29$ \( 1 - 9 T + 29 T^{2} \)
$31$ \( 1 + 10 T + 31 T^{2} \)
$37$ \( 1 - 10 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 - 2 T + 43 T^{2} \)
$47$ \( 1 - 6 T + 47 T^{2} \)
$53$ \( 1 - 12 T + 53 T^{2} \)
$59$ \( 1 - 4 T + 59 T^{2} \)
$61$ \( 1 - 6 T + 61 T^{2} \)
$67$ \( 1 - 5 T + 67 T^{2} \)
$71$ \( 1 - 2 T + 71 T^{2} \)
$73$ \( 1 - 4 T + 73 T^{2} \)
$79$ \( 1 + 11 T + 79 T^{2} \)
$83$ \( 1 - 7 T + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 - 7 T + 97 T^{2} \)
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