Properties

Label 2166.2.a.d
Level $2166$
Weight $2$
Character orbit 2166.a
Self dual yes
Analytic conductor $17.296$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.2955970778\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} - q^{8} + q^{9} - 2q^{10} - 4q^{11} + q^{12} - 2q^{13} + 2q^{15} + q^{16} - 6q^{17} - q^{18} + 2q^{20} + 4q^{22} - 4q^{23} - q^{24} - q^{25} + 2q^{26} + q^{27} + 2q^{29} - 2q^{30} - 4q^{31} - q^{32} - 4q^{33} + 6q^{34} + q^{36} - 10q^{37} - 2q^{39} - 2q^{40} - 10q^{41} + 4q^{43} - 4q^{44} + 2q^{45} + 4q^{46} - 4q^{47} + q^{48} - 7q^{49} + q^{50} - 6q^{51} - 2q^{52} + 10q^{53} - q^{54} - 8q^{55} - 2q^{58} - 12q^{59} + 2q^{60} + 14q^{61} + 4q^{62} + q^{64} - 4q^{65} + 4q^{66} + 12q^{67} - 6q^{68} - 4q^{69} - 8q^{71} - q^{72} - 6q^{73} + 10q^{74} - q^{75} + 2q^{78} + 4q^{79} + 2q^{80} + q^{81} + 10q^{82} + 12q^{83} - 12q^{85} - 4q^{86} + 2q^{87} + 4q^{88} + 6q^{89} - 2q^{90} - 4q^{92} - 4q^{93} + 4q^{94} - q^{96} - 10q^{97} + 7q^{98} - 4q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(19\) \(-1\)

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 2166.2.a.d 1
3.b odd 2 1 6498.2.a.p 1
19.b odd 2 1 114.2.a.b 1
57.d even 2 1 342.2.a.b 1
76.d even 2 1 912.2.a.k 1
95.d odd 2 1 2850.2.a.j 1
95.g even 4 2 2850.2.d.b 2
133.c even 2 1 5586.2.a.y 1
152.b even 2 1 3648.2.a.c 1
152.g odd 2 1 3648.2.a.x 1
228.b odd 2 1 2736.2.a.d 1
285.b even 2 1 8550.2.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.b 1 19.b odd 2 1
342.2.a.b 1 57.d even 2 1
912.2.a.k 1 76.d even 2 1
2166.2.a.d 1 1.a even 1 1 trivial
2736.2.a.d 1 228.b odd 2 1
2850.2.a.j 1 95.d odd 2 1
2850.2.d.b 2 95.g even 4 2
3648.2.a.c 1 152.b even 2 1
3648.2.a.x 1 152.g odd 2 1
5586.2.a.y 1 133.c even 2 1
6498.2.a.p 1 3.b odd 2 1
8550.2.a.ba 1 285.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5} - 2 \)
\( T_{7} \)
\( T_{13} + 2 \)
\( T_{29} - 2 \)

Hecke characteristic polynomials

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