Properties

Label 2166.2.a.f
Level $2166$
Weight $2$
Character orbit 2166.a
Self dual yes
Analytic conductor $17.296$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} - 3q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} - 3q^{7} + q^{8} + q^{9} - 4q^{10} + 2q^{11} - q^{12} - 7q^{13} - 3q^{14} + 4q^{15} + q^{16} + q^{18} - 4q^{20} + 3q^{21} + 2q^{22} - 4q^{23} - q^{24} + 11q^{25} - 7q^{26} - q^{27} - 3q^{28} + 4q^{29} + 4q^{30} + q^{31} + q^{32} - 2q^{33} + 12q^{35} + q^{36} + 7q^{37} + 7q^{39} - 4q^{40} + 4q^{41} + 3q^{42} + 7q^{43} + 2q^{44} - 4q^{45} - 4q^{46} + 2q^{47} - q^{48} + 2q^{49} + 11q^{50} - 7q^{52} - 4q^{53} - q^{54} - 8q^{55} - 3q^{56} + 4q^{58} - 6q^{59} + 4q^{60} - q^{61} + q^{62} - 3q^{63} + q^{64} + 28q^{65} - 2q^{66} + 3q^{67} + 4q^{69} + 12q^{70} + 2q^{71} + q^{72} - 3q^{73} + 7q^{74} - 11q^{75} - 6q^{77} + 7q^{78} + 5q^{79} - 4q^{80} + q^{81} + 4q^{82} - 12q^{83} + 3q^{84} + 7q^{86} - 4q^{87} + 2q^{88} + 18q^{89} - 4q^{90} + 21q^{91} - 4q^{92} - q^{93} + 2q^{94} - q^{96} + 10q^{97} + 2q^{98} + 2q^{99} + O(q^{100}) \)

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 −4.00000 −1.00000 −3.00000 1.00000 1.00000 −4.00000
\(n\): 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.f 1
3.b odd 2 1 6498.2.a.l 1
19.b odd 2 1 2166.2.a.c 1
19.c even 3 2 114.2.e.a 2
57.d even 2 1 6498.2.a.x 1
57.h odd 6 2 342.2.g.d 2
76.g odd 6 2 912.2.q.d 2
228.m even 6 2 2736.2.s.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.a 2 19.c even 3 2
342.2.g.d 2 57.h odd 6 2
912.2.q.d 2 76.g odd 6 2
2166.2.a.c 1 19.b odd 2 1
2166.2.a.f 1 1.a even 1 1 trivial
2736.2.s.c 2 228.m even 6 2
6498.2.a.l 1 3.b odd 2 1
6498.2.a.x 1 57.d even 2 1

Hecke kernels

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

\( T_{5} + 4 \)
\( T_{7} + 3 \)
\( T_{13} + 7 \)
\( T_{29} - 4 \)

Hecke characteristic polynomials

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