Properties

Label 2166.2.a.h
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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} - 2q^{11} + q^{12} + 3q^{13} + q^{14} + q^{16} + 4q^{17} + q^{18} + q^{21} - 2q^{22} + 4q^{23} + q^{24} - 5q^{25} + 3q^{26} + q^{27} + q^{28} + 3q^{31} + q^{32} - 2q^{33} + 4q^{34} + q^{36} + 5q^{37} + 3q^{39} - 4q^{41} + q^{42} - 9q^{43} - 2q^{44} + 4q^{46} + 10q^{47} + q^{48} - 6q^{49} - 5q^{50} + 4q^{51} + 3q^{52} + 4q^{53} + q^{54} + q^{56} + 14q^{59} + 11q^{61} + 3q^{62} + q^{63} + q^{64} - 2q^{66} - 3q^{67} + 4q^{68} + 4q^{69} - 14q^{71} + q^{72} - 11q^{73} + 5q^{74} - 5q^{75} - 2q^{77} + 3q^{78} - q^{79} + q^{81} - 4q^{82} + 8q^{83} + q^{84} - 9q^{86} - 2q^{88} + 14q^{89} + 3q^{91} + 4q^{92} + 3q^{93} + 10q^{94} + q^{96} - 2q^{97} - 6q^{98} - 2q^{99} + O(q^{100}) \)

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 0 1.00000 1.00000 1.00000 1.00000 0
\(n\): 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.h 1
3.b odd 2 1 6498.2.a.g 1
19.b odd 2 1 2166.2.a.b 1
19.d odd 6 2 114.2.e.b 2
57.d even 2 1 6498.2.a.u 1
57.f even 6 2 342.2.g.c 2
76.f even 6 2 912.2.q.b 2
228.n odd 6 2 2736.2.s.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.b 2 19.d odd 6 2
342.2.g.c 2 57.f even 6 2
912.2.q.b 2 76.f even 6 2
2166.2.a.b 1 19.b odd 2 1
2166.2.a.h 1 1.a even 1 1 trivial
2736.2.s.k 2 228.n odd 6 2
6498.2.a.g 1 3.b odd 2 1
6498.2.a.u 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5} \)
\( T_{7} - 1 \)
\( T_{13} - 3 \)
\( T_{29} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( -1 + T \)
$11$ \( 2 + T \)
$13$ \( -3 + T \)
$17$ \( -4 + T \)
$19$ \( T \)
$23$ \( -4 + T \)
$29$ \( T \)
$31$ \( -3 + T \)
$37$ \( -5 + T \)
$41$ \( 4 + T \)
$43$ \( 9 + T \)
$47$ \( -10 + T \)
$53$ \( -4 + T \)
$59$ \( -14 + T \)
$61$ \( -11 + T \)
$67$ \( 3 + T \)
$71$ \( 14 + T \)
$73$ \( 11 + T \)
$79$ \( 1 + T \)
$83$ \( -8 + T \)
$89$ \( -14 + T \)
$97$ \( 2 + T \)
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