Properties

Label 2166.2.a.a
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} - 4q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} + q^{6} - 4q^{7} - q^{8} + q^{9} - q^{12} + 4q^{13} + 4q^{14} + q^{16} + 6q^{17} - q^{18} + 4q^{21} - 6q^{23} + q^{24} - 5q^{25} - 4q^{26} - q^{27} - 4q^{28} - 6q^{29} - 2q^{31} - q^{32} - 6q^{34} + q^{36} + 4q^{37} - 4q^{39} - 6q^{41} - 4q^{42} - 4q^{43} + 6q^{46} + 6q^{47} - q^{48} + 9q^{49} + 5q^{50} - 6q^{51} + 4q^{52} - 6q^{53} + q^{54} + 4q^{56} + 6q^{58} + 12q^{59} + 14q^{61} + 2q^{62} - 4q^{63} + q^{64} - 8q^{67} + 6q^{68} + 6q^{69} - q^{72} + 14q^{73} - 4q^{74} + 5q^{75} + 4q^{78} + 10q^{79} + q^{81} + 6q^{82} - 12q^{83} + 4q^{84} + 4q^{86} + 6q^{87} + 6q^{89} - 16q^{91} - 6q^{92} + 2q^{93} - 6q^{94} + q^{96} + 10q^{97} - 9q^{98} + O(q^{100}) \)

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 −4.00000 −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.a 1
3.b odd 2 1 6498.2.a.t 1
19.b odd 2 1 114.2.a.c 1
57.d even 2 1 342.2.a.c 1
76.d even 2 1 912.2.a.c 1
95.d odd 2 1 2850.2.a.g 1
95.g even 4 2 2850.2.d.p 2
133.c even 2 1 5586.2.a.u 1
152.b even 2 1 3648.2.a.bc 1
152.g odd 2 1 3648.2.a.i 1
228.b odd 2 1 2736.2.a.o 1
285.b even 2 1 8550.2.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.c 1 19.b odd 2 1
342.2.a.c 1 57.d even 2 1
912.2.a.c 1 76.d even 2 1
2166.2.a.a 1 1.a even 1 1 trivial
2736.2.a.o 1 228.b odd 2 1
2850.2.a.g 1 95.d odd 2 1
2850.2.d.p 2 95.g even 4 2
3648.2.a.i 1 152.g odd 2 1
3648.2.a.bc 1 152.b even 2 1
5586.2.a.u 1 133.c even 2 1
6498.2.a.t 1 3.b odd 2 1
8550.2.a.bj 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5} \)
\( T_{7} + 4 \)
\( T_{13} - 4 \)
\( T_{29} + 6 \)

Hecke characteristic polynomials

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