Properties

Label 2166.4.a.g
Level $2166$
Weight $4$
Character orbit 2166.a
Self dual yes
Analytic conductor $127.798$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.798137072\)
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

\(f(q)\) \(=\) \( q + 2q^{2} - 3q^{3} + 4q^{4} + 12q^{5} - 6q^{6} + 4q^{7} + 8q^{8} + 9q^{9} + O(q^{10}) \) \( q + 2q^{2} - 3q^{3} + 4q^{4} + 12q^{5} - 6q^{6} + 4q^{7} + 8q^{8} + 9q^{9} + 24q^{10} + 8q^{11} - 12q^{12} + 24q^{13} + 8q^{14} - 36q^{15} + 16q^{16} + 62q^{17} + 18q^{18} + 48q^{20} - 12q^{21} + 16q^{22} + 194q^{23} - 24q^{24} + 19q^{25} + 48q^{26} - 27q^{27} + 16q^{28} - 102q^{29} - 72q^{30} - 18q^{31} + 32q^{32} - 24q^{33} + 124q^{34} + 48q^{35} + 36q^{36} + 296q^{37} - 72q^{39} + 96q^{40} - 134q^{41} - 24q^{42} - 60q^{43} + 32q^{44} + 108q^{45} + 388q^{46} - 226q^{47} - 48q^{48} - 327q^{49} + 38q^{50} - 186q^{51} + 96q^{52} + 362q^{53} - 54q^{54} + 96q^{55} + 32q^{56} - 204q^{58} + 316q^{59} - 144q^{60} + 134q^{61} - 36q^{62} + 36q^{63} + 64q^{64} + 288q^{65} - 48q^{66} + 240q^{67} + 248q^{68} - 582q^{69} + 96q^{70} + 800q^{71} + 72q^{72} - 578q^{73} + 592q^{74} - 57q^{75} + 32q^{77} - 144q^{78} - 1078q^{79} + 192q^{80} + 81q^{81} - 268q^{82} + 940q^{83} - 48q^{84} + 744q^{85} - 120q^{86} + 306q^{87} + 64q^{88} - 170q^{89} + 216q^{90} + 96q^{91} + 776q^{92} + 54q^{93} - 452q^{94} - 96q^{96} - 206q^{97} - 654q^{98} + 72q^{99} + O(q^{100}) \)

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
2.00000 −3.00000 4.00000 12.0000 −6.00000 4.00000 8.00000 9.00000 24.0000
\(n\): 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 2166.4.a.g 1
19.b odd 2 1 114.4.a.c 1
57.d even 2 1 342.4.a.b 1
76.d even 2 1 912.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.a.c 1 19.b odd 2 1
342.4.a.b 1 57.d even 2 1
912.4.a.d 1 76.d even 2 1
2166.4.a.g 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5} - 12 \)
\( T_{13} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( 3 + T \)
$5$ \( -12 + T \)
$7$ \( -4 + T \)
$11$ \( -8 + T \)
$13$ \( -24 + T \)
$17$ \( -62 + T \)
$19$ \( T \)
$23$ \( -194 + T \)
$29$ \( 102 + T \)
$31$ \( 18 + T \)
$37$ \( -296 + T \)
$41$ \( 134 + T \)
$43$ \( 60 + T \)
$47$ \( 226 + T \)
$53$ \( -362 + T \)
$59$ \( -316 + T \)
$61$ \( -134 + T \)
$67$ \( -240 + T \)
$71$ \( -800 + T \)
$73$ \( 578 + T \)
$79$ \( 1078 + T \)
$83$ \( -940 + T \)
$89$ \( 170 + T \)
$97$ \( 206 + T \)
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