Properties

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

Newform invariants

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

\(f(q)\) \(=\) \( q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{5} - 6q^{6} + 19q^{7} - 8q^{8} + 9q^{9} + O(q^{10}) \) \( q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{5} - 6q^{6} + 19q^{7} - 8q^{8} + 9q^{9} + 12q^{10} + 32q^{11} + 12q^{12} + 81q^{13} - 38q^{14} - 18q^{15} + 16q^{16} - 124q^{17} - 18q^{18} - 24q^{20} + 57q^{21} - 64q^{22} + 98q^{23} - 24q^{24} - 89q^{25} - 162q^{26} + 27q^{27} + 76q^{28} - 300q^{29} + 36q^{30} - 225q^{31} - 32q^{32} + 96q^{33} + 248q^{34} - 114q^{35} + 36q^{36} - 293q^{37} + 243q^{39} + 48q^{40} + 176q^{41} - 114q^{42} - 111q^{43} + 128q^{44} - 54q^{45} - 196q^{46} - 550q^{47} + 48q^{48} + 18q^{49} + 178q^{50} - 372q^{51} + 324q^{52} - 482q^{53} - 54q^{54} - 192q^{55} - 152q^{56} + 600q^{58} - 496q^{59} - 72q^{60} + 155q^{61} + 450q^{62} + 171q^{63} + 64q^{64} - 486q^{65} - 192q^{66} + 465q^{67} - 496q^{68} + 294q^{69} + 228q^{70} - 110q^{71} - 72q^{72} + 817q^{73} + 586q^{74} - 267q^{75} + 608q^{77} - 486q^{78} + 259q^{79} - 96q^{80} + 81q^{81} - 352q^{82} - 56q^{83} + 228q^{84} + 744q^{85} + 222q^{86} - 900q^{87} - 256q^{88} + 308q^{89} + 108q^{90} + 1539q^{91} + 392q^{92} - 675q^{93} + 1100q^{94} - 96q^{96} - 1150q^{97} - 36q^{98} + 288q^{99} + O(q^{100}) \)

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 −6.00000 −6.00000 19.0000 −8.00000 9.00000 12.0000
\(n\): 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.b 1
19.b odd 2 1 2166.4.a.e 1
19.c even 3 2 114.4.e.b 2
57.h odd 6 2 342.4.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.b 2 19.c even 3 2
342.4.g.a 2 57.h odd 6 2
2166.4.a.b 1 1.a even 1 1 trivial
2166.4.a.e 1 19.b odd 2 1

Hecke kernels

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

\( T_{5} + 6 \)
\( T_{13} - 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( -3 + T \)
$5$ \( 6 + T \)
$7$ \( -19 + T \)
$11$ \( -32 + T \)
$13$ \( -81 + T \)
$17$ \( 124 + T \)
$19$ \( T \)
$23$ \( -98 + T \)
$29$ \( 300 + T \)
$31$ \( 225 + T \)
$37$ \( 293 + T \)
$41$ \( -176 + T \)
$43$ \( 111 + T \)
$47$ \( 550 + T \)
$53$ \( 482 + T \)
$59$ \( 496 + T \)
$61$ \( -155 + T \)
$67$ \( -465 + T \)
$71$ \( 110 + T \)
$73$ \( -817 + T \)
$79$ \( -259 + T \)
$83$ \( 56 + T \)
$89$ \( -308 + T \)
$97$ \( 1150 + T \)
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