Properties

Label 2166.4.a.f
Level $2166$
Weight $4$
Character orbit 2166.a
Self dual yes
Analytic conductor $127.798$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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

\(f(q)\) \(=\) \( q + 2q^{2} - 3q^{3} + 4q^{4} + 2q^{5} - 6q^{6} - 21q^{7} + 8q^{8} + 9q^{9} + O(q^{10}) \) \( q + 2q^{2} - 3q^{3} + 4q^{4} + 2q^{5} - 6q^{6} - 21q^{7} + 8q^{8} + 9q^{9} + 4q^{10} - 40q^{11} - 12q^{12} - 17q^{13} - 42q^{14} - 6q^{15} + 16q^{16} + 36q^{17} + 18q^{18} + 8q^{20} + 63q^{21} - 80q^{22} + 74q^{23} - 24q^{24} - 121q^{25} - 34q^{26} - 27q^{27} - 84q^{28} - 100q^{29} - 12q^{30} - 103q^{31} + 32q^{32} + 120q^{33} + 72q^{34} - 42q^{35} + 36q^{36} - 187q^{37} + 51q^{39} + 16q^{40} + 128q^{41} + 126q^{42} + 121q^{43} - 160q^{44} + 18q^{45} + 148q^{46} + 410q^{47} - 48q^{48} + 98q^{49} - 242q^{50} - 108q^{51} - 68q^{52} - 230q^{53} - 54q^{54} - 80q^{55} - 168q^{56} - 200q^{58} + 744q^{59} - 24q^{60} - 277q^{61} - 206q^{62} - 189q^{63} + 64q^{64} - 34q^{65} + 240q^{66} + 231q^{67} + 144q^{68} - 222q^{69} - 84q^{70} - 578q^{71} + 72q^{72} + 609q^{73} - 374q^{74} + 363q^{75} + 840q^{77} + 102q^{78} - 1259q^{79} + 32q^{80} + 81q^{81} + 256q^{82} - 696q^{83} + 252q^{84} + 72q^{85} + 242q^{86} + 300q^{87} - 320q^{88} + 612q^{89} + 36q^{90} + 357q^{91} + 296q^{92} + 309q^{93} + 820q^{94} - 96q^{96} + 1550q^{97} + 196q^{98} - 360q^{99} + O(q^{100}) \)

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 2.00000 −6.00000 −21.0000 8.00000 9.00000 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 2166.4.a.f 1
19.b odd 2 1 2166.4.a.c 1
19.d odd 6 2 114.4.e.a 2
57.f even 6 2 342.4.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.a 2 19.d odd 6 2
342.4.g.b 2 57.f even 6 2
2166.4.a.c 1 19.b odd 2 1
2166.4.a.f 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5} - 2 \)
\( T_{13} + 17 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( 3 + T \)
$5$ \( -2 + T \)
$7$ \( 21 + T \)
$11$ \( 40 + T \)
$13$ \( 17 + T \)
$17$ \( -36 + T \)
$19$ \( T \)
$23$ \( -74 + T \)
$29$ \( 100 + T \)
$31$ \( 103 + T \)
$37$ \( 187 + T \)
$41$ \( -128 + T \)
$43$ \( -121 + T \)
$47$ \( -410 + T \)
$53$ \( 230 + T \)
$59$ \( -744 + T \)
$61$ \( 277 + T \)
$67$ \( -231 + T \)
$71$ \( 578 + T \)
$73$ \( -609 + T \)
$79$ \( 1259 + T \)
$83$ \( 696 + T \)
$89$ \( -612 + T \)
$97$ \( -1550 + T \)
show more
show less