Properties

Label 2166.4.a.h
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

\(f(q)\) \(=\) \( q + 2q^{2} + 3q^{3} + 4q^{4} - 19q^{5} + 6q^{6} + 9q^{7} + 8q^{8} + 9q^{9} + O(q^{10}) \) \( q + 2q^{2} + 3q^{3} + 4q^{4} - 19q^{5} + 6q^{6} + 9q^{7} + 8q^{8} + 9q^{9} - 38q^{10} - 13q^{11} + 12q^{12} - 38q^{13} + 18q^{14} - 57q^{15} + 16q^{16} + 99q^{17} + 18q^{18} - 76q^{20} + 27q^{21} - 26q^{22} + 68q^{23} + 24q^{24} + 236q^{25} - 76q^{26} + 27q^{27} + 36q^{28} - 130q^{29} - 114q^{30} - 262q^{31} + 32q^{32} - 39q^{33} + 198q^{34} - 171q^{35} + 36q^{36} + 296q^{37} - 114q^{39} - 152q^{40} + 8q^{41} + 54q^{42} + 73q^{43} - 52q^{44} - 171q^{45} + 136q^{46} - 271q^{47} + 48q^{48} - 262q^{49} + 472q^{50} + 297q^{51} - 152q^{52} + 502q^{53} + 54q^{54} + 247q^{55} + 72q^{56} - 260q^{58} - 540q^{59} - 228q^{60} + 587q^{61} - 524q^{62} + 81q^{63} + 64q^{64} + 722q^{65} - 78q^{66} - 684q^{67} + 396q^{68} + 204q^{69} - 342q^{70} - 992q^{71} + 72q^{72} - 507q^{73} + 592q^{74} + 708q^{75} - 117q^{77} - 228q^{78} - 980q^{79} - 304q^{80} + 81q^{81} + 16q^{82} - 492q^{83} + 108q^{84} - 1881q^{85} + 146q^{86} - 390q^{87} - 104q^{88} - 810q^{89} - 342q^{90} - 342q^{91} + 272q^{92} - 786q^{93} - 542q^{94} + 96q^{96} + 1046q^{97} - 524q^{98} - 117q^{99} + O(q^{100}) \)

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 −19.0000 6.00000 9.00000 8.00000 9.00000 −38.0000
\(n\): 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.h 1
19.b odd 2 1 114.4.a.a 1
57.d even 2 1 342.4.a.e 1
76.d even 2 1 912.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.a.a 1 19.b odd 2 1
342.4.a.e 1 57.d even 2 1
912.4.a.e 1 76.d even 2 1
2166.4.a.h 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5} + 19 \)
\( T_{13} + 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( -3 + T \)
$5$ \( 19 + T \)
$7$ \( -9 + T \)
$11$ \( 13 + T \)
$13$ \( 38 + T \)
$17$ \( -99 + T \)
$19$ \( T \)
$23$ \( -68 + T \)
$29$ \( 130 + T \)
$31$ \( 262 + T \)
$37$ \( -296 + T \)
$41$ \( -8 + T \)
$43$ \( -73 + T \)
$47$ \( 271 + T \)
$53$ \( -502 + T \)
$59$ \( 540 + T \)
$61$ \( -587 + T \)
$67$ \( 684 + T \)
$71$ \( 992 + T \)
$73$ \( 507 + T \)
$79$ \( 980 + T \)
$83$ \( 492 + T \)
$89$ \( 810 + T \)
$97$ \( -1046 + T \)
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