Properties

Label 2166.4.a.d
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

\(f(q)\) \(=\) \( q + 2q^{2} - 3q^{3} + 4q^{4} - 7q^{5} - 6q^{6} - 15q^{7} + 8q^{8} + 9q^{9} + O(q^{10}) \) \( q + 2q^{2} - 3q^{3} + 4q^{4} - 7q^{5} - 6q^{6} - 15q^{7} + 8q^{8} + 9q^{9} - 14q^{10} - 49q^{11} - 12q^{12} - 14q^{13} - 30q^{14} + 21q^{15} + 16q^{16} - 33q^{17} + 18q^{18} - 28q^{20} + 45q^{21} - 98q^{22} - 148q^{23} - 24q^{24} - 76q^{25} - 28q^{26} - 27q^{27} - 60q^{28} + 278q^{29} + 42q^{30} - 94q^{31} + 32q^{32} + 147q^{33} - 66q^{34} + 105q^{35} + 36q^{36} - 160q^{37} + 42q^{39} - 56q^{40} - 400q^{41} + 90q^{42} + 73q^{43} - 196q^{44} - 63q^{45} - 296q^{46} + 173q^{47} - 48q^{48} - 118q^{49} - 152q^{50} + 99q^{51} - 56q^{52} - 170q^{53} - 54q^{54} + 343q^{55} - 120q^{56} + 556q^{58} + 12q^{59} + 84q^{60} + 419q^{61} - 188q^{62} - 135q^{63} + 64q^{64} + 98q^{65} + 294q^{66} - 444q^{67} - 132q^{68} + 444q^{69} + 210q^{70} + 952q^{71} + 72q^{72} - 27q^{73} - 320q^{74} + 228q^{75} + 735q^{77} + 84q^{78} + 556q^{79} - 112q^{80} + 81q^{81} - 800q^{82} - 276q^{83} + 180q^{84} + 231q^{85} + 146q^{86} - 834q^{87} - 392q^{88} - 1386q^{89} - 126q^{90} + 210q^{91} - 592q^{92} + 282q^{93} + 346q^{94} - 96q^{96} - 130q^{97} - 236q^{98} - 441q^{99} + O(q^{100}) \)

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 −7.00000 −6.00000 −15.0000 8.00000 9.00000 −14.0000
\(n\): 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.d 1
19.b odd 2 1 114.4.a.b 1
57.d even 2 1 342.4.a.c 1
76.d even 2 1 912.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.a.b 1 19.b odd 2 1
342.4.a.c 1 57.d even 2 1
912.4.a.b 1 76.d even 2 1
2166.4.a.d 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5} + 7 \)
\( T_{13} + 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( 3 + T \)
$5$ \( 7 + T \)
$7$ \( 15 + T \)
$11$ \( 49 + T \)
$13$ \( 14 + T \)
$17$ \( 33 + T \)
$19$ \( T \)
$23$ \( 148 + T \)
$29$ \( -278 + T \)
$31$ \( 94 + T \)
$37$ \( 160 + T \)
$41$ \( 400 + T \)
$43$ \( -73 + T \)
$47$ \( -173 + T \)
$53$ \( 170 + T \)
$59$ \( -12 + T \)
$61$ \( -419 + T \)
$67$ \( 444 + T \)
$71$ \( -952 + T \)
$73$ \( 27 + T \)
$79$ \( -556 + T \)
$83$ \( 276 + T \)
$89$ \( 1386 + T \)
$97$ \( 130 + T \)
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