Properties

Label 2166.2.a.r
Level $2166$
Weight $2$
Character orbit 2166.a
Self dual yes
Analytic conductor $17.296$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.2955970778\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + ( -2 + \beta_{1} ) q^{5} - q^{6} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + ( -2 + \beta_{1} ) q^{5} - q^{6} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{7} + q^{8} + q^{9} + ( -2 + \beta_{1} ) q^{10} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{11} - q^{12} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{13} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{14} + ( 2 - \beta_{1} ) q^{15} + q^{16} + ( -2 + \beta_{2} ) q^{17} + q^{18} + ( -2 + \beta_{1} ) q^{20} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{21} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{22} + ( 2 - 3 \beta_{1} + 6 \beta_{2} ) q^{23} - q^{24} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{25} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{26} - q^{27} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{28} + ( 2 + \beta_{1} - \beta_{2} ) q^{29} + ( 2 - \beta_{1} ) q^{30} + ( 5 + \beta_{1} - \beta_{2} ) q^{31} + q^{32} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{33} + ( -2 + \beta_{2} ) q^{34} + ( -7 + 4 \beta_{1} ) q^{35} + q^{36} + ( -1 - 4 \beta_{1} ) q^{37} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{39} + ( -2 + \beta_{1} ) q^{40} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{41} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{42} + ( 3 + 5 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{44} + ( -2 + \beta_{1} ) q^{45} + ( 2 - 3 \beta_{1} + 6 \beta_{2} ) q^{46} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{47} - q^{48} + ( 8 + \beta_{1} + \beta_{2} ) q^{49} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{50} + ( 2 - \beta_{2} ) q^{51} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{52} + ( 9 + 2 \beta_{1} ) q^{53} - q^{54} + ( 4 - 8 \beta_{1} + 7 \beta_{2} ) q^{55} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{56} + ( 2 + \beta_{1} - \beta_{2} ) q^{58} + ( 6 + 4 \beta_{1} + \beta_{2} ) q^{59} + ( 2 - \beta_{1} ) q^{60} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{61} + ( 5 + \beta_{1} - \beta_{2} ) q^{62} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{63} + q^{64} + ( -4 - 2 \beta_{1} + 5 \beta_{2} ) q^{65} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{66} + ( 4 - 5 \beta_{1} + 11 \beta_{2} ) q^{67} + ( -2 + \beta_{2} ) q^{68} + ( -2 + 3 \beta_{1} - 6 \beta_{2} ) q^{69} + ( -7 + 4 \beta_{1} ) q^{70} + ( 7 + 5 \beta_{1} - 2 \beta_{2} ) q^{71} + q^{72} + ( 5 + \beta_{1} - 3 \beta_{2} ) q^{73} + ( -1 - 4 \beta_{1} ) q^{74} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{75} + ( -7 + 6 \beta_{1} - 10 \beta_{2} ) q^{77} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{78} + ( 5 + 4 \beta_{1} - 5 \beta_{2} ) q^{79} + ( -2 + \beta_{1} ) q^{80} + q^{81} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{82} + ( -1 - \beta_{1} + 4 \beta_{2} ) q^{83} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{84} + ( 5 - \beta_{1} - 2 \beta_{2} ) q^{85} + ( 3 + 5 \beta_{1} - 2 \beta_{2} ) q^{86} + ( -2 - \beta_{1} + \beta_{2} ) q^{87} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{88} + ( 1 + 7 \beta_{1} - 7 \beta_{2} ) q^{89} + ( -2 + \beta_{1} ) q^{90} + ( 5 + 2 \beta_{1} - 8 \beta_{2} ) q^{91} + ( 2 - 3 \beta_{1} + 6 \beta_{2} ) q^{92} + ( -5 - \beta_{1} + \beta_{2} ) q^{93} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{94} - q^{96} + ( 4 - 7 \beta_{1} + 8 \beta_{2} ) q^{97} + ( 8 + \beta_{1} + \beta_{2} ) q^{98} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 6q^{5} - 3q^{6} + 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 6q^{5} - 3q^{6} + 3q^{7} + 3q^{8} + 3q^{9} - 6q^{10} - 3q^{12} + 6q^{13} + 3q^{14} + 6q^{15} + 3q^{16} - 6q^{17} + 3q^{18} - 6q^{20} - 3q^{21} + 6q^{23} - 3q^{24} + 3q^{25} + 6q^{26} - 3q^{27} + 3q^{28} + 6q^{29} + 6q^{30} + 15q^{31} + 3q^{32} - 6q^{34} - 21q^{35} + 3q^{36} - 3q^{37} - 6q^{39} - 6q^{40} + 9q^{41} - 3q^{42} + 9q^{43} - 6q^{45} + 6q^{46} - 9q^{47} - 3q^{48} + 24q^{49} + 3q^{50} + 6q^{51} + 6q^{52} + 27q^{53} - 3q^{54} + 12q^{55} + 3q^{56} + 6q^{58} + 18q^{59} + 6q^{60} + 3q^{61} + 15q^{62} + 3q^{63} + 3q^{64} - 12q^{65} + 12q^{67} - 6q^{68} - 6q^{69} - 21q^{70} + 21q^{71} + 3q^{72} + 15q^{73} - 3q^{74} - 3q^{75} - 21q^{77} - 6q^{78} + 15q^{79} - 6q^{80} + 3q^{81} + 9q^{82} - 3q^{83} - 3q^{84} + 15q^{85} + 9q^{86} - 6q^{87} + 3q^{89} - 6q^{90} + 15q^{91} + 6q^{92} - 15q^{93} - 9q^{94} - 3q^{96} + 12q^{97} + 24q^{98} + O(q^{100}) \)

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
−1.53209
−0.347296
1.87939
1.00000 −1.00000 1.00000 −3.53209 −1.00000 3.71688 1.00000 1.00000 −3.53209
1.2 1.00000 −1.00000 1.00000 −2.34730 −1.00000 3.57398 1.00000 1.00000 −2.34730
1.3 1.00000 −1.00000 1.00000 −0.120615 −1.00000 −4.29086 1.00000 1.00000 −0.120615
\(n\): 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 2166.2.a.r 3
3.b odd 2 1 6498.2.a.bp 3
19.b odd 2 1 2166.2.a.p 3
19.e even 9 2 114.2.i.c 6
57.d even 2 1 6498.2.a.bu 3
57.l odd 18 2 342.2.u.b 6
76.l odd 18 2 912.2.bo.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.c 6 19.e even 9 2
342.2.u.b 6 57.l odd 18 2
912.2.bo.d 6 76.l odd 18 2
2166.2.a.p 3 19.b odd 2 1
2166.2.a.r 3 1.a even 1 1 trivial
6498.2.a.bp 3 3.b odd 2 1
6498.2.a.bu 3 57.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5}^{3} + 6 T_{5}^{2} + 9 T_{5} + 1 \)
\( T_{7}^{3} - 3 T_{7}^{2} - 18 T_{7} + 57 \)
\( T_{13}^{3} - 6 T_{13}^{2} + 3 T_{13} + 1 \)
\( T_{29}^{3} - 6 T_{29}^{2} + 9 T_{29} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 1 + 9 T + 6 T^{2} + T^{3} \)
$7$ \( 57 - 18 T - 3 T^{2} + T^{3} \)
$11$ \( 37 - 21 T + T^{3} \)
$13$ \( 1 + 3 T - 6 T^{2} + T^{3} \)
$17$ \( 3 + 9 T + 6 T^{2} + T^{3} \)
$19$ \( T^{3} \)
$23$ \( 397 - 69 T - 6 T^{2} + T^{3} \)
$29$ \( -1 + 9 T - 6 T^{2} + T^{3} \)
$31$ \( -109 + 72 T - 15 T^{2} + T^{3} \)
$37$ \( 17 - 45 T + 3 T^{2} + T^{3} \)
$41$ \( 53 + 6 T - 9 T^{2} + T^{3} \)
$43$ \( 251 - 30 T - 9 T^{2} + T^{3} \)
$47$ \( -71 - 12 T + 9 T^{2} + T^{3} \)
$53$ \( -629 + 231 T - 27 T^{2} + T^{3} \)
$59$ \( -9 + 45 T - 18 T^{2} + T^{3} \)
$61$ \( -17 - 18 T - 3 T^{2} + T^{3} \)
$67$ \( 2649 - 225 T - 12 T^{2} + T^{3} \)
$71$ \( 163 + 90 T - 21 T^{2} + T^{3} \)
$73$ \( -57 + 54 T - 15 T^{2} + T^{3} \)
$79$ \( 181 + 12 T - 15 T^{2} + T^{3} \)
$83$ \( 51 - 36 T + 3 T^{2} + T^{3} \)
$89$ \( 489 - 144 T - 3 T^{2} + T^{3} \)
$97$ \( 467 - 123 T - 12 T^{2} + T^{3} \)
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