Properties

Label 2166.2.a.s
Level $2166$
Weight $2$
Character orbit 2166.a
Self dual yes
Analytic conductor $17.296$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

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
1.87939
−0.347296
−1.53209
1.00000 −1.00000 1.00000 −2.22668 −1.00000 −2.65270 1.00000 1.00000 −2.22668
1.2 1.00000 −1.00000 1.00000 −1.18479 −1.00000 −1.46791 1.00000 1.00000 −1.18479
1.3 1.00000 −1.00000 1.00000 3.41147 −1.00000 −4.87939 1.00000 1.00000 3.41147
\(n\): 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.s 3
3.b odd 2 1 6498.2.a.bm 3
19.b odd 2 1 2166.2.a.q 3
19.f odd 18 2 114.2.i.a 6
57.d even 2 1 6498.2.a.br 3
57.j even 18 2 342.2.u.e 6
76.k even 18 2 912.2.bo.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.a 6 19.f odd 18 2
342.2.u.e 6 57.j even 18 2
912.2.bo.a 6 76.k even 18 2
2166.2.a.q 3 19.b odd 2 1
2166.2.a.s 3 1.a even 1 1 trivial
6498.2.a.bm 3 3.b odd 2 1
6498.2.a.br 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5}^{3} - 9 T_{5} - 9 \)
\( T_{7}^{3} + 9 T_{7}^{2} + 24 T_{7} + 19 \)
\( T_{13}^{3} - 21 T_{13} - 37 \)
\( T_{29}^{3} + 18 T_{29}^{2} + 99 T_{29} + 171 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -9 - 9 T + T^{3} \)
$7$ \( 19 + 24 T + 9 T^{2} + T^{3} \)
$11$ \( 9 - 9 T + T^{3} \)
$13$ \( -37 - 21 T + T^{3} \)
$17$ \( 9 - 9 T + T^{3} \)
$19$ \( T^{3} \)
$23$ \( 9 - 9 T + T^{3} \)
$29$ \( 171 + 99 T + 18 T^{2} + T^{3} \)
$31$ \( -541 - 66 T + 9 T^{2} + T^{3} \)
$37$ \( -1 + 15 T + 9 T^{2} + T^{3} \)
$41$ \( -459 - 54 T + 9 T^{2} + T^{3} \)
$43$ \( -179 - 12 T + 9 T^{2} + T^{3} \)
$47$ \( 153 - 36 T - 9 T^{2} + T^{3} \)
$53$ \( -9 - 9 T + 9 T^{2} + T^{3} \)
$59$ \( 81 + 81 T + 18 T^{2} + T^{3} \)
$61$ \( 307 - 30 T - 9 T^{2} + T^{3} \)
$67$ \( 53 + 87 T + 18 T^{2} + T^{3} \)
$71$ \( 513 + 216 T + 27 T^{2} + T^{3} \)
$73$ \( -233 - 66 T + 9 T^{2} + T^{3} \)
$79$ \( 53 + 186 T + 27 T^{2} + T^{3} \)
$83$ \( 639 - 144 T - 9 T^{2} + T^{3} \)
$89$ \( 657 + 234 T + 27 T^{2} + T^{3} \)
$97$ \( 1637 - 195 T - 12 T^{2} + T^{3} \)
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