Properties

Label 2166.2.a.t
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: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + ( - \beta_1 - 2) q^{5} + q^{6} + ( - \beta_{2} + 2 \beta_1 - 1) q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + ( - \beta_1 - 2) q^{5} + q^{6} + ( - \beta_{2} + 2 \beta_1 - 1) q^{7} + q^{8} + q^{9} + ( - \beta_1 - 2) q^{10} + (\beta_1 - 4) q^{11} + q^{12} + (4 \beta_{2} - 3 \beta_1) q^{13} + ( - \beta_{2} + 2 \beta_1 - 1) q^{14} + ( - \beta_1 - 2) q^{15} + q^{16} + ( - \beta_{2} - 4) q^{17} + q^{18} + ( - \beta_1 - 2) q^{20} + ( - \beta_{2} + 2 \beta_1 - 1) q^{21} + (\beta_1 - 4) q^{22} + ( - 2 \beta_{2} - \beta_1 - 4) q^{23} + q^{24} + (\beta_{2} + 4 \beta_1 + 1) q^{25} + (4 \beta_{2} - 3 \beta_1) q^{26} + q^{27} + ( - \beta_{2} + 2 \beta_1 - 1) q^{28} + ( - 3 \beta_{2} - \beta_1 - 2) q^{29} + ( - \beta_1 - 2) q^{30} + (\beta_{2} + \beta_1 - 1) q^{31} + q^{32} + (\beta_1 - 4) q^{33} + ( - \beta_{2} - 4) q^{34} + ( - 2 \beta_1 - 1) q^{35} + q^{36} + ( - 2 \beta_{2} + 2 \beta_1 + 1) q^{37} + (4 \beta_{2} - 3 \beta_1) q^{39} + ( - \beta_1 - 2) q^{40} + (3 \beta_{2} - 3) q^{41} + ( - \beta_{2} + 2 \beta_1 - 1) q^{42} + ( - 3 \beta_1 - 1) q^{43} + (\beta_1 - 4) q^{44} + ( - \beta_1 - 2) q^{45} + ( - 2 \beta_{2} - \beta_1 - 4) q^{46} + (\beta_{2} + 3 \beta_1 - 5) q^{47} + q^{48} + (5 \beta_{2} - 7 \beta_1) q^{49} + (\beta_{2} + 4 \beta_1 + 1) q^{50} + ( - \beta_{2} - 4) q^{51} + (4 \beta_{2} - 3 \beta_1) q^{52} + (4 \beta_{2} + 2 \beta_1 - 1) q^{53} + q^{54} + ( - \beta_{2} + 2 \beta_1 + 6) q^{55} + ( - \beta_{2} + 2 \beta_1 - 1) q^{56} + ( - 3 \beta_{2} - \beta_1 - 2) q^{58} + ( - 5 \beta_{2} - 2 \beta_1) q^{59} + ( - \beta_1 - 2) q^{60} + ( - 3 \beta_{2} + 3 \beta_1 - 1) q^{61} + (\beta_{2} + \beta_1 - 1) q^{62} + ( - \beta_{2} + 2 \beta_1 - 1) q^{63} + q^{64} + ( - 5 \beta_{2} + 2 \beta_1 + 2) q^{65} + (\beta_1 - 4) q^{66} + (3 \beta_{2} + \beta_1 - 2) q^{67} + ( - \beta_{2} - 4) q^{68} + ( - 2 \beta_{2} - \beta_1 - 4) q^{69} + ( - 2 \beta_1 - 1) q^{70} + (3 \beta_1 - 3) q^{71} + q^{72} + ( - 7 \beta_{2} - 3 \beta_1 + 1) q^{73} + ( - 2 \beta_{2} + 2 \beta_1 + 1) q^{74} + (\beta_{2} + 4 \beta_1 + 1) q^{75} + (6 \beta_{2} - 10 \beta_1 + 7) q^{77} + (4 \beta_{2} - 3 \beta_1) q^{78} + ( - 5 \beta_{2} + 3) q^{79} + ( - \beta_1 - 2) q^{80} + q^{81} + (3 \beta_{2} - 3) q^{82} + ( - 6 \beta_{2} + 11 \beta_1 + 1) q^{83} + ( - \beta_{2} + 2 \beta_1 - 1) q^{84} + (2 \beta_{2} + 5 \beta_1 + 9) q^{85} + ( - 3 \beta_1 - 1) q^{86} + ( - 3 \beta_{2} - \beta_1 - 2) q^{87} + (\beta_1 - 4) q^{88} + (11 \beta_{2} - 5 \beta_1 + 1) q^{89} + ( - \beta_1 - 2) q^{90} + ( - 6 \beta_{2} + 10 \beta_1 - 9) q^{91} + ( - 2 \beta_{2} - \beta_1 - 4) q^{92} + (\beta_{2} + \beta_1 - 1) q^{93} + (\beta_{2} + 3 \beta_1 - 5) q^{94} + q^{96} + ( - 8 \beta_{2} + \beta_1 - 4) q^{97} + (5 \beta_{2} - 7 \beta_1) q^{98} + (\beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 6 q^{5} + 3 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 6 q^{5} + 3 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 6 q^{10} - 12 q^{11} + 3 q^{12} - 3 q^{14} - 6 q^{15} + 3 q^{16} - 12 q^{17} + 3 q^{18} - 6 q^{20} - 3 q^{21} - 12 q^{22} - 12 q^{23} + 3 q^{24} + 3 q^{25} + 3 q^{27} - 3 q^{28} - 6 q^{29} - 6 q^{30} - 3 q^{31} + 3 q^{32} - 12 q^{33} - 12 q^{34} - 3 q^{35} + 3 q^{36} + 3 q^{37} - 6 q^{40} - 9 q^{41} - 3 q^{42} - 3 q^{43} - 12 q^{44} - 6 q^{45} - 12 q^{46} - 15 q^{47} + 3 q^{48} + 3 q^{50} - 12 q^{51} - 3 q^{53} + 3 q^{54} + 18 q^{55} - 3 q^{56} - 6 q^{58} - 6 q^{60} - 3 q^{61} - 3 q^{62} - 3 q^{63} + 3 q^{64} + 6 q^{65} - 12 q^{66} - 6 q^{67} - 12 q^{68} - 12 q^{69} - 3 q^{70} - 9 q^{71} + 3 q^{72} + 3 q^{73} + 3 q^{74} + 3 q^{75} + 21 q^{77} + 9 q^{79} - 6 q^{80} + 3 q^{81} - 9 q^{82} + 3 q^{83} - 3 q^{84} + 27 q^{85} - 3 q^{86} - 6 q^{87} - 12 q^{88} + 3 q^{89} - 6 q^{90} - 27 q^{91} - 12 q^{92} - 3 q^{93} - 15 q^{94} + 3 q^{96} - 12 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

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 −3.87939 1.00000 1.22668 1.00000 1.00000 −3.87939
1.2 1.00000 1.00000 1.00000 −1.65270 1.00000 0.184793 1.00000 1.00000 −1.65270
1.3 1.00000 1.00000 1.00000 −0.467911 1.00000 −4.41147 1.00000 1.00000 −0.467911
\(n\): 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.t 3
3.b odd 2 1 6498.2.a.bo 3
19.b odd 2 1 2166.2.a.n 3
19.e even 9 2 114.2.i.b 6
57.d even 2 1 6498.2.a.bt 3
57.l odd 18 2 342.2.u.d 6
76.l odd 18 2 912.2.bo.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.b 6 19.e even 9 2
342.2.u.d 6 57.l odd 18 2
912.2.bo.c 6 76.l odd 18 2
2166.2.a.n 3 19.b odd 2 1
2166.2.a.t 3 1.a even 1 1 trivial
6498.2.a.bo 3 3.b odd 2 1
6498.2.a.bt 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5}^{3} + 6T_{5}^{2} + 9T_{5} + 3 \) Copy content Toggle raw display
\( T_{7}^{3} + 3T_{7}^{2} - 6T_{7} + 1 \) Copy content Toggle raw display
\( T_{13}^{3} - 39T_{13} + 19 \) Copy content Toggle raw display
\( T_{29}^{3} + 6T_{29}^{2} - 27T_{29} - 51 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 6 T^{2} + 9 T + 3 \) Copy content Toggle raw display
$7$ \( T^{3} + 3 T^{2} - 6 T + 1 \) Copy content Toggle raw display
$11$ \( T^{3} + 12 T^{2} + 45 T + 51 \) Copy content Toggle raw display
$13$ \( T^{3} - 39T + 19 \) Copy content Toggle raw display
$17$ \( T^{3} + 12 T^{2} + 45 T + 51 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 12 T^{2} + 27 T - 3 \) Copy content Toggle raw display
$29$ \( T^{3} + 6 T^{2} - 27 T - 51 \) Copy content Toggle raw display
$31$ \( T^{3} + 3 T^{2} - 6 T - 17 \) Copy content Toggle raw display
$37$ \( T^{3} - 3 T^{2} - 9 T + 19 \) Copy content Toggle raw display
$41$ \( T^{3} + 9T^{2} - 27 \) Copy content Toggle raw display
$43$ \( T^{3} + 3 T^{2} - 24 T + 1 \) Copy content Toggle raw display
$47$ \( T^{3} + 15 T^{2} + 36 T - 159 \) Copy content Toggle raw display
$53$ \( T^{3} + 3 T^{2} - 81 T - 219 \) Copy content Toggle raw display
$59$ \( T^{3} - 117T + 153 \) Copy content Toggle raw display
$61$ \( T^{3} + 3 T^{2} - 24 T + 1 \) Copy content Toggle raw display
$67$ \( T^{3} + 6 T^{2} - 27 T - 89 \) Copy content Toggle raw display
$71$ \( T^{3} + 9T^{2} - 81 \) Copy content Toggle raw display
$73$ \( T^{3} - 3 T^{2} - 234 T + 739 \) Copy content Toggle raw display
$79$ \( T^{3} - 9 T^{2} - 48 T + 73 \) Copy content Toggle raw display
$83$ \( T^{3} - 3 T^{2} - 270 T + 1893 \) Copy content Toggle raw display
$89$ \( T^{3} - 3 T^{2} - 270 T + 1893 \) Copy content Toggle raw display
$97$ \( T^{3} + 12 T^{2} - 123 T - 1277 \) Copy content Toggle raw display
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