Properties

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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + ( - \beta_{2} - \beta_1) q^{5} + q^{6} + ( - \beta_1 - 1) q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + ( - \beta_{2} - \beta_1) q^{5} + q^{6} + ( - \beta_1 - 1) q^{7} + q^{8} + q^{9} + ( - \beta_{2} - \beta_1) q^{10} + (\beta_{2} - \beta_1) q^{11} + q^{12} + (\beta_{2} - \beta_1 + 2) q^{13} + ( - \beta_1 - 1) q^{14} + ( - \beta_{2} - \beta_1) q^{15} + q^{16} + (2 \beta_{2} + \beta_1 + 2) q^{17} + q^{18} + ( - \beta_{2} - \beta_1) q^{20} + ( - \beta_1 - 1) q^{21} + (\beta_{2} - \beta_1) q^{22} + ( - 3 \beta_{2} + 5 \beta_1 + 2) q^{23} + q^{24} + (3 \beta_1 + 1) q^{25} + (\beta_{2} - \beta_1 + 2) q^{26} + q^{27} + ( - \beta_1 - 1) q^{28} + ( - 3 \beta_{2} - 2 \beta_1 + 2) q^{29} + ( - \beta_{2} - \beta_1) q^{30} + ( - 3 \beta_{2} + 3) q^{31} + q^{32} + (\beta_{2} - \beta_1) q^{33} + (2 \beta_{2} + \beta_1 + 2) q^{34} + (2 \beta_{2} + 2 \beta_1 + 3) q^{35} + q^{36} + ( - 2 \beta_{2} + 6 \beta_1 + 3) q^{37} + (\beta_{2} - \beta_1 + 2) q^{39} + ( - \beta_{2} - \beta_1) q^{40} + (4 \beta_{2} - 3 \beta_1 + 3) q^{41} + ( - \beta_1 - 1) q^{42} + ( - 3 \beta_{2} + \beta_1 + 5) q^{43} + (\beta_{2} - \beta_1) q^{44} + ( - \beta_{2} - \beta_1) q^{45} + ( - 3 \beta_{2} + 5 \beta_1 + 2) q^{46} + (3 \beta_{2} + 1) q^{47} + q^{48} + (\beta_{2} + 2 \beta_1 - 4) q^{49} + (3 \beta_1 + 1) q^{50} + (2 \beta_{2} + \beta_1 + 2) q^{51} + (\beta_{2} - \beta_1 + 2) q^{52} + ( - 8 \beta_{2} + 6 \beta_1 - 1) q^{53} + q^{54} + (2 \beta_{2} - \beta_1) q^{55} + ( - \beta_1 - 1) q^{56} + ( - 3 \beta_{2} - 2 \beta_1 + 2) q^{58} + (6 \beta_{2} - 3 \beta_1 - 4) q^{59} + ( - \beta_{2} - \beta_1) q^{60} + ( - \beta_{2} - 11) q^{61} + ( - 3 \beta_{2} + 3) q^{62} + ( - \beta_1 - 1) q^{63} + q^{64} - 3 \beta_1 q^{65} + (\beta_{2} - \beta_1) q^{66} + (3 \beta_{2} - 4 \beta_1 + 4) q^{67} + (2 \beta_{2} + \beta_1 + 2) q^{68} + ( - 3 \beta_{2} + 5 \beta_1 + 2) q^{69} + (2 \beta_{2} + 2 \beta_1 + 3) q^{70} + ( - \beta_{2} + 7 \beta_1 + 5) q^{71} + q^{72} + (5 \beta_{2} + 2 \beta_1 + 1) q^{73} + ( - 2 \beta_{2} + 6 \beta_1 + 3) q^{74} + (3 \beta_1 + 1) q^{75} + q^{77} + (\beta_{2} - \beta_1 + 2) q^{78} + (10 \beta_{2} - 5 \beta_1 + 5) q^{79} + ( - \beta_{2} - \beta_1) q^{80} + q^{81} + (4 \beta_{2} - 3 \beta_1 + 3) q^{82} + (3 \beta_{2} - \beta_1 + 9) q^{83} + ( - \beta_1 - 1) q^{84} + ( - \beta_{2} - 7 \beta_1 - 9) q^{85} + ( - 3 \beta_{2} + \beta_1 + 5) q^{86} + ( - 3 \beta_{2} - 2 \beta_1 + 2) q^{87} + (\beta_{2} - \beta_1) q^{88} + (7 \beta_{2} - 4 \beta_1 - 5) q^{89} + ( - \beta_{2} - \beta_1) q^{90} + ( - 2 \beta_1 - 1) q^{91} + ( - 3 \beta_{2} + 5 \beta_1 + 2) q^{92} + ( - 3 \beta_{2} + 3) q^{93} + (3 \beta_{2} + 1) q^{94} + q^{96} + ( - \beta_{2} + \beta_1 - 4) q^{97} + (\beta_{2} + 2 \beta_1 - 4) q^{98} + (\beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{12} + 6 q^{13} - 3 q^{14} + 3 q^{16} + 6 q^{17} + 3 q^{18} - 3 q^{21} + 6 q^{23} + 3 q^{24} + 3 q^{25} + 6 q^{26} + 3 q^{27} - 3 q^{28} + 6 q^{29} + 9 q^{31} + 3 q^{32} + 6 q^{34} + 9 q^{35} + 3 q^{36} + 9 q^{37} + 6 q^{39} + 9 q^{41} - 3 q^{42} + 15 q^{43} + 6 q^{46} + 3 q^{47} + 3 q^{48} - 12 q^{49} + 3 q^{50} + 6 q^{51} + 6 q^{52} - 3 q^{53} + 3 q^{54} - 3 q^{56} + 6 q^{58} - 12 q^{59} - 33 q^{61} + 9 q^{62} - 3 q^{63} + 3 q^{64} + 12 q^{67} + 6 q^{68} + 6 q^{69} + 9 q^{70} + 15 q^{71} + 3 q^{72} + 3 q^{73} + 9 q^{74} + 3 q^{75} + 3 q^{77} + 6 q^{78} + 15 q^{79} + 3 q^{81} + 9 q^{82} + 27 q^{83} - 3 q^{84} - 27 q^{85} + 15 q^{86} + 6 q^{87} - 15 q^{89} - 3 q^{91} + 6 q^{92} + 9 q^{93} + 3 q^{94} + 3 q^{96} - 12 q^{97} - 12 q^{98}+O(q^{100}) \) 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
−1.53209
−0.347296
1.00000 1.00000 1.00000 −3.41147 1.00000 −2.87939 1.00000 1.00000 −3.41147
1.2 1.00000 1.00000 1.00000 1.18479 1.00000 0.532089 1.00000 1.00000 1.18479
1.3 1.00000 1.00000 1.00000 2.22668 1.00000 −0.652704 1.00000 1.00000 2.22668
\(n\): 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.u 3
3.b odd 2 1 6498.2.a.bn 3
19.b odd 2 1 2166.2.a.o 3
19.f odd 18 2 114.2.i.d 6
57.d even 2 1 6498.2.a.bs 3
57.j even 18 2 342.2.u.a 6
76.k even 18 2 912.2.bo.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.d 6 19.f odd 18 2
342.2.u.a 6 57.j even 18 2
912.2.bo.f 6 76.k even 18 2
2166.2.a.o 3 19.b odd 2 1
2166.2.a.u 3 1.a even 1 1 trivial
6498.2.a.bn 3 3.b odd 2 1
6498.2.a.bs 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5}^{3} - 9T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{3} + 3T_{7}^{2} - 1 \) Copy content Toggle raw display
\( T_{13}^{3} - 6T_{13}^{2} + 9T_{13} - 3 \) Copy content Toggle raw display
\( T_{29}^{3} - 6T_{29}^{2} - 45T_{29} + 213 \) 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
$5$ \( T^{3} - 9T + 9 \) Copy content Toggle raw display
$7$ \( T^{3} + 3T^{2} - 1 \) Copy content Toggle raw display
$11$ \( T^{3} - 3T - 1 \) Copy content Toggle raw display
$13$ \( T^{3} - 6 T^{2} + 9 T - 3 \) Copy content Toggle raw display
$17$ \( T^{3} - 6 T^{2} - 9 T + 17 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 6 T^{2} - 45 T + 269 \) Copy content Toggle raw display
$29$ \( T^{3} - 6 T^{2} - 45 T + 213 \) Copy content Toggle raw display
$31$ \( T^{3} - 9T^{2} + 27 \) Copy content Toggle raw display
$37$ \( T^{3} - 9 T^{2} - 57 T + 361 \) Copy content Toggle raw display
$41$ \( T^{3} - 9 T^{2} - 12 T + 109 \) Copy content Toggle raw display
$43$ \( T^{3} - 15 T^{2} + 54 T - 57 \) Copy content Toggle raw display
$47$ \( T^{3} - 3 T^{2} - 24 T + 53 \) Copy content Toggle raw display
$53$ \( T^{3} + 3 T^{2} - 153 T - 307 \) Copy content Toggle raw display
$59$ \( T^{3} + 12 T^{2} - 33 T - 17 \) Copy content Toggle raw display
$61$ \( T^{3} + 33 T^{2} + 360 T + 1297 \) Copy content Toggle raw display
$67$ \( T^{3} - 12 T^{2} + 9 T + 3 \) Copy content Toggle raw display
$71$ \( T^{3} - 15 T^{2} - 54 T + 449 \) Copy content Toggle raw display
$73$ \( T^{3} - 3 T^{2} - 114 T - 37 \) Copy content Toggle raw display
$79$ \( T^{3} - 15 T^{2} - 150 T + 2125 \) Copy content Toggle raw display
$83$ \( T^{3} - 27 T^{2} + 222 T - 503 \) Copy content Toggle raw display
$89$ \( T^{3} + 15 T^{2} - 36 T - 107 \) Copy content Toggle raw display
$97$ \( T^{3} + 12 T^{2} + 45 T + 53 \) Copy content Toggle raw display
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