Properties

Label 2166.2.a.k
Level $2166$
Weight $2$
Character orbit 2166.a
Self dual yes
Analytic conductor $17.296$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−1.00000 1.00000 1.00000 −1.61803 −1.00000 0 −1.00000 1.00000 1.61803
1.2 −1.00000 1.00000 1.00000 0.618034 −1.00000 0 −1.00000 1.00000 −0.618034
\(n\): 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.k 2
3.b odd 2 1 6498.2.a.bj 2
19.b odd 2 1 2166.2.a.l yes 2
57.d even 2 1 6498.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2166.2.a.k 2 1.a even 1 1 trivial
2166.2.a.l yes 2 19.b odd 2 1
6498.2.a.bd 2 57.d even 2 1
6498.2.a.bj 2 3.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + T_{5} - 1 \)
\( T_{7} \)
\( T_{13}^{2} - T_{13} - 1 \)
\( T_{29}^{2} + 11 T_{29} + 19 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -1 - T + T^{2} \)
$17$ \( -29 - 3 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( 19 + 11 T + T^{2} \)
$31$ \( 16 - 12 T + T^{2} \)
$37$ \( -5 - 5 T + T^{2} \)
$41$ \( -5 - 5 T + T^{2} \)
$43$ \( 16 + 12 T + T^{2} \)
$47$ \( -64 + 8 T + T^{2} \)
$53$ \( 55 + 15 T + T^{2} \)
$59$ \( -16 + 4 T + T^{2} \)
$61$ \( 71 + 17 T + T^{2} \)
$67$ \( -176 - 4 T + T^{2} \)
$71$ \( -16 - 4 T + T^{2} \)
$73$ \( -209 - 3 T + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( 131 + 23 T + T^{2} \)
$97$ \( -145 - 5 T + T^{2} \)
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