Properties

Label 2166.2.a.w
Level $2166$
Weight $2$
Character orbit 2166.a
Self dual yes
Analytic conductor $17.296$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \(x^{4} - 5 x^{2} + 5\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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
−1.90211
−1.17557
1.17557
1.90211
−1.00000 1.00000 1.00000 0.0978870 −1.00000 2.61803 −1.00000 1.00000 −0.0978870
1.2 −1.00000 1.00000 1.00000 0.824429 −1.00000 0.381966 −1.00000 1.00000 −0.824429
1.3 −1.00000 1.00000 1.00000 3.17557 −1.00000 0.381966 −1.00000 1.00000 −3.17557
1.4 −1.00000 1.00000 1.00000 3.90211 −1.00000 2.61803 −1.00000 1.00000 −3.90211
\(n\): 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.w 4
3.b odd 2 1 6498.2.a.by 4
19.b odd 2 1 2166.2.a.x yes 4
57.d even 2 1 6498.2.a.bv 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2166.2.a.w 4 1.a even 1 1 trivial
2166.2.a.x yes 4 19.b odd 2 1
6498.2.a.bv 4 57.d even 2 1
6498.2.a.by 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5}^{4} - 8 T_{5}^{3} + 19 T_{5}^{2} - 12 T_{5} + 1 \)
\( T_{7}^{2} - 3 T_{7} + 1 \)
\( T_{13}^{4} - 4 T_{13}^{3} - 24 T_{13}^{2} + 16 T_{13} + 16 \)
\( T_{29}^{4} + 10 T_{29}^{3} - 5 T_{29}^{2} - 210 T_{29} - 395 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 1 - 12 T + 19 T^{2} - 8 T^{3} + T^{4} \)
$7$ \( ( 1 - 3 T + T^{2} )^{2} \)
$11$ \( -79 - 8 T + 39 T^{2} - 12 T^{3} + T^{4} \)
$13$ \( 16 + 16 T - 24 T^{2} - 4 T^{3} + T^{4} \)
$17$ \( -304 + 256 T - 44 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( -64 + 32 T + 24 T^{2} - 12 T^{3} + T^{4} \)
$29$ \( -395 - 210 T - 5 T^{2} + 10 T^{3} + T^{4} \)
$31$ \( -1264 + 608 T - 56 T^{2} - 8 T^{3} + T^{4} \)
$37$ \( 976 - 288 T - 56 T^{2} + 8 T^{3} + T^{4} \)
$41$ \( 1616 + 368 T - 76 T^{2} - 8 T^{3} + T^{4} \)
$43$ \( -944 - 656 T - 104 T^{2} + 4 T^{3} + T^{4} \)
$47$ \( 976 + 336 T - 104 T^{2} - 4 T^{3} + T^{4} \)
$53$ \( -395 + 210 T - 5 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( -2179 + 1014 T - 109 T^{2} - 6 T^{3} + T^{4} \)
$61$ \( 1616 + 16 T - 104 T^{2} - 4 T^{3} + T^{4} \)
$67$ \( -3904 - 1632 T - 136 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( 3280 + 640 T - 200 T^{2} + T^{4} \)
$73$ \( 505 - 210 T - 65 T^{2} + 10 T^{3} + T^{4} \)
$79$ \( 461 + 688 T + 299 T^{2} + 32 T^{3} + T^{4} \)
$83$ \( 4201 + 508 T - 161 T^{2} - 8 T^{3} + T^{4} \)
$89$ \( 16 - 96 T + 76 T^{2} - 16 T^{3} + T^{4} \)
$97$ \( 10496 - 2688 T - 336 T^{2} + 8 T^{3} + T^{4} \)
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