# 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 $1$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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