# Properties

 Label 2166.2.a.n Level $2166$ Weight $2$ Character orbit 2166.a Self dual yes Analytic conductor $17.296$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# 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: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 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.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + ( - \beta_1 - 2) q^{5} + q^{6} + ( - \beta_{2} + 2 \beta_1 - 1) q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + (-b1 - 2) * q^5 + q^6 + (-b2 + 2*b1 - 1) * q^7 - q^8 + q^9 $$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})$$ q - q^2 - q^3 + q^4 + (-b1 - 2) * q^5 + q^6 + (-b2 + 2*b1 - 1) * q^7 - q^8 + q^9 + (b1 + 2) * q^10 + (b1 - 4) * q^11 - q^12 + (-4*b2 + 3*b1) * q^13 + (b2 - 2*b1 + 1) * q^14 + (b1 + 2) * q^15 + q^16 + (-b2 - 4) * q^17 - q^18 + (-b1 - 2) * q^20 + (b2 - 2*b1 + 1) * q^21 + (-b1 + 4) * q^22 + (-2*b2 - b1 - 4) * q^23 + q^24 + (b2 + 4*b1 + 1) * q^25 + (4*b2 - 3*b1) * q^26 - q^27 + (-b2 + 2*b1 - 1) * q^28 + (3*b2 + b1 + 2) * q^29 + (-b1 - 2) * q^30 + (-b2 - b1 + 1) * q^31 - q^32 + (-b1 + 4) * q^33 + (b2 + 4) * q^34 + (-2*b1 - 1) * q^35 + q^36 + (2*b2 - 2*b1 - 1) * q^37 + (4*b2 - 3*b1) * q^39 + (b1 + 2) * q^40 + (-3*b2 + 3) * q^41 + (-b2 + 2*b1 - 1) * q^42 + (-3*b1 - 1) * q^43 + (b1 - 4) * q^44 + (-b1 - 2) * q^45 + (2*b2 + b1 + 4) * q^46 + (b2 + 3*b1 - 5) * q^47 - q^48 + (5*b2 - 7*b1) * q^49 + (-b2 - 4*b1 - 1) * q^50 + (b2 + 4) * q^51 + (-4*b2 + 3*b1) * q^52 + (-4*b2 - 2*b1 + 1) * q^53 + q^54 + (-b2 + 2*b1 + 6) * q^55 + (b2 - 2*b1 + 1) * q^56 + (-3*b2 - b1 - 2) * q^58 + (5*b2 + 2*b1) * q^59 + (b1 + 2) * q^60 + (-3*b2 + 3*b1 - 1) * q^61 + (b2 + b1 - 1) * q^62 + (-b2 + 2*b1 - 1) * q^63 + q^64 + (5*b2 - 2*b1 - 2) * q^65 + (b1 - 4) * q^66 + (-3*b2 - b1 + 2) * q^67 + (-b2 - 4) * q^68 + (2*b2 + b1 + 4) * q^69 + (2*b1 + 1) * q^70 + (-3*b1 + 3) * q^71 - q^72 + (-7*b2 - 3*b1 + 1) * q^73 + (-2*b2 + 2*b1 + 1) * q^74 + (-b2 - 4*b1 - 1) * q^75 + (6*b2 - 10*b1 + 7) * q^77 + (-4*b2 + 3*b1) * q^78 + (5*b2 - 3) * q^79 + (-b1 - 2) * q^80 + q^81 + (3*b2 - 3) * q^82 + (-6*b2 + 11*b1 + 1) * q^83 + (b2 - 2*b1 + 1) * q^84 + (2*b2 + 5*b1 + 9) * q^85 + (3*b1 + 1) * q^86 + (-3*b2 - b1 - 2) * q^87 + (-b1 + 4) * q^88 + (-11*b2 + 5*b1 - 1) * q^89 + (b1 + 2) * q^90 + (6*b2 - 10*b1 + 9) * q^91 + (-2*b2 - b1 - 4) * q^92 + (b2 + b1 - 1) * q^93 + (-b2 - 3*b1 + 5) * q^94 + q^96 + (8*b2 - b1 + 4) * q^97 + (-5*b2 + 7*b1) * q^98 + (b1 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} + 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 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 $$3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} + 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9} + 6 q^{10} - 12 q^{11} - 3 q^{12} + 3 q^{14} + 6 q^{15} + 3 q^{16} - 12 q^{17} - 3 q^{18} - 6 q^{20} + 3 q^{21} + 12 q^{22} - 12 q^{23} + 3 q^{24} + 3 q^{25} - 3 q^{27} - 3 q^{28} + 6 q^{29} - 6 q^{30} + 3 q^{31} - 3 q^{32} + 12 q^{33} + 12 q^{34} - 3 q^{35} + 3 q^{36} - 3 q^{37} + 6 q^{40} + 9 q^{41} - 3 q^{42} - 3 q^{43} - 12 q^{44} - 6 q^{45} + 12 q^{46} - 15 q^{47} - 3 q^{48} - 3 q^{50} + 12 q^{51} + 3 q^{53} + 3 q^{54} + 18 q^{55} + 3 q^{56} - 6 q^{58} + 6 q^{60} - 3 q^{61} - 3 q^{62} - 3 q^{63} + 3 q^{64} - 6 q^{65} - 12 q^{66} + 6 q^{67} - 12 q^{68} + 12 q^{69} + 3 q^{70} + 9 q^{71} - 3 q^{72} + 3 q^{73} + 3 q^{74} - 3 q^{75} + 21 q^{77} - 9 q^{79} - 6 q^{80} + 3 q^{81} - 9 q^{82} + 3 q^{83} + 3 q^{84} + 27 q^{85} + 3 q^{86} - 6 q^{87} + 12 q^{88} - 3 q^{89} + 6 q^{90} + 27 q^{91} - 12 q^{92} - 3 q^{93} + 15 q^{94} + 3 q^{96} + 12 q^{97} - 12 q^{99}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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: 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.n 3
3.b odd 2 1 6498.2.a.bt 3
19.b odd 2 1 2166.2.a.t 3
19.f odd 18 2 114.2.i.b 6
57.d even 2 1 6498.2.a.bo 3
57.j even 18 2 342.2.u.d 6
76.k even 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.f odd 18 2
342.2.u.d 6 57.j even 18 2
912.2.bo.c 6 76.k even 18 2
2166.2.a.n 3 1.a even 1 1 trivial
2166.2.a.t 3 19.b odd 2 1
6498.2.a.bo 3 57.d even 2 1
6498.2.a.bt 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2166))$$:

 $$T_{5}^{3} + 6T_{5}^{2} + 9T_{5} + 3$$ T5^3 + 6*T5^2 + 9*T5 + 3 $$T_{7}^{3} + 3T_{7}^{2} - 6T_{7} + 1$$ T7^3 + 3*T7^2 - 6*T7 + 1 $$T_{13}^{3} - 39T_{13} - 19$$ T13^3 - 39*T13 - 19 $$T_{29}^{3} - 6T_{29}^{2} - 27T_{29} + 51$$ T29^3 - 6*T29^2 - 27*T29 + 51

## Hecke characteristic polynomials

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