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 $1$

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: $$x^{3} - 3 x - 1$$ 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.

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

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 −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: 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.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2166))$$:

 $$T_{5}^{3} - 9 T_{5} + 9$$ $$T_{7}^{3} + 3 T_{7}^{2} - 1$$ $$T_{13}^{3} - 6 T_{13}^{2} + 9 T_{13} - 3$$ $$T_{29}^{3} - 6 T_{29}^{2} - 45 T_{29} + 213$$

Hecke characteristic polynomials

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