# 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 $1$

# 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: $$4$$ 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: 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.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

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