# Properties

 Label 2166.4.a.s Level $2166$ Weight $4$ Character orbit 2166.a Self dual yes Analytic conductor $127.798$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2166 = 2 \cdot 3 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2166.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$127.798137072$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1524.1 Defining polynomial: $$x^{3} - x^{2} - 7 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 3$$ 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.

 $$f(q)$$ $$=$$ $$q -2 q^{2} -3 q^{3} + 4 q^{4} + ( 1 - \beta_{2} ) q^{5} + 6 q^{6} + ( -6 - 2 \beta_{1} + \beta_{2} ) q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q -2 q^{2} -3 q^{3} + 4 q^{4} + ( 1 - \beta_{2} ) q^{5} + 6 q^{6} + ( -6 - 2 \beta_{1} + \beta_{2} ) q^{7} -8 q^{8} + 9 q^{9} + ( -2 + 2 \beta_{2} ) q^{10} + ( -18 + \beta_{1} + 2 \beta_{2} ) q^{11} -12 q^{12} + ( 25 - 2 \beta_{1} ) q^{13} + ( 12 + 4 \beta_{1} - 2 \beta_{2} ) q^{14} + ( -3 + 3 \beta_{2} ) q^{15} + 16 q^{16} + ( -17 - \beta_{1} + 3 \beta_{2} ) q^{17} -18 q^{18} + ( 4 - 4 \beta_{2} ) q^{20} + ( 18 + 6 \beta_{1} - 3 \beta_{2} ) q^{21} + ( 36 - 2 \beta_{1} - 4 \beta_{2} ) q^{22} + ( -81 + 10 \beta_{1} + 5 \beta_{2} ) q^{23} + 24 q^{24} + ( 80 + 7 \beta_{1} - 11 \beta_{2} ) q^{25} + ( -50 + 4 \beta_{1} ) q^{26} -27 q^{27} + ( -24 - 8 \beta_{1} + 4 \beta_{2} ) q^{28} + ( -3 - 13 \beta_{1} + \beta_{2} ) q^{29} + ( 6 - 6 \beta_{2} ) q^{30} + ( 33 - 3 \beta_{1} + 8 \beta_{2} ) q^{31} -32 q^{32} + ( 54 - 3 \beta_{1} - 6 \beta_{2} ) q^{33} + ( 34 + 2 \beta_{1} - 6 \beta_{2} ) q^{34} + ( -104 + 11 \beta_{1} + 18 \beta_{2} ) q^{35} + 36 q^{36} + ( 94 - 13 \beta_{1} + 23 \beta_{2} ) q^{37} + ( -75 + 6 \beta_{1} ) q^{39} + ( -8 + 8 \beta_{2} ) q^{40} + ( 4 - 26 \beta_{1} + 4 \beta_{2} ) q^{41} + ( -36 - 12 \beta_{1} + 6 \beta_{2} ) q^{42} + ( -108 - 18 \beta_{1} - 7 \beta_{2} ) q^{43} + ( -72 + 4 \beta_{1} + 8 \beta_{2} ) q^{44} + ( 9 - 9 \beta_{2} ) q^{45} + ( 162 - 20 \beta_{1} - 10 \beta_{2} ) q^{46} + ( -260 - 18 \beta_{1} + 14 \beta_{2} ) q^{47} -48 q^{48} + ( 377 - 13 \beta_{1} + 11 \beta_{2} ) q^{49} + ( -160 - 14 \beta_{1} + 22 \beta_{2} ) q^{50} + ( 51 + 3 \beta_{1} - 9 \beta_{2} ) q^{51} + ( 100 - 8 \beta_{1} ) q^{52} + ( -30 + 13 \beta_{1} - 28 \beta_{2} ) q^{53} + 54 q^{54} + ( -479 - 23 \beta_{1} + 37 \beta_{2} ) q^{55} + ( 48 + 16 \beta_{1} - 8 \beta_{2} ) q^{56} + ( 6 + 26 \beta_{1} - 2 \beta_{2} ) q^{58} + ( 315 + 24 \beta_{1} - 9 \beta_{2} ) q^{59} + ( -12 + 12 \beta_{2} ) q^{60} + ( -123 + 2 \beta_{1} - 30 \beta_{2} ) q^{61} + ( -66 + 6 \beta_{1} - 16 \beta_{2} ) q^{62} + ( -54 - 18 \beta_{1} + 9 \beta_{2} ) q^{63} + 64 q^{64} + ( 131 + 18 \beta_{1} - 23 \beta_{2} ) q^{65} + ( -108 + 6 \beta_{1} + 12 \beta_{2} ) q^{66} + ( 29 + 5 \beta_{1} - 26 \beta_{2} ) q^{67} + ( -68 - 4 \beta_{1} + 12 \beta_{2} ) q^{68} + ( 243 - 30 \beta_{1} - 15 \beta_{2} ) q^{69} + ( 208 - 22 \beta_{1} - 36 \beta_{2} ) q^{70} + ( 335 - \beta_{1} - 31 \beta_{2} ) q^{71} -72 q^{72} + ( 35 - 40 \beta_{1} - 14 \beta_{2} ) q^{73} + ( -188 + 26 \beta_{1} - 46 \beta_{2} ) q^{74} + ( -240 - 21 \beta_{1} + 33 \beta_{2} ) q^{75} + ( 11 + 16 \beta_{1} - 69 \beta_{2} ) q^{77} + ( 150 - 12 \beta_{1} ) q^{78} + ( -101 + 43 \beta_{1} - 18 \beta_{2} ) q^{79} + ( 16 - 16 \beta_{2} ) q^{80} + 81 q^{81} + ( -8 + 52 \beta_{1} - 8 \beta_{2} ) q^{82} + ( -709 - 17 \beta_{1} - 21 \beta_{2} ) q^{83} + ( 72 + 24 \beta_{1} - 12 \beta_{2} ) q^{84} + ( -576 - 12 \beta_{1} + 48 \beta_{2} ) q^{85} + ( 216 + 36 \beta_{1} + 14 \beta_{2} ) q^{86} + ( 9 + 39 \beta_{1} - 3 \beta_{2} ) q^{87} + ( 144 - 8 \beta_{1} - 16 \beta_{2} ) q^{88} + ( 380 + \beta_{1} - 24 \beta_{2} ) q^{89} + ( -18 + 18 \beta_{2} ) q^{90} + ( 436 - 62 \beta_{1} + 59 \beta_{2} ) q^{91} + ( -324 + 40 \beta_{1} + 20 \beta_{2} ) q^{92} + ( -99 + 9 \beta_{1} - 24 \beta_{2} ) q^{93} + ( 520 + 36 \beta_{1} - 28 \beta_{2} ) q^{94} + 96 q^{96} + ( 463 - 7 \beta_{1} - 7 \beta_{2} ) q^{97} + ( -754 + 26 \beta_{1} - 22 \beta_{2} ) q^{98} + ( -162 + 9 \beta_{1} + 18 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 6q^{2} - 9q^{3} + 12q^{4} + 2q^{5} + 18q^{6} - 17q^{7} - 24q^{8} + 27q^{9} + O(q^{10})$$ $$3q - 6q^{2} - 9q^{3} + 12q^{4} + 2q^{5} + 18q^{6} - 17q^{7} - 24q^{8} + 27q^{9} - 4q^{10} - 52q^{11} - 36q^{12} + 75q^{13} + 34q^{14} - 6q^{15} + 48q^{16} - 48q^{17} - 54q^{18} + 8q^{20} + 51q^{21} + 104q^{22} - 238q^{23} + 72q^{24} + 229q^{25} - 150q^{26} - 81q^{27} - 68q^{28} - 8q^{29} + 12q^{30} + 107q^{31} - 96q^{32} + 156q^{33} + 96q^{34} - 294q^{35} + 108q^{36} + 305q^{37} - 225q^{39} - 16q^{40} + 16q^{41} - 102q^{42} - 331q^{43} - 208q^{44} + 18q^{45} + 476q^{46} - 766q^{47} - 144q^{48} + 1142q^{49} - 458q^{50} + 144q^{51} + 300q^{52} - 118q^{53} + 162q^{54} - 1400q^{55} + 136q^{56} + 16q^{58} + 936q^{59} - 24q^{60} - 399q^{61} - 214q^{62} - 153q^{63} + 192q^{64} + 370q^{65} - 312q^{66} + 61q^{67} - 192q^{68} + 714q^{69} + 588q^{70} + 974q^{71} - 216q^{72} + 91q^{73} - 610q^{74} - 687q^{75} - 36q^{77} + 450q^{78} - 321q^{79} + 32q^{80} + 243q^{81} - 32q^{82} - 2148q^{83} + 204q^{84} - 1680q^{85} + 662q^{86} + 24q^{87} + 416q^{88} + 1116q^{89} - 36q^{90} + 1367q^{91} - 952q^{92} - 321q^{93} + 1532q^{94} + 288q^{96} + 1382q^{97} - 2284q^{98} - 468q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$6 \nu - 2$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{2} - 2 \nu - 19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 2$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{2} + \beta_{1} + 59$$$$)/12$$

## 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
 3.13264 −2.27307 0.140435
−2.00000 −3.00000 4.00000 −12.9884 6.00000 −25.6033 −8.00000 9.00000 25.9768
1.2 −2.00000 −3.00000 4.00000 −5.21359 6.00000 31.4905 −8.00000 9.00000 10.4272
1.3 −2.00000 −3.00000 4.00000 20.2020 6.00000 −22.8872 −8.00000 9.00000 −40.4040
 $$n$$: 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 2166.4.a.s 3
19.b odd 2 1 2166.4.a.w 3
19.c even 3 2 114.4.e.e 6
57.h odd 6 2 342.4.g.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.e 6 19.c even 3 2
342.4.g.g 6 57.h odd 6 2
2166.4.a.s 3 1.a even 1 1 trivial
2166.4.a.w 3 19.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2166))$$:

 $$T_{5}^{3} - 2 T_{5}^{2} - 300 T_{5} - 1368$$ $$T_{13}^{3} - 75 T_{13}^{2} + 819 T_{13} + 13207$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{3}$$
$3$ $$( 3 + T )^{3}$$
$5$ $$-1368 - 300 T - 2 T^{2} + T^{3}$$
$7$ $$-18453 - 941 T + 17 T^{2} + T^{3}$$
$11$ $$-32688 - 888 T + 52 T^{2} + T^{3}$$
$13$ $$13207 + 819 T - 75 T^{2} + T^{3}$$
$17$ $$10368 - 1728 T + 48 T^{2} + T^{3}$$
$19$ $$T^{3}$$
$23$ $$-6104664 - 23052 T + 238 T^{2} + T^{3}$$
$29$ $$-306432 - 42816 T + 8 T^{2} + T^{3}$$
$31$ $$1435247 - 14005 T - 107 T^{2} + T^{3}$$
$37$ $$28900349 - 125173 T - 305 T^{2} + T^{3}$$
$41$ $$-7007616 - 166560 T - 16 T^{2} + T^{3}$$
$43$ $$3121601 - 83941 T + 331 T^{2} + T^{3}$$
$47$ $$-20196504 + 91308 T + 766 T^{2} + T^{3}$$
$53$ $$-40793976 - 217980 T + 118 T^{2} + T^{3}$$
$59$ $$31669488 + 150120 T - 936 T^{2} + T^{3}$$
$61$ $$-78172163 - 209589 T + 399 T^{2} + T^{3}$$
$67$ $$-27605943 - 188261 T - 61 T^{2} + T^{3}$$
$71$ $$16989912 + 21420 T - 974 T^{2} + T^{3}$$
$73$ $$167210439 - 568301 T - 91 T^{2} + T^{3}$$
$79$ $$63732523 - 427581 T + 321 T^{2} + T^{3}$$
$83$ $$211415616 + 1271664 T + 2148 T^{2} + T^{3}$$
$89$ $$-11038032 + 245160 T - 1116 T^{2} + T^{3}$$
$97$ $$-79278088 + 601100 T - 1382 T^{2} + T^{3}$$