# Properties

 Label 2166.4.a.r 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: 3.3.373564.1 Defining polynomial: $$x^{3} - 106 x - 348$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} -3 q^{3} + 4 q^{4} + ( -2 + \beta_{2} ) q^{5} + 6 q^{6} + ( -4 + \beta_{1} + \beta_{2} ) q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q -2 q^{2} -3 q^{3} + 4 q^{4} + ( -2 + \beta_{2} ) q^{5} + 6 q^{6} + ( -4 + \beta_{1} + \beta_{2} ) q^{7} -8 q^{8} + 9 q^{9} + ( 4 - 2 \beta_{2} ) q^{10} + ( 26 - 3 \beta_{1} - \beta_{2} ) q^{11} -12 q^{12} + ( 16 - \beta_{1} - 4 \beta_{2} ) q^{13} + ( 8 - 2 \beta_{1} - 2 \beta_{2} ) q^{14} + ( 6 - 3 \beta_{2} ) q^{15} + 16 q^{16} + ( 14 - \beta_{1} + 3 \beta_{2} ) q^{17} -18 q^{18} + ( -8 + 4 \beta_{2} ) q^{20} + ( 12 - 3 \beta_{1} - 3 \beta_{2} ) q^{21} + ( -52 + 6 \beta_{1} + 2 \beta_{2} ) q^{22} + ( 6 + 7 \beta_{1} + 2 \beta_{2} ) q^{23} + 24 q^{24} + ( 95 - 9 \beta_{1} - 3 \beta_{2} ) q^{25} + ( -32 + 2 \beta_{1} + 8 \beta_{2} ) q^{26} -27 q^{27} + ( -16 + 4 \beta_{1} + 4 \beta_{2} ) q^{28} + ( 94 - 4 \beta_{1} - 10 \beta_{2} ) q^{29} + ( -12 + 6 \beta_{2} ) q^{30} + ( 20 - 5 \beta_{1} + 4 \beta_{2} ) q^{31} -32 q^{32} + ( -78 + 9 \beta_{1} + 3 \beta_{2} ) q^{33} + ( -28 + 2 \beta_{1} - 6 \beta_{2} ) q^{34} + ( 152 - 11 \beta_{1} - 17 \beta_{2} ) q^{35} + 36 q^{36} + ( -92 - \beta_{1} - 4 \beta_{2} ) q^{37} + ( -48 + 3 \beta_{1} + 12 \beta_{2} ) q^{39} + ( 16 - 8 \beta_{2} ) q^{40} + ( -10 - 14 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -24 + 6 \beta_{1} + 6 \beta_{2} ) q^{42} + ( 60 + 11 \beta_{1} - 7 \beta_{2} ) q^{43} + ( 104 - 12 \beta_{1} - 4 \beta_{2} ) q^{44} + ( -18 + 9 \beta_{2} ) q^{45} + ( -12 - 14 \beta_{1} - 4 \beta_{2} ) q^{46} + ( 166 - 2 \beta_{1} + 9 \beta_{2} ) q^{47} -48 q^{48} + ( 25 - 5 \beta_{1} - 23 \beta_{2} ) q^{49} + ( -190 + 18 \beta_{1} + 6 \beta_{2} ) q^{50} + ( -42 + 3 \beta_{1} - 9 \beta_{2} ) q^{51} + ( 64 - 4 \beta_{1} - 16 \beta_{2} ) q^{52} + ( -74 - 10 \beta_{1} + 14 \beta_{2} ) q^{53} + 54 q^{54} + ( -52 + 15 \beta_{1} + 63 \beta_{2} ) q^{55} + ( 32 - 8 \beta_{1} - 8 \beta_{2} ) q^{56} + ( -188 + 8 \beta_{1} + 20 \beta_{2} ) q^{58} + ( 100 + 14 \beta_{1} + 44 \beta_{2} ) q^{59} + ( 24 - 12 \beta_{2} ) q^{60} + ( 290 + \beta_{1} - 23 \beta_{2} ) q^{61} + ( -40 + 10 \beta_{1} - 8 \beta_{2} ) q^{62} + ( -36 + 9 \beta_{1} + 9 \beta_{2} ) q^{63} + 64 q^{64} + ( -824 + 38 \beta_{1} + 32 \beta_{2} ) q^{65} + ( 156 - 18 \beta_{1} - 6 \beta_{2} ) q^{66} + ( 80 + 22 \beta_{1} + 28 \beta_{2} ) q^{67} + ( 56 - 4 \beta_{1} + 12 \beta_{2} ) q^{68} + ( -18 - 21 \beta_{1} - 6 \beta_{2} ) q^{69} + ( -304 + 22 \beta_{1} + 34 \beta_{2} ) q^{70} + ( 120 - 18 \beta_{1} + 24 \beta_{2} ) q^{71} -72 q^{72} + ( 82 + 33 \beta_{1} - 21 \beta_{2} ) q^{73} + ( 184 + 2 \beta_{1} + 8 \beta_{2} ) q^{74} + ( -285 + 27 \beta_{1} + 9 \beta_{2} ) q^{75} + ( -872 + 11 \beta_{1} + 53 \beta_{2} ) q^{77} + ( 96 - 6 \beta_{1} - 24 \beta_{2} ) q^{78} + ( -192 + 9 \beta_{1} ) q^{79} + ( -32 + 16 \beta_{2} ) q^{80} + 81 q^{81} + ( 20 + 28 \beta_{1} + 4 \beta_{2} ) q^{82} + ( 478 + 12 \beta_{1} + 58 \beta_{2} ) q^{83} + ( 48 - 12 \beta_{1} - 12 \beta_{2} ) q^{84} + ( 692 - 25 \beta_{1} + 23 \beta_{2} ) q^{85} + ( -120 - 22 \beta_{1} + 14 \beta_{2} ) q^{86} + ( -282 + 12 \beta_{1} + 30 \beta_{2} ) q^{87} + ( -208 + 24 \beta_{1} + 8 \beta_{2} ) q^{88} + ( 270 - 60 \beta_{1} + 6 \beta_{2} ) q^{89} + ( 36 - 18 \beta_{2} ) q^{90} + ( -848 + 44 \beta_{1} + 80 \beta_{2} ) q^{91} + ( 24 + 28 \beta_{1} + 8 \beta_{2} ) q^{92} + ( -60 + 15 \beta_{1} - 12 \beta_{2} ) q^{93} + ( -332 + 4 \beta_{1} - 18 \beta_{2} ) q^{94} + 96 q^{96} + ( 1332 + 6 \beta_{1} + 12 \beta_{2} ) q^{97} + ( -50 + 10 \beta_{1} + 46 \beta_{2} ) q^{98} + ( 234 - 27 \beta_{1} - 9 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 6q^{2} - 9q^{3} + 12q^{4} - 5q^{5} + 18q^{6} - 11q^{7} - 24q^{8} + 27q^{9} + O(q^{10})$$ $$3q - 6q^{2} - 9q^{3} + 12q^{4} - 5q^{5} + 18q^{6} - 11q^{7} - 24q^{8} + 27q^{9} + 10q^{10} + 77q^{11} - 36q^{12} + 44q^{13} + 22q^{14} + 15q^{15} + 48q^{16} + 45q^{17} - 54q^{18} - 20q^{20} + 33q^{21} - 154q^{22} + 20q^{23} + 72q^{24} + 282q^{25} - 88q^{26} - 81q^{27} - 44q^{28} + 272q^{29} - 30q^{30} + 64q^{31} - 96q^{32} - 231q^{33} - 90q^{34} + 439q^{35} + 108q^{36} - 280q^{37} - 132q^{39} + 40q^{40} - 32q^{41} - 66q^{42} + 173q^{43} + 308q^{44} - 45q^{45} - 40q^{46} + 507q^{47} - 144q^{48} + 52q^{49} - 564q^{50} - 135q^{51} + 176q^{52} - 208q^{53} + 162q^{54} - 93q^{55} + 88q^{56} - 544q^{58} + 344q^{59} + 60q^{60} + 847q^{61} - 128q^{62} - 99q^{63} + 192q^{64} - 2440q^{65} + 462q^{66} + 268q^{67} + 180q^{68} - 60q^{69} - 878q^{70} + 384q^{71} - 216q^{72} + 225q^{73} + 560q^{74} - 846q^{75} - 2563q^{77} + 264q^{78} - 576q^{79} - 80q^{80} + 243q^{81} + 64q^{82} + 1492q^{83} + 132q^{84} + 2099q^{85} - 346q^{86} - 816q^{87} - 616q^{88} + 816q^{89} + 90q^{90} - 2464q^{91} + 80q^{92} - 192q^{93} - 1014q^{94} + 288q^{96} + 4008q^{97} - 104q^{98} + 693q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 106 x - 348$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - 6 \nu - 70$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} + 3 \beta_{1} + 70$$

## 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
 −3.80115 11.6558 −7.85460
−2.00000 −3.00000 4.00000 −18.3722 6.00000 −27.9745 −8.00000 9.00000 36.7444
1.2 −2.00000 −3.00000 4.00000 −4.03900 6.00000 17.2725 −8.00000 9.00000 8.07799
1.3 −2.00000 −3.00000 4.00000 17.4112 6.00000 −0.298020 −8.00000 9.00000 −34.8224
 $$n$$: 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.r 3
19.b odd 2 1 2166.4.a.v yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2166.4.a.r 3 1.a even 1 1 trivial
2166.4.a.v yes 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2166))$$:

 $$T_{5}^{3} + 5 T_{5}^{2} - 316 T_{5} - 1292$$ $$T_{13}^{3} - 44 T_{13}^{2} - 4056 T_{13} + 3456$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{3}$$
$3$ $$( 3 + T )^{3}$$
$5$ $$-1292 - 316 T + 5 T^{2} + T^{3}$$
$7$ $$-144 - 480 T + 11 T^{2} + T^{3}$$
$11$ $$146684 - 1480 T - 77 T^{2} + T^{3}$$
$13$ $$3456 - 4056 T - 44 T^{2} + T^{3}$$
$17$ $$-37332 - 3352 T - 45 T^{2} + T^{3}$$
$19$ $$T^{3}$$
$23$ $$-859984 - 18748 T - 20 T^{2} + T^{3}$$
$29$ $$227088 - 5436 T - 272 T^{2} + T^{3}$$
$31$ $$-137952 - 18984 T - 64 T^{2} + T^{3}$$
$37$ $$311904 + 21432 T + 280 T^{2} + T^{3}$$
$41$ $$7344432 - 77676 T + 32 T^{2} + T^{3}$$
$43$ $$7482032 - 74776 T - 173 T^{2} + T^{3}$$
$47$ $$-1272732 + 53612 T - 507 T^{2} + T^{3}$$
$53$ $$-27060048 - 123468 T + 208 T^{2} + T^{3}$$
$59$ $$179632512 - 531120 T - 344 T^{2} + T^{3}$$
$61$ $$40865796 + 61896 T - 847 T^{2} + T^{3}$$
$67$ $$81247104 - 295104 T - 268 T^{2} + T^{3}$$
$71$ $$-41202432 - 373536 T - 384 T^{2} + T^{3}$$
$73$ $$132019148 - 745896 T - 225 T^{2} + T^{3}$$
$79$ $$-1545696 + 76248 T + 576 T^{2} + T^{3}$$
$83$ $$509389312 - 251404 T - 1492 T^{2} + T^{3}$$
$89$ $$952098192 - 1398204 T - 816 T^{2} + T^{3}$$
$97$ $$-2320071552 + 5309136 T - 4008 T^{2} + T^{3}$$