# Properties

 Label 2166.4.a.n Level $2166$ Weight $4$ Character orbit 2166.a Self dual yes Analytic conductor $127.798$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ 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 $$\beta = 3\sqrt{17}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 3 q^{3} + 4 q^{4} + ( 9 + \beta ) q^{5} -6 q^{6} + ( 2 - 2 \beta ) q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q -2 q^{2} + 3 q^{3} + 4 q^{4} + ( 9 + \beta ) q^{5} -6 q^{6} + ( 2 - 2 \beta ) q^{7} -8 q^{8} + 9 q^{9} + ( -18 - 2 \beta ) q^{10} + ( 30 + 2 \beta ) q^{11} + 12 q^{12} + ( -29 + 3 \beta ) q^{13} + ( -4 + 4 \beta ) q^{14} + ( 27 + 3 \beta ) q^{15} + 16 q^{16} + ( 48 + 2 \beta ) q^{17} -18 q^{18} + ( 36 + 4 \beta ) q^{20} + ( 6 - 6 \beta ) q^{21} + ( -60 - 4 \beta ) q^{22} + ( -15 - \beta ) q^{23} -24 q^{24} + ( 109 + 18 \beta ) q^{25} + ( 58 - 6 \beta ) q^{26} + 27 q^{27} + ( 8 - 8 \beta ) q^{28} + ( 126 + 8 \beta ) q^{29} + ( -54 - 6 \beta ) q^{30} + ( 169 + 11 \beta ) q^{31} -32 q^{32} + ( 90 + 6 \beta ) q^{33} + ( -96 - 4 \beta ) q^{34} + ( -288 - 16 \beta ) q^{35} + 36 q^{36} + ( 79 + 7 \beta ) q^{37} + ( -87 + 9 \beta ) q^{39} + ( -72 - 8 \beta ) q^{40} -30 \beta q^{41} + ( -12 + 12 \beta ) q^{42} + ( -124 - 8 \beta ) q^{43} + ( 120 + 8 \beta ) q^{44} + ( 81 + 9 \beta ) q^{45} + ( 30 + 2 \beta ) q^{46} + ( -405 - 11 \beta ) q^{47} + 48 q^{48} + ( 273 - 8 \beta ) q^{49} + ( -218 - 36 \beta ) q^{50} + ( 144 + 6 \beta ) q^{51} + ( -116 + 12 \beta ) q^{52} + ( -42 + 40 \beta ) q^{53} -54 q^{54} + ( 576 + 48 \beta ) q^{55} + ( -16 + 16 \beta ) q^{56} + ( -252 - 16 \beta ) q^{58} + ( 252 - 48 \beta ) q^{59} + ( 108 + 12 \beta ) q^{60} + ( 290 - 44 \beta ) q^{61} + ( -338 - 22 \beta ) q^{62} + ( 18 - 18 \beta ) q^{63} + 64 q^{64} + ( 198 - 2 \beta ) q^{65} + ( -180 - 12 \beta ) q^{66} + ( 70 - 26 \beta ) q^{67} + ( 192 + 8 \beta ) q^{68} + ( -45 - 3 \beta ) q^{69} + ( 576 + 32 \beta ) q^{70} + ( 240 - 48 \beta ) q^{71} -72 q^{72} + ( 308 + 14 \beta ) q^{73} + ( -158 - 14 \beta ) q^{74} + ( 327 + 54 \beta ) q^{75} + ( -552 - 56 \beta ) q^{77} + ( 174 - 18 \beta ) q^{78} + ( -101 - 39 \beta ) q^{79} + ( 144 + 16 \beta ) q^{80} + 81 q^{81} + 60 \beta q^{82} + ( -276 - 20 \beta ) q^{83} + ( 24 - 24 \beta ) q^{84} + ( 738 + 66 \beta ) q^{85} + ( 248 + 16 \beta ) q^{86} + ( 378 + 24 \beta ) q^{87} + ( -240 - 16 \beta ) q^{88} + ( 462 - 4 \beta ) q^{89} + ( -162 - 18 \beta ) q^{90} + ( -976 + 64 \beta ) q^{91} + ( -60 - 4 \beta ) q^{92} + ( 507 + 33 \beta ) q^{93} + ( 810 + 22 \beta ) q^{94} -96 q^{96} + ( -20 + 114 \beta ) q^{97} + ( -546 + 16 \beta ) q^{98} + ( 270 + 18 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 6q^{3} + 8q^{4} + 18q^{5} - 12q^{6} + 4q^{7} - 16q^{8} + 18q^{9} + O(q^{10})$$ $$2q - 4q^{2} + 6q^{3} + 8q^{4} + 18q^{5} - 12q^{6} + 4q^{7} - 16q^{8} + 18q^{9} - 36q^{10} + 60q^{11} + 24q^{12} - 58q^{13} - 8q^{14} + 54q^{15} + 32q^{16} + 96q^{17} - 36q^{18} + 72q^{20} + 12q^{21} - 120q^{22} - 30q^{23} - 48q^{24} + 218q^{25} + 116q^{26} + 54q^{27} + 16q^{28} + 252q^{29} - 108q^{30} + 338q^{31} - 64q^{32} + 180q^{33} - 192q^{34} - 576q^{35} + 72q^{36} + 158q^{37} - 174q^{39} - 144q^{40} - 24q^{42} - 248q^{43} + 240q^{44} + 162q^{45} + 60q^{46} - 810q^{47} + 96q^{48} + 546q^{49} - 436q^{50} + 288q^{51} - 232q^{52} - 84q^{53} - 108q^{54} + 1152q^{55} - 32q^{56} - 504q^{58} + 504q^{59} + 216q^{60} + 580q^{61} - 676q^{62} + 36q^{63} + 128q^{64} + 396q^{65} - 360q^{66} + 140q^{67} + 384q^{68} - 90q^{69} + 1152q^{70} + 480q^{71} - 144q^{72} + 616q^{73} - 316q^{74} + 654q^{75} - 1104q^{77} + 348q^{78} - 202q^{79} + 288q^{80} + 162q^{81} - 552q^{83} + 48q^{84} + 1476q^{85} + 496q^{86} + 756q^{87} - 480q^{88} + 924q^{89} - 324q^{90} - 1952q^{91} - 120q^{92} + 1014q^{93} + 1620q^{94} - 192q^{96} - 40q^{97} - 1092q^{98} + 540q^{99} + O(q^{100})$$

## 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
 −1.56155 2.56155
−2.00000 3.00000 4.00000 −3.36932 −6.00000 26.7386 −8.00000 9.00000 6.73863
1.2 −2.00000 3.00000 4.00000 21.3693 −6.00000 −22.7386 −8.00000 9.00000 −42.7386
 $$n$$: 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.n 2
19.b odd 2 1 114.4.a.e 2
57.d even 2 1 342.4.a.f 2
76.d even 2 1 912.4.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.a.e 2 19.b odd 2 1
342.4.a.f 2 57.d even 2 1
912.4.a.m 2 76.d even 2 1
2166.4.a.n 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2166))$$:

 $$T_{5}^{2} - 18 T_{5} - 72$$ $$T_{13}^{2} + 58 T_{13} - 536$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$-72 - 18 T + T^{2}$$
$7$ $$-608 - 4 T + T^{2}$$
$11$ $$288 - 60 T + T^{2}$$
$13$ $$-536 + 58 T + T^{2}$$
$17$ $$1692 - 96 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$72 + 30 T + T^{2}$$
$29$ $$6084 - 252 T + T^{2}$$
$31$ $$10048 - 338 T + T^{2}$$
$37$ $$-1256 - 158 T + T^{2}$$
$41$ $$-137700 + T^{2}$$
$43$ $$5584 + 248 T + T^{2}$$
$47$ $$145512 + 810 T + T^{2}$$
$53$ $$-243036 + 84 T + T^{2}$$
$59$ $$-289008 - 504 T + T^{2}$$
$61$ $$-212108 - 580 T + T^{2}$$
$67$ $$-98528 - 140 T + T^{2}$$
$71$ $$-294912 - 480 T + T^{2}$$
$73$ $$64876 - 616 T + T^{2}$$
$79$ $$-222512 + 202 T + T^{2}$$
$83$ $$14976 + 552 T + T^{2}$$
$89$ $$210996 - 924 T + T^{2}$$
$97$ $$-1987988 + 40 T + T^{2}$$