# Properties

 Label 2166.4.a.j Level $2166$ Weight $4$ Character orbit 2166.a Self dual yes Analytic conductor $127.798$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} -3 q^{3} + 4 q^{4} + ( -4 - 5 \beta ) q^{5} + 6 q^{6} + ( -12 + 7 \beta ) q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q -2 q^{2} -3 q^{3} + 4 q^{4} + ( -4 - 5 \beta ) q^{5} + 6 q^{6} + ( -12 + 7 \beta ) q^{7} -8 q^{8} + 9 q^{9} + ( 8 + 10 \beta ) q^{10} + ( 3 + 8 \beta ) q^{11} -12 q^{12} + ( -12 + 5 \beta ) q^{13} + ( 24 - 14 \beta ) q^{14} + ( 12 + 15 \beta ) q^{15} + 16 q^{16} + ( 11 - 34 \beta ) q^{17} -18 q^{18} + ( -16 - 20 \beta ) q^{20} + ( 36 - 21 \beta ) q^{21} + ( -6 - 16 \beta ) q^{22} + ( -35 + 16 \beta ) q^{23} + 24 q^{24} + ( -84 + 65 \beta ) q^{25} + ( 24 - 10 \beta ) q^{26} -27 q^{27} + ( -48 + 28 \beta ) q^{28} + ( 22 + 86 \beta ) q^{29} + ( -24 - 30 \beta ) q^{30} + ( 200 - 161 \beta ) q^{31} -32 q^{32} + ( -9 - 24 \beta ) q^{33} + ( -22 + 68 \beta ) q^{34} + ( 13 - 3 \beta ) q^{35} + 36 q^{36} + ( 154 - 160 \beta ) q^{37} + ( 36 - 15 \beta ) q^{39} + ( 32 + 40 \beta ) q^{40} + ( -128 - 75 \beta ) q^{41} + ( -72 + 42 \beta ) q^{42} + ( -159 + 94 \beta ) q^{43} + ( 12 + 32 \beta ) q^{44} + ( -36 - 45 \beta ) q^{45} + ( 70 - 32 \beta ) q^{46} + ( 176 - 19 \beta ) q^{47} -48 q^{48} + ( -150 - 119 \beta ) q^{49} + ( 168 - 130 \beta ) q^{50} + ( -33 + 102 \beta ) q^{51} + ( -48 + 20 \beta ) q^{52} + ( 217 + 332 \beta ) q^{53} + 54 q^{54} + ( -52 - 87 \beta ) q^{55} + ( 96 - 56 \beta ) q^{56} + ( -44 - 172 \beta ) q^{58} + ( 18 + 424 \beta ) q^{59} + ( 48 + 60 \beta ) q^{60} + ( 372 - 250 \beta ) q^{61} + ( -400 + 322 \beta ) q^{62} + ( -108 + 63 \beta ) q^{63} + 64 q^{64} + ( 23 + 15 \beta ) q^{65} + ( 18 + 48 \beta ) q^{66} + ( 131 - 129 \beta ) q^{67} + ( 44 - 136 \beta ) q^{68} + ( 105 - 48 \beta ) q^{69} + ( -26 + 6 \beta ) q^{70} + ( 352 - 245 \beta ) q^{71} -72 q^{72} + ( 161 + 74 \beta ) q^{73} + ( -308 + 320 \beta ) q^{74} + ( 252 - 195 \beta ) q^{75} + ( 20 - 19 \beta ) q^{77} + ( -72 + 30 \beta ) q^{78} + ( -375 - 250 \beta ) q^{79} + ( -64 - 80 \beta ) q^{80} + 81 q^{81} + ( 256 + 150 \beta ) q^{82} + ( 69 - 212 \beta ) q^{83} + ( 144 - 84 \beta ) q^{84} + ( 126 + 251 \beta ) q^{85} + ( 318 - 188 \beta ) q^{86} + ( -66 - 258 \beta ) q^{87} + ( -24 - 64 \beta ) q^{88} + ( 404 + 372 \beta ) q^{89} + ( 72 + 90 \beta ) q^{90} + ( 179 - 109 \beta ) q^{91} + ( -140 + 64 \beta ) q^{92} + ( -600 + 483 \beta ) q^{93} + ( -352 + 38 \beta ) q^{94} + 96 q^{96} + ( 1030 - 122 \beta ) q^{97} + ( 300 + 238 \beta ) q^{98} + ( 27 + 72 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} - 6q^{3} + 8q^{4} - 13q^{5} + 12q^{6} - 17q^{7} - 16q^{8} + 18q^{9} + O(q^{10})$$ $$2q - 4q^{2} - 6q^{3} + 8q^{4} - 13q^{5} + 12q^{6} - 17q^{7} - 16q^{8} + 18q^{9} + 26q^{10} + 14q^{11} - 24q^{12} - 19q^{13} + 34q^{14} + 39q^{15} + 32q^{16} - 12q^{17} - 36q^{18} - 52q^{20} + 51q^{21} - 28q^{22} - 54q^{23} + 48q^{24} - 103q^{25} + 38q^{26} - 54q^{27} - 68q^{28} + 130q^{29} - 78q^{30} + 239q^{31} - 64q^{32} - 42q^{33} + 24q^{34} + 23q^{35} + 72q^{36} + 148q^{37} + 57q^{39} + 104q^{40} - 331q^{41} - 102q^{42} - 224q^{43} + 56q^{44} - 117q^{45} + 108q^{46} + 333q^{47} - 96q^{48} - 419q^{49} + 206q^{50} + 36q^{51} - 76q^{52} + 766q^{53} + 108q^{54} - 191q^{55} + 136q^{56} - 260q^{58} + 460q^{59} + 156q^{60} + 494q^{61} - 478q^{62} - 153q^{63} + 128q^{64} + 61q^{65} + 84q^{66} + 133q^{67} - 48q^{68} + 162q^{69} - 46q^{70} + 459q^{71} - 144q^{72} + 396q^{73} - 296q^{74} + 309q^{75} + 21q^{77} - 114q^{78} - 1000q^{79} - 208q^{80} + 162q^{81} + 662q^{82} - 74q^{83} + 204q^{84} + 503q^{85} + 448q^{86} - 390q^{87} - 112q^{88} + 1180q^{89} + 234q^{90} + 249q^{91} - 216q^{92} - 717q^{93} - 666q^{94} + 192q^{96} + 1938q^{97} + 838q^{98} + 126q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
−2.00000 −3.00000 4.00000 −12.0902 6.00000 −0.673762 −8.00000 9.00000 24.1803
1.2 −2.00000 −3.00000 4.00000 −0.909830 6.00000 −16.3262 −8.00000 9.00000 1.81966
 $$n$$: 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.j 2
19.b odd 2 1 2166.4.a.p yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2166.4.a.j 2 1.a even 1 1 trivial
2166.4.a.p yes 2 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2166))$$:

 $$T_{5}^{2} + 13 T_{5} + 11$$ $$T_{13}^{2} + 19 T_{13} + 59$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$11 + 13 T + T^{2}$$
$7$ $$11 + 17 T + T^{2}$$
$11$ $$-31 - 14 T + T^{2}$$
$13$ $$59 + 19 T + T^{2}$$
$17$ $$-1409 + 12 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$409 + 54 T + T^{2}$$
$29$ $$-5020 - 130 T + T^{2}$$
$31$ $$-18121 - 239 T + T^{2}$$
$37$ $$-26524 - 148 T + T^{2}$$
$41$ $$20359 + 331 T + T^{2}$$
$43$ $$1499 + 224 T + T^{2}$$
$47$ $$27271 - 333 T + T^{2}$$
$53$ $$8909 - 766 T + T^{2}$$
$59$ $$-171820 - 460 T + T^{2}$$
$61$ $$-17116 - 494 T + T^{2}$$
$67$ $$-16379 - 133 T + T^{2}$$
$71$ $$-22361 - 459 T + T^{2}$$
$73$ $$32359 - 396 T + T^{2}$$
$79$ $$171875 + 1000 T + T^{2}$$
$83$ $$-54811 + 74 T + T^{2}$$
$89$ $$175120 - 1180 T + T^{2}$$
$97$ $$920356 - 1938 T + T^{2}$$