# Properties

 Label 2166.4.a.p 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 13 q^{5} + 12 q^{6} - 17 q^{7} + 16 q^{8} + 18 q^{9} + O(q^{10})$$ $$2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 13 q^{5} + 12 q^{6} - 17 q^{7} + 16 q^{8} + 18 q^{9} - 26 q^{10} + 14 q^{11} + 24 q^{12} + 19 q^{13} - 34 q^{14} - 39 q^{15} + 32 q^{16} - 12 q^{17} + 36 q^{18} - 52 q^{20} - 51 q^{21} + 28 q^{22} - 54 q^{23} + 48 q^{24} - 103 q^{25} + 38 q^{26} + 54 q^{27} - 68 q^{28} - 130 q^{29} - 78 q^{30} - 239 q^{31} + 64 q^{32} + 42 q^{33} - 24 q^{34} + 23 q^{35} + 72 q^{36} - 148 q^{37} + 57 q^{39} - 104 q^{40} + 331 q^{41} - 102 q^{42} - 224 q^{43} + 56 q^{44} - 117 q^{45} - 108 q^{46} + 333 q^{47} + 96 q^{48} - 419 q^{49} - 206 q^{50} - 36 q^{51} + 76 q^{52} - 766 q^{53} + 108 q^{54} - 191 q^{55} - 136 q^{56} - 260 q^{58} - 460 q^{59} - 156 q^{60} + 494 q^{61} - 478 q^{62} - 153 q^{63} + 128 q^{64} - 61 q^{65} + 84 q^{66} - 133 q^{67} - 48 q^{68} - 162 q^{69} + 46 q^{70} - 459 q^{71} + 144 q^{72} + 396 q^{73} - 296 q^{74} - 309 q^{75} + 21 q^{77} + 114 q^{78} + 1000 q^{79} - 208 q^{80} + 162 q^{81} + 662 q^{82} - 74 q^{83} - 204 q^{84} + 503 q^{85} - 448 q^{86} - 390 q^{87} + 112 q^{88} - 1180 q^{89} - 234 q^{90} - 249 q^{91} - 216 q^{92} - 717 q^{93} + 666 q^{94} + 192 q^{96} - 1938 q^{97} - 838 q^{98} + 126 q^{99} + O(q^{100})$$

## 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.p yes 2
19.b odd 2 1 2166.4.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2166.4.a.j 2 19.b odd 2 1
2166.4.a.p yes 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2166))$$:

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

## Hecke characteristic polynomials

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