# Properties

 Label 2166.2.a.f Level $2166$ Weight $2$ Character orbit 2166.a Self dual yes Analytic conductor $17.296$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$17.2955970778$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} - 3q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} - 3q^{7} + q^{8} + q^{9} - 4q^{10} + 2q^{11} - q^{12} - 7q^{13} - 3q^{14} + 4q^{15} + q^{16} + q^{18} - 4q^{20} + 3q^{21} + 2q^{22} - 4q^{23} - q^{24} + 11q^{25} - 7q^{26} - q^{27} - 3q^{28} + 4q^{29} + 4q^{30} + q^{31} + q^{32} - 2q^{33} + 12q^{35} + q^{36} + 7q^{37} + 7q^{39} - 4q^{40} + 4q^{41} + 3q^{42} + 7q^{43} + 2q^{44} - 4q^{45} - 4q^{46} + 2q^{47} - q^{48} + 2q^{49} + 11q^{50} - 7q^{52} - 4q^{53} - q^{54} - 8q^{55} - 3q^{56} + 4q^{58} - 6q^{59} + 4q^{60} - q^{61} + q^{62} - 3q^{63} + q^{64} + 28q^{65} - 2q^{66} + 3q^{67} + 4q^{69} + 12q^{70} + 2q^{71} + q^{72} - 3q^{73} + 7q^{74} - 11q^{75} - 6q^{77} + 7q^{78} + 5q^{79} - 4q^{80} + q^{81} + 4q^{82} - 12q^{83} + 3q^{84} + 7q^{86} - 4q^{87} + 2q^{88} + 18q^{89} - 4q^{90} + 21q^{91} - 4q^{92} - q^{93} + 2q^{94} - q^{96} + 10q^{97} + 2q^{98} + 2q^{99} + O(q^{100})$$

## 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
 0
1.00000 −1.00000 1.00000 −4.00000 −1.00000 −3.00000 1.00000 1.00000 −4.00000
 $$n$$: 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 2166.2.a.f 1
3.b odd 2 1 6498.2.a.l 1
19.b odd 2 1 2166.2.a.c 1
19.c even 3 2 114.2.e.a 2
57.d even 2 1 6498.2.a.x 1
57.h odd 6 2 342.2.g.d 2
76.g odd 6 2 912.2.q.d 2
228.m even 6 2 2736.2.s.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.a 2 19.c even 3 2
342.2.g.d 2 57.h odd 6 2
912.2.q.d 2 76.g odd 6 2
2166.2.a.c 1 19.b odd 2 1
2166.2.a.f 1 1.a even 1 1 trivial
2736.2.s.c 2 228.m even 6 2
6498.2.a.l 1 3.b odd 2 1
6498.2.a.x 1 57.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2166))$$:

 $$T_{5} + 4$$ $$T_{7} + 3$$ $$T_{13} + 7$$ $$T_{29} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$4 + T$$
$7$ $$3 + T$$
$11$ $$-2 + T$$
$13$ $$7 + T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$4 + T$$
$29$ $$-4 + T$$
$31$ $$-1 + T$$
$37$ $$-7 + T$$
$41$ $$-4 + T$$
$43$ $$-7 + T$$
$47$ $$-2 + T$$
$53$ $$4 + T$$
$59$ $$6 + T$$
$61$ $$1 + T$$
$67$ $$-3 + T$$
$71$ $$-2 + T$$
$73$ $$3 + T$$
$79$ $$-5 + T$$
$83$ $$12 + T$$
$89$ $$-18 + T$$
$97$ $$-10 + T$$