# Properties

 Label 187.2.a.c Level 187 Weight 2 Character orbit 187.a Self dual Yes Analytic conductor 1.493 Analytic rank 1 Dimension 2 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$187 = 11 \cdot 17$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 187.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$1.4932025178$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} -\beta q^{3} + ( 2 - 2 \beta ) q^{4} + ( -2 + \beta ) q^{5} + ( -3 + \beta ) q^{6} -2 q^{7} + ( -6 + 2 \beta ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} -\beta q^{3} + ( 2 - 2 \beta ) q^{4} + ( -2 + \beta ) q^{5} + ( -3 + \beta ) q^{6} -2 q^{7} + ( -6 + 2 \beta ) q^{8} + ( 5 - 3 \beta ) q^{10} + q^{11} + ( 6 - 2 \beta ) q^{12} + ( -5 - \beta ) q^{13} + ( 2 - 2 \beta ) q^{14} + ( -3 + 2 \beta ) q^{15} + ( 8 - 4 \beta ) q^{16} + q^{17} + ( -1 + 3 \beta ) q^{19} + ( -10 + 6 \beta ) q^{20} + 2 \beta q^{21} + ( -1 + \beta ) q^{22} + ( -2 + \beta ) q^{23} + ( -6 + 6 \beta ) q^{24} + ( 2 - 4 \beta ) q^{25} + ( 2 - 4 \beta ) q^{26} + 3 \beta q^{27} + ( -4 + 4 \beta ) q^{28} + ( -3 - \beta ) q^{29} + ( 9 - 5 \beta ) q^{30} + ( 4 + \beta ) q^{31} + ( -8 + 8 \beta ) q^{32} -\beta q^{33} + ( -1 + \beta ) q^{34} + ( 4 - 2 \beta ) q^{35} + ( -2 + \beta ) q^{37} + ( 10 - 4 \beta ) q^{38} + ( 3 + 5 \beta ) q^{39} + ( 18 - 10 \beta ) q^{40} + ( 6 - 2 \beta ) q^{41} + ( 6 - 2 \beta ) q^{42} -2 q^{43} + ( 2 - 2 \beta ) q^{44} + ( 5 - 3 \beta ) q^{46} + ( -6 - 4 \beta ) q^{47} + ( 12 - 8 \beta ) q^{48} -3 q^{49} + ( -14 + 6 \beta ) q^{50} -\beta q^{51} + ( -4 + 8 \beta ) q^{52} + ( -6 - 2 \beta ) q^{53} + ( 9 - 3 \beta ) q^{54} + ( -2 + \beta ) q^{55} + ( 12 - 4 \beta ) q^{56} + ( -9 + \beta ) q^{57} -2 \beta q^{58} + 3 q^{59} + ( -18 + 10 \beta ) q^{60} + ( -4 - 4 \beta ) q^{61} + ( -1 + 3 \beta ) q^{62} + ( 16 - 8 \beta ) q^{64} + ( 7 - 3 \beta ) q^{65} + ( -3 + \beta ) q^{66} + q^{67} + ( 2 - 2 \beta ) q^{68} + ( -3 + 2 \beta ) q^{69} + ( -10 + 6 \beta ) q^{70} + ( 2 + \beta ) q^{71} + ( 9 + 3 \beta ) q^{73} + ( 5 - 3 \beta ) q^{74} + ( 12 - 2 \beta ) q^{75} + ( -20 + 8 \beta ) q^{76} -2 q^{77} + ( 12 - 2 \beta ) q^{78} + ( -3 - 3 \beta ) q^{79} + ( -28 + 16 \beta ) q^{80} -9 q^{81} + ( -12 + 8 \beta ) q^{82} + ( 2 - 8 \beta ) q^{83} + ( -12 + 4 \beta ) q^{84} + ( -2 + \beta ) q^{85} + ( 2 - 2 \beta ) q^{86} + ( 3 + 3 \beta ) q^{87} + ( -6 + 2 \beta ) q^{88} + ( -5 + 4 \beta ) q^{89} + ( 10 + 2 \beta ) q^{91} + ( -10 + 6 \beta ) q^{92} + ( -3 - 4 \beta ) q^{93} + ( -6 - 2 \beta ) q^{94} + ( 11 - 7 \beta ) q^{95} + ( -24 + 8 \beta ) q^{96} + ( 14 - \beta ) q^{97} + ( 3 - 3 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 4q^{4} - 4q^{5} - 6q^{6} - 4q^{7} - 12q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 4q^{4} - 4q^{5} - 6q^{6} - 4q^{7} - 12q^{8} + 10q^{10} + 2q^{11} + 12q^{12} - 10q^{13} + 4q^{14} - 6q^{15} + 16q^{16} + 2q^{17} - 2q^{19} - 20q^{20} - 2q^{22} - 4q^{23} - 12q^{24} + 4q^{25} + 4q^{26} - 8q^{28} - 6q^{29} + 18q^{30} + 8q^{31} - 16q^{32} - 2q^{34} + 8q^{35} - 4q^{37} + 20q^{38} + 6q^{39} + 36q^{40} + 12q^{41} + 12q^{42} - 4q^{43} + 4q^{44} + 10q^{46} - 12q^{47} + 24q^{48} - 6q^{49} - 28q^{50} - 8q^{52} - 12q^{53} + 18q^{54} - 4q^{55} + 24q^{56} - 18q^{57} + 6q^{59} - 36q^{60} - 8q^{61} - 2q^{62} + 32q^{64} + 14q^{65} - 6q^{66} + 2q^{67} + 4q^{68} - 6q^{69} - 20q^{70} + 4q^{71} + 18q^{73} + 10q^{74} + 24q^{75} - 40q^{76} - 4q^{77} + 24q^{78} - 6q^{79} - 56q^{80} - 18q^{81} - 24q^{82} + 4q^{83} - 24q^{84} - 4q^{85} + 4q^{86} + 6q^{87} - 12q^{88} - 10q^{89} + 20q^{91} - 20q^{92} - 6q^{93} - 12q^{94} + 22q^{95} - 48q^{96} + 28q^{97} + 6q^{98} + O(q^{100})$$

## 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
 −1.73205 1.73205
−2.73205 1.73205 5.46410 −3.73205 −4.73205 −2.00000 −9.46410 0 10.1962
1.2 0.732051 −1.73205 −1.46410 −0.267949 −1.26795 −2.00000 −2.53590 0 −0.196152
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$-1$$
$$17$$ $$-1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(187))$$:

 $$T_{2}^{2} + 2 T_{2} - 2$$ $$T_{3}^{2} - 3$$