# Properties

 Label 187.2.a.d Level 187 Weight 2 Character orbit 187.a Self dual Yes Analytic conductor 1.493 Analytic rank 0 Dimension 2 CM No Inner twists 1

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## 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + ( -1 + \beta ) q^{3} + 2 q^{4} + ( 1 - \beta ) q^{5} + ( -2 + 2 \beta ) q^{6} + ( 2 - \beta ) q^{7} + ( 2 - \beta ) q^{9} +O(q^{10})$$ $$q + 2 q^{2} + ( -1 + \beta ) q^{3} + 2 q^{4} + ( 1 - \beta ) q^{5} + ( -2 + 2 \beta ) q^{6} + ( 2 - \beta ) q^{7} + ( 2 - \beta ) q^{9} + ( 2 - 2 \beta ) q^{10} + q^{11} + ( -2 + 2 \beta ) q^{12} + ( 4 - 2 \beta ) q^{14} + ( -5 + \beta ) q^{15} -4 q^{16} - q^{17} + ( 4 - 2 \beta ) q^{18} + ( -4 + 2 \beta ) q^{19} + ( 2 - 2 \beta ) q^{20} + ( -6 + 2 \beta ) q^{21} + 2 q^{22} + ( -1 + 3 \beta ) q^{23} -\beta q^{25} + ( -3 - \beta ) q^{27} + ( 4 - 2 \beta ) q^{28} + ( 8 - \beta ) q^{29} + ( -10 + 2 \beta ) q^{30} + ( -1 + \beta ) q^{31} -8 q^{32} + ( -1 + \beta ) q^{33} -2 q^{34} + ( 6 - 2 \beta ) q^{35} + ( 4 - 2 \beta ) q^{36} + ( 3 - \beta ) q^{37} + ( -8 + 4 \beta ) q^{38} + ( -4 + 3 \beta ) q^{41} + ( -12 + 4 \beta ) q^{42} + 2 q^{43} + 2 q^{44} + ( 6 - 2 \beta ) q^{45} + ( -2 + 6 \beta ) q^{46} + ( -4 + 3 \beta ) q^{47} + ( 4 - 4 \beta ) q^{48} + ( 1 - 3 \beta ) q^{49} -2 \beta q^{50} + ( 1 - \beta ) q^{51} + ( 6 - \beta ) q^{53} + ( -6 - 2 \beta ) q^{54} + ( 1 - \beta ) q^{55} + ( 12 - 4 \beta ) q^{57} + ( 16 - 2 \beta ) q^{58} -3 q^{59} + ( -10 + 2 \beta ) q^{60} + ( -4 - 2 \beta ) q^{61} + ( -2 + 2 \beta ) q^{62} + ( 8 - 3 \beta ) q^{63} -8 q^{64} + ( -2 + 2 \beta ) q^{66} + ( 1 - 2 \beta ) q^{67} -2 q^{68} + ( 13 - \beta ) q^{69} + ( 12 - 4 \beta ) q^{70} + ( 5 - 3 \beta ) q^{71} + ( 4 + 3 \beta ) q^{73} + ( 6 - 2 \beta ) q^{74} -4 q^{75} + ( -8 + 4 \beta ) q^{76} + ( 2 - \beta ) q^{77} + ( -6 + 6 \beta ) q^{79} + ( -4 + 4 \beta ) q^{80} -7 q^{81} + ( -8 + 6 \beta ) q^{82} + ( 2 - 4 \beta ) q^{83} + ( -12 + 4 \beta ) q^{84} + ( -1 + \beta ) q^{85} + 4 q^{86} + ( -12 + 8 \beta ) q^{87} + ( 7 - 6 \beta ) q^{89} + ( 12 - 4 \beta ) q^{90} + ( -2 + 6 \beta ) q^{92} + ( 5 - \beta ) q^{93} + ( -8 + 6 \beta ) q^{94} + ( -12 + 4 \beta ) q^{95} + ( 8 - 8 \beta ) q^{96} + ( -7 - 3 \beta ) q^{97} + ( 2 - 6 \beta ) q^{98} + ( 2 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} - q^{3} + 4q^{4} + q^{5} - 2q^{6} + 3q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 4q^{2} - q^{3} + 4q^{4} + q^{5} - 2q^{6} + 3q^{7} + 3q^{9} + 2q^{10} + 2q^{11} - 2q^{12} + 6q^{14} - 9q^{15} - 8q^{16} - 2q^{17} + 6q^{18} - 6q^{19} + 2q^{20} - 10q^{21} + 4q^{22} + q^{23} - q^{25} - 7q^{27} + 6q^{28} + 15q^{29} - 18q^{30} - q^{31} - 16q^{32} - q^{33} - 4q^{34} + 10q^{35} + 6q^{36} + 5q^{37} - 12q^{38} - 5q^{41} - 20q^{42} + 4q^{43} + 4q^{44} + 10q^{45} + 2q^{46} - 5q^{47} + 4q^{48} - q^{49} - 2q^{50} + q^{51} + 11q^{53} - 14q^{54} + q^{55} + 20q^{57} + 30q^{58} - 6q^{59} - 18q^{60} - 10q^{61} - 2q^{62} + 13q^{63} - 16q^{64} - 2q^{66} - 4q^{68} + 25q^{69} + 20q^{70} + 7q^{71} + 11q^{73} + 10q^{74} - 8q^{75} - 12q^{76} + 3q^{77} - 6q^{79} - 4q^{80} - 14q^{81} - 10q^{82} - 20q^{84} - q^{85} + 8q^{86} - 16q^{87} + 8q^{89} + 20q^{90} + 2q^{92} + 9q^{93} - 10q^{94} - 20q^{95} + 8q^{96} - 17q^{97} - 2q^{98} + 3q^{99} + O(q^{100})$$

## 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 −2.56155 2.00000 2.56155 −5.12311 3.56155 0 3.56155 5.12311
1.2 2.00000 1.56155 2.00000 −1.56155 3.12311 −0.561553 0 −0.561553 −3.12311
 $$n$$: 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(187))$$:

 $$T_{2} - 2$$ $$T_{3}^{2} + T_{3} - 4$$