# Properties

 Label 187.2.g.e Level 187 Weight 2 Character orbit 187.g Analytic conductor 1.493 Analytic rank 0 Dimension 8 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$187 = 11 \cdot 17$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 187.g (of order $$5$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.4932025178$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.13140625.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{5} ) q^{2} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{3} + ( 1 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{4} + ( -1 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( 1 + \beta_{7} ) q^{6} + ( -\beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{7} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{8} + ( 1 - \beta_{2} - \beta_{3} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{5} ) q^{2} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{3} + ( 1 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{4} + ( -1 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( 1 + \beta_{7} ) q^{6} + ( -\beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{7} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{8} + ( 1 - \beta_{2} - \beta_{3} + \beta_{6} ) q^{9} + ( 1 + \beta_{1} + \beta_{2} - \beta_{5} ) q^{10} + ( 1 - 3 \beta_{3} - \beta_{4} + \beta_{7} ) q^{11} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{7} ) q^{12} + ( 4 \beta_{1} + 3 \beta_{2} + \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{13} + ( 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{14} + ( -2 - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{15} + ( 2 + \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{16} -\beta_{4} q^{17} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{18} + ( -2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{19} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{20} + ( 1 - \beta_{5} ) q^{21} + ( -5 \beta_{1} - 2 \beta_{2} + 3 \beta_{5} + 3 \beta_{6} ) q^{22} + ( -4 + 4 \beta_{3} - 4 \beta_{5} - 4 \beta_{7} ) q^{23} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{24} + ( \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{25} + ( -4 - 3 \beta_{1} + 5 \beta_{3} - 4 \beta_{4} + \beta_{5} + \beta_{6} ) q^{26} + ( 2 - \beta_{2} + \beta_{4} + \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{27} + ( 2 + 2 \beta_{2} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{28} + ( 4 - 2 \beta_{1} - 4 \beta_{3} + 4 \beta_{4} ) q^{29} + ( -\beta_{2} - \beta_{7} ) q^{30} + ( 3 + 3 \beta_{1} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{31} + ( -4 + \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{32} + ( -1 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{33} + ( -\beta_{1} - \beta_{2} + \beta_{5} ) q^{34} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{35} + ( \beta_{1} - 2 \beta_{2} - \beta_{6} - 2 \beta_{7} ) q^{36} + ( -3 - 3 \beta_{1} - \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{37} + ( -5 - 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 5 \beta_{7} ) q^{38} + ( \beta_{2} + 5 \beta_{4} - \beta_{5} - \beta_{6} ) q^{39} + ( 2 + \beta_{1} + 2 \beta_{4} ) q^{40} + ( 2 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} + 4 \beta_{7} ) q^{41} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{42} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} + 2 \beta_{7} ) q^{43} + ( -2 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{6} - 3 \beta_{7} ) q^{44} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} ) q^{45} + ( -4 + 4 \beta_{1} + 4 \beta_{3} - 4 \beta_{5} ) q^{46} + ( 7 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 7 \beta_{6} + 4 \beta_{7} ) q^{47} + ( 2 + \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{48} + ( -1 - 3 \beta_{2} - 5 \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{49} + ( -1 + 3 \beta_{2} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{50} + ( -1 + \beta_{1} - \beta_{4} ) q^{51} + ( 4 \beta_{1} - 5 \beta_{3} + 5 \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{52} + ( -3 - \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + \beta_{5} + 4 \beta_{6} ) q^{53} + ( 5 - 4 \beta_{3} - \beta_{5} + 4 \beta_{7} ) q^{54} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} + 4 \beta_{4} - 4 \beta_{6} + 3 \beta_{7} ) q^{55} + ( -1 + \beta_{3} + 3 \beta_{5} - \beta_{7} ) q^{56} + ( 7 + 2 \beta_{2} - 7 \beta_{3} + 7 \beta_{4} - 2 \beta_{6} + 7 \beta_{7} ) q^{57} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{6} ) q^{58} + ( -6 + \beta_{1} + 7 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{59} + ( -3 - \beta_{2} - 6 \beta_{4} + \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{60} + ( -3 - 3 \beta_{2} + 3 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{61} + ( -3 - 8 \beta_{1} + 6 \beta_{3} - 3 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} ) q^{62} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{63} + ( 9 \beta_{1} + 7 \beta_{2} + 2 \beta_{4} - 9 \beta_{5} - 7 \beta_{6} + 2 \beta_{7} ) q^{64} + ( -2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{5} - 4 \beta_{7} ) q^{65} + ( 3 - \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{66} + ( -3 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} ) q^{67} + ( \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{68} + ( -4 \beta_{1} + 4 \beta_{6} - 8 \beta_{7} ) q^{69} + ( -1 - 3 \beta_{1} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{70} + ( -4 - 3 \beta_{2} - 11 \beta_{4} + 3 \beta_{5} - 4 \beta_{7} ) q^{71} + ( 3 - \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{72} + ( 7 - \beta_{1} + 7 \beta_{4} ) q^{73} + ( 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} - 4 \beta_{7} ) q^{74} + ( 3 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{75} + ( 5 + 5 \beta_{1} + 5 \beta_{2} - 7 \beta_{3} - 5 \beta_{5} + 7 \beta_{7} ) q^{76} + ( -2 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + \beta_{6} - \beta_{7} ) q^{77} + ( -1 + 4 \beta_{1} + 4 \beta_{2} - 5 \beta_{5} ) q^{78} + ( -4 - 7 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} - 9 \beta_{4} + 7 \beta_{5} + 8 \beta_{6} - 9 \beta_{7} ) q^{79} + ( -\beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 6 \beta_{7} ) q^{80} + ( 3 + \beta_{1} + 3 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} ) q^{81} + ( -2 + \beta_{2} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{82} + ( 4 + 5 \beta_{2} - 3 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{83} + ( 2 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{84} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{85} + ( -6 \beta_{1} - 4 \beta_{2} + \beta_{4} + 6 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{86} + ( 6 - 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{5} ) q^{87} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{88} + ( 2 + 3 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - 7 \beta_{5} - 5 \beta_{7} ) q^{89} + ( 2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{90} + ( -2 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{91} + ( 4 - 4 \beta_{1} + 4 \beta_{4} ) q^{92} + ( -2 - 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{93} + ( 5 + 2 \beta_{2} - 7 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} ) q^{94} + ( 5 + 3 \beta_{1} + 5 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{95} + ( -\beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{6} + 4 \beta_{7} ) q^{96} + ( 6 - 5 \beta_{1} - 6 \beta_{3} + 9 \beta_{4} + 5 \beta_{5} + 9 \beta_{7} ) q^{97} + ( 6 - \beta_{1} - \beta_{2} - 3 \beta_{3} + 3 \beta_{7} ) q^{98} + ( 3 + 4 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - q^{2} + 3q^{3} + 5q^{4} - 3q^{5} + 6q^{6} + 3q^{7} + 6q^{8} + 5q^{9} + O(q^{10})$$ $$8q - q^{2} + 3q^{3} + 5q^{4} - 3q^{5} + 6q^{6} + 3q^{7} + 6q^{8} + 5q^{9} + 12q^{10} + 2q^{11} + 2q^{12} + 3q^{13} - 10q^{15} + 7q^{16} + 2q^{17} + 4q^{18} - 5q^{19} - 4q^{20} + 6q^{21} - 9q^{22} - 24q^{23} - 7q^{24} - 3q^{25} - 19q^{26} + 3q^{27} + 16q^{28} + 10q^{29} - q^{30} + 17q^{31} - 24q^{32} - 8q^{33} - 4q^{34} + 6q^{35} - q^{36} - 13q^{37} - 38q^{38} - 11q^{39} + 15q^{40} - 22q^{41} - 9q^{42} + 8q^{43} - 5q^{44} - 24q^{45} - 20q^{46} + 9q^{47} + 13q^{48} + q^{49} - q^{50} - 3q^{51} - 18q^{52} - 23q^{53} + 22q^{54} - 12q^{55} + 2q^{56} + 16q^{57} - 6q^{58} - 35q^{59} - q^{60} - 19q^{61} - 6q^{62} - 4q^{63} + 8q^{64} + 10q^{65} + 24q^{66} - 10q^{67} + 5q^{68} + 12q^{69} + 3q^{70} - 5q^{71} + 19q^{72} + 39q^{73} - 7q^{74} + 6q^{75} + 32q^{76} - 3q^{77} + 6q^{78} - 3q^{79} + 21q^{80} + 7q^{81} - 5q^{82} + 29q^{83} + 8q^{84} - 2q^{85} - 14q^{86} + 32q^{87} + 9q^{88} + 40q^{89} + 9q^{90} + 21q^{91} + 12q^{92} - 14q^{93} + 40q^{94} + 23q^{95} - 12q^{96} - 5q^{97} + 30q^{98} + 20q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} - 3 \nu^{5} - 4 \nu^{3} - 7 \nu^{2} - 12 \nu - 7$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 7 \nu^{5} + 20 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} + 6 \nu + 9$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{6} - 9 \nu^{5} + 12 \nu^{4} - 16 \nu^{3} + 13 \nu^{2} - 10 \nu - 1$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 10 \nu^{6} - 17 \nu^{5} + 8 \nu^{4} - 4 \nu^{3} - 13 \nu^{2} - 8 \nu - 5$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} - 12 \nu^{6} + 23 \nu^{5} - 20 \nu^{4} + 16 \nu^{3} + \nu^{2} + 6 \nu - 1$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 18 \nu^{6} - 35 \nu^{5} + 32 \nu^{4} - 28 \nu^{3} - 11 \nu^{2} - 12 \nu - 7$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 4 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{7} - 6 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} - 1$$ $$\nu^{6}$$ $$=$$ $$-16 \beta_{6} - 16 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 7 \beta_{1} - 6$$ $$\nu^{7}$$ $$=$$ $$-16 \beta_{7} - 51 \beta_{6} - 29 \beta_{5} - 23 \beta_{4} + 29 \beta_{2} - 16$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/187\mathbb{Z}\right)^\times$$.

 $$n$$ $$35$$ $$122$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
69.1
 0.418926 − 1.28932i −0.227943 + 0.701538i −0.386111 − 0.280526i 1.69513 + 1.23158i 0.418926 + 1.28932i −0.227943 − 0.701538i −0.386111 + 0.280526i 1.69513 − 1.23158i
−1.77460 + 1.28932i −0.227943 0.701538i 0.868820 2.67395i −1.17784 0.855749i 1.30902 + 0.951057i −0.986854 + 3.03722i 0.550106 + 1.69305i 1.98685 1.44353i 3.19353
69.2 0.965584 0.701538i 0.418926 + 1.28932i −0.177837 + 0.547326i −0.131180 0.0953077i 1.30902 + 0.951057i 0.0598032 0.184055i 0.949894 + 2.92347i 0.940197 0.683093i −0.193527
86.1 −0.0911485 + 0.280526i 1.69513 1.23158i 1.54765 + 1.12443i −0.738630 2.27327i 0.190983 + 0.587785i 0.570387 + 0.414410i −0.933758 + 0.678415i 0.429613 1.32221i 0.705037
86.2 0.400166 1.23158i −0.386111 + 0.280526i 0.261370 + 0.189896i 0.547647 + 1.68548i 0.190983 + 0.587785i 1.85666 + 1.34895i 2.43376 1.76823i −0.856664 + 2.63654i 2.29496
103.1 −1.77460 1.28932i −0.227943 + 0.701538i 0.868820 + 2.67395i −1.17784 + 0.855749i 1.30902 0.951057i −0.986854 3.03722i 0.550106 1.69305i 1.98685 + 1.44353i 3.19353
103.2 0.965584 + 0.701538i 0.418926 1.28932i −0.177837 0.547326i −0.131180 + 0.0953077i 1.30902 0.951057i 0.0598032 + 0.184055i 0.949894 2.92347i 0.940197 + 0.683093i −0.193527
137.1 −0.0911485 0.280526i 1.69513 + 1.23158i 1.54765 1.12443i −0.738630 + 2.27327i 0.190983 0.587785i 0.570387 0.414410i −0.933758 0.678415i 0.429613 + 1.32221i 0.705037
137.2 0.400166 + 1.23158i −0.386111 0.280526i 0.261370 0.189896i 0.547647 1.68548i 0.190983 0.587785i 1.85666 1.34895i 2.43376 + 1.76823i −0.856664 2.63654i 2.29496
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 137.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
11.c Even 1 yes

## Hecke kernels

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

 $$T_{2}^{8} + T_{2}^{7} - T_{2}^{5} + 9 T_{2}^{4} - 11 T_{2}^{3} + 10 T_{2}^{2} + T_{2} + 1$$ $$T_{3}^{8} - \cdots$$