# Properties

 Label 187.2.g.a Level 187 Weight 2 Character orbit 187.g Analytic conductor 1.493 Analytic rank 1 Dimension 4 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: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character Values

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

 $$n$$ $$35$$ $$122$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$ $$1$$

## 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
 0.809017 + 0.587785i −0.309017 − 0.951057i 0.809017 − 0.587785i −0.309017 + 0.951057i
−0.500000 + 0.363271i −0.690983 2.12663i −0.500000 + 1.53884i −1.80902 1.31433i 1.11803 + 0.812299i −0.881966 + 2.71441i −0.690983 2.12663i −1.61803 + 1.17557i 1.38197
86.1 −0.500000 + 1.53884i −1.80902 + 1.31433i −0.500000 0.363271i −0.690983 2.12663i −1.11803 3.44095i −3.11803 2.26538i −1.80902 + 1.31433i 0.618034 1.90211i 3.61803
103.1 −0.500000 0.363271i −0.690983 + 2.12663i −0.500000 1.53884i −1.80902 + 1.31433i 1.11803 0.812299i −0.881966 2.71441i −0.690983 + 2.12663i −1.61803 1.17557i 1.38197
137.1 −0.500000 1.53884i −1.80902 1.31433i −0.500000 + 0.363271i −0.690983 + 2.12663i −1.11803 + 3.44095i −3.11803 + 2.26538i −1.80902 1.31433i 0.618034 + 1.90211i 3.61803
 $$n$$: e.g. 2-40 or 990-1000 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 $$S_{2}^{\mathrm{new}}(187, [\chi])$$:

 $$T_{2}^{4} + 2 T_{2}^{3} + 4 T_{2}^{2} + 3 T_{2} + 1$$ $$T_{3}^{4} + 5 T_{3}^{3} + 15 T_{3}^{2} + 25 T_{3} + 25$$