# Properties

 Label 187.2.g.c Level 187 Weight 2 Character orbit 187.g Analytic conductor 1.493 Analytic rank 0 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ 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 primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
1.80902 1.31433i −0.809017 2.48990i 0.927051 2.85317i 1.61803 + 1.17557i −4.73607 3.44095i −0.500000 + 1.53884i −0.690983 2.12663i −3.11803 + 2.26538i 4.47214
86.1 0.690983 2.12663i 0.309017 0.224514i −2.42705 1.76336i −0.618034 1.90211i −0.263932 0.812299i −0.500000 0.363271i −1.80902 + 1.31433i −0.881966 + 2.71441i −4.47214
103.1 1.80902 + 1.31433i −0.809017 + 2.48990i 0.927051 + 2.85317i 1.61803 1.17557i −4.73607 + 3.44095i −0.500000 1.53884i −0.690983 + 2.12663i −3.11803 2.26538i 4.47214
137.1 0.690983 + 2.12663i 0.309017 + 0.224514i −2.42705 + 1.76336i −0.618034 + 1.90211i −0.263932 + 0.812299i −0.500000 + 0.363271i −1.80902 1.31433i −0.881966 2.71441i −4.47214
 $$n$$: 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 $$S_{2}^{\mathrm{new}}(187, [\chi])$$:

 $$T_{2}^{4} - 5 T_{2}^{3} + 15 T_{2}^{2} - 25 T_{2} + 25$$ $$T_{3}^{4} + T_{3}^{3} + 6 T_{3}^{2} - 4 T_{3} + 1$$