# Properties

 Label 187.2.g.b 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: $$\Z[a_1, a_2]$$ 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.

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

## 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
0.500000 0.363271i 0.118034 + 0.363271i −0.500000 + 1.53884i 0.309017 + 0.224514i 0.190983 + 0.138757i −0.881966 + 2.71441i 0.690983 + 2.12663i 2.30902 1.67760i 0.236068
86.1 0.500000 1.53884i −2.11803 + 1.53884i −0.500000 0.363271i −0.809017 2.48990i 1.30902 + 4.02874i −3.11803 2.26538i 1.80902 1.31433i 1.19098 3.66547i −4.23607
103.1 0.500000 + 0.363271i 0.118034 0.363271i −0.500000 1.53884i 0.309017 0.224514i 0.190983 0.138757i −0.881966 2.71441i 0.690983 2.12663i 2.30902 + 1.67760i 0.236068
137.1 0.500000 + 1.53884i −2.11803 1.53884i −0.500000 + 0.363271i −0.809017 + 2.48990i 1.30902 4.02874i −3.11803 + 2.26538i 1.80902 + 1.31433i 1.19098 + 3.66547i −4.23607
 $$n$$: 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 $$S_{2}^{\mathrm{new}}(187, [\chi])$$:

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