# Properties

 Label 187.2.g.d 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: 8.0.324000000.3 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 9 x^{4} + 27 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/9$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/9$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/27$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$27 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$27 \beta_{7}$$

## Character Values

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

 $$n$$ $$35$$ $$122$$ $$\chi(n)$$ $$\beta_{6}$$ $$1$$

## 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
 −0.535233 + 1.64728i 0.535233 − 1.64728i −1.40126 − 1.01807i 1.40126 + 1.01807i −0.535233 − 1.64728i 0.535233 + 1.64728i −1.40126 + 1.01807i 1.40126 − 1.01807i
−0.500000 + 0.363271i −0.344250 1.05949i −0.500000 + 1.53884i −0.901259 0.654803i 0.557008 + 0.404690i 1.34425 4.13718i −0.690983 2.12663i 1.42303 1.03389i 0.688500
69.2 −0.500000 + 0.363271i 0.726216 + 2.23506i −0.500000 + 1.53884i 1.90126 + 1.38135i −1.17504 0.853718i 0.273784 0.842620i −0.690983 2.12663i −2.04107 + 1.48292i −1.45243
86.1 −0.500000 + 1.53884i −0.0922415 + 0.0670174i −0.500000 0.363271i −0.0352331 0.108436i −0.0570084 0.175454i 1.09224 + 0.793560i −1.80902 + 1.31433i −0.923034 + 2.84081i 0.184483
86.2 −0.500000 + 1.53884i 2.71028 1.96913i −0.500000 0.363271i 1.03523 + 3.18612i 1.67504 + 5.15525i −1.71028 1.24259i −1.80902 + 1.31433i 2.54107 7.82060i −5.42055
103.1 −0.500000 0.363271i −0.344250 + 1.05949i −0.500000 1.53884i −0.901259 + 0.654803i 0.557008 0.404690i 1.34425 + 4.13718i −0.690983 + 2.12663i 1.42303 + 1.03389i 0.688500
103.2 −0.500000 0.363271i 0.726216 2.23506i −0.500000 1.53884i 1.90126 1.38135i −1.17504 + 0.853718i 0.273784 + 0.842620i −0.690983 + 2.12663i −2.04107 1.48292i −1.45243
137.1 −0.500000 1.53884i −0.0922415 0.0670174i −0.500000 + 0.363271i −0.0352331 + 0.108436i −0.0570084 + 0.175454i 1.09224 0.793560i −1.80902 1.31433i −0.923034 2.84081i 0.184483
137.2 −0.500000 1.53884i 2.71028 + 1.96913i −0.500000 + 0.363271i 1.03523 3.18612i 1.67504 5.15525i −1.71028 + 1.24259i −1.80902 1.31433i 2.54107 + 7.82060i −5.42055
 $$n$$: 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 $$S_{2}^{\mathrm{new}}(187, [\chi])$$:

 $$T_{2}^{4} + 2 T_{2}^{3} + 4 T_{2}^{2} + 3 T_{2} + 1$$ $$T_{3}^{8} - \cdots$$