Properties

Label 187.2.a.e
Level 187
Weight 2
Character orbit 187.a
Self dual Yes
Analytic conductor 1.493
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 187 = 11 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 187.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.4932025178\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.48119
0.311108
2.17009
−2.48119 −2.67513 4.15633 −0.518806 6.63752 3.35026 −5.35026 4.15633 1.28726
1.2 −0.688892 1.21432 −1.52543 −2.31111 −0.836535 −4.42864 2.42864 −1.52543 1.59210
1.3 1.17009 −1.53919 −0.630898 −4.17009 −1.80098 1.07838 −3.07838 −0.630898 −4.87936
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)
\(17\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(187))\):

\( T_{2}^{3} + 2 T_{2}^{2} - 2 T_{2} - 2 \)
\( T_{3}^{3} + 3 T_{3}^{2} - T_{3} - 5 \)