Properties

Label 187.2.a.f
Level 187
Weight 2
Character orbit 187.a
Self dual Yes
Analytic conductor 1.493
Analytic rank 0
Dimension 4
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.33844.1
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6 x^{2} + 2 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 6 \beta_{1} + 2\)

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
−2.09787
−0.452661
0.752785
2.79774
−2.09787 2.14452 2.40105 3.09787 −4.49891 0 −0.841339 1.59895 −6.49891
1.2 −0.452661 −2.96565 −1.79510 1.45266 1.34244 0 1.71789 5.79510 −0.657564
1.3 0.752785 2.90402 −1.43332 0.247215 2.18610 0 −2.58455 5.43332 0.186100
1.4 2.79774 −1.08288 5.82737 −1.79774 −3.02962 0 10.7080 −1.82737 −5.02962
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(187))\):

\( T_{2}^{4} - T_{2}^{3} - 6 T_{2}^{2} + 2 T_{2} + 2 \)
\( T_{3}^{4} - T_{3}^{3} - 11 T_{3}^{2} + 9 T_{3} + 20 \)