Properties

Label 187.2.a.c
Level 187
Weight 2
Character orbit 187.a
Self dual Yes
Analytic conductor 1.493
Analytic rank 1
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−2.73205 1.73205 5.46410 −3.73205 −4.73205 −2.00000 −9.46410 0 10.1962
1.2 0.732051 −1.73205 −1.46410 −0.267949 −1.26795 −2.00000 −2.53590 0 −0.196152
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(187))\):

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