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

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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:

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 \(S_{2}^{\mathrm{new}}(187, [\chi])\):

\( T_{2}^{4} - 2 T_{2}^{3} + 4 T_{2}^{2} - 3 T_{2} + 1 \)
\( T_{3}^{4} + 4 T_{3}^{3} + 6 T_{3}^{2} - T_{3} + 1 \)