Properties

Label 187.2.e.a
Level 187
Weight 2
Character orbit 187.e
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.e (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.4932025178\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

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

\(n\) \(35\) \(122\)
\(\chi(n)\) \(1\) \(\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
2.00000i −2.41421 + 2.41421i −2.00000 2.00000 2.00000i 4.82843 + 4.82843i −0.707107 0.707107i 0 8.65685i −4.00000 4.00000i
89.2 2.00000i 0.414214 0.414214i −2.00000 2.00000 2.00000i −0.828427 0.828427i 0.707107 + 0.707107i 0 2.65685i −4.00000 4.00000i
166.1 2.00000i −2.41421 2.41421i −2.00000 2.00000 + 2.00000i 4.82843 4.82843i −0.707107 + 0.707107i 0 8.65685i −4.00000 + 4.00000i
166.2 2.00000i 0.414214 + 0.414214i −2.00000 2.00000 + 2.00000i −0.828427 + 0.828427i 0.707107 0.707107i 0 2.65685i −4.00000 + 4.00000i
\(n\): 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
17.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(187, [\chi])\).