Properties

Label 187.2.d.a
Level 187
Weight 2
Character orbit 187.d
Analytic conductor 1.493
Analytic rank 0
Dimension 16
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 187 = 11 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 187.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.4932025178\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 21 x^{14} + 172 x^{12} + 700 x^{10} + 1492 x^{8} + 1620 x^{6} + 840 x^{4} + 196 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 3 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{15} - 21 \nu^{13} - 172 \nu^{11} - 700 \nu^{9} - 1492 \nu^{7} - 1620 \nu^{5} - 840 \nu^{3} - 188 \nu \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{15} - 41 \nu^{13} - 322 \nu^{11} - 1212 \nu^{9} - 2201 \nu^{7} - 1622 \nu^{5} - 206 \nu^{3} + 38 \nu \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{14} - 41 \nu^{12} - 325 \nu^{10} - 1263 \nu^{8} - 2507 \nu^{6} - 2390 \nu^{4} - 920 \nu^{2} - 100 \)\()/6\)
\(\beta_{6}\)\(=\)\((\)\( -5 \nu^{15} - 101 \nu^{13} - 780 \nu^{11} - 2888 \nu^{9} - 5204 \nu^{7} - 3980 \nu^{5} - 808 \nu^{3} + 4 \nu \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -7 \nu^{14} - 145 \nu^{12} - 1160 \nu^{10} - 4524 \nu^{8} - 8878 \nu^{6} - 8092 \nu^{4} - 2824 \nu^{2} - 296 \)\()/12\)
\(\beta_{8}\)\(=\)\((\)\( 13 \nu^{14} + 265 \nu^{12} + 2078 \nu^{10} + 7908 \nu^{8} + 15058 \nu^{6} + 13180 \nu^{4} + 4300 \nu^{2} + 416 \)\()/12\)
\(\beta_{9}\)\(=\)\((\)\( -3 \nu^{15} + 6 \nu^{14} - 61 \nu^{13} + 124 \nu^{12} - 474 \nu^{11} + 990 \nu^{10} - 1760 \nu^{9} + 3860 \nu^{8} - 3144 \nu^{7} + 7616 \nu^{6} - 2284 \nu^{5} + 7092 \nu^{4} - 312 \nu^{3} + 2632 \nu^{2} + 60 \nu + 304 \)\()/8\)
\(\beta_{10}\)\(=\)\((\)\( -37 \nu^{15} - 763 \nu^{13} - 6074 \nu^{11} - 23580 \nu^{9} - 46156 \nu^{7} - 42184 \nu^{5} - 14896 \nu^{3} - 1604 \nu \)\()/24\)
\(\beta_{11}\)\(=\)\((\)\( -9 \nu^{15} - 34 \nu^{14} - 183 \nu^{13} - 700 \nu^{12} - 1422 \nu^{11} - 5558 \nu^{10} - 5280 \nu^{9} - 21480 \nu^{8} - 9432 \nu^{7} - 41680 \nu^{6} - 6852 \nu^{5} - 37324 \nu^{4} - 936 \nu^{3} - 12400 \nu^{2} + 180 \nu - 1136 \)\()/24\)
\(\beta_{12}\)\(=\)\((\)\( 5 \nu^{14} + 103 \nu^{12} + 819 \nu^{10} + 3176 \nu^{8} + 6213 \nu^{6} + 5682 \nu^{4} + 2012 \nu^{2} + 216 \)\()/2\)
\(\beta_{13}\)\(=\)\((\)\( 19 \nu^{14} + 391 \nu^{12} + 3104 \nu^{10} + 12003 \nu^{8} + 23347 \nu^{6} + 21058 \nu^{4} + 7162 \nu^{2} + 710 \)\()/6\)
\(\beta_{14}\)\(=\)\((\)\( -47 \nu^{15} - 968 \nu^{13} - 7693 \nu^{11} - 29796 \nu^{9} - 58118 \nu^{7} - 52754 \nu^{5} - 18272 \nu^{3} - 1876 \nu \)\()/12\)
\(\beta_{15}\)\(=\)\((\)\( -97 \nu^{15} - 1999 \nu^{13} - 15902 \nu^{11} - 61692 \nu^{9} - 120712 \nu^{7} - 110368 \nu^{5} - 39040 \nu^{3} - 4220 \nu \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{15} - \beta_{14} - \beta_{3} - 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{12} + \beta_{11} - \beta_{9} + \beta_{5} - 7 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(-10 \beta_{15} + 9 \beta_{14} + 3 \beta_{10} - \beta_{4} + 7 \beta_{3} + 20 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-\beta_{13} - 10 \beta_{12} - 10 \beta_{11} + 10 \beta_{9} + \beta_{8} - 3 \beta_{7} - 11 \beta_{5} + 46 \beta_{2} - 85\)
\(\nu^{7}\)\(=\)\(80 \beta_{15} - 65 \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 43 \beta_{10} + \beta_{9} - 2 \beta_{6} + 11 \beta_{4} - 45 \beta_{3} - 110 \beta_{1} - 1\)
\(\nu^{8}\)\(=\)\(11 \beta_{13} + 80 \beta_{12} + 76 \beta_{11} - 76 \beta_{9} - 13 \beta_{8} + 43 \beta_{7} + 93 \beta_{5} - 302 \beta_{2} + 511\)
\(\nu^{9}\)\(=\)\(-594 \beta_{15} + 441 \beta_{14} - 19 \beta_{13} + 19 \beta_{12} - 19 \beta_{11} + 429 \beta_{10} - 19 \beta_{9} + 32 \beta_{6} - 87 \beta_{4} + 283 \beta_{3} + 640 \beta_{1} + 19\)
\(\nu^{10}\)\(=\)\(-87 \beta_{13} - 594 \beta_{12} - 528 \beta_{11} + 528 \beta_{9} + 119 \beta_{8} - 425 \beta_{7} - 719 \beta_{5} + 1996 \beta_{2} - 3189\)
\(\nu^{11}\)\(=\)\(4262 \beta_{15} - 2933 \beta_{14} + 223 \beta_{13} - 223 \beta_{12} + 223 \beta_{11} - 3681 \beta_{10} + 223 \beta_{9} - 342 \beta_{6} + 615 \beta_{4} - 1773 \beta_{3} - 3872 \beta_{1} - 223\)
\(\nu^{12}\)\(=\)\(615 \beta_{13} + 4262 \beta_{12} + 3548 \beta_{11} - 3548 \beta_{9} - 957 \beta_{8} + 3607 \beta_{7} + 5323 \beta_{5} - 13286 \beta_{2} + 20415\)
\(\nu^{13}\)\(=\)\(-30026 \beta_{15} + 19425 \beta_{14} - 2117 \beta_{13} + 2117 \beta_{12} - 2117 \beta_{11} + 29145 \beta_{10} - 2117 \beta_{9} + 3074 \beta_{6} - 4163 \beta_{4} + 11169 \beta_{3} + 24116 \beta_{1} + 2117\)
\(\nu^{14}\)\(=\)\(-4163 \beta_{13} - 30026 \beta_{12} - 23588 \beta_{11} + 23588 \beta_{9} + 7237 \beta_{8} - 28275 \beta_{7} - 38423 \beta_{5} + 88970 \beta_{2} - 133039\)
\(\nu^{15}\)\(=\)\(209282 \beta_{15} - 128909 \beta_{14} + 17909 \beta_{13} - 17909 \beta_{12} + 17909 \beta_{11} - 219917 \beta_{10} + 17909 \beta_{9} - 25146 \beta_{6} + 27751 \beta_{4} - 71061 \beta_{3} - 153560 \beta_{1} - 17909\)

Character Values

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

\(n\) \(35\) \(122\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
2.61227i
2.61227i
1.77835i
1.77835i
1.19202i
1.19202i
0.622470i
0.622470i
0.420099i
0.420099i
0.651198i
0.651198i
1.89309i
1.89309i
2.24072i
2.24072i
−2.61227 1.70629i 4.82394 0.846648i 4.45730i 3.51966i −7.37688 0.0885594 2.21167i
67.2 −2.61227 1.70629i 4.82394 0.846648i 4.45730i 3.51966i −7.37688 0.0885594 2.21167i
67.3 −1.77835 2.30556i 1.16253 0.346288i 4.10009i 3.32341i 1.48932 −2.31560 0.615822i
67.4 −1.77835 2.30556i 1.16253 0.346288i 4.10009i 3.32341i 1.48932 −2.31560 0.615822i
67.5 −1.19202 1.00943i −0.579089 1.48581i 1.20326i 1.55528i 3.07432 1.98105 1.77111i
67.6 −1.19202 1.00943i −0.579089 1.48581i 1.20326i 1.55528i 3.07432 1.98105 1.77111i
67.7 −0.622470 2.95438i −1.61253 3.59054i 1.83901i 3.45035i 2.24869 −5.72834 2.23500i
67.8 −0.622470 2.95438i −1.61253 3.59054i 1.83901i 3.45035i 2.24869 −5.72834 2.23500i
67.9 0.420099 0.0890922i −1.82352 3.34068i 0.0374276i 2.13044i −1.60626 2.99206 1.40342i
67.10 0.420099 0.0890922i −1.82352 3.34068i 0.0374276i 2.13044i −1.60626 2.99206 1.40342i
67.11 0.651198 2.48452i −1.57594 1.42007i 1.61791i 5.16858i −2.32864 −3.17284 0.924744i
67.12 0.651198 2.48452i −1.57594 1.42007i 1.61791i 5.16858i −2.32864 −3.17284 0.924744i
67.13 1.89309 1.04073i 1.58380 1.83662i 1.97021i 1.61788i −0.787912 1.91687 3.47689i
67.14 1.89309 1.04073i 1.58380 1.83662i 1.97021i 1.61788i −0.787912 1.91687 3.47689i
67.15 2.24072 2.96003i 3.02082 2.34815i 6.63259i 2.31788i 2.28736 −5.76177 5.26153i
67.16 2.24072 2.96003i 3.02082 2.34815i 6.63259i 2.31788i 2.28736 −5.76177 5.26153i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.16
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
17.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(187, [\chi])\).