Properties

Label 150.2.g.c
Level 150
Weight 2
Character orbit 150.g
Analytic conductor 1.198
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 150.g (of order \(5\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.1064390625.3
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(2\) \(x^{7}\mathstrut -\mathstrut \) \(3\) \(x^{6}\mathstrut -\mathstrut \) \(5\) \(x^{5}\mathstrut +\mathstrut \) \(36\) \(x^{4}\mathstrut -\mathstrut \) \(35\) \(x^{3}\mathstrut +\mathstrut \) \(23\) \(x^{2}\mathstrut -\mathstrut \) \(171\) \(x\mathstrut +\mathstrut \) \(361\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -27571 \nu^{7} + 156 \nu^{6} - 50800 \nu^{5} + 197116 \nu^{4} - 261151 \nu^{3} + 1281772 \nu^{2} - 1105295 \nu + 4276691 \)\()/4238558\)
\(\beta_{3}\)\(=\)\((\)\( -45697 \nu^{7} - 18958 \nu^{6} + 87368 \nu^{5} + 389890 \nu^{4} - 351515 \nu^{3} + 1275844 \nu^{2} - 687257 \nu + 5880747 \)\()/4238558\)
\(\beta_{4}\)\(=\)\((\)\( 2894 \nu^{7} + 7027 \nu^{6} - 3119 \nu^{5} - 38495 \nu^{4} - 16673 \nu^{3} + 24798 \nu^{2} + 23050 \nu - 523849 \)\()/223082\)
\(\beta_{5}\)\(=\)\((\)\( 60697 \nu^{7} + 29865 \nu^{6} - 107003 \nu^{5} - 736723 \nu^{4} + 989612 \nu^{3} + 38090 \nu^{2} + 939005 \nu - 13281190 \)\()/4238558\)
\(\beta_{6}\)\(=\)\((\)\( -72436 \nu^{7} - 26109 \nu^{6} + 188067 \nu^{5} + 265983 \nu^{4} - 1082509 \nu^{3} + 501044 \nu^{2} + 3422970 \nu + 7026371 \)\()/4238558\)
\(\beta_{7}\)\(=\)\((\)\( 5808 \nu^{7} + 2617 \nu^{6} - 8495 \nu^{5} - 68083 \nu^{4} + 17029 \nu^{3} - 19146 \nu^{2} + 101760 \nu - 868243 \)\()/223082\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\)
\(\nu^{3}\)\(=\)\(-\)\(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\)
\(\nu^{4}\)\(=\)\(-\)\(8\) \(\beta_{6}\mathstrut -\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut -\mathstrut \) \(12\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(39\) \(\beta_{3}\mathstrut -\mathstrut \) \(39\) \(\beta_{2}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(12\)
\(\nu^{6}\)\(=\)\(-\)\(39\) \(\beta_{7}\mathstrut +\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(39\) \(\beta_{4}\mathstrut -\mathstrut \) \(26\) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)
\(\nu^{7}\)\(=\)\(71\) \(\beta_{7}\mathstrut -\mathstrut \) \(100\) \(\beta_{6}\mathstrut -\mathstrut \) \(101\) \(\beta_{5}\mathstrut +\mathstrut \) \(164\) \(\beta_{3}\mathstrut -\mathstrut \) \(101\) \(\beta_{2}\mathstrut +\mathstrut \) \(71\) \(\beta_{1}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.815575 1.64827i
1.31557 + 1.28500i
−1.86886 1.45788i
2.36886 0.0809628i
−1.86886 + 1.45788i
2.36886 + 0.0809628i
−0.815575 + 1.64827i
1.31557 1.28500i
−0.309017 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i −2.12459 0.697217i 0.809017 + 0.587785i −4.25729 0.809017 + 0.587785i 0.309017 0.951057i −0.00655751 + 2.23606i
31.2 −0.309017 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.00655751 + 2.23606i 0.809017 + 0.587785i 2.63925 0.809017 + 0.587785i 0.309017 0.951057i 2.12459 0.697217i
61.1 0.809017 0.587785i 0.309017 0.951057i 0.309017 0.951057i −2.05984 0.870094i −0.309017 0.951057i 2.92807 −0.309017 0.951057i −0.809017 0.587785i −2.17787 + 0.506822i
61.2 0.809017 0.587785i 0.309017 0.951057i 0.309017 0.951057i 2.17787 + 0.506822i −0.309017 0.951057i −2.31003 −0.309017 0.951057i −0.809017 0.587785i 2.05984 0.870094i
91.1 0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i −2.05984 + 0.870094i −0.309017 + 0.951057i 2.92807 −0.309017 + 0.951057i −0.809017 + 0.587785i −2.17787 0.506822i
91.2 0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i 2.17787 0.506822i −0.309017 + 0.951057i −2.31003 −0.309017 + 0.951057i −0.809017 + 0.587785i 2.05984 + 0.870094i
121.1 −0.309017 + 0.951057i −0.809017 0.587785i −0.809017 0.587785i −2.12459 + 0.697217i 0.809017 0.587785i −4.25729 0.809017 0.587785i 0.309017 + 0.951057i −0.00655751 2.23606i
121.2 −0.309017 + 0.951057i −0.809017 0.587785i −0.809017 0.587785i 0.00655751 2.23606i 0.809017 0.587785i 2.63925 0.809017 0.587785i 0.309017 + 0.951057i 2.12459 + 0.697217i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
25.d Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7}^{4} \) \(\mathstrut +\mathstrut T_{7}^{3} \) \(\mathstrut -\mathstrut 19 T_{7}^{2} \) \(\mathstrut -\mathstrut 4 T_{7} \) \(\mathstrut +\mathstrut 76 \) acting on \(S_{2}^{\mathrm{new}}(150, [\chi])\).