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

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

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

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 \( T_{7}^{4} + T_{7}^{3} - 19 T_{7}^{2} - 4 T_{7} + 76 \) acting on \(S_{2}^{\mathrm{new}}(150, [\chi])\).