Properties

Label 150.2.g.b
Level 150
Weight 2
Character orbit 150.g
Analytic conductor 1.198
Analytic rank 0
Dimension 4
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: \(4\)
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.

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

Character Values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 0.951057i 0.809017 0.587785i −0.809017 + 0.587785i 1.80902 + 1.31433i −0.809017 0.587785i 2.00000 0.809017 + 0.587785i 0.309017 0.951057i 0.690983 2.12663i
61.1 0.809017 0.587785i −0.309017 + 0.951057i 0.309017 0.951057i 0.690983 + 2.12663i 0.309017 + 0.951057i 2.00000 −0.309017 0.951057i −0.809017 0.587785i 1.80902 + 1.31433i
91.1 0.809017 + 0.587785i −0.309017 0.951057i 0.309017 + 0.951057i 0.690983 2.12663i 0.309017 0.951057i 2.00000 −0.309017 + 0.951057i −0.809017 + 0.587785i 1.80902 1.31433i
121.1 −0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 0.587785i 1.80902 1.31433i −0.809017 + 0.587785i 2.00000 0.809017 0.587785i 0.309017 + 0.951057i 0.690983 + 2.12663i
\(n\): 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
25.d Even 1 yes

Hecke kernels

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