Properties

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

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{2} \) \( + ( -2 \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{3} \) \( -\zeta_{24}^{6} q^{4} \) \( + ( 2 - \zeta_{24}^{4} ) q^{6} \) \( + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7} \) \( + \zeta_{24}^{3} q^{8} \) \( + 3 \zeta_{24}^{2} q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{2} \) \( + ( -2 \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{3} \) \( -\zeta_{24}^{6} q^{4} \) \( + ( 2 - \zeta_{24}^{4} ) q^{6} \) \( + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7} \) \( + \zeta_{24}^{3} q^{8} \) \( + 3 \zeta_{24}^{2} q^{9} \) \( + ( -3 + 6 \zeta_{24}^{4} ) q^{11} \) \( + ( -\zeta_{24} + 2 \zeta_{24}^{5} ) q^{12} \) \( + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{14} \) \(- q^{16}\) \( + ( 3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{17} \) \( + ( -3 \zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{18} \) \( -\zeta_{24}^{6} q^{19} \) \( -6 \zeta_{24}^{4} q^{21} \) \( + ( -3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{22} \) \( + 6 \zeta_{24}^{3} q^{23} \) \( + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{24} \) \( + ( -3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{27} \) \( + ( 2 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{28} \) \( -2 q^{31} \) \( + ( \zeta_{24} - \zeta_{24}^{5} ) q^{32} \) \( -9 \zeta_{24}^{7} q^{33} \) \( + 3 \zeta_{24}^{6} q^{34} \) \( + ( 3 - 3 \zeta_{24}^{4} ) q^{36} \) \( + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{37} \) \( + \zeta_{24}^{3} q^{38} \) \( + ( 3 - 6 \zeta_{24}^{4} ) q^{41} \) \( + 6 \zeta_{24} q^{42} \) \( + ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{43} \) \( + ( 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{44} \) \( -6 q^{46} \) \( + ( 2 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{48} \) \( + 5 \zeta_{24}^{6} q^{49} \) \( + ( -6 + 3 \zeta_{24}^{4} ) q^{51} \) \( -6 \zeta_{24}^{3} q^{53} \) \( + ( 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{54} \) \( + ( -2 + 4 \zeta_{24}^{4} ) q^{56} \) \( + ( -\zeta_{24} + 2 \zeta_{24}^{5} ) q^{57} \) \( + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{59} \) \( + 14 q^{61} \) \( + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{62} \) \( + ( 6 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{63} \) \( + \zeta_{24}^{6} q^{64} \) \( + 9 \zeta_{24}^{4} q^{66} \) \( + ( 3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{67} \) \( -3 \zeta_{24}^{3} q^{68} \) \( + ( -6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{69} \) \( + 3 \zeta_{24}^{5} q^{72} \) \( + ( -5 \zeta_{24}^{3} + 10 \zeta_{24}^{7} ) q^{73} \) \( + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{74} \) \(- q^{76}\) \( + ( -18 \zeta_{24} + 18 \zeta_{24}^{5} ) q^{77} \) \( -14 \zeta_{24}^{6} q^{79} \) \( + 9 \zeta_{24}^{4} q^{81} \) \( + ( 3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{82} \) \( -3 \zeta_{24}^{3} q^{83} \) \( + ( -6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{84} \) \( + ( 2 - 4 \zeta_{24}^{4} ) q^{86} \) \( + ( -3 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{88} \) \( + ( 18 \zeta_{24}^{2} - 9 \zeta_{24}^{6} ) q^{89} \) \( + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{92} \) \( + ( 4 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{93} \) \( + ( -2 + \zeta_{24}^{4} ) q^{96} \) \( + ( 4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{97} \) \( -5 \zeta_{24}^{3} q^{98} \) \( + ( -9 \zeta_{24}^{2} + 18 \zeta_{24}^{6} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 12q^{6} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 12q^{6} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 24q^{21} \) \(\mathstrut -\mathstrut 16q^{31} \) \(\mathstrut +\mathstrut 12q^{36} \) \(\mathstrut -\mathstrut 48q^{46} \) \(\mathstrut -\mathstrut 36q^{51} \) \(\mathstrut +\mathstrut 112q^{61} \) \(\mathstrut +\mathstrut 36q^{66} \) \(\mathstrut -\mathstrut 8q^{76} \) \(\mathstrut +\mathstrut 36q^{81} \) \(\mathstrut -\mathstrut 12q^{96} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 + 0.258819i
−0.707107 + 0.707107i −1.67303 0.448288i 1.00000i 0 1.50000 0.866025i 2.44949 + 2.44949i 0.707107 + 0.707107i 2.59808 + 1.50000i 0
107.2 −0.707107 + 0.707107i −0.448288 1.67303i 1.00000i 0 1.50000 + 0.866025i −2.44949 2.44949i 0.707107 + 0.707107i −2.59808 + 1.50000i 0
107.3 0.707107 0.707107i 0.448288 + 1.67303i 1.00000i 0 1.50000 + 0.866025i 2.44949 + 2.44949i −0.707107 0.707107i −2.59808 + 1.50000i 0
107.4 0.707107 0.707107i 1.67303 + 0.448288i 1.00000i 0 1.50000 0.866025i −2.44949 2.44949i −0.707107 0.707107i 2.59808 + 1.50000i 0
143.1 −0.707107 0.707107i −1.67303 + 0.448288i 1.00000i 0 1.50000 + 0.866025i 2.44949 2.44949i 0.707107 0.707107i 2.59808 1.50000i 0
143.2 −0.707107 0.707107i −0.448288 + 1.67303i 1.00000i 0 1.50000 0.866025i −2.44949 + 2.44949i 0.707107 0.707107i −2.59808 1.50000i 0
143.3 0.707107 + 0.707107i 0.448288 1.67303i 1.00000i 0 1.50000 0.866025i 2.44949 2.44949i −0.707107 + 0.707107i −2.59808 1.50000i 0
143.4 0.707107 + 0.707107i 1.67303 0.448288i 1.00000i 0 1.50000 + 0.866025i −2.44949 + 2.44949i −0.707107 + 0.707107i 2.59808 1.50000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.4
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
5.b Even 1 yes
5.c Odd 2 yes
15.d Odd 1 yes
15.e Even 2 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7}^{4} \) \(\mathstrut +\mathstrut 144 \) acting on \(S_{2}^{\mathrm{new}}(150, [\chi])\).