Properties

Label 150.2.h.a
Level 150
Weight 2
Character orbit 150.h
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.h (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(\zeta_{20}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.951057 + 0.309017i
0.951057 0.309017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.951057 + 0.309017i −0.587785 0.809017i 0.809017 0.587785i −2.06909 + 0.847859i 0.809017 + 0.587785i 4.07768i −0.587785 + 0.809017i −0.309017 + 0.951057i 1.70582 1.44575i
19.2 0.951057 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i −0.166977 + 2.22982i 0.809017 + 0.587785i 2.07768i 0.587785 0.809017i −0.309017 + 0.951057i 0.530249 + 2.17229i
79.1 −0.951057 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i −2.06909 0.847859i 0.809017 0.587785i 4.07768i −0.587785 0.809017i −0.309017 0.951057i 1.70582 + 1.44575i
79.2 0.951057 + 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i −0.166977 2.22982i 0.809017 0.587785i 2.07768i 0.587785 + 0.809017i −0.309017 0.951057i 0.530249 2.17229i
109.1 −0.587785 + 0.809017i 0.951057 0.309017i −0.309017 0.951057i 0.530249 + 2.17229i −0.309017 + 0.951057i 0.273457i 0.951057 + 0.309017i 0.809017 0.587785i −2.06909 0.847859i
109.2 0.587785 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i 1.70582 1.44575i −0.309017 + 0.951057i 1.72654i −0.951057 0.309017i 0.809017 0.587785i −0.166977 2.22982i
139.1 −0.587785 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 0.530249 2.17229i −0.309017 0.951057i 0.273457i 0.951057 0.309017i 0.809017 + 0.587785i −2.06909 + 0.847859i
139.2 0.587785 + 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i 1.70582 + 1.44575i −0.309017 0.951057i 1.72654i −0.951057 + 0.309017i 0.809017 + 0.587785i −0.166977 + 2.22982i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.2
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{7}^{8} + 24 T_{7}^{6} + 136 T_{7}^{4} + 224 T_{7}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(150, [\chi])\).