Properties

Label 462.2.k.f
Level $462$
Weight $2$
Character orbit 462.k
Analytic conductor $3.689$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
−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
−0.965926 0.258819i
−0.866025 0.500000i 0.292893 1.70711i 0.500000 + 0.866025i 0.758819 1.31431i −1.10721 + 1.33195i −1.48356 + 2.19067i 1.00000i −2.82843 1.00000i −1.31431 + 0.758819i
89.2 −0.866025 0.500000i 1.70711 0.292893i 0.500000 + 0.866025i 0.241181 0.417738i −1.62484 0.599900i 1.48356 2.19067i 1.00000i 2.82843 1.00000i −0.417738 + 0.241181i
89.3 0.866025 + 0.500000i 0.292893 + 1.70711i 0.500000 + 0.866025i −0.465926 + 0.807007i −0.599900 + 1.62484i 2.19067 + 1.48356i 1.00000i −2.82843 + 1.00000i −0.807007 + 0.465926i
89.4 0.866025 + 0.500000i 1.70711 + 0.292893i 0.500000 + 0.866025i 1.46593 2.53906i 1.33195 + 1.10721i −2.19067 1.48356i 1.00000i 2.82843 + 1.00000i 2.53906 1.46593i
353.1 −0.866025 + 0.500000i 0.292893 + 1.70711i 0.500000 0.866025i 0.758819 + 1.31431i −1.10721 1.33195i −1.48356 2.19067i 1.00000i −2.82843 + 1.00000i −1.31431 0.758819i
353.2 −0.866025 + 0.500000i 1.70711 + 0.292893i 0.500000 0.866025i 0.241181 + 0.417738i −1.62484 + 0.599900i 1.48356 + 2.19067i 1.00000i 2.82843 + 1.00000i −0.417738 0.241181i
353.3 0.866025 0.500000i 0.292893 1.70711i 0.500000 0.866025i −0.465926 0.807007i −0.599900 1.62484i 2.19067 1.48356i 1.00000i −2.82843 1.00000i −0.807007 0.465926i
353.4 0.866025 0.500000i 1.70711 0.292893i 0.500000 0.866025i 1.46593 + 2.53906i 1.33195 1.10721i −2.19067 + 1.48356i 1.00000i 2.82843 1.00000i 2.53906 + 1.46593i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.k.f yes 8
3.b odd 2 1 462.2.k.e 8
7.d odd 6 1 462.2.k.e 8
21.g even 6 1 inner 462.2.k.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.k.e 8 3.b odd 2 1
462.2.k.e 8 7.d odd 6 1
462.2.k.f yes 8 1.a even 1 1 trivial
462.2.k.f yes 8 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\(T_{5}^{8} - \cdots\)
\(T_{17}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( ( 9 - 12 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$5$ \( 4 - 8 T + 20 T^{2} - 8 T^{3} + 22 T^{4} - 16 T^{5} + 14 T^{6} - 4 T^{7} + T^{8} \)
$7$ \( 2401 + 71 T^{4} + T^{8} \)
$11$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$13$ \( ( 9 + 30 T^{2} + T^{4} )^{2} \)
$17$ \( 141376 + 150400 T + 107360 T^{2} + 40960 T^{3} + 11224 T^{4} + 2000 T^{5} + 260 T^{6} + 20 T^{7} + T^{8} \)
$19$ \( 4 + 24 T + 52 T^{2} + 24 T^{3} - 42 T^{4} - 24 T^{5} + 46 T^{6} + 12 T^{7} + T^{8} \)
$23$ \( 4 - 24 T + 36 T^{2} + 72 T^{3} - 10 T^{4} - 72 T^{5} + 54 T^{6} - 12 T^{7} + T^{8} \)
$29$ \( 187489 + 49836 T^{2} + 3782 T^{4} + 108 T^{6} + T^{8} \)
$31$ \( 153664 - 263424 T + 170912 T^{2} - 34944 T^{3} + 408 T^{4} + 624 T^{5} - 4 T^{6} - 12 T^{7} + T^{8} \)
$37$ \( 145924 - 32088 T + 29212 T^{2} + 4872 T^{3} + 2982 T^{4} + 168 T^{5} + 58 T^{6} + T^{8} \)
$41$ \( ( -3554 - 620 T + 74 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$43$ \( ( -8 - 16 T - 4 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$47$ \( ( 784 + 336 T + 116 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$53$ \( 19909444 - 17723064 T + 7177588 T^{2} - 1707960 T^{3} + 259878 T^{4} - 25800 T^{5} + 1630 T^{6} - 60 T^{7} + T^{8} \)
$59$ \( 625 - 2500 T + 10700 T^{2} + 2600 T^{3} + 1159 T^{4} + 88 T^{5} + 44 T^{6} + 4 T^{7} + T^{8} \)
$61$ \( 4977361 + 1097652 T - 146874 T^{2} - 50184 T^{3} + 6731 T^{4} + 3672 T^{5} + 534 T^{6} + 36 T^{7} + T^{8} \)
$67$ \( 5329 + 20148 T + 76468 T^{2} + 648 T^{3} + 3255 T^{4} - 504 T^{5} + 148 T^{6} - 12 T^{7} + T^{8} \)
$71$ \( 8836 + 6840 T^{2} + 1088 T^{4} + 60 T^{6} + T^{8} \)
$73$ \( 3694084 + 714984 T - 372868 T^{2} - 81096 T^{3} + 55398 T^{4} - 10464 T^{5} + 986 T^{6} - 48 T^{7} + T^{8} \)
$79$ \( 12794929 - 3805928 T + 1747340 T^{2} + 125776 T^{3} + 34519 T^{4} + 752 T^{5} + 236 T^{6} + 8 T^{7} + T^{8} \)
$83$ \( ( 3384 - 2160 T + 444 T^{2} - 36 T^{3} + T^{4} )^{2} \)
$89$ \( 583696 - 470624 T + 293888 T^{2} - 75104 T^{3} + 15772 T^{4} - 784 T^{5} + 128 T^{6} - 4 T^{7} + T^{8} \)
$97$ \( 2618266561 + 55187580 T^{2} + 370886 T^{4} + 1020 T^{6} + T^{8} \)
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