# Properties

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

# Related objects

## 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.

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

## 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
 0.965926 − 0.258819i −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
−0.866025 0.500000i −1.33195 1.10721i 0.500000 + 0.866025i 0.465926 0.807007i 0.599900 + 1.62484i 2.19067 + 1.48356i 1.00000i 0.548188 + 2.94949i −0.807007 + 0.465926i
89.2 −0.866025 0.500000i 0.599900 1.62484i 0.500000 + 0.866025i −1.46593 + 2.53906i −1.33195 + 1.10721i −2.19067 1.48356i 1.00000i −2.28024 1.94949i 2.53906 1.46593i
89.3 0.866025 + 0.500000i 1.10721 1.33195i 0.500000 + 0.866025i −0.241181 + 0.417738i 1.62484 0.599900i 1.48356 2.19067i 1.00000i −0.548188 2.94949i −0.417738 + 0.241181i
89.4 0.866025 + 0.500000i 1.62484 + 0.599900i 0.500000 + 0.866025i −0.758819 + 1.31431i 1.10721 + 1.33195i −1.48356 + 2.19067i 1.00000i 2.28024 + 1.94949i −1.31431 + 0.758819i
353.1 −0.866025 + 0.500000i −1.33195 + 1.10721i 0.500000 0.866025i 0.465926 + 0.807007i 0.599900 1.62484i 2.19067 1.48356i 1.00000i 0.548188 2.94949i −0.807007 0.465926i
353.2 −0.866025 + 0.500000i 0.599900 + 1.62484i 0.500000 0.866025i −1.46593 2.53906i −1.33195 1.10721i −2.19067 + 1.48356i 1.00000i −2.28024 + 1.94949i 2.53906 + 1.46593i
353.3 0.866025 0.500000i 1.10721 + 1.33195i 0.500000 0.866025i −0.241181 0.417738i 1.62484 + 0.599900i 1.48356 + 2.19067i 1.00000i −0.548188 + 2.94949i −0.417738 0.241181i
353.4 0.866025 0.500000i 1.62484 0.599900i 0.500000 0.866025i −0.758819 1.31431i 1.10721 1.33195i −1.48356 2.19067i 1.00000i 2.28024 1.94949i −1.31431 0.758819i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.e 8
3.b odd 2 1 462.2.k.f yes 8
7.d odd 6 1 462.2.k.f yes 8
21.g even 6 1 inner 462.2.k.e 8

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

## 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

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$81 - 108 T + 72 T^{2} - 24 T^{3} + 7 T^{4} - 8 T^{5} + 8 T^{6} - 4 T^{7} + T^{8}$$
$5$ $$4 + 8 T + 20 T^{2} + 8 T^{3} + 22 T^{4} + 16 T^{5} + 14 T^{6} + 4 T^{7} + T^{8}$$
$7$ $$2401 + 71 T^{4} + T^{8}$$
$11$ $$( 1 - T^{2} + T^{4} )^{2}$$
$13$ $$( 9 + 30 T^{2} + T^{4} )^{2}$$
$17$ $$141376 - 150400 T + 107360 T^{2} - 40960 T^{3} + 11224 T^{4} - 2000 T^{5} + 260 T^{6} - 20 T^{7} + T^{8}$$
$19$ $$4 + 24 T + 52 T^{2} + 24 T^{3} - 42 T^{4} - 24 T^{5} + 46 T^{6} + 12 T^{7} + T^{8}$$
$23$ $$4 + 24 T + 36 T^{2} - 72 T^{3} - 10 T^{4} + 72 T^{5} + 54 T^{6} + 12 T^{7} + T^{8}$$
$29$ $$187489 + 49836 T^{2} + 3782 T^{4} + 108 T^{6} + T^{8}$$
$31$ $$153664 - 263424 T + 170912 T^{2} - 34944 T^{3} + 408 T^{4} + 624 T^{5} - 4 T^{6} - 12 T^{7} + T^{8}$$
$37$ $$145924 - 32088 T + 29212 T^{2} + 4872 T^{3} + 2982 T^{4} + 168 T^{5} + 58 T^{6} + T^{8}$$
$41$ $$( -3554 + 620 T + 74 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$43$ $$( -8 - 16 T - 4 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$47$ $$( 784 - 336 T + 116 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$53$ $$19909444 + 17723064 T + 7177588 T^{2} + 1707960 T^{3} + 259878 T^{4} + 25800 T^{5} + 1630 T^{6} + 60 T^{7} + T^{8}$$
$59$ $$625 + 2500 T + 10700 T^{2} - 2600 T^{3} + 1159 T^{4} - 88 T^{5} + 44 T^{6} - 4 T^{7} + T^{8}$$
$61$ $$4977361 + 1097652 T - 146874 T^{2} - 50184 T^{3} + 6731 T^{4} + 3672 T^{5} + 534 T^{6} + 36 T^{7} + T^{8}$$
$67$ $$5329 + 20148 T + 76468 T^{2} + 648 T^{3} + 3255 T^{4} - 504 T^{5} + 148 T^{6} - 12 T^{7} + T^{8}$$
$71$ $$8836 + 6840 T^{2} + 1088 T^{4} + 60 T^{6} + T^{8}$$
$73$ $$3694084 + 714984 T - 372868 T^{2} - 81096 T^{3} + 55398 T^{4} - 10464 T^{5} + 986 T^{6} - 48 T^{7} + T^{8}$$
$79$ $$12794929 - 3805928 T + 1747340 T^{2} + 125776 T^{3} + 34519 T^{4} + 752 T^{5} + 236 T^{6} + 8 T^{7} + T^{8}$$
$83$ $$( 3384 + 2160 T + 444 T^{2} + 36 T^{3} + T^{4} )^{2}$$
$89$ $$583696 + 470624 T + 293888 T^{2} + 75104 T^{3} + 15772 T^{4} + 784 T^{5} + 128 T^{6} + 4 T^{7} + T^{8}$$
$97$ $$2618266561 + 55187580 T^{2} + 370886 T^{4} + 1020 T^{6} + T^{8}$$