Properties

Label 462.2.k.d
Level $462$
Weight $2$
Character orbit 462.k
Analytic conductor $3.689$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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.

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

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{2}\) \(1\)

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 −1.50000 + 0.866025i 0.500000 + 0.866025i −0.358719 + 0.621320i 1.73205 2.62132 + 0.358719i 1.00000i 1.50000 2.59808i 0.621320 0.358719i
89.2 −0.866025 0.500000i −1.50000 + 0.866025i 0.500000 + 0.866025i 2.09077 3.62132i 1.73205 −1.62132 2.09077i 1.00000i 1.50000 2.59808i −3.62132 + 2.09077i
89.3 0.866025 + 0.500000i −1.50000 + 0.866025i 0.500000 + 0.866025i −2.09077 + 3.62132i −1.73205 −1.62132 2.09077i 1.00000i 1.50000 2.59808i −3.62132 + 2.09077i
89.4 0.866025 + 0.500000i −1.50000 + 0.866025i 0.500000 + 0.866025i 0.358719 0.621320i −1.73205 2.62132 + 0.358719i 1.00000i 1.50000 2.59808i 0.621320 0.358719i
353.1 −0.866025 + 0.500000i −1.50000 0.866025i 0.500000 0.866025i −0.358719 0.621320i 1.73205 2.62132 0.358719i 1.00000i 1.50000 + 2.59808i 0.621320 + 0.358719i
353.2 −0.866025 + 0.500000i −1.50000 0.866025i 0.500000 0.866025i 2.09077 + 3.62132i 1.73205 −1.62132 + 2.09077i 1.00000i 1.50000 + 2.59808i −3.62132 2.09077i
353.3 0.866025 0.500000i −1.50000 0.866025i 0.500000 0.866025i −2.09077 3.62132i −1.73205 −1.62132 + 2.09077i 1.00000i 1.50000 + 2.59808i −3.62132 2.09077i
353.4 0.866025 0.500000i −1.50000 0.866025i 0.500000 0.866025i 0.358719 + 0.621320i −1.73205 2.62132 0.358719i 1.00000i 1.50000 + 2.59808i 0.621320 + 0.358719i
\(n\): 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
3.b odd 2 1 inner
7.d odd 6 1 inner
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.d 8
3.b odd 2 1 inner 462.2.k.d 8
7.d odd 6 1 inner 462.2.k.d 8
21.g even 6 1 inner 462.2.k.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.k.d 8 1.a even 1 1 trivial
462.2.k.d 8 3.b odd 2 1 inner
462.2.k.d 8 7.d odd 6 1 inner
462.2.k.d 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 \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\( T_{5}^{8} + 18 T_{5}^{6} + 315 T_{5}^{4} + 162 T_{5}^{2} + 81 \)
\( T_{17}^{8} + 54 T_{17}^{6} + 2475 T_{17}^{4} + 23814 T_{17}^{2} + 194481 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( ( 3 + 3 T + T^{2} )^{4} \)
$5$ \( 81 + 162 T^{2} + 315 T^{4} + 18 T^{6} + T^{8} \)
$7$ \( ( 49 - 14 T - 3 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$11$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$13$ \( ( 12 + T^{2} )^{4} \)
$17$ \( 194481 + 23814 T^{2} + 2475 T^{4} + 54 T^{6} + T^{8} \)
$19$ \( ( 144 - 144 T + 36 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$23$ \( 6561 - 4374 T^{2} + 2835 T^{4} - 54 T^{6} + T^{8} \)
$29$ \( ( 72 + T^{2} )^{4} \)
$31$ \( ( 144 - 144 T + 36 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$37$ \( ( 16 + 4 T + T^{2} )^{4} \)
$41$ \( ( -27 + T^{2} )^{4} \)
$43$ \( ( -56 - 8 T + T^{2} )^{4} \)
$47$ \( 81 + 162 T^{2} + 315 T^{4} + 18 T^{6} + T^{8} \)
$53$ \( ( 5184 - 72 T^{2} + T^{4} )^{2} \)
$59$ \( 49787136 + 1524096 T^{2} + 39600 T^{4} + 216 T^{6} + T^{8} \)
$61$ \( ( 9 - 18 T + 9 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$67$ \( ( 2209 - 470 T + 147 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$71$ \( ( 36 + T^{2} )^{4} \)
$73$ \( ( 144 - 144 T + 36 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$79$ \( ( 289 + 34 T + 21 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$83$ \( ( 441 - 54 T^{2} + T^{4} )^{2} \)
$89$ \( 331776 + 82944 T^{2} + 20160 T^{4} + 144 T^{6} + T^{8} \)
$97$ \( ( 9 + 102 T^{2} + T^{4} )^{2} \)
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