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 $2$

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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: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
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.

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

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:

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 \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\( T_{5}^{8} + 4T_{5}^{7} + 14T_{5}^{6} + 16T_{5}^{5} + 22T_{5}^{4} + 8T_{5}^{3} + 20T_{5}^{2} + 8T_{5} + 4 \) Copy content Toggle raw display
\( T_{17}^{8} - 20 T_{17}^{7} + 260 T_{17}^{6} - 2000 T_{17}^{5} + 11224 T_{17}^{4} - 40960 T_{17}^{3} + 107360 T_{17}^{2} - 150400 T_{17} + 141376 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + 8 T^{6} - 8 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + 14 T^{6} + 16 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{8} + 71T^{4} + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 30 T^{2} + 9)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 20 T^{7} + 260 T^{6} + \cdots + 141376 \) Copy content Toggle raw display
$19$ \( T^{8} + 12 T^{7} + 46 T^{6} - 24 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + 54 T^{6} + 72 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( T^{8} + 108 T^{6} + 3782 T^{4} + \cdots + 187489 \) Copy content Toggle raw display
$31$ \( T^{8} - 12 T^{7} - 4 T^{6} + \cdots + 153664 \) Copy content Toggle raw display
$37$ \( T^{8} + 58 T^{6} + 168 T^{5} + \cdots + 145924 \) Copy content Toggle raw display
$41$ \( (T^{4} - 20 T^{3} + 74 T^{2} + 620 T - 3554)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} - 4 T^{2} - 16 T - 8)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 12 T^{3} + 116 T^{2} - 336 T + 784)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 60 T^{7} + 1630 T^{6} + \cdots + 19909444 \) Copy content Toggle raw display
$59$ \( T^{8} - 4 T^{7} + 44 T^{6} - 88 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$61$ \( T^{8} + 36 T^{7} + 534 T^{6} + \cdots + 4977361 \) Copy content Toggle raw display
$67$ \( T^{8} - 12 T^{7} + 148 T^{6} + \cdots + 5329 \) Copy content Toggle raw display
$71$ \( T^{8} + 60 T^{6} + 1088 T^{4} + \cdots + 8836 \) Copy content Toggle raw display
$73$ \( T^{8} - 48 T^{7} + 986 T^{6} + \cdots + 3694084 \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} + 236 T^{6} + \cdots + 12794929 \) Copy content Toggle raw display
$83$ \( (T^{4} + 36 T^{3} + 444 T^{2} + 2160 T + 3384)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 4 T^{7} + 128 T^{6} + \cdots + 583696 \) Copy content Toggle raw display
$97$ \( T^{8} + 1020 T^{6} + \cdots + 2618266561 \) Copy content Toggle raw display
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