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: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{2} + ( - \beta_{2} - 1) q^{3} + ( - \beta_{2} + 1) q^{4} + (\beta_{5} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{3} + 2 \beta_1) q^{6} + ( - \beta_{7} + \beta_{2}) q^{7} + \beta_{3} q^{8} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1) q^{2} + ( - \beta_{2} - 1) q^{3} + ( - \beta_{2} + 1) q^{4} + (\beta_{5} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{3} + 2 \beta_1) q^{6} + ( - \beta_{7} + \beta_{2}) q^{7} + \beta_{3} q^{8} + 3 \beta_{2} q^{9} + ( - \beta_{4} - \beta_{2} - 1) q^{10} - \beta_1 q^{11} + (\beta_{2} - 2) q^{12} + ( - 4 \beta_{2} + 2) q^{13} + (\beta_{6} - \beta_{5} - \beta_1) q^{14} + (\beta_{6} - 2 \beta_{5} - 3 \beta_{3}) q^{15} - \beta_{2} q^{16} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} - \beta_1) q^{17} - 3 \beta_1 q^{18} + ( - 2 \beta_{7} + 2 \beta_{2} - 4) q^{19} + (\beta_{6} - \beta_{3} + 2 \beta_1) q^{20} + (\beta_{7} + \beta_{4} - 2 \beta_{2} + 1) q^{21} + q^{22} + ( - 2 \beta_{6} + \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{23} + ( - 2 \beta_{3} + \beta_1) q^{24} + ( - 4 \beta_{7} + 2 \beta_{4} + 4 \beta_{2} - 4) q^{25} + (2 \beta_{3} + 2 \beta_1) q^{26} + ( - 6 \beta_{2} + 3) q^{27} + ( - \beta_{7} + \beta_{4} + 1) q^{28} + ( - 2 \beta_{6} + 4 \beta_{5}) q^{29} + ( - \beta_{7} + 2 \beta_{4} + 3 \beta_{2}) q^{30} + (2 \beta_{4} - 2 \beta_{2} - 2) q^{31} + \beta_1 q^{32} + (\beta_{3} + \beta_1) q^{33} + ( - 2 \beta_{7} + 2 \beta_{4} - 2 \beta_{2} + 1) q^{34} + ( - 2 \beta_{6} + 2 \beta_{3} - 7 \beta_1) q^{35} + 3 q^{36} - 4 \beta_{2} q^{37} + (2 \beta_{6} - 2 \beta_{5} - 4 \beta_{3} + 2 \beta_1) q^{38} + (6 \beta_{2} - 6) q^{39} + ( - \beta_{7} + \beta_{2} - 2) q^{40} + (3 \beta_{3} - 6 \beta_1) q^{41} + ( - 2 \beta_{6} + \beta_{5} + \beta_{3} + \beta_1) q^{42} + ( - 2 \beta_{7} - 2 \beta_{4} + 4) q^{43} + (\beta_{3} - \beta_1) q^{44} + ( - 3 \beta_{6} + 3 \beta_{5} + 6 \beta_{3} - 3 \beta_1) q^{45} + (2 \beta_{7} - \beta_{4} + 3 \beta_{2} - 3) q^{46} + (\beta_{5} + \beta_{3} + \beta_1) q^{47} + (2 \beta_{2} - 1) q^{48} + ( - 2 \beta_{4} - 5 \beta_{2} + 5) q^{49} + (2 \beta_{6} - 4 \beta_{5} - 4 \beta_{3}) q^{50} + ( - 4 \beta_{6} + 2 \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{51} + ( - 2 \beta_{2} - 2) q^{52} + (2 \beta_{6} + 2 \beta_{5}) q^{53} + (3 \beta_{3} + 3 \beta_1) q^{54} + (\beta_{7} - \beta_{4} - 2 \beta_{2} + 1) q^{55} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{56} + (2 \beta_{7} + 2 \beta_{4} + 6) q^{57} + (2 \beta_{7} - 4 \beta_{4}) q^{58} + ( - 4 \beta_{6} + 4 \beta_{5} - 4 \beta_{3} + 2 \beta_1) q^{59} + ( - \beta_{6} - \beta_{5} - 3 \beta_1) q^{60} + (\beta_{7} + \beta_{2} - 2) q^{61} + ( - 2 \beta_{6} - 2 \beta_{3} + 4 \beta_1) q^{62} + ( - 3 \beta_{4} + 3 \beta_{2} - 3) q^{63} - q^{64} + (4 \beta_{6} - 2 \beta_{5} - 6 \beta_{3} + 6 \beta_1) q^{65} + ( - \beta_{2} - 1) q^{66} + (4 \beta_{7} - 2 \beta_{4} + 5 \beta_{2} - 5) q^{67} + ( - 2 \beta_{5} + \beta_{3} + \beta_1) q^{68} + (3 \beta_{6} + 3 \beta_{3} - 6 \beta_1) q^{69} + (2 \beta_{7} - 2 \beta_{2} + 7) q^{70} + 6 \beta_{3} q^{71} + (3 \beta_{3} - 3 \beta_1) q^{72} + ( - 2 \beta_{4} - 2 \beta_{2} - 2) q^{73} + 4 \beta_1 q^{74} + (6 \beta_{7} - 4 \beta_{2} + 8) q^{75} + ( - 2 \beta_{7} + 2 \beta_{4} + 4 \beta_{2} - 2) q^{76} + (\beta_{6} - \beta_{3}) q^{77} - 6 \beta_{3} q^{78} + ( - \beta_{7} + 2 \beta_{4} + \beta_{2}) q^{79} + (\beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_1) q^{80} + (9 \beta_{2} - 9) q^{81} + ( - 3 \beta_{2} + 6) q^{82} + ( - 2 \beta_{6} - \beta_{3} + 2 \beta_1) q^{83} + (2 \beta_{7} - \beta_{4} - \beta_{2} - 1) q^{84} + (\beta_{7} + \beta_{4} + 9) q^{85} + (4 \beta_{6} - 2 \beta_{5} + 4 \beta_{3} - 4 \beta_1) q^{86} + (6 \beta_{6} - 6 \beta_{5}) q^{87} + ( - \beta_{2} + 1) q^{88} + (2 \beta_{5} - 4 \beta_{3} - 4 \beta_1) q^{89} + (3 \beta_{7} - 3 \beta_{4} - 6 \beta_{2} + 3) q^{90} + ( - 2 \beta_{7} + 4 \beta_{4} - 2 \beta_{2} + 4) q^{91} + ( - \beta_{6} + 2 \beta_{5} - 3 \beta_{3}) q^{92} + (2 \beta_{7} - 4 \beta_{4} + 6 \beta_{2}) q^{93} + ( - \beta_{4} - \beta_{2} - 1) q^{94} + ( - 4 \beta_{6} - 4 \beta_{5} - 18 \beta_1) q^{95} + ( - \beta_{3} - \beta_1) q^{96} + (2 \beta_{7} - 2 \beta_{4} + 6 \beta_{2} - 3) q^{97} + (2 \beta_{6} + 5 \beta_{3}) q^{98} - 3 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} + 4 q^{4} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} + 4 q^{4} + 4 q^{7} + 12 q^{9} - 12 q^{10} - 12 q^{12} - 4 q^{16} - 24 q^{19} + 8 q^{22} - 16 q^{25} + 8 q^{28} + 12 q^{30} - 24 q^{31} + 24 q^{36} - 16 q^{37} - 24 q^{39} - 12 q^{40} + 32 q^{43} - 12 q^{46} + 20 q^{49} - 24 q^{52} + 48 q^{57} - 12 q^{61} - 12 q^{63} - 8 q^{64} - 12 q^{66} - 20 q^{67} + 48 q^{70} - 24 q^{73} + 48 q^{75} + 4 q^{79} - 36 q^{81} + 36 q^{82} - 12 q^{84} + 72 q^{85} + 4 q^{88} + 24 q^{91} + 24 q^{93} - 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} - 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} - \beta_{5} + 2\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} + \beta_{5} + 2\beta_{4} ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(-1\) \(\beta_{2}\) \(1\)

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

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

Hecke characteristic polynomials

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