Properties

Label 462.2.k.b
Level $462$
Weight $2$
Character orbit 462.k
Analytic conductor $3.689$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{12}^{2}\) \(1\)

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
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i −1.73205 + 3.00000i 1.73205i 2.00000 1.73205i 1.00000i −1.50000 + 2.59808i 3.00000 1.73205i
89.2 0.866025 + 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i 1.73205 3.00000i 1.73205i 2.00000 1.73205i 1.00000i −1.50000 + 2.59808i 3.00000 1.73205i
353.1 −0.866025 + 0.500000i −0.866025 + 1.50000i 0.500000 0.866025i −1.73205 3.00000i 1.73205i 2.00000 + 1.73205i 1.00000i −1.50000 2.59808i 3.00000 + 1.73205i
353.2 0.866025 0.500000i 0.866025 1.50000i 0.500000 0.866025i 1.73205 + 3.00000i 1.73205i 2.00000 + 1.73205i 1.00000i −1.50000 2.59808i 3.00000 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
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.b 4
3.b odd 2 1 inner 462.2.k.b 4
7.d odd 6 1 inner 462.2.k.b 4
21.g even 6 1 inner 462.2.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.k.b 4 1.a even 1 1 trivial
462.2.k.b 4 3.b odd 2 1 inner
462.2.k.b 4 7.d odd 6 1 inner
462.2.k.b 4 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 \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\( T_{5}^{4} + 12 T_{5}^{2} + 144 \)
\( T_{17}^{4} + 27 T_{17}^{2} + 729 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( 144 + 12 T^{2} + T^{4} \)
$7$ \( ( 7 - 4 T + T^{2} )^{2} \)
$11$ \( 1 - T^{2} + T^{4} \)
$13$ \( ( 12 + T^{2} )^{2} \)
$17$ \( 729 + 27 T^{2} + T^{4} \)
$19$ \( ( 27 + 9 T + T^{2} )^{2} \)
$23$ \( 81 - 9 T^{2} + T^{4} \)
$29$ \( ( 81 + T^{2} )^{2} \)
$31$ \( ( 12 + 6 T + T^{2} )^{2} \)
$37$ \( ( 49 + 7 T + T^{2} )^{2} \)
$41$ \( ( -48 + T^{2} )^{2} \)
$43$ \( ( 5 + T )^{4} \)
$47$ \( 21609 + 147 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( 9 + 3 T^{2} + T^{4} \)
$61$ \( ( 12 + 6 T + T^{2} )^{2} \)
$67$ \( ( 64 + 8 T + T^{2} )^{2} \)
$71$ \( ( 81 + T^{2} )^{2} \)
$73$ \( ( 192 - 24 T + T^{2} )^{2} \)
$79$ \( ( 256 - 16 T + T^{2} )^{2} \)
$83$ \( ( -12 + T^{2} )^{2} \)
$89$ \( 144 + 12 T^{2} + T^{4} \)
$97$ \( ( 3 + T^{2} )^{2} \)
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