Properties

Label 462.2.g.c
Level $462$
Weight $2$
Character orbit 462.g
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.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
1.00000i −1.73205 −1.00000 0 1.73205i 2.00000 1.73205i 1.00000i 3.00000 0
419.2 1.00000i 1.73205 −1.00000 0 1.73205i 2.00000 + 1.73205i 1.00000i 3.00000 0
419.3 1.00000i −1.73205 −1.00000 0 1.73205i 2.00000 + 1.73205i 1.00000i 3.00000 0
419.4 1.00000i 1.73205 −1.00000 0 1.73205i 2.00000 1.73205i 1.00000i 3.00000 0
\(n\): 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.g.c 4
3.b odd 2 1 inner 462.2.g.c 4
7.b odd 2 1 inner 462.2.g.c 4
21.c even 2 1 inner 462.2.g.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.g.c 4 1.a even 1 1 trivial
462.2.g.c 4 3.b odd 2 1 inner
462.2.g.c 4 7.b odd 2 1 inner
462.2.g.c 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 7 - 4 T + T^{2} )^{2} \)
$11$ \( ( 1 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( -12 + T^{2} )^{2} \)
$19$ \( ( 12 + T^{2} )^{2} \)
$23$ \( ( 36 + T^{2} )^{2} \)
$29$ \( ( 36 + T^{2} )^{2} \)
$31$ \( ( 12 + T^{2} )^{2} \)
$37$ \( ( 2 + T )^{4} \)
$41$ \( ( -12 + T^{2} )^{2} \)
$43$ \( ( 2 + T )^{4} \)
$47$ \( ( -108 + T^{2} )^{2} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( -108 + T^{2} )^{2} \)
$61$ \( ( 48 + T^{2} )^{2} \)
$67$ \( ( 4 + T )^{4} \)
$71$ \( ( 36 + T^{2} )^{2} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( -8 + T )^{4} \)
$83$ \( ( -108 + T^{2} )^{2} \)
$89$ \( ( -192 + T^{2} )^{2} \)
$97$ \( ( 48 + T^{2} )^{2} \)
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