Properties

Label 462.2.g.b
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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

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
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
1.00000i −1.22474 1.22474i −1.00000 −2.44949 −1.22474 + 1.22474i −1.00000 2.44949i 1.00000i 3.00000i 2.44949i
419.2 1.00000i 1.22474 + 1.22474i −1.00000 2.44949 1.22474 1.22474i −1.00000 + 2.44949i 1.00000i 3.00000i 2.44949i
419.3 1.00000i −1.22474 + 1.22474i −1.00000 −2.44949 −1.22474 1.22474i −1.00000 + 2.44949i 1.00000i 3.00000i 2.44949i
419.4 1.00000i 1.22474 1.22474i −1.00000 2.44949 1.22474 + 1.22474i −1.00000 2.44949i 1.00000i 3.00000i 2.44949i
\(n\): 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.b 4
3.b odd 2 1 inner 462.2.g.b 4
7.b odd 2 1 inner 462.2.g.b 4
21.c even 2 1 inner 462.2.g.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.g.b 4 1.a even 1 1 trivial
462.2.g.b 4 3.b odd 2 1 inner
462.2.g.b 4 7.b odd 2 1 inner
462.2.g.b 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 6 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 9 + T^{4} \)
$5$ \( ( -6 + T^{2} )^{2} \)
$7$ \( ( 7 + 2 T + T^{2} )^{2} \)
$11$ \( ( 1 + T^{2} )^{2} \)
$13$ \( ( 6 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( 54 + T^{2} )^{2} \)
$23$ \( ( 36 + T^{2} )^{2} \)
$29$ \( ( 36 + T^{2} )^{2} \)
$31$ \( ( 24 + T^{2} )^{2} \)
$37$ \( ( -10 + T )^{4} \)
$41$ \( ( -96 + T^{2} )^{2} \)
$43$ \( ( 8 + T )^{4} \)
$47$ \( ( -96 + T^{2} )^{2} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( -6 + T^{2} )^{2} \)
$61$ \( ( 54 + T^{2} )^{2} \)
$67$ \( ( 4 + T )^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 24 + T^{2} )^{2} \)
$79$ \( ( 10 + T )^{4} \)
$83$ \( ( -6 + T^{2} )^{2} \)
$89$ \( ( -96 + T^{2} )^{2} \)
$97$ \( ( 24 + T^{2} )^{2} \)
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