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: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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 \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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