Properties

Label 462.2.g.d
Level $462$
Weight $2$
Character orbit 462.g
Analytic conductor $3.689$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

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

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
0.618034i
1.61803i
0.618034i
1.61803i
1.00000i −0.618034 + 1.61803i −1.00000 −1.23607 1.61803 + 0.618034i 2.61803 0.381966i 1.00000i −2.23607 2.00000i 1.23607i
419.2 1.00000i 1.61803 0.618034i −1.00000 3.23607 −0.618034 1.61803i 0.381966 2.61803i 1.00000i 2.23607 2.00000i 3.23607i
419.3 1.00000i −0.618034 1.61803i −1.00000 −1.23607 1.61803 0.618034i 2.61803 + 0.381966i 1.00000i −2.23607 + 2.00000i 1.23607i
419.4 1.00000i 1.61803 + 0.618034i −1.00000 3.23607 −0.618034 + 1.61803i 0.381966 + 2.61803i 1.00000i 2.23607 + 2.00000i 3.23607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 2 T_{5} - 4 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 9 - 6 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$5$ \( ( -4 - 2 T + T^{2} )^{2} \)
$7$ \( 49 - 42 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$11$ \( ( 1 + T^{2} )^{2} \)
$13$ \( ( 36 + T^{2} )^{2} \)
$17$ \( ( 20 + 10 T + T^{2} )^{2} \)
$19$ \( ( 4 + T^{2} )^{2} \)
$23$ \( 16 + 12 T^{2} + T^{4} \)
$29$ \( 16 + 72 T^{2} + T^{4} \)
$31$ \( 16 + 28 T^{2} + T^{4} \)
$37$ \( ( -20 + T^{2} )^{2} \)
$41$ \( ( -124 + 2 T + T^{2} )^{2} \)
$43$ \( ( -76 - 4 T + T^{2} )^{2} \)
$47$ \( ( 16 + 12 T + T^{2} )^{2} \)
$53$ \( 1936 + 92 T^{2} + T^{4} \)
$59$ \( ( -4 + 2 T + T^{2} )^{2} \)
$61$ \( 1936 + 168 T^{2} + T^{4} \)
$67$ \( ( 16 - 12 T + T^{2} )^{2} \)
$71$ \( 10000 + 300 T^{2} + T^{4} \)
$73$ \( 13456 + 268 T^{2} + T^{4} \)
$79$ \( ( 20 - 10 T + T^{2} )^{2} \)
$83$ \( ( 6 + T )^{4} \)
$89$ \( ( -64 - 8 T + T^{2} )^{2} \)
$97$ \( 256 + 48 T^{2} + T^{4} \)
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