Properties

Label 462.2.g.e
Level $462$
Weight $2$
Character orbit 462.g
Analytic conductor $3.689$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.342102016.5
Defining polynomial: \(x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} + 2 \nu^{3} \)\()/16\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + 3 \nu^{5} + 5 \nu^{4} + 2 \nu^{3} - 2 \nu^{2} + 8 \)\()/16\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 3 \nu^{4} + 8 \nu^{3} - 10 \nu^{2} + 24 \nu - 8 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} + \nu^{5} - 3 \nu^{4} - 6 \nu^{3} - 10 \nu^{2} + 8 \nu - 8 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - \nu^{5} - 3 \nu^{4} + 6 \nu^{3} - 10 \nu^{2} - 8 \nu - 8 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 3 \nu^{4} + 6 \nu^{2} - 8 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} - 3 \nu^{5} + 5 \nu^{4} - 2 \nu^{3} - 2 \nu^{2} + 8 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} + \beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - \beta_{5} - \beta_{4}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{2} - 4\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-4 \beta_{7} - \beta_{5} + 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + 5 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-6 \beta_{7} - 7 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 6 \beta_{2} - 4\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-4 \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{3} + 4 \beta_{2} - 21 \beta_{1}\)\()/2\)

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
1.17915 0.780776i
−0.599676 + 1.28078i
0.599676 + 1.28078i
−1.17915 0.780776i
1.17915 + 0.780776i
−0.599676 1.28078i
0.599676 1.28078i
−1.17915 + 0.780776i
1.00000i −1.51022 + 0.848071i −1.00000 1.69614 0.848071 + 1.51022i −2.56155 + 0.662153i 1.00000i 1.56155 2.56155i 1.69614i
419.2 1.00000i −0.468213 1.66757i −1.00000 −3.33513 −1.66757 + 0.468213i 1.56155 + 2.13578i 1.00000i −2.56155 + 1.56155i 3.33513i
419.3 1.00000i 0.468213 + 1.66757i −1.00000 3.33513 1.66757 0.468213i 1.56155 2.13578i 1.00000i −2.56155 + 1.56155i 3.33513i
419.4 1.00000i 1.51022 0.848071i −1.00000 −1.69614 −0.848071 1.51022i −2.56155 0.662153i 1.00000i 1.56155 2.56155i 1.69614i
419.5 1.00000i −1.51022 0.848071i −1.00000 1.69614 0.848071 1.51022i −2.56155 0.662153i 1.00000i 1.56155 + 2.56155i 1.69614i
419.6 1.00000i −0.468213 + 1.66757i −1.00000 −3.33513 −1.66757 0.468213i 1.56155 2.13578i 1.00000i −2.56155 1.56155i 3.33513i
419.7 1.00000i 0.468213 1.66757i −1.00000 3.33513 1.66757 + 0.468213i 1.56155 + 2.13578i 1.00000i −2.56155 1.56155i 3.33513i
419.8 1.00000i 1.51022 + 0.848071i −1.00000 −1.69614 −0.848071 + 1.51022i −2.56155 + 0.662153i 1.00000i 1.56155 + 2.56155i 1.69614i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.8
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.e 8
3.b odd 2 1 inner 462.2.g.e 8
7.b odd 2 1 inner 462.2.g.e 8
21.c even 2 1 inner 462.2.g.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.g.e 8 1.a even 1 1 trivial
462.2.g.e 8 3.b odd 2 1 inner
462.2.g.e 8 7.b odd 2 1 inner
462.2.g.e 8 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 14 T_{5}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( 81 + 18 T^{2} + 2 T^{4} + 2 T^{6} + T^{8} \)
$5$ \( ( 32 - 14 T^{2} + T^{4} )^{2} \)
$7$ \( ( 49 + 14 T - 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$11$ \( ( 1 + T^{2} )^{4} \)
$13$ \( ( 512 + 46 T^{2} + T^{4} )^{2} \)
$17$ \( ( 128 - 28 T^{2} + T^{4} )^{2} \)
$19$ \( ( 1352 + 74 T^{2} + T^{4} )^{2} \)
$23$ \( ( 36 + T^{2} )^{4} \)
$29$ \( ( 4 + T^{2} )^{4} \)
$31$ \( ( 32 + 20 T^{2} + T^{4} )^{2} \)
$37$ \( ( -68 + T^{2} )^{4} \)
$41$ \( ( 2048 - 92 T^{2} + T^{4} )^{2} \)
$43$ \( ( 32 - 14 T + T^{2} )^{4} \)
$47$ \( ( 2048 - 92 T^{2} + T^{4} )^{2} \)
$53$ \( ( 4 + T^{2} )^{4} \)
$59$ \( ( 17672 - 266 T^{2} + T^{4} )^{2} \)
$61$ \( ( 32 + 14 T^{2} + T^{4} )^{2} \)
$67$ \( ( 12 + T )^{8} \)
$71$ \( ( 20736 + 324 T^{2} + T^{4} )^{2} \)
$73$ \( ( 8192 + 184 T^{2} + T^{4} )^{2} \)
$79$ \( ( 104 + 22 T + T^{2} )^{4} \)
$83$ \( ( 1352 - 74 T^{2} + T^{4} )^{2} \)
$89$ \( ( 8192 - 224 T^{2} + T^{4} )^{2} \)
$97$ \( ( 128 + 40 T^{2} + T^{4} )^{2} \)
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