Properties

Label 460.2.g.b
Level $460$
Weight $2$
Character orbit 460.g
Analytic conductor $3.673$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.207360000.1
Defining polynomial: \(x^{8} + 6 x^{6} + 32 x^{4} + 24 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 6 x^{6} + 32 x^{4} + 24 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} - 72 \)\()/32\)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{7} + 32 \nu^{5} + 160 \nu^{3} + 120 \nu \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{6} + 16 \nu^{4} + 96 \nu^{2} + 40 \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( -7 \nu^{7} - 32 \nu^{5} - 160 \nu^{3} + 88 \nu \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( 11 \nu^{7} + 64 \nu^{5} + 352 \nu^{3} + 264 \nu \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 6 \nu^{4} - 28 \nu^{2} - 12 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -13 \nu^{7} - 64 \nu^{5} - 352 \nu^{3} + 200 \nu \)\()/64\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{5} - \beta_{4} - \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + 3 \beta_{3} - \beta_{1} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{5} + 2 \beta_{4} - 2 \beta_{2}\)
\(\nu^{4}\)\(=\)\(-3 \beta_{6} - 7 \beta_{3} - 3 \beta_{1} - 7\)
\(\nu^{5}\)\(=\)\(-10 \beta_{5} + 22 \beta_{2}\)
\(\nu^{6}\)\(=\)\(32 \beta_{1} + 72\)
\(\nu^{7}\)\(=\)\(26 \beta_{7} + 26 \beta_{5} - 58 \beta_{4} - 58 \beta_{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(277\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
459.1
−0.437016 + 0.756934i
1.14412 1.98168i
−0.437016 0.756934i
1.14412 + 1.98168i
0.437016 + 0.756934i
−1.14412 1.98168i
0.437016 0.756934i
−1.14412 + 1.98168i
−0.707107 1.22474i 2.82843 −1.00000 + 1.73205i −2.23607 −2.00000 3.46410i 3.87298i 2.82843 5.00000 1.58114 + 2.73861i
459.2 −0.707107 1.22474i 2.82843 −1.00000 + 1.73205i 2.23607 −2.00000 3.46410i 3.87298i 2.82843 5.00000 −1.58114 2.73861i
459.3 −0.707107 + 1.22474i 2.82843 −1.00000 1.73205i −2.23607 −2.00000 + 3.46410i 3.87298i 2.82843 5.00000 1.58114 2.73861i
459.4 −0.707107 + 1.22474i 2.82843 −1.00000 1.73205i 2.23607 −2.00000 + 3.46410i 3.87298i 2.82843 5.00000 −1.58114 + 2.73861i
459.5 0.707107 1.22474i −2.82843 −1.00000 1.73205i −2.23607 −2.00000 + 3.46410i 3.87298i −2.82843 5.00000 −1.58114 + 2.73861i
459.6 0.707107 1.22474i −2.82843 −1.00000 1.73205i 2.23607 −2.00000 + 3.46410i 3.87298i −2.82843 5.00000 1.58114 2.73861i
459.7 0.707107 + 1.22474i −2.82843 −1.00000 + 1.73205i −2.23607 −2.00000 3.46410i 3.87298i −2.82843 5.00000 −1.58114 2.73861i
459.8 0.707107 + 1.22474i −2.82843 −1.00000 + 1.73205i 2.23607 −2.00000 3.46410i 3.87298i −2.82843 5.00000 1.58114 + 2.73861i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 459.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner
115.c odd 2 1 inner
460.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.g.b 8
4.b odd 2 1 inner 460.2.g.b 8
5.b even 2 1 inner 460.2.g.b 8
20.d odd 2 1 inner 460.2.g.b 8
23.b odd 2 1 inner 460.2.g.b 8
92.b even 2 1 inner 460.2.g.b 8
115.c odd 2 1 inner 460.2.g.b 8
460.g even 2 1 inner 460.2.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.g.b 8 1.a even 1 1 trivial
460.2.g.b 8 4.b odd 2 1 inner
460.2.g.b 8 5.b even 2 1 inner
460.2.g.b 8 20.d odd 2 1 inner
460.2.g.b 8 23.b odd 2 1 inner
460.2.g.b 8 92.b even 2 1 inner
460.2.g.b 8 115.c odd 2 1 inner
460.2.g.b 8 460.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8 \) acting on \(S_{2}^{\mathrm{new}}(460, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$3$ \( ( -8 + T^{2} )^{4} \)
$5$ \( ( -5 + T^{2} )^{4} \)
$7$ \( ( 15 + T^{2} )^{4} \)
$11$ \( T^{8} \)
$13$ \( ( 24 + T^{2} )^{4} \)
$17$ \( ( -5 + T^{2} )^{4} \)
$19$ \( ( -40 + T^{2} )^{4} \)
$23$ \( ( 529 + 14 T^{2} + T^{4} )^{2} \)
$29$ \( ( -3 + T )^{8} \)
$31$ \( ( 3 + T^{2} )^{4} \)
$37$ \( ( -45 + T^{2} )^{4} \)
$41$ \( ( 9 + T )^{8} \)
$43$ \( ( 60 + T^{2} )^{4} \)
$47$ \( ( -32 + T^{2} )^{4} \)
$53$ \( ( -5 + T^{2} )^{4} \)
$59$ \( ( 75 + T^{2} )^{4} \)
$61$ \( ( 120 + T^{2} )^{4} \)
$67$ \( ( 135 + T^{2} )^{4} \)
$71$ \( ( 3 + T^{2} )^{4} \)
$73$ \( ( 24 + T^{2} )^{4} \)
$79$ \( ( -160 + T^{2} )^{4} \)
$83$ \( ( 15 + T^{2} )^{4} \)
$89$ \( ( 120 + T^{2} )^{4} \)
$97$ \( ( -180 + T^{2} )^{4} \)
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