Properties

Label 460.2.g.a
Level $460$
Weight $2$
Character orbit 460.g
Analytic conductor $3.673$
Analytic rank $0$
Dimension $4$
CM discriminant -20
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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} -\beta_{2} q^{3} + 2 q^{4} + \beta_{3} q^{5} + 2 q^{6} + ( -2 \beta_{1} - \beta_{2} ) q^{7} -2 \beta_{2} q^{8} - q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} -\beta_{2} q^{3} + 2 q^{4} + \beta_{3} q^{5} + 2 q^{6} + ( -2 \beta_{1} - \beta_{2} ) q^{7} -2 \beta_{2} q^{8} - q^{9} + ( 2 \beta_{1} + \beta_{2} ) q^{10} -2 \beta_{2} q^{12} -2 \beta_{3} q^{14} + ( 2 \beta_{1} + \beta_{2} ) q^{15} + 4 q^{16} + \beta_{2} q^{18} + 2 \beta_{3} q^{20} -2 \beta_{3} q^{21} + ( 3 \beta_{1} + \beta_{2} ) q^{23} + 4 q^{24} -5 q^{25} + 4 \beta_{2} q^{27} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{28} -6 q^{29} + 2 \beta_{3} q^{30} -4 \beta_{2} q^{32} -5 \beta_{2} q^{35} -2 q^{36} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{40} -12 q^{41} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{42} + ( -2 \beta_{1} - \beta_{2} ) q^{43} -\beta_{3} q^{45} + ( 1 + 3 \beta_{3} ) q^{46} -7 \beta_{2} q^{47} -4 \beta_{2} q^{48} -3 q^{49} + 5 \beta_{2} q^{50} -8 q^{54} -4 \beta_{3} q^{56} + 6 \beta_{2} q^{58} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{60} -6 \beta_{3} q^{61} + ( 2 \beta_{1} + \beta_{2} ) q^{63} + 8 q^{64} + ( -10 \beta_{1} - 5 \beta_{2} ) q^{67} + ( 1 + 3 \beta_{3} ) q^{69} + 10 q^{70} + 2 \beta_{2} q^{72} + 5 \beta_{2} q^{75} + 4 \beta_{3} q^{80} -5 q^{81} + 12 \beta_{2} q^{82} + ( 6 \beta_{1} + 3 \beta_{2} ) q^{83} -4 \beta_{3} q^{84} -2 \beta_{3} q^{86} + 6 \beta_{2} q^{87} + 8 \beta_{3} q^{89} + ( -2 \beta_{1} - \beta_{2} ) q^{90} + ( 6 \beta_{1} + 2 \beta_{2} ) q^{92} + 14 q^{94} + 8 q^{96} + 3 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + 8q^{6} - 4q^{9} + O(q^{10}) \) \( 4q + 8q^{4} + 8q^{6} - 4q^{9} + 16q^{16} + 16q^{24} - 20q^{25} - 24q^{29} - 8q^{36} - 48q^{41} + 4q^{46} - 12q^{49} - 32q^{54} + 32q^{64} + 4q^{69} + 40q^{70} - 20q^{81} + 56q^{94} + 32q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 9\):

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

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
−0.707107 + 1.58114i
−0.707107 1.58114i
0.707107 1.58114i
0.707107 + 1.58114i
−1.41421 −1.41421 2.00000 2.23607i 2.00000 3.16228i −2.82843 −1.00000 3.16228i
459.2 −1.41421 −1.41421 2.00000 2.23607i 2.00000 3.16228i −2.82843 −1.00000 3.16228i
459.3 1.41421 1.41421 2.00000 2.23607i 2.00000 3.16228i 2.82843 −1.00000 3.16228i
459.4 1.41421 1.41421 2.00000 2.23607i 2.00000 3.16228i 2.82843 −1.00000 3.16228i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 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.a 4
4.b odd 2 1 inner 460.2.g.a 4
5.b even 2 1 inner 460.2.g.a 4
20.d odd 2 1 CM 460.2.g.a 4
23.b odd 2 1 inner 460.2.g.a 4
92.b even 2 1 inner 460.2.g.a 4
115.c odd 2 1 inner 460.2.g.a 4
460.g even 2 1 inner 460.2.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.g.a 4 1.a even 1 1 trivial
460.2.g.a 4 4.b odd 2 1 inner
460.2.g.a 4 5.b even 2 1 inner
460.2.g.a 4 20.d odd 2 1 CM
460.2.g.a 4 23.b odd 2 1 inner
460.2.g.a 4 92.b even 2 1 inner
460.2.g.a 4 115.c odd 2 1 inner
460.2.g.a 4 460.g even 2 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{2} \)
$3$ \( ( -2 + T^{2} )^{2} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( ( 10 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 529 + 44 T^{2} + T^{4} \)
$29$ \( ( 6 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 12 + T )^{4} \)
$43$ \( ( 10 + T^{2} )^{2} \)
$47$ \( ( -98 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 180 + T^{2} )^{2} \)
$67$ \( ( 250 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 90 + T^{2} )^{2} \)
$89$ \( ( 320 + T^{2} )^{2} \)
$97$ \( T^{4} \)
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