Properties

Label 460.2.i.a
Level $460$
Weight $2$
Character orbit 460.i
Analytic conductor $3.673$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.11574317056.3
Defining polynomial: \(x^{8} + 45 x^{4} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{3} ) q^{3} + ( -\beta_{1} + \beta_{6} ) q^{5} -\beta_{2} q^{7} -\beta_{3} q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{3} ) q^{3} + ( -\beta_{1} + \beta_{6} ) q^{5} -\beta_{2} q^{7} -\beta_{3} q^{9} + ( \beta_{1} + \beta_{2} ) q^{11} + ( 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{13} + ( \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} ) q^{15} + ( -\beta_{2} + 2 \beta_{6} - 2 \beta_{7} ) q^{17} + ( -\beta_{1} + \beta_{2} - 2 \beta_{7} ) q^{19} + ( \beta_{1} + \beta_{2} ) q^{21} + ( -2 + 2 \beta_{1} - 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{23} + ( -2 - \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{25} + ( -4 + 4 \beta_{3} ) q^{27} + ( -5 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{29} + ( 1 + 3 \beta_{4} + 3 \beta_{5} ) q^{31} -2 \beta_{1} q^{33} + ( -2 - \beta_{3} + \beta_{4} + \beta_{5} ) q^{35} + ( -\beta_{2} + 2 \beta_{6} - 2 \beta_{7} ) q^{37} + ( 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{39} + ( 5 + \beta_{4} + \beta_{5} ) q^{41} + ( -4 \beta_{1} - \beta_{6} - \beta_{7} ) q^{43} + ( -\beta_{2} - \beta_{7} ) q^{45} + ( -1 + \beta_{3} - 4 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{47} + ( -4 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{49} + ( \beta_{1} + \beta_{2} - 4 \beta_{6} ) q^{51} + ( \beta_{1} + 2 \beta_{6} + 2 \beta_{7} ) q^{53} + ( 1 + 3 \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{55} + ( -2 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{57} + ( -\beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{59} + ( 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{6} ) q^{61} + \beta_{1} q^{63} + ( 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{6} + \beta_{7} ) q^{65} + ( -3 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{67} + ( -2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{69} + ( 5 - \beta_{4} - \beta_{5} ) q^{71} + ( 3 + 3 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( 1 + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{75} + ( 3 - 3 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{77} + ( -3 \beta_{1} + 3 \beta_{2} + 6 \beta_{7} ) q^{79} + 5 q^{81} -\beta_{1} q^{83} + ( -6 + 7 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{85} + ( -5 + 5 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{87} + ( -5 \beta_{1} + 5 \beta_{2} + 8 \beta_{7} ) q^{89} + ( 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{6} ) q^{91} + ( -1 - \beta_{3} - 6 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{93} + ( 5 + 5 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{95} + ( 6 \beta_{2} - 5 \beta_{6} + 5 \beta_{7} ) q^{97} + ( -\beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{3} + O(q^{10}) \) \( 8q - 8q^{3} - 4q^{13} - 14q^{23} - 12q^{25} - 32q^{27} - 4q^{31} - 20q^{35} + 36q^{41} + 12q^{55} + 44q^{71} + 32q^{73} + 28q^{77} + 40q^{81} - 44q^{85} - 44q^{87} + 4q^{93} + 44q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 45 x^{4} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 47 \nu^{3} \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 47 \nu^{2} \)\()/14\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{6} + 2 \nu^{5} + 2 \nu^{4} + 127 \nu^{2} + 94 \nu + 38 \)\()/28\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{6} - 2 \nu^{5} + 2 \nu^{4} - 127 \nu^{2} - 94 \nu + 38 \)\()/28\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{7} + 2 \nu^{5} + 315 \nu^{3} + 94 \nu \)\()/28\)
\(\beta_{7}\)\(=\)\((\)\( -7 \nu^{7} + 2 \nu^{5} - 315 \nu^{3} + 94 \nu \)\()/28\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{6} + 7 \beta_{2}\)
\(\nu^{4}\)\(=\)\(7 \beta_{5} + 7 \beta_{4} - 19\)
\(\nu^{5}\)\(=\)\(7 \beta_{7} + 7 \beta_{6} - 47 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-47 \beta_{7} - 47 \beta_{6} - 47 \beta_{5} + 47 \beta_{4} + 127 \beta_{3}\)
\(\nu^{7}\)\(=\)\(-47 \beta_{7} + 47 \beta_{6} - 315 \beta_{2}\)

Character values

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

\(n\) \(231\) \(277\) \(281\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1
1.83051 + 1.83051i
0.386289 + 0.386289i
−0.386289 0.386289i
−1.83051 1.83051i
1.83051 1.83051i
0.386289 0.386289i
−0.386289 + 0.386289i
−1.83051 + 1.83051i
0 −1.00000 + 1.00000i 0 −1.83051 1.28422i 0 1.83051 1.83051i 0 1.00000i 0
137.2 0 −1.00000 + 1.00000i 0 −0.386289 + 2.20245i 0 0.386289 0.386289i 0 1.00000i 0
137.3 0 −1.00000 + 1.00000i 0 0.386289 2.20245i 0 −0.386289 + 0.386289i 0 1.00000i 0
137.4 0 −1.00000 + 1.00000i 0 1.83051 + 1.28422i 0 −1.83051 + 1.83051i 0 1.00000i 0
413.1 0 −1.00000 1.00000i 0 −1.83051 + 1.28422i 0 1.83051 + 1.83051i 0 1.00000i 0
413.2 0 −1.00000 1.00000i 0 −0.386289 2.20245i 0 0.386289 + 0.386289i 0 1.00000i 0
413.3 0 −1.00000 1.00000i 0 0.386289 + 2.20245i 0 −0.386289 0.386289i 0 1.00000i 0
413.4 0 −1.00000 1.00000i 0 1.83051 1.28422i 0 −1.83051 1.83051i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 413.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.i.a 8
5.b even 2 1 2300.2.i.c 8
5.c odd 4 1 inner 460.2.i.a 8
5.c odd 4 1 2300.2.i.c 8
23.b odd 2 1 inner 460.2.i.a 8
115.c odd 2 1 2300.2.i.c 8
115.e even 4 1 inner 460.2.i.a 8
115.e even 4 1 2300.2.i.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.i.a 8 1.a even 1 1 trivial
460.2.i.a 8 5.c odd 4 1 inner
460.2.i.a 8 23.b odd 2 1 inner
460.2.i.a 8 115.e even 4 1 inner
2300.2.i.c 8 5.b even 2 1
2300.2.i.c 8 5.c odd 4 1
2300.2.i.c 8 115.c odd 2 1
2300.2.i.c 8 115.e even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 2 + 2 T + T^{2} )^{4} \)
$5$ \( 625 + 150 T^{2} + 18 T^{4} + 6 T^{6} + T^{8} \)
$7$ \( 4 + 45 T^{4} + T^{8} \)
$11$ \( ( 8 + 14 T^{2} + T^{4} )^{2} \)
$13$ \( ( 400 - 40 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$17$ \( 2500 + 2109 T^{4} + T^{8} \)
$19$ \( ( 800 - 58 T^{2} + T^{4} )^{2} \)
$23$ \( 279841 + 170338 T + 51842 T^{2} + 8694 T^{3} + 1442 T^{4} + 378 T^{5} + 98 T^{6} + 14 T^{7} + T^{8} \)
$29$ \( ( 400 + 81 T^{2} + T^{4} )^{2} \)
$31$ \( ( -92 + T + T^{2} )^{4} \)
$37$ \( 2500 + 2109 T^{4} + T^{8} \)
$41$ \( ( 10 - 9 T + T^{2} )^{4} \)
$43$ \( 17909824 + 16500 T^{4} + T^{8} \)
$47$ \( ( 6724 + T^{4} )^{2} \)
$53$ \( 1119364 + 4125 T^{4} + T^{8} \)
$59$ \( ( 100 + 21 T^{2} + T^{4} )^{2} \)
$61$ \( ( 8192 + 202 T^{2} + T^{4} )^{2} \)
$67$ \( 48469444 + 13965 T^{4} + T^{8} \)
$71$ \( ( 20 - 11 T + T^{2} )^{4} \)
$73$ \( ( 2500 + 800 T + 128 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$79$ \( ( 10368 - 234 T^{2} + T^{4} )^{2} \)
$83$ \( 4 + 45 T^{4} + T^{8} \)
$89$ \( ( 55112 - 478 T^{2} + T^{4} )^{2} \)
$97$ \( 945685504 + 69540 T^{4} + T^{8} \)
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