Properties

Label 4560.2.d.e
Level $4560$
Weight $2$
Character orbit 4560.d
Analytic conductor $36.412$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9821011968.3
Defining polynomial: \(x^{6} + 20 x^{4} + 118 x^{2} + 192\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 20 x^{4} + 118 x^{2} + 192\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 12 \nu^{3} + 22 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 7 \)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} - 16 \nu^{3} - 58 \nu \)\()/4\)
\(\beta_{5}\)\(=\)\( \nu^{4} + 13 \nu^{2} + 32 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 7\)
\(\nu^{3}\)\(=\)\(-\beta_{4} - 2 \beta_{2} - 9 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} - 13 \beta_{3} + 59\)
\(\nu^{5}\)\(=\)\(12 \beta_{4} + 32 \beta_{2} + 86 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(1711\) \(1921\) \(2737\) \(3041\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
3.24237i
2.60808i
1.63858i
1.63858i
2.60808i
3.24237i
0 −1.00000 0 −1.00000 0 3.24237i 0 1.00000 0
2431.2 0 −1.00000 0 −1.00000 0 2.60808i 0 1.00000 0
2431.3 0 −1.00000 0 −1.00000 0 1.63858i 0 1.00000 0
2431.4 0 −1.00000 0 −1.00000 0 1.63858i 0 1.00000 0
2431.5 0 −1.00000 0 −1.00000 0 2.60808i 0 1.00000 0
2431.6 0 −1.00000 0 −1.00000 0 3.24237i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2431.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4560.2.d.e 6
4.b odd 2 1 4560.2.d.g yes 6
19.b odd 2 1 4560.2.d.g yes 6
76.d even 2 1 inner 4560.2.d.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4560.2.d.e 6 1.a even 1 1 trivial
4560.2.d.e 6 76.d even 2 1 inner
4560.2.d.g yes 6 4.b odd 2 1
4560.2.d.g yes 6 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4560, [\chi])\):

\( T_{7}^{6} + 20 T_{7}^{4} + 118 T_{7}^{2} + 192 \)
\( T_{31}^{3} + 6 T_{31}^{2} - 66 T_{31} - 404 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( ( 1 + T )^{6} \)
$7$ \( 192 + 118 T^{2} + 20 T^{4} + T^{6} \)
$11$ \( 108 + 142 T^{2} + 38 T^{4} + T^{6} \)
$13$ \( 192 + 118 T^{2} + 20 T^{4} + T^{6} \)
$17$ \( ( -12 - 14 T + 2 T^{2} + T^{3} )^{2} \)
$19$ \( 6859 - 722 T - 57 T^{2} - 52 T^{3} - 3 T^{4} - 2 T^{5} + T^{6} \)
$23$ \( 3072 + 2176 T^{2} + 98 T^{4} + T^{6} \)
$29$ \( 108 + 142 T^{2} + 38 T^{4} + T^{6} \)
$31$ \( ( -404 - 66 T + 6 T^{2} + T^{3} )^{2} \)
$37$ \( 48 + 742 T^{2} + 92 T^{4} + T^{6} \)
$41$ \( 108 + 270 T^{2} + 54 T^{4} + T^{6} \)
$43$ \( 48 + 742 T^{2} + 92 T^{4} + T^{6} \)
$47$ \( 768 + 4672 T^{2} + 146 T^{4} + T^{6} \)
$53$ \( 3072 + 2176 T^{2} + 98 T^{4} + T^{6} \)
$59$ \( ( 96 - 56 T - 4 T^{2} + T^{3} )^{2} \)
$61$ \( ( -256 - 78 T + T^{3} )^{2} \)
$67$ \( ( 96 - 96 T - 2 T^{2} + T^{3} )^{2} \)
$71$ \( ( 96 - 56 T - 4 T^{2} + T^{3} )^{2} \)
$73$ \( ( -976 - 204 T + T^{3} )^{2} \)
$79$ \( ( 128 + 44 T - 20 T^{2} + T^{3} )^{2} \)
$83$ \( 84672 + 6912 T^{2} + 162 T^{4} + T^{6} \)
$89$ \( 57132 + 6750 T^{2} + 198 T^{4} + T^{6} \)
$97$ \( 4853952 + 103734 T^{2} + 612 T^{4} + T^{6} \)
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