Properties

Label 4560.2.d.h
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: 6.0.792772608.2
Defining polynomial: \(x^{6} + 22 x^{4} + 62 x^{2} + 12\)
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.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + q^{9} + \beta_{1} q^{11} + ( -\beta_{1} - \beta_{2} ) q^{13} + q^{15} + ( -1 - \beta_{3} + \beta_{4} ) q^{17} + ( 1 + \beta_{2} - \beta_{3} ) q^{19} + ( -\beta_{1} - \beta_{2} ) q^{21} + ( -\beta_{2} + \beta_{5} ) q^{23} + q^{25} + q^{27} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{29} + ( 2 + \beta_{3} ) q^{31} + \beta_{1} q^{33} + ( -\beta_{1} - \beta_{2} ) q^{35} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{37} + ( -\beta_{1} - \beta_{2} ) q^{39} + ( \beta_{1} - 2 \beta_{5} ) q^{41} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{43} + q^{45} + ( -\beta_{2} + 3 \beta_{5} ) q^{47} + ( -2 - 3 \beta_{3} - \beta_{4} ) q^{49} + ( -1 - \beta_{3} + \beta_{4} ) q^{51} + ( -\beta_{2} + \beta_{5} ) q^{53} + \beta_{1} q^{55} + ( 1 + \beta_{2} - \beta_{3} ) q^{57} + ( -2 - 2 \beta_{3} - 2 \beta_{4} ) q^{59} + ( -2 - 3 \beta_{3} - 2 \beta_{4} ) q^{61} + ( -\beta_{1} - \beta_{2} ) q^{63} + ( -\beta_{1} - \beta_{2} ) q^{65} + ( 3 + 2 \beta_{3} + \beta_{4} ) q^{67} + ( -\beta_{2} + \beta_{5} ) q^{69} + ( 2 + 2 \beta_{3} + 2 \beta_{4} ) q^{71} + ( -1 - 2 \beta_{3} - \beta_{4} ) q^{73} + q^{75} + ( 4 + \beta_{3} ) q^{77} + ( -1 - \beta_{4} ) q^{79} + q^{81} + ( -2 \beta_{1} - 3 \beta_{2} - \beta_{5} ) q^{83} + ( -1 - \beta_{3} + \beta_{4} ) q^{85} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{87} + ( -3 \beta_{1} - 4 \beta_{2} - \beta_{5} ) q^{89} + ( -9 - 3 \beta_{3} - \beta_{4} ) q^{91} + ( 2 + \beta_{3} ) q^{93} + ( 1 + \beta_{2} - \beta_{3} ) q^{95} + ( -\beta_{1} - \beta_{2} - 2 \beta_{5} ) q^{97} + \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} + 6q^{5} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} + 6q^{5} + 6q^{9} + 6q^{15} - 4q^{17} + 6q^{19} + 6q^{25} + 6q^{27} + 12q^{31} + 6q^{45} - 14q^{49} - 4q^{51} + 6q^{57} - 16q^{59} - 16q^{61} + 20q^{67} + 16q^{71} - 8q^{73} + 6q^{75} + 24q^{77} - 8q^{79} + 6q^{81} - 4q^{85} - 56q^{91} + 12q^{93} + 6q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 22 x^{4} + 62 x^{2} + 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 26 \nu^{3} + 114 \nu \)\()/26\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{4} - 39 \nu^{2} - 46 \)\()/13\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} + 26 \nu^{2} + 75 \)\()/13\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{5} + 65 \nu^{3} + 160 \nu \)\()/13\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{4} + \beta_{3} - 8\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + 6 \beta_{2} - 14 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-39 \beta_{4} - 26 \beta_{3} + 133\)
\(\nu^{5}\)\(=\)\(26 \beta_{5} - 130 \beta_{2} + 250 \beta_{1}\)

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
1.75167i
0.457038i
4.32698i
4.32698i
0.457038i
1.75167i
0 1.00000 0 1.00000 0 4.69162i 0 1.00000 0
2431.2 0 1.00000 0 1.00000 0 2.36627i 0 1.00000 0
2431.3 0 1.00000 0 1.00000 0 0.624069i 0 1.00000 0
2431.4 0 1.00000 0 1.00000 0 0.624069i 0 1.00000 0
2431.5 0 1.00000 0 1.00000 0 2.36627i 0 1.00000 0
2431.6 0 1.00000 0 1.00000 0 4.69162i 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.h yes 6
4.b odd 2 1 4560.2.d.f 6
19.b odd 2 1 4560.2.d.f 6
76.d even 2 1 inner 4560.2.d.h yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4560.2.d.f 6 4.b odd 2 1
4560.2.d.f 6 19.b odd 2 1
4560.2.d.h yes 6 1.a even 1 1 trivial
4560.2.d.h yes 6 76.d even 2 1 inner

Hecke kernels

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

\( T_{7}^{6} + 28 T_{7}^{4} + 134 T_{7}^{2} + 48 \)
\( T_{31}^{3} - 6 T_{31}^{2} - 2 T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( -1 + T )^{6} \)
$5$ \( ( -1 + T )^{6} \)
$7$ \( 48 + 134 T^{2} + 28 T^{4} + T^{6} \)
$11$ \( 12 + 62 T^{2} + 22 T^{4} + T^{6} \)
$13$ \( 48 + 134 T^{2} + 28 T^{4} + T^{6} \)
$17$ \( ( -156 - 46 T + 2 T^{2} + T^{3} )^{2} \)
$19$ \( 6859 - 2166 T + 247 T^{2} + 4 T^{3} + 13 T^{4} - 6 T^{5} + T^{6} \)
$23$ \( 768 + 320 T^{2} + 34 T^{4} + T^{6} \)
$29$ \( 164268 + 9086 T^{2} + 166 T^{4} + T^{6} \)
$31$ \( ( 4 - 2 T - 6 T^{2} + T^{3} )^{2} \)
$37$ \( 1728 + 566 T^{2} + 52 T^{4} + T^{6} \)
$41$ \( 137388 + 9662 T^{2} + 182 T^{4} + T^{6} \)
$43$ \( 1728 + 566 T^{2} + 52 T^{4} + T^{6} \)
$47$ \( 27648 + 10880 T^{2} + 242 T^{4} + T^{6} \)
$53$ \( 768 + 320 T^{2} + 34 T^{4} + T^{6} \)
$59$ \( ( -768 - 104 T + 8 T^{2} + T^{3} )^{2} \)
$61$ \( ( -688 - 158 T + 8 T^{2} + T^{3} )^{2} \)
$67$ \( ( 128 - 32 T - 10 T^{2} + T^{3} )^{2} \)
$71$ \( ( 768 - 104 T - 8 T^{2} + T^{3} )^{2} \)
$73$ \( ( -32 - 60 T + 4 T^{2} + T^{3} )^{2} \)
$79$ \( ( -32 - 20 T + 4 T^{2} + T^{3} )^{2} \)
$83$ \( 292032 + 20864 T^{2} + 274 T^{4} + T^{6} \)
$89$ \( 629292 + 53966 T^{2} + 454 T^{4} + T^{6} \)
$97$ \( 3888 + 1862 T^{2} + 172 T^{4} + T^{6} \)
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