Properties

Label 4560.2.d.b
Level $4560$
Weight $2$
Character orbit 4560.d
Analytic conductor $36.412$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} + q^{9} + ( 2 - 4 \zeta_{6} ) q^{11} - q^{15} + ( 5 - 2 \zeta_{6} ) q^{19} + ( -2 + 4 \zeta_{6} ) q^{23} + q^{25} - q^{27} + ( -4 + 8 \zeta_{6} ) q^{29} + 2 q^{31} + ( -2 + 4 \zeta_{6} ) q^{33} + ( -4 + 8 \zeta_{6} ) q^{37} + ( 2 - 4 \zeta_{6} ) q^{41} + ( 6 - 12 \zeta_{6} ) q^{43} + q^{45} + ( 2 - 4 \zeta_{6} ) q^{47} + 7 q^{49} + ( 2 - 4 \zeta_{6} ) q^{55} + ( -5 + 2 \zeta_{6} ) q^{57} + 6 q^{59} + 10 q^{61} -4 q^{67} + ( 2 - 4 \zeta_{6} ) q^{69} -14 q^{73} - q^{75} -14 q^{79} + q^{81} + ( 8 - 16 \zeta_{6} ) q^{83} + ( 4 - 8 \zeta_{6} ) q^{87} + ( -6 + 12 \zeta_{6} ) q^{89} -2 q^{93} + ( 5 - 2 \zeta_{6} ) q^{95} + ( 4 - 8 \zeta_{6} ) q^{97} + ( 2 - 4 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} + 2q^{9} - 2q^{15} + 8q^{19} + 2q^{25} - 2q^{27} + 4q^{31} + 2q^{45} + 14q^{49} - 8q^{57} + 12q^{59} + 20q^{61} - 8q^{67} - 28q^{73} - 2q^{75} - 28q^{79} + 2q^{81} - 4q^{93} + 8q^{95} + O(q^{100}) \)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.00000 0 1.00000 0 0 0 1.00000 0
2431.2 0 −1.00000 0 1.00000 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
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.b 2
4.b odd 2 1 4560.2.d.d yes 2
19.b odd 2 1 4560.2.d.d yes 2
76.d even 2 1 inner 4560.2.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4560.2.d.b 2 1.a even 1 1 trivial
4560.2.d.b 2 76.d even 2 1 inner
4560.2.d.d yes 2 4.b odd 2 1
4560.2.d.d yes 2 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 \(S_{2}^{\mathrm{new}}(4560, [\chi])\):

\( T_{7} \)
\( T_{31} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( 12 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( 19 - 8 T + T^{2} \)
$23$ \( 12 + T^{2} \)
$29$ \( 48 + T^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( 48 + T^{2} \)
$41$ \( 12 + T^{2} \)
$43$ \( 108 + T^{2} \)
$47$ \( 12 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( ( -6 + T )^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 14 + T )^{2} \)
$79$ \( ( 14 + T )^{2} \)
$83$ \( 192 + T^{2} \)
$89$ \( 108 + T^{2} \)
$97$ \( 48 + T^{2} \)
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