# Properties

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

# Related objects

## 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
 0.5 − 0.866025i 0.5 + 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.d yes 2
4.b odd 2 1 4560.2.d.b 2
19.b odd 2 1 4560.2.d.b 2
76.d even 2 1 inner 4560.2.d.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4560.2.d.b 2 4.b odd 2 1
4560.2.d.b 2 19.b odd 2 1
4560.2.d.d yes 2 1.a even 1 1 trivial
4560.2.d.d yes 2 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 $$S_{2}^{\mathrm{new}}(4560, [\chi])$$:

 $$T_{7}$$ $$T_{31} + 2$$

## Hecke characteristic polynomials

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