# Properties

 Label 2520.1.ek.a Level $2520$ Weight $1$ Character orbit 2520.ek Analytic conductor $1.258$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2520.ek (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.25764383184$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.3630312000.4

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + \zeta_{6} q^{5} -\zeta_{6} q^{7} + q^{8} +O(q^{10})$$ $$q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + \zeta_{6} q^{5} -\zeta_{6} q^{7} + q^{8} - q^{10} + ( -1 + \zeta_{6}^{2} ) q^{11} + q^{14} + \zeta_{6}^{2} q^{16} -\zeta_{6}^{2} q^{20} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{22} + \zeta_{6}^{2} q^{25} + \zeta_{6}^{2} q^{28} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{29} + ( -1 + \zeta_{6}^{2} ) q^{31} -\zeta_{6} q^{32} -\zeta_{6}^{2} q^{35} + \zeta_{6} q^{40} + ( 1 + \zeta_{6} ) q^{44} + \zeta_{6}^{2} q^{49} -\zeta_{6} q^{50} -\zeta_{6} q^{53} + ( -1 - \zeta_{6} ) q^{55} -\zeta_{6} q^{56} + ( -1 - \zeta_{6} ) q^{58} + \zeta_{6} q^{59} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{62} + q^{64} + \zeta_{6} q^{70} + 2 \zeta_{6} q^{73} + ( 1 + \zeta_{6} ) q^{77} + \zeta_{6}^{2} q^{79} - q^{80} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{83} + ( -1 + \zeta_{6}^{2} ) q^{88} - q^{97} -\zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + q^{5} - q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} + q^{5} - q^{7} + 2q^{8} - 2q^{10} - 3q^{11} + 2q^{14} - q^{16} + q^{20} - q^{25} - q^{28} - 3q^{31} - q^{32} + q^{35} + q^{40} + 3q^{44} - q^{49} - q^{50} - q^{53} - 3q^{55} - q^{56} - 3q^{58} + q^{59} + 2q^{64} + q^{70} + 2q^{73} + 3q^{77} - q^{79} - 2q^{80} - 3q^{88} - 2q^{97} - q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$1081$$ $$1261$$ $$2017$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}^{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
829.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 −1.00000
1909.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 + 0.866025i 0 −0.500000 0.866025i 1.00000 0 −1.00000
 $$n$$: 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
105.p even 6 1 inner
280.bk odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.1.ek.a 2
3.b odd 2 1 2520.1.ek.d yes 2
5.b even 2 1 2520.1.ek.c yes 2
7.d odd 6 1 2520.1.ek.b yes 2
8.b even 2 1 2520.1.ek.d yes 2
15.d odd 2 1 2520.1.ek.b yes 2
21.g even 6 1 2520.1.ek.c yes 2
24.h odd 2 1 CM 2520.1.ek.a 2
35.i odd 6 1 2520.1.ek.d yes 2
40.f even 2 1 2520.1.ek.b yes 2
56.j odd 6 1 2520.1.ek.c yes 2
105.p even 6 1 inner 2520.1.ek.a 2
120.i odd 2 1 2520.1.ek.c yes 2
168.ba even 6 1 2520.1.ek.b yes 2
280.bk odd 6 1 inner 2520.1.ek.a 2
840.cb even 6 1 2520.1.ek.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.1.ek.a 2 1.a even 1 1 trivial
2520.1.ek.a 2 24.h odd 2 1 CM
2520.1.ek.a 2 105.p even 6 1 inner
2520.1.ek.a 2 280.bk odd 6 1 inner
2520.1.ek.b yes 2 7.d odd 6 1
2520.1.ek.b yes 2 15.d odd 2 1
2520.1.ek.b yes 2 40.f even 2 1
2520.1.ek.b yes 2 168.ba even 6 1
2520.1.ek.c yes 2 5.b even 2 1
2520.1.ek.c yes 2 21.g even 6 1
2520.1.ek.c yes 2 56.j odd 6 1
2520.1.ek.c yes 2 120.i odd 2 1
2520.1.ek.d yes 2 3.b odd 2 1
2520.1.ek.d yes 2 8.b even 2 1
2520.1.ek.d yes 2 35.i odd 6 1
2520.1.ek.d yes 2 840.cb even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2520, [\chi])$$:

 $$T_{11}^{2} + 3 T_{11} + 3$$ $$T_{53}^{2} + T_{53} + 1$$

## Hecke characteristic polynomials

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