# Properties

 Label 2520.1.ef.c Level $2520$ Weight $1$ Character orbit 2520.ef Analytic conductor $1.258$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -40 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.25764383184$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.829785600.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{12}^{2} q^{2} + \zeta_{12}^{4} q^{4} + \zeta_{12}^{2} q^{5} -\zeta_{12}^{3} q^{7} + q^{8} +O(q^{10})$$ $$q -\zeta_{12}^{2} q^{2} + \zeta_{12}^{4} q^{4} + \zeta_{12}^{2} q^{5} -\zeta_{12}^{3} q^{7} + q^{8} -\zeta_{12}^{4} q^{10} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{11} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{13} + \zeta_{12}^{5} q^{14} -\zeta_{12}^{2} q^{16} -\zeta_{12}^{2} q^{19} - q^{20} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{22} -\zeta_{12}^{2} q^{23} + \zeta_{12}^{4} q^{25} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{26} + \zeta_{12} q^{28} + \zeta_{12}^{4} q^{32} -\zeta_{12}^{5} q^{35} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{37} + \zeta_{12}^{4} q^{38} + \zeta_{12}^{2} q^{40} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{41} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{44} + \zeta_{12}^{4} q^{46} + \zeta_{12}^{2} q^{47} - q^{49} + q^{50} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{52} -\zeta_{12}^{4} q^{53} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{55} -\zeta_{12}^{3} q^{56} + q^{64} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{65} -\zeta_{12} q^{70} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{74} + q^{76} + ( -1 - \zeta_{12}^{2} ) q^{77} -\zeta_{12}^{4} q^{80} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{82} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{88} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{91} + q^{92} -\zeta_{12}^{4} q^{94} -\zeta_{12}^{4} q^{95} + \zeta_{12}^{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} + 2q^{5} + 4q^{8} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} + 2q^{5} + 4q^{8} + 2q^{10} - 2q^{16} - 2q^{19} - 4q^{20} - 2q^{23} - 2q^{25} - 2q^{32} - 2q^{38} + 2q^{40} - 2q^{46} + 2q^{47} - 4q^{49} + 4q^{50} + 2q^{53} + 4q^{64} + 4q^{76} - 6q^{77} + 2q^{80} + 4q^{92} + 2q^{94} + 2q^{95} + 2q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$281$$ $$631$$ $$1081$$ $$1261$$ $$2017$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
739.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 1.00000i 1.00000 0 0.500000 0.866025i
739.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 1.00000i 1.00000 0 0.500000 0.866025i
2179.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 1.00000i 1.00000 0 0.500000 + 0.866025i
2179.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 1.00000i 1.00000 0 0.500000 + 0.866025i
 $$n$$: 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
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
7.c even 3 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
105.o odd 6 1 inner
168.v even 6 1 inner
280.bi 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.ef.c 4
3.b odd 2 1 2520.1.ef.d yes 4
5.b even 2 1 2520.1.ef.d yes 4
7.c even 3 1 inner 2520.1.ef.c 4
8.d odd 2 1 2520.1.ef.d yes 4
15.d odd 2 1 inner 2520.1.ef.c 4
21.h odd 6 1 2520.1.ef.d yes 4
24.f even 2 1 inner 2520.1.ef.c 4
35.j even 6 1 2520.1.ef.d yes 4
40.e odd 2 1 CM 2520.1.ef.c 4
56.k odd 6 1 2520.1.ef.d yes 4
105.o odd 6 1 inner 2520.1.ef.c 4
120.m even 2 1 2520.1.ef.d yes 4
168.v even 6 1 inner 2520.1.ef.c 4
280.bi odd 6 1 inner 2520.1.ef.c 4
840.cv even 6 1 2520.1.ef.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.1.ef.c 4 1.a even 1 1 trivial
2520.1.ef.c 4 7.c even 3 1 inner
2520.1.ef.c 4 15.d odd 2 1 inner
2520.1.ef.c 4 24.f even 2 1 inner
2520.1.ef.c 4 40.e odd 2 1 CM
2520.1.ef.c 4 105.o odd 6 1 inner
2520.1.ef.c 4 168.v even 6 1 inner
2520.1.ef.c 4 280.bi odd 6 1 inner
2520.1.ef.d yes 4 3.b odd 2 1
2520.1.ef.d yes 4 5.b even 2 1
2520.1.ef.d yes 4 8.d odd 2 1
2520.1.ef.d yes 4 21.h odd 6 1
2520.1.ef.d yes 4 35.j even 6 1
2520.1.ef.d yes 4 56.k odd 6 1
2520.1.ef.d yes 4 120.m even 2 1
2520.1.ef.d yes 4 840.cv even 6 1

## Hecke kernels

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

 $$T_{11}^{4} + 3 T_{11}^{2} + 9$$ $$T_{13}^{2} - 3$$ $$T_{23}^{2} + T_{23} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$9 + 3 T^{2} + T^{4}$$
$13$ $$( -3 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( 1 + T + T^{2} )^{2}$$
$23$ $$( 1 + T + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$9 + 3 T^{2} + T^{4}$$
$41$ $$( -3 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$( 1 - T + T^{2} )^{2}$$
$53$ $$( 1 - T + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$