# Properties

 Label 1560.1.dk.a Level $1560$ Weight $1$ Character orbit 1560.dk Analytic conductor $0.779$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -120 Inner twists $8$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1560.dk (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.778541419707$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.16039857600.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{4} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{5} + \zeta_{12}^{5} q^{6} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + z * q^2 + z^4 * q^3 + z^2 * q^4 + z^3 * q^5 + z^5 * q^6 + z^3 * q^8 - z^2 * q^9 $$q + \zeta_{12} q^{2} + \zeta_{12}^{4} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{5} + \zeta_{12}^{5} q^{6} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{2} q^{9} + \zeta_{12}^{4} q^{10} + \zeta_{12} q^{11} - q^{12} - q^{13} - \zeta_{12} q^{15} + \zeta_{12}^{4} q^{16} - \zeta_{12}^{3} q^{18} + \zeta_{12}^{5} q^{20} + \zeta_{12}^{2} q^{22} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{23} - \zeta_{12} q^{24} - q^{25} - \zeta_{12} q^{26} + q^{27} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{29} - \zeta_{12}^{2} q^{30} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{31} + \zeta_{12}^{5} q^{32} + \zeta_{12}^{5} q^{33} - \zeta_{12}^{4} q^{36} + ( - \zeta_{12}^{2} - 1) q^{37} - \zeta_{12}^{4} q^{39} - q^{40} + \zeta_{12}^{2} q^{43} + \zeta_{12}^{3} q^{44} - \zeta_{12}^{5} q^{45} + ( - \zeta_{12}^{4} + 1) q^{46} + \zeta_{12}^{3} q^{47} - \zeta_{12}^{2} q^{48} - \zeta_{12}^{4} q^{49} - \zeta_{12} q^{50} - \zeta_{12}^{2} q^{52} + \zeta_{12} q^{54} + \zeta_{12}^{4} q^{55} + ( - \zeta_{12}^{4} + 1) q^{58} - \zeta_{12}^{5} q^{59} - \zeta_{12}^{3} q^{60} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{62} - q^{64} - \zeta_{12}^{3} q^{65} - q^{66} + (\zeta_{12}^{3} + \zeta_{12}) q^{69} - \zeta_{12}^{5} q^{72} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{74} - \zeta_{12}^{4} q^{75} - \zeta_{12}^{5} q^{78} - q^{79} - \zeta_{12} q^{80} + \zeta_{12}^{4} q^{81} + \zeta_{12}^{3} q^{86} + (\zeta_{12}^{3} + \zeta_{12}) q^{87} + \zeta_{12}^{4} q^{88} + q^{90} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{92} + ( - \zeta_{12}^{2} - 1) q^{93} + \zeta_{12}^{4} q^{94} - \zeta_{12}^{3} q^{96} - \zeta_{12}^{5} q^{98} - \zeta_{12}^{3} q^{99} +O(q^{100})$$ q + z * q^2 + z^4 * q^3 + z^2 * q^4 + z^3 * q^5 + z^5 * q^6 + z^3 * q^8 - z^2 * q^9 + z^4 * q^10 + z * q^11 - q^12 - q^13 - z * q^15 + z^4 * q^16 - z^3 * q^18 + z^5 * q^20 + z^2 * q^22 + (-z^5 - z^3) * q^23 - z * q^24 - q^25 - z * q^26 + q^27 + (-z^5 - z^3) * q^29 - z^2 * q^30 + (z^4 + z^2) * q^31 + z^5 * q^32 + z^5 * q^33 - z^4 * q^36 + (-z^2 - 1) * q^37 - z^4 * q^39 - q^40 + z^2 * q^43 + z^3 * q^44 - z^5 * q^45 + (-z^4 + 1) * q^46 + z^3 * q^47 - z^2 * q^48 - z^4 * q^49 - z * q^50 - z^2 * q^52 + z * q^54 + z^4 * q^55 + (-z^4 + 1) * q^58 - z^5 * q^59 - z^3 * q^60 + (z^5 + z^3) * q^62 - q^64 - z^3 * q^65 - q^66 + (z^3 + z) * q^69 - z^5 * q^72 + (-z^3 - z) * q^74 - z^4 * q^75 - z^5 * q^78 - q^79 - z * q^80 + z^4 * q^81 + z^3 * q^86 + (z^3 + z) * q^87 + z^4 * q^88 + q^90 + (-z^5 + z) * q^92 + (-z^2 - 1) * q^93 + z^4 * q^94 - z^3 * q^96 - z^5 * q^98 - z^3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 2 q^{4} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 2 * q^4 - 2 * q^9 $$4 q - 2 q^{3} + 2 q^{4} - 2 q^{9} - 2 q^{10} - 4 q^{12} - 4 q^{13} - 2 q^{16} + 2 q^{22} - 4 q^{25} + 4 q^{27} - 2 q^{30} + 2 q^{36} - 6 q^{37} + 2 q^{39} - 4 q^{40} + 2 q^{43} + 6 q^{46} - 2 q^{48} + 2 q^{49} - 2 q^{52} - 2 q^{55} + 6 q^{58} - 4 q^{64} - 4 q^{66} + 2 q^{75} - 4 q^{79} - 2 q^{81} - 2 q^{88} + 4 q^{90} - 6 q^{93} - 2 q^{94}+O(q^{100})$$ 4 * q - 2 * q^3 + 2 * q^4 - 2 * q^9 - 2 * q^10 - 4 * q^12 - 4 * q^13 - 2 * q^16 + 2 * q^22 - 4 * q^25 + 4 * q^27 - 2 * q^30 + 2 * q^36 - 6 * q^37 + 2 * q^39 - 4 * q^40 + 2 * q^43 + 6 * q^46 - 2 * q^48 + 2 * q^49 - 2 * q^52 - 2 * q^55 + 6 * q^58 - 4 * q^64 - 4 * q^66 + 2 * q^75 - 4 * q^79 - 2 * q^81 - 2 * q^88 + 4 * q^90 - 6 * q^93 - 2 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$$.

 $$n$$ $$391$$ $$521$$ $$781$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1109.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000i 0.866025 0.500000i 0 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
1109.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000i −0.866025 + 0.500000i 0 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
1349.1 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 1.00000i 0.866025 + 0.500000i 0 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
1349.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 1.00000i −0.866025 0.500000i 0 1.00000i −0.500000 + 0.866025i −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
120.i odd 2 1 CM by $$\Q(\sqrt{-30})$$
3.b odd 2 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner
40.f even 2 1 inner
520.bp even 6 1 inner
1560.dk odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.1.dk.a 4
3.b odd 2 1 inner 1560.1.dk.a 4
5.b even 2 1 1560.1.dk.b yes 4
8.b even 2 1 1560.1.dk.b yes 4
13.e even 6 1 inner 1560.1.dk.a 4
15.d odd 2 1 1560.1.dk.b yes 4
24.h odd 2 1 1560.1.dk.b yes 4
39.h odd 6 1 inner 1560.1.dk.a 4
40.f even 2 1 inner 1560.1.dk.a 4
65.l even 6 1 1560.1.dk.b yes 4
104.s even 6 1 1560.1.dk.b yes 4
120.i odd 2 1 CM 1560.1.dk.a 4
195.y odd 6 1 1560.1.dk.b yes 4
312.bg odd 6 1 1560.1.dk.b yes 4
520.bp even 6 1 inner 1560.1.dk.a 4
1560.dk odd 6 1 inner 1560.1.dk.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.1.dk.a 4 1.a even 1 1 trivial
1560.1.dk.a 4 3.b odd 2 1 inner
1560.1.dk.a 4 13.e even 6 1 inner
1560.1.dk.a 4 39.h odd 6 1 inner
1560.1.dk.a 4 40.f even 2 1 inner
1560.1.dk.a 4 120.i odd 2 1 CM
1560.1.dk.a 4 520.bp even 6 1 inner
1560.1.dk.a 4 1560.dk odd 6 1 inner
1560.1.dk.b yes 4 5.b even 2 1
1560.1.dk.b yes 4 8.b even 2 1
1560.1.dk.b yes 4 15.d odd 2 1
1560.1.dk.b yes 4 24.h odd 2 1
1560.1.dk.b yes 4 65.l even 6 1
1560.1.dk.b yes 4 104.s even 6 1
1560.1.dk.b yes 4 195.y odd 6 1
1560.1.dk.b yes 4 312.bg odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{37}^{2} + 3T_{37} + 3$$ acting on $$S_{1}^{\mathrm{new}}(1560, [\chi])$$.

## Hecke characteristic polynomials

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