# Properties

 Label 1560.1.y.c Level $1560$ Weight $1$ Character orbit 1560.y Self dual yes Analytic conductor $0.779$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -39, -1560, 40 Inner twists $4$

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## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$0.778541419707$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{10}, \sqrt{-39})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.60840.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^5 - q^6 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - q^{12} + q^{13} + q^{15} + q^{16} + q^{18} - q^{20} - q^{24} + q^{25} + q^{26} - q^{27} + q^{30} + q^{32} + q^{36} - q^{39} - q^{40} + 2 q^{41} + 2 q^{43} - q^{45} - q^{48} - q^{49} + q^{50} + q^{52} - q^{54} + q^{60} + q^{64} - q^{65} - 2 q^{71} + q^{72} - q^{75} - q^{78} - 2 q^{79} - q^{80} + q^{81} + 2 q^{82} - 2 q^{83} + 2 q^{86} - 2 q^{89} - q^{90} - q^{96} - q^{98}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^5 - q^6 + q^8 + q^9 - q^10 - q^12 + q^13 + q^15 + q^16 + q^18 - q^20 - q^24 + q^25 + q^26 - q^27 + q^30 + q^32 + q^36 - q^39 - q^40 + 2 * q^41 + 2 * q^43 - q^45 - q^48 - q^49 + q^50 + q^52 - q^54 + q^60 + q^64 - q^65 - 2 * q^71 + q^72 - q^75 - q^78 - 2 * q^79 - q^80 + q^81 + 2 * q^82 - 2 * q^83 + 2 * q^86 - 2 * q^89 - q^90 - q^96 - q^98

## Character values

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

 $$n$$ $$391$$ $$521$$ $$781$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
389.1
 0
1.00000 −1.00000 1.00000 −1.00000 −1.00000 0 1.00000 1.00000 −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
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
40.f even 2 1 RM by $$\Q(\sqrt{10})$$
1560.y odd 2 1 CM by $$\Q(\sqrt{-390})$$

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1560, [\chi])$$:

 $$T_{11}$$ T11 $$T_{41} - 2$$ T41 - 2 $$T_{43} - 2$$ T43 - 2

## Hecke characteristic polynomials

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