# Properties

 Label 1560.1.y.e Level $1560$ Weight $1$ Character orbit 1560.y Analytic conductor $0.779$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -39, -120, 520 Inner twists $8$

# Related objects

## 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: no Analytic conductor: $$0.778541419707$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-30}, \sqrt{-39})$$ Artin image: $D_4:C_2$ Artin field: Galois closure of 8.0.3701505600.2

## $q$-expansion

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

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

## Character values

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

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

## 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
 1.00000i − 1.00000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 1.00000 −1.00000
389.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 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 $$\Q(\sqrt{-39})$$
120.i odd 2 1 CM by $$\Q(\sqrt{-30})$$
520.p even 2 1 RM by $$\Q(\sqrt{130})$$
3.b odd 2 1 inner
13.b even 2 1 inner
40.f even 2 1 inner
1560.y odd 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{11}^{2} + 4$$ T11^2 + 4 $$T_{41}$$ T41 $$T_{43} + 2$$ T43 + 2

## Hecke characteristic polynomials

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