# Properties

 Label 1560.1.cs.e Level $1560$ Weight $1$ Character orbit 1560.cs Analytic conductor $0.779$ Analytic rank $0$ Dimension $8$ Projective image $D_{8}$ CM discriminant -39 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.cs (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.778541419707$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ x^8 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.2.49353408000000.10

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{16}^{3} q^{2} + \zeta_{16}^{2} q^{3} + \zeta_{16}^{6} q^{4} - \zeta_{16}^{5} q^{5} + \zeta_{16}^{5} q^{6} - \zeta_{16} q^{8} + \zeta_{16}^{4} q^{9} +O(q^{10})$$ q + z^3 * q^2 + z^2 * q^3 + z^6 * q^4 - z^5 * q^5 + z^5 * q^6 - z * q^8 + z^4 * q^9 $$q + \zeta_{16}^{3} q^{2} + \zeta_{16}^{2} q^{3} + \zeta_{16}^{6} q^{4} - \zeta_{16}^{5} q^{5} + \zeta_{16}^{5} q^{6} - \zeta_{16} q^{8} + \zeta_{16}^{4} q^{9} + q^{10} + ( - \zeta_{16}^{7} + \zeta_{16}) q^{11} - q^{12} + \zeta_{16}^{6} q^{13} - \zeta_{16}^{7} q^{15} - \zeta_{16}^{4} q^{16} + \zeta_{16}^{7} q^{18} + \zeta_{16}^{3} q^{20} + (\zeta_{16}^{4} + \zeta_{16}^{2}) q^{22} - \zeta_{16}^{3} q^{24} - \zeta_{16}^{2} q^{25} - \zeta_{16} q^{26} + \zeta_{16}^{6} q^{27} + \zeta_{16}^{2} q^{30} - \zeta_{16}^{7} q^{32} + (\zeta_{16}^{3} + \zeta_{16}) q^{33} - \zeta_{16}^{2} q^{36} - q^{39} + \zeta_{16}^{6} q^{40} + (\zeta_{16}^{5} - \zeta_{16}^{3}) q^{41} + ( - \zeta_{16}^{4} - 1) q^{43} + (\zeta_{16}^{7} + \zeta_{16}^{5}) q^{44} + \zeta_{16} q^{45} + (\zeta_{16}^{7} - \zeta_{16}^{5}) q^{47} - \zeta_{16}^{6} q^{48} + \zeta_{16}^{4} q^{49} - \zeta_{16}^{5} q^{50} - \zeta_{16}^{4} q^{52} - \zeta_{16} q^{54} + ( - \zeta_{16}^{6} - \zeta_{16}^{4}) q^{55} + (\zeta_{16}^{7} + \zeta_{16}) q^{59} + \zeta_{16}^{5} q^{60} + (\zeta_{16}^{6} + \zeta_{16}^{2}) q^{61} + \zeta_{16}^{2} q^{64} + \zeta_{16}^{3} q^{65} + (\zeta_{16}^{6} + \zeta_{16}^{4}) q^{66} + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{71} - \zeta_{16}^{5} q^{72} - \zeta_{16}^{4} q^{75} - \zeta_{16}^{3} q^{78} + ( - \zeta_{16}^{6} + \zeta_{16}^{2}) q^{79} - \zeta_{16} q^{80} - q^{81} + ( - \zeta_{16}^{6} - 1) q^{82} + ( - \zeta_{16}^{3} - \zeta_{16}) q^{83} + ( - \zeta_{16}^{7} - \zeta_{16}^{3}) q^{86} + ( - \zeta_{16}^{2} - 1) q^{88} + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{89} + \zeta_{16}^{4} q^{90} + ( - \zeta_{16}^{2} + 1) q^{94} + \zeta_{16} q^{96} + \zeta_{16}^{7} q^{98} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{99} +O(q^{100})$$ q + z^3 * q^2 + z^2 * q^3 + z^6 * q^4 - z^5 * q^5 + z^5 * q^6 - z * q^8 + z^4 * q^9 + q^10 + (-z^7 + z) * q^11 - q^12 + z^6 * q^13 - z^7 * q^15 - z^4 * q^16 + z^7 * q^18 + z^3 * q^20 + (z^4 + z^2) * q^22 - z^3 * q^24 - z^2 * q^25 - z * q^26 + z^6 * q^27 + z^2 * q^30 - z^7 * q^32 + (z^3 + z) * q^33 - z^2 * q^36 - q^39 + z^6 * q^40 + (z^5 - z^3) * q^41 + (-z^4 - 1) * q^43 + (z^7 + z^5) * q^44 + z * q^45 + (z^7 - z^5) * q^47 - z^6 * q^48 + z^4 * q^49 - z^5 * q^50 - z^4 * q^52 - z * q^54 + (-z^6 - z^4) * q^55 + (z^7 + z) * q^59 + z^5 * q^60 + (z^6 + z^2) * q^61 + z^2 * q^64 + z^3 * q^65 + (z^6 + z^4) * q^66 + (-z^5 - z^3) * q^71 - z^5 * q^72 - z^4 * q^75 - z^3 * q^78 + (-z^6 + z^2) * q^79 - z * q^80 - q^81 + (-z^6 - 1) * q^82 + (-z^3 - z) * q^83 + (-z^7 - z^3) * q^86 + (-z^2 - 1) * q^88 + (-z^5 - z^3) * q^89 + z^4 * q^90 + (-z^2 + 1) * q^94 + z * q^96 + z^7 * q^98 + (z^5 + z^3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 8 q^{10} - 8 q^{12} - 8 q^{39} - 8 q^{43} - 8 q^{81} - 8 q^{82} - 8 q^{88} + 8 q^{94}+O(q^{100})$$ 8 * q + 8 * q^10 - 8 * q^12 - 8 * q^39 - 8 * q^43 - 8 * q^81 - 8 * q^82 - 8 * q^88 + 8 * q^94

## Character values

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

 $$n$$ $$391$$ $$521$$ $$781$$ $$937$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$-\zeta_{16}^{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
467.1
 0.382683 + 0.923880i −0.923880 + 0.382683i 0.923880 − 0.382683i −0.382683 − 0.923880i 0.382683 − 0.923880i −0.923880 − 0.382683i 0.923880 + 0.382683i −0.382683 + 0.923880i
−0.923880 0.382683i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.923880 + 0.382683i 0.923880 0.382683i 0 −0.382683 0.923880i 1.00000i 1.00000
467.2 −0.382683 + 0.923880i 0.707107 0.707107i −0.707107 0.707107i −0.382683 0.923880i 0.382683 + 0.923880i 0 0.923880 0.382683i 1.00000i 1.00000
467.3 0.382683 0.923880i 0.707107 0.707107i −0.707107 0.707107i 0.382683 + 0.923880i −0.382683 0.923880i 0 −0.923880 + 0.382683i 1.00000i 1.00000
467.4 0.923880 + 0.382683i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.923880 0.382683i −0.923880 + 0.382683i 0 0.382683 + 0.923880i 1.00000i 1.00000
1403.1 −0.923880 + 0.382683i −0.707107 0.707107i 0.707107 0.707107i −0.923880 0.382683i 0.923880 + 0.382683i 0 −0.382683 + 0.923880i 1.00000i 1.00000
1403.2 −0.382683 0.923880i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.382683 + 0.923880i 0.382683 0.923880i 0 0.923880 + 0.382683i 1.00000i 1.00000
1403.3 0.382683 + 0.923880i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.382683 0.923880i −0.382683 + 0.923880i 0 −0.923880 0.382683i 1.00000i 1.00000
1403.4 0.923880 0.382683i −0.707107 0.707107i 0.707107 0.707107i 0.923880 + 0.382683i −0.923880 0.382683i 0 0.382683 0.923880i 1.00000i 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1403.4 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 $$\Q(\sqrt{-39})$$
3.b odd 2 1 inner
13.b even 2 1 inner
40.k even 4 1 inner
120.q odd 4 1 inner
520.bc even 4 1 inner
1560.cs odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.1.cs.e yes 8
3.b odd 2 1 inner 1560.1.cs.e yes 8
5.c odd 4 1 1560.1.cs.d 8
8.d odd 2 1 1560.1.cs.d 8
13.b even 2 1 inner 1560.1.cs.e yes 8
15.e even 4 1 1560.1.cs.d 8
24.f even 2 1 1560.1.cs.d 8
39.d odd 2 1 CM 1560.1.cs.e yes 8
40.k even 4 1 inner 1560.1.cs.e yes 8
65.h odd 4 1 1560.1.cs.d 8
104.h odd 2 1 1560.1.cs.d 8
120.q odd 4 1 inner 1560.1.cs.e yes 8
195.s even 4 1 1560.1.cs.d 8
312.h even 2 1 1560.1.cs.d 8
520.bc even 4 1 inner 1560.1.cs.e yes 8
1560.cs odd 4 1 inner 1560.1.cs.e yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.1.cs.d 8 5.c odd 4 1
1560.1.cs.d 8 8.d odd 2 1
1560.1.cs.d 8 15.e even 4 1
1560.1.cs.d 8 24.f even 2 1
1560.1.cs.d 8 65.h odd 4 1
1560.1.cs.d 8 104.h odd 2 1
1560.1.cs.d 8 195.s even 4 1
1560.1.cs.d 8 312.h even 2 1
1560.1.cs.e yes 8 1.a even 1 1 trivial
1560.1.cs.e yes 8 3.b odd 2 1 inner
1560.1.cs.e yes 8 13.b even 2 1 inner
1560.1.cs.e yes 8 39.d odd 2 1 CM
1560.1.cs.e yes 8 40.k even 4 1 inner
1560.1.cs.e yes 8 120.q odd 4 1 inner
1560.1.cs.e yes 8 520.bc even 4 1 inner
1560.1.cs.e yes 8 1560.cs odd 4 1 inner

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11}^{4} - 4T_{11}^{2} + 2$$ T11^4 - 4*T11^2 + 2 $$T_{17}$$ T17 $$T_{103}^{2} - 2T_{103} + 2$$ T103^2 - 2*T103 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 1$$
$3$ $$(T^{4} + 1)^{2}$$
$5$ $$T^{8} + 1$$
$7$ $$T^{8}$$
$11$ $$(T^{4} - 4 T^{2} + 2)^{2}$$
$13$ $$(T^{4} + 1)^{2}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$(T^{4} - 4 T^{2} + 2)^{2}$$
$43$ $$(T^{2} + 2 T + 2)^{4}$$
$47$ $$T^{8} + 12T^{4} + 4$$
$53$ $$T^{8}$$
$59$ $$(T^{4} + 4 T^{2} + 2)^{2}$$
$61$ $$(T^{2} + 2)^{4}$$
$67$ $$T^{8}$$
$71$ $$(T^{4} + 4 T^{2} + 2)^{2}$$
$73$ $$T^{8}$$
$79$ $$(T^{2} - 2)^{4}$$
$83$ $$T^{8} + 12T^{4} + 4$$
$89$ $$(T^{4} + 4 T^{2} + 2)^{2}$$
$97$ $$T^{8}$$