# Properties

 Label 1560.1.cs.b Level $1560$ Weight $1$ Character orbit 1560.cs Analytic conductor $0.779$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -104 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.1521000.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{8} q^{2} + q^{3} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{5} - \zeta_{8} q^{6} - \zeta_{8}^{3} q^{8} + q^{9} +O(q^{10})$$ q - z * q^2 + q^3 + z^2 * q^4 - z^3 * q^5 - z * q^6 - z^3 * q^8 + q^9 $$q - \zeta_{8} q^{2} + q^{3} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{5} - \zeta_{8} q^{6} - \zeta_{8}^{3} q^{8} + q^{9} - q^{10} + \zeta_{8}^{2} q^{12} - \zeta_{8} q^{13} - \zeta_{8}^{3} q^{15} - q^{16} + ( - \zeta_{8}^{2} - 1) q^{17} - \zeta_{8} q^{18} + \zeta_{8} q^{20} - \zeta_{8}^{3} q^{24} - \zeta_{8}^{2} q^{25} + \zeta_{8}^{2} q^{26} + q^{27} - q^{30} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{31} + \zeta_{8} q^{32} + (\zeta_{8}^{3} + \zeta_{8}) q^{34} + \zeta_{8}^{2} q^{36} + \zeta_{8}^{3} q^{37} - \zeta_{8} q^{39} - \zeta_{8}^{2} q^{40} + (\zeta_{8}^{2} + 1) q^{43} - \zeta_{8}^{3} q^{45} - q^{48} + \zeta_{8}^{2} q^{49} + \zeta_{8}^{3} q^{50} + ( - \zeta_{8}^{2} - 1) q^{51} - \zeta_{8}^{3} q^{52} - \zeta_{8} q^{54} + \zeta_{8} q^{60} + ( - \zeta_{8}^{2} - 1) q^{62} - \zeta_{8}^{2} q^{64} - q^{65} + ( - \zeta_{8}^{2} + 1) q^{68} + (\zeta_{8}^{3} + \zeta_{8}) q^{71} - \zeta_{8}^{3} q^{72} + 2 q^{74} - \zeta_{8}^{2} q^{75} + \zeta_{8}^{2} q^{78} + \zeta_{8}^{3} q^{80} + q^{81} + (\zeta_{8}^{3} - \zeta_{8}) q^{85} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{86} - q^{90} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{93} + \zeta_{8} q^{96} - \zeta_{8}^{3} q^{98} +O(q^{100})$$ q - z * q^2 + q^3 + z^2 * q^4 - z^3 * q^5 - z * q^6 - z^3 * q^8 + q^9 - q^10 + z^2 * q^12 - z * q^13 - z^3 * q^15 - q^16 + (-z^2 - 1) * q^17 - z * q^18 + z * q^20 - z^3 * q^24 - z^2 * q^25 + z^2 * q^26 + q^27 - q^30 + (-z^3 + z) * q^31 + z * q^32 + (z^3 + z) * q^34 + z^2 * q^36 + z^3 * q^37 - z * q^39 - z^2 * q^40 + (z^2 + 1) * q^43 - z^3 * q^45 - q^48 + z^2 * q^49 + z^3 * q^50 + (-z^2 - 1) * q^51 - z^3 * q^52 - z * q^54 + z * q^60 + (-z^2 - 1) * q^62 - z^2 * q^64 - q^65 + (-z^2 + 1) * q^68 + (z^3 + z) * q^71 - z^3 * q^72 + 2 * q^74 - z^2 * q^75 + z^2 * q^78 + z^3 * q^80 + q^81 + (z^3 - z) * q^85 + (-z^3 - z) * q^86 - q^90 + (-z^3 + z) * q^93 + z * q^96 - z^3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 4 * q^9 $$4 q + 4 q^{3} + 4 q^{9} - 4 q^{10} - 4 q^{16} - 4 q^{17} + 4 q^{27} - 4 q^{30} + 4 q^{43} - 4 q^{48} - 4 q^{51} - 4 q^{62} - 4 q^{65} + 4 q^{68} + 8 q^{74} + 4 q^{81} - 4 q^{90}+O(q^{100})$$ 4 * q + 4 * q^3 + 4 * q^9 - 4 * q^10 - 4 * q^16 - 4 * q^17 + 4 * q^27 - 4 * q^30 + 4 * q^43 - 4 * q^48 - 4 * q^51 - 4 * q^62 - 4 * q^65 + 4 * q^68 + 8 * q^74 + 4 * q^81 - 4 * q^90

## Character values

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

 $$n$$ $$391$$ $$521$$ $$781$$ $$937$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$-\zeta_{8}^{2}$$ $$-1$$

## 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
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
−0.707107 + 0.707107i 1.00000 1.00000i 0.707107 + 0.707107i −0.707107 + 0.707107i 0 0.707107 + 0.707107i 1.00000 −1.00000
467.2 0.707107 0.707107i 1.00000 1.00000i −0.707107 0.707107i 0.707107 0.707107i 0 −0.707107 0.707107i 1.00000 −1.00000
1403.1 −0.707107 0.707107i 1.00000 1.00000i 0.707107 0.707107i −0.707107 0.707107i 0 0.707107 0.707107i 1.00000 −1.00000
1403.2 0.707107 + 0.707107i 1.00000 1.00000i −0.707107 + 0.707107i 0.707107 + 0.707107i 0 −0.707107 + 0.707107i 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
104.h odd 2 1 CM by $$\Q(\sqrt{-26})$$
8.d odd 2 1 inner
13.b even 2 1 inner
15.e even 4 1 inner
120.q odd 4 1 inner
195.s 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.b yes 4
3.b odd 2 1 1560.1.cs.a 4
5.c odd 4 1 1560.1.cs.a 4
8.d odd 2 1 inner 1560.1.cs.b yes 4
13.b even 2 1 inner 1560.1.cs.b yes 4
15.e even 4 1 inner 1560.1.cs.b yes 4
24.f even 2 1 1560.1.cs.a 4
39.d odd 2 1 1560.1.cs.a 4
40.k even 4 1 1560.1.cs.a 4
65.h odd 4 1 1560.1.cs.a 4
104.h odd 2 1 CM 1560.1.cs.b yes 4
120.q odd 4 1 inner 1560.1.cs.b yes 4
195.s even 4 1 inner 1560.1.cs.b yes 4
312.h even 2 1 1560.1.cs.a 4
520.bc even 4 1 1560.1.cs.a 4
1560.cs odd 4 1 inner 1560.1.cs.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.1.cs.a 4 3.b odd 2 1
1560.1.cs.a 4 5.c odd 4 1
1560.1.cs.a 4 24.f even 2 1
1560.1.cs.a 4 39.d odd 2 1
1560.1.cs.a 4 40.k even 4 1
1560.1.cs.a 4 65.h odd 4 1
1560.1.cs.a 4 312.h even 2 1
1560.1.cs.a 4 520.bc even 4 1
1560.1.cs.b yes 4 1.a even 1 1 trivial
1560.1.cs.b yes 4 8.d odd 2 1 inner
1560.1.cs.b yes 4 13.b even 2 1 inner
1560.1.cs.b yes 4 15.e even 4 1 inner
1560.1.cs.b yes 4 104.h odd 2 1 CM
1560.1.cs.b yes 4 120.q odd 4 1 inner
1560.1.cs.b yes 4 195.s even 4 1 inner
1560.1.cs.b yes 4 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 $$T_{11}$$ T11 $$T_{17}^{2} + 2T_{17} + 2$$ T17^2 + 2*T17 + 2 $$T_{103}$$ T103

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 1$$
$3$ $$(T - 1)^{4}$$
$5$ $$T^{4} + 1$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 1$$
$17$ $$(T^{2} + 2 T + 2)^{2}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} - 2)^{2}$$
$37$ $$T^{4} + 16$$
$41$ $$T^{4}$$
$43$ $$(T^{2} - 2 T + 2)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$(T^{2} + 2)^{2}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$