# Properties

 Label 1120.1.p.a Level $1120$ Weight $1$ Character orbit 1120.p Analytic conductor $0.559$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -20 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1120 = 2^{5} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1120.p (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.558952814149$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.19600.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{8}^{3} + \zeta_{8}) q^{3} + \zeta_{8}^{2} q^{5} + \zeta_{8}^{3} q^{7} + q^{9}+O(q^{10})$$ q + (-z^3 + z) * q^3 + z^2 * q^5 + z^3 * q^7 + q^9 $$q + ( - \zeta_{8}^{3} + \zeta_{8}) q^{3} + \zeta_{8}^{2} q^{5} + \zeta_{8}^{3} q^{7} + q^{9} + (\zeta_{8}^{3} + \zeta_{8}) q^{15} + (\zeta_{8}^{2} - 1) q^{21} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{23} - q^{25} + q^{29} - \zeta_{8} q^{35} + \zeta_{8}^{2} q^{41} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{43} + \zeta_{8}^{2} q^{45} + (\zeta_{8}^{3} - \zeta_{8}) q^{47} - \zeta_{8}^{2} q^{49} - \zeta_{8}^{2} q^{61} + \zeta_{8}^{3} q^{63} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{67} - \zeta_{8}^{2} q^{69} + (\zeta_{8}^{3} - \zeta_{8}) q^{75} - q^{81} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{83} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{87} +O(q^{100})$$ q + (-z^3 + z) * q^3 + z^2 * q^5 + z^3 * q^7 + q^9 + (z^3 + z) * q^15 + (z^2 - 1) * q^21 + (-z^3 - z) * q^23 - q^25 + q^29 - z * q^35 + z^2 * q^41 + (-z^3 - z) * q^43 + z^2 * q^45 + (z^3 - z) * q^47 - z^2 * q^49 - z^2 * q^61 + z^3 * q^63 + (-z^3 - z) * q^67 - z^2 * q^69 + (z^3 - z) * q^75 - q^81 + (-z^3 + z) * q^83 + (-2*z^3 + 2*z) * q^87 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^9 $$4 q + 4 q^{9} - 4 q^{21} - 4 q^{25} + 8 q^{29} - 4 q^{81}+O(q^{100})$$ 4 * q + 4 * q^9 - 4 * q^21 - 4 * q^25 + 8 * q^29 - 4 * q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$421$$ $$801$$ $$897$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
769.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
0 −1.41421 0 1.00000i 0 0.707107 + 0.707107i 0 1.00000 0
769.2 0 −1.41421 0 1.00000i 0 0.707107 0.707107i 0 1.00000 0
769.3 0 1.41421 0 1.00000i 0 −0.707107 0.707107i 0 1.00000 0
769.4 0 1.41421 0 1.00000i 0 −0.707107 + 0.707107i 0 1.00000 0
 $$n$$: 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.1.p.a 4
4.b odd 2 1 inner 1120.1.p.a 4
5.b even 2 1 inner 1120.1.p.a 4
7.b odd 2 1 inner 1120.1.p.a 4
8.b even 2 1 2240.1.p.e 4
8.d odd 2 1 2240.1.p.e 4
20.d odd 2 1 CM 1120.1.p.a 4
28.d even 2 1 inner 1120.1.p.a 4
35.c odd 2 1 inner 1120.1.p.a 4
40.e odd 2 1 2240.1.p.e 4
40.f even 2 1 2240.1.p.e 4
56.e even 2 1 2240.1.p.e 4
56.h odd 2 1 2240.1.p.e 4
140.c even 2 1 inner 1120.1.p.a 4
280.c odd 2 1 2240.1.p.e 4
280.n even 2 1 2240.1.p.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.1.p.a 4 1.a even 1 1 trivial
1120.1.p.a 4 4.b odd 2 1 inner
1120.1.p.a 4 5.b even 2 1 inner
1120.1.p.a 4 7.b odd 2 1 inner
1120.1.p.a 4 20.d odd 2 1 CM
1120.1.p.a 4 28.d even 2 1 inner
1120.1.p.a 4 35.c odd 2 1 inner
1120.1.p.a 4 140.c even 2 1 inner
2240.1.p.e 4 8.b even 2 1
2240.1.p.e 4 8.d odd 2 1
2240.1.p.e 4 40.e odd 2 1
2240.1.p.e 4 40.f even 2 1
2240.1.p.e 4 56.e even 2 1
2240.1.p.e 4 56.h odd 2 1
2240.1.p.e 4 280.c odd 2 1
2240.1.p.e 4 280.n even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1120, [\chi])$$.

## Hecke characteristic polynomials

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