# Properties

 Label 1120.3.c.a Level $1120$ Weight $3$ Character orbit 1120.c Self dual yes Analytic conductor $30.518$ Analytic rank $0$ Dimension $1$ CM discriminant -280 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1120 = 2^{5} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1120.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.5177896084$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 5q^{5} - 7q^{7} + 9q^{9} + O(q^{10})$$ $$q - 5q^{5} - 7q^{7} + 9q^{9} - 6q^{17} - 18q^{19} + 25q^{25} + 35q^{35} - 66q^{37} + 54q^{43} - 45q^{45} + 66q^{47} + 49q^{49} - 34q^{53} + 62q^{59} + 102q^{61} - 63q^{63} + 6q^{67} + 138q^{71} + 106q^{73} + 122q^{79} + 81q^{81} + 30q^{85} + 90q^{95} - 166q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 0
0 0 0 −5.00000 0 −7.00000 0 9.00000 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
280.c odd 2 1 CM by $$\Q(\sqrt{-70})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.3.c.a 1
4.b odd 2 1 280.3.c.c yes 1
5.b even 2 1 1120.3.c.b 1
7.b odd 2 1 1120.3.c.d 1
8.b even 2 1 1120.3.c.c 1
8.d odd 2 1 280.3.c.b yes 1
20.d odd 2 1 280.3.c.a 1
28.d even 2 1 280.3.c.d yes 1
35.c odd 2 1 1120.3.c.c 1
40.e odd 2 1 280.3.c.d yes 1
40.f even 2 1 1120.3.c.d 1
56.e even 2 1 280.3.c.a 1
56.h odd 2 1 1120.3.c.b 1
140.c even 2 1 280.3.c.b yes 1
280.c odd 2 1 CM 1120.3.c.a 1
280.n even 2 1 280.3.c.c yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.3.c.a 1 20.d odd 2 1
280.3.c.a 1 56.e even 2 1
280.3.c.b yes 1 8.d odd 2 1
280.3.c.b yes 1 140.c even 2 1
280.3.c.c yes 1 4.b odd 2 1
280.3.c.c yes 1 280.n even 2 1
280.3.c.d yes 1 28.d even 2 1
280.3.c.d yes 1 40.e odd 2 1
1120.3.c.a 1 1.a even 1 1 trivial
1120.3.c.a 1 280.c odd 2 1 CM
1120.3.c.b 1 5.b even 2 1
1120.3.c.b 1 56.h odd 2 1
1120.3.c.c 1 8.b even 2 1
1120.3.c.c 1 35.c odd 2 1
1120.3.c.d 1 7.b odd 2 1
1120.3.c.d 1 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1120, [\chi])$$:

 $$T_{3}$$ $$T_{17} + 6$$ $$T_{19} + 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$5 + T$$
$7$ $$7 + T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$6 + T$$
$19$ $$18 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$66 + T$$
$41$ $$T$$
$43$ $$-54 + T$$
$47$ $$-66 + T$$
$53$ $$34 + T$$
$59$ $$-62 + T$$
$61$ $$-102 + T$$
$67$ $$-6 + T$$
$71$ $$-138 + T$$
$73$ $$-106 + T$$
$79$ $$-122 + T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$166 + T$$