# Properties

 Label 2240.1.p.a Level $2240$ Weight $1$ Character orbit 2240.p Self dual yes Analytic conductor $1.118$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -35 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2240.p (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.11790562830$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.140.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.2508800.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} - q^{7}+O(q^{10})$$ q - q^3 - q^5 - q^7 $$q - q^{3} - q^{5} - q^{7} - q^{11} + q^{13} + q^{15} - q^{17} + q^{21} + q^{25} + q^{27} + q^{29} + q^{33} + q^{35} - q^{39} + q^{47} + q^{49} + q^{51} + q^{55} - q^{65} - 2 q^{71} + 2 q^{73} - q^{75} + q^{77} + q^{79} - q^{81} + 2 q^{83} + q^{85} - q^{87} - q^{91} - q^{97}+O(q^{100})$$ q - q^3 - q^5 - q^7 - q^11 + q^13 + q^15 - q^17 + q^21 + q^25 + q^27 + q^29 + q^33 + q^35 - q^39 + q^47 + q^49 + q^51 + q^55 - q^65 - 2 * q^71 + 2 * q^73 - q^75 + q^77 + q^79 - q^81 + 2 * q^83 + q^85 - q^87 - q^91 - q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$$.

 $$n$$ $$897$$ $$1471$$ $$1541$$ $$1921$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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
0 −1.00000 0 −1.00000 0 −1.00000 0 0 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.1.p.a 1
4.b odd 2 1 2240.1.p.c 1
5.b even 2 1 2240.1.p.d 1
7.b odd 2 1 2240.1.p.d 1
8.b even 2 1 560.1.p.b 1
8.d odd 2 1 140.1.h.a 1
20.d odd 2 1 2240.1.p.b 1
24.f even 2 1 1260.1.p.a 1
28.d even 2 1 2240.1.p.b 1
35.c odd 2 1 CM 2240.1.p.a 1
40.e odd 2 1 140.1.h.b yes 1
40.f even 2 1 560.1.p.a 1
40.i odd 4 2 2800.1.f.c 2
40.k even 4 2 700.1.d.a 2
56.e even 2 1 140.1.h.b yes 1
56.h odd 2 1 560.1.p.a 1
56.j odd 6 2 3920.1.br.b 2
56.k odd 6 2 980.1.n.b 2
56.m even 6 2 980.1.n.a 2
56.p even 6 2 3920.1.br.a 2
120.m even 2 1 1260.1.p.b 1
140.c even 2 1 2240.1.p.c 1
168.e odd 2 1 1260.1.p.b 1
280.c odd 2 1 560.1.p.b 1
280.n even 2 1 140.1.h.a 1
280.s even 4 2 2800.1.f.c 2
280.y odd 4 2 700.1.d.a 2
280.ba even 6 2 980.1.n.b 2
280.bf even 6 2 3920.1.br.b 2
280.bi odd 6 2 980.1.n.a 2
280.bk odd 6 2 3920.1.br.a 2
840.b odd 2 1 1260.1.p.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.h.a 1 8.d odd 2 1
140.1.h.a 1 280.n even 2 1
140.1.h.b yes 1 40.e odd 2 1
140.1.h.b yes 1 56.e even 2 1
560.1.p.a 1 40.f even 2 1
560.1.p.a 1 56.h odd 2 1
560.1.p.b 1 8.b even 2 1
560.1.p.b 1 280.c odd 2 1
700.1.d.a 2 40.k even 4 2
700.1.d.a 2 280.y odd 4 2
980.1.n.a 2 56.m even 6 2
980.1.n.a 2 280.bi odd 6 2
980.1.n.b 2 56.k odd 6 2
980.1.n.b 2 280.ba even 6 2
1260.1.p.a 1 24.f even 2 1
1260.1.p.a 1 840.b odd 2 1
1260.1.p.b 1 120.m even 2 1
1260.1.p.b 1 168.e odd 2 1
2240.1.p.a 1 1.a even 1 1 trivial
2240.1.p.a 1 35.c odd 2 1 CM
2240.1.p.b 1 20.d odd 2 1
2240.1.p.b 1 28.d even 2 1
2240.1.p.c 1 4.b odd 2 1
2240.1.p.c 1 140.c even 2 1
2240.1.p.d 1 5.b even 2 1
2240.1.p.d 1 7.b odd 2 1
2800.1.f.c 2 40.i odd 4 2
2800.1.f.c 2 280.s even 4 2
3920.1.br.a 2 56.p even 6 2
3920.1.br.a 2 280.bk odd 6 2
3920.1.br.b 2 56.j odd 6 2
3920.1.br.b 2 280.bf even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2240, [\chi])$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{11} + 1$$ T11 + 1

## Hecke characteristic polynomials

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