# Properties

 Label 2800.1.f.c Level $2800$ Weight $1$ Character orbit 2800.f Analytic conductor $1.397$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -35 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2800.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.39738203537$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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: $C_4\times S_3$ Artin field: Galois closure of 12.4.307328000000000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + i q^{3} + i q^{7}+O(q^{10})$$ q + z * q^3 + z * q^7 $$q + i q^{3} + i q^{7} + q^{11} - i q^{13} + i q^{17} - q^{21} + i q^{27} + q^{29} + i q^{33} + q^{39} - i q^{47} - q^{49} - q^{51} - q^{71} + i q^{73} + i q^{77} - q^{79} - q^{81} - i q^{83} + i q^{87} + q^{91} + i q^{97} +O(q^{100})$$ q + z * q^3 + z * q^7 + q^11 - z * q^13 + z * q^17 - q^21 + z * q^27 + q^29 + z * q^33 + q^39 - z * q^47 - q^49 - q^51 - q^71 + z * q^73 + z * q^77 - q^79 - q^81 - z * q^83 + z * q^87 + q^91 + z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 2 q^{11} - 2 q^{21} + 2 q^{29} + 2 q^{39} - 2 q^{49} - 2 q^{51} - 4 q^{71} - 2 q^{79} - 2 q^{81} + 2 q^{91}+O(q^{100})$$ 2 * q + 2 * q^11 - 2 * q^21 + 2 * q^29 + 2 * q^39 - 2 * q^49 - 2 * q^51 - 4 * q^71 - 2 * q^79 - 2 * q^81 + 2 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$801$$ $$2101$$ $$2577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2001.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 1.00000i 0 0 0
2001.2 0 1.00000i 0 0 0 1.00000i 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 $$\Q(\sqrt{-35})$$
5.b even 2 1 inner
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.1.f.c 2
4.b odd 2 1 700.1.d.a 2
5.b even 2 1 inner 2800.1.f.c 2
5.c odd 4 1 560.1.p.a 1
5.c odd 4 1 560.1.p.b 1
7.b odd 2 1 inner 2800.1.f.c 2
20.d odd 2 1 700.1.d.a 2
20.e even 4 1 140.1.h.a 1
20.e even 4 1 140.1.h.b yes 1
28.d even 2 1 700.1.d.a 2
35.c odd 2 1 CM 2800.1.f.c 2
35.f even 4 1 560.1.p.a 1
35.f even 4 1 560.1.p.b 1
35.k even 12 2 3920.1.br.a 2
35.k even 12 2 3920.1.br.b 2
35.l odd 12 2 3920.1.br.a 2
35.l odd 12 2 3920.1.br.b 2
40.i odd 4 1 2240.1.p.a 1
40.i odd 4 1 2240.1.p.d 1
40.k even 4 1 2240.1.p.b 1
40.k even 4 1 2240.1.p.c 1
60.l odd 4 1 1260.1.p.a 1
60.l odd 4 1 1260.1.p.b 1
140.c even 2 1 700.1.d.a 2
140.j odd 4 1 140.1.h.a 1
140.j odd 4 1 140.1.h.b yes 1
140.w even 12 2 980.1.n.a 2
140.w even 12 2 980.1.n.b 2
140.x odd 12 2 980.1.n.a 2
140.x odd 12 2 980.1.n.b 2
280.s even 4 1 2240.1.p.a 1
280.s even 4 1 2240.1.p.d 1
280.y odd 4 1 2240.1.p.b 1
280.y odd 4 1 2240.1.p.c 1
420.w even 4 1 1260.1.p.a 1
420.w even 4 1 1260.1.p.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.h.a 1 20.e even 4 1
140.1.h.a 1 140.j odd 4 1
140.1.h.b yes 1 20.e even 4 1
140.1.h.b yes 1 140.j odd 4 1
560.1.p.a 1 5.c odd 4 1
560.1.p.a 1 35.f even 4 1
560.1.p.b 1 5.c odd 4 1
560.1.p.b 1 35.f even 4 1
700.1.d.a 2 4.b odd 2 1
700.1.d.a 2 20.d odd 2 1
700.1.d.a 2 28.d even 2 1
700.1.d.a 2 140.c even 2 1
980.1.n.a 2 140.w even 12 2
980.1.n.a 2 140.x odd 12 2
980.1.n.b 2 140.w even 12 2
980.1.n.b 2 140.x odd 12 2
1260.1.p.a 1 60.l odd 4 1
1260.1.p.a 1 420.w even 4 1
1260.1.p.b 1 60.l odd 4 1
1260.1.p.b 1 420.w even 4 1
2240.1.p.a 1 40.i odd 4 1
2240.1.p.a 1 280.s even 4 1
2240.1.p.b 1 40.k even 4 1
2240.1.p.b 1 280.y odd 4 1
2240.1.p.c 1 40.k even 4 1
2240.1.p.c 1 280.y odd 4 1
2240.1.p.d 1 40.i odd 4 1
2240.1.p.d 1 280.s even 4 1
2800.1.f.c 2 1.a even 1 1 trivial
2800.1.f.c 2 5.b even 2 1 inner
2800.1.f.c 2 7.b odd 2 1 inner
2800.1.f.c 2 35.c odd 2 1 CM
3920.1.br.a 2 35.k even 12 2
3920.1.br.a 2 35.l odd 12 2
3920.1.br.b 2 35.k even 12 2
3920.1.br.b 2 35.l odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2800, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{23}$$ T23

## Hecke characteristic polynomials

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