# Properties

 Label 2800.1.f.b Level $2800$ Weight $1$ Character orbit 2800.f Self dual yes Analytic conductor $1.397$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -7 Inner twists $2$

# 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: yes Analytic conductor: $$1.39738203537$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 175) Projective image $$D_{3}$$ Projective field Galois closure of 3.1.175.1 Artin image $D_6$ Artin field Galois closure of 6.0.9800000.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{7} + q^{9} + O(q^{10})$$ $$q + q^{7} + q^{9} + q^{11} - q^{23} - q^{29} + q^{37} - q^{43} + q^{49} - 2q^{53} + q^{63} - q^{67} + q^{71} + q^{77} + q^{79} + q^{81} + q^{99} + O(q^{100})$$

## 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
 0
0 0 0 0 0 1.00000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.1.f.b 1
4.b odd 2 1 175.1.d.b yes 1
5.b even 2 1 2800.1.f.a 1
5.c odd 4 2 2800.1.p.a 2
7.b odd 2 1 CM 2800.1.f.b 1
12.b even 2 1 1575.1.h.a 1
20.d odd 2 1 175.1.d.a 1
20.e even 4 2 175.1.c.a 2
28.d even 2 1 175.1.d.b yes 1
28.f even 6 2 1225.1.i.a 2
28.g odd 6 2 1225.1.i.a 2
35.c odd 2 1 2800.1.f.a 1
35.f even 4 2 2800.1.p.a 2
60.h even 2 1 1575.1.h.c 1
60.l odd 4 2 1575.1.e.a 2
84.h odd 2 1 1575.1.h.a 1
140.c even 2 1 175.1.d.a 1
140.j odd 4 2 175.1.c.a 2
140.p odd 6 2 1225.1.i.b 2
140.s even 6 2 1225.1.i.b 2
140.w even 12 4 1225.1.j.a 4
140.x odd 12 4 1225.1.j.a 4
420.o odd 2 1 1575.1.h.c 1
420.w even 4 2 1575.1.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.1.c.a 2 20.e even 4 2
175.1.c.a 2 140.j odd 4 2
175.1.d.a 1 20.d odd 2 1
175.1.d.a 1 140.c even 2 1
175.1.d.b yes 1 4.b odd 2 1
175.1.d.b yes 1 28.d even 2 1
1225.1.i.a 2 28.f even 6 2
1225.1.i.a 2 28.g odd 6 2
1225.1.i.b 2 140.p odd 6 2
1225.1.i.b 2 140.s even 6 2
1225.1.j.a 4 140.w even 12 4
1225.1.j.a 4 140.x odd 12 4
1575.1.e.a 2 60.l odd 4 2
1575.1.e.a 2 420.w even 4 2
1575.1.h.a 1 12.b even 2 1
1575.1.h.a 1 84.h odd 2 1
1575.1.h.c 1 60.h even 2 1
1575.1.h.c 1 420.o odd 2 1
2800.1.f.a 1 5.b even 2 1
2800.1.f.a 1 35.c odd 2 1
2800.1.f.b 1 1.a even 1 1 trivial
2800.1.f.b 1 7.b odd 2 1 CM
2800.1.p.a 2 5.c odd 4 2
2800.1.p.a 2 35.f even 4 2

## Hecke kernels

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

 $$T_{3}$$ $$T_{23} + 1$$

## Hecke characteristic polynomials

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