# Properties

 Label 2800.1.p.a Level $2800$ Weight $1$ Character orbit 2800.p Analytic conductor $1.397$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -7 Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2800.p (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: $$\Z[a_1, \ldots, a_{7}]$$ 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

## $q$-expansion

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2449.1
 1.00000i − 1.00000i
0 0 0 0 0 1.00000i 0 −1.00000 0
2449.2 0 0 0 0 0 1.00000i 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 $$\Q(\sqrt{-7})$$
5.b even 2 1 inner
35.c 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.p.a 2
4.b odd 2 1 175.1.c.a 2
5.b even 2 1 inner 2800.1.p.a 2
5.c odd 4 1 2800.1.f.a 1
5.c odd 4 1 2800.1.f.b 1
7.b odd 2 1 CM 2800.1.p.a 2
12.b even 2 1 1575.1.e.a 2
20.d odd 2 1 175.1.c.a 2
20.e even 4 1 175.1.d.a 1
20.e even 4 1 175.1.d.b yes 1
28.d even 2 1 175.1.c.a 2
28.f even 6 2 1225.1.j.a 4
28.g odd 6 2 1225.1.j.a 4
35.c odd 2 1 inner 2800.1.p.a 2
35.f even 4 1 2800.1.f.a 1
35.f even 4 1 2800.1.f.b 1
60.h even 2 1 1575.1.e.a 2
60.l odd 4 1 1575.1.h.a 1
60.l odd 4 1 1575.1.h.c 1
84.h odd 2 1 1575.1.e.a 2
140.c even 2 1 175.1.c.a 2
140.j odd 4 1 175.1.d.a 1
140.j odd 4 1 175.1.d.b yes 1
140.p odd 6 2 1225.1.j.a 4
140.s even 6 2 1225.1.j.a 4
140.w even 12 2 1225.1.i.a 2
140.w even 12 2 1225.1.i.b 2
140.x odd 12 2 1225.1.i.a 2
140.x odd 12 2 1225.1.i.b 2
420.o odd 2 1 1575.1.e.a 2
420.w even 4 1 1575.1.h.a 1
420.w even 4 1 1575.1.h.c 1

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2800, [\chi])$$.

## Hecke characteristic polynomials

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