# Properties

 Label 2800.2.g.b Level $2800$ Weight $2$ Character orbit 2800.g Analytic conductor $22.358$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$22.3581125660$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + i q^{7} -6 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} + i q^{7} -6 q^{9} + 5 q^{11} -5 i q^{13} + 7 i q^{17} -2 q^{19} -3 q^{21} + 2 i q^{23} -9 i q^{27} -7 q^{29} -4 q^{31} + 15 i q^{33} + 6 i q^{37} + 15 q^{39} -12 q^{41} + 2 i q^{43} + i q^{47} - q^{49} -21 q^{51} -6 i q^{57} -4 q^{59} + 4 q^{61} -6 i q^{63} + 8 i q^{67} -6 q^{69} + 6 i q^{73} + 5 i q^{77} -3 q^{79} + 9 q^{81} + 4 i q^{83} -21 i q^{87} + 5 q^{91} -12 i q^{93} -13 i q^{97} -30 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 12q^{9} + O(q^{10})$$ $$2q - 12q^{9} + 10q^{11} - 4q^{19} - 6q^{21} - 14q^{29} - 8q^{31} + 30q^{39} - 24q^{41} - 2q^{49} - 42q^{51} - 8q^{59} + 8q^{61} - 12q^{69} - 6q^{79} + 18q^{81} + 10q^{91} - 60q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 3.00000i 0 0 0 1.00000i 0 −6.00000 0
449.2 0 3.00000i 0 0 0 1.00000i 0 −6.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.2.g.b 2
4.b odd 2 1 1400.2.g.a 2
5.b even 2 1 inner 2800.2.g.b 2
5.c odd 4 1 560.2.a.f 1
5.c odd 4 1 2800.2.a.c 1
15.e even 4 1 5040.2.a.a 1
20.d odd 2 1 1400.2.g.a 2
20.e even 4 1 280.2.a.a 1
20.e even 4 1 1400.2.a.n 1
35.f even 4 1 3920.2.a.c 1
40.i odd 4 1 2240.2.a.a 1
40.k even 4 1 2240.2.a.z 1
60.l odd 4 1 2520.2.a.i 1
140.j odd 4 1 1960.2.a.o 1
140.j odd 4 1 9800.2.a.a 1
140.w even 12 2 1960.2.q.o 2
140.x odd 12 2 1960.2.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.a 1 20.e even 4 1
560.2.a.f 1 5.c odd 4 1
1400.2.a.n 1 20.e even 4 1
1400.2.g.a 2 4.b odd 2 1
1400.2.g.a 2 20.d odd 2 1
1960.2.a.o 1 140.j odd 4 1
1960.2.q.a 2 140.x odd 12 2
1960.2.q.o 2 140.w even 12 2
2240.2.a.a 1 40.i odd 4 1
2240.2.a.z 1 40.k even 4 1
2520.2.a.i 1 60.l odd 4 1
2800.2.a.c 1 5.c odd 4 1
2800.2.g.b 2 1.a even 1 1 trivial
2800.2.g.b 2 5.b even 2 1 inner
3920.2.a.c 1 35.f even 4 1
5040.2.a.a 1 15.e even 4 1
9800.2.a.a 1 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2800, [\chi])$$:

 $$T_{3}^{2} + 9$$ $$T_{11} - 5$$ $$T_{13}^{2} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$25 + T^{2}$$
$17$ $$49 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$( 7 + T )^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$( 12 + T )^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$1 + T^{2}$$
$53$ $$T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( -4 + T )^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( 3 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$169 + T^{2}$$