# Properties

 Label 2800.2.g.n 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 70) 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.

 $$f(q)$$ $$=$$ $$q + i q^{7} + 3 q^{9} +O(q^{10})$$ $$q + i q^{7} + 3 q^{9} -4 q^{11} + 6 i q^{13} + 2 i q^{17} -6 q^{29} -8 q^{31} -10 i q^{37} + 2 q^{41} + 4 i q^{43} -8 i q^{47} - q^{49} + 2 i q^{53} -8 q^{59} -14 q^{61} + 3 i q^{63} + 12 i q^{67} + 16 q^{71} -2 i q^{73} -4 i q^{77} -8 q^{79} + 9 q^{81} + 8 i q^{83} -10 q^{89} -6 q^{91} + 2 i q^{97} -12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{9} + O(q^{10})$$ $$2q + 6q^{9} - 8q^{11} - 12q^{29} - 16q^{31} + 4q^{41} - 2q^{49} - 16q^{59} - 28q^{61} + 32q^{71} - 16q^{79} + 18q^{81} - 20q^{89} - 12q^{91} - 24q^{99} + O(q^{100})$$

## 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 0 0 0 0 1.00000i 0 3.00000 0
449.2 0 0 0 0 0 1.00000i 0 3.00000 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.n 2
4.b odd 2 1 350.2.c.b 2
5.b even 2 1 inner 2800.2.g.n 2
5.c odd 4 1 560.2.a.d 1
5.c odd 4 1 2800.2.a.m 1
12.b even 2 1 3150.2.g.c 2
15.e even 4 1 5040.2.a.bm 1
20.d odd 2 1 350.2.c.b 2
20.e even 4 1 70.2.a.a 1
20.e even 4 1 350.2.a.b 1
28.d even 2 1 2450.2.c.k 2
35.f even 4 1 3920.2.a.t 1
40.i odd 4 1 2240.2.a.q 1
40.k even 4 1 2240.2.a.n 1
60.h even 2 1 3150.2.g.c 2
60.l odd 4 1 630.2.a.d 1
60.l odd 4 1 3150.2.a.bj 1
140.c even 2 1 2450.2.c.k 2
140.j odd 4 1 490.2.a.h 1
140.j odd 4 1 2450.2.a.l 1
140.w even 12 2 490.2.e.d 2
140.x odd 12 2 490.2.e.c 2
220.i odd 4 1 8470.2.a.j 1
420.w even 4 1 4410.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 20.e even 4 1
350.2.a.b 1 20.e even 4 1
350.2.c.b 2 4.b odd 2 1
350.2.c.b 2 20.d odd 2 1
490.2.a.h 1 140.j odd 4 1
490.2.e.c 2 140.x odd 12 2
490.2.e.d 2 140.w even 12 2
560.2.a.d 1 5.c odd 4 1
630.2.a.d 1 60.l odd 4 1
2240.2.a.n 1 40.k even 4 1
2240.2.a.q 1 40.i odd 4 1
2450.2.a.l 1 140.j odd 4 1
2450.2.c.k 2 28.d even 2 1
2450.2.c.k 2 140.c even 2 1
2800.2.a.m 1 5.c odd 4 1
2800.2.g.n 2 1.a even 1 1 trivial
2800.2.g.n 2 5.b even 2 1 inner
3150.2.a.bj 1 60.l odd 4 1
3150.2.g.c 2 12.b even 2 1
3150.2.g.c 2 60.h even 2 1
3920.2.a.t 1 35.f even 4 1
4410.2.a.b 1 420.w even 4 1
5040.2.a.bm 1 15.e even 4 1
8470.2.a.j 1 220.i odd 4 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{11} + 4$$ $$T_{13}^{2} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$( 14 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$( -16 + T )^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$64 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$4 + T^{2}$$