# Properties

 Label 2800.2.g.e Level $2800$ Weight $2$ Character orbit 2800.g Analytic conductor $22.358$ Analytic rank $1$ 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: $$1$$ 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 700) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} + i q^{7} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} + i q^{7} - q^{9} -3 q^{11} -4 i q^{13} + 2 q^{19} -2 q^{21} + 3 i q^{23} + 4 i q^{27} -9 q^{29} -8 q^{31} -6 i q^{33} -5 i q^{37} + 8 q^{39} -6 q^{41} -11 i q^{43} + 6 i q^{47} - q^{49} + 6 i q^{53} + 4 i q^{57} -10 q^{61} -i q^{63} + 5 i q^{67} -6 q^{69} -15 q^{71} -10 i q^{73} -3 i q^{77} -7 q^{79} -11 q^{81} -12 i q^{83} -18 i q^{87} + 12 q^{89} + 4 q^{91} -16 i q^{93} -8 i q^{97} + 3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{9} - 6 q^{11} + 4 q^{19} - 4 q^{21} - 18 q^{29} - 16 q^{31} + 16 q^{39} - 12 q^{41} - 2 q^{49} - 20 q^{61} - 12 q^{69} - 30 q^{71} - 14 q^{79} - 22 q^{81} + 24 q^{89} + 8 q^{91} + 6 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 1.00000i 0 −1.00000 0
449.2 0 2.00000i 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
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.e 2
4.b odd 2 1 700.2.e.b 2
5.b even 2 1 inner 2800.2.g.e 2
5.c odd 4 1 2800.2.a.e 1
5.c odd 4 1 2800.2.a.ba 1
12.b even 2 1 6300.2.k.d 2
20.d odd 2 1 700.2.e.b 2
20.e even 4 1 700.2.a.c 1
20.e even 4 1 700.2.a.i yes 1
28.d even 2 1 4900.2.e.g 2
60.h even 2 1 6300.2.k.d 2
60.l odd 4 1 6300.2.a.e 1
60.l odd 4 1 6300.2.a.s 1
140.c even 2 1 4900.2.e.g 2
140.j odd 4 1 4900.2.a.f 1
140.j odd 4 1 4900.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.a.c 1 20.e even 4 1
700.2.a.i yes 1 20.e even 4 1
700.2.e.b 2 4.b odd 2 1
700.2.e.b 2 20.d odd 2 1
2800.2.a.e 1 5.c odd 4 1
2800.2.a.ba 1 5.c odd 4 1
2800.2.g.e 2 1.a even 1 1 trivial
2800.2.g.e 2 5.b even 2 1 inner
4900.2.a.f 1 140.j odd 4 1
4900.2.a.t 1 140.j odd 4 1
4900.2.e.g 2 28.d even 2 1
4900.2.e.g 2 140.c even 2 1
6300.2.a.e 1 60.l odd 4 1
6300.2.a.s 1 60.l odd 4 1
6300.2.k.d 2 12.b even 2 1
6300.2.k.d 2 60.h even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ $$T_{11} + 3$$ $$T_{13}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$9 + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$25 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$121 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$25 + T^{2}$$
$71$ $$( 15 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( 7 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -12 + T )^{2}$$
$97$ $$64 + T^{2}$$