# Properties

 Label 2800.2.g.f 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: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1400) 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 $$q + 2 i q^{3} + i q^{7} - q^{9} - q^{11} - 4 i q^{13} + 6 q^{19} - 2 q^{21} + 3 i q^{23} + 4 i q^{27} + 3 q^{29} - 2 i q^{33} + 9 i q^{37} + 8 q^{39} + 2 q^{41} + 9 i q^{43} + 6 i q^{47} - q^{49} - 6 i q^{53} + 12 i q^{57} + 8 q^{59} - 10 q^{61} - i q^{63} + i q^{67} - 6 q^{69} + 7 q^{71} + 2 i q^{73} - i q^{77} - 9 q^{79} - 11 q^{81} + 12 i q^{83} + 6 i q^{87} + 4 q^{89} + 4 q^{91} + 16 i q^{97} + q^{99} +O(q^{100})$$ q + 2*i * q^3 + i * q^7 - q^9 - q^11 - 4*i * q^13 + 6 * q^19 - 2 * q^21 + 3*i * q^23 + 4*i * q^27 + 3 * q^29 - 2*i * q^33 + 9*i * q^37 + 8 * q^39 + 2 * q^41 + 9*i * q^43 + 6*i * q^47 - q^49 - 6*i * q^53 + 12*i * q^57 + 8 * q^59 - 10 * q^61 - i * q^63 + i * q^67 - 6 * q^69 + 7 * q^71 + 2*i * q^73 - i * q^77 - 9 * q^79 - 11 * q^81 + 12*i * q^83 + 6*i * q^87 + 4 * q^89 + 4 * q^91 + 16*i * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 2 q^{11} + 12 q^{19} - 4 q^{21} + 6 q^{29} + 16 q^{39} + 4 q^{41} - 2 q^{49} + 16 q^{59} - 20 q^{61} - 12 q^{69} + 14 q^{71} - 18 q^{79} - 22 q^{81} + 8 q^{89} + 8 q^{91} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 - 2 * q^11 + 12 * q^19 - 4 * q^21 + 6 * q^29 + 16 * q^39 + 4 * q^41 - 2 * q^49 + 16 * q^59 - 20 * q^61 - 12 * q^69 + 14 * q^71 - 18 * q^79 - 22 * q^81 + 8 * q^89 + 8 * q^91 + 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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.f 2
4.b odd 2 1 1400.2.g.c 2
5.b even 2 1 inner 2800.2.g.f 2
5.c odd 4 1 2800.2.a.f 1
5.c odd 4 1 2800.2.a.bb 1
20.d odd 2 1 1400.2.g.c 2
20.e even 4 1 1400.2.a.c 1
20.e even 4 1 1400.2.a.l yes 1
140.j odd 4 1 9800.2.a.h 1
140.j odd 4 1 9800.2.a.bk 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.a.c 1 20.e even 4 1
1400.2.a.l yes 1 20.e even 4 1
1400.2.g.c 2 4.b odd 2 1
1400.2.g.c 2 20.d odd 2 1
2800.2.a.f 1 5.c odd 4 1
2800.2.a.bb 1 5.c odd 4 1
2800.2.g.f 2 1.a even 1 1 trivial
2800.2.g.f 2 5.b even 2 1 inner
9800.2.a.h 1 140.j odd 4 1
9800.2.a.bk 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 $$S_{2}^{\mathrm{new}}(2800, [\chi])$$:

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{11} + 1$$ T11 + 1 $$T_{13}^{2} + 16$$ T13^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2}$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2} + 9$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 81$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 81$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 8)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 1$$
$71$ $$(T - 7)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T + 9)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T - 4)^{2}$$
$97$ $$T^{2} + 256$$