# Properties

 Label 2800.2.g.p 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 56) 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 $$q + i q^{7} + 3 q^{9} + 4 q^{11} - 2 i q^{13} - 6 i q^{17} + 8 q^{19} - 6 q^{29} - 8 q^{31} - 2 i q^{37} + 2 q^{41} - 4 i q^{43} + 8 i q^{47} - q^{49} - 6 i q^{53} - 6 q^{61} + 3 i q^{63} + 4 i q^{67} + 8 q^{71} - 10 i q^{73} + 4 i q^{77} + 16 q^{79} + 9 q^{81} + 8 i q^{83} + 6 q^{89} + 2 q^{91} - 6 i q^{97} + 12 q^{99} +O(q^{100})$$ q + i * q^7 + 3 * q^9 + 4 * q^11 - 2*i * q^13 - 6*i * q^17 + 8 * q^19 - 6 * q^29 - 8 * q^31 - 2*i * q^37 + 2 * q^41 - 4*i * q^43 + 8*i * q^47 - q^49 - 6*i * q^53 - 6 * q^61 + 3*i * q^63 + 4*i * q^67 + 8 * q^71 - 10*i * q^73 + 4*i * q^77 + 16 * q^79 + 9 * q^81 + 8*i * q^83 + 6 * q^89 + 2 * q^91 - 6*i * q^97 + 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{9}+O(q^{10})$$ 2 * q + 6 * q^9 $$2 q + 6 q^{9} + 8 q^{11} + 16 q^{19} - 12 q^{29} - 16 q^{31} + 4 q^{41} - 2 q^{49} - 12 q^{61} + 16 q^{71} + 32 q^{79} + 18 q^{81} + 12 q^{89} + 4 q^{91} + 24 q^{99}+O(q^{100})$$ 2 * q + 6 * q^9 + 8 * q^11 + 16 * q^19 - 12 * q^29 - 16 * q^31 + 4 * q^41 - 2 * q^49 - 12 * q^61 + 16 * q^71 + 32 * q^79 + 18 * q^81 + 12 * q^89 + 4 * q^91 + 24 * 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 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.p 2
4.b odd 2 1 1400.2.g.g 2
5.b even 2 1 inner 2800.2.g.p 2
5.c odd 4 1 112.2.a.b 1
5.c odd 4 1 2800.2.a.p 1
15.e even 4 1 1008.2.a.d 1
20.d odd 2 1 1400.2.g.g 2
20.e even 4 1 56.2.a.a 1
20.e even 4 1 1400.2.a.g 1
35.f even 4 1 784.2.a.e 1
35.k even 12 2 784.2.i.g 2
35.l odd 12 2 784.2.i.e 2
40.i odd 4 1 448.2.a.e 1
40.k even 4 1 448.2.a.d 1
60.l odd 4 1 504.2.a.c 1
80.i odd 4 1 1792.2.b.d 2
80.j even 4 1 1792.2.b.i 2
80.s even 4 1 1792.2.b.i 2
80.t odd 4 1 1792.2.b.d 2
105.k odd 4 1 7056.2.a.bo 1
120.q odd 4 1 4032.2.a.bb 1
120.w even 4 1 4032.2.a.bk 1
140.j odd 4 1 392.2.a.d 1
140.j odd 4 1 9800.2.a.u 1
140.w even 12 2 392.2.i.c 2
140.x odd 12 2 392.2.i.d 2
220.i odd 4 1 6776.2.a.g 1
260.p even 4 1 9464.2.a.c 1
280.s even 4 1 3136.2.a.p 1
280.y odd 4 1 3136.2.a.q 1
420.w even 4 1 3528.2.a.x 1
420.bp odd 12 2 3528.2.s.t 2
420.br even 12 2 3528.2.s.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 20.e even 4 1
112.2.a.b 1 5.c odd 4 1
392.2.a.d 1 140.j odd 4 1
392.2.i.c 2 140.w even 12 2
392.2.i.d 2 140.x odd 12 2
448.2.a.d 1 40.k even 4 1
448.2.a.e 1 40.i odd 4 1
504.2.a.c 1 60.l odd 4 1
784.2.a.e 1 35.f even 4 1
784.2.i.e 2 35.l odd 12 2
784.2.i.g 2 35.k even 12 2
1008.2.a.d 1 15.e even 4 1
1400.2.a.g 1 20.e even 4 1
1400.2.g.g 2 4.b odd 2 1
1400.2.g.g 2 20.d odd 2 1
1792.2.b.d 2 80.i odd 4 1
1792.2.b.d 2 80.t odd 4 1
1792.2.b.i 2 80.j even 4 1
1792.2.b.i 2 80.s even 4 1
2800.2.a.p 1 5.c odd 4 1
2800.2.g.p 2 1.a even 1 1 trivial
2800.2.g.p 2 5.b even 2 1 inner
3136.2.a.p 1 280.s even 4 1
3136.2.a.q 1 280.y odd 4 1
3528.2.a.x 1 420.w even 4 1
3528.2.s.e 2 420.br even 12 2
3528.2.s.t 2 420.bp odd 12 2
4032.2.a.bb 1 120.q odd 4 1
4032.2.a.bk 1 120.w even 4 1
6776.2.a.g 1 220.i odd 4 1
7056.2.a.bo 1 105.k odd 4 1
9464.2.a.c 1 260.p even 4 1
9800.2.a.u 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 $$S_{2}^{\mathrm{new}}(2800, [\chi])$$:

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

## Hecke characteristic polynomials

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