# Properties

 Label 2800.2.g.i 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 350) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} -i q^{7} + 2 q^{9} +O(q^{10})$$ $$q + i q^{3} -i q^{7} + 2 q^{9} -3 q^{11} -2 i q^{13} + 3 i q^{17} -7 q^{19} + q^{21} + 5 i q^{27} + 6 q^{29} + 4 q^{31} -3 i q^{33} + 8 i q^{37} + 2 q^{39} -9 q^{41} + 8 i q^{43} + 6 i q^{47} - q^{49} -3 q^{51} + 12 i q^{53} -7 i q^{57} + 12 q^{59} -10 q^{61} -2 i q^{63} + 7 i q^{67} -6 q^{71} -5 i q^{73} + 3 i q^{77} + 14 q^{79} + q^{81} -9 i q^{83} + 6 i q^{87} + 15 q^{89} -2 q^{91} + 4 i q^{93} -10 i q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{9} + O(q^{10})$$ $$2q + 4q^{9} - 6q^{11} - 14q^{19} + 2q^{21} + 12q^{29} + 8q^{31} + 4q^{39} - 18q^{41} - 2q^{49} - 6q^{51} + 24q^{59} - 20q^{61} - 12q^{71} + 28q^{79} + 2q^{81} + 30q^{89} - 4q^{91} - 12q^{99} + O(q^{100})$$

## 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 1.00000i 0 0 0 1.00000i 0 2.00000 0
449.2 0 1.00000i 0 0 0 1.00000i 0 2.00000 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.i 2
4.b odd 2 1 350.2.c.c 2
5.b even 2 1 inner 2800.2.g.i 2
5.c odd 4 1 2800.2.a.h 1
5.c odd 4 1 2800.2.a.x 1
12.b even 2 1 3150.2.g.f 2
20.d odd 2 1 350.2.c.c 2
20.e even 4 1 350.2.a.a 1
20.e even 4 1 350.2.a.e yes 1
28.d even 2 1 2450.2.c.h 2
60.h even 2 1 3150.2.g.f 2
60.l odd 4 1 3150.2.a.m 1
60.l odd 4 1 3150.2.a.x 1
140.c even 2 1 2450.2.c.h 2
140.j odd 4 1 2450.2.a.m 1
140.j odd 4 1 2450.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.a 1 20.e even 4 1
350.2.a.e yes 1 20.e even 4 1
350.2.c.c 2 4.b odd 2 1
350.2.c.c 2 20.d odd 2 1
2450.2.a.m 1 140.j odd 4 1
2450.2.a.x 1 140.j odd 4 1
2450.2.c.h 2 28.d even 2 1
2450.2.c.h 2 140.c even 2 1
2800.2.a.h 1 5.c odd 4 1
2800.2.a.x 1 5.c odd 4 1
2800.2.g.i 2 1.a even 1 1 trivial
2800.2.g.i 2 5.b even 2 1 inner
3150.2.a.m 1 60.l odd 4 1
3150.2.a.x 1 60.l odd 4 1
3150.2.g.f 2 12.b even 2 1
3150.2.g.f 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 $$S_{2}^{\mathrm{new}}(2800, [\chi])$$:

 $$T_{3}^{2} + 1$$ $$T_{11} + 3$$ $$T_{13}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( 7 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( 9 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$144 + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$49 + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$25 + T^{2}$$
$79$ $$( -14 + T )^{2}$$
$83$ $$81 + T^{2}$$
$89$ $$( -15 + T )^{2}$$
$97$ $$100 + T^{2}$$