# Properties

 Label 2800.2.a.x Level $2800$ Weight $2$ Character orbit 2800.a Self dual yes Analytic conductor $22.358$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(1,2800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$22.3581125660$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 350) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 1.00000 0 0 0 1.00000 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$+1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.2.a.x 1
4.b odd 2 1 350.2.a.a 1
5.b even 2 1 2800.2.a.h 1
5.c odd 4 2 2800.2.g.i 2
12.b even 2 1 3150.2.a.x 1
20.d odd 2 1 350.2.a.e yes 1
20.e even 4 2 350.2.c.c 2
28.d even 2 1 2450.2.a.m 1
60.h even 2 1 3150.2.a.m 1
60.l odd 4 2 3150.2.g.f 2
140.c even 2 1 2450.2.a.x 1
140.j odd 4 2 2450.2.c.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.a 1 4.b odd 2 1
350.2.a.e yes 1 20.d odd 2 1
350.2.c.c 2 20.e even 4 2
2450.2.a.m 1 28.d even 2 1
2450.2.a.x 1 140.c even 2 1
2450.2.c.h 2 140.j odd 4 2
2800.2.a.h 1 5.b even 2 1
2800.2.a.x 1 1.a even 1 1 trivial
2800.2.g.i 2 5.c odd 4 2
3150.2.a.m 1 60.h even 2 1
3150.2.a.x 1 12.b even 2 1
3150.2.g.f 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2800))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{11} + 3$$ T11 + 3 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T + 3$$
$13$ $$T + 2$$
$17$ $$T + 3$$
$19$ $$T - 7$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T - 4$$
$37$ $$T + 8$$
$41$ $$T + 9$$
$43$ $$T - 8$$
$47$ $$T + 6$$
$53$ $$T - 12$$
$59$ $$T + 12$$
$61$ $$T + 10$$
$67$ $$T + 7$$
$71$ $$T + 6$$
$73$ $$T + 5$$
$79$ $$T + 14$$
$83$ $$T + 9$$
$89$ $$T + 15$$
$97$ $$T - 10$$