Properties

 Label 2800.2.a.m Level $2800$ Weight $2$ Character orbit 2800.a Self dual yes Analytic conductor $22.358$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) 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)

 $$f(q)$$ $$=$$ $$q - q^{7} - 3 q^{9}+O(q^{10})$$ q - q^7 - 3 * q^9 $$q - q^{7} - 3 q^{9} - 4 q^{11} + 6 q^{13} - 2 q^{17} + 6 q^{29} - 8 q^{31} + 10 q^{37} + 2 q^{41} + 4 q^{43} + 8 q^{47} + q^{49} + 2 q^{53} + 8 q^{59} - 14 q^{61} + 3 q^{63} - 12 q^{67} + 16 q^{71} - 2 q^{73} + 4 q^{77} + 8 q^{79} + 9 q^{81} + 8 q^{83} + 10 q^{89} - 6 q^{91} - 2 q^{97} + 12 q^{99}+O(q^{100})$$ q - q^7 - 3 * q^9 - 4 * q^11 + 6 * q^13 - 2 * q^17 + 6 * q^29 - 8 * q^31 + 10 * q^37 + 2 * q^41 + 4 * q^43 + 8 * q^47 + q^49 + 2 * q^53 + 8 * q^59 - 14 * q^61 + 3 * q^63 - 12 * q^67 + 16 * q^71 - 2 * q^73 + 4 * q^77 + 8 * q^79 + 9 * q^81 + 8 * q^83 + 10 * q^89 - 6 * q^91 - 2 * q^97 + 12 * 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 0 0 0 0 −1.00000 0 −3.00000 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.m 1
4.b odd 2 1 350.2.a.b 1
5.b even 2 1 560.2.a.d 1
5.c odd 4 2 2800.2.g.n 2
12.b even 2 1 3150.2.a.bj 1
15.d odd 2 1 5040.2.a.bm 1
20.d odd 2 1 70.2.a.a 1
20.e even 4 2 350.2.c.b 2
28.d even 2 1 2450.2.a.l 1
35.c odd 2 1 3920.2.a.t 1
40.e odd 2 1 2240.2.a.n 1
40.f even 2 1 2240.2.a.q 1
60.h even 2 1 630.2.a.d 1
60.l odd 4 2 3150.2.g.c 2
140.c even 2 1 490.2.a.h 1
140.j odd 4 2 2450.2.c.k 2
140.p odd 6 2 490.2.e.d 2
140.s even 6 2 490.2.e.c 2
220.g even 2 1 8470.2.a.j 1
420.o odd 2 1 4410.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 20.d odd 2 1
350.2.a.b 1 4.b odd 2 1
350.2.c.b 2 20.e even 4 2
490.2.a.h 1 140.c even 2 1
490.2.e.c 2 140.s even 6 2
490.2.e.d 2 140.p odd 6 2
560.2.a.d 1 5.b even 2 1
630.2.a.d 1 60.h even 2 1
2240.2.a.n 1 40.e odd 2 1
2240.2.a.q 1 40.f even 2 1
2450.2.a.l 1 28.d even 2 1
2450.2.c.k 2 140.j odd 4 2
2800.2.a.m 1 1.a even 1 1 trivial
2800.2.g.n 2 5.c odd 4 2
3150.2.a.bj 1 12.b even 2 1
3150.2.g.c 2 60.l odd 4 2
3920.2.a.t 1 35.c odd 2 1
4410.2.a.b 1 420.o odd 2 1
5040.2.a.bm 1 15.d odd 2 1
8470.2.a.j 1 220.g even 2 1

Hecke kernels

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

 $$T_{3}$$ T3 $$T_{11} + 4$$ T11 + 4 $$T_{13} - 6$$ T13 - 6

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T + 4$$
$13$ $$T - 6$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T + 8$$
$37$ $$T - 10$$
$41$ $$T - 2$$
$43$ $$T - 4$$
$47$ $$T - 8$$
$53$ $$T - 2$$
$59$ $$T - 8$$
$61$ $$T + 14$$
$67$ $$T + 12$$
$71$ $$T - 16$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T - 8$$
$89$ $$T - 10$$
$97$ $$T + 2$$