# Properties

 Label 1600.2.a.m Level $1600$ Weight $2$ Character orbit 1600.a Self dual yes Analytic conductor $12.776$ Analytic rank $1$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.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: $$12.7760643234$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 3 q^{9}+O(q^{10})$$ q - 3 * q^9 $$q - 3 q^{9} + 4 q^{13} - 8 q^{17} - 10 q^{29} + 12 q^{37} - 10 q^{41} - 7 q^{49} - 4 q^{53} - 10 q^{61} - 16 q^{73} + 9 q^{81} - 10 q^{89} + 8 q^{97}+O(q^{100})$$ q - 3 * q^9 + 4 * q^13 - 8 * q^17 - 10 * q^29 + 12 * q^37 - 10 * q^41 - 7 * q^49 - 4 * q^53 - 10 * q^61 - 16 * q^73 + 9 * q^81 - 10 * q^89 + 8 * q^97

## 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 0 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
$$2$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.a.m 1
4.b odd 2 1 CM 1600.2.a.m 1
5.b even 2 1 1600.2.a.l 1
5.c odd 4 2 320.2.c.a 2
8.b even 2 1 800.2.a.e 1
8.d odd 2 1 800.2.a.e 1
15.e even 4 2 2880.2.f.n 2
20.d odd 2 1 1600.2.a.l 1
20.e even 4 2 320.2.c.a 2
24.f even 2 1 7200.2.a.y 1
24.h odd 2 1 7200.2.a.y 1
40.e odd 2 1 800.2.a.f 1
40.f even 2 1 800.2.a.f 1
40.i odd 4 2 160.2.c.a 2
40.k even 4 2 160.2.c.a 2
60.l odd 4 2 2880.2.f.n 2
80.i odd 4 2 1280.2.f.d 2
80.j even 4 2 1280.2.f.c 2
80.s even 4 2 1280.2.f.d 2
80.t odd 4 2 1280.2.f.c 2
120.i odd 2 1 7200.2.a.bb 1
120.m even 2 1 7200.2.a.bb 1
120.q odd 4 2 1440.2.f.c 2
120.w even 4 2 1440.2.f.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.c.a 2 40.i odd 4 2
160.2.c.a 2 40.k even 4 2
320.2.c.a 2 5.c odd 4 2
320.2.c.a 2 20.e even 4 2
800.2.a.e 1 8.b even 2 1
800.2.a.e 1 8.d odd 2 1
800.2.a.f 1 40.e odd 2 1
800.2.a.f 1 40.f even 2 1
1280.2.f.c 2 80.j even 4 2
1280.2.f.c 2 80.t odd 4 2
1280.2.f.d 2 80.i odd 4 2
1280.2.f.d 2 80.s even 4 2
1440.2.f.c 2 120.q odd 4 2
1440.2.f.c 2 120.w even 4 2
1600.2.a.l 1 5.b even 2 1
1600.2.a.l 1 20.d odd 2 1
1600.2.a.m 1 1.a even 1 1 trivial
1600.2.a.m 1 4.b odd 2 1 CM
2880.2.f.n 2 15.e even 4 2
2880.2.f.n 2 60.l odd 4 2
7200.2.a.y 1 24.f even 2 1
7200.2.a.y 1 24.h odd 2 1
7200.2.a.bb 1 120.i odd 2 1
7200.2.a.bb 1 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1600))$$:

 $$T_{3}$$ T3 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

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