# Properties

 Label 1600.2.a.n Level $1600$ Weight $2$ Character orbit 1600.a Self dual yes Analytic conductor $12.776$ Analytic rank $0$ 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 # Warning: the index may be different

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

 Self dual: yes Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 32) 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 $$q - 3 q^{9} + 6 q^{13} - 2 q^{17} + 10 q^{29} - 2 q^{37} + 10 q^{41} - 7 q^{49} + 14 q^{53} + 10 q^{61} + 6 q^{73} + 9 q^{81} + 10 q^{89} - 18 q^{97}+O(q^{100})$$ q - 3 * q^9 + 6 * q^13 - 2 * q^17 + 10 * q^29 - 2 * q^37 + 10 * q^41 - 7 * q^49 + 14 * q^53 + 10 * q^61 + 6 * q^73 + 9 * q^81 + 10 * q^89 - 18 * 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.n 1
4.b odd 2 1 CM 1600.2.a.n 1
5.b even 2 1 64.2.a.a 1
5.c odd 4 2 1600.2.c.l 2
8.b even 2 1 800.2.a.d 1
8.d odd 2 1 800.2.a.d 1
15.d odd 2 1 576.2.a.c 1
20.d odd 2 1 64.2.a.a 1
20.e even 4 2 1600.2.c.l 2
24.f even 2 1 7200.2.a.v 1
24.h odd 2 1 7200.2.a.v 1
35.c odd 2 1 3136.2.a.m 1
40.e odd 2 1 32.2.a.a 1
40.f even 2 1 32.2.a.a 1
40.i odd 4 2 800.2.c.e 2
40.k even 4 2 800.2.c.e 2
55.d odd 2 1 7744.2.a.v 1
60.h even 2 1 576.2.a.c 1
80.k odd 4 2 256.2.b.b 2
80.q even 4 2 256.2.b.b 2
120.i odd 2 1 288.2.a.d 1
120.m even 2 1 288.2.a.d 1
120.q odd 4 2 7200.2.f.m 2
120.w even 4 2 7200.2.f.m 2
140.c even 2 1 3136.2.a.m 1
160.y odd 8 4 1024.2.e.j 4
160.z even 8 4 1024.2.e.j 4
220.g even 2 1 7744.2.a.v 1
240.t even 4 2 2304.2.d.j 2
240.bm odd 4 2 2304.2.d.j 2
280.c odd 2 1 1568.2.a.e 1
280.n even 2 1 1568.2.a.e 1
280.ba even 6 2 1568.2.i.f 2
280.bf even 6 2 1568.2.i.g 2
280.bi odd 6 2 1568.2.i.g 2
280.bk odd 6 2 1568.2.i.f 2
360.z odd 6 2 2592.2.i.t 2
360.bd even 6 2 2592.2.i.e 2
360.bh odd 6 2 2592.2.i.e 2
360.bk even 6 2 2592.2.i.t 2
440.c even 2 1 3872.2.a.f 1
440.o odd 2 1 3872.2.a.f 1
520.b odd 2 1 5408.2.a.g 1
520.p even 2 1 5408.2.a.g 1
680.h even 2 1 9248.2.a.f 1
680.k odd 2 1 9248.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 40.e odd 2 1
32.2.a.a 1 40.f even 2 1
64.2.a.a 1 5.b even 2 1
64.2.a.a 1 20.d odd 2 1
256.2.b.b 2 80.k odd 4 2
256.2.b.b 2 80.q even 4 2
288.2.a.d 1 120.i odd 2 1
288.2.a.d 1 120.m even 2 1
576.2.a.c 1 15.d odd 2 1
576.2.a.c 1 60.h even 2 1
800.2.a.d 1 8.b even 2 1
800.2.a.d 1 8.d odd 2 1
800.2.c.e 2 40.i odd 4 2
800.2.c.e 2 40.k even 4 2
1024.2.e.j 4 160.y odd 8 4
1024.2.e.j 4 160.z even 8 4
1568.2.a.e 1 280.c odd 2 1
1568.2.a.e 1 280.n even 2 1
1568.2.i.f 2 280.ba even 6 2
1568.2.i.f 2 280.bk odd 6 2
1568.2.i.g 2 280.bf even 6 2
1568.2.i.g 2 280.bi odd 6 2
1600.2.a.n 1 1.a even 1 1 trivial
1600.2.a.n 1 4.b odd 2 1 CM
1600.2.c.l 2 5.c odd 4 2
1600.2.c.l 2 20.e even 4 2
2304.2.d.j 2 240.t even 4 2
2304.2.d.j 2 240.bm odd 4 2
2592.2.i.e 2 360.bd even 6 2
2592.2.i.e 2 360.bh odd 6 2
2592.2.i.t 2 360.z odd 6 2
2592.2.i.t 2 360.bk even 6 2
3136.2.a.m 1 35.c odd 2 1
3136.2.a.m 1 140.c even 2 1
3872.2.a.f 1 440.c even 2 1
3872.2.a.f 1 440.o odd 2 1
5408.2.a.g 1 520.b odd 2 1
5408.2.a.g 1 520.p even 2 1
7200.2.a.v 1 24.f even 2 1
7200.2.a.v 1 24.h odd 2 1
7200.2.f.m 2 120.q odd 4 2
7200.2.f.m 2 120.w even 4 2
7744.2.a.v 1 55.d odd 2 1
7744.2.a.v 1 220.g even 2 1
9248.2.a.f 1 680.h even 2 1
9248.2.a.f 1 680.k odd 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 $$T_{13} - 6$$ T13 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 6$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 10$$
$31$ $$T$$
$37$ $$T + 2$$
$41$ $$T - 10$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T - 14$$
$59$ $$T$$
$61$ $$T - 10$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T - 6$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 10$$
$97$ $$T + 18$$