# Properties

 Label 1600.2.a.p Level $1600$ Weight $2$ Character orbit 1600.a Self dual yes Analytic conductor $12.776$ 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] = [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 50) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 - 2 * q^7 - 2 * q^9 $$q + q^{3} - 2 q^{7} - 2 q^{9} - 3 q^{11} + 4 q^{13} - 3 q^{17} + 5 q^{19} - 2 q^{21} - 6 q^{23} - 5 q^{27} - 2 q^{31} - 3 q^{33} - 2 q^{37} + 4 q^{39} - 3 q^{41} - 4 q^{43} - 12 q^{47} - 3 q^{49} - 3 q^{51} - 6 q^{53} + 5 q^{57} - 2 q^{61} + 4 q^{63} - 13 q^{67} - 6 q^{69} - 12 q^{71} + 11 q^{73} + 6 q^{77} + 10 q^{79} + q^{81} - 9 q^{83} + 15 q^{89} - 8 q^{91} - 2 q^{93} + 2 q^{97} + 6 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^7 - 2 * q^9 - 3 * q^11 + 4 * q^13 - 3 * q^17 + 5 * q^19 - 2 * q^21 - 6 * q^23 - 5 * q^27 - 2 * q^31 - 3 * q^33 - 2 * q^37 + 4 * q^39 - 3 * q^41 - 4 * q^43 - 12 * q^47 - 3 * q^49 - 3 * q^51 - 6 * q^53 + 5 * q^57 - 2 * q^61 + 4 * q^63 - 13 * q^67 - 6 * q^69 - 12 * q^71 + 11 * q^73 + 6 * q^77 + 10 * q^79 + q^81 - 9 * q^83 + 15 * q^89 - 8 * q^91 - 2 * q^93 + 2 * 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 −2.00000 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
$$2$$ $$-1$$
$$5$$ $$-1$$

## 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 1600.2.a.p 1
4.b odd 2 1 1600.2.a.j 1
5.b even 2 1 1600.2.a.i 1
5.c odd 4 2 1600.2.c.h 2
8.b even 2 1 400.2.a.d 1
8.d odd 2 1 50.2.a.a 1
20.d odd 2 1 1600.2.a.q 1
20.e even 4 2 1600.2.c.i 2
24.f even 2 1 450.2.a.g 1
24.h odd 2 1 3600.2.a.l 1
40.e odd 2 1 50.2.a.b yes 1
40.f even 2 1 400.2.a.f 1
40.i odd 4 2 400.2.c.c 2
40.k even 4 2 50.2.b.a 2
56.e even 2 1 2450.2.a.g 1
88.g even 2 1 6050.2.a.bi 1
104.h odd 2 1 8450.2.a.v 1
120.i odd 2 1 3600.2.a.bc 1
120.m even 2 1 450.2.a.c 1
120.q odd 4 2 450.2.c.c 2
120.w even 4 2 3600.2.f.f 2
280.n even 2 1 2450.2.a.bd 1
280.y odd 4 2 2450.2.c.m 2
440.c even 2 1 6050.2.a.h 1
520.b odd 2 1 8450.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.a.a 1 8.d odd 2 1
50.2.a.b yes 1 40.e odd 2 1
50.2.b.a 2 40.k even 4 2
400.2.a.d 1 8.b even 2 1
400.2.a.f 1 40.f even 2 1
400.2.c.c 2 40.i odd 4 2
450.2.a.c 1 120.m even 2 1
450.2.a.g 1 24.f even 2 1
450.2.c.c 2 120.q odd 4 2
1600.2.a.i 1 5.b even 2 1
1600.2.a.j 1 4.b odd 2 1
1600.2.a.p 1 1.a even 1 1 trivial
1600.2.a.q 1 20.d odd 2 1
1600.2.c.h 2 5.c odd 4 2
1600.2.c.i 2 20.e even 4 2
2450.2.a.g 1 56.e even 2 1
2450.2.a.bd 1 280.n even 2 1
2450.2.c.m 2 280.y odd 4 2
3600.2.a.l 1 24.h odd 2 1
3600.2.a.bc 1 120.i odd 2 1
3600.2.f.f 2 120.w even 4 2
6050.2.a.h 1 440.c even 2 1
6050.2.a.bi 1 88.g even 2 1
8450.2.a.d 1 520.b odd 2 1
8450.2.a.v 1 104.h odd 2 1

## Hecke kernels

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

 $$T_{3} - 1$$ T3 - 1 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 3$$ T11 + 3 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

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