# Properties

 Label 1600.2.a.bd Level $1600$ Weight $2$ Character orbit 1600.a Self dual yes Analytic conductor $12.776$ Analytic rank $0$ Dimension $2$ CM discriminant -20 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + (\beta - 3) q^{7} + (2 \beta + 3) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + (b - 3) * q^7 + (2*b + 3) * q^9 $$q + (\beta + 1) q^{3} + (\beta - 3) q^{7} + (2 \beta + 3) q^{9} + ( - 2 \beta + 2) q^{21} + (3 \beta - 1) q^{23} + (2 \beta + 10) q^{27} + 6 q^{29} - 2 \beta q^{41} + (\beta + 9) q^{43} + ( - 3 \beta - 7) q^{47} + ( - 6 \beta + 7) q^{49} + 6 \beta q^{61} + ( - 3 \beta + 1) q^{63} + ( - 5 \beta + 3) q^{67} + (2 \beta + 14) q^{69} + (6 \beta + 11) q^{81} + ( - 3 \beta - 11) q^{83} + (6 \beta + 6) q^{87} + 6 q^{89} +O(q^{100})$$ q + (b + 1) * q^3 + (b - 3) * q^7 + (2*b + 3) * q^9 + (-2*b + 2) * q^21 + (3*b - 1) * q^23 + (2*b + 10) * q^27 + 6 * q^29 - 2*b * q^41 + (b + 9) * q^43 + (-3*b - 7) * q^47 + (-6*b + 7) * q^49 + 6*b * q^61 + (-3*b + 1) * q^63 + (-5*b + 3) * q^67 + (2*b + 14) * q^69 + (6*b + 11) * q^81 + (-3*b - 11) * q^83 + (6*b + 6) * q^87 + 6 * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 6 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 6 * q^7 + 6 * q^9 $$2 q + 2 q^{3} - 6 q^{7} + 6 q^{9} + 4 q^{21} - 2 q^{23} + 20 q^{27} + 12 q^{29} + 18 q^{43} - 14 q^{47} + 14 q^{49} + 2 q^{63} + 6 q^{67} + 28 q^{69} + 22 q^{81} - 22 q^{83} + 12 q^{87} + 12 q^{89}+O(q^{100})$$ 2 * q + 2 * q^3 - 6 * q^7 + 6 * q^9 + 4 * q^21 - 2 * q^23 + 20 * q^27 + 12 * q^29 + 18 * q^43 - 14 * q^47 + 14 * q^49 + 2 * q^63 + 6 * q^67 + 28 * q^69 + 22 * q^81 - 22 * q^83 + 12 * q^87 + 12 * q^89

## 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
 −0.618034 1.61803
0 −1.23607 0 0 0 −5.23607 0 −1.47214 0
1.2 0 3.23607 0 0 0 −0.763932 0 7.47214 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{2} - 2T_{3} - 4$$ T3^2 - 2*T3 - 4 $$T_{7}^{2} + 6T_{7} + 4$$ T7^2 + 6*T7 + 4 $$T_{11}$$ T11 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 6T + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 2T - 44$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2} - 20$$
$43$ $$T^{2} - 18T + 76$$
$47$ $$T^{2} + 14T + 4$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 180$$
$67$ $$T^{2} - 6T - 116$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 22T + 76$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2}$$