# Properties

 Label 1600.2.a.ba Level $1600$ Weight $2$ Character orbit 1600.a Self dual yes Analytic conductor $12.776$ Analytic rank $1$ Dimension $2$ 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: $$2$$ 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 800) 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)

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

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + 2 \beta q^{7} + 2 q^{9} +O(q^{10})$$ q - b * q^3 + 2*b * q^7 + 2 * q^9 $$q - \beta q^{3} + 2 \beta q^{7} + 2 q^{9} - \beta q^{11} - 4 q^{13} - 7 q^{17} + 3 \beta q^{19} - 10 q^{21} - 2 \beta q^{23} + \beta q^{27} + 2 \beta q^{31} + 5 q^{33} - 2 q^{37} + 4 \beta q^{39} + 5 q^{41} - 4 \beta q^{47} + 13 q^{49} + 7 \beta q^{51} - 6 q^{53} - 15 q^{57} - 4 \beta q^{59} - 10 q^{61} + 4 \beta q^{63} + \beta q^{67} + 10 q^{69} - 4 \beta q^{71} - 9 q^{73} - 10 q^{77} - 2 \beta q^{79} - 11 q^{81} + 5 \beta q^{83} - 5 q^{89} - 8 \beta q^{91} - 10 q^{93} + 2 q^{97} - 2 \beta q^{99} +O(q^{100})$$ q - b * q^3 + 2*b * q^7 + 2 * q^9 - b * q^11 - 4 * q^13 - 7 * q^17 + 3*b * q^19 - 10 * q^21 - 2*b * q^23 + b * q^27 + 2*b * q^31 + 5 * q^33 - 2 * q^37 + 4*b * q^39 + 5 * q^41 - 4*b * q^47 + 13 * q^49 + 7*b * q^51 - 6 * q^53 - 15 * q^57 - 4*b * q^59 - 10 * q^61 + 4*b * q^63 + b * q^67 + 10 * q^69 - 4*b * q^71 - 9 * q^73 - 10 * q^77 - 2*b * q^79 - 11 * q^81 + 5*b * q^83 - 5 * q^89 - 8*b * q^91 - 10 * q^93 + 2 * q^97 - 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} - 8 q^{13} - 14 q^{17} - 20 q^{21} + 10 q^{33} - 4 q^{37} + 10 q^{41} + 26 q^{49} - 12 q^{53} - 30 q^{57} - 20 q^{61} + 20 q^{69} - 18 q^{73} - 20 q^{77} - 22 q^{81} - 10 q^{89} - 20 q^{93} + 4 q^{97}+O(q^{100})$$ 2 * q + 4 * q^9 - 8 * q^13 - 14 * q^17 - 20 * q^21 + 10 * q^33 - 4 * q^37 + 10 * q^41 + 26 * q^49 - 12 * q^53 - 30 * q^57 - 20 * q^61 + 20 * q^69 - 18 * q^73 - 20 * q^77 - 22 * q^81 - 10 * q^89 - 20 * q^93 + 4 * 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
 1.61803 −0.618034
0 −2.23607 0 0 0 4.47214 0 2.00000 0
1.2 0 2.23607 0 0 0 −4.47214 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.a.ba 2
4.b odd 2 1 inner 1600.2.a.ba 2
5.b even 2 1 1600.2.a.bb 2
5.c odd 4 2 1600.2.c.o 4
8.b even 2 1 800.2.a.l yes 2
8.d odd 2 1 800.2.a.l yes 2
20.d odd 2 1 1600.2.a.bb 2
20.e even 4 2 1600.2.c.o 4
24.f even 2 1 7200.2.a.cn 2
24.h odd 2 1 7200.2.a.cn 2
40.e odd 2 1 800.2.a.k 2
40.f even 2 1 800.2.a.k 2
40.i odd 4 2 800.2.c.g 4
40.k even 4 2 800.2.c.g 4
120.i odd 2 1 7200.2.a.cf 2
120.m even 2 1 7200.2.a.cf 2
120.q odd 4 2 7200.2.f.bg 4
120.w even 4 2 7200.2.f.bg 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.a.k 2 40.e odd 2 1
800.2.a.k 2 40.f even 2 1
800.2.a.l yes 2 8.b even 2 1
800.2.a.l yes 2 8.d odd 2 1
800.2.c.g 4 40.i odd 4 2
800.2.c.g 4 40.k even 4 2
1600.2.a.ba 2 1.a even 1 1 trivial
1600.2.a.ba 2 4.b odd 2 1 inner
1600.2.a.bb 2 5.b even 2 1
1600.2.a.bb 2 20.d odd 2 1
1600.2.c.o 4 5.c odd 4 2
1600.2.c.o 4 20.e even 4 2
7200.2.a.cf 2 120.i odd 2 1
7200.2.a.cf 2 120.m even 2 1
7200.2.a.cn 2 24.f even 2 1
7200.2.a.cn 2 24.h odd 2 1
7200.2.f.bg 4 120.q odd 4 2
7200.2.f.bg 4 120.w even 4 2

## Hecke kernels

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

 $$T_{3}^{2} - 5$$ T3^2 - 5 $$T_{7}^{2} - 20$$ T7^2 - 20 $$T_{11}^{2} - 5$$ T11^2 - 5 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 5$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 20$$
$11$ $$T^{2} - 5$$
$13$ $$(T + 4)^{2}$$
$17$ $$(T + 7)^{2}$$
$19$ $$T^{2} - 45$$
$23$ $$T^{2} - 20$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 20$$
$37$ $$(T + 2)^{2}$$
$41$ $$(T - 5)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} - 80$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} - 80$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} - 5$$
$71$ $$T^{2} - 80$$
$73$ $$(T + 9)^{2}$$
$79$ $$T^{2} - 20$$
$83$ $$T^{2} - 125$$
$89$ $$(T + 5)^{2}$$
$97$ $$(T - 2)^{2}$$