# Properties

 Label 1600.2.a.bc Level $1600$ Weight $2$ Character orbit 1600.a Self dual yes Analytic conductor $12.776$ Analytic rank $0$ Dimension $2$ CM no 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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: $\mathrm{SU}(2)$

## $q$-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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + \beta q^{7} + 5 q^{9}+O(q^{10})$$ q + b * q^3 + b * q^7 + 5 * q^9 $$q + \beta q^{3} + \beta q^{7} + 5 q^{9} + 2 \beta q^{11} - 2 q^{13} - 2 q^{17} + 8 q^{21} - \beta q^{23} + 2 \beta q^{27} - 6 q^{29} + 2 \beta q^{31} + 16 q^{33} - 10 q^{37} - 2 \beta q^{39} + 2 q^{41} - 3 \beta q^{43} + \beta q^{47} + q^{49} - 2 \beta q^{51} + 6 q^{53} - 4 \beta q^{59} + 2 q^{61} + 5 \beta q^{63} - \beta q^{67} - 8 q^{69} + 2 \beta q^{71} + 6 q^{73} + 16 q^{77} + 4 \beta q^{79} + q^{81} + \beta q^{83} - 6 \beta q^{87} + 10 q^{89} - 2 \beta q^{91} + 16 q^{93} - 2 q^{97} + 10 \beta q^{99} +O(q^{100})$$ q + b * q^3 + b * q^7 + 5 * q^9 + 2*b * q^11 - 2 * q^13 - 2 * q^17 + 8 * q^21 - b * q^23 + 2*b * q^27 - 6 * q^29 + 2*b * q^31 + 16 * q^33 - 10 * q^37 - 2*b * q^39 + 2 * q^41 - 3*b * q^43 + b * q^47 + q^49 - 2*b * q^51 + 6 * q^53 - 4*b * q^59 + 2 * q^61 + 5*b * q^63 - b * q^67 - 8 * q^69 + 2*b * q^71 + 6 * q^73 + 16 * q^77 + 4*b * q^79 + q^81 + b * q^83 - 6*b * q^87 + 10 * q^89 - 2*b * q^91 + 16 * q^93 - 2 * q^97 + 10*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{9}+O(q^{10})$$ 2 * q + 10 * q^9 $$2 q + 10 q^{9} - 4 q^{13} - 4 q^{17} + 16 q^{21} - 12 q^{29} + 32 q^{33} - 20 q^{37} + 4 q^{41} + 2 q^{49} + 12 q^{53} + 4 q^{61} - 16 q^{69} + 12 q^{73} + 32 q^{77} + 2 q^{81} + 20 q^{89} + 32 q^{93} - 4 q^{97}+O(q^{100})$$ 2 * q + 10 * q^9 - 4 * q^13 - 4 * q^17 + 16 * q^21 - 12 * q^29 + 32 * q^33 - 20 * q^37 + 4 * q^41 + 2 * q^49 + 12 * q^53 + 4 * q^61 - 16 * q^69 + 12 * q^73 + 32 * q^77 + 2 * q^81 + 20 * q^89 + 32 * 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
 −1.41421 1.41421
0 −2.82843 0 0 0 −2.82843 0 5.00000 0
1.2 0 2.82843 0 0 0 2.82843 0 5.00000 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.bc 2
4.b odd 2 1 inner 1600.2.a.bc 2
5.b even 2 1 320.2.a.g 2
5.c odd 4 2 1600.2.c.n 4
8.b even 2 1 800.2.a.m 2
8.d odd 2 1 800.2.a.m 2
15.d odd 2 1 2880.2.a.bk 2
20.d odd 2 1 320.2.a.g 2
20.e even 4 2 1600.2.c.n 4
24.f even 2 1 7200.2.a.cm 2
24.h odd 2 1 7200.2.a.cm 2
40.e odd 2 1 160.2.a.c 2
40.f even 2 1 160.2.a.c 2
40.i odd 4 2 800.2.c.f 4
40.k even 4 2 800.2.c.f 4
60.h even 2 1 2880.2.a.bk 2
80.k odd 4 2 1280.2.d.l 4
80.q even 4 2 1280.2.d.l 4
120.i odd 2 1 1440.2.a.o 2
120.m even 2 1 1440.2.a.o 2
120.q odd 4 2 7200.2.f.bh 4
120.w even 4 2 7200.2.f.bh 4
280.c odd 2 1 7840.2.a.bf 2
280.n even 2 1 7840.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.c 2 40.e odd 2 1
160.2.a.c 2 40.f even 2 1
320.2.a.g 2 5.b even 2 1
320.2.a.g 2 20.d odd 2 1
800.2.a.m 2 8.b even 2 1
800.2.a.m 2 8.d odd 2 1
800.2.c.f 4 40.i odd 4 2
800.2.c.f 4 40.k even 4 2
1280.2.d.l 4 80.k odd 4 2
1280.2.d.l 4 80.q even 4 2
1440.2.a.o 2 120.i odd 2 1
1440.2.a.o 2 120.m even 2 1
1600.2.a.bc 2 1.a even 1 1 trivial
1600.2.a.bc 2 4.b odd 2 1 inner
1600.2.c.n 4 5.c odd 4 2
1600.2.c.n 4 20.e even 4 2
2880.2.a.bk 2 15.d odd 2 1
2880.2.a.bk 2 60.h even 2 1
7200.2.a.cm 2 24.f even 2 1
7200.2.a.cm 2 24.h odd 2 1
7200.2.f.bh 4 120.q odd 4 2
7200.2.f.bh 4 120.w even 4 2
7840.2.a.bf 2 280.c odd 2 1
7840.2.a.bf 2 280.n even 2 1

## Hecke kernels

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

 $$T_{3}^{2} - 8$$ T3^2 - 8 $$T_{7}^{2} - 8$$ T7^2 - 8 $$T_{11}^{2} - 32$$ T11^2 - 32 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 8$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 8$$
$11$ $$T^{2} - 32$$
$13$ $$(T + 2)^{2}$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 8$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} - 32$$
$37$ $$(T + 10)^{2}$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} - 72$$
$47$ $$T^{2} - 8$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 128$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} - 8$$
$71$ $$T^{2} - 32$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2} - 128$$
$83$ $$T^{2} - 8$$
$89$ $$(T - 10)^{2}$$
$97$ $$(T + 2)^{2}$$