# Properties

 Label 1040.2.bg.a Level $1040$ Weight $2$ Character orbit 1040.bg Analytic conductor $8.304$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(577,1040)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1040.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1040 = 2^{4} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1040.bg (of order $$4$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$8.30444181021$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - i - 1) q^{3} + ( - 2 i - 1) q^{5} - 2 i q^{7} - i q^{9} +O(q^{10})$$ q + (-i - 1) * q^3 + (-2*i - 1) * q^5 - 2*i * q^7 - i * q^9 $$q + ( - i - 1) q^{3} + ( - 2 i - 1) q^{5} - 2 i q^{7} - i q^{9} + ( - i + 1) q^{11} + ( - 2 i + 3) q^{13} + (3 i - 1) q^{15} + (i + 1) q^{17} + ( - 5 i + 5) q^{19} + (2 i - 2) q^{21} + (3 i - 3) q^{23} + (4 i - 3) q^{25} + (4 i - 4) q^{27} + ( - 5 i - 5) q^{31} - 2 q^{33} + (2 i - 4) q^{35} + ( - i - 5) q^{39} + ( - 7 i - 7) q^{41} + ( - i + 1) q^{43} + (i - 2) q^{45} + 6 i q^{47} + 3 q^{49} - 2 i q^{51} + (5 i + 5) q^{53} + ( - i - 3) q^{55} - 10 q^{57} + (7 i + 7) q^{59} - 14 q^{61} - 2 q^{63} + ( - 4 i - 7) q^{65} + 4 q^{67} + 6 q^{69} + ( - i - 1) q^{71} - 10 q^{73} + ( - i + 7) q^{75} + ( - 2 i - 2) q^{77} - 2 i q^{79} + 5 q^{81} - 6 i q^{83} + ( - 3 i + 1) q^{85} + (5 i + 5) q^{89} + ( - 6 i - 4) q^{91} + 10 i q^{93} + ( - 5 i - 15) q^{95} + 2 q^{97} + ( - i - 1) q^{99} +O(q^{100})$$ q + (-i - 1) * q^3 + (-2*i - 1) * q^5 - 2*i * q^7 - i * q^9 + (-i + 1) * q^11 + (-2*i + 3) * q^13 + (3*i - 1) * q^15 + (i + 1) * q^17 + (-5*i + 5) * q^19 + (2*i - 2) * q^21 + (3*i - 3) * q^23 + (4*i - 3) * q^25 + (4*i - 4) * q^27 + (-5*i - 5) * q^31 - 2 * q^33 + (2*i - 4) * q^35 + (-i - 5) * q^39 + (-7*i - 7) * q^41 + (-i + 1) * q^43 + (i - 2) * q^45 + 6*i * q^47 + 3 * q^49 - 2*i * q^51 + (5*i + 5) * q^53 + (-i - 3) * q^55 - 10 * q^57 + (7*i + 7) * q^59 - 14 * q^61 - 2 * q^63 + (-4*i - 7) * q^65 + 4 * q^67 + 6 * q^69 + (-i - 1) * q^71 - 10 * q^73 + (-i + 7) * q^75 + (-2*i - 2) * q^77 - 2*i * q^79 + 5 * q^81 - 6*i * q^83 + (-3*i + 1) * q^85 + (5*i + 5) * q^89 + (-6*i - 4) * q^91 + 10*i * q^93 + (-5*i - 15) * q^95 + 2 * q^97 + (-i - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^5 $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} + 10 q^{19} - 4 q^{21} - 6 q^{23} - 6 q^{25} - 8 q^{27} - 10 q^{31} - 4 q^{33} - 8 q^{35} - 10 q^{39} - 14 q^{41} + 2 q^{43} - 4 q^{45} + 6 q^{49} + 10 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 28 q^{61} - 4 q^{63} - 14 q^{65} + 8 q^{67} + 12 q^{69} - 2 q^{71} - 20 q^{73} + 14 q^{75} - 4 q^{77} + 10 q^{81} + 2 q^{85} + 10 q^{89} - 8 q^{91} - 30 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^11 + 6 * q^13 - 2 * q^15 + 2 * q^17 + 10 * q^19 - 4 * q^21 - 6 * q^23 - 6 * q^25 - 8 * q^27 - 10 * q^31 - 4 * q^33 - 8 * q^35 - 10 * q^39 - 14 * q^41 + 2 * q^43 - 4 * q^45 + 6 * q^49 + 10 * q^53 - 6 * q^55 - 20 * q^57 + 14 * q^59 - 28 * q^61 - 4 * q^63 - 14 * q^65 + 8 * q^67 + 12 * q^69 - 2 * q^71 - 20 * q^73 + 14 * q^75 - 4 * q^77 + 10 * q^81 + 2 * q^85 + 10 * q^89 - 8 * q^91 - 30 * q^95 + 4 * q^97 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$417$$ $$561$$ $$911$$ $$\chi(n)$$ $$1$$ $$-i$$ $$i$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 − 1.00000i 1.00000i
0 −1.00000 + 1.00000i 0 −1.00000 + 2.00000i 0 2.00000i 0 1.00000i 0
593.1 0 −1.00000 1.00000i 0 −1.00000 2.00000i 0 2.00000i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.bg.a 2
4.b odd 2 1 65.2.k.a yes 2
5.c odd 4 1 1040.2.cd.b 2
12.b even 2 1 585.2.w.b 2
13.d odd 4 1 1040.2.cd.b 2
20.d odd 2 1 325.2.k.a 2
20.e even 4 1 65.2.f.a 2
20.e even 4 1 325.2.f.a 2
52.b odd 2 1 845.2.k.a 2
52.f even 4 1 65.2.f.a 2
52.f even 4 1 845.2.f.a 2
52.i odd 6 2 845.2.o.b 4
52.j odd 6 2 845.2.o.a 4
52.l even 12 2 845.2.t.a 4
52.l even 12 2 845.2.t.b 4
60.l odd 4 1 585.2.n.c 2
65.k even 4 1 inner 1040.2.bg.a 2
156.l odd 4 1 585.2.n.c 2
260.l odd 4 1 325.2.k.a 2
260.l odd 4 1 845.2.k.a 2
260.p even 4 1 845.2.f.a 2
260.s odd 4 1 65.2.k.a yes 2
260.u even 4 1 325.2.f.a 2
260.be odd 12 2 845.2.o.a 4
260.bg even 12 2 845.2.t.b 4
260.bj even 12 2 845.2.t.a 4
260.bl odd 12 2 845.2.o.b 4
780.bn even 4 1 585.2.w.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.a 2 20.e even 4 1
65.2.f.a 2 52.f even 4 1
65.2.k.a yes 2 4.b odd 2 1
65.2.k.a yes 2 260.s odd 4 1
325.2.f.a 2 20.e even 4 1
325.2.f.a 2 260.u even 4 1
325.2.k.a 2 20.d odd 2 1
325.2.k.a 2 260.l odd 4 1
585.2.n.c 2 60.l odd 4 1
585.2.n.c 2 156.l odd 4 1
585.2.w.b 2 12.b even 2 1
585.2.w.b 2 780.bn even 4 1
845.2.f.a 2 52.f even 4 1
845.2.f.a 2 260.p even 4 1
845.2.k.a 2 52.b odd 2 1
845.2.k.a 2 260.l odd 4 1
845.2.o.a 4 52.j odd 6 2
845.2.o.a 4 260.be odd 12 2
845.2.o.b 4 52.i odd 6 2
845.2.o.b 4 260.bl odd 12 2
845.2.t.a 4 52.l even 12 2
845.2.t.a 4 260.bj even 12 2
845.2.t.b 4 52.l even 12 2
845.2.t.b 4 260.bg even 12 2
1040.2.bg.a 2 1.a even 1 1 trivial
1040.2.bg.a 2 65.k even 4 1 inner
1040.2.cd.b 2 5.c odd 4 1
1040.2.cd.b 2 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1040, [\chi])$$:

 $$T_{3}^{2} + 2T_{3} + 2$$ T3^2 + 2*T3 + 2 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}^{2} - 2T_{11} + 2$$ T11^2 - 2*T11 + 2 $$T_{19}^{2} - 10T_{19} + 50$$ T19^2 - 10*T19 + 50

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 2$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} - 2T + 2$$
$13$ $$T^{2} - 6T + 13$$
$17$ $$T^{2} - 2T + 2$$
$19$ $$T^{2} - 10T + 50$$
$23$ $$T^{2} + 6T + 18$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 10T + 50$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 14T + 98$$
$43$ $$T^{2} - 2T + 2$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} - 10T + 50$$
$59$ $$T^{2} - 14T + 98$$
$61$ $$(T + 14)^{2}$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} + 2T + 2$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} + 4$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} - 10T + 50$$
$97$ $$(T - 2)^{2}$$