# Properties

 Label 1040.2.a.f Level $1040$ Weight $2$ Character orbit 1040.a Self dual yes Analytic conductor $8.304$ Analytic rank $0$ 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] = [1040,2,Mod(1,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1040 = 2^{4} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1040.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: $$8.30444181021$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} - q^{5} + 4 q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^3 - q^5 + 4 * q^7 + q^9 $$q + 2 q^{3} - q^{5} + 4 q^{7} + q^{9} - 2 q^{11} - q^{13} - 2 q^{15} + 2 q^{17} + 6 q^{19} + 8 q^{21} + 6 q^{23} + q^{25} - 4 q^{27} + 2 q^{29} + 10 q^{31} - 4 q^{33} - 4 q^{35} - 2 q^{37} - 2 q^{39} - 6 q^{41} - 10 q^{43} - q^{45} - 4 q^{47} + 9 q^{49} + 4 q^{51} + 2 q^{53} + 2 q^{55} + 12 q^{57} - 6 q^{59} + 2 q^{61} + 4 q^{63} + q^{65} + 4 q^{67} + 12 q^{69} - 6 q^{71} - 6 q^{73} + 2 q^{75} - 8 q^{77} + 12 q^{79} - 11 q^{81} + 16 q^{83} - 2 q^{85} + 4 q^{87} + 2 q^{89} - 4 q^{91} + 20 q^{93} - 6 q^{95} - 2 q^{97} - 2 q^{99}+O(q^{100})$$ q + 2 * q^3 - q^5 + 4 * q^7 + q^9 - 2 * q^11 - q^13 - 2 * q^15 + 2 * q^17 + 6 * q^19 + 8 * q^21 + 6 * q^23 + q^25 - 4 * q^27 + 2 * q^29 + 10 * q^31 - 4 * q^33 - 4 * q^35 - 2 * q^37 - 2 * q^39 - 6 * q^41 - 10 * q^43 - q^45 - 4 * q^47 + 9 * q^49 + 4 * q^51 + 2 * q^53 + 2 * q^55 + 12 * q^57 - 6 * q^59 + 2 * q^61 + 4 * q^63 + q^65 + 4 * q^67 + 12 * q^69 - 6 * q^71 - 6 * q^73 + 2 * q^75 - 8 * q^77 + 12 * q^79 - 11 * q^81 + 16 * q^83 - 2 * q^85 + 4 * q^87 + 2 * q^89 - 4 * q^91 + 20 * q^93 - 6 * q^95 - 2 * q^97 - 2 * 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 2.00000 0 −1.00000 0 4.00000 0 1.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 1040.2.a.f 1
3.b odd 2 1 9360.2.a.ca 1
4.b odd 2 1 65.2.a.a 1
5.b even 2 1 5200.2.a.d 1
8.b even 2 1 4160.2.a.f 1
8.d odd 2 1 4160.2.a.q 1
12.b even 2 1 585.2.a.h 1
20.d odd 2 1 325.2.a.d 1
20.e even 4 2 325.2.b.b 2
28.d even 2 1 3185.2.a.e 1
44.c even 2 1 7865.2.a.c 1
52.b odd 2 1 845.2.a.a 1
52.f even 4 2 845.2.c.a 2
52.i odd 6 2 845.2.e.a 2
52.j odd 6 2 845.2.e.b 2
52.l even 12 4 845.2.m.b 4
60.h even 2 1 2925.2.a.f 1
60.l odd 4 2 2925.2.c.h 2
156.h even 2 1 7605.2.a.f 1
260.g odd 2 1 4225.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.a 1 4.b odd 2 1
325.2.a.d 1 20.d odd 2 1
325.2.b.b 2 20.e even 4 2
585.2.a.h 1 12.b even 2 1
845.2.a.a 1 52.b odd 2 1
845.2.c.a 2 52.f even 4 2
845.2.e.a 2 52.i odd 6 2
845.2.e.b 2 52.j odd 6 2
845.2.m.b 4 52.l even 12 4
1040.2.a.f 1 1.a even 1 1 trivial
2925.2.a.f 1 60.h even 2 1
2925.2.c.h 2 60.l odd 4 2
3185.2.a.e 1 28.d even 2 1
4160.2.a.f 1 8.b even 2 1
4160.2.a.q 1 8.d odd 2 1
4225.2.a.g 1 260.g odd 2 1
5200.2.a.d 1 5.b even 2 1
7605.2.a.f 1 156.h even 2 1
7865.2.a.c 1 44.c even 2 1
9360.2.a.ca 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1040))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T + 1$$
$7$ $$T - 4$$
$11$ $$T + 2$$
$13$ $$T + 1$$
$17$ $$T - 2$$
$19$ $$T - 6$$
$23$ $$T - 6$$
$29$ $$T - 2$$
$31$ $$T - 10$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T + 10$$
$47$ $$T + 4$$
$53$ $$T - 2$$
$59$ $$T + 6$$
$61$ $$T - 2$$
$67$ $$T - 4$$
$71$ $$T + 6$$
$73$ $$T + 6$$
$79$ $$T - 12$$
$83$ $$T - 16$$
$89$ $$T - 2$$
$97$ $$T + 2$$