# Properties

 Label 40.4.a.a Level $40$ Weight $4$ Character orbit 40.a Self dual yes Analytic conductor $2.360$ Analytic rank $1$ 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] = [40,4,Mod(1,40)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(40, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("40.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$40 = 2^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 40.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: $$2.36007640023$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q - 6 q^{3} - 5 q^{5} - 34 q^{7} + 9 q^{9}+O(q^{10})$$ q - 6 * q^3 - 5 * q^5 - 34 * q^7 + 9 * q^9 $$q - 6 q^{3} - 5 q^{5} - 34 q^{7} + 9 q^{9} + 16 q^{11} + 58 q^{13} + 30 q^{15} - 70 q^{17} + 4 q^{19} + 204 q^{21} - 134 q^{23} + 25 q^{25} + 108 q^{27} - 242 q^{29} + 100 q^{31} - 96 q^{33} + 170 q^{35} - 438 q^{37} - 348 q^{39} - 138 q^{41} + 178 q^{43} - 45 q^{45} + 22 q^{47} + 813 q^{49} + 420 q^{51} + 162 q^{53} - 80 q^{55} - 24 q^{57} - 268 q^{59} + 250 q^{61} - 306 q^{63} - 290 q^{65} + 422 q^{67} + 804 q^{69} - 852 q^{71} + 306 q^{73} - 150 q^{75} - 544 q^{77} - 456 q^{79} - 891 q^{81} + 434 q^{83} + 350 q^{85} + 1452 q^{87} - 726 q^{89} - 1972 q^{91} - 600 q^{93} - 20 q^{95} + 1378 q^{97} + 144 q^{99}+O(q^{100})$$ q - 6 * q^3 - 5 * q^5 - 34 * q^7 + 9 * q^9 + 16 * q^11 + 58 * q^13 + 30 * q^15 - 70 * q^17 + 4 * q^19 + 204 * q^21 - 134 * q^23 + 25 * q^25 + 108 * q^27 - 242 * q^29 + 100 * q^31 - 96 * q^33 + 170 * q^35 - 438 * q^37 - 348 * q^39 - 138 * q^41 + 178 * q^43 - 45 * q^45 + 22 * q^47 + 813 * q^49 + 420 * q^51 + 162 * q^53 - 80 * q^55 - 24 * q^57 - 268 * q^59 + 250 * q^61 - 306 * q^63 - 290 * q^65 + 422 * q^67 + 804 * q^69 - 852 * q^71 + 306 * q^73 - 150 * q^75 - 544 * q^77 - 456 * q^79 - 891 * q^81 + 434 * q^83 + 350 * q^85 + 1452 * q^87 - 726 * q^89 - 1972 * q^91 - 600 * q^93 - 20 * q^95 + 1378 * q^97 + 144 * 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 −6.00000 0 −5.00000 0 −34.0000 0 9.00000 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

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 40.4.a.a 1
3.b odd 2 1 360.4.a.h 1
4.b odd 2 1 80.4.a.e 1
5.b even 2 1 200.4.a.i 1
5.c odd 4 2 200.4.c.c 2
7.b odd 2 1 1960.4.a.h 1
8.b even 2 1 320.4.a.l 1
8.d odd 2 1 320.4.a.c 1
12.b even 2 1 720.4.a.bd 1
15.d odd 2 1 1800.4.a.bi 1
15.e even 4 2 1800.4.f.j 2
16.e even 4 2 1280.4.d.p 2
16.f odd 4 2 1280.4.d.a 2
20.d odd 2 1 400.4.a.e 1
20.e even 4 2 400.4.c.f 2
40.e odd 2 1 1600.4.a.br 1
40.f even 2 1 1600.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.a 1 1.a even 1 1 trivial
80.4.a.e 1 4.b odd 2 1
200.4.a.i 1 5.b even 2 1
200.4.c.c 2 5.c odd 4 2
320.4.a.c 1 8.d odd 2 1
320.4.a.l 1 8.b even 2 1
360.4.a.h 1 3.b odd 2 1
400.4.a.e 1 20.d odd 2 1
400.4.c.f 2 20.e even 4 2
720.4.a.bd 1 12.b even 2 1
1280.4.d.a 2 16.f odd 4 2
1280.4.d.p 2 16.e even 4 2
1600.4.a.j 1 40.f even 2 1
1600.4.a.br 1 40.e odd 2 1
1800.4.a.bi 1 15.d odd 2 1
1800.4.f.j 2 15.e even 4 2
1960.4.a.h 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 6$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(40))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 6$$
$5$ $$T + 5$$
$7$ $$T + 34$$
$11$ $$T - 16$$
$13$ $$T - 58$$
$17$ $$T + 70$$
$19$ $$T - 4$$
$23$ $$T + 134$$
$29$ $$T + 242$$
$31$ $$T - 100$$
$37$ $$T + 438$$
$41$ $$T + 138$$
$43$ $$T - 178$$
$47$ $$T - 22$$
$53$ $$T - 162$$
$59$ $$T + 268$$
$61$ $$T - 250$$
$67$ $$T - 422$$
$71$ $$T + 852$$
$73$ $$T - 306$$
$79$ $$T + 456$$
$83$ $$T - 434$$
$89$ $$T + 726$$
$97$ $$T - 1378$$