# Properties

 Label 5408.2.a.u Level $5408$ Weight $2$ Character orbit 5408.a Self dual yes Analytic conductor $43.183$ 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] = [5408,2,Mod(1,5408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5408, 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("5408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5408 = 2^{5} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5408.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: $$43.1830974131$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 416) 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 2 \beta q^{5} - 3 q^{7}+O(q^{10})$$ q + b * q^3 + 2*b * q^5 - 3 * q^7 $$q + \beta q^{3} + 2 \beta q^{5} - 3 q^{7} - 5 q^{11} + 6 q^{15} + 7 q^{17} - 5 q^{19} - 3 \beta q^{21} - 3 \beta q^{23} + 7 q^{25} - 3 \beta q^{27} - 5 q^{29} - 2 q^{31} - 5 \beta q^{33} - 6 \beta q^{35} - 3 \beta q^{37} - \beta q^{41} - 3 \beta q^{43} + 4 q^{47} + 2 q^{49} + 7 \beta q^{51} + 4 q^{53} - 10 \beta q^{55} - 5 \beta q^{57} - 7 q^{59} + 3 q^{61} + 3 q^{67} - 9 q^{69} - 7 q^{71} - 2 \beta q^{73} + 7 \beta q^{75} + 15 q^{77} + 2 \beta q^{79} - 9 q^{81} + 14 q^{83} + 14 \beta q^{85} - 5 \beta q^{87} - \beta q^{89} - 2 \beta q^{93} - 10 \beta q^{95} - 5 \beta q^{97} +O(q^{100})$$ q + b * q^3 + 2*b * q^5 - 3 * q^7 - 5 * q^11 + 6 * q^15 + 7 * q^17 - 5 * q^19 - 3*b * q^21 - 3*b * q^23 + 7 * q^25 - 3*b * q^27 - 5 * q^29 - 2 * q^31 - 5*b * q^33 - 6*b * q^35 - 3*b * q^37 - b * q^41 - 3*b * q^43 + 4 * q^47 + 2 * q^49 + 7*b * q^51 + 4 * q^53 - 10*b * q^55 - 5*b * q^57 - 7 * q^59 + 3 * q^61 + 3 * q^67 - 9 * q^69 - 7 * q^71 - 2*b * q^73 + 7*b * q^75 + 15 * q^77 + 2*b * q^79 - 9 * q^81 + 14 * q^83 + 14*b * q^85 - 5*b * q^87 - b * q^89 - 2*b * q^93 - 10*b * q^95 - 5*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{7}+O(q^{10})$$ 2 * q - 6 * q^7 $$2 q - 6 q^{7} - 10 q^{11} + 12 q^{15} + 14 q^{17} - 10 q^{19} + 14 q^{25} - 10 q^{29} - 4 q^{31} + 8 q^{47} + 4 q^{49} + 8 q^{53} - 14 q^{59} + 6 q^{61} + 6 q^{67} - 18 q^{69} - 14 q^{71} + 30 q^{77} - 18 q^{81} + 28 q^{83}+O(q^{100})$$ 2 * q - 6 * q^7 - 10 * q^11 + 12 * q^15 + 14 * q^17 - 10 * q^19 + 14 * q^25 - 10 * q^29 - 4 * q^31 + 8 * q^47 + 4 * q^49 + 8 * q^53 - 14 * q^59 + 6 * q^61 + 6 * q^67 - 18 * q^69 - 14 * q^71 + 30 * q^77 - 18 * q^81 + 28 * q^83

## 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
 −1.73205 1.73205
0 −1.73205 0 −3.46410 0 −3.00000 0 0 0
1.2 0 1.73205 0 3.46410 0 −3.00000 0 0 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
$$2$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5408.2.a.u 2
4.b odd 2 1 5408.2.a.z 2
13.b even 2 1 5408.2.a.z 2
13.f odd 12 2 416.2.w.a 4
52.b odd 2 1 inner 5408.2.a.u 2
52.l even 12 2 416.2.w.a 4
104.u even 12 2 832.2.w.e 4
104.x odd 12 2 832.2.w.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.w.a 4 13.f odd 12 2
416.2.w.a 4 52.l even 12 2
832.2.w.e 4 104.u even 12 2
832.2.w.e 4 104.x odd 12 2
5408.2.a.u 2 1.a even 1 1 trivial
5408.2.a.u 2 52.b odd 2 1 inner
5408.2.a.z 2 4.b odd 2 1
5408.2.a.z 2 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5408))$$:

 $$T_{3}^{2} - 3$$ T3^2 - 3 $$T_{5}^{2} - 12$$ T5^2 - 12 $$T_{7} + 3$$ T7 + 3 $$T_{37}^{2} - 27$$ T37^2 - 27

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3$$
$5$ $$T^{2} - 12$$
$7$ $$(T + 3)^{2}$$
$11$ $$(T + 5)^{2}$$
$13$ $$T^{2}$$
$17$ $$(T - 7)^{2}$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} - 27$$
$29$ $$(T + 5)^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} - 27$$
$41$ $$T^{2} - 3$$
$43$ $$T^{2} - 27$$
$47$ $$(T - 4)^{2}$$
$53$ $$(T - 4)^{2}$$
$59$ $$(T + 7)^{2}$$
$61$ $$(T - 3)^{2}$$
$67$ $$(T - 3)^{2}$$
$71$ $$(T + 7)^{2}$$
$73$ $$T^{2} - 12$$
$79$ $$T^{2} - 12$$
$83$ $$(T - 14)^{2}$$
$89$ $$T^{2} - 3$$
$97$ $$T^{2} - 75$$