# Properties

 Label 5408.2.a.bg Level $5408$ Weight $2$ Character orbit 5408.a Self dual yes Analytic conductor $43.183$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.7488.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 2x + 1$$ x^4 - 2*x^3 - 4*x^2 + 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - 2 q^{5} + (\beta_{3} + \beta_1) q^{7} + ( - 2 \beta_{2} + 2) q^{9}+O(q^{10})$$ q + b1 * q^3 - 2 * q^5 + (b3 + b1) * q^7 + (-2*b2 + 2) * q^9 $$q + \beta_1 q^{3} - 2 q^{5} + (\beta_{3} + \beta_1) q^{7} + ( - 2 \beta_{2} + 2) q^{9} + (\beta_{3} - \beta_1) q^{11} - 2 \beta_1 q^{15} + (2 \beta_{2} - 3) q^{17} + ( - \beta_{3} - \beta_1) q^{19} + (\beta_{2} + 4) q^{21} + ( - 2 \beta_{3} + \beta_1) q^{23} - q^{25} + ( - 2 \beta_{3} + \beta_1) q^{27} + 3 q^{29} + ( - 2 \beta_{3} - 2 \beta_1) q^{31} + (5 \beta_{2} - 6) q^{33} + ( - 2 \beta_{3} - 2 \beta_1) q^{35} + ( - \beta_{2} - 6) q^{37} + ( - 3 \beta_{2} - 2) q^{41} - 3 \beta_1 q^{43} + (4 \beta_{2} - 4) q^{45} + ( - 2 \beta_{3} - 4 \beta_1) q^{47} + (6 \beta_{2} + 4) q^{49} + (2 \beta_{3} - 5 \beta_1) q^{51} + ( - 2 \beta_{2} + 6) q^{53} + ( - 2 \beta_{3} + 2 \beta_1) q^{55} + ( - \beta_{2} - 4) q^{57} + (\beta_{3} + 5 \beta_1) q^{59} + 3 q^{61} - 2 \beta_{3} q^{63} + ( - \beta_{3} - \beta_1) q^{67} + ( - 8 \beta_{2} + 7) q^{69} + (3 \beta_{3} - 3 \beta_1) q^{71} - 6 q^{73} - \beta_1 q^{75} + (4 \beta_{2} + 3) q^{77} + (2 \beta_{3} - 2 \beta_1) q^{79} + ( - 2 \beta_{2} + 1) q^{81} + (2 \beta_{3} - 2 \beta_1) q^{83} + ( - 4 \beta_{2} + 6) q^{85} + 3 \beta_1 q^{87} + (3 \beta_{2} - 4) q^{89} + ( - 2 \beta_{2} - 8) q^{93} + (2 \beta_{3} + 2 \beta_1) q^{95} + ( - \beta_{2} - 12) q^{97} + (2 \beta_{3} - 8 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 - 2 * q^5 + (b3 + b1) * q^7 + (-2*b2 + 2) * q^9 + (b3 - b1) * q^11 - 2*b1 * q^15 + (2*b2 - 3) * q^17 + (-b3 - b1) * q^19 + (b2 + 4) * q^21 + (-2*b3 + b1) * q^23 - q^25 + (-2*b3 + b1) * q^27 + 3 * q^29 + (-2*b3 - 2*b1) * q^31 + (5*b2 - 6) * q^33 + (-2*b3 - 2*b1) * q^35 + (-b2 - 6) * q^37 + (-3*b2 - 2) * q^41 - 3*b1 * q^43 + (4*b2 - 4) * q^45 + (-2*b3 - 4*b1) * q^47 + (6*b2 + 4) * q^49 + (2*b3 - 5*b1) * q^51 + (-2*b2 + 6) * q^53 + (-2*b3 + 2*b1) * q^55 + (-b2 - 4) * q^57 + (b3 + 5*b1) * q^59 + 3 * q^61 - 2*b3 * q^63 + (-b3 - b1) * q^67 + (-8*b2 + 7) * q^69 + (3*b3 - 3*b1) * q^71 - 6 * q^73 - b1 * q^75 + (4*b2 + 3) * q^77 + (2*b3 - 2*b1) * q^79 + (-2*b2 + 1) * q^81 + (2*b3 - 2*b1) * q^83 + (-4*b2 + 6) * q^85 + 3*b1 * q^87 + (3*b2 - 4) * q^89 + (-2*b2 - 8) * q^93 + (2*b3 + 2*b1) * q^95 + (-b2 - 12) * q^97 + (2*b3 - 8*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{5} + 8 q^{9}+O(q^{10})$$ 4 * q - 8 * q^5 + 8 * q^9 $$4 q - 8 q^{5} + 8 q^{9} - 12 q^{17} + 16 q^{21} - 4 q^{25} + 12 q^{29} - 24 q^{33} - 24 q^{37} - 8 q^{41} - 16 q^{45} + 16 q^{49} + 24 q^{53} - 16 q^{57} + 12 q^{61} + 28 q^{69} - 24 q^{73} + 12 q^{77} + 4 q^{81} + 24 q^{85} - 16 q^{89} - 32 q^{93} - 48 q^{97}+O(q^{100})$$ 4 * q - 8 * q^5 + 8 * q^9 - 12 * q^17 + 16 * q^21 - 4 * q^25 + 12 * q^29 - 24 * q^33 - 24 * q^37 - 8 * q^41 - 16 * q^45 + 16 * q^49 + 24 * q^53 - 16 * q^57 + 12 * q^61 + 28 * q^69 - 24 * q^73 + 12 * q^77 + 4 * q^81 + 24 * q^85 - 16 * q^89 - 32 * q^93 - 48 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 4x^{2} + 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2\nu - 2$$ v^2 - 2*v - 2 $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 3\nu + 1$$ v^3 - 2*v^2 - 3*v + 1 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 2\nu^{2} + 5\nu - 2$$ -v^3 + 2*v^2 + 5*v - 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + 1 ) / 2$$ (b3 + b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta _1 + 3$$ b3 + b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$( 7\beta_{3} + 9\beta_{2} + 4\beta _1 + 13 ) / 2$$ (7*b3 + 9*b2 + 4*b1 + 13) / 2

## 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
 0.698857 −0.326909 3.05896 −1.43091
0 −2.90931 0 −2.00000 0 −0.779548 0 5.46410 0
1.2 0 −1.23931 0 −2.00000 0 −4.62518 0 −1.46410 0
1.3 0 1.23931 0 −2.00000 0 4.62518 0 −1.46410 0
1.4 0 2.90931 0 −2.00000 0 0.779548 0 5.46410 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
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 5408.2.a.bg 4
4.b odd 2 1 inner 5408.2.a.bg 4
13.b even 2 1 5408.2.a.bk 4
13.f odd 12 2 416.2.w.c 8
52.b odd 2 1 5408.2.a.bk 4
52.l even 12 2 416.2.w.c 8
104.u even 12 2 832.2.w.h 8
104.x odd 12 2 832.2.w.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.w.c 8 13.f odd 12 2
416.2.w.c 8 52.l even 12 2
832.2.w.h 8 104.u even 12 2
832.2.w.h 8 104.x odd 12 2
5408.2.a.bg 4 1.a even 1 1 trivial
5408.2.a.bg 4 4.b odd 2 1 inner
5408.2.a.bk 4 13.b even 2 1
5408.2.a.bk 4 52.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5408))$$:

 $$T_{3}^{4} - 10T_{3}^{2} + 13$$ T3^4 - 10*T3^2 + 13 $$T_{5} + 2$$ T5 + 2 $$T_{7}^{4} - 22T_{7}^{2} + 13$$ T7^4 - 22*T7^2 + 13 $$T_{37}^{2} + 12T_{37} + 33$$ T37^2 + 12*T37 + 33

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 10T^{2} + 13$$
$5$ $$(T + 2)^{4}$$
$7$ $$T^{4} - 22T^{2} + 13$$
$11$ $$T^{4} - 30T^{2} + 117$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 6 T - 3)^{2}$$
$19$ $$T^{4} - 22T^{2} + 13$$
$23$ $$T^{4} - 82T^{2} + 1573$$
$29$ $$(T - 3)^{4}$$
$31$ $$T^{4} - 88T^{2} + 208$$
$37$ $$(T^{2} + 12 T + 33)^{2}$$
$41$ $$(T^{2} + 4 T - 23)^{2}$$
$43$ $$T^{4} - 90T^{2} + 1053$$
$47$ $$T^{4} - 192T^{2} + 7488$$
$53$ $$(T^{2} - 12 T + 24)^{2}$$
$59$ $$T^{4} - 246 T^{2} + 14157$$
$61$ $$(T - 3)^{4}$$
$67$ $$T^{4} - 22T^{2} + 13$$
$71$ $$T^{4} - 270T^{2} + 9477$$
$73$ $$(T + 6)^{4}$$
$79$ $$T^{4} - 120T^{2} + 1872$$
$83$ $$T^{4} - 120T^{2} + 1872$$
$89$ $$(T^{2} + 8 T - 11)^{2}$$
$97$ $$(T^{2} + 24 T + 141)^{2}$$