# Properties

 Label 5408.2.a.bm Level $5408$ Weight $2$ Character orbit 5408.a Self dual yes Analytic conductor $43.183$ Analytic rank $1$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.134509248.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 15x^{4} + 60x^{2} - 48$$ x^6 - 15*x^4 + 60*x^2 - 48 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{3} - \beta_1) q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b3 - b1) * q^5 + (-b2 - 1) * q^7 + (b2 + 2) * q^9 $$q + \beta_1 q^{3} + (\beta_{3} - \beta_1) q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} + 2) q^{9} + (\beta_{2} + 1) q^{11} + (\beta_{4} - \beta_{2} - 4) q^{15} - q^{17} + ( - \beta_{4} - 1) q^{19} + ( - \beta_{5} - \beta_{3} - 3 \beta_1) q^{21} + ( - 4 \beta_{3} + \beta_1) q^{23} + ( - 2 \beta_{4} + \beta_{2} + 1) q^{25} + (\beta_{5} + \beta_{3} + \beta_1) q^{27} + (\beta_{4} - \beta_{2} + 1) q^{29} + (\beta_{4} - \beta_{2} - 4) q^{31} + (\beta_{5} + \beta_{3} + 3 \beta_1) q^{33} + 2 \beta_1 q^{35} + \beta_{5} q^{37} + ( - \beta_{3} - 2 \beta_1) q^{41} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{43} + (\beta_{3} - 3 \beta_1) q^{45} + ( - \beta_{4} - \beta_{2} + 2) q^{47} + (2 \beta_{4} + \beta_{2} + 4) q^{49} - \beta_1 q^{51} + (\beta_{4} - 4) q^{53} - 2 \beta_1 q^{55} + ( - \beta_{5} - 5 \beta_{3} - \beta_1) q^{57} + ( - 2 \beta_{4} + \beta_{2} - 1) q^{59} + (\beta_{4} + \beta_{2} - 3) q^{61} + ( - 2 \beta_{4} - 2 \beta_{2} - 12) q^{63} + (\beta_{4} + 2 \beta_{2} - 5) q^{67} + ( - 4 \beta_{4} + \beta_{2} + 1) q^{69} + (\beta_{4} - 3) q^{71} + (\beta_{5} - 3 \beta_1) q^{73} + ( - \beta_{5} - 9 \beta_{3} + 3 \beta_1) q^{75} + ( - 2 \beta_{4} - \beta_{2} - 11) q^{77} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{79} + (2 \beta_{4} - 1) q^{81} + (2 \beta_{2} - 6) q^{83} + ( - \beta_{3} + \beta_1) q^{85} + (4 \beta_{3} - \beta_1) q^{87} + ( - \beta_{5} + 3 \beta_{3} - 5 \beta_1) q^{89} + (4 \beta_{3} - 6 \beta_1) q^{93} + (\beta_{5} + 5 \beta_{3} - 2 \beta_1) q^{95} + ( - \beta_{5} - 5 \beta_{3} - \beta_1) q^{97} + (2 \beta_{4} + 2 \beta_{2} + 12) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b3 - b1) * q^5 + (-b2 - 1) * q^7 + (b2 + 2) * q^9 + (b2 + 1) * q^11 + (b4 - b2 - 4) * q^15 - q^17 + (-b4 - 1) * q^19 + (-b5 - b3 - 3*b1) * q^21 + (-4*b3 + b1) * q^23 + (-2*b4 + b2 + 1) * q^25 + (b5 + b3 + b1) * q^27 + (b4 - b2 + 1) * q^29 + (b4 - b2 - 4) * q^31 + (b5 + b3 + 3*b1) * q^33 + 2*b1 * q^35 + b5 * q^37 + (-b3 - 2*b1) * q^41 + (-b5 - b3 + b1) * q^43 + (b3 - 3*b1) * q^45 + (-b4 - b2 + 2) * q^47 + (2*b4 + b2 + 4) * q^49 - b1 * q^51 + (b4 - 4) * q^53 - 2*b1 * q^55 + (-b5 - 5*b3 - b1) * q^57 + (-2*b4 + b2 - 1) * q^59 + (b4 + b2 - 3) * q^61 + (-2*b4 - 2*b2 - 12) * q^63 + (b4 + 2*b2 - 5) * q^67 + (-4*b4 + b2 + 1) * q^69 + (b4 - 3) * q^71 + (b5 - 3*b1) * q^73 + (-b5 - 9*b3 + 3*b1) * q^75 + (-2*b4 - b2 - 11) * q^77 + (-2*b5 - 2*b3 + 2*b1) * q^79 + (2*b4 - 1) * q^81 + (2*b2 - 6) * q^83 + (-b3 + b1) * q^85 + (4*b3 - b1) * q^87 + (-b5 + 3*b3 - 5*b1) * q^89 + (4*b3 - 6*b1) * q^93 + (b5 + 5*b3 - 2*b1) * q^95 + (-b5 - 5*b3 - b1) * q^97 + (2*b4 + 2*b2 + 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{7} + 12 q^{9}+O(q^{10})$$ 6 * q - 6 * q^7 + 12 * q^9 $$6 q - 6 q^{7} + 12 q^{9} + 6 q^{11} - 24 q^{15} - 6 q^{17} - 6 q^{19} + 6 q^{25} + 6 q^{29} - 24 q^{31} + 12 q^{47} + 24 q^{49} - 24 q^{53} - 6 q^{59} - 18 q^{61} - 72 q^{63} - 30 q^{67} + 6 q^{69} - 18 q^{71} - 66 q^{77} - 6 q^{81} - 36 q^{83} + 72 q^{99}+O(q^{100})$$ 6 * q - 6 * q^7 + 12 * q^9 + 6 * q^11 - 24 * q^15 - 6 * q^17 - 6 * q^19 + 6 * q^25 + 6 * q^29 - 24 * q^31 + 12 * q^47 + 24 * q^49 - 24 * q^53 - 6 * q^59 - 18 * q^61 - 72 * q^63 - 30 * q^67 + 6 * q^69 - 18 * q^71 - 66 * q^77 - 6 * q^81 - 36 * q^83 + 72 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 15x^{4} + 60x^{2} - 48$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - 11\nu^{3} + 24\nu ) / 8$$ (v^5 - 11*v^3 + 24*v) / 8 $$\beta_{4}$$ $$=$$ $$( \nu^{4} - 9\nu^{2} + 10 ) / 2$$ (v^4 - 9*v^2 + 10) / 2 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + 19\nu^{3} - 80\nu ) / 8$$ (-v^5 + 19*v^3 - 80*v) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ b2 + 5 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{3} + 7\beta_1$$ b5 + b3 + 7*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{4} + 9\beta_{2} + 35$$ 2*b4 + 9*b2 + 35 $$\nu^{5}$$ $$=$$ $$11\beta_{5} + 19\beta_{3} + 53\beta_1$$ 11*b5 + 19*b3 + 53*b1

## 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
 −2.96724 −2.26572 −1.03053 1.03053 2.26572 2.96724
0 −2.96724 0 1.23519 0 −4.80451 0 5.80451 0
1.2 0 −2.26572 0 3.99777 0 −1.13349 0 2.13349 0
1.3 0 −1.03053 0 −0.701519 0 2.93800 0 −1.93800 0
1.4 0 1.03053 0 0.701519 0 2.93800 0 −1.93800 0
1.5 0 2.26572 0 −3.99777 0 −1.13349 0 2.13349 0
1.6 0 2.96724 0 −1.23519 0 −4.80451 0 5.80451 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.bm 6
4.b odd 2 1 5408.2.a.bn 6
13.b even 2 1 5408.2.a.bn 6
13.f odd 12 2 416.2.w.d 12
52.b odd 2 1 inner 5408.2.a.bm 6
52.l even 12 2 416.2.w.d 12
104.u even 12 2 832.2.w.j 12
104.x odd 12 2 832.2.w.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.w.d 12 13.f odd 12 2
416.2.w.d 12 52.l even 12 2
832.2.w.j 12 104.u even 12 2
832.2.w.j 12 104.x odd 12 2
5408.2.a.bm 6 1.a even 1 1 trivial
5408.2.a.bm 6 52.b odd 2 1 inner
5408.2.a.bn 6 4.b odd 2 1
5408.2.a.bn 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5408))$$:

 $$T_{3}^{6} - 15T_{3}^{4} + 60T_{3}^{2} - 48$$ T3^6 - 15*T3^4 + 60*T3^2 - 48 $$T_{5}^{6} - 18T_{5}^{4} + 33T_{5}^{2} - 12$$ T5^6 - 18*T5^4 + 33*T5^2 - 12 $$T_{7}^{3} + 3T_{7}^{2} - 12T_{7} - 16$$ T7^3 + 3*T7^2 - 12*T7 - 16 $$T_{37}^{6} - 81T_{37}^{4} + 1275T_{37}^{2} - 5043$$ T37^6 - 81*T37^4 + 1275*T37^2 - 5043

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 15 T^{4} + \cdots - 48$$
$5$ $$T^{6} - 18 T^{4} + \cdots - 12$$
$7$ $$(T^{3} + 3 T^{2} - 12 T - 16)^{2}$$
$11$ $$(T^{3} - 3 T^{2} - 12 T + 16)^{2}$$
$13$ $$T^{6}$$
$17$ $$(T + 1)^{6}$$
$19$ $$(T^{3} + 3 T^{2} - 18 T - 36)^{2}$$
$23$ $$T^{6} - 135 T^{4} + \cdots - 46128$$
$29$ $$(T^{3} - 3 T^{2} - 21 T + 31)^{2}$$
$31$ $$(T^{3} + 12 T^{2} + \cdots - 24)^{2}$$
$37$ $$T^{6} - 81 T^{4} + \cdots - 5043$$
$41$ $$T^{6} - 81 T^{4} + \cdots - 6627$$
$43$ $$T^{6} - 99 T^{4} + \cdots - 12288$$
$47$ $$(T^{3} - 6 T^{2} + \cdots + 208)^{2}$$
$53$ $$(T^{3} + 12 T^{2} + 27 T - 4)^{2}$$
$59$ $$(T^{3} + 3 T^{2} + \cdots - 320)^{2}$$
$61$ $$(T^{3} + 9 T^{2} + \cdots - 237)^{2}$$
$67$ $$(T^{3} + 15 T^{2} + \cdots - 788)^{2}$$
$71$ $$(T^{3} + 9 T^{2} + 6 T - 20)^{2}$$
$73$ $$T^{6} - 234 T^{4} + \cdots - 288300$$
$79$ $$T^{6} - 396 T^{4} + \cdots - 786432$$
$83$ $$(T^{3} + 18 T^{2} + \cdots - 128)^{2}$$
$89$ $$T^{6} - 435 T^{4} + \cdots - 2157312$$
$97$ $$T^{6} - 315 T^{4} + \cdots - 62208$$