# Properties

 Label 5472.2.a.bs Level $5472$ Weight $2$ Character orbit 5472.a Self dual yes Analytic conductor $43.694$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(1,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, 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("5472.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5472 = 2^{5} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5472.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: $$43.6941399860$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.15317.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 2$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 608) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1) q^{5} + \beta_{3} q^{7}+O(q^{10})$$ q + (b2 - b1) * q^5 + b3 * q^7 $$q + (\beta_{2} - \beta_1) q^{5} + \beta_{3} q^{7} + ( - \beta_{2} + \beta_1 - 2) q^{11} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{13} + (2 \beta_{3} - \beta_1 - 1) q^{17} + q^{19} + (\beta_{3} - \beta_{2} + \beta_1 + 2) q^{23} + (2 \beta_{3} + \beta_{2} - \beta_1 + 5) q^{25} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{29} + (2 \beta_{3} - 2 \beta_{2} - 2) q^{31} + (3 \beta_{2} - \beta_1) q^{35} + ( - 2 \beta_{3} + 2 \beta_{2} + 4) q^{37} - 2 \beta_1 q^{41} + ( - \beta_{2} - \beta_1) q^{43} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{47} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{49} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{53} + ( - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 10) q^{55} + (\beta_{2} - 2 \beta_1 - 5) q^{59} + ( - 3 \beta_{2} + \beta_1 - 2) q^{61} + ( - 2 \beta_{3} + 8 \beta_{2} - 2 \beta_1 + 6) q^{65} + ( - \beta_{2} - 2 \beta_1 + 1) q^{67} + (2 \beta_{3} + 8) q^{71} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{73} + ( - 2 \beta_{3} - 3 \beta_{2} + \beta_1) q^{77} + ( - 2 \beta_{3} + 4 \beta_{2} + 4) q^{79} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{83} + (7 \beta_{2} - \beta_1 + 8) q^{85} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 6) q^{89} + (4 \beta_{3} - \beta_{2} + 3) q^{91} + (\beta_{2} - \beta_1) q^{95} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{97}+O(q^{100})$$ q + (b2 - b1) * q^5 + b3 * q^7 + (-b2 + b1 - 2) * q^11 + (b3 - b2 - b1 + 2) * q^13 + (2*b3 - b1 - 1) * q^17 + q^19 + (b3 - b2 + b1 + 2) * q^23 + (2*b3 + b2 - b1 + 5) * q^25 + (-b3 + b2 + b1 - 2) * q^29 + (2*b3 - 2*b2 - 2) * q^31 + (3*b2 - b1) * q^35 + (-2*b3 + 2*b2 + 4) * q^37 - 2*b1 * q^41 + (-b2 - b1) * q^43 + (-2*b3 + 3*b2 + b1 + 2) * q^47 + (-2*b3 - 2*b2 - b1) * q^49 + (-b3 - b2 - b1 - 2) * q^53 + (-2*b3 - 3*b2 + 3*b1 - 10) * q^55 + (b2 - 2*b1 - 5) * q^59 + (-3*b2 + b1 - 2) * q^61 + (-2*b3 + 8*b2 - 2*b1 + 6) * q^65 + (-b2 - 2*b1 + 1) * q^67 + (2*b3 + 8) * q^71 + (-2*b3 + 2*b2 + b1 + 3) * q^73 + (-2*b3 - 3*b2 + b1) * q^77 + (-2*b3 + 4*b2 + 4) * q^79 + (2*b3 + 2*b2 - 2*b1) * q^83 + (7*b2 - b1 + 8) * q^85 + (2*b3 + 2*b2 + 2*b1 + 6) * q^89 + (4*b3 - b2 + 3) * q^91 + (b2 - b1) * q^95 + (-2*b2 + 2*b1 - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{5} - q^{7}+O(q^{10})$$ 4 * q - q^5 - q^7 $$4 q - q^{5} - q^{7} - 7 q^{11} + 10 q^{13} - 5 q^{17} + 4 q^{19} + 8 q^{23} + 17 q^{25} - 10 q^{29} - 6 q^{31} - 5 q^{35} + 14 q^{37} + 2 q^{41} + 3 q^{43} + 3 q^{47} + 7 q^{49} - 4 q^{53} - 35 q^{55} - 20 q^{59} - 3 q^{61} + 12 q^{65} + 8 q^{67} + 30 q^{71} + 9 q^{73} + 7 q^{77} + 10 q^{79} - 4 q^{83} + 19 q^{85} + 16 q^{89} + 10 q^{91} - q^{95} - 6 q^{97}+O(q^{100})$$ 4 * q - q^5 - q^7 - 7 * q^11 + 10 * q^13 - 5 * q^17 + 4 * q^19 + 8 * q^23 + 17 * q^25 - 10 * q^29 - 6 * q^31 - 5 * q^35 + 14 * q^37 + 2 * q^41 + 3 * q^43 + 3 * q^47 + 7 * q^49 - 4 * q^53 - 35 * q^55 - 20 * q^59 - 3 * q^61 + 12 * q^65 + 8 * q^67 + 30 * q^71 + 9 * q^73 + 7 * q^77 + 10 * q^79 - 4 * q^83 + 19 * q^85 + 16 * q^89 + 10 * q^91 - q^95 - 6 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 4x^{2} + 5x + 2$$ :

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

## 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
 −0.329727 2.69353 1.32973 −1.69353
0 0 0 −3.40617 0 2.50407 0 0 0
1.2 0 0 0 −2.64453 0 −0.180969 0 0 0
1.3 0 0 0 0.844614 0 −5.06562 0 0 0
1.4 0 0 0 4.20608 0 1.74252 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$-1$$

## 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 5472.2.a.bs 4
3.b odd 2 1 608.2.a.j yes 4
4.b odd 2 1 5472.2.a.bt 4
12.b even 2 1 608.2.a.i 4
24.f even 2 1 1216.2.a.x 4
24.h odd 2 1 1216.2.a.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.a.i 4 12.b even 2 1
608.2.a.j yes 4 3.b odd 2 1
1216.2.a.w 4 24.h odd 2 1
1216.2.a.x 4 24.f even 2 1
5472.2.a.bs 4 1.a even 1 1 trivial
5472.2.a.bt 4 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5472))$$:

 $$T_{5}^{4} + T_{5}^{3} - 18T_{5}^{2} - 24T_{5} + 32$$ T5^4 + T5^3 - 18*T5^2 - 24*T5 + 32 $$T_{7}^{4} + T_{7}^{3} - 17T_{7}^{2} + 19T_{7} + 4$$ T7^4 + T7^3 - 17*T7^2 + 19*T7 + 4 $$T_{11}^{4} + 7T_{11}^{3} - 28T_{11} + 16$$ T11^4 + 7*T11^3 - 28*T11 + 16 $$T_{13}^{4} - 10T_{13}^{3} + 7T_{13}^{2} + 158T_{13} - 344$$ T13^4 - 10*T13^3 + 7*T13^2 + 158*T13 - 344 $$T_{23}^{4} - 8T_{23}^{3} - 17T_{23}^{2} + 132T_{23} + 64$$ T23^4 - 8*T23^3 - 17*T23^2 + 132*T23 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + T^{3} - 18 T^{2} - 24 T + 32$$
$7$ $$T^{4} + T^{3} - 17 T^{2} + 19 T + 4$$
$11$ $$T^{4} + 7 T^{3} - 28 T + 16$$
$13$ $$T^{4} - 10 T^{3} + 7 T^{2} + 158 T - 344$$
$17$ $$T^{4} + 5 T^{3} - 51 T^{2} - 141 T + 698$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} - 8 T^{3} - 17 T^{2} + 132 T + 64$$
$29$ $$T^{4} + 10 T^{3} + 7 T^{2} - 158 T - 344$$
$31$ $$T^{4} + 6 T^{3} - 56 T^{2} - 416 T - 512$$
$37$ $$T^{4} - 14 T^{3} + 4 T^{2} + \cdots - 1504$$
$41$ $$T^{4} - 2 T^{3} - 72 T^{2} + 192 T + 512$$
$43$ $$T^{4} - 3 T^{3} - 32 T^{2} + 96 T - 64$$
$47$ $$T^{4} - 3 T^{3} - 108 T^{2} + 176 T - 64$$
$53$ $$T^{4} + 4 T^{3} - 69 T^{2} - 384 T - 508$$
$59$ $$T^{4} + 20 T^{3} + 85 T^{2} + \cdots - 152$$
$61$ $$T^{4} + 3 T^{3} - 66 T^{2} - 28 T + 344$$
$67$ $$T^{4} - 8 T^{3} - 75 T^{2} + 602 T - 824$$
$71$ $$T^{4} - 30 T^{3} + 268 T^{2} + \cdots - 2432$$
$73$ $$T^{4} - 9 T^{3} - 47 T^{2} + 309 T + 2$$
$79$ $$T^{4} - 10 T^{3} - 100 T^{2} + \cdots - 3136$$
$83$ $$T^{4} + 4 T^{3} - 116 T^{2} + \cdots - 1024$$
$89$ $$T^{4} - 16 T^{3} - 204 T^{2} + \cdots - 15296$$
$97$ $$T^{4} + 6 T^{3} - 60 T^{2} - 88 T + 608$$