# Properties

 Label 3240.2.a.t Level $3240$ Weight $2$ Character orbit 3240.a Self dual yes Analytic conductor $25.872$ 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] = [3240,2,Mod(1,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, 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("3240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3240 = 2^{3} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3240.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: $$25.8715302549$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.62352.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 11x^{2} + 12x + 24$$ x^4 - 2*x^3 - 11*x^2 + 12*x + 24 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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)

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.

 $$f(q)$$ $$=$$ $$q - q^{5} + ( - \beta_{2} + \beta_1) q^{7}+O(q^{10})$$ q - q^5 + (-b2 + b1) * q^7 $$q - q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + \beta_1 - 2) q^{11} + (\beta_{3} - \beta_{2} + \beta_1) q^{13} + ( - \beta_{3} + 2 \beta_{2}) q^{17} + (2 \beta_1 - 1) q^{19} + (\beta_1 - 3) q^{23} + q^{25} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{29} + ( - \beta_1 + 4) q^{31} + (\beta_{2} - \beta_1) q^{35} + ( - \beta_{3} - 2 \beta_1 + 4) q^{37} + (2 \beta_{3} - 3 \beta_{2}) q^{41} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 6) q^{43} + (\beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{47} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{49}+ \cdots + ( - \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 4) q^{97}+O(q^{100})$$ q - q^5 + (-b2 + b1) * q^7 + (-b3 + b1 - 2) * q^11 + (b3 - b2 + b1) * q^13 + (-b3 + 2*b2) * q^17 + (2*b1 - 1) * q^19 + (b1 - 3) * q^23 + q^25 + (b3 - 2*b2 - b1 + 2) * q^29 + (-b1 + 4) * q^31 + (b2 - b1) * q^35 + (-b3 - 2*b1 + 4) * q^37 + (2*b3 - 3*b2) * q^41 + (-b3 - 2*b2 - 2*b1 + 6) * q^43 + (b3 - 4*b2 - b1 + 1) * q^47 + (-2*b3 + 2*b2 - b1 + 2) * q^49 + (b3 + 4*b2 - b1 + 3) * q^53 + (b3 - b1 + 2) * q^55 - b2 * q^59 + (2*b3 + 2*b2 + 2) * q^61 + (-b3 + b2 - b1) * q^65 + (-2*b3 + 4*b2 + 2*b1 + 4) * q^67 + (b3 + 2*b2 + 3*b1 - 8) * q^71 + (b3 - 2*b2 + 2*b1) * q^73 + (-3*b3 + 6) * q^77 + (-2*b2 + 4) * q^79 + (b3 - 2*b2 - 2*b1) * q^83 + (b3 - 2*b2) * q^85 + (b3 + 6*b2 + b1 + 2) * q^89 + (6*b2 - 3*b1 + 9) * q^91 + (-2*b1 + 1) * q^95 + (-b3 + 4*b2 + 2*b1 - 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5} + 2 q^{7}+O(q^{10})$$ 4 * q - 4 * q^5 + 2 * q^7 $$4 q - 4 q^{5} + 2 q^{7} - 4 q^{11} + 2 q^{17} - 10 q^{23} + 4 q^{25} + 4 q^{29} + 14 q^{31} - 2 q^{35} + 14 q^{37} - 4 q^{41} + 22 q^{43} + 10 q^{49} + 8 q^{53} + 4 q^{55} + 4 q^{61} + 24 q^{67} - 28 q^{71} + 2 q^{73} + 30 q^{77} + 16 q^{79} - 6 q^{83} - 2 q^{85} + 8 q^{89} + 30 q^{91} - 10 q^{97}+O(q^{100})$$ 4 * q - 4 * q^5 + 2 * q^7 - 4 * q^11 + 2 * q^17 - 10 * q^23 + 4 * q^25 + 4 * q^29 + 14 * q^31 - 2 * q^35 + 14 * q^37 - 4 * q^41 + 22 * q^43 + 10 * q^49 + 8 * q^53 + 4 * q^55 + 4 * q^61 + 24 * q^67 - 28 * q^71 + 2 * q^73 + 30 * q^77 + 16 * q^79 - 6 * q^83 - 2 * q^85 + 8 * q^89 + 30 * q^91 - 10 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 11x^{2} + 12x + 24$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - \nu - 6 ) / 2$$ (v^2 - v - 6) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 8\nu ) / 2$$ (v^3 - v^2 - 8*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2} + \beta _1 + 6$$ 2*b2 + b1 + 6 $$\nu^{3}$$ $$=$$ $$2\beta_{3} + 2\beta_{2} + 9\beta _1 + 6$$ 2*b3 + 2*b2 + 9*b1 + 6

## 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
 −2.61675 −1.16910 3.61675 2.16910
0 0 0 −1.00000 0 −4.34880 0 0 0
1.2 0 0 0 −1.00000 0 0.562950 0 0 0
1.3 0 0 0 −1.00000 0 1.88469 0 0 0
1.4 0 0 0 −1.00000 0 3.90115 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$+1$$

## 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 3240.2.a.t 4
3.b odd 2 1 3240.2.a.v yes 4
4.b odd 2 1 6480.2.a.by 4
9.c even 3 2 3240.2.q.bh 8
9.d odd 6 2 3240.2.q.bg 8
12.b even 2 1 6480.2.a.ca 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.a.t 4 1.a even 1 1 trivial
3240.2.a.v yes 4 3.b odd 2 1
3240.2.q.bg 8 9.d odd 6 2
3240.2.q.bh 8 9.c even 3 2
6480.2.a.by 4 4.b odd 2 1
6480.2.a.ca 4 12.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3240))$$:

 $$T_{7}^{4} - 2T_{7}^{3} - 17T_{7}^{2} + 42T_{7} - 18$$ T7^4 - 2*T7^3 - 17*T7^2 + 42*T7 - 18 $$T_{11}^{4} + 4T_{11}^{3} - 35T_{11}^{2} - 144T_{11} - 108$$ T11^4 + 4*T11^3 - 35*T11^2 - 144*T11 - 108 $$T_{17}^{4} - 2T_{17}^{3} - 38T_{17}^{2} + 120T_{17} - 72$$ T17^4 - 2*T17^3 - 38*T17^2 + 120*T17 - 72

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T + 1)^{4}$$
$7$ $$T^{4} - 2 T^{3} + \cdots - 18$$
$11$ $$T^{4} + 4 T^{3} + \cdots - 108$$
$13$ $$T^{4} - 39 T^{2} + \cdots + 216$$
$17$ $$T^{4} - 2 T^{3} + \cdots - 72$$
$19$ $$T^{4} - 50T^{2} + 433$$
$23$ $$T^{4} + 10 T^{3} + \cdots - 12$$
$29$ $$T^{4} - 4 T^{3} + \cdots - 48$$
$31$ $$T^{4} - 14 T^{3} + \cdots + 24$$
$37$ $$T^{4} - 14 T^{3} + \cdots - 1152$$
$41$ $$T^{4} + 4 T^{3} + \cdots + 69$$
$43$ $$T^{4} - 22 T^{3} + \cdots - 6768$$
$47$ $$T^{4} - 113 T^{2} + \cdots - 74$$
$53$ $$T^{4} - 8 T^{3} + \cdots - 498$$
$59$ $$(T^{2} - 3)^{2}$$
$61$ $$T^{4} - 4 T^{3} + \cdots - 1152$$
$67$ $$T^{4} - 24 T^{3} + \cdots - 18944$$
$71$ $$T^{4} + 28 T^{3} + \cdots - 16428$$
$73$ $$T^{4} - 2 T^{3} + \cdots - 552$$
$79$ $$(T^{2} - 8 T + 4)^{2}$$
$83$ $$T^{4} + 6 T^{3} + \cdots - 72$$
$89$ $$T^{4} - 8 T^{3} + \cdots + 11344$$
$97$ $$T^{4} + 10 T^{3} + \cdots - 32$$