# Properties

 Label 3240.2.a.h Level $3240$ Weight $2$ Character orbit 3240.a Self dual yes Analytic conductor $25.872$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + \beta q^{7}+O(q^{10})$$ q - q^5 + b * q^7 $$q - q^{5} + \beta q^{7} + (\beta - 3) q^{11} + (\beta + 2) q^{13} - 2 q^{17} + q^{19} + ( - \beta + 4) q^{23} + q^{25} + (\beta - 3) q^{29} + ( - \beta + 3) q^{31} - \beta q^{35} + (2 \beta - 4) q^{37} + (2 \beta - 1) q^{41} + 4 q^{43} + ( - \beta - 2) q^{47} + (\beta + 7) q^{49} + ( - \beta - 4) q^{53} + ( - \beta + 3) q^{55} + 13 q^{59} + ( - 2 \beta + 2) q^{61} + ( - \beta - 2) q^{65} + ( - 2 \beta - 6) q^{67} + ( - \beta + 5) q^{71} + ( - 4 \beta + 2) q^{73} + ( - 2 \beta + 14) q^{77} + ( - 2 \beta + 4) q^{79} + ( - 2 \beta - 4) q^{83} + 2 q^{85} + (3 \beta - 3) q^{89} + (3 \beta + 14) q^{91} - q^{95} + 16 q^{97} +O(q^{100})$$ q - q^5 + b * q^7 + (b - 3) * q^11 + (b + 2) * q^13 - 2 * q^17 + q^19 + (-b + 4) * q^23 + q^25 + (b - 3) * q^29 + (-b + 3) * q^31 - b * q^35 + (2*b - 4) * q^37 + (2*b - 1) * q^41 + 4 * q^43 + (-b - 2) * q^47 + (b + 7) * q^49 + (-b - 4) * q^53 + (-b + 3) * q^55 + 13 * q^59 + (-2*b + 2) * q^61 + (-b - 2) * q^65 + (-2*b - 6) * q^67 + (-b + 5) * q^71 + (-4*b + 2) * q^73 + (-2*b + 14) * q^77 + (-2*b + 4) * q^79 + (-2*b - 4) * q^83 + 2 * q^85 + (3*b - 3) * q^89 + (3*b + 14) * q^91 - q^95 + 16 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + q^7 $$2 q - 2 q^{5} + q^{7} - 5 q^{11} + 5 q^{13} - 4 q^{17} + 2 q^{19} + 7 q^{23} + 2 q^{25} - 5 q^{29} + 5 q^{31} - q^{35} - 6 q^{37} + 8 q^{43} - 5 q^{47} + 15 q^{49} - 9 q^{53} + 5 q^{55} + 26 q^{59} + 2 q^{61} - 5 q^{65} - 14 q^{67} + 9 q^{71} + 26 q^{77} + 6 q^{79} - 10 q^{83} + 4 q^{85} - 3 q^{89} + 31 q^{91} - 2 q^{95} + 32 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + q^7 - 5 * q^11 + 5 * q^13 - 4 * q^17 + 2 * q^19 + 7 * q^23 + 2 * q^25 - 5 * q^29 + 5 * q^31 - q^35 - 6 * q^37 + 8 * q^43 - 5 * q^47 + 15 * q^49 - 9 * q^53 + 5 * q^55 + 26 * q^59 + 2 * q^61 - 5 * q^65 - 14 * q^67 + 9 * q^71 + 26 * q^77 + 6 * q^79 - 10 * q^83 + 4 * q^85 - 3 * q^89 + 31 * q^91 - 2 * q^95 + 32 * q^97

## 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
 −3.27492 4.27492
0 0 0 −1.00000 0 −3.27492 0 0 0
1.2 0 0 0 −1.00000 0 4.27492 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.h 2
3.b odd 2 1 3240.2.a.n yes 2
4.b odd 2 1 6480.2.a.bf 2
9.c even 3 2 3240.2.q.be 4
9.d odd 6 2 3240.2.q.y 4
12.b even 2 1 6480.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.a.h 2 1.a even 1 1 trivial
3240.2.a.n yes 2 3.b odd 2 1
3240.2.q.y 4 9.d odd 6 2
3240.2.q.be 4 9.c even 3 2
6480.2.a.bf 2 4.b odd 2 1
6480.2.a.bm 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3240))$$:

 $$T_{7}^{2} - T_{7} - 14$$ T7^2 - T7 - 14 $$T_{11}^{2} + 5T_{11} - 8$$ T11^2 + 5*T11 - 8 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - T - 14$$
$11$ $$T^{2} + 5T - 8$$
$13$ $$T^{2} - 5T - 8$$
$17$ $$(T + 2)^{2}$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 7T - 2$$
$29$ $$T^{2} + 5T - 8$$
$31$ $$T^{2} - 5T - 8$$
$37$ $$T^{2} + 6T - 48$$
$41$ $$T^{2} - 57$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 5T - 8$$
$53$ $$T^{2} + 9T + 6$$
$59$ $$(T - 13)^{2}$$
$61$ $$T^{2} - 2T - 56$$
$67$ $$T^{2} + 14T - 8$$
$71$ $$T^{2} - 9T + 6$$
$73$ $$T^{2} - 228$$
$79$ $$T^{2} - 6T - 48$$
$83$ $$T^{2} + 10T - 32$$
$89$ $$T^{2} + 3T - 126$$
$97$ $$(T - 16)^{2}$$