# Properties

 Label 3040.2.a.n Level $3040$ Weight $2$ Character orbit 3040.a Self dual yes Analytic conductor $24.275$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,2,Mod(1,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, 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("3040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3040 = 2^{5} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3040.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: $$24.2745222145$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{3} - q^{5} + (\beta_{2} - 1) q^{7} + (\beta_{2} - 2 \beta_1 + 2) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^3 - q^5 + (b2 - 1) * q^7 + (b2 - 2*b1 + 2) * q^9 $$q + ( - \beta_1 + 1) q^{3} - q^{5} + (\beta_{2} - 1) q^{7} + (\beta_{2} - 2 \beta_1 + 2) q^{9} + (\beta_{2} + 1) q^{11} + (\beta_{2} - \beta_1 + 2) q^{13} + (\beta_1 - 1) q^{15} + 2 q^{17} - q^{19} - 2 q^{21} + ( - \beta_{2} - 3) q^{23} + q^{25} + (2 \beta_{2} - 2 \beta_1 + 6) q^{27} + 2 \beta_1 q^{29} - 2 \beta_1 q^{31} - 2 \beta_1 q^{33} + ( - \beta_{2} + 1) q^{35} + ( - \beta_{2} + \beta_1) q^{37} + (\beta_{2} - 4 \beta_1 + 5) q^{39} + ( - 2 \beta_1 + 6) q^{41} + ( - \beta_{2} - 2 \beta_1 + 3) q^{43} + ( - \beta_{2} + 2 \beta_1 - 2) q^{45} + (\beta_{2} - 1) q^{47} + ( - 4 \beta_{2} + 2 \beta_1 - 1) q^{49} + ( - 2 \beta_1 + 2) q^{51} + (\beta_{2} - \beta_1 + 4) q^{53} + ( - \beta_{2} - 1) q^{55} + (\beta_1 - 1) q^{57} + ( - 4 \beta_1 + 2) q^{59} + (3 \beta_{2} + 2 \beta_1 + 1) q^{61} + ( - 3 \beta_{2} + 2 \beta_1 + 1) q^{63} + ( - \beta_{2} + \beta_1 - 2) q^{65} + ( - 2 \beta_{2} + 5 \beta_1 - 1) q^{67} + (4 \beta_1 - 2) q^{69} + ( - 2 \beta_1 - 8) q^{71} + 2 \beta_1 q^{73} + ( - \beta_1 + 1) q^{75} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{77} + 2 q^{79} + ( - \beta_{2} - 4 \beta_1 + 6) q^{81} + (\beta_{2} - 2 \beta_1 + 5) q^{83} - 2 q^{85} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{87} + (2 \beta_1 + 8) q^{89} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{91} + (2 \beta_{2} - 2 \beta_1 + 8) q^{93} + q^{95} + ( - \beta_{2} - \beta_1 + 6) q^{97} + ( - \beta_{2} - 2 \beta_1 + 5) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^3 - q^5 + (b2 - 1) * q^7 + (b2 - 2*b1 + 2) * q^9 + (b2 + 1) * q^11 + (b2 - b1 + 2) * q^13 + (b1 - 1) * q^15 + 2 * q^17 - q^19 - 2 * q^21 + (-b2 - 3) * q^23 + q^25 + (2*b2 - 2*b1 + 6) * q^27 + 2*b1 * q^29 - 2*b1 * q^31 - 2*b1 * q^33 + (-b2 + 1) * q^35 + (-b2 + b1) * q^37 + (b2 - 4*b1 + 5) * q^39 + (-2*b1 + 6) * q^41 + (-b2 - 2*b1 + 3) * q^43 + (-b2 + 2*b1 - 2) * q^45 + (b2 - 1) * q^47 + (-4*b2 + 2*b1 - 1) * q^49 + (-2*b1 + 2) * q^51 + (b2 - b1 + 4) * q^53 + (-b2 - 1) * q^55 + (b1 - 1) * q^57 + (-4*b1 + 2) * q^59 + (3*b2 + 2*b1 + 1) * q^61 + (-3*b2 + 2*b1 + 1) * q^63 + (-b2 + b1 - 2) * q^65 + (-2*b2 + 5*b1 - 1) * q^67 + (4*b1 - 2) * q^69 + (-2*b1 - 8) * q^71 + 2*b1 * q^73 + (-b1 + 1) * q^75 + (-2*b2 + 2*b1 + 4) * q^77 + 2 * q^79 + (-b2 - 4*b1 + 6) * q^81 + (b2 - 2*b1 + 5) * q^83 - 2 * q^85 + (-2*b2 + 2*b1 - 8) * q^87 + (2*b1 + 8) * q^89 + (-2*b2 + 2*b1 + 2) * q^91 + (2*b2 - 2*b1 + 8) * q^93 + q^95 + (-b2 - b1 + 6) * q^97 + (-b2 - 2*b1 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 - 3 * q^5 - 4 * q^7 + 3 * q^9 $$3 q + 2 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 2 q^{15} + 6 q^{17} - 3 q^{19} - 6 q^{21} - 8 q^{23} + 3 q^{25} + 14 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{33} + 4 q^{35} + 2 q^{37} + 10 q^{39} + 16 q^{41} + 8 q^{43} - 3 q^{45} - 4 q^{47} + 3 q^{49} + 4 q^{51} + 10 q^{53} - 2 q^{55} - 2 q^{57} + 2 q^{59} + 2 q^{61} + 8 q^{63} - 4 q^{65} + 4 q^{67} - 2 q^{69} - 26 q^{71} + 2 q^{73} + 2 q^{75} + 16 q^{77} + 6 q^{79} + 15 q^{81} + 12 q^{83} - 6 q^{85} - 20 q^{87} + 26 q^{89} + 10 q^{91} + 20 q^{93} + 3 q^{95} + 18 q^{97} + 14 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 - 3 * q^5 - 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - 2 * q^15 + 6 * q^17 - 3 * q^19 - 6 * q^21 - 8 * q^23 + 3 * q^25 + 14 * q^27 + 2 * q^29 - 2 * q^31 - 2 * q^33 + 4 * q^35 + 2 * q^37 + 10 * q^39 + 16 * q^41 + 8 * q^43 - 3 * q^45 - 4 * q^47 + 3 * q^49 + 4 * q^51 + 10 * q^53 - 2 * q^55 - 2 * q^57 + 2 * q^59 + 2 * q^61 + 8 * q^63 - 4 * q^65 + 4 * q^67 - 2 * q^69 - 26 * q^71 + 2 * q^73 + 2 * q^75 + 16 * q^77 + 6 * q^79 + 15 * q^81 + 12 * q^83 - 6 * q^85 - 20 * q^87 + 26 * q^89 + 10 * q^91 + 20 * q^93 + 3 * q^95 + 18 * q^97 + 14 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4

## 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
 2.51414 0.571993 −2.08613
0 −1.51414 0 −1.00000 0 1.32088 0 −0.707389 0
1.2 0 0.428007 0 −1.00000 0 −4.67282 0 −2.81681 0
1.3 0 3.08613 0 −1.00000 0 −0.648061 0 6.52420 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$1$$

## 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 3040.2.a.n yes 3
4.b odd 2 1 3040.2.a.k 3
8.b even 2 1 6080.2.a.bp 3
8.d odd 2 1 6080.2.a.bz 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.a.k 3 4.b odd 2 1
3040.2.a.n yes 3 1.a even 1 1 trivial
6080.2.a.bp 3 8.b even 2 1
6080.2.a.bz 3 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3040))$$:

 $$T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 2$$ T3^3 - 2*T3^2 - 4*T3 + 2 $$T_{7}^{3} + 4T_{7}^{2} - 4T_{7} - 4$$ T7^3 + 4*T7^2 - 4*T7 - 4 $$T_{11}^{3} - 2T_{11}^{2} - 8T_{11} + 12$$ T11^3 - 2*T11^2 - 8*T11 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 2 T^{2} + \cdots + 2$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + 4 T^{2} + \cdots - 4$$
$11$ $$T^{3} - 2 T^{2} + \cdots + 12$$
$13$ $$T^{3} - 4 T^{2} + \cdots + 18$$
$17$ $$(T - 2)^{3}$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} + 8 T^{2} + \cdots - 12$$
$29$ $$T^{3} - 2 T^{2} + \cdots + 24$$
$31$ $$T^{3} + 2 T^{2} + \cdots - 24$$
$37$ $$T^{3} - 2 T^{2} + \cdots + 2$$
$41$ $$T^{3} - 16 T^{2} + \cdots - 48$$
$43$ $$T^{3} - 8 T^{2} + \cdots + 164$$
$47$ $$T^{3} + 4 T^{2} + \cdots - 4$$
$53$ $$T^{3} - 10 T^{2} + \cdots + 6$$
$59$ $$T^{3} - 2 T^{2} + \cdots - 24$$
$61$ $$T^{3} - 2 T^{2} + \cdots - 244$$
$67$ $$T^{3} - 4 T^{2} + \cdots + 774$$
$71$ $$T^{3} + 26 T^{2} + \cdots + 456$$
$73$ $$T^{3} - 2 T^{2} + \cdots + 24$$
$79$ $$(T - 2)^{3}$$
$83$ $$T^{3} - 12 T^{2} + \cdots - 4$$
$89$ $$T^{3} - 26 T^{2} + \cdots - 456$$
$97$ $$T^{3} - 18 T^{2} + \cdots - 82$$