# Properties

 Label 3040.2.a.r Level $3040$ Weight $2$ Character orbit 3040.a Self dual yes Analytic conductor $24.275$ Analytic rank $1$ 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] = [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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.17428.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ x^4 - x^3 - 6*x^2 + 4*x + 6 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + q^{5} + ( - \beta_{2} - 1) q^{7} + \beta_{2} q^{9}+O(q^{10})$$ q - b1 * q^3 + q^5 + (-b2 - 1) * q^7 + b2 * q^9 $$q - \beta_1 q^{3} + q^{5} + ( - \beta_{2} - 1) q^{7} + \beta_{2} q^{9} + ( - \beta_{3} + \beta_{2} - 2) q^{11} + (\beta_{3} - \beta_{2} + \beta_1) q^{13} - \beta_1 q^{15} + ( - \beta_{3} + 2 \beta_1 - 1) q^{17} - q^{19} + (\beta_{3} + 2 \beta_1 + 1) q^{21} + (2 \beta_{3} + \beta_{2} - 1) q^{23} + q^{25} + ( - \beta_{3} + 2 \beta_1 - 1) q^{27} + (\beta_{3} + 1) q^{29} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{31} + 2 \beta_{2} q^{33} + ( - \beta_{2} - 1) q^{35} + ( - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{37} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{39} - 2 \beta_1 q^{41} + (\beta_{3} + \beta_{2}) q^{43} + \beta_{2} q^{45} + ( - \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{47} + (\beta_{3} + 2 \beta_{2} - 2) q^{49} + (\beta_{3} - 5) q^{51} + ( - \beta_{3} + 5 \beta_{2} - \beta_1 + 2) q^{53} + ( - \beta_{3} + \beta_{2} - 2) q^{55} + \beta_1 q^{57} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{59} + ( - \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{61} + ( - \beta_{3} - \beta_{2} - 4) q^{63} + (\beta_{3} - \beta_{2} + \beta_1) q^{65} + (2 \beta_{3} + 4 \beta_{2} - \beta_1) q^{67} + ( - 3 \beta_{3} - 4 \beta_{2} + \cdots - 3) q^{69}+ \cdots + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 + 4) q^{99}+O(q^{100})$$ q - b1 * q^3 + q^5 + (-b2 - 1) * q^7 + b2 * q^9 + (-b3 + b2 - 2) * q^11 + (b3 - b2 + b1) * q^13 - b1 * q^15 + (-b3 + 2*b1 - 1) * q^17 - q^19 + (b3 + 2*b1 + 1) * q^21 + (2*b3 + b2 - 1) * q^23 + q^25 + (-b3 + 2*b1 - 1) * q^27 + (b3 + 1) * q^29 + (-2*b2 + 2*b1 - 4) * q^31 + 2*b2 * q^33 + (-b2 - 1) * q^35 + (-2*b3 - b2 + 3*b1 - 3) * q^37 + (-3*b2 + 2*b1 - 3) * q^39 - 2*b1 * q^41 + (b3 + b2) * q^43 + b2 * q^45 + (-b3 + 3*b2 - 2*b1) * q^47 + (b3 + 2*b2 - 2) * q^49 + (b3 - 5) * q^51 + (-b3 + 5*b2 - b1 + 2) * q^53 + (-b3 + b2 - 2) * q^55 + b1 * q^57 + (b3 - 2*b2 + 2*b1 - 1) * q^59 + (-b3 - 5*b2 + 2*b1) * q^61 + (-b3 - b2 - 4) * q^63 + (b3 - b2 + b1) * q^65 + (2*b3 + 4*b2 - b1) * q^67 + (-3*b3 - 4*b2 + 2*b1 - 3) * q^69 + (-2*b3 + 2*b2 - 2*b1 - 6) * q^71 + (-b3 - 4*b1 + 3) * q^73 - b1 * q^75 + (2*b2 + 2*b1 - 2) * q^77 + (2*b3 + 2*b2 - 6) * q^79 + (b3 - 3*b2 - 5) * q^81 + (-b3 - b2 - 4) * q^83 + (-b3 + 2*b1 - 1) * q^85 + (-b3 - 2*b2 - 1) * q^87 + (-2*b3 - 2*b1) * q^89 + (-b3 - 4*b1 + 3) * q^91 + (2*b3 - 2*b2 + 6*b1 - 4) * q^93 - q^95 + (-b2 + b1 + 3) * q^97 + (b3 - 3*b2 - 2*b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{3} + 4 q^{5} - 5 q^{7} + q^{9}+O(q^{10})$$ 4 * q - q^3 + 4 * q^5 - 5 * q^7 + q^9 $$4 q - q^{3} + 4 q^{5} - 5 q^{7} + q^{9} - 6 q^{11} - q^{13} - q^{15} - q^{17} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 4 q^{25} - q^{27} + 3 q^{29} - 16 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} - 13 q^{39} - 2 q^{41} + q^{45} + 2 q^{47} - 7 q^{49} - 21 q^{51} + 13 q^{53} - 6 q^{55} + q^{57} - 5 q^{59} - 2 q^{61} - 16 q^{63} - q^{65} + q^{67} - 11 q^{69} - 22 q^{71} + 9 q^{73} - q^{75} - 4 q^{77} - 24 q^{79} - 24 q^{81} - 16 q^{83} - q^{85} - 5 q^{87} + 9 q^{91} - 14 q^{93} - 4 q^{95} + 12 q^{97} + 10 q^{99}+O(q^{100})$$ 4 * q - q^3 + 4 * q^5 - 5 * q^7 + q^9 - 6 * q^11 - q^13 - q^15 - q^17 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 4 * q^25 - q^27 + 3 * q^29 - 16 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 - 13 * q^39 - 2 * q^41 + q^45 + 2 * q^47 - 7 * q^49 - 21 * q^51 + 13 * q^53 - 6 * q^55 + q^57 - 5 * q^59 - 2 * q^61 - 16 * q^63 - q^65 + q^67 - 11 * q^69 - 22 * q^71 + 9 * q^73 - q^75 - 4 * q^77 - 24 * q^79 - 24 * q^81 - 16 * q^83 - q^85 - 5 * q^87 + 9 * q^91 - 14 * q^93 - 4 * q^95 + 12 * q^97 + 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu - 1$$ v^3 - 4*v - 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta _1 + 1$$ b3 + 4*b1 + 1

## 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
 2.36865 1.52616 −0.787711 −2.10710
0 −2.36865 0 1.00000 0 −3.61050 0 2.61050 0
1.2 0 −1.52616 0 1.00000 0 −0.329157 0 −0.670843 0
1.3 0 0.787711 0 1.00000 0 1.37951 0 −2.37951 0
1.4 0 2.10710 0 1.00000 0 −2.43986 0 1.43986 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.r 4
4.b odd 2 1 3040.2.a.t yes 4
8.b even 2 1 6080.2.a.cf 4
8.d odd 2 1 6080.2.a.cd 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.a.r 4 1.a even 1 1 trivial
3040.2.a.t yes 4 4.b odd 2 1
6080.2.a.cd 4 8.d odd 2 1
6080.2.a.cf 4 8.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3040))$$:

 $$T_{3}^{4} + T_{3}^{3} - 6T_{3}^{2} - 4T_{3} + 6$$ T3^4 + T3^3 - 6*T3^2 - 4*T3 + 6 $$T_{7}^{4} + 5T_{7}^{3} + 2T_{7}^{2} - 12T_{7} - 4$$ T7^4 + 5*T7^3 + 2*T7^2 - 12*T7 - 4 $$T_{11}^{4} + 6T_{11}^{3} - 4T_{11}^{2} - 20T_{11} + 16$$ T11^4 + 6*T11^3 - 4*T11^2 - 20*T11 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + T^{3} - 6 T^{2} + \cdots + 6$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} + 5 T^{3} + \cdots - 4$$
$11$ $$T^{4} + 6 T^{3} + \cdots + 16$$
$13$ $$T^{4} + T^{3} + \cdots + 62$$
$17$ $$T^{4} + T^{3} + \cdots + 72$$
$19$ $$(T + 1)^{4}$$
$23$ $$T^{4} + 5 T^{3} + \cdots - 12$$
$29$ $$T^{4} - 3 T^{3} + \cdots + 24$$
$31$ $$T^{4} + 16 T^{3} + \cdots - 16$$
$37$ $$T^{4} + 8 T^{3} + \cdots - 1684$$
$41$ $$T^{4} + 2 T^{3} + \cdots + 96$$
$43$ $$T^{4} - 24 T^{2} + \cdots - 8$$
$47$ $$T^{4} - 2 T^{3} + \cdots + 32$$
$53$ $$T^{4} - 13 T^{3} + \cdots - 942$$
$59$ $$T^{4} + 5 T^{3} + \cdots + 8$$
$61$ $$T^{4} + 2 T^{3} + \cdots + 9104$$
$67$ $$T^{4} - T^{3} + \cdots + 3362$$
$71$ $$T^{4} + 22 T^{3} + \cdots - 2272$$
$73$ $$T^{4} - 9 T^{3} + \cdots - 248$$
$79$ $$T^{4} + 24 T^{3} + \cdots - 3632$$
$83$ $$T^{4} + 16 T^{3} + \cdots - 24$$
$89$ $$T^{4} - 96 T^{2} + \cdots + 592$$
$97$ $$T^{4} - 12 T^{3} + \cdots - 36$$