# Properties

 Label 3040.2.a.s 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: 4.4.78292.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 10x^{2} + 8x + 18$$ x^4 - x^3 - 10*x^2 + 8*x + 18 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q + b1 * q^3 - q^5 + (-b2 - 1) * q^7 + (b2 + 2) * q^9 $$q + \beta_1 q^{3} - q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} + 2) q^{9} + ( - \beta_{3} - \beta_1 - 1) q^{11} + (\beta_{3} + 1) q^{13} - \beta_1 q^{15} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{17} + q^{19} + ( - \beta_{3} + \beta_{2} - 3 \beta_1) q^{21} + ( - \beta_{2} - 5) q^{23} + q^{25} + (\beta_{3} - \beta_{2} + \beta_1) q^{27} + (\beta_{3} - \beta_{2} - \beta_1) q^{29} + ( - 2 \beta_{3} - 2 \beta_1) q^{31} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{33}+ \cdots + ( - 3 \beta_{3} - 2 \beta_{2} + \cdots - 1) q^{99}+O(q^{100})$$ q + b1 * q^3 - q^5 + (-b2 - 1) * q^7 + (b2 + 2) * q^9 + (-b3 - b1 - 1) * q^11 + (b3 + 1) * q^13 - b1 * q^15 + (b3 + b2 + b1 - 2) * q^17 + q^19 + (-b3 + b2 - 3*b1) * q^21 + (-b2 - 5) * q^23 + q^25 + (b3 - b2 + b1) * q^27 + (b3 - b2 - b1) * q^29 + (-2*b3 - 2*b1) * q^31 + (-2*b3 - 2*b2 - 2*b1 - 2) * q^33 + (b2 + 1) * q^35 + (-b2 - b1 + 3) * q^37 + (2*b3 + b2 + 2*b1 - 3) * q^39 + (-2*b3 + 2*b2 - 2*b1) * q^41 + (b3 + 2*b2 - b1 + 1) * q^43 + (-b2 - 2) * q^45 + (b3 + 2*b2 + b1 - 7) * q^47 + (b3 + b2 - b1 + 1) * q^49 + (3*b3 + b2 + b1 + 2) * q^51 + (b3 - 2*b1 - 1) * q^53 + (b3 + b1 + 1) * q^55 + b1 * q^57 + (-3*b3 + b2 - b1 - 2) * q^59 + (-b3 - b1 - 1) * q^61 + (-b3 - 2*b2 + b1 - 9) * q^63 + (-b3 - 1) * q^65 + (-2*b3 - 2*b2 - b1 + 6) * q^67 + (-b3 + b2 - 7*b1) * q^69 - 2*b2 * q^71 + (3*b3 + b2 + b1) * q^73 + b1 * q^75 + (2*b3 + 2*b2 + 4*b1) * q^77 + (2*b2 + 2*b1) * q^79 + (b3 - b1 - 4) * q^81 + (b3 + 2*b2 - b1 - 3) * q^83 + (-b3 - b2 - b1 + 2) * q^85 + (b3 + b2 - b1 - 8) * q^87 + (-2*b3 - 2*b2 + 6) * q^89 + (-b3 - 3*b2 - b1) * q^91 + (-4*b3 - 4*b2 - 2*b1 - 4) * q^93 - q^95 + (-2*b3 - 3*b2 + 3*b1 - 1) * q^97 + (-3*b3 - 2*b2 - 5*b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{3} - 4 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10})$$ 4 * q + q^3 - 4 * q^5 - 5 * q^7 + 9 * q^9 $$4 q + q^{3} - 4 q^{5} - 5 q^{7} + 9 q^{9} - 6 q^{11} + 5 q^{13} - q^{15} - 5 q^{17} + 4 q^{19} - 3 q^{21} - 21 q^{23} + 4 q^{25} + q^{27} - q^{29} - 4 q^{31} - 14 q^{33} + 5 q^{35} + 10 q^{37} - 7 q^{39} - 2 q^{41} + 6 q^{43} - 9 q^{45} - 24 q^{47} + 5 q^{49} + 13 q^{51} - 5 q^{53} + 6 q^{55} + q^{57} - 11 q^{59} - 6 q^{61} - 38 q^{63} - 5 q^{65} + 19 q^{67} - 7 q^{69} - 2 q^{71} + 5 q^{73} + q^{75} + 8 q^{77} + 4 q^{79} - 16 q^{81} - 10 q^{83} + 5 q^{85} - 31 q^{87} + 20 q^{89} - 5 q^{91} - 26 q^{93} - 4 q^{95} - 6 q^{97} - 14 q^{99}+O(q^{100})$$ 4 * q + q^3 - 4 * q^5 - 5 * q^7 + 9 * q^9 - 6 * q^11 + 5 * q^13 - q^15 - 5 * q^17 + 4 * q^19 - 3 * q^21 - 21 * q^23 + 4 * q^25 + q^27 - q^29 - 4 * q^31 - 14 * q^33 + 5 * q^35 + 10 * q^37 - 7 * q^39 - 2 * q^41 + 6 * q^43 - 9 * q^45 - 24 * q^47 + 5 * q^49 + 13 * q^51 - 5 * q^53 + 6 * q^55 + q^57 - 11 * q^59 - 6 * q^61 - 38 * q^63 - 5 * q^65 + 19 * q^67 - 7 * q^69 - 2 * q^71 + 5 * q^73 + q^75 + 8 * q^77 + 4 * q^79 - 16 * q^81 - 10 * q^83 + 5 * q^85 - 31 * q^87 + 20 * q^89 - 5 * q^91 - 26 * q^93 - 4 * q^95 - 6 * q^97 - 14 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 10x^{2} + 8x + 18$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5 $$\beta_{3}$$ $$=$$ $$\nu^{3} + \nu^{2} - 7\nu - 5$$ v^3 + v^2 - 7*v - 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ b2 + 5 $$\nu^{3}$$ $$=$$ $$\beta_{3} - \beta_{2} + 7\beta_1$$ b3 - b2 + 7*b1

## 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
 −2.78678 −1.09502 2.19719 2.68461
0 −2.78678 0 −1.00000 0 −3.76616 0 4.76616 0
1.2 0 −1.09502 0 −1.00000 0 2.80094 0 −1.80094 0
1.3 0 2.19719 0 −1.00000 0 −0.827631 0 1.82763 0
1.4 0 2.68461 0 −1.00000 0 −3.20715 0 4.20715 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.s yes 4
4.b odd 2 1 3040.2.a.q 4
8.b even 2 1 6080.2.a.ce 4
8.d odd 2 1 6080.2.a.cg 4

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

 $$T_{3}^{4} - T_{3}^{3} - 10T_{3}^{2} + 8T_{3} + 18$$ T3^4 - T3^3 - 10*T3^2 + 8*T3 + 18 $$T_{7}^{4} + 5T_{7}^{3} - 4T_{7}^{2} - 40T_{7} - 28$$ T7^4 + 5*T7^3 - 4*T7^2 - 40*T7 - 28 $$T_{11}^{4} + 6T_{11}^{3} - 8T_{11}^{2} - 28T_{11} + 32$$ T11^4 + 6*T11^3 - 8*T11^2 - 28*T11 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{3} + \cdots + 18$$
$5$ $$(T + 1)^{4}$$
$7$ $$T^{4} + 5 T^{3} + \cdots - 28$$
$11$ $$T^{4} + 6 T^{3} + \cdots + 32$$
$13$ $$T^{4} - 5 T^{3} + \cdots - 86$$
$17$ $$T^{4} + 5 T^{3} + \cdots - 168$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} + 21 T^{3} + \cdots + 324$$
$29$ $$T^{4} + T^{3} + \cdots + 72$$
$31$ $$T^{4} + 4 T^{3} + \cdots + 752$$
$37$ $$T^{4} - 10 T^{3} + \cdots - 44$$
$41$ $$T^{4} + 2 T^{3} + \cdots + 3456$$
$43$ $$T^{4} - 6 T^{3} + \cdots + 1048$$
$47$ $$T^{4} + 24 T^{3} + \cdots - 1376$$
$53$ $$T^{4} + 5 T^{3} + \cdots + 726$$
$59$ $$T^{4} + 11 T^{3} + \cdots + 2312$$
$61$ $$T^{4} + 6 T^{3} + \cdots + 32$$
$67$ $$T^{4} - 19 T^{3} + \cdots - 1778$$
$71$ $$T^{4} + 2 T^{3} + \cdots + 64$$
$73$ $$T^{4} - 5 T^{3} + \cdots - 872$$
$79$ $$T^{4} - 4 T^{3} + \cdots + 16$$
$83$ $$T^{4} + 10 T^{3} + \cdots + 648$$
$89$ $$T^{4} - 20 T^{3} + \cdots + 432$$
$97$ $$T^{4} + 6 T^{3} + \cdots + 3108$$