# Properties

 Label 3040.2.a.x Level $3040$ Weight $2$ Character orbit 3040.a Self dual yes Analytic conductor $24.275$ Analytic rank $1$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.387268.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2$$ x^5 - x^4 - 7*x^3 + 4*x^2 + 12*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{3} - q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{4} - \beta_{3} - \beta_{2} + 2) q^{9}+O(q^{10})$$ q + (b1 + 1) * q^3 - q^5 + (-b4 - 1) * q^7 + (b4 - b3 - b2 + 2) * q^9 $$q + (\beta_1 + 1) q^{3} - q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{4} - \beta_{3} - \beta_{2} + 2) q^{9} + (\beta_{4} + \beta_{2} - \beta_1) q^{11} + ( - \beta_{4} + \beta_{3} - 1) q^{13} + ( - \beta_1 - 1) q^{15} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{17} - q^{19} + ( - 2 \beta_{4} + \beta_{2} - \beta_1 - 3) q^{21} + (\beta_{4} - 2 \beta_{2} - 2 \beta_1 - 1) q^{23} + q^{25} + (2 \beta_{4} - \beta_{3} - 4 \beta_{2} + \cdots + 5) q^{27}+ \cdots + ( - \beta_{4} + \beta_{2} - 3 \beta_1 - 4) q^{99}+O(q^{100})$$ q + (b1 + 1) * q^3 - q^5 + (-b4 - 1) * q^7 + (b4 - b3 - b2 + 2) * q^9 + (b4 + b2 - b1) * q^11 + (-b4 + b3 - 1) * q^13 + (-b1 - 1) * q^15 + (b3 + 2*b2 + b1 - 3) * q^17 - q^19 + (-2*b4 + b2 - b1 - 3) * q^21 + (b4 - 2*b2 - 2*b1 - 1) * q^23 + q^25 + (2*b4 - b3 - 4*b2 - b1 + 5) * q^27 + (2*b4 - 2*b3 - b2 - 3*b1 - 1) * q^29 + (2*b3 - 2) * q^31 + (2*b3 + 2*b2 + 2*b1 - 4) * q^33 + (b4 + 1) * q^35 + (-b4 + b3 - 3*b2 + b1) * q^37 + (-b4 + 2*b2 - 2*b1 - 5) * q^39 + (-2*b1 - 2) * q^41 + (-b4 - 2*b3 - b2 - 3*b1 + 2) * q^43 + (-b4 + b3 + b2 - 2) * q^45 + (b4 + 3*b2 + b1 - 4) * q^47 + (-b3 + 3*b1 + 2) * q^49 + (b3 + 4*b2 - 3*b1 - 5) * q^51 + (-b4 - b3 + 4*b2 + 2*b1 - 5) * q^53 + (-b4 - b2 + b1) * q^55 + (-b1 - 1) * q^57 + (-b3 - 3*b1 + 1) * q^59 + (-b4 + b2 + b1) * q^61 + (-3*b4 + 2*b3 + 5*b2 - b1 - 10) * q^63 + (b4 - b3 + 1) * q^65 + (-2*b4 + 2*b2 + b1 + 1) * q^67 + (2*b4 - 3*b2 - b1 - 3) * q^69 + (4*b4 - 4*b3 - 2*b1 + 4) * q^71 + (b3 - 2*b2 - b1 - 5) * q^73 + (b1 + 1) * q^75 + (2*b4 - 4*b2 - 2*b1 - 4) * q^77 + (2*b3 - 2*b2 - 2) * q^79 + (3*b4 - 7*b2 + 3*b1 + 9) * q^81 + (-3*b4 + 2*b3 - 3*b2 - b1 + 2) * q^83 + (-b3 - 2*b2 - b1 + 3) * q^85 + (2*b3 - 3*b2 + 3*b1 - 3) * q^87 + (2*b2 + 4*b1 - 6) * q^89 + (2*b4 - b3 - 2*b2 + b1 + 7) * q^91 + (2*b4 + 2*b2 - 4*b1 - 6) * q^93 + q^95 + (-b4 - 3*b3 - b2 - 3*b1 - 4) * q^97 + (-b4 + b2 - 3*b1 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 4 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10})$$ 5 * q + 4 * q^3 - 5 * q^5 - 4 * q^7 + 7 * q^9 $$5 q + 4 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9} + 2 q^{11} - 4 q^{13} - 4 q^{15} - 12 q^{17} - 5 q^{19} - 10 q^{21} - 8 q^{23} + 5 q^{25} + 16 q^{27} - 6 q^{29} - 10 q^{31} - 18 q^{33} + 4 q^{35} - 6 q^{37} - 18 q^{39} - 8 q^{41} + 12 q^{43} - 7 q^{45} - 16 q^{47} + 7 q^{49} - 14 q^{51} - 18 q^{53} - 2 q^{55} - 4 q^{57} + 8 q^{59} + 2 q^{61} - 36 q^{63} + 4 q^{65} + 10 q^{67} - 22 q^{69} + 18 q^{71} - 28 q^{73} + 4 q^{75} - 28 q^{77} - 14 q^{79} + 25 q^{81} + 8 q^{83} + 12 q^{85} - 24 q^{87} - 30 q^{89} + 28 q^{91} - 24 q^{93} + 5 q^{95} - 18 q^{97} - 14 q^{99}+O(q^{100})$$ 5 * q + 4 * q^3 - 5 * q^5 - 4 * q^7 + 7 * q^9 + 2 * q^11 - 4 * q^13 - 4 * q^15 - 12 * q^17 - 5 * q^19 - 10 * q^21 - 8 * q^23 + 5 * q^25 + 16 * q^27 - 6 * q^29 - 10 * q^31 - 18 * q^33 + 4 * q^35 - 6 * q^37 - 18 * q^39 - 8 * q^41 + 12 * q^43 - 7 * q^45 - 16 * q^47 + 7 * q^49 - 14 * q^51 - 18 * q^53 - 2 * q^55 - 4 * q^57 + 8 * q^59 + 2 * q^61 - 36 * q^63 + 4 * q^65 + 10 * q^67 - 22 * q^69 + 18 * q^71 - 28 * q^73 + 4 * q^75 - 28 * q^77 - 14 * q^79 + 25 * q^81 + 8 * q^83 + 12 * q^85 - 24 * q^87 - 30 * q^89 + 28 * q^91 - 24 * q^93 + 5 * q^95 - 18 * q^97 - 14 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu + 2$$ v^3 - v^2 - 3*v + 2 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 5\nu - 2$$ -v^3 + v^2 + 5*v - 2 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 3\nu^{2} + 6\nu - 1$$ v^4 - 2*v^3 - 3*v^2 + 6*v - 1
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 2\beta _1 + 6 ) / 2$$ (b3 + b2 + 2*b1 + 6) / 2 $$\nu^{3}$$ $$=$$ $$2\beta_{3} + 3\beta_{2} + \beta _1 + 1$$ 2*b3 + 3*b2 + b1 + 1 $$\nu^{4}$$ $$=$$ $$( 2\beta_{4} + 5\beta_{3} + 9\beta_{2} + 10\beta _1 + 24 ) / 2$$ (2*b4 + 5*b3 + 9*b2 + 10*b1 + 24) / 2

## 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
 0.160536 1.81079 2.42170 −1.50372 −1.88930
0 −2.13476 0 −1.00000 0 −0.878290 0 1.55722 0
1.2 0 −0.531822 0 −1.00000 0 0.0955795 0 −2.71717 0
1.3 0 1.44292 0 −1.00000 0 −2.92540 0 −0.917992 0
1.4 0 1.76491 0 −1.00000 0 3.89255 0 0.114895 0
1.5 0 3.45876 0 −1.00000 0 −4.18444 0 8.96304 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.x yes 5
4.b odd 2 1 3040.2.a.u 5
8.b even 2 1 6080.2.a.ci 5
8.d odd 2 1 6080.2.a.cl 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.a.u 5 4.b odd 2 1
3040.2.a.x yes 5 1.a even 1 1 trivial
6080.2.a.ci 5 8.b even 2 1
6080.2.a.cl 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3040))$$:

 $$T_{3}^{5} - 4T_{3}^{4} - 3T_{3}^{3} + 20T_{3}^{2} - 8T_{3} - 10$$ T3^5 - 4*T3^4 - 3*T3^3 + 20*T3^2 - 8*T3 - 10 $$T_{7}^{5} + 4T_{7}^{4} - 13T_{7}^{3} - 60T_{7}^{2} - 36T_{7} + 4$$ T7^5 + 4*T7^4 - 13*T7^3 - 60*T7^2 - 36*T7 + 4 $$T_{11}^{5} - 2T_{11}^{4} - 32T_{11}^{3} + 44T_{11}^{2} + 208T_{11} + 64$$ T11^5 - 2*T11^4 - 32*T11^3 + 44*T11^2 + 208*T11 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - 4 T^{4} + \cdots - 10$$
$5$ $$(T + 1)^{5}$$
$7$ $$T^{5} + 4 T^{4} + \cdots + 4$$
$11$ $$T^{5} - 2 T^{4} + \cdots + 64$$
$13$ $$T^{5} + 4 T^{4} + \cdots + 2$$
$17$ $$T^{5} + 12 T^{4} + \cdots - 1192$$
$19$ $$(T + 1)^{5}$$
$23$ $$T^{5} + 8 T^{4} + \cdots - 772$$
$29$ $$T^{5} + 6 T^{4} + \cdots + 5480$$
$31$ $$T^{5} + 10 T^{4} + \cdots + 1888$$
$37$ $$T^{5} + 6 T^{4} + \cdots + 2216$$
$41$ $$T^{5} + 8 T^{4} + \cdots + 320$$
$43$ $$T^{5} - 12 T^{4} + \cdots + 1328$$
$47$ $$T^{5} + 16 T^{4} + \cdots - 6592$$
$53$ $$T^{5} + 18 T^{4} + \cdots + 28486$$
$59$ $$T^{5} - 8 T^{4} + \cdots - 392$$
$61$ $$T^{5} - 2 T^{4} + \cdots + 64$$
$67$ $$T^{5} - 10 T^{4} + \cdots + 50$$
$71$ $$T^{5} - 18 T^{4} + \cdots - 22016$$
$73$ $$T^{5} + 28 T^{4} + \cdots - 5480$$
$79$ $$T^{5} + 14 T^{4} + \cdots + 3296$$
$83$ $$T^{5} - 8 T^{4} + \cdots + 5840$$
$89$ $$T^{5} + 30 T^{4} + \cdots - 736$$
$97$ $$T^{5} + 18 T^{4} + \cdots + 100552$$