# Properties

 Label 3040.2.a.e Level $3040$ Weight $2$ Character orbit 3040.a Self dual yes Analytic conductor $24.275$ Analytic rank $1$ 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] = [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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - q^{5} + (\beta + 2) q^{7} + (\beta + 1) q^{9} +O(q^{10})$$ q - b * q^3 - q^5 + (b + 2) * q^7 + (b + 1) * q^9 $$q - \beta q^{3} - q^{5} + (\beta + 2) q^{7} + (\beta + 1) q^{9} - 4 q^{11} + \beta q^{13} + \beta q^{15} + ( - \beta + 2) q^{17} - q^{19} + ( - 3 \beta - 4) q^{21} + ( - 3 \beta + 2) q^{23} + q^{25} + (\beta - 4) q^{27} + (\beta - 6) q^{29} - 2 \beta q^{31} + 4 \beta q^{33} + ( - \beta - 2) q^{35} + ( - 2 \beta + 4) q^{37} + ( - \beta - 4) q^{39} + (4 \beta - 2) q^{41} - 2 q^{43} + ( - \beta - 1) q^{45} + (2 \beta - 2) q^{47} + (5 \beta + 1) q^{49} + ( - \beta + 4) q^{51} + (\beta + 4) q^{53} + 4 q^{55} + \beta q^{57} + 5 \beta q^{59} + (2 \beta - 10) q^{61} + (4 \beta + 6) q^{63} - \beta q^{65} + ( - \beta - 8) q^{67} + (\beta + 12) q^{69} - 8 q^{71} + (5 \beta + 2) q^{73} - \beta q^{75} + ( - 4 \beta - 8) q^{77} + 2 \beta q^{79} - 7 q^{81} - 14 q^{83} + (\beta - 2) q^{85} + (5 \beta - 4) q^{87} - 10 q^{89} + (3 \beta + 4) q^{91} + (2 \beta + 8) q^{93} + q^{95} + ( - 2 \beta + 4) q^{97} + ( - 4 \beta - 4) q^{99} +O(q^{100})$$ q - b * q^3 - q^5 + (b + 2) * q^7 + (b + 1) * q^9 - 4 * q^11 + b * q^13 + b * q^15 + (-b + 2) * q^17 - q^19 + (-3*b - 4) * q^21 + (-3*b + 2) * q^23 + q^25 + (b - 4) * q^27 + (b - 6) * q^29 - 2*b * q^31 + 4*b * q^33 + (-b - 2) * q^35 + (-2*b + 4) * q^37 + (-b - 4) * q^39 + (4*b - 2) * q^41 - 2 * q^43 + (-b - 1) * q^45 + (2*b - 2) * q^47 + (5*b + 1) * q^49 + (-b + 4) * q^51 + (b + 4) * q^53 + 4 * q^55 + b * q^57 + 5*b * q^59 + (2*b - 10) * q^61 + (4*b + 6) * q^63 - b * q^65 + (-b - 8) * q^67 + (b + 12) * q^69 - 8 * q^71 + (5*b + 2) * q^73 - b * q^75 + (-4*b - 8) * q^77 + 2*b * q^79 - 7 * q^81 - 14 * q^83 + (b - 2) * q^85 + (5*b - 4) * q^87 - 10 * q^89 + (3*b + 4) * q^91 + (2*b + 8) * q^93 + q^95 + (-2*b + 4) * q^97 + (-4*b - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 3 * q^9 $$2 q - q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9} - 8 q^{11} + q^{13} + q^{15} + 3 q^{17} - 2 q^{19} - 11 q^{21} + q^{23} + 2 q^{25} - 7 q^{27} - 11 q^{29} - 2 q^{31} + 4 q^{33} - 5 q^{35} + 6 q^{37} - 9 q^{39} - 4 q^{43} - 3 q^{45} - 2 q^{47} + 7 q^{49} + 7 q^{51} + 9 q^{53} + 8 q^{55} + q^{57} + 5 q^{59} - 18 q^{61} + 16 q^{63} - q^{65} - 17 q^{67} + 25 q^{69} - 16 q^{71} + 9 q^{73} - q^{75} - 20 q^{77} + 2 q^{79} - 14 q^{81} - 28 q^{83} - 3 q^{85} - 3 q^{87} - 20 q^{89} + 11 q^{91} + 18 q^{93} + 2 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 3 * q^9 - 8 * q^11 + q^13 + q^15 + 3 * q^17 - 2 * q^19 - 11 * q^21 + q^23 + 2 * q^25 - 7 * q^27 - 11 * q^29 - 2 * q^31 + 4 * q^33 - 5 * q^35 + 6 * q^37 - 9 * q^39 - 4 * q^43 - 3 * q^45 - 2 * q^47 + 7 * q^49 + 7 * q^51 + 9 * q^53 + 8 * q^55 + q^57 + 5 * q^59 - 18 * q^61 + 16 * q^63 - q^65 - 17 * q^67 + 25 * q^69 - 16 * q^71 + 9 * q^73 - q^75 - 20 * q^77 + 2 * q^79 - 14 * q^81 - 28 * q^83 - 3 * q^85 - 3 * q^87 - 20 * q^89 + 11 * q^91 + 18 * q^93 + 2 * q^95 + 6 * q^97 - 12 * q^99

## 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
 2.56155 −1.56155
0 −2.56155 0 −1.00000 0 4.56155 0 3.56155 0
1.2 0 1.56155 0 −1.00000 0 0.438447 0 −0.561553 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.e 2
4.b odd 2 1 3040.2.a.h yes 2
8.b even 2 1 6080.2.a.bi 2
8.d odd 2 1 6080.2.a.bc 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 4$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 5T + 2$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} - T - 4$$
$17$ $$T^{2} - 3T - 2$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - T - 38$$
$29$ $$T^{2} + 11T + 26$$
$31$ $$T^{2} + 2T - 16$$
$37$ $$T^{2} - 6T - 8$$
$41$ $$T^{2} - 68$$
$43$ $$(T + 2)^{2}$$
$47$ $$T^{2} + 2T - 16$$
$53$ $$T^{2} - 9T + 16$$
$59$ $$T^{2} - 5T - 100$$
$61$ $$T^{2} + 18T + 64$$
$67$ $$T^{2} + 17T + 68$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} - 9T - 86$$
$79$ $$T^{2} - 2T - 16$$
$83$ $$(T + 14)^{2}$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} - 6T - 8$$