# Properties

 Label 3040.2.a.w Level $3040$ Weight $2$ Character orbit 3040.a Self dual yes Analytic conductor $24.275$ Analytic rank $0$ 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: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.2363492.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 11x^{3} - 6x^{2} + 14x + 10$$ x^5 - x^4 - 11*x^3 - 6*x^2 + 14*x + 10 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + q^{5} - \beta_{2} q^{7} + ( - \beta_{3} + 4) q^{9}+O(q^{10})$$ q - b1 * q^3 + q^5 - b2 * q^7 + (-b3 + 4) * q^9 $$q - \beta_1 q^{3} + q^{5} - \beta_{2} q^{7} + ( - \beta_{3} + 4) q^{9} + ( - \beta_{4} + \beta_{3} + 2) q^{11} + ( - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{13} - \beta_1 q^{15} + ( - \beta_{4} + 1) q^{17} + q^{19} + (3 \beta_{4} - \beta_{3} - \beta_{2} + 2) q^{21} - \beta_{2} q^{23} + q^{25} + (\beta_{4} - 2 \beta_{3} - 2 \beta_1 + 1) q^{27} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{29} + ( - 2 \beta_{4} + 2 \beta_{3} - 2) q^{31} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{33} - \beta_{2} q^{35} + ( - 2 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{37} + (4 \beta_{4} + \beta_{2} - 2 \beta_1 - 4) q^{39} + (2 \beta_{3} + 2) q^{41} + ( - \beta_{4} - \beta_{3}) q^{43} + ( - \beta_{3} + 4) q^{45} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots - 6) q^{47}+ \cdots + ( - 5 \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{99}+O(q^{100})$$ q - b1 * q^3 + q^5 - b2 * q^7 + (-b3 + 4) * q^9 + (-b4 + b3 + 2) * q^11 + (-b4 - b2 + b1 + 1) * q^13 - b1 * q^15 + (-b4 + 1) * q^17 + q^19 + (3*b4 - b3 - b2 + 2) * q^21 - b2 * q^23 + q^25 + (b4 - 2*b3 - 2*b1 + 1) * q^27 + (-b4 - b3 + b2 + 2) * q^29 + (-2*b4 + 2*b3 - 2) * q^31 + (2*b3 + 2*b2 - 2*b1) * q^33 - b2 * q^35 + (-2*b4 + b3 + b1 + 1) * q^37 + (4*b4 + b2 - 2*b1 - 4) * q^39 + (2*b3 + 2) * q^41 + (-b4 - b3) * q^43 + (-b3 + 4) * q^45 + (-b4 + b3 + 2*b2 - 2*b1 - 6) * q^47 + (b4 + 2*b2 - 2*b1 + 2) * q^49 + (b4 + 2*b2 - 2*b1 + 1) * q^51 + (b4 - b2 + b1 - 3) * q^53 + (-b4 + b3 + 2) * q^55 - b1 * q^57 + (-b4 - 2*b3 - 2*b2 + 2*b1 + 3) * q^59 + (b4 + b3 + 2*b2) * q^61 + (b4 - 3*b3 - 4*b2) * q^63 + (-b4 - b2 + b1 + 1) * q^65 + (4*b4 - 3*b1 + 2) * q^67 + (3*b4 - b3 - b2 + 2) * q^69 - 2*b3 * q^71 + (b4 - 2*b3 - 2*b1 - 3) * q^73 - b1 * q^75 + (2*b4 + 2*b3 - 2*b2 + 2*b1) * q^77 + (-4*b4 + 2*b2 + 2*b1 - 2) * q^79 + (b4 - 3*b3 - 2*b2 - 2*b1 + 3) * q^81 + (3*b4 + 3*b3 - 4) * q^83 + (-b4 + 1) * q^85 + (-b4 - b3 + 3*b2 - 4*b1) * q^87 + (-2*b4 + 2*b3 - 2*b2 + 2*b1 + 6) * q^89 + (b4 + 2*b2 + 7) * q^91 + (4*b3 + 4*b2 + 2*b1) * q^93 + q^95 + (-b3 + 2*b2 - b1 - 3) * q^97 + (-5*b4 + b3 + 2*b2 + 2*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 19 q^{9}+O(q^{10})$$ 5 * q + 2 * q^3 + 5 * q^5 - 2 * q^7 + 19 * q^9 $$5 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 19 q^{9} + 10 q^{11} + 2 q^{15} + 4 q^{17} + 5 q^{19} + 10 q^{21} - 2 q^{23} + 5 q^{25} + 8 q^{27} + 10 q^{29} - 10 q^{31} + 10 q^{33} - 2 q^{35} + 2 q^{37} - 10 q^{39} + 12 q^{41} - 2 q^{43} + 19 q^{45} - 22 q^{47} + 19 q^{49} + 14 q^{51} - 18 q^{53} + 10 q^{55} + 2 q^{57} + 4 q^{59} + 6 q^{61} - 10 q^{63} + 20 q^{67} + 10 q^{69} - 2 q^{71} - 12 q^{73} + 2 q^{75} - 4 q^{77} - 14 q^{79} + 13 q^{81} - 14 q^{83} + 4 q^{85} + 12 q^{87} + 22 q^{89} + 40 q^{91} + 8 q^{93} + 5 q^{95} - 10 q^{97} + 6 q^{99}+O(q^{100})$$ 5 * q + 2 * q^3 + 5 * q^5 - 2 * q^7 + 19 * q^9 + 10 * q^11 + 2 * q^15 + 4 * q^17 + 5 * q^19 + 10 * q^21 - 2 * q^23 + 5 * q^25 + 8 * q^27 + 10 * q^29 - 10 * q^31 + 10 * q^33 - 2 * q^35 + 2 * q^37 - 10 * q^39 + 12 * q^41 - 2 * q^43 + 19 * q^45 - 22 * q^47 + 19 * q^49 + 14 * q^51 - 18 * q^53 + 10 * q^55 + 2 * q^57 + 4 * q^59 + 6 * q^61 - 10 * q^63 + 20 * q^67 + 10 * q^69 - 2 * q^71 - 12 * q^73 + 2 * q^75 - 4 * q^77 - 14 * q^79 + 13 * q^81 - 14 * q^83 + 4 * q^85 + 12 * q^87 + 22 * q^89 + 40 * q^91 + 8 * q^93 + 5 * q^95 - 10 * q^97 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 11x^{3} - 6x^{2} + 14x + 10$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{4} - 3\nu^{3} - 6\nu^{2} + 7\nu + 6$$ v^4 - 3*v^3 - 6*v^2 + 7*v + 6 $$\beta_{2}$$ $$=$$ $$-\nu^{4} + 2\nu^{3} + 9\nu^{2} - 4\nu - 10$$ -v^4 + 2*v^3 + 9*v^2 - 4*v - 10 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 9\nu^{2} + 2\nu + 11$$ v^4 - 2*v^3 - 9*v^2 + 2*v + 11 $$\beta_{4}$$ $$=$$ $$2\nu^{4} - 5\nu^{3} - 14\nu^{2} + 8\nu + 13$$ 2*v^4 - 5*v^3 - 14*v^2 + 8*v + 13
 $$\nu$$ $$=$$ $$( -\beta_{3} - \beta_{2} + 1 ) / 2$$ (-b3 - b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{4} - 3\beta_{3} - \beta_{2} - 2\beta _1 + 9 ) / 2$$ (2*b4 - 3*b3 - b2 - 2*b1 + 9) / 2 $$\nu^{3}$$ $$=$$ $$3\beta_{4} - 6\beta_{3} - 4\beta_{2} - 4\beta _1 + 11$$ 3*b4 - 6*b3 - 4*b2 - 4*b1 + 11 $$\nu^{4}$$ $$=$$ $$( 30\beta_{4} - 47\beta_{3} - 23\beta_{2} - 34\beta _1 + 101 ) / 2$$ (30*b4 - 47*b3 - 23*b2 - 34*b1 + 101) / 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
 1.19018 −1.82337 −0.778695 −1.50919 3.92107
0 −2.78093 0 1.00000 0 0.646760 0 4.73359 0
1.2 0 −2.52807 0 1.00000 0 −4.03789 0 3.39116 0
1.3 0 1.30486 0 1.00000 0 2.73996 0 −1.29735 0
1.4 0 2.73033 0 1.00000 0 −4.47310 0 4.45473 0
1.5 0 3.27382 0 1.00000 0 3.12428 0 7.71786 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.w yes 5
4.b odd 2 1 3040.2.a.v 5
8.b even 2 1 6080.2.a.cj 5
8.d odd 2 1 6080.2.a.ck 5

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

 $$T_{3}^{5} - 2T_{3}^{4} - 15T_{3}^{3} + 26T_{3}^{2} + 56T_{3} - 82$$ T3^5 - 2*T3^4 - 15*T3^3 + 26*T3^2 + 56*T3 - 82 $$T_{7}^{5} + 2T_{7}^{4} - 25T_{7}^{3} - 18T_{7}^{2} + 176T_{7} - 100$$ T7^5 + 2*T7^4 - 25*T7^3 - 18*T7^2 + 176*T7 - 100 $$T_{11}^{5} - 10T_{11}^{4} + 12T_{11}^{3} + 100T_{11}^{2} - 112T_{11} - 320$$ T11^5 - 10*T11^4 + 12*T11^3 + 100*T11^2 - 112*T11 - 320

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - 2 T^{4} + \cdots - 82$$
$5$ $$(T - 1)^{5}$$
$7$ $$T^{5} + 2 T^{4} + \cdots - 100$$
$11$ $$T^{5} - 10 T^{4} + \cdots - 320$$
$13$ $$T^{5} - 35 T^{3} + \cdots - 34$$
$17$ $$T^{5} - 4 T^{4} + \cdots - 8$$
$19$ $$(T - 1)^{5}$$
$23$ $$T^{5} + 2 T^{4} + \cdots - 100$$
$29$ $$T^{5} - 10 T^{4} + \cdots + 712$$
$31$ $$T^{5} + 10 T^{4} + \cdots + 32$$
$37$ $$T^{5} - 2 T^{4} + \cdots - 2056$$
$41$ $$T^{5} - 12 T^{4} + \cdots + 128$$
$43$ $$T^{5} + 2 T^{4} + \cdots - 1008$$
$47$ $$T^{5} + 22 T^{4} + \cdots - 43840$$
$53$ $$T^{5} + 18 T^{4} + \cdots + 2$$
$59$ $$T^{5} - 4 T^{4} + \cdots + 30200$$
$61$ $$T^{5} - 6 T^{4} + \cdots - 2816$$
$67$ $$T^{5} - 20 T^{4} + \cdots - 81334$$
$71$ $$T^{5} + 2 T^{4} + \cdots - 128$$
$73$ $$T^{5} + 12 T^{4} + \cdots - 136$$
$79$ $$T^{5} + 14 T^{4} + \cdots + 215776$$
$83$ $$T^{5} + 14 T^{4} + \cdots + 417232$$
$89$ $$T^{5} - 22 T^{4} + \cdots + 7264$$
$97$ $$T^{5} + 10 T^{4} + \cdots + 3160$$