# Properties

 Label 3040.2.j.a Level $3040$ Weight $2$ Character orbit 3040.j Analytic conductor $24.275$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,2,Mod(2431,3040)]

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([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3040.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3040 = 2^{5} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3040.j (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$24.2745222145$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{5} + (\beta_{2} - \beta_1) q^{7} - 3 q^{9}+O(q^{10})$$ q + q^5 + (b2 - b1) * q^7 - 3 * q^9 $$q + q^{5} + (\beta_{2} - \beta_1) q^{7} - 3 q^{9} + ( - \beta_{2} + \beta_1) q^{11} + 2 q^{17} + (\beta_{2} + 2 \beta_1) q^{19} + ( - 3 \beta_{2} + 3 \beta_1) q^{23} + q^{25} - \beta_{3} q^{29} + ( - 2 \beta_{2} - 2 \beta_1) q^{31} + (\beta_{2} - \beta_1) q^{35} + 2 \beta_{3} q^{41} + ( - 3 \beta_{2} + 3 \beta_1) q^{43} - 3 q^{45} + ( - 3 \beta_{2} + 3 \beta_1) q^{47} + 3 q^{49} + 2 \beta_{3} q^{53} + ( - \beta_{2} + \beta_1) q^{55} + (\beta_{2} + \beta_1) q^{59} + 14 q^{61} + ( - 3 \beta_{2} + 3 \beta_1) q^{63} + (2 \beta_{2} + 2 \beta_1) q^{67} + (4 \beta_{2} + 4 \beta_1) q^{71} + 10 q^{73} + 4 q^{77} + ( - 4 \beta_{2} - 4 \beta_1) q^{79} + 9 q^{81} + ( - 3 \beta_{2} + 3 \beta_1) q^{83} + 2 q^{85} - 4 \beta_{3} q^{89} + (\beta_{2} + 2 \beta_1) q^{95} + \beta_{3} q^{97} + (3 \beta_{2} - 3 \beta_1) q^{99}+O(q^{100})$$ q + q^5 + (b2 - b1) * q^7 - 3 * q^9 + (-b2 + b1) * q^11 + 2 * q^17 + (b2 + 2*b1) * q^19 + (-3*b2 + 3*b1) * q^23 + q^25 - b3 * q^29 + (-2*b2 - 2*b1) * q^31 + (b2 - b1) * q^35 + 2*b3 * q^41 + (-3*b2 + 3*b1) * q^43 - 3 * q^45 + (-3*b2 + 3*b1) * q^47 + 3 * q^49 + 2*b3 * q^53 + (-b2 + b1) * q^55 + (b2 + b1) * q^59 + 14 * q^61 + (-3*b2 + 3*b1) * q^63 + (2*b2 + 2*b1) * q^67 + (4*b2 + 4*b1) * q^71 + 10 * q^73 + 4 * q^77 + (-4*b2 - 4*b1) * q^79 + 9 * q^81 + (-3*b2 + 3*b1) * q^83 + 2 * q^85 - 4*b3 * q^89 + (b2 + 2*b1) * q^95 + b3 * q^97 + (3*b2 - 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5} - 12 q^{9}+O(q^{10})$$ 4 * q + 4 * q^5 - 12 * q^9 $$4 q + 4 q^{5} - 12 q^{9} + 8 q^{17} + 4 q^{25} - 12 q^{45} + 12 q^{49} + 56 q^{61} + 40 q^{73} + 16 q^{77} + 36 q^{81} + 8 q^{85}+O(q^{100})$$ 4 * q + 4 * q^5 - 12 * q^9 + 8 * q^17 + 4 * q^25 - 12 * q^45 + 12 * q^49 + 56 * q^61 + 40 * q^73 + 16 * q^77 + 36 * q^81 + 8 * q^85

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8}$$ -v^3 - v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ -v^3 + v^2 + v $$\beta_{3}$$ $$=$$ $$2\zeta_{8}^{3} + 2\zeta_{8}$$ 2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta_1 ) / 4$$ (b3 + b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( \beta_{3} - \beta_{2} - \beta_1 ) / 4$$ (b3 - b2 - b1) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$1217$$ $$1921$$ $$2661$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2431.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
0 0 0 1.00000 0 2.00000i 0 −3.00000 0
2431.2 0 0 0 1.00000 0 2.00000i 0 −3.00000 0
2431.3 0 0 0 1.00000 0 2.00000i 0 −3.00000 0
2431.4 0 0 0 1.00000 0 2.00000i 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.2.j.a 4
4.b odd 2 1 inner 3040.2.j.a 4
19.b odd 2 1 inner 3040.2.j.a 4
76.d even 2 1 inner 3040.2.j.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.j.a 4 1.a even 1 1 trivial
3040.2.j.a 4 4.b odd 2 1 inner
3040.2.j.a 4 19.b odd 2 1 inner
3040.2.j.a 4 76.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(3040, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T - 1)^{4}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} + 4)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T - 2)^{4}$$
$19$ $$T^{4} - 34T^{2} + 361$$
$23$ $$(T^{2} + 36)^{2}$$
$29$ $$(T^{2} + 8)^{2}$$
$31$ $$(T^{2} - 32)^{2}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 32)^{2}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$(T^{2} + 36)^{2}$$
$53$ $$(T^{2} + 32)^{2}$$
$59$ $$(T^{2} - 8)^{2}$$
$61$ $$(T - 14)^{4}$$
$67$ $$(T^{2} - 32)^{2}$$
$71$ $$(T^{2} - 128)^{2}$$
$73$ $$(T - 10)^{4}$$
$79$ $$(T^{2} - 128)^{2}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$(T^{2} + 128)^{2}$$
$97$ $$(T^{2} + 8)^{2}$$