# Properties

 Label 3024.2.t.d Level $3024$ Weight $2$ Character orbit 3024.t Analytic conductor $24.147$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(289,3024)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 2]))

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

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

chi := DirichletCharacter("3024.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.t (of order $$3$$, degree $$2$$, not 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: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + ( - 3 \zeta_{6} + 1) q^{7}+O(q^{10})$$ q + q^5 + (-3*z + 1) * q^7 $$q + q^{5} + ( - 3 \zeta_{6} + 1) q^{7} + 5 q^{11} + ( - 5 \zeta_{6} + 5) q^{13} + ( - 3 \zeta_{6} + 3) q^{17} + \zeta_{6} q^{19} + 3 q^{23} - 4 q^{25} - \zeta_{6} q^{29} + ( - 3 \zeta_{6} + 1) q^{35} - 3 \zeta_{6} q^{37} + (5 \zeta_{6} - 5) q^{41} - \zeta_{6} q^{43} + (3 \zeta_{6} - 8) q^{49} + (9 \zeta_{6} - 9) q^{53} + 5 q^{55} + ( - 14 \zeta_{6} + 14) q^{61} + ( - 5 \zeta_{6} + 5) q^{65} + 4 \zeta_{6} q^{67} - 12 q^{71} + (3 \zeta_{6} - 3) q^{73} + ( - 15 \zeta_{6} + 5) q^{77} + ( - 8 \zeta_{6} + 8) q^{79} + 9 \zeta_{6} q^{83} + ( - 3 \zeta_{6} + 3) q^{85} - 13 \zeta_{6} q^{89} + ( - 5 \zeta_{6} - 10) q^{91} + \zeta_{6} q^{95} + 9 \zeta_{6} q^{97} +O(q^{100})$$ q + q^5 + (-3*z + 1) * q^7 + 5 * q^11 + (-5*z + 5) * q^13 + (-3*z + 3) * q^17 + z * q^19 + 3 * q^23 - 4 * q^25 - z * q^29 + (-3*z + 1) * q^35 - 3*z * q^37 + (5*z - 5) * q^41 - z * q^43 + (3*z - 8) * q^49 + (9*z - 9) * q^53 + 5 * q^55 + (-14*z + 14) * q^61 + (-5*z + 5) * q^65 + 4*z * q^67 - 12 * q^71 + (3*z - 3) * q^73 + (-15*z + 5) * q^77 + (-8*z + 8) * q^79 + 9*z * q^83 + (-3*z + 3) * q^85 - 13*z * q^89 + (-5*z - 10) * q^91 + z * q^95 + 9*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 - q^7 $$2 q + 2 q^{5} - q^{7} + 10 q^{11} + 5 q^{13} + 3 q^{17} + q^{19} + 6 q^{23} - 8 q^{25} - q^{29} - q^{35} - 3 q^{37} - 5 q^{41} - q^{43} - 13 q^{49} - 9 q^{53} + 10 q^{55} + 14 q^{61} + 5 q^{65} + 4 q^{67} - 24 q^{71} - 3 q^{73} - 5 q^{77} + 8 q^{79} + 9 q^{83} + 3 q^{85} - 13 q^{89} - 25 q^{91} + q^{95} + 9 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - q^7 + 10 * q^11 + 5 * q^13 + 3 * q^17 + q^19 + 6 * q^23 - 8 * q^25 - q^29 - q^35 - 3 * q^37 - 5 * q^41 - q^43 - 13 * q^49 - 9 * q^53 + 10 * q^55 + 14 * q^61 + 5 * q^65 + 4 * q^67 - 24 * q^71 - 3 * q^73 - 5 * q^77 + 8 * q^79 + 9 * q^83 + 3 * q^85 - 13 * q^89 - 25 * q^91 + q^95 + 9 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 1.00000 0 −0.500000 + 2.59808i 0 0 0
1873.1 0 0 0 1.00000 0 −0.500000 2.59808i 0 0 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
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.t.d 2
3.b odd 2 1 1008.2.t.d 2
4.b odd 2 1 189.2.g.a 2
7.c even 3 1 3024.2.q.b 2
9.c even 3 1 3024.2.q.b 2
9.d odd 6 1 1008.2.q.c 2
12.b even 2 1 63.2.g.a 2
21.h odd 6 1 1008.2.q.c 2
28.d even 2 1 1323.2.g.a 2
28.f even 6 1 1323.2.f.b 2
28.f even 6 1 1323.2.h.a 2
28.g odd 6 1 189.2.h.a 2
28.g odd 6 1 1323.2.f.a 2
36.f odd 6 1 189.2.h.a 2
36.f odd 6 1 567.2.e.b 2
36.h even 6 1 63.2.h.a yes 2
36.h even 6 1 567.2.e.a 2
63.g even 3 1 inner 3024.2.t.d 2
63.n odd 6 1 1008.2.t.d 2
84.h odd 2 1 441.2.g.a 2
84.j odd 6 1 441.2.f.a 2
84.j odd 6 1 441.2.h.a 2
84.n even 6 1 63.2.h.a yes 2
84.n even 6 1 441.2.f.b 2
252.n even 6 1 1323.2.g.a 2
252.n even 6 1 3969.2.a.a 1
252.o even 6 1 63.2.g.a 2
252.o even 6 1 3969.2.a.d 1
252.r odd 6 1 441.2.f.a 2
252.s odd 6 1 441.2.h.a 2
252.u odd 6 1 567.2.e.b 2
252.u odd 6 1 1323.2.f.a 2
252.bb even 6 1 441.2.f.b 2
252.bb even 6 1 567.2.e.a 2
252.bi even 6 1 1323.2.h.a 2
252.bj even 6 1 1323.2.f.b 2
252.bl odd 6 1 189.2.g.a 2
252.bl odd 6 1 3969.2.a.c 1
252.bn odd 6 1 441.2.g.a 2
252.bn odd 6 1 3969.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 12.b even 2 1
63.2.g.a 2 252.o even 6 1
63.2.h.a yes 2 36.h even 6 1
63.2.h.a yes 2 84.n even 6 1
189.2.g.a 2 4.b odd 2 1
189.2.g.a 2 252.bl odd 6 1
189.2.h.a 2 28.g odd 6 1
189.2.h.a 2 36.f odd 6 1
441.2.f.a 2 84.j odd 6 1
441.2.f.a 2 252.r odd 6 1
441.2.f.b 2 84.n even 6 1
441.2.f.b 2 252.bb even 6 1
441.2.g.a 2 84.h odd 2 1
441.2.g.a 2 252.bn odd 6 1
441.2.h.a 2 84.j odd 6 1
441.2.h.a 2 252.s odd 6 1
567.2.e.a 2 36.h even 6 1
567.2.e.a 2 252.bb even 6 1
567.2.e.b 2 36.f odd 6 1
567.2.e.b 2 252.u odd 6 1
1008.2.q.c 2 9.d odd 6 1
1008.2.q.c 2 21.h odd 6 1
1008.2.t.d 2 3.b odd 2 1
1008.2.t.d 2 63.n odd 6 1
1323.2.f.a 2 28.g odd 6 1
1323.2.f.a 2 252.u odd 6 1
1323.2.f.b 2 28.f even 6 1
1323.2.f.b 2 252.bj even 6 1
1323.2.g.a 2 28.d even 2 1
1323.2.g.a 2 252.n even 6 1
1323.2.h.a 2 28.f even 6 1
1323.2.h.a 2 252.bi even 6 1
3024.2.q.b 2 7.c even 3 1
3024.2.q.b 2 9.c even 3 1
3024.2.t.d 2 1.a even 1 1 trivial
3024.2.t.d 2 63.g even 3 1 inner
3969.2.a.a 1 252.n even 6 1
3969.2.a.c 1 252.bl odd 6 1
3969.2.a.d 1 252.o even 6 1
3969.2.a.f 1 252.bn odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3024, [\chi])$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{11} - 5$$ T11 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + T + 7$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2} - 5T + 25$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} - T + 1$$
$23$ $$(T - 3)^{2}$$
$29$ $$T^{2} + T + 1$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$T^{2} + 5T + 25$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 9T + 81$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 14T + 196$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 3T + 9$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$T^{2} - 9T + 81$$
$89$ $$T^{2} + 13T + 169$$
$97$ $$T^{2} - 9T + 81$$