# Properties

 Label 3024.2.r.g Level $3024$ Weight $2$ Character orbit 3024.r Analytic conductor $24.147$ Analytic rank $0$ Dimension $6$ 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(1009,3024)]

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, 2, 0]))

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

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

chi := DirichletCharacter("3024.1009");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.r (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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 19*x^2 - 12*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{4} - \beta_{3} - \beta_{2}) q^{5} + (\beta_{4} - 1) q^{7}+O(q^{10})$$ q + (-2*b4 - b3 - b2) * q^5 + (b4 - 1) * q^7 $$q + ( - 2 \beta_{4} - \beta_{3} - \beta_{2}) q^{5} + (\beta_{4} - 1) q^{7} + (2 \beta_{5} - \beta_{4} + \beta_{2} + 1) q^{11} - \beta_{4} q^{13} + ( - \beta_{3} + \beta_1 + 4) q^{17} + (\beta_{3} - \beta_1 + 1) q^{19} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{23} + ( - \beta_{5} + 3 \beta_{4} + 2 \beta_{2} - 3) q^{25} - \beta_{5} q^{29} + (2 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{31} + (\beta_{3} + 2) q^{35} + 3 \beta_{3} q^{37} + ( - 7 \beta_{4} + \beta_{3} + \beta_{2}) q^{41} + (4 \beta_{5} + \beta_{2}) q^{43} + ( - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{2} + 3) q^{47} - \beta_{4} q^{49} + ( - 2 \beta_{3} - \beta_1 + 5) q^{53} + ( - \beta_{3} - 5 \beta_1) q^{55} + ( - \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{59} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2} + 1) q^{61} + (2 \beta_{4} + \beta_{2} - 2) q^{65} + ( - 7 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + 7 \beta_1) q^{67} + ( - 2 \beta_{3} - \beta_1 + 2) q^{71} + ( - 5 \beta_{3} - \beta_1 - 1) q^{73} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{77} + ( - 5 \beta_{5} - 3 \beta_{4} + \beta_{2} + 3) q^{79} + (\beta_{5} - 4 \beta_{4} + \beta_{2} + 4) q^{83} + (2 \beta_{5} - 5 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{85} + (\beta_{3} - 6 \beta_1 - 1) q^{89} + q^{91} + ( - 2 \beta_{5} - 5 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{95} + ( - 5 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 2) q^{97}+O(q^{100})$$ q + (-2*b4 - b3 - b2) * q^5 + (b4 - 1) * q^7 + (2*b5 - b4 + b2 + 1) * q^11 - b4 * q^13 + (-b3 + b1 + 4) * q^17 + (b3 - b1 + 1) * q^19 + (b5 - b4 - 2*b3 - 2*b2 - b1) * q^23 + (-b5 + 3*b4 + 2*b2 - 3) * q^25 - b5 * q^29 + (2*b5 - 2*b4 - b3 - b2 - 2*b1) * q^31 + (b3 + 2) * q^35 + 3*b3 * q^37 + (-7*b4 + b3 + b2) * q^41 + (4*b5 + b2) * q^43 + (-3*b5 - 3*b4 - 3*b2 + 3) * q^47 - b4 * q^49 + (-2*b3 - b1 + 5) * q^53 + (-b3 - 5*b1) * q^55 + (-b5 + 4*b4 + 2*b3 + 2*b2 + b1) * q^59 + (-b5 - b4 + 2*b2 + 1) * q^61 + (2*b4 + b2 - 2) * q^65 + (-7*b5 + 2*b4 - b3 - b2 + 7*b1) * q^67 + (-2*b3 - b1 + 2) * q^71 + (-5*b3 - b1 - 1) * q^73 + (-2*b5 + b4 - b3 - b2 + 2*b1) * q^77 + (-5*b5 - 3*b4 + b2 + 3) * q^79 + (b5 - 4*b4 + b2 + 4) * q^83 + (2*b5 - 5*b4 - 4*b3 - 4*b2 - 2*b1) * q^85 + (b3 - 6*b1 - 1) * q^89 + q^91 + (-2*b5 - 5*b4 - b3 - b2 + 2*b1) * q^95 + (-5*b5 + 2*b4 - 2*b2 - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 5 q^{5} - 3 q^{7}+O(q^{10})$$ 6 * q - 5 * q^5 - 3 * q^7 $$6 q - 5 q^{5} - 3 q^{7} + 2 q^{11} - 3 q^{13} + 24 q^{17} + 6 q^{19} - 6 q^{25} + q^{29} - 3 q^{31} + 10 q^{35} - 6 q^{37} - 22 q^{41} - 3 q^{43} + 9 q^{47} - 3 q^{49} + 36 q^{53} + 12 q^{55} + 9 q^{59} + 6 q^{61} - 5 q^{65} + 18 q^{71} + 6 q^{73} + 2 q^{77} + 15 q^{79} + 12 q^{83} - 9 q^{85} + 4 q^{89} + 6 q^{91} - 16 q^{95} - 3 q^{97}+O(q^{100})$$ 6 * q - 5 * q^5 - 3 * q^7 + 2 * q^11 - 3 * q^13 + 24 * q^17 + 6 * q^19 - 6 * q^25 + q^29 - 3 * q^31 + 10 * q^35 - 6 * q^37 - 22 * q^41 - 3 * q^43 + 9 * q^47 - 3 * q^49 + 36 * q^53 + 12 * q^55 + 9 * q^59 + 6 * q^61 - 5 * q^65 + 18 * q^71 + 6 * q^73 + 2 * q^77 + 15 * q^79 + 12 * q^83 - 9 * q^85 + 4 * q^89 + 6 * q^91 - 16 * q^95 - 3 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 2$$ v^2 - v + 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3$$ (-v^5 + v^4 - 8*v^3 + 5*v^2 - 18*v + 6) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3$$ v^4 - 2*v^3 + 6*v^2 - 5*v + 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3$$ (-2*v^5 + 5*v^4 - 16*v^3 + 19*v^2 - 21*v + 9) / 3 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3$$ (2*v^5 - 5*v^4 + 19*v^3 - 22*v^2 + 30*v - 9) / 3
 $$\nu$$ $$=$$ $$( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3$$ (-2*b5 - b4 - b3 - 2*b2 + b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3$$ (-2*b5 - b4 - b3 - 2*b2 + 4*b1 - 4) / 3 $$\nu^{3}$$ $$=$$ $$( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3$$ (7*b5 + 5*b4 + 2*b3 + 4*b2 + b1 - 10) / 3 $$\nu^{4}$$ $$=$$ $$( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3$$ (16*b5 + 11*b4 + 8*b3 + 10*b2 - 17*b1 + 5) / 3 $$\nu^{5}$$ $$=$$ $$( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3$$ (-14*b5 - 16*b4 + 5*b3 - 5*b2 - 23*b1 + 47) / 3

## Character values

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

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1009.1
 0.5 + 0.224437i 0.5 + 2.05195i 0.5 − 1.41036i 0.5 − 0.224437i 0.5 − 2.05195i 0.5 + 1.41036i
0 0 0 −1.79418 + 3.10761i 0 −0.500000 0.866025i 0 0 0
1009.2 0 0 0 −1.29679 + 2.24611i 0 −0.500000 0.866025i 0 0 0
1009.3 0 0 0 0.590972 1.02359i 0 −0.500000 0.866025i 0 0 0
2017.1 0 0 0 −1.79418 3.10761i 0 −0.500000 + 0.866025i 0 0 0
2017.2 0 0 0 −1.29679 2.24611i 0 −0.500000 + 0.866025i 0 0 0
2017.3 0 0 0 0.590972 + 1.02359i 0 −0.500000 + 0.866025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1009.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c 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.r.g 6
3.b odd 2 1 1008.2.r.k 6
4.b odd 2 1 189.2.f.a 6
9.c even 3 1 inner 3024.2.r.g 6
9.c even 3 1 9072.2.a.cd 3
9.d odd 6 1 1008.2.r.k 6
9.d odd 6 1 9072.2.a.bq 3
12.b even 2 1 63.2.f.b 6
28.d even 2 1 1323.2.f.c 6
28.f even 6 1 1323.2.g.b 6
28.f even 6 1 1323.2.h.e 6
28.g odd 6 1 1323.2.g.c 6
28.g odd 6 1 1323.2.h.d 6
36.f odd 6 1 189.2.f.a 6
36.f odd 6 1 567.2.a.g 3
36.h even 6 1 63.2.f.b 6
36.h even 6 1 567.2.a.d 3
84.h odd 2 1 441.2.f.d 6
84.j odd 6 1 441.2.g.d 6
84.j odd 6 1 441.2.h.b 6
84.n even 6 1 441.2.g.e 6
84.n even 6 1 441.2.h.c 6
252.n even 6 1 1323.2.h.e 6
252.o even 6 1 441.2.h.c 6
252.r odd 6 1 441.2.g.d 6
252.s odd 6 1 441.2.f.d 6
252.s odd 6 1 3969.2.a.m 3
252.u odd 6 1 1323.2.g.c 6
252.bb even 6 1 441.2.g.e 6
252.bi even 6 1 1323.2.f.c 6
252.bi even 6 1 3969.2.a.p 3
252.bj even 6 1 1323.2.g.b 6
252.bl odd 6 1 1323.2.h.d 6
252.bn odd 6 1 441.2.h.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 12.b even 2 1
63.2.f.b 6 36.h even 6 1
189.2.f.a 6 4.b odd 2 1
189.2.f.a 6 36.f odd 6 1
441.2.f.d 6 84.h odd 2 1
441.2.f.d 6 252.s odd 6 1
441.2.g.d 6 84.j odd 6 1
441.2.g.d 6 252.r odd 6 1
441.2.g.e 6 84.n even 6 1
441.2.g.e 6 252.bb even 6 1
441.2.h.b 6 84.j odd 6 1
441.2.h.b 6 252.bn odd 6 1
441.2.h.c 6 84.n even 6 1
441.2.h.c 6 252.o even 6 1
567.2.a.d 3 36.h even 6 1
567.2.a.g 3 36.f odd 6 1
1008.2.r.k 6 3.b odd 2 1
1008.2.r.k 6 9.d odd 6 1
1323.2.f.c 6 28.d even 2 1
1323.2.f.c 6 252.bi even 6 1
1323.2.g.b 6 28.f even 6 1
1323.2.g.b 6 252.bj even 6 1
1323.2.g.c 6 28.g odd 6 1
1323.2.g.c 6 252.u odd 6 1
1323.2.h.d 6 28.g odd 6 1
1323.2.h.d 6 252.bl odd 6 1
1323.2.h.e 6 28.f even 6 1
1323.2.h.e 6 252.n even 6 1
3024.2.r.g 6 1.a even 1 1 trivial
3024.2.r.g 6 9.c even 3 1 inner
3969.2.a.m 3 252.s odd 6 1
3969.2.a.p 3 252.bi even 6 1
9072.2.a.bq 3 9.d odd 6 1
9072.2.a.cd 3 9.c even 3 1

## Hecke kernels

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

 $$T_{5}^{6} + 5T_{5}^{5} + 23T_{5}^{4} + 32T_{5}^{3} + 59T_{5}^{2} - 22T_{5} + 121$$ T5^6 + 5*T5^5 + 23*T5^4 + 32*T5^3 + 59*T5^2 - 22*T5 + 121 $$T_{11}^{6} - 2T_{11}^{5} + 23T_{11}^{4} - 56T_{11}^{3} + 455T_{11}^{2} - 893T_{11} + 2209$$ T11^6 - 2*T11^5 + 23*T11^4 - 56*T11^3 + 455*T11^2 - 893*T11 + 2209

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 5 T^{5} + 23 T^{4} + 32 T^{3} + \cdots + 121$$
$7$ $$(T^{2} + T + 1)^{3}$$
$11$ $$T^{6} - 2 T^{5} + 23 T^{4} + \cdots + 2209$$
$13$ $$(T^{2} + T + 1)^{3}$$
$17$ $$(T^{3} - 12 T^{2} + 39 T - 27)^{2}$$
$19$ $$(T^{3} - 3 T^{2} - 6 T + 7)^{2}$$
$23$ $$T^{6} + 33 T^{4} - 18 T^{3} + 1089 T^{2} + \cdots + 81$$
$29$ $$T^{6} - T^{5} + 5 T^{4} + 2 T^{3} + 17 T^{2} + \cdots + 1$$
$31$ $$T^{6} + 3 T^{5} + 33 T^{4} - 126 T^{3} + \cdots + 729$$
$37$ $$(T^{3} + 3 T^{2} - 54 T + 81)^{2}$$
$41$ $$T^{6} + 22 T^{5} + 329 T^{4} + \cdots + 124609$$
$43$ $$T^{6} + 3 T^{5} + 75 T^{4} + \cdots + 14641$$
$47$ $$T^{6} - 9 T^{5} + 135 T^{4} + \cdots + 35721$$
$53$ $$(T^{3} - 18 T^{2} + 75 T - 9)^{2}$$
$59$ $$T^{6} - 9 T^{5} + 87 T^{4} + \cdots + 3969$$
$61$ $$T^{6} - 6 T^{5} + 57 T^{4} + \cdots + 4489$$
$67$ $$T^{6} + 207 T^{4} - 1366 T^{3} + \cdots + 466489$$
$71$ $$(T^{3} - 9 T^{2} - 6 T + 81)^{2}$$
$73$ $$(T^{3} - 3 T^{2} - 168 T - 243)^{2}$$
$79$ $$T^{6} - 15 T^{5} + 273 T^{4} + \cdots + 591361$$
$83$ $$T^{6} - 12 T^{5} + 105 T^{4} + \cdots + 729$$
$89$ $$(T^{3} - 2 T^{2} - 151 T - 379)^{2}$$
$97$ $$T^{6} + 3 T^{5} + 123 T^{4} + \cdots + 363609$$