# Properties

 Label 3024.2.t.h Level $3024$ Weight $2$ Character orbit 3024.t 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(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: $$6$$ 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 126) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{5} + ( - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{7}+O(q^{10})$$ q - b3 * q^5 + (-b5 - b2 + b1 + 1) * q^7 $$q - \beta_{3} q^{5} + ( - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{7} - \beta_{3} q^{11} + (2 \beta_{5} - 3 \beta_{4} + \beta_{2} + 3) q^{13} + (2 \beta_{5} - 2 \beta_{4} + 2) q^{17} + ( - 3 \beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{19} + ( - \beta_1 + 2) q^{23} + ( - 2 \beta_{3} + \beta_1 - 1) q^{25} + (3 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{29} + ( - \beta_{5} - 6 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{31} + (3 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{35} + \beta_{4} q^{37} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} - 1) q^{41} + (3 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{43} + (3 \beta_{5} + 3 \beta_{4} + 3 \beta_{2} - 3) q^{47} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1) q^{49} + (\beta_{5} + 5 \beta_{4} + \beta_{2} - 5) q^{53} + ( - 2 \beta_{3} + \beta_1 + 4) q^{55} + ( - \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{59} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_{2} + 3) q^{61} + ( - \beta_{5} - 2 \beta_{4} + 5 \beta_{2} + 2) q^{65} + (\beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - \beta_1) q^{67} + ( - 2 \beta_{3} + 7 \beta_1 + 4) q^{71} + (\beta_{5} - 8 \beta_{4} - 4 \beta_{2} + 8) q^{73} + (3 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{77} + (4 \beta_{5} - \beta_{2}) q^{79} + (2 \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{83} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 2) q^{85} + (4 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1) q^{89} + (3 \beta_{5} - \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - \beta_1 + 8) q^{91} + (3 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{95} + (2 \beta_{5} + 8 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{97}+O(q^{100})$$ q - b3 * q^5 + (-b5 - b2 + b1 + 1) * q^7 - b3 * q^11 + (2*b5 - 3*b4 + b2 + 3) * q^13 + (2*b5 - 2*b4 + 2) * q^17 + (-3*b5 + 2*b4 + 3*b1) * q^19 + (-b1 + 2) * q^23 + (-2*b3 + b1 - 1) * q^25 + (3*b5 - 2*b3 - 2*b2 - 3*b1) * q^29 + (-b5 - 6*b4 + b3 + b2 + b1) * q^31 + (3*b4 - b3 - 2*b2 - b1 - 4) * q^35 + b4 * q^37 + (-b5 + b4 + 2*b2 - 1) * q^41 + (3*b5 + 2*b4 + 3*b3 + 3*b2 - 3*b1) * q^43 + (3*b5 + 3*b4 + 3*b2 - 3) * q^47 + (-b4 + 2*b3 + b2 + 3*b1) * q^49 + (b5 + 5*b4 + b2 - 5) * q^53 + (-2*b3 + b1 + 4) * q^55 + (-b5 - 5*b4 - 2*b3 - 2*b2 + b1) * q^59 + (-b5 - 3*b4 - 2*b2 + 3) * q^61 + (-b5 - 2*b4 + 5*b2 + 2) * q^65 + (b5 - 2*b4 - 4*b3 - 4*b2 - b1) * q^67 + (-2*b3 + 7*b1 + 4) * q^71 + (b5 - 8*b4 - 4*b2 + 8) * q^73 + (3*b4 - b3 - 2*b2 - b1 - 4) * q^77 + (4*b5 - b2) * q^79 + (2*b5 + b4 + 3*b3 + 3*b2 - 2*b1) * q^83 + (-2*b5 + 2*b4 + 2*b2 - 2) * q^85 + (4*b5 + 2*b4 + b3 + b2 - 4*b1) * q^89 + (3*b5 - b4 - 3*b3 - 4*b2 - b1 + 8) * q^91 + (3*b5 - 3*b4 - 2*b3 - 2*b2 - 3*b1) * q^95 + (2*b5 + 8*b4 - 2*b3 - 2*b2 - 2*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{5} + 4 q^{7}+O(q^{10})$$ 6 * q + 2 * q^5 + 4 * q^7 $$6 q + 2 q^{5} + 4 q^{7} + 2 q^{11} + 8 q^{13} + 4 q^{17} + 3 q^{19} + 14 q^{23} - 4 q^{25} + 5 q^{29} - 20 q^{31} - 13 q^{35} + 3 q^{37} + 6 q^{43} - 9 q^{47} - 12 q^{49} - 15 q^{53} + 26 q^{55} - 14 q^{59} + 8 q^{61} + 12 q^{65} - q^{67} + 14 q^{71} + 19 q^{73} - 13 q^{77} - 5 q^{79} + 2 q^{83} - 2 q^{85} + 9 q^{89} + 46 q^{91} - 4 q^{95} + 28 q^{97}+O(q^{100})$$ 6 * q + 2 * q^5 + 4 * q^7 + 2 * q^11 + 8 * q^13 + 4 * q^17 + 3 * q^19 + 14 * q^23 - 4 * q^25 + 5 * q^29 - 20 * q^31 - 13 * q^35 + 3 * q^37 + 6 * q^43 - 9 * q^47 - 12 * q^49 - 15 * q^53 + 26 * q^55 - 14 * q^59 + 8 * q^61 + 12 * q^65 - q^67 + 14 * q^71 + 19 * q^73 - 13 * q^77 - 5 * q^79 + 2 * q^83 - 2 * q^85 + 9 * q^89 + 46 * q^91 - 4 * q^95 + 28 * 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 $$\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$$.

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

## 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 + 0.224437i 0.5 + 2.05195i 0.5 − 1.41036i 0.5 − 0.224437i 0.5 − 2.05195i 0.5 + 1.41036i
0 0 0 −1.58836 0 2.64400 0.0963576i 0 0 0
289.2 0 0 0 −0.593579 0 0.0665372 + 2.64491i 0 0 0
289.3 0 0 0 3.18194 0 −0.710533 2.54856i 0 0 0
1873.1 0 0 0 −1.58836 0 2.64400 + 0.0963576i 0 0 0
1873.2 0 0 0 −0.593579 0 0.0665372 2.64491i 0 0 0
1873.3 0 0 0 3.18194 0 −0.710533 + 2.54856i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.h 6
3.b odd 2 1 1008.2.t.h 6
4.b odd 2 1 378.2.h.c 6
7.c even 3 1 3024.2.q.g 6
9.c even 3 1 3024.2.q.g 6
9.d odd 6 1 1008.2.q.g 6
12.b even 2 1 126.2.h.d yes 6
21.h odd 6 1 1008.2.q.g 6
28.d even 2 1 2646.2.h.o 6
28.f even 6 1 2646.2.e.p 6
28.f even 6 1 2646.2.f.m 6
28.g odd 6 1 378.2.e.d 6
28.g odd 6 1 2646.2.f.l 6
36.f odd 6 1 378.2.e.d 6
36.f odd 6 1 1134.2.g.l 6
36.h even 6 1 126.2.e.c 6
36.h even 6 1 1134.2.g.m 6
63.g even 3 1 inner 3024.2.t.h 6
63.n odd 6 1 1008.2.t.h 6
84.h odd 2 1 882.2.h.p 6
84.j odd 6 1 882.2.e.o 6
84.j odd 6 1 882.2.f.o 6
84.n even 6 1 126.2.e.c 6
84.n even 6 1 882.2.f.n 6
252.n even 6 1 2646.2.h.o 6
252.n even 6 1 7938.2.a.bz 3
252.o even 6 1 126.2.h.d yes 6
252.o even 6 1 7938.2.a.bv 3
252.r odd 6 1 882.2.f.o 6
252.s odd 6 1 882.2.e.o 6
252.u odd 6 1 1134.2.g.l 6
252.u odd 6 1 2646.2.f.l 6
252.bb even 6 1 882.2.f.n 6
252.bb even 6 1 1134.2.g.m 6
252.bi even 6 1 2646.2.e.p 6
252.bj even 6 1 2646.2.f.m 6
252.bl odd 6 1 378.2.h.c 6
252.bl odd 6 1 7938.2.a.ca 3
252.bn odd 6 1 882.2.h.p 6
252.bn odd 6 1 7938.2.a.bw 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 36.h even 6 1
126.2.e.c 6 84.n even 6 1
126.2.h.d yes 6 12.b even 2 1
126.2.h.d yes 6 252.o even 6 1
378.2.e.d 6 28.g odd 6 1
378.2.e.d 6 36.f odd 6 1
378.2.h.c 6 4.b odd 2 1
378.2.h.c 6 252.bl odd 6 1
882.2.e.o 6 84.j odd 6 1
882.2.e.o 6 252.s odd 6 1
882.2.f.n 6 84.n even 6 1
882.2.f.n 6 252.bb even 6 1
882.2.f.o 6 84.j odd 6 1
882.2.f.o 6 252.r odd 6 1
882.2.h.p 6 84.h odd 2 1
882.2.h.p 6 252.bn odd 6 1
1008.2.q.g 6 9.d odd 6 1
1008.2.q.g 6 21.h odd 6 1
1008.2.t.h 6 3.b odd 2 1
1008.2.t.h 6 63.n odd 6 1
1134.2.g.l 6 36.f odd 6 1
1134.2.g.l 6 252.u odd 6 1
1134.2.g.m 6 36.h even 6 1
1134.2.g.m 6 252.bb even 6 1
2646.2.e.p 6 28.f even 6 1
2646.2.e.p 6 252.bi even 6 1
2646.2.f.l 6 28.g odd 6 1
2646.2.f.l 6 252.u odd 6 1
2646.2.f.m 6 28.f even 6 1
2646.2.f.m 6 252.bj even 6 1
2646.2.h.o 6 28.d even 2 1
2646.2.h.o 6 252.n even 6 1
3024.2.q.g 6 7.c even 3 1
3024.2.q.g 6 9.c even 3 1
3024.2.t.h 6 1.a even 1 1 trivial
3024.2.t.h 6 63.g even 3 1 inner
7938.2.a.bv 3 252.o even 6 1
7938.2.a.bw 3 252.bn odd 6 1
7938.2.a.bz 3 252.n even 6 1
7938.2.a.ca 3 252.bl odd 6 1

## Hecke kernels

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

 $$T_{5}^{3} - T_{5}^{2} - 6T_{5} - 3$$ T5^3 - T5^2 - 6*T5 - 3 $$T_{11}^{3} - T_{11}^{2} - 6T_{11} - 3$$ T11^3 - T11^2 - 6*T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$(T^{3} - T^{2} - 6 T - 3)^{2}$$
$7$ $$T^{6} - 4 T^{5} + 14 T^{4} - 55 T^{3} + \cdots + 343$$
$11$ $$(T^{3} - T^{2} - 6 T - 3)^{2}$$
$13$ $$T^{6} - 8 T^{5} + 63 T^{4} + \cdots + 4761$$
$17$ $$T^{6} - 4 T^{5} + 28 T^{4} + 240 T^{2} + \cdots + 576$$
$19$ $$T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 2401$$
$23$ $$(T^{3} - 7 T^{2} + 12 T - 3)^{2}$$
$29$ $$T^{6} - 5 T^{5} + 91 T^{4} + \cdots + 131769$$
$31$ $$T^{6} + 20 T^{5} + 279 T^{4} + \cdots + 40401$$
$37$ $$(T^{2} - T + 1)^{3}$$
$41$ $$T^{6} + 33 T^{4} - 18 T^{3} + 1089 T^{2} + \cdots + 81$$
$43$ $$T^{6} - 6 T^{5} + 105 T^{4} + \cdots + 16129$$
$47$ $$T^{6} + 9 T^{5} + 135 T^{4} + \cdots + 35721$$
$53$ $$T^{6} + 15 T^{5} + 159 T^{4} + \cdots + 6561$$
$59$ $$T^{6} + 14 T^{5} + 157 T^{4} + \cdots + 3969$$
$61$ $$T^{6} - 8 T^{5} + 69 T^{4} + \cdots + 8649$$
$67$ $$T^{6} + T^{5} + 113 T^{4} + \cdots + 44521$$
$71$ $$(T^{3} - 7 T^{2} - 198 T + 1593)^{2}$$
$73$ $$T^{6} - 19 T^{5} + 353 T^{4} + \cdots + 398161$$
$79$ $$T^{6} + 5 T^{5} + 99 T^{4} + \cdots + 103041$$
$83$ $$T^{6} - 2 T^{5} + 67 T^{4} + \cdots + 21609$$
$89$ $$T^{6} - 9 T^{5} + 123 T^{4} + 396 T^{3} + \cdots + 81$$
$97$ $$T^{6} - 28 T^{5} + 572 T^{4} + \cdots + 61504$$