# Properties

 Label 3528.2.s.bk Level $3528$ Weight $2$ Character orbit 3528.s Analytic conductor $28.171$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,2,Mod(361,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("3528.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3528 = 2^{3} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3528.s (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: $$28.1712218331$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ x^4 - x^3 - 4*x^2 - 5*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 168) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5}+O(q^{10})$$ q + b3 * q^5 $$q + \beta_{3} q^{5} + (\beta_{3} - \beta_{2}) q^{11} + (\beta_{2} - 3) q^{13} - 4 \beta_1 q^{17} + (\beta_{3} - 3 \beta_1 - 3) q^{19} + (4 \beta_1 + 4) q^{23} + (\beta_{3} - \beta_{2} + 9 \beta_1) q^{25} + (\beta_{2} - 2) q^{29} + \beta_1 q^{31} + ( - \beta_{3} - \beta_1 - 1) q^{37} + (2 \beta_{2} + 2) q^{41} + ( - \beta_{2} - 3) q^{43} + (6 \beta_1 + 6) q^{47} + (\beta_{3} - \beta_{2} - 6 \beta_1) q^{53} + ( - \beta_{2} - 14) q^{55} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{59} + (10 \beta_1 + 10) q^{61} + ( - 2 \beta_{3} + 14 \beta_1 + 14) q^{65} + (\beta_{3} - \beta_{2} + 3 \beta_1) q^{67} - 2 q^{71} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{73} + (2 \beta_{3} - 5 \beta_1 - 5) q^{79} + ( - \beta_{2} + 4) q^{83} + 4 \beta_{2} q^{85} + ( - 2 \beta_{3} + 4 \beta_1 + 4) q^{89} + ( - 2 \beta_{3} + 2 \beta_{2} + 14 \beta_1) q^{95} + ( - \beta_{2} - 12) q^{97}+O(q^{100})$$ q + b3 * q^5 + (b3 - b2) * q^11 + (b2 - 3) * q^13 - 4*b1 * q^17 + (b3 - 3*b1 - 3) * q^19 + (4*b1 + 4) * q^23 + (b3 - b2 + 9*b1) * q^25 + (b2 - 2) * q^29 + b1 * q^31 + (-b3 - b1 - 1) * q^37 + (2*b2 + 2) * q^41 + (-b2 - 3) * q^43 + (6*b1 + 6) * q^47 + (b3 - b2 - 6*b1) * q^53 + (-b2 - 14) * q^55 + (b3 - b2 + 2*b1) * q^59 + (10*b1 + 10) * q^61 + (-2*b3 + 14*b1 + 14) * q^65 + (b3 - b2 + 3*b1) * q^67 - 2 * q^71 + (-b3 + b2 + b1) * q^73 + (2*b3 - 5*b1 - 5) * q^79 + (-b2 + 4) * q^83 + 4*b2 * q^85 + (-2*b3 + 4*b1 + 4) * q^89 + (-2*b3 + 2*b2 + 14*b1) * q^95 + (-b2 - 12) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{5}+O(q^{10})$$ 4 * q + q^5 $$4 q + q^{5} - q^{11} - 10 q^{13} + 8 q^{17} - 5 q^{19} + 8 q^{23} - 19 q^{25} - 6 q^{29} - 2 q^{31} - 3 q^{37} + 12 q^{41} - 14 q^{43} + 12 q^{47} + 11 q^{53} - 58 q^{55} - 5 q^{59} + 20 q^{61} + 26 q^{65} - 7 q^{67} - 8 q^{71} - q^{73} - 8 q^{79} + 14 q^{83} + 8 q^{85} + 6 q^{89} - 26 q^{95} - 50 q^{97}+O(q^{100})$$ 4 * q + q^5 - q^11 - 10 * q^13 + 8 * q^17 - 5 * q^19 + 8 * q^23 - 19 * q^25 - 6 * q^29 - 2 * q^31 - 3 * q^37 + 12 * q^41 - 14 * q^43 + 12 * q^47 + 11 * q^53 - 58 * q^55 - 5 * q^59 + 20 * q^61 + 26 * q^65 - 7 * q^67 - 8 * q^71 - q^73 - 8 * q^79 + 14 * q^83 + 8 * q^85 + 6 * q^89 - 26 * q^95 - 50 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20$$ (v^3 + 4*v^2 - 4*v - 25) / 20 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5$$ (-v^3 + v^2 + 9*v + 5) / 5 $$\beta_{3}$$ $$=$$ $$( 3\nu^{3} + 2\nu^{2} + 8\nu - 25 ) / 10$$ (3*v^3 + 2*v^2 + 8*v - 25) / 10
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3$$ (b3 + b2 - 2*b1 - 1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 14\beta _1 + 13 ) / 3$$ (-b3 + 2*b2 + 14*b1 + 13) / 3 $$\nu^{3}$$ $$=$$ $$( 8\beta_{3} - 4\beta_{2} - 4\beta _1 + 19 ) / 3$$ (8*b3 - 4*b2 - 4*b1 + 19) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{1}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −1.63746 − 1.52274i 2.13746 + 0.656712i −1.63746 + 1.52274i 2.13746 − 0.656712i
0 0 0 −1.63746 2.83616i 0 0 0 0 0
361.2 0 0 0 2.13746 + 3.70219i 0 0 0 0 0
3313.1 0 0 0 −1.63746 + 2.83616i 0 0 0 0 0
3313.2 0 0 0 2.13746 3.70219i 0 0 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.2.s.bk 4
3.b odd 2 1 1176.2.q.l 4
7.b odd 2 1 504.2.s.i 4
7.c even 3 1 3528.2.a.bd 2
7.c even 3 1 inner 3528.2.s.bk 4
7.d odd 6 1 504.2.s.i 4
7.d odd 6 1 3528.2.a.bk 2
12.b even 2 1 2352.2.q.bf 4
21.c even 2 1 168.2.q.c 4
21.g even 6 1 168.2.q.c 4
21.g even 6 1 1176.2.a.k 2
21.h odd 6 1 1176.2.a.n 2
21.h odd 6 1 1176.2.q.l 4
28.d even 2 1 1008.2.s.r 4
28.f even 6 1 1008.2.s.r 4
28.f even 6 1 7056.2.a.cu 2
28.g odd 6 1 7056.2.a.ch 2
84.h odd 2 1 336.2.q.g 4
84.j odd 6 1 336.2.q.g 4
84.j odd 6 1 2352.2.a.bf 2
84.n even 6 1 2352.2.a.ba 2
84.n even 6 1 2352.2.q.bf 4
168.e odd 2 1 1344.2.q.x 4
168.i even 2 1 1344.2.q.w 4
168.s odd 6 1 9408.2.a.dj 2
168.v even 6 1 9408.2.a.dw 2
168.ba even 6 1 1344.2.q.w 4
168.ba even 6 1 9408.2.a.ec 2
168.be odd 6 1 1344.2.q.x 4
168.be odd 6 1 9408.2.a.dp 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 21.c even 2 1
168.2.q.c 4 21.g even 6 1
336.2.q.g 4 84.h odd 2 1
336.2.q.g 4 84.j odd 6 1
504.2.s.i 4 7.b odd 2 1
504.2.s.i 4 7.d odd 6 1
1008.2.s.r 4 28.d even 2 1
1008.2.s.r 4 28.f even 6 1
1176.2.a.k 2 21.g even 6 1
1176.2.a.n 2 21.h odd 6 1
1176.2.q.l 4 3.b odd 2 1
1176.2.q.l 4 21.h odd 6 1
1344.2.q.w 4 168.i even 2 1
1344.2.q.w 4 168.ba even 6 1
1344.2.q.x 4 168.e odd 2 1
1344.2.q.x 4 168.be odd 6 1
2352.2.a.ba 2 84.n even 6 1
2352.2.a.bf 2 84.j odd 6 1
2352.2.q.bf 4 12.b even 2 1
2352.2.q.bf 4 84.n even 6 1
3528.2.a.bd 2 7.c even 3 1
3528.2.a.bk 2 7.d odd 6 1
3528.2.s.bk 4 1.a even 1 1 trivial
3528.2.s.bk 4 7.c even 3 1 inner
7056.2.a.ch 2 28.g odd 6 1
7056.2.a.cu 2 28.f even 6 1
9408.2.a.dj 2 168.s odd 6 1
9408.2.a.dp 2 168.be odd 6 1
9408.2.a.dw 2 168.v even 6 1
9408.2.a.ec 2 168.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3528, [\chi])$$:

 $$T_{5}^{4} - T_{5}^{3} + 15T_{5}^{2} + 14T_{5} + 196$$ T5^4 - T5^3 + 15*T5^2 + 14*T5 + 196 $$T_{11}^{4} + T_{11}^{3} + 15T_{11}^{2} - 14T_{11} + 196$$ T11^4 + T11^3 + 15*T11^2 - 14*T11 + 196 $$T_{13}^{2} + 5T_{13} - 8$$ T13^2 + 5*T13 - 8 $$T_{23}^{2} - 4T_{23} + 16$$ T23^2 - 4*T23 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - T^{3} + \cdots + 196$$
$7$ $$T^{4}$$
$11$ $$T^{4} + T^{3} + \cdots + 196$$
$13$ $$(T^{2} + 5 T - 8)^{2}$$
$17$ $$(T^{2} - 4 T + 16)^{2}$$
$19$ $$T^{4} + 5 T^{3} + \cdots + 64$$
$23$ $$(T^{2} - 4 T + 16)^{2}$$
$29$ $$(T^{2} + 3 T - 12)^{2}$$
$31$ $$(T^{2} + T + 1)^{2}$$
$37$ $$T^{4} + 3 T^{3} + \cdots + 144$$
$41$ $$(T^{2} - 6 T - 48)^{2}$$
$43$ $$(T^{2} + 7 T - 2)^{2}$$
$47$ $$(T^{2} - 6 T + 36)^{2}$$
$53$ $$T^{4} - 11 T^{3} + \cdots + 256$$
$59$ $$T^{4} + 5 T^{3} + \cdots + 64$$
$61$ $$(T^{2} - 10 T + 100)^{2}$$
$67$ $$T^{4} + 7 T^{3} + \cdots + 4$$
$71$ $$(T + 2)^{4}$$
$73$ $$T^{4} + T^{3} + \cdots + 196$$
$79$ $$T^{4} + 8 T^{3} + \cdots + 1681$$
$83$ $$(T^{2} - 7 T - 2)^{2}$$
$89$ $$T^{4} - 6 T^{3} + \cdots + 2304$$
$97$ $$(T^{2} + 25 T + 142)^{2}$$