# Properties

 Label 3528.2.s.j Level $3528$ Weight $2$ Character orbit 3528.s Analytic conductor $28.171$ 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] = [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 # Warning: the index may be different

gp: f = lf \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$28.1712218331$$ 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) 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.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{5} +O(q^{10})$$ q - 2*z * q^5 $$q - 2 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{11} - 2 q^{13} + ( - 2 \zeta_{6} + 2) q^{17} + 4 \zeta_{6} q^{19} - 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - 6 q^{29} + (8 \zeta_{6} - 8) q^{31} - 6 \zeta_{6} q^{37} + 6 q^{41} + 4 q^{43} + (2 \zeta_{6} - 2) q^{53} - 8 q^{55} + ( - 4 \zeta_{6} + 4) q^{59} + 2 \zeta_{6} q^{61} + 4 \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{67} - 8 q^{71} + (10 \zeta_{6} - 10) q^{73} + 8 \zeta_{6} q^{79} + 4 q^{83} - 4 q^{85} - 6 \zeta_{6} q^{89} + ( - 8 \zeta_{6} + 8) q^{95} + 2 q^{97} +O(q^{100})$$ q - 2*z * q^5 + (-4*z + 4) * q^11 - 2 * q^13 + (-2*z + 2) * q^17 + 4*z * q^19 - 8*z * q^23 + (-z + 1) * q^25 - 6 * q^29 + (8*z - 8) * q^31 - 6*z * q^37 + 6 * q^41 + 4 * q^43 + (2*z - 2) * q^53 - 8 * q^55 + (-4*z + 4) * q^59 + 2*z * q^61 + 4*z * q^65 + (-4*z + 4) * q^67 - 8 * q^71 + (10*z - 10) * q^73 + 8*z * q^79 + 4 * q^83 - 4 * q^85 - 6*z * q^89 + (-8*z + 8) * q^95 + 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^5 $$2 q - 2 q^{5} + 4 q^{11} - 4 q^{13} + 2 q^{17} + 4 q^{19} - 8 q^{23} + q^{25} - 12 q^{29} - 8 q^{31} - 6 q^{37} + 12 q^{41} + 8 q^{43} - 2 q^{53} - 16 q^{55} + 4 q^{59} + 2 q^{61} + 4 q^{65} + 4 q^{67} - 16 q^{71} - 10 q^{73} + 8 q^{79} + 8 q^{83} - 8 q^{85} - 6 q^{89} + 8 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + 4 * q^11 - 4 * q^13 + 2 * q^17 + 4 * q^19 - 8 * q^23 + q^25 - 12 * q^29 - 8 * q^31 - 6 * q^37 + 12 * q^41 + 8 * q^43 - 2 * q^53 - 16 * q^55 + 4 * q^59 + 2 * q^61 + 4 * q^65 + 4 * q^67 - 16 * q^71 - 10 * q^73 + 8 * q^79 + 8 * q^83 - 8 * q^85 - 6 * q^89 + 8 * q^95 + 4 * q^97

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −1.00000 1.73205i 0 0 0 0 0
3313.1 0 0 0 −1.00000 + 1.73205i 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.j 2
3.b odd 2 1 1176.2.q.i 2
7.b odd 2 1 3528.2.s.y 2
7.c even 3 1 72.2.a.a 1
7.c even 3 1 inner 3528.2.s.j 2
7.d odd 6 1 3528.2.a.d 1
7.d odd 6 1 3528.2.s.y 2
12.b even 2 1 2352.2.q.l 2
21.c even 2 1 1176.2.q.a 2
21.g even 6 1 1176.2.a.i 1
21.g even 6 1 1176.2.q.a 2
21.h odd 6 1 24.2.a.a 1
21.h odd 6 1 1176.2.q.i 2
28.f even 6 1 7056.2.a.q 1
28.g odd 6 1 144.2.a.b 1
35.j even 6 1 1800.2.a.m 1
35.l odd 12 2 1800.2.f.c 2
56.k odd 6 1 576.2.a.b 1
56.p even 6 1 576.2.a.d 1
63.g even 3 1 648.2.i.b 2
63.h even 3 1 648.2.i.b 2
63.j odd 6 1 648.2.i.g 2
63.n odd 6 1 648.2.i.g 2
77.h odd 6 1 8712.2.a.u 1
84.h odd 2 1 2352.2.q.r 2
84.j odd 6 1 2352.2.a.i 1
84.j odd 6 1 2352.2.q.r 2
84.n even 6 1 48.2.a.a 1
84.n even 6 1 2352.2.q.l 2
105.o odd 6 1 600.2.a.h 1
105.x even 12 2 600.2.f.e 2
112.u odd 12 2 2304.2.d.k 2
112.w even 12 2 2304.2.d.i 2
140.p odd 6 1 3600.2.a.v 1
140.w even 12 2 3600.2.f.r 2
168.s odd 6 1 192.2.a.d 1
168.v even 6 1 192.2.a.b 1
168.ba even 6 1 9408.2.a.h 1
168.be odd 6 1 9408.2.a.cc 1
231.l even 6 1 2904.2.a.c 1
252.o even 6 1 1296.2.i.m 2
252.u odd 6 1 1296.2.i.e 2
252.bb even 6 1 1296.2.i.m 2
252.bl odd 6 1 1296.2.i.e 2
273.w odd 6 1 4056.2.a.i 1
273.cd even 12 2 4056.2.c.e 2
336.bt odd 12 2 768.2.d.e 2
336.bu even 12 2 768.2.d.d 2
357.q odd 6 1 6936.2.a.p 1
399.w even 6 1 8664.2.a.j 1
420.ba even 6 1 1200.2.a.d 1
420.bp odd 12 2 1200.2.f.b 2
840.cg odd 6 1 4800.2.a.q 1
840.cv even 6 1 4800.2.a.cc 1
840.dc even 12 2 4800.2.f.d 2
840.dp odd 12 2 4800.2.f.bg 2
924.z odd 6 1 5808.2.a.s 1
1092.by even 6 1 8112.2.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 21.h odd 6 1
48.2.a.a 1 84.n even 6 1
72.2.a.a 1 7.c even 3 1
144.2.a.b 1 28.g odd 6 1
192.2.a.b 1 168.v even 6 1
192.2.a.d 1 168.s odd 6 1
576.2.a.b 1 56.k odd 6 1
576.2.a.d 1 56.p even 6 1
600.2.a.h 1 105.o odd 6 1
600.2.f.e 2 105.x even 12 2
648.2.i.b 2 63.g even 3 1
648.2.i.b 2 63.h even 3 1
648.2.i.g 2 63.j odd 6 1
648.2.i.g 2 63.n odd 6 1
768.2.d.d 2 336.bu even 12 2
768.2.d.e 2 336.bt odd 12 2
1176.2.a.i 1 21.g even 6 1
1176.2.q.a 2 21.c even 2 1
1176.2.q.a 2 21.g even 6 1
1176.2.q.i 2 3.b odd 2 1
1176.2.q.i 2 21.h odd 6 1
1200.2.a.d 1 420.ba even 6 1
1200.2.f.b 2 420.bp odd 12 2
1296.2.i.e 2 252.u odd 6 1
1296.2.i.e 2 252.bl odd 6 1
1296.2.i.m 2 252.o even 6 1
1296.2.i.m 2 252.bb even 6 1
1800.2.a.m 1 35.j even 6 1
1800.2.f.c 2 35.l odd 12 2
2304.2.d.i 2 112.w even 12 2
2304.2.d.k 2 112.u odd 12 2
2352.2.a.i 1 84.j odd 6 1
2352.2.q.l 2 12.b even 2 1
2352.2.q.l 2 84.n even 6 1
2352.2.q.r 2 84.h odd 2 1
2352.2.q.r 2 84.j odd 6 1
2904.2.a.c 1 231.l even 6 1
3528.2.a.d 1 7.d odd 6 1
3528.2.s.j 2 1.a even 1 1 trivial
3528.2.s.j 2 7.c even 3 1 inner
3528.2.s.y 2 7.b odd 2 1
3528.2.s.y 2 7.d odd 6 1
3600.2.a.v 1 140.p odd 6 1
3600.2.f.r 2 140.w even 12 2
4056.2.a.i 1 273.w odd 6 1
4056.2.c.e 2 273.cd even 12 2
4800.2.a.q 1 840.cg odd 6 1
4800.2.a.cc 1 840.cv even 6 1
4800.2.f.d 2 840.dc even 12 2
4800.2.f.bg 2 840.dp odd 12 2
5808.2.a.s 1 924.z odd 6 1
6936.2.a.p 1 357.q odd 6 1
7056.2.a.q 1 28.f even 6 1
8112.2.a.be 1 1092.by even 6 1
8664.2.a.j 1 399.w even 6 1
8712.2.a.u 1 77.h odd 6 1
9408.2.a.h 1 168.ba even 6 1
9408.2.a.cc 1 168.be odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{11}^{2} - 4T_{11} + 16$$ T11^2 - 4*T11 + 16 $$T_{13} + 2$$ T13 + 2 $$T_{23}^{2} + 8T_{23} + 64$$ T23^2 + 8*T23 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 4T + 16$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2} + 8T + 64$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 8T + 64$$
$37$ $$T^{2} + 6T + 36$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 2T + 4$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 10T + 100$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$(T - 2)^{2}$$