# Properties

 Label 3528.1.ce.a Level $3528$ Weight $1$ Character orbit 3528.ce Analytic conductor $1.761$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -8 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,1,Mod(2419,3528)]

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([3, 3, 4, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3528.2419");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3528 = 2^{3} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3528.ce (of order $$6$$, 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: $$1.76070136457$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.648.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.99574272.2

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + q^{2} + \zeta_{6}^{2} q^{3} + q^{4} + \zeta_{6}^{2} q^{6} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + q^2 + z^2 * q^3 + q^4 + z^2 * q^6 + q^8 - z * q^9 $$q + q^{2} + \zeta_{6}^{2} q^{3} + q^{4} + \zeta_{6}^{2} q^{6} + q^{8} - \zeta_{6} q^{9} - \zeta_{6}^{2} q^{11} + \zeta_{6}^{2} q^{12} + q^{16} + \zeta_{6} q^{17} - \zeta_{6} q^{18} - \zeta_{6}^{2} q^{19} - \zeta_{6}^{2} q^{22} + \zeta_{6}^{2} q^{24} + \zeta_{6}^{2} q^{25} + q^{27} + q^{32} + \zeta_{6} q^{33} + \zeta_{6} q^{34} - \zeta_{6} q^{36} - \zeta_{6}^{2} q^{38} - \zeta_{6}^{2} q^{41} + \zeta_{6} q^{43} - \zeta_{6}^{2} q^{44} + \zeta_{6}^{2} q^{48} + \zeta_{6}^{2} q^{50} - q^{51} + q^{54} + \zeta_{6} q^{57} - q^{59} + q^{64} + \zeta_{6} q^{66} - q^{67} + \zeta_{6} q^{68} - \zeta_{6} q^{72} + \zeta_{6} q^{73} - \zeta_{6} q^{75} - \zeta_{6}^{2} q^{76} + \zeta_{6}^{2} q^{81} - \zeta_{6}^{2} q^{82} - \zeta_{6} q^{83} + \zeta_{6} q^{86} - \zeta_{6}^{2} q^{88} + \zeta_{6}^{2} q^{89} + \zeta_{6}^{2} q^{96} + \zeta_{6} q^{97} - q^{99} +O(q^{100})$$ q + q^2 + z^2 * q^3 + q^4 + z^2 * q^6 + q^8 - z * q^9 - z^2 * q^11 + z^2 * q^12 + q^16 + z * q^17 - z * q^18 - z^2 * q^19 - z^2 * q^22 + z^2 * q^24 + z^2 * q^25 + q^27 + q^32 + z * q^33 + z * q^34 - z * q^36 - z^2 * q^38 - z^2 * q^41 + z * q^43 - z^2 * q^44 + z^2 * q^48 + z^2 * q^50 - q^51 + q^54 + z * q^57 - q^59 + q^64 + z * q^66 - q^67 + z * q^68 - z * q^72 + z * q^73 - z * q^75 - z^2 * q^76 + z^2 * q^81 - z^2 * q^82 - z * q^83 + z * q^86 - z^2 * q^88 + z^2 * q^89 + z^2 * q^96 + z * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{6} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - q^6 + 2 * q^8 - q^9 $$2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{6} + 2 q^{8} - q^{9} + q^{11} - q^{12} + 2 q^{16} + q^{17} - q^{18} + q^{19} + q^{22} - q^{24} - q^{25} + 2 q^{27} + 2 q^{32} + q^{33} + q^{34} - q^{36} + q^{38} + q^{41} + q^{43} + q^{44} - q^{48} - q^{50} - 2 q^{51} + 2 q^{54} + q^{57} - 2 q^{59} + 2 q^{64} + q^{66} - 2 q^{67} + q^{68} - q^{72} + q^{73} - q^{75} + q^{76} - q^{81} + q^{82} - 2 q^{83} + q^{86} + q^{88} - 2 q^{89} - q^{96} + q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - q^6 + 2 * q^8 - q^9 + q^11 - q^12 + 2 * q^16 + q^17 - q^18 + q^19 + q^22 - q^24 - q^25 + 2 * q^27 + 2 * q^32 + q^33 + q^34 - q^36 + q^38 + q^41 + q^43 + q^44 - q^48 - q^50 - 2 * q^51 + 2 * q^54 + q^57 - 2 * q^59 + 2 * q^64 + q^66 - 2 * q^67 + q^68 - q^72 + q^73 - q^75 + q^76 - q^81 + q^82 - 2 * q^83 + q^86 + q^88 - 2 * q^89 - q^96 + q^97 - 2 * q^99

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$\zeta_{6}^{2}$$ $$\zeta_{6}^{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2419.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 −0.500000 0.866025i 1.00000 0 −0.500000 0.866025i 0 1.00000 −0.500000 + 0.866025i 0
3019.1 1.00000 −0.500000 + 0.866025i 1.00000 0 −0.500000 + 0.866025i 0 1.00000 −0.500000 0.866025i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
63.h even 3 1 inner
504.ce odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.ce.a 2
7.b odd 2 1 3528.1.ce.b 2
7.c even 3 1 72.1.p.a 2
7.c even 3 1 3528.1.ba.b 2
7.d odd 6 1 3528.1.ba.a 2
7.d odd 6 1 3528.1.cg.a 2
8.d odd 2 1 CM 3528.1.ce.a 2
9.c even 3 1 3528.1.ba.b 2
21.h odd 6 1 216.1.p.a 2
28.g odd 6 1 288.1.t.a 2
35.j even 6 1 1800.1.bk.d 2
35.l odd 12 2 1800.1.ba.b 4
56.e even 2 1 3528.1.ce.b 2
56.k odd 6 1 72.1.p.a 2
56.k odd 6 1 3528.1.ba.b 2
56.m even 6 1 3528.1.ba.a 2
56.m even 6 1 3528.1.cg.a 2
56.p even 6 1 288.1.t.a 2
63.g even 3 1 72.1.p.a 2
63.h even 3 1 648.1.b.b 1
63.h even 3 1 inner 3528.1.ce.a 2
63.j odd 6 1 648.1.b.a 1
63.k odd 6 1 3528.1.cg.a 2
63.l odd 6 1 3528.1.ba.a 2
63.n odd 6 1 216.1.p.a 2
63.t odd 6 1 3528.1.ce.b 2
72.p odd 6 1 3528.1.ba.b 2
84.n even 6 1 864.1.t.a 2
112.u odd 12 2 2304.1.o.c 4
112.w even 12 2 2304.1.o.c 4
168.s odd 6 1 864.1.t.a 2
168.v even 6 1 216.1.p.a 2
252.o even 6 1 864.1.t.a 2
252.u odd 6 1 2592.1.b.b 1
252.bb even 6 1 2592.1.b.a 1
252.bl odd 6 1 288.1.t.a 2
280.bi odd 6 1 1800.1.bk.d 2
280.br even 12 2 1800.1.ba.b 4
315.bo even 6 1 1800.1.bk.d 2
315.ch odd 12 2 1800.1.ba.b 4
504.w even 6 1 288.1.t.a 2
504.ba odd 6 1 72.1.p.a 2
504.be even 6 1 3528.1.ba.a 2
504.bf even 6 1 3528.1.ce.b 2
504.bi odd 6 1 2592.1.b.a 1
504.bt even 6 1 648.1.b.a 1
504.ce odd 6 1 648.1.b.b 1
504.ce odd 6 1 inner 3528.1.ce.a 2
504.cq even 6 1 2592.1.b.b 1
504.cy even 6 1 216.1.p.a 2
504.cz even 6 1 3528.1.cg.a 2
504.db odd 6 1 864.1.t.a 2
1008.do even 12 2 2304.1.o.c 4
1008.dz odd 12 2 2304.1.o.c 4
2520.gj odd 6 1 1800.1.bk.d 2
2520.jd even 12 2 1800.1.ba.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 7.c even 3 1
72.1.p.a 2 56.k odd 6 1
72.1.p.a 2 63.g even 3 1
72.1.p.a 2 504.ba odd 6 1
216.1.p.a 2 21.h odd 6 1
216.1.p.a 2 63.n odd 6 1
216.1.p.a 2 168.v even 6 1
216.1.p.a 2 504.cy even 6 1
288.1.t.a 2 28.g odd 6 1
288.1.t.a 2 56.p even 6 1
288.1.t.a 2 252.bl odd 6 1
288.1.t.a 2 504.w even 6 1
648.1.b.a 1 63.j odd 6 1
648.1.b.a 1 504.bt even 6 1
648.1.b.b 1 63.h even 3 1
648.1.b.b 1 504.ce odd 6 1
864.1.t.a 2 84.n even 6 1
864.1.t.a 2 168.s odd 6 1
864.1.t.a 2 252.o even 6 1
864.1.t.a 2 504.db odd 6 1
1800.1.ba.b 4 35.l odd 12 2
1800.1.ba.b 4 280.br even 12 2
1800.1.ba.b 4 315.ch odd 12 2
1800.1.ba.b 4 2520.jd even 12 2
1800.1.bk.d 2 35.j even 6 1
1800.1.bk.d 2 280.bi odd 6 1
1800.1.bk.d 2 315.bo even 6 1
1800.1.bk.d 2 2520.gj odd 6 1
2304.1.o.c 4 112.u odd 12 2
2304.1.o.c 4 112.w even 12 2
2304.1.o.c 4 1008.do even 12 2
2304.1.o.c 4 1008.dz odd 12 2
2592.1.b.a 1 252.bb even 6 1
2592.1.b.a 1 504.bi odd 6 1
2592.1.b.b 1 252.u odd 6 1
2592.1.b.b 1 504.cq even 6 1
3528.1.ba.a 2 7.d odd 6 1
3528.1.ba.a 2 56.m even 6 1
3528.1.ba.a 2 63.l odd 6 1
3528.1.ba.a 2 504.be even 6 1
3528.1.ba.b 2 7.c even 3 1
3528.1.ba.b 2 9.c even 3 1
3528.1.ba.b 2 56.k odd 6 1
3528.1.ba.b 2 72.p odd 6 1
3528.1.ce.a 2 1.a even 1 1 trivial
3528.1.ce.a 2 8.d odd 2 1 CM
3528.1.ce.a 2 63.h even 3 1 inner
3528.1.ce.a 2 504.ce odd 6 1 inner
3528.1.ce.b 2 7.b odd 2 1
3528.1.ce.b 2 56.e even 2 1
3528.1.ce.b 2 63.t odd 6 1
3528.1.ce.b 2 504.bf even 6 1
3528.1.cg.a 2 7.d odd 6 1
3528.1.cg.a 2 56.m even 6 1
3528.1.cg.a 2 63.k odd 6 1
3528.1.cg.a 2 504.cz even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3528, [\chi])$$:

 $$T_{5}$$ T5 $$T_{11}^{2} - T_{11} + 1$$ T11^2 - T11 + 1 $$T_{17}^{2} - T_{17} + 1$$ T17^2 - T17 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - T + 1$$
$13$ $$T^{2}$$
$17$ $$T^{2} - T + 1$$
$19$ $$T^{2} - T + 1$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2} - T + 1$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$(T + 1)^{2}$$
$61$ $$T^{2}$$
$67$ $$(T + 1)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - T + 1$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 2T + 4$$
$89$ $$T^{2} + 2T + 4$$
$97$ $$T^{2} - T + 1$$