# Properties

 Label 3528.1.cg.a Level $3528$ Weight $1$ Character orbit 3528.cg 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(2059,3528)]

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, 0]))

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

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

chi := DirichletCharacter("3528.2059");

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.cg (of order $$6$$, degree $$2$$, 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: $$1.76070136457$$ 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: $$\Z[a_1, a_2]$$ 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: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2059.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i 0 1.00000 −0.500000 0.866025i 0
3235.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 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 $$\Q(\sqrt{-2})$$
9.c even 3 1 inner
72.p 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.cg.a 2
7.b odd 2 1 72.1.p.a 2
7.c even 3 1 3528.1.ba.a 2
7.c even 3 1 3528.1.ce.b 2
7.d odd 6 1 3528.1.ba.b 2
7.d odd 6 1 3528.1.ce.a 2
8.d odd 2 1 CM 3528.1.cg.a 2
9.c even 3 1 inner 3528.1.cg.a 2
21.c even 2 1 216.1.p.a 2
28.d even 2 1 288.1.t.a 2
35.c odd 2 1 1800.1.bk.d 2
35.f even 4 2 1800.1.ba.b 4
56.e even 2 1 72.1.p.a 2
56.h odd 2 1 288.1.t.a 2
56.k odd 6 1 3528.1.ba.a 2
56.k odd 6 1 3528.1.ce.b 2
56.m even 6 1 3528.1.ba.b 2
56.m even 6 1 3528.1.ce.a 2
63.g even 3 1 3528.1.ce.b 2
63.h even 3 1 3528.1.ba.a 2
63.k odd 6 1 3528.1.ce.a 2
63.l odd 6 1 72.1.p.a 2
63.l odd 6 1 648.1.b.b 1
63.o even 6 1 216.1.p.a 2
63.o even 6 1 648.1.b.a 1
63.t odd 6 1 3528.1.ba.b 2
72.p odd 6 1 inner 3528.1.cg.a 2
84.h odd 2 1 864.1.t.a 2
112.j even 4 2 2304.1.o.c 4
112.l odd 4 2 2304.1.o.c 4
168.e odd 2 1 216.1.p.a 2
168.i even 2 1 864.1.t.a 2
252.s odd 6 1 864.1.t.a 2
252.s odd 6 1 2592.1.b.a 1
252.bi even 6 1 288.1.t.a 2
252.bi even 6 1 2592.1.b.b 1
280.n even 2 1 1800.1.bk.d 2
280.y odd 4 2 1800.1.ba.b 4
315.bg odd 6 1 1800.1.bk.d 2
315.cb even 12 2 1800.1.ba.b 4
504.ba odd 6 1 3528.1.ce.b 2
504.be even 6 1 72.1.p.a 2
504.be even 6 1 648.1.b.b 1
504.bf even 6 1 3528.1.ba.b 2
504.bn odd 6 1 288.1.t.a 2
504.bn odd 6 1 2592.1.b.b 1
504.cc even 6 1 864.1.t.a 2
504.cc even 6 1 2592.1.b.a 1
504.ce odd 6 1 3528.1.ba.a 2
504.co odd 6 1 216.1.p.a 2
504.co odd 6 1 648.1.b.a 1
504.cz even 6 1 3528.1.ce.a 2
1008.dk even 12 2 2304.1.o.c 4
1008.dn odd 12 2 2304.1.o.c 4
2520.dy even 6 1 1800.1.bk.d 2
2520.hw odd 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.b odd 2 1
72.1.p.a 2 56.e even 2 1
72.1.p.a 2 63.l odd 6 1
72.1.p.a 2 504.be even 6 1
216.1.p.a 2 21.c even 2 1
216.1.p.a 2 63.o even 6 1
216.1.p.a 2 168.e odd 2 1
216.1.p.a 2 504.co odd 6 1
288.1.t.a 2 28.d even 2 1
288.1.t.a 2 56.h odd 2 1
288.1.t.a 2 252.bi even 6 1
288.1.t.a 2 504.bn odd 6 1
648.1.b.a 1 63.o even 6 1
648.1.b.a 1 504.co odd 6 1
648.1.b.b 1 63.l odd 6 1
648.1.b.b 1 504.be even 6 1
864.1.t.a 2 84.h odd 2 1
864.1.t.a 2 168.i even 2 1
864.1.t.a 2 252.s odd 6 1
864.1.t.a 2 504.cc even 6 1
1800.1.ba.b 4 35.f even 4 2
1800.1.ba.b 4 280.y odd 4 2
1800.1.ba.b 4 315.cb even 12 2
1800.1.ba.b 4 2520.hw odd 12 2
1800.1.bk.d 2 35.c odd 2 1
1800.1.bk.d 2 280.n even 2 1
1800.1.bk.d 2 315.bg odd 6 1
1800.1.bk.d 2 2520.dy even 6 1
2304.1.o.c 4 112.j even 4 2
2304.1.o.c 4 112.l odd 4 2
2304.1.o.c 4 1008.dk even 12 2
2304.1.o.c 4 1008.dn odd 12 2
2592.1.b.a 1 252.s odd 6 1
2592.1.b.a 1 504.cc even 6 1
2592.1.b.b 1 252.bi even 6 1
2592.1.b.b 1 504.bn odd 6 1
3528.1.ba.a 2 7.c even 3 1
3528.1.ba.a 2 56.k odd 6 1
3528.1.ba.a 2 63.h even 3 1
3528.1.ba.a 2 504.ce odd 6 1
3528.1.ba.b 2 7.d odd 6 1
3528.1.ba.b 2 56.m even 6 1
3528.1.ba.b 2 63.t odd 6 1
3528.1.ba.b 2 504.bf even 6 1
3528.1.ce.a 2 7.d odd 6 1
3528.1.ce.a 2 56.m even 6 1
3528.1.ce.a 2 63.k odd 6 1
3528.1.ce.a 2 504.cz even 6 1
3528.1.ce.b 2 7.c even 3 1
3528.1.ce.b 2 56.k odd 6 1
3528.1.ce.b 2 63.g even 3 1
3528.1.ce.b 2 504.ba odd 6 1
3528.1.cg.a 2 1.a even 1 1 trivial
3528.1.cg.a 2 8.d odd 2 1 CM
3528.1.cg.a 2 9.c even 3 1 inner
3528.1.cg.a 2 72.p odd 6 1 inner

## 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 $$T_{17} - 1$$ T17 - 1

## Hecke characteristic polynomials

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