# Properties

 Label 3724.1.bc.b Level $3724$ Weight $1$ Character orbit 3724.bc Analytic conductor $1.859$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3724,1,Mod(569,3724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3724, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("3724.569");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3724 = 2^{2} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3724.bc (of order $$6$$, 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: $$1.85851810705$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.76.1 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^5 - z * q^9 $$q - \zeta_{6} q^{5} - \zeta_{6} q^{9} - \zeta_{6}^{2} q^{11} + \zeta_{6}^{2} q^{17} + \zeta_{6} q^{19} - 2 \zeta_{6} q^{23} - q^{43} + \zeta_{6}^{2} q^{45} - \zeta_{6} q^{47} - q^{55} - \zeta_{6} q^{61} + \zeta_{6}^{2} q^{73} + \zeta_{6}^{2} q^{81} - 2 q^{83} + q^{85} - \zeta_{6}^{2} q^{95} - q^{99} +O(q^{100})$$ q - z * q^5 - z * q^9 - z^2 * q^11 + z^2 * q^17 + z * q^19 - 2*z * q^23 - q^43 + z^2 * q^45 - z * q^47 - q^55 - z * q^61 + z^2 * q^73 + z^2 * q^81 - 2 * q^83 + q^85 - z^2 * q^95 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} - q^{9}+O(q^{10})$$ 2 * q - q^5 - q^9 $$2 q - q^{5} - q^{9} + q^{11} - q^{17} + q^{19} - 2 q^{23} - 2 q^{43} - q^{45} - q^{47} - 2 q^{55} - q^{61} - q^{73} - q^{81} - 4 q^{83} + 2 q^{85} + q^{95} - 2 q^{99}+O(q^{100})$$ 2 * q - q^5 - q^9 + q^11 - q^17 + q^19 - 2 * q^23 - 2 * q^43 - q^45 - q^47 - 2 * q^55 - q^61 - q^73 - q^81 - 4 * q^83 + 2 * q^85 + q^95 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3724\mathbb{Z}\right)^\times$$.

 $$n$$ $$1863$$ $$3041$$ $$3137$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
569.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 0
949.1 0 0 0 −0.500000 0.866025i 0 0 0 −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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
7.c even 3 1 inner
133.r odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3724.1.bc.b 2
7.b odd 2 1 3724.1.bc.c 2
7.c even 3 1 3724.1.e.c 1
7.c even 3 1 inner 3724.1.bc.b 2
7.d odd 6 1 76.1.c.a 1
7.d odd 6 1 3724.1.bc.c 2
19.b odd 2 1 CM 3724.1.bc.b 2
21.g even 6 1 684.1.h.a 1
28.f even 6 1 304.1.e.a 1
35.i odd 6 1 1900.1.e.a 1
35.k even 12 2 1900.1.g.a 2
56.j odd 6 1 1216.1.e.a 1
56.m even 6 1 1216.1.e.b 1
84.j odd 6 1 2736.1.o.b 1
133.c even 2 1 3724.1.bc.c 2
133.i even 6 1 1444.1.h.a 2
133.k odd 6 1 1444.1.h.a 2
133.o even 6 1 76.1.c.a 1
133.o even 6 1 3724.1.bc.c 2
133.r odd 6 1 3724.1.e.c 1
133.r odd 6 1 inner 3724.1.bc.b 2
133.s even 6 1 1444.1.h.a 2
133.t odd 6 1 1444.1.h.a 2
133.x odd 18 3 1444.1.j.a 6
133.z odd 18 3 1444.1.j.a 6
133.bb even 18 3 1444.1.j.a 6
133.bf even 18 3 1444.1.j.a 6
399.s odd 6 1 684.1.h.a 1
532.bh odd 6 1 304.1.e.a 1
665.y even 6 1 1900.1.e.a 1
665.ca odd 12 2 1900.1.g.a 2
1064.bi odd 6 1 1216.1.e.b 1
1064.cf even 6 1 1216.1.e.a 1
1596.bl even 6 1 2736.1.o.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 7.d odd 6 1
76.1.c.a 1 133.o even 6 1
304.1.e.a 1 28.f even 6 1
304.1.e.a 1 532.bh odd 6 1
684.1.h.a 1 21.g even 6 1
684.1.h.a 1 399.s odd 6 1
1216.1.e.a 1 56.j odd 6 1
1216.1.e.a 1 1064.cf even 6 1
1216.1.e.b 1 56.m even 6 1
1216.1.e.b 1 1064.bi odd 6 1
1444.1.h.a 2 133.i even 6 1
1444.1.h.a 2 133.k odd 6 1
1444.1.h.a 2 133.s even 6 1
1444.1.h.a 2 133.t odd 6 1
1444.1.j.a 6 133.x odd 18 3
1444.1.j.a 6 133.z odd 18 3
1444.1.j.a 6 133.bb even 18 3
1444.1.j.a 6 133.bf even 18 3
1900.1.e.a 1 35.i odd 6 1
1900.1.e.a 1 665.y even 6 1
1900.1.g.a 2 35.k even 12 2
1900.1.g.a 2 665.ca odd 12 2
2736.1.o.b 1 84.j odd 6 1
2736.1.o.b 1 1596.bl even 6 1
3724.1.e.c 1 7.c even 3 1
3724.1.e.c 1 133.r odd 6 1
3724.1.bc.b 2 1.a even 1 1 trivial
3724.1.bc.b 2 7.c even 3 1 inner
3724.1.bc.b 2 19.b odd 2 1 CM
3724.1.bc.b 2 133.r odd 6 1 inner
3724.1.bc.c 2 7.b odd 2 1
3724.1.bc.c 2 7.d odd 6 1
3724.1.bc.c 2 133.c even 2 1
3724.1.bc.c 2 133.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3724, [\chi])$$:

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

## Hecke characteristic polynomials

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