# Properties

 Label 3332.1.o.e Level $3332$ Weight $1$ Character orbit 3332.o Analytic conductor $1.663$ Analytic rank $0$ Dimension $4$ Projective image $D_{2}$ CM/RM discs -4, -119, 476 Inner twists $16$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(67,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4, 3]))

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

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

chi := DirichletCharacter("3332.67");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.o (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: $$1.66288462209$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{119})$$ Artin image: $C_3\times D_4:C_2$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{2} q^{2} + \zeta_{12}^{4} q^{4} + \zeta_{12}^{5} q^{5} + q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ q - z^2 * q^2 + z^4 * q^4 + z^5 * q^5 + q^8 + z^2 * q^9 $$q - \zeta_{12}^{2} q^{2} + \zeta_{12}^{4} q^{4} + \zeta_{12}^{5} q^{5} + q^{8} + \zeta_{12}^{2} q^{9} + 2 \zeta_{12} q^{10} - \zeta_{12}^{2} q^{16} + \zeta_{12} q^{17} - \zeta_{12}^{4} q^{18} - 2 \zeta_{12}^{3} q^{20} - 3 \zeta_{12}^{4} q^{25} + \zeta_{12}^{4} q^{32} - \zeta_{12}^{3} q^{34} - q^{36} + 2 \zeta_{12}^{5} q^{40} + \zeta_{12}^{3} q^{41} - 2 \zeta_{12} q^{45} - 3 q^{50} + \zeta_{12}^{4} q^{53} + \zeta_{12}^{5} q^{61} + q^{64} + \zeta_{12}^{5} q^{68} + \zeta_{12}^{2} q^{72} - \zeta_{12} q^{73} + 2 \zeta_{12} q^{80} + \zeta_{12}^{4} q^{81} - 2 \zeta_{12}^{5} q^{82} - 2 q^{85} + 2 \zeta_{12}^{3} q^{90} - \zeta_{12}^{3} q^{97} +O(q^{100})$$ q - z^2 * q^2 + z^4 * q^4 + z^5 * q^5 + q^8 + z^2 * q^9 + 2*z * q^10 - z^2 * q^16 + z * q^17 - z^4 * q^18 - 2*z^3 * q^20 - 3*z^4 * q^25 + z^4 * q^32 - z^3 * q^34 - q^36 + 2*z^5 * q^40 + z^3 * q^41 - 2*z * q^45 - 3 * q^50 + z^4 * q^53 + z^5 * q^61 + q^64 + z^5 * q^68 + z^2 * q^72 - z * q^73 + 2*z * q^80 + z^4 * q^81 - 2*z^5 * q^82 - 2 * q^85 + 2*z^3 * q^90 - z^3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} + 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8 + 2 * q^9 $$4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} + 2 q^{9} - 2 q^{16} + 2 q^{18} + 6 q^{25} - 2 q^{32} - 4 q^{36} - 12 q^{50} - 4 q^{53} + 4 q^{64} + 2 q^{72} - 2 q^{81} - 8 q^{85}+O(q^{100})$$ 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8 + 2 * q^9 - 2 * q^16 + 2 * q^18 + 6 * q^25 - 2 * q^32 - 4 * q^36 - 12 * q^50 - 4 * q^53 + 4 * q^64 + 2 * q^72 - 2 * q^81 - 8 * q^85

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −1.73205 + 1.00000i 0 0 1.00000 0.500000 + 0.866025i 1.73205 + 1.00000i
67.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.73205 1.00000i 0 0 1.00000 0.500000 + 0.866025i −1.73205 1.00000i
2039.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.73205 1.00000i 0 0 1.00000 0.500000 0.866025i 1.73205 1.00000i
2039.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.73205 + 1.00000i 0 0 1.00000 0.500000 0.866025i −1.73205 + 1.00000i
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
119.d odd 2 1 CM by $$\Q(\sqrt{-119})$$
476.e even 2 1 RM by $$\Q(\sqrt{119})$$
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
17.b even 2 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
68.d odd 2 1 inner
119.h odd 6 1 inner
119.j even 6 1 inner
476.o odd 6 1 inner
476.q even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.o.e 4
4.b odd 2 1 CM 3332.1.o.e 4
7.b odd 2 1 inner 3332.1.o.e 4
7.c even 3 1 3332.1.g.h 2
7.c even 3 1 inner 3332.1.o.e 4
7.d odd 6 1 3332.1.g.h 2
7.d odd 6 1 inner 3332.1.o.e 4
17.b even 2 1 inner 3332.1.o.e 4
28.d even 2 1 inner 3332.1.o.e 4
28.f even 6 1 3332.1.g.h 2
28.f even 6 1 inner 3332.1.o.e 4
28.g odd 6 1 3332.1.g.h 2
28.g odd 6 1 inner 3332.1.o.e 4
68.d odd 2 1 inner 3332.1.o.e 4
119.d odd 2 1 CM 3332.1.o.e 4
119.h odd 6 1 3332.1.g.h 2
119.h odd 6 1 inner 3332.1.o.e 4
119.j even 6 1 3332.1.g.h 2
119.j even 6 1 inner 3332.1.o.e 4
476.e even 2 1 RM 3332.1.o.e 4
476.o odd 6 1 3332.1.g.h 2
476.o odd 6 1 inner 3332.1.o.e 4
476.q even 6 1 3332.1.g.h 2
476.q even 6 1 inner 3332.1.o.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.g.h 2 7.c even 3 1
3332.1.g.h 2 7.d odd 6 1
3332.1.g.h 2 28.f even 6 1
3332.1.g.h 2 28.g odd 6 1
3332.1.g.h 2 119.h odd 6 1
3332.1.g.h 2 119.j even 6 1
3332.1.g.h 2 476.o odd 6 1
3332.1.g.h 2 476.q even 6 1
3332.1.o.e 4 1.a even 1 1 trivial
3332.1.o.e 4 4.b odd 2 1 CM
3332.1.o.e 4 7.b odd 2 1 inner
3332.1.o.e 4 7.c even 3 1 inner
3332.1.o.e 4 7.d odd 6 1 inner
3332.1.o.e 4 17.b even 2 1 inner
3332.1.o.e 4 28.d even 2 1 inner
3332.1.o.e 4 28.f even 6 1 inner
3332.1.o.e 4 28.g odd 6 1 inner
3332.1.o.e 4 68.d odd 2 1 inner
3332.1.o.e 4 119.d odd 2 1 CM
3332.1.o.e 4 119.h odd 6 1 inner
3332.1.o.e 4 119.j even 6 1 inner
3332.1.o.e 4 476.e even 2 1 RM
3332.1.o.e 4 476.o odd 6 1 inner
3332.1.o.e 4 476.q even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5}^{4} - 4T_{5}^{2} + 16$$ T5^4 - 4*T5^2 + 16 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 4T^{2} + 16$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4} - T^{2} + 1$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 4)^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 2 T + 4)^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4} - 4T^{2} + 16$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 4T^{2} + 16$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$(T^{2} + 4)^{2}$$