# Properties

 Label 3332.1.bc.d Level $3332$ Weight $1$ Character orbit 3332.bc Analytic conductor $1.663$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -4 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(667,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 4, 9]))

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

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

chi := DirichletCharacter("3332.667");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.bc (of order $$12$$, degree $$4$$, 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: $$4$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.962948.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{5} + \zeta_{12}^{3} q^{8} - \zeta_{12} q^{9} +O(q^{10})$$ q + z * q^2 + z^2 * q^4 + (-z^4 + z) * q^5 + z^3 * q^8 - z * q^9 $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{5} + \zeta_{12}^{3} q^{8} - \zeta_{12} q^{9} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{10} + q^{13} + \zeta_{12}^{4} q^{16} + \zeta_{12}^{5} q^{17} - \zeta_{12}^{2} q^{18} + (\zeta_{12}^{3} + 1) q^{20} - \zeta_{12}^{5} q^{25} + 2 \zeta_{12} q^{26} + ( - \zeta_{12}^{3} - 1) q^{29} + \zeta_{12}^{5} q^{32} - q^{34} - \zeta_{12}^{3} q^{36} + (\zeta_{12}^{4} - \zeta_{12}) q^{37} + (\zeta_{12}^{4} + \zeta_{12}) q^{40} + ( - \zeta_{12}^{3} + 1) q^{41} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{45} + q^{50} + 2 \zeta_{12}^{2} q^{52} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{58} + (\zeta_{12}^{4} + \zeta_{12}) q^{61} - q^{64} + ( - 2 \zeta_{12}^{4} + 2 \zeta_{12}) q^{65} - \zeta_{12} q^{68} - \zeta_{12}^{4} q^{72} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{73} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{74} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{80} + \zeta_{12}^{2} q^{81} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{82} + (\zeta_{12}^{3} - 1) q^{85} - \zeta_{12}^{4} q^{89} + ( - \zeta_{12}^{3} - 1) q^{90} + ( - \zeta_{12}^{3} - 1) q^{97} +O(q^{100})$$ q + z * q^2 + z^2 * q^4 + (-z^4 + z) * q^5 + z^3 * q^8 - z * q^9 + (-z^5 + z^2) * q^10 + q^13 + z^4 * q^16 + z^5 * q^17 - z^2 * q^18 + (z^3 + 1) * q^20 - z^5 * q^25 + 2*z * q^26 + (-z^3 - 1) * q^29 + z^5 * q^32 - q^34 - z^3 * q^36 + (z^4 - z) * q^37 + (z^4 + z) * q^40 + (-z^3 + 1) * q^41 + (z^5 - z^2) * q^45 + q^50 + 2*z^2 * q^52 + (-z^4 - z) * q^58 + (z^4 + z) * q^61 - q^64 + (-2*z^4 + 2*z) * q^65 - z * q^68 - z^4 * q^72 + (z^5 + z^2) * q^73 + (z^5 - z^2) * q^74 + (z^5 + z^2) * q^80 + z^2 * q^81 + (-z^4 + z) * q^82 + (z^3 - 1) * q^85 - z^4 * q^89 + (-z^3 - 1) * q^90 + (-z^3 - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 2 q^{5}+O(q^{10})$$ 4 * q + 2 * q^4 + 2 * q^5 $$4 q + 2 q^{4} + 2 q^{5} + 2 q^{10} + 8 q^{13} - 2 q^{16} - 2 q^{18} + 4 q^{20} - 4 q^{29} - 4 q^{34} - 2 q^{37} - 2 q^{40} + 4 q^{41} - 2 q^{45} + 4 q^{50} + 4 q^{52} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 4 q^{65} + 2 q^{72} + 2 q^{73} - 2 q^{74} + 2 q^{80} + 2 q^{81} + 2 q^{82} - 4 q^{85} + 4 q^{89} - 4 q^{90} - 4 q^{97}+O(q^{100})$$ 4 * q + 2 * q^4 + 2 * q^5 + 2 * q^10 + 8 * q^13 - 2 * q^16 - 2 * q^18 + 4 * q^20 - 4 * q^29 - 4 * q^34 - 2 * q^37 - 2 * q^40 + 4 * q^41 - 2 * q^45 + 4 * q^50 + 4 * q^52 + 2 * q^58 - 2 * q^61 - 4 * q^64 + 4 * q^65 + 2 * q^72 + 2 * q^73 - 2 * q^74 + 2 * q^80 + 2 * q^81 + 2 * q^82 - 4 * q^85 + 4 * q^89 - 4 * q^90 - 4 * q^97

## Character values

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

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$\zeta_{12}^{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
667.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i −0.366025 1.36603i 0 0 1.00000i 0.866025 + 0.500000i −0.366025 + 1.36603i
863.1 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.36603 0.366025i 0 0 1.00000i −0.866025 0.500000i 1.36603 + 0.366025i
2027.1 0.866025 0.500000i 0 0.500000 0.866025i 1.36603 + 0.366025i 0 0 1.00000i −0.866025 + 0.500000i 1.36603 0.366025i
2223.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.366025 + 1.36603i 0 0 1.00000i 0.866025 0.500000i −0.366025 1.36603i
 $$n$$: 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 $$\Q(\sqrt{-1})$$
7.c even 3 1 inner
17.c even 4 1 inner
28.g odd 6 1 inner
68.f odd 4 1 inner
119.n even 12 1 inner
476.bb odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.bc.d 4
4.b odd 2 1 CM 3332.1.bc.d 4
7.b odd 2 1 3332.1.bc.a 4
7.c even 3 1 3332.1.m.a 2
7.c even 3 1 inner 3332.1.bc.d 4
7.d odd 6 1 3332.1.m.c yes 2
7.d odd 6 1 3332.1.bc.a 4
17.c even 4 1 inner 3332.1.bc.d 4
28.d even 2 1 3332.1.bc.a 4
28.f even 6 1 3332.1.m.c yes 2
28.f even 6 1 3332.1.bc.a 4
28.g odd 6 1 3332.1.m.a 2
28.g odd 6 1 inner 3332.1.bc.d 4
68.f odd 4 1 inner 3332.1.bc.d 4
119.f odd 4 1 3332.1.bc.a 4
119.m odd 12 1 3332.1.m.c yes 2
119.m odd 12 1 3332.1.bc.a 4
119.n even 12 1 3332.1.m.a 2
119.n even 12 1 inner 3332.1.bc.d 4
476.k even 4 1 3332.1.bc.a 4
476.z even 12 1 3332.1.m.c yes 2
476.z even 12 1 3332.1.bc.a 4
476.bb odd 12 1 3332.1.m.a 2
476.bb odd 12 1 inner 3332.1.bc.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.m.a 2 7.c even 3 1
3332.1.m.a 2 28.g odd 6 1
3332.1.m.a 2 119.n even 12 1
3332.1.m.a 2 476.bb odd 12 1
3332.1.m.c yes 2 7.d odd 6 1
3332.1.m.c yes 2 28.f even 6 1
3332.1.m.c yes 2 119.m odd 12 1
3332.1.m.c yes 2 476.z even 12 1
3332.1.bc.a 4 7.b odd 2 1
3332.1.bc.a 4 7.d odd 6 1
3332.1.bc.a 4 28.d even 2 1
3332.1.bc.a 4 28.f even 6 1
3332.1.bc.a 4 119.f odd 4 1
3332.1.bc.a 4 119.m odd 12 1
3332.1.bc.a 4 476.k even 4 1
3332.1.bc.a 4 476.z even 12 1
3332.1.bc.d 4 1.a even 1 1 trivial
3332.1.bc.d 4 4.b odd 2 1 CM
3332.1.bc.d 4 7.c even 3 1 inner
3332.1.bc.d 4 17.c even 4 1 inner
3332.1.bc.d 4 28.g odd 6 1 inner
3332.1.bc.d 4 68.f odd 4 1 inner
3332.1.bc.d 4 119.n even 12 1 inner
3332.1.bc.d 4 476.bb odd 12 1 inner

## Hecke kernels

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

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

## Hecke characteristic polynomials

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