# Properties

 Label 3332.1.ce.d Level $3332$ Weight $1$ Character orbit 3332.ce Analytic conductor $1.663$ Analytic rank $0$ Dimension $16$ Projective image $D_{16}$ 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(31,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(48))

chi = DirichletCharacter(H, H._module([24, 8, 27]))

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

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

chi := DirichletCharacter("3332.31");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.ce (of order $$48$$, degree $$16$$, 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: $$16$$ Coefficient field: $$\Q(\zeta_{48})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - x^{8} + 1$$ x^16 - x^8 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{16}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{48} q^{2} + \zeta_{48}^{2} q^{4} + (\zeta_{48}^{10} + \zeta_{48}) q^{5} - \zeta_{48}^{3} q^{8} - \zeta_{48}^{7} q^{9} +O(q^{10})$$ q - z * q^2 + z^2 * q^4 + (z^10 + z) * q^5 - z^3 * q^8 - z^7 * q^9 $$q - \zeta_{48} q^{2} + \zeta_{48}^{2} q^{4} + (\zeta_{48}^{10} + \zeta_{48}) q^{5} - \zeta_{48}^{3} q^{8} - \zeta_{48}^{7} q^{9} + ( - \zeta_{48}^{11} - \zeta_{48}^{2}) q^{10} + ( - \zeta_{48}^{12} + 1) q^{13} + \zeta_{48}^{4} q^{16} - \zeta_{48}^{17} q^{17} + \zeta_{48}^{8} q^{18} + (\zeta_{48}^{12} + \zeta_{48}^{3}) q^{20} + (\zeta_{48}^{20} + \cdots + \zeta_{48}^{2}) q^{25} + \cdots + ( - \zeta_{48}^{18} + \zeta_{48}^{9}) q^{97} +O(q^{100})$$ q - z * q^2 + z^2 * q^4 + (z^10 + z) * q^5 - z^3 * q^8 - z^7 * q^9 + (-z^11 - z^2) * q^10 + (-z^12 + 1) * q^13 + z^4 * q^16 - z^17 * q^17 + z^8 * q^18 + (z^12 + z^3) * q^20 + (z^20 + z^11 + z^2) * q^25 + (z^13 - z) * q^26 + (-z^21 - z^6) * q^29 - z^5 * q^32 + z^18 * q^34 - z^9 * q^36 + (z^16 + z^7) * q^37 + (-z^13 - z^4) * q^40 + (-z^12 - z^9) * q^41 + (-z^17 - z^8) * q^45 + (-z^21 - z^12 - z^3) * q^50 + (-z^14 + z^2) * q^52 + (-z^20 - z^14) * q^53 + (z^22 + z^7) * q^58 + (z^4 - z) * q^61 + z^6 * q^64 + (-z^22 - z^13 + z^10 + z) * q^65 - z^19 * q^68 + z^10 * q^72 + (z^14 - z^5) * q^73 + (-z^17 - z^8) * q^74 + (z^14 + z^5) * q^80 + z^14 * q^81 + (z^13 + z^10) * q^82 + (-z^18 + z^3) * q^85 + 2*z^22 * q^89 + (z^18 + z^9) * q^90 + (-z^18 + z^9) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q+O(q^{10})$$ 16 * q $$16 q + 16 q^{13} + 8 q^{18} - 8 q^{37} - 8 q^{45} - 8 q^{74}+O(q^{100})$$ 16 * q + 16 * q^13 + 8 * q^18 - 8 * q^37 - 8 * q^45 - 8 * q^74

## Character values

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

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-\zeta_{48}^{15}$$ $$-\zeta_{48}^{16}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −0.793353 − 0.608761i −0.793353 + 0.608761i 0.130526 + 0.991445i 0.130526 − 0.991445i 0.991445 + 0.130526i −0.991445 + 0.130526i −0.991445 − 0.130526i −0.130526 + 0.991445i 0.793353 − 0.608761i 0.991445 − 0.130526i 0.608761 + 0.793353i −0.130526 − 0.991445i 0.793353 + 0.608761i −0.608761 + 0.793353i −0.608761 − 0.793353i 0.608761 − 0.793353i
0.793353 + 0.608761i 0 0.258819 + 0.965926i 0.172572 0.349942i 0 0 −0.382683 + 0.923880i −0.130526 0.991445i 0.349942 0.172572i
215.1 0.793353 0.608761i 0 0.258819 0.965926i 0.172572 + 0.349942i 0 0 −0.382683 0.923880i −0.130526 + 0.991445i 0.349942 + 0.172572i
227.1 −0.130526 0.991445i 0 −0.965926 + 0.258819i −0.128293 + 1.95737i 0 0 0.382683 + 0.923880i 0.793353 + 0.608761i 1.95737 0.128293i
411.1 −0.130526 + 0.991445i 0 −0.965926 0.258819i −0.128293 1.95737i 0 0 0.382683 0.923880i 0.793353 0.608761i 1.95737 + 0.128293i
607.1 −0.991445 0.130526i 0 0.965926 + 0.258819i 1.25026 + 1.09645i 0 0 −0.923880 0.382683i −0.608761 0.793353i −1.09645 1.25026i
619.1 0.991445 0.130526i 0 0.965926 0.258819i −0.732626 0.835400i 0 0 0.923880 0.382683i 0.608761 0.793353i −0.835400 0.732626i
1195.1 0.991445 + 0.130526i 0 0.965926 + 0.258819i −0.732626 + 0.835400i 0 0 0.923880 + 0.382683i 0.608761 + 0.793353i −0.835400 + 0.732626i
1391.1 0.130526 0.991445i 0 −0.965926 0.258819i −0.389345 + 0.0255190i 0 0 −0.382683 + 0.923880i −0.793353 + 0.608761i −0.0255190 + 0.389345i
1587.1 −0.793353 + 0.608761i 0 0.258819 0.965926i 1.75928 0.867580i 0 0 0.382683 + 0.923880i 0.130526 0.991445i −0.867580 + 1.75928i
1795.1 −0.991445 + 0.130526i 0 0.965926 0.258819i 1.25026 1.09645i 0 0 −0.923880 + 0.382683i −0.608761 + 0.793353i −1.09645 + 1.25026i
1979.1 −0.608761 0.793353i 0 −0.258819 + 0.965926i −0.357164 + 1.05217i 0 0 0.923880 0.382683i −0.991445 0.130526i 1.05217 0.357164i
2187.1 0.130526 + 0.991445i 0 −0.965926 + 0.258819i −0.389345 0.0255190i 0 0 −0.382683 0.923880i −0.793353 0.608761i −0.0255190 0.389345i
2383.1 −0.793353 0.608761i 0 0.258819 + 0.965926i 1.75928 + 0.867580i 0 0 0.382683 0.923880i 0.130526 + 0.991445i −0.867580 1.75928i
2579.1 0.608761 0.793353i 0 −0.258819 0.965926i −1.57469 + 0.534534i 0 0 −0.923880 0.382683i 0.991445 0.130526i −0.534534 + 1.57469i
3155.1 0.608761 + 0.793353i 0 −0.258819 + 0.965926i −1.57469 0.534534i 0 0 −0.923880 + 0.382683i 0.991445 + 0.130526i −0.534534 1.57469i
3167.1 −0.608761 + 0.793353i 0 −0.258819 0.965926i −0.357164 1.05217i 0 0 0.923880 + 0.382683i −0.991445 + 0.130526i 1.05217 + 0.357164i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.1 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
28.g odd 6 1 inner
119.p even 16 1 inner
119.s even 48 1 inner
476.bf odd 16 1 inner
476.bk odd 48 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.ce.d 16
4.b odd 2 1 CM 3332.1.ce.d 16
7.b odd 2 1 3332.1.ce.b 16
7.c even 3 1 3332.1.bn.c yes 8
7.c even 3 1 inner 3332.1.ce.d 16
7.d odd 6 1 3332.1.bn.a 8
7.d odd 6 1 3332.1.ce.b 16
17.e odd 16 1 3332.1.ce.b 16
28.d even 2 1 3332.1.ce.b 16
28.f even 6 1 3332.1.bn.a 8
28.f even 6 1 3332.1.ce.b 16
28.g odd 6 1 3332.1.bn.c yes 8
28.g odd 6 1 inner 3332.1.ce.d 16
68.i even 16 1 3332.1.ce.b 16
119.p even 16 1 inner 3332.1.ce.d 16
119.s even 48 1 3332.1.bn.c yes 8
119.s even 48 1 inner 3332.1.ce.d 16
119.t odd 48 1 3332.1.bn.a 8
119.t odd 48 1 3332.1.ce.b 16
476.bf odd 16 1 inner 3332.1.ce.d 16
476.bk odd 48 1 3332.1.bn.c yes 8
476.bk odd 48 1 inner 3332.1.ce.d 16
476.bm even 48 1 3332.1.bn.a 8
476.bm even 48 1 3332.1.ce.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.bn.a 8 7.d odd 6 1
3332.1.bn.a 8 28.f even 6 1
3332.1.bn.a 8 119.t odd 48 1
3332.1.bn.a 8 476.bm even 48 1
3332.1.bn.c yes 8 7.c even 3 1
3332.1.bn.c yes 8 28.g odd 6 1
3332.1.bn.c yes 8 119.s even 48 1
3332.1.bn.c yes 8 476.bk odd 48 1
3332.1.ce.b 16 7.b odd 2 1
3332.1.ce.b 16 7.d odd 6 1
3332.1.ce.b 16 17.e odd 16 1
3332.1.ce.b 16 28.d even 2 1
3332.1.ce.b 16 28.f even 6 1
3332.1.ce.b 16 68.i even 16 1
3332.1.ce.b 16 119.t odd 48 1
3332.1.ce.b 16 476.bm even 48 1
3332.1.ce.d 16 1.a even 1 1 trivial
3332.1.ce.d 16 4.b odd 2 1 CM
3332.1.ce.d 16 7.c even 3 1 inner
3332.1.ce.d 16 28.g odd 6 1 inner
3332.1.ce.d 16 119.p even 16 1 inner
3332.1.ce.d 16 119.s even 48 1 inner
3332.1.ce.d 16 476.bf odd 16 1 inner
3332.1.ce.d 16 476.bk odd 48 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} - 2 T_{5}^{12} + 16 T_{5}^{11} + 40 T_{5}^{10} + 8 T_{5}^{9} + 2 T_{5}^{8} + 32 T_{5}^{7} + \cdots + 4$$ acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - T^{8} + 1$$
$3$ $$T^{16}$$
$5$ $$T^{16} - 2 T^{12} + \cdots + 4$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$(T^{2} - 2 T + 2)^{8}$$
$17$ $$T^{16} - T^{8} + 1$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$(T^{8} + 8 T^{5} + 2 T^{4} + \cdots + 2)^{2}$$
$31$ $$T^{16}$$
$37$ $$T^{16} + 8 T^{15} + \cdots + 4$$
$41$ $$(T^{8} + 4 T^{6} + 6 T^{4} + \cdots + 2)^{2}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$(T^{8} - 2 T^{6} + 8 T^{5} + \cdots + 4)^{2}$$
$59$ $$T^{16}$$
$61$ $$T^{16} - 4 T^{14} + \cdots + 4$$
$67$ $$T^{16}$$
$71$ $$T^{16}$$
$73$ $$T^{16} + 16 T^{13} + \cdots + 4$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$(T^{8} - 16 T^{4} + 256)^{2}$$
$97$ $$(T^{8} + 2 T^{4} - 16 T^{3} + \cdots + 2)^{2}$$