# Properties

 Label 3328.1.v.b Level $3328$ Weight $1$ Character orbit 3328.v Analytic conductor $1.661$ Analytic rank $0$ Dimension $4$ Projective image $D_{3}$ CM discriminant -4 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3328,1,Mod(1407,3328)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3328, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("3328.1407");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3328 = 2^{8} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3328.v (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: $$1.66088836204$$ 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 52) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.676.1 Artin image: $S_3\times C_{12}$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{3} q^{5} - \zeta_{12}^{4} q^{9} +O(q^{10})$$ q - z^3 * q^5 - z^4 * q^9 $$q - \zeta_{12}^{3} q^{5} - \zeta_{12}^{4} q^{9} + \zeta_{12}^{5} q^{13} - \zeta_{12}^{4} q^{17} - \zeta_{12}^{5} q^{29} + \zeta_{12}^{5} q^{37} - \zeta_{12}^{2} q^{41} - \zeta_{12} q^{45} - \zeta_{12}^{2} q^{49} - \zeta_{12}^{3} q^{53} - \zeta_{12} q^{61} + \zeta_{12}^{2} q^{65} + q^{73} - \zeta_{12}^{2} q^{81} - \zeta_{12} q^{85} + \zeta_{12}^{2} q^{89} + \zeta_{12}^{4} q^{97} +O(q^{100})$$ q - z^3 * q^5 - z^4 * q^9 + z^5 * q^13 - z^4 * q^17 - z^5 * q^29 + z^5 * q^37 - z^2 * q^41 - z * q^45 - z^2 * q^49 - z^3 * q^53 - z * q^61 + z^2 * q^65 + q^73 - z^2 * q^81 - z * q^85 + z^2 * q^89 + z^4 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^9 $$4 q + 2 q^{9} + 2 q^{17} - 2 q^{41} - 2 q^{49} + 2 q^{65} + 4 q^{73} - 2 q^{81} + 4 q^{89} - 4 q^{97}+O(q^{100})$$ 4 * q + 2 * q^9 + 2 * q^17 - 2 * q^41 - 2 * q^49 + 2 * q^65 + 4 * q^73 - 2 * q^81 + 4 * q^89 - 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$769$$ $$1535$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1407.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 0 0 1.00000i 0 0 0 0.500000 0.866025i 0
1407.2 0 0 0 1.00000i 0 0 0 0.500000 0.866025i 0
2687.1 0 0 0 1.00000i 0 0 0 0.500000 + 0.866025i 0
2687.2 0 0 0 1.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
8.b even 2 1 inner
8.d odd 2 1 inner
13.c even 3 1 inner
52.j odd 6 1 inner
104.n odd 6 1 inner
104.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3328.1.v.b 4
4.b odd 2 1 CM 3328.1.v.b 4
8.b even 2 1 inner 3328.1.v.b 4
8.d odd 2 1 inner 3328.1.v.b 4
13.c even 3 1 inner 3328.1.v.b 4
16.e even 4 1 52.1.j.a 2
16.e even 4 1 832.1.bb.a 2
16.f odd 4 1 52.1.j.a 2
16.f odd 4 1 832.1.bb.a 2
48.i odd 4 1 468.1.br.a 2
48.k even 4 1 468.1.br.a 2
52.j odd 6 1 inner 3328.1.v.b 4
80.i odd 4 1 1300.1.w.a 4
80.j even 4 1 1300.1.w.a 4
80.k odd 4 1 1300.1.bc.a 2
80.q even 4 1 1300.1.bc.a 2
80.s even 4 1 1300.1.w.a 4
80.t odd 4 1 1300.1.w.a 4
104.n odd 6 1 inner 3328.1.v.b 4
104.r even 6 1 inner 3328.1.v.b 4
112.j even 4 1 2548.1.bn.a 2
112.l odd 4 1 2548.1.bn.a 2
112.u odd 12 1 2548.1.q.b 2
112.u odd 12 1 2548.1.bi.b 2
112.v even 12 1 2548.1.q.a 2
112.v even 12 1 2548.1.bi.a 2
112.w even 12 1 2548.1.q.b 2
112.w even 12 1 2548.1.bi.b 2
112.x odd 12 1 2548.1.q.a 2
112.x odd 12 1 2548.1.bi.a 2
208.l even 4 1 676.1.i.a 4
208.m odd 4 1 676.1.i.a 4
208.o odd 4 1 676.1.j.a 2
208.p even 4 1 676.1.j.a 2
208.r odd 4 1 676.1.i.a 4
208.s even 4 1 676.1.i.a 4
208.be odd 12 1 676.1.b.a 2
208.be odd 12 1 676.1.i.a 4
208.bf even 12 1 676.1.b.a 2
208.bf even 12 1 676.1.i.a 4
208.bg odd 12 1 52.1.j.a 2
208.bg odd 12 1 676.1.c.b 1
208.bg odd 12 1 832.1.bb.a 2
208.bh even 12 1 676.1.c.a 1
208.bh even 12 1 676.1.j.a 2
208.bi odd 12 1 676.1.c.a 1
208.bi odd 12 1 676.1.j.a 2
208.bj even 12 1 52.1.j.a 2
208.bj even 12 1 676.1.c.b 1
208.bj even 12 1 832.1.bb.a 2
208.bk even 12 1 676.1.b.a 2
208.bk even 12 1 676.1.i.a 4
208.bl odd 12 1 676.1.b.a 2
208.bl odd 12 1 676.1.i.a 4
624.cl even 12 1 468.1.br.a 2
624.cw odd 12 1 468.1.br.a 2
1040.dp even 12 1 1300.1.w.a 4
1040.dr odd 12 1 1300.1.w.a 4
1040.ec even 12 1 1300.1.bc.a 2
1040.fb odd 12 1 1300.1.bc.a 2
1040.fh even 12 1 1300.1.w.a 4
1040.fj odd 12 1 1300.1.w.a 4
1456.fh even 12 1 2548.1.bi.b 2
1456.fj odd 12 1 2548.1.bi.a 2
1456.fk odd 12 1 2548.1.bi.b 2
1456.fm even 12 1 2548.1.bi.a 2
1456.fs odd 12 1 2548.1.q.a 2
1456.fx odd 12 1 2548.1.bn.a 2
1456.fy even 12 1 2548.1.q.b 2
1456.gb even 12 1 2548.1.q.a 2
1456.ge even 12 1 2548.1.bn.a 2
1456.gh odd 12 1 2548.1.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 16.e even 4 1
52.1.j.a 2 16.f odd 4 1
52.1.j.a 2 208.bg odd 12 1
52.1.j.a 2 208.bj even 12 1
468.1.br.a 2 48.i odd 4 1
468.1.br.a 2 48.k even 4 1
468.1.br.a 2 624.cl even 12 1
468.1.br.a 2 624.cw odd 12 1
676.1.b.a 2 208.be odd 12 1
676.1.b.a 2 208.bf even 12 1
676.1.b.a 2 208.bk even 12 1
676.1.b.a 2 208.bl odd 12 1
676.1.c.a 1 208.bh even 12 1
676.1.c.a 1 208.bi odd 12 1
676.1.c.b 1 208.bg odd 12 1
676.1.c.b 1 208.bj even 12 1
676.1.i.a 4 208.l even 4 1
676.1.i.a 4 208.m odd 4 1
676.1.i.a 4 208.r odd 4 1
676.1.i.a 4 208.s even 4 1
676.1.i.a 4 208.be odd 12 1
676.1.i.a 4 208.bf even 12 1
676.1.i.a 4 208.bk even 12 1
676.1.i.a 4 208.bl odd 12 1
676.1.j.a 2 208.o odd 4 1
676.1.j.a 2 208.p even 4 1
676.1.j.a 2 208.bh even 12 1
676.1.j.a 2 208.bi odd 12 1
832.1.bb.a 2 16.e even 4 1
832.1.bb.a 2 16.f odd 4 1
832.1.bb.a 2 208.bg odd 12 1
832.1.bb.a 2 208.bj even 12 1
1300.1.w.a 4 80.i odd 4 1
1300.1.w.a 4 80.j even 4 1
1300.1.w.a 4 80.s even 4 1
1300.1.w.a 4 80.t odd 4 1
1300.1.w.a 4 1040.dp even 12 1
1300.1.w.a 4 1040.dr odd 12 1
1300.1.w.a 4 1040.fh even 12 1
1300.1.w.a 4 1040.fj odd 12 1
1300.1.bc.a 2 80.k odd 4 1
1300.1.bc.a 2 80.q even 4 1
1300.1.bc.a 2 1040.ec even 12 1
1300.1.bc.a 2 1040.fb odd 12 1
2548.1.q.a 2 112.v even 12 1
2548.1.q.a 2 112.x odd 12 1
2548.1.q.a 2 1456.fs odd 12 1
2548.1.q.a 2 1456.gb even 12 1
2548.1.q.b 2 112.u odd 12 1
2548.1.q.b 2 112.w even 12 1
2548.1.q.b 2 1456.fy even 12 1
2548.1.q.b 2 1456.gh odd 12 1
2548.1.bi.a 2 112.v even 12 1
2548.1.bi.a 2 112.x odd 12 1
2548.1.bi.a 2 1456.fj odd 12 1
2548.1.bi.a 2 1456.fm even 12 1
2548.1.bi.b 2 112.u odd 12 1
2548.1.bi.b 2 112.w even 12 1
2548.1.bi.b 2 1456.fh even 12 1
2548.1.bi.b 2 1456.fk odd 12 1
2548.1.bn.a 2 112.j even 4 1
2548.1.bn.a 2 112.l odd 4 1
2548.1.bn.a 2 1456.fx odd 12 1
2548.1.bn.a 2 1456.ge even 12 1
3328.1.v.b 4 1.a even 1 1 trivial
3328.1.v.b 4 4.b odd 2 1 CM
3328.1.v.b 4 8.b even 2 1 inner
3328.1.v.b 4 8.d odd 2 1 inner
3328.1.v.b 4 13.c even 3 1 inner
3328.1.v.b 4 52.j odd 6 1 inner
3328.1.v.b 4 104.n odd 6 1 inner
3328.1.v.b 4 104.r even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{1}^{\mathrm{new}}(3328, [\chi])$$.

## Hecke characteristic polynomials

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