# Properties

 Label 1911.1.w.a Level $1911$ Weight $1$ Character orbit 1911.w Analytic conductor $0.954$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -3, -39, 13 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1911,1,Mod(116,1911)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1911, 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("1911.116");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1911.w (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: $$0.953713239142$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{13})$$ Artin image: $C_3\times D_4$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{3} + \zeta_{6} q^{4} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ q - z * q^3 + z * q^4 + z^2 * q^9 $$q - \zeta_{6} q^{3} + \zeta_{6} q^{4} + \zeta_{6}^{2} q^{9} - \zeta_{6}^{2} q^{12} + q^{13} + \zeta_{6}^{2} q^{16} + \zeta_{6} q^{25} + q^{27} - q^{36} - \zeta_{6} q^{39} + q^{43} + q^{48} + \zeta_{6} q^{52} + \zeta_{6}^{2} q^{61} - q^{64} - \zeta_{6}^{2} q^{75} - \zeta_{6}^{2} q^{79} - \zeta_{6} q^{81} +O(q^{100})$$ q - z * q^3 + z * q^4 + z^2 * q^9 - z^2 * q^12 + q^13 + z^2 * q^16 + z * q^25 + q^27 - q^36 - z * q^39 + q^43 + q^48 + z * q^52 + z^2 * q^61 - q^64 - z^2 * q^75 - z^2 * q^79 - z * q^81 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + q^{4} - q^{9}+O(q^{10})$$ 2 * q - q^3 + q^4 - q^9 $$2 q - q^{3} + q^{4} - q^{9} + q^{12} + 2 q^{13} - q^{16} + q^{25} + 2 q^{27} - 2 q^{36} - q^{39} + 4 q^{43} + 2 q^{48} + q^{52} - 2 q^{61} - 2 q^{64} + q^{75} + 2 q^{79} - q^{81}+O(q^{100})$$ 2 * q - q^3 + q^4 - q^9 + q^12 + 2 * q^13 - q^16 + q^25 + 2 * q^27 - 2 * q^36 - q^39 + 4 * q^43 + 2 * q^48 + q^52 - 2 * q^61 - 2 * q^64 + q^75 + 2 * q^79 - q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times$$.

 $$n$$ $$638$$ $$1471$$ $$1522$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{6}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 + 0.866025i 0.500000 0.866025i 0 0 0 0 −0.500000 0.866025i 0
1598.1 0 −0.500000 0.866025i 0.500000 + 0.866025i 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.b even 2 1 RM by $$\Q(\sqrt{13})$$
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
7.c even 3 1 inner
21.h odd 6 1 inner
91.r even 6 1 inner
273.w odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.1.w.a 2
3.b odd 2 1 CM 1911.1.w.a 2
7.b odd 2 1 1911.1.w.b 2
7.c even 3 1 1911.1.h.a 1
7.c even 3 1 inner 1911.1.w.a 2
7.d odd 6 1 39.1.d.a 1
7.d odd 6 1 1911.1.w.b 2
13.b even 2 1 RM 1911.1.w.a 2
21.c even 2 1 1911.1.w.b 2
21.g even 6 1 39.1.d.a 1
21.g even 6 1 1911.1.w.b 2
21.h odd 6 1 1911.1.h.a 1
21.h odd 6 1 inner 1911.1.w.a 2
28.f even 6 1 624.1.l.a 1
35.i odd 6 1 975.1.g.a 1
35.k even 12 2 975.1.e.a 2
39.d odd 2 1 CM 1911.1.w.a 2
56.j odd 6 1 2496.1.l.b 1
56.m even 6 1 2496.1.l.a 1
63.i even 6 1 1053.1.n.b 2
63.k odd 6 1 1053.1.n.b 2
63.s even 6 1 1053.1.n.b 2
63.t odd 6 1 1053.1.n.b 2
84.j odd 6 1 624.1.l.a 1
91.b odd 2 1 1911.1.w.b 2
91.l odd 6 1 507.1.h.a 2
91.m odd 6 1 507.1.h.a 2
91.p odd 6 1 507.1.h.a 2
91.r even 6 1 1911.1.h.a 1
91.r even 6 1 inner 1911.1.w.a 2
91.s odd 6 1 39.1.d.a 1
91.s odd 6 1 1911.1.w.b 2
91.v odd 6 1 507.1.h.a 2
91.w even 12 2 507.1.i.a 2
91.ba even 12 2 507.1.i.a 2
91.bb even 12 2 507.1.c.a 1
105.p even 6 1 975.1.g.a 1
105.w odd 12 2 975.1.e.a 2
168.ba even 6 1 2496.1.l.b 1
168.be odd 6 1 2496.1.l.a 1
273.g even 2 1 1911.1.w.b 2
273.r even 6 1 507.1.h.a 2
273.w odd 6 1 1911.1.h.a 1
273.w odd 6 1 inner 1911.1.w.a 2
273.y even 6 1 507.1.h.a 2
273.ba even 6 1 39.1.d.a 1
273.ba even 6 1 1911.1.w.b 2
273.bf even 6 1 507.1.h.a 2
273.br even 6 1 507.1.h.a 2
273.bs odd 12 2 507.1.i.a 2
273.cb odd 12 2 507.1.c.a 1
273.ch odd 12 2 507.1.i.a 2
364.x even 6 1 624.1.l.a 1
455.bf odd 6 1 975.1.g.a 1
455.df even 12 2 975.1.e.a 2
728.bv odd 6 1 2496.1.l.b 1
728.cy even 6 1 2496.1.l.a 1
819.bf odd 6 1 1053.1.n.b 2
819.bt even 6 1 1053.1.n.b 2
819.dv odd 6 1 1053.1.n.b 2
819.eg even 6 1 1053.1.n.b 2
1092.ct odd 6 1 624.1.l.a 1
1365.dd even 6 1 975.1.g.a 1
1365.fr odd 12 2 975.1.e.a 2
2184.de odd 6 1 2496.1.l.a 1
2184.ds even 6 1 2496.1.l.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.1.d.a 1 7.d odd 6 1
39.1.d.a 1 21.g even 6 1
39.1.d.a 1 91.s odd 6 1
39.1.d.a 1 273.ba even 6 1
507.1.c.a 1 91.bb even 12 2
507.1.c.a 1 273.cb odd 12 2
507.1.h.a 2 91.l odd 6 1
507.1.h.a 2 91.m odd 6 1
507.1.h.a 2 91.p odd 6 1
507.1.h.a 2 91.v odd 6 1
507.1.h.a 2 273.r even 6 1
507.1.h.a 2 273.y even 6 1
507.1.h.a 2 273.bf even 6 1
507.1.h.a 2 273.br even 6 1
507.1.i.a 2 91.w even 12 2
507.1.i.a 2 91.ba even 12 2
507.1.i.a 2 273.bs odd 12 2
507.1.i.a 2 273.ch odd 12 2
624.1.l.a 1 28.f even 6 1
624.1.l.a 1 84.j odd 6 1
624.1.l.a 1 364.x even 6 1
624.1.l.a 1 1092.ct odd 6 1
975.1.e.a 2 35.k even 12 2
975.1.e.a 2 105.w odd 12 2
975.1.e.a 2 455.df even 12 2
975.1.e.a 2 1365.fr odd 12 2
975.1.g.a 1 35.i odd 6 1
975.1.g.a 1 105.p even 6 1
975.1.g.a 1 455.bf odd 6 1
975.1.g.a 1 1365.dd even 6 1
1053.1.n.b 2 63.i even 6 1
1053.1.n.b 2 63.k odd 6 1
1053.1.n.b 2 63.s even 6 1
1053.1.n.b 2 63.t odd 6 1
1053.1.n.b 2 819.bf odd 6 1
1053.1.n.b 2 819.bt even 6 1
1053.1.n.b 2 819.dv odd 6 1
1053.1.n.b 2 819.eg even 6 1
1911.1.h.a 1 7.c even 3 1
1911.1.h.a 1 21.h odd 6 1
1911.1.h.a 1 91.r even 6 1
1911.1.h.a 1 273.w odd 6 1
1911.1.w.a 2 1.a even 1 1 trivial
1911.1.w.a 2 3.b odd 2 1 CM
1911.1.w.a 2 7.c even 3 1 inner
1911.1.w.a 2 13.b even 2 1 RM
1911.1.w.a 2 21.h odd 6 1 inner
1911.1.w.a 2 39.d odd 2 1 CM
1911.1.w.a 2 91.r even 6 1 inner
1911.1.w.a 2 273.w odd 6 1 inner
1911.1.w.b 2 7.b odd 2 1
1911.1.w.b 2 7.d odd 6 1
1911.1.w.b 2 21.c even 2 1
1911.1.w.b 2 21.g even 6 1
1911.1.w.b 2 91.b odd 2 1
1911.1.w.b 2 91.s odd 6 1
1911.1.w.b 2 273.g even 2 1
1911.1.w.b 2 273.ba even 6 1
2496.1.l.a 1 56.m even 6 1
2496.1.l.a 1 168.be odd 6 1
2496.1.l.a 1 728.cy even 6 1
2496.1.l.a 1 2184.de odd 6 1
2496.1.l.b 1 56.j odd 6 1
2496.1.l.b 1 168.ba even 6 1
2496.1.l.b 1 728.bv odd 6 1
2496.1.l.b 1 2184.ds even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1911, [\chi])$$:

 $$T_{2}$$ T2 $$T_{61}^{2} + 2T_{61} + 4$$ T61^2 + 2*T61 + 4 $$T_{199}^{2} + 2T_{199} + 4$$ T199^2 + 2*T199 + 4

## Hecke characteristic polynomials

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