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 $8$

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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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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
0.500000 0.866025i
0.500000 + 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:

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 \(S_{1}^{\mathrm{new}}(1911, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{61}^{2} + 2T_{61} + 4 \) Copy content Toggle raw display
\( T_{199}^{2} + 2T_{199} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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