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Properties

Label 441.1.m.a
Level $441$
Weight $1$
Character orbit 441.m
Analytic conductor $0.220$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -3, -7, 21
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,1,Mod(19,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.19");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 441.m (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: \(0.220087670571\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\)
Artin image: $C_3\times D_4$
Artin field: Galois closure of 12.0.794280046581.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{4} - \zeta_{6} q^{16} + \zeta_{6}^{2} q^{25} + 2 \zeta_{6} q^{37} - 2 q^{43} - q^{64} + 2 \zeta_{6}^{2} q^{67} - 2 \zeta_{6} q^{79} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{4} - q^{16} - q^{25} + 2 q^{37} - 4 q^{43} - 2 q^{64} - 2 q^{67} - 2 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 0.500000 + 0.866025i 0 0 0 0 0 0
325.1 0 0 0.500000 − 0.866025i 0 0 0 0 0 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 \(\Q(\sqrt{-3}) \)
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
21.c even 2 1 RM by \(\Q(\sqrt{21}) \)
7.c even 3 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.1.m.a 2
3.b odd 2 1 CM 441.1.m.a 2
7.b odd 2 1 CM 441.1.m.a 2
7.c even 3 1 63.1.d.a ✓ 1
7.c even 3 1 inner 441.1.m.a 2
7.d odd 6 1 63.1.d.a ✓ 1
7.d odd 6 1 inner 441.1.m.a 2
9.c even 3 1 3969.1.k.b 2
9.c even 3 1 3969.1.t.c 2
9.d odd 6 1 3969.1.k.b 2
9.d odd 6 1 3969.1.t.c 2
21.c even 2 1 RM 441.1.m.a 2
21.g even 6 1 63.1.d.a ✓ 1
21.g even 6 1 inner 441.1.m.a 2
21.h odd 6 1 63.1.d.a ✓ 1
21.h odd 6 1 inner 441.1.m.a 2
28.f even 6 1 1008.1.f.a 1
28.g odd 6 1 1008.1.f.a 1
35.i odd 6 1 1575.1.h.b 1
35.j even 6 1 1575.1.h.b 1
35.k even 12 2 1575.1.e.b 2
35.l odd 12 2 1575.1.e.b 2
63.g even 3 1 567.1.l.b 2
63.g even 3 1 3969.1.t.c 2
63.h even 3 1 567.1.l.b 2
63.h even 3 1 3969.1.k.b 2
63.i even 6 1 567.1.l.b 2
63.i even 6 1 3969.1.k.b 2
63.j odd 6 1 567.1.l.b 2
63.j odd 6 1 3969.1.k.b 2
63.k odd 6 1 567.1.l.b 2
63.k odd 6 1 3969.1.t.c 2
63.l odd 6 1 3969.1.k.b 2
63.l odd 6 1 3969.1.t.c 2
63.n odd 6 1 567.1.l.b 2
63.n odd 6 1 3969.1.t.c 2
63.o even 6 1 3969.1.k.b 2
63.o even 6 1 3969.1.t.c 2
63.s even 6 1 567.1.l.b 2
63.s even 6 1 3969.1.t.c 2
63.t odd 6 1 567.1.l.b 2
63.t odd 6 1 3969.1.k.b 2
84.j odd 6 1 1008.1.f.a 1
84.n even 6 1 1008.1.f.a 1
105.o odd 6 1 1575.1.h.b 1
105.p even 6 1 1575.1.h.b 1
105.w odd 12 2 1575.1.e.b 2
105.x even 12 2 1575.1.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.1.d.a ✓ 1 7.c even 3 1
63.1.d.a ✓ 1 7.d odd 6 1
63.1.d.a ✓ 1 21.g even 6 1
63.1.d.a ✓ 1 21.h odd 6 1
441.1.m.a 2 1.a even 1 1 trivial
441.1.m.a 2 3.b odd 2 1 CM
441.1.m.a 2 7.b odd 2 1 CM
441.1.m.a 2 7.c even 3 1 inner
441.1.m.a 2 7.d odd 6 1 inner
441.1.m.a 2 21.c even 2 1 RM
441.1.m.a 2 21.g even 6 1 inner
441.1.m.a 2 21.h odd 6 1 inner
567.1.l.b 2 63.g even 3 1
567.1.l.b 2 63.h even 3 1
567.1.l.b 2 63.i even 6 1
567.1.l.b 2 63.j odd 6 1
567.1.l.b 2 63.k odd 6 1
567.1.l.b 2 63.n odd 6 1
567.1.l.b 2 63.s even 6 1
567.1.l.b 2 63.t odd 6 1
1008.1.f.a 1 28.f even 6 1
1008.1.f.a 1 28.g odd 6 1
1008.1.f.a 1 84.j odd 6 1
1008.1.f.a 1 84.n even 6 1
1575.1.e.b 2 35.k even 12 2
1575.1.e.b 2 35.l odd 12 2
1575.1.e.b 2 105.w odd 12 2
1575.1.e.b 2 105.x even 12 2
1575.1.h.b 1 35.i odd 6 1
1575.1.h.b 1 35.j even 6 1
1575.1.h.b 1 105.o odd 6 1
1575.1.h.b 1 105.p even 6 1
3969.1.k.b 2 9.c even 3 1
3969.1.k.b 2 9.d odd 6 1
3969.1.k.b 2 63.h even 3 1
3969.1.k.b 2 63.i even 6 1
3969.1.k.b 2 63.j odd 6 1
3969.1.k.b 2 63.l odd 6 1
3969.1.k.b 2 63.o even 6 1
3969.1.k.b 2 63.t odd 6 1
3969.1.t.c 2 9.c even 3 1
3969.1.t.c 2 9.d odd 6 1
3969.1.t.c 2 63.g even 3 1
3969.1.t.c 2 63.k odd 6 1
3969.1.t.c 2 63.l odd 6 1
3969.1.t.c 2 63.n odd 6 1
3969.1.t.c 2 63.o even 6 1
3969.1.t.c 2 63.s even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) 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
$13$ \( T^{2} \) 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
$37$ \( T^{2} - 2T + 4 \) 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
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) 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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