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Properties

Label 1984.1.e.b
Level $1984$
Weight $1$
Character orbit 1984.e
Self dual yes
Analytic conductor $0.990$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -31
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1984,1,Mod(1921,1984)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1984, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1984.1921");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1984 = 2^{6} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1984.e (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.990144985064\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.31.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.492032.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + q^{7} + q^{9} - q^{19} - q^{31} + q^{35} - q^{41} + q^{45} - 2 q^{47} - q^{59} + q^{63} + 2 q^{67} + q^{71} + q^{81} - q^{95} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1984\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\) \(1861\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1921.1
0
0 0 0 1.00000 0 1.00000 0 1.00000 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
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1984.1.e.b 1
4.b odd 2 1 1984.1.e.a 1
8.b even 2 1 496.1.e.a 1
8.d odd 2 1 31.1.b.a ✓ 1
24.f even 2 1 279.1.d.b 1
31.b odd 2 1 CM 1984.1.e.b 1
40.e odd 2 1 775.1.d.b 1
40.k even 4 2 775.1.c.a 2
56.e even 2 1 1519.1.c.a 1
56.k odd 6 2 1519.1.n.b 2
56.m even 6 2 1519.1.n.a 2
72.l even 6 2 2511.1.m.a 2
72.p odd 6 2 2511.1.m.e 2
88.g even 2 1 3751.1.d.b 1
88.k even 10 4 3751.1.t.a 4
88.l odd 10 4 3751.1.t.c 4
124.d even 2 1 1984.1.e.a 1
248.b even 2 1 31.1.b.a ✓ 1
248.g odd 2 1 496.1.e.a 1
248.m odd 6 2 961.1.e.a 2
248.q even 6 2 961.1.e.a 2
248.s odd 10 4 961.1.f.a 4
248.v even 10 4 961.1.f.a 4
248.bb even 30 8 961.1.h.a 8
248.be odd 30 8 961.1.h.a 8
744.m odd 2 1 279.1.d.b 1
1240.o even 2 1 775.1.d.b 1
1240.s odd 4 2 775.1.c.a 2
1736.n odd 2 1 1519.1.c.a 1
1736.bh odd 6 2 1519.1.n.a 2
1736.ct even 6 2 1519.1.n.b 2
2232.bp odd 6 2 2511.1.m.a 2
2232.cq even 6 2 2511.1.m.e 2
2728.e odd 2 1 3751.1.d.b 1
2728.eb odd 10 4 3751.1.t.a 4
2728.ef even 10 4 3751.1.t.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a ✓ 1 8.d odd 2 1
31.1.b.a ✓ 1 248.b even 2 1
279.1.d.b 1 24.f even 2 1
279.1.d.b 1 744.m odd 2 1
496.1.e.a 1 8.b even 2 1
496.1.e.a 1 248.g odd 2 1
775.1.c.a 2 40.k even 4 2
775.1.c.a 2 1240.s odd 4 2
775.1.d.b 1 40.e odd 2 1
775.1.d.b 1 1240.o even 2 1
961.1.e.a 2 248.m odd 6 2
961.1.e.a 2 248.q even 6 2
961.1.f.a 4 248.s odd 10 4
961.1.f.a 4 248.v even 10 4
961.1.h.a 8 248.bb even 30 8
961.1.h.a 8 248.be odd 30 8
1519.1.c.a 1 56.e even 2 1
1519.1.c.a 1 1736.n odd 2 1
1519.1.n.a 2 56.m even 6 2
1519.1.n.a 2 1736.bh odd 6 2
1519.1.n.b 2 56.k odd 6 2
1519.1.n.b 2 1736.ct even 6 2
1984.1.e.a 1 4.b odd 2 1
1984.1.e.a 1 124.d even 2 1
1984.1.e.b 1 1.a even 1 1 trivial
1984.1.e.b 1 31.b odd 2 1 CM
2511.1.m.a 2 72.l even 6 2
2511.1.m.a 2 2232.bp odd 6 2
2511.1.m.e 2 72.p odd 6 2
2511.1.m.e 2 2232.cq even 6 2
3751.1.d.b 1 88.g even 2 1
3751.1.d.b 1 2728.e odd 2 1
3751.1.t.a 4 88.k even 10 4
3751.1.t.a 4 2728.eb odd 10 4
3751.1.t.c 4 88.l odd 10 4
3751.1.t.c 4 2728.ef even 10 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1984, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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