Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2001,1,Mod(68,2001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(28))
chi = DirichletCharacter(H, H._module([14, 14, 23]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2001.68");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2001 = 3 \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2001.bf (of order \(28\), degree \(12\), 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: | \(0.998629090279\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\Q(\zeta_{28})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{28}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{28} - \cdots)\) |
$q$-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.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2001\mathbb{Z}\right)^\times\).
| \(n\) | \(553\) | \(668\) | \(1132\) |
| \(\chi(n)\) | \(\zeta_{28}^{3}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 68.1 |
|
−0.900969 | − | 1.43388i | 0.974928 | − | 0.222521i | −0.810394 | + | 1.68280i | 0 | −1.19745 | − | 1.19745i | 0 | 1.46028 | − | 0.164534i | 0.900969 | − | 0.433884i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 137.1 | 0.623490 | + | 0.218169i | −0.433884 | + | 0.900969i | −0.440689 | − | 0.351438i | 0 | −0.467085 | + | 0.467085i | 0 | −0.549531 | − | 0.874573i | −0.623490 | − | 0.781831i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 206.1 | −0.900969 | + | 1.43388i | 0.974928 | + | 0.222521i | −0.810394 | − | 1.68280i | 0 | −1.19745 | + | 1.19745i | 0 | 1.46028 | + | 0.164534i | 0.900969 | + | 0.433884i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 275.1 | −0.222521 | + | 0.0250721i | 0.781831 | − | 0.623490i | −0.926041 | + | 0.211363i | 0 | −0.158342 | + | 0.158342i | 0 | 0.412127 | − | 0.144209i | 0.222521 | − | 0.974928i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 482.1 | 0.623490 | − | 0.218169i | −0.433884 | − | 0.900969i | −0.440689 | + | 0.351438i | 0 | −0.467085 | − | 0.467085i | 0 | −0.549531 | + | 0.874573i | −0.623490 | + | 0.781831i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 620.1 | 0.623490 | + | 1.78183i | 0.433884 | + | 0.900969i | −2.00435 | + | 1.59842i | 0 | −1.33485 | + | 1.33485i | 0 | −2.49939 | − | 1.57047i | −0.623490 | + | 0.781831i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 827.1 | −0.222521 | − | 1.97493i | −0.781831 | + | 0.623490i | −2.87590 | + | 0.656405i | 0 | 1.40532 | + | 1.40532i | 0 | 1.27989 | + | 3.65773i | 0.222521 | − | 0.974928i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 896.1 | −0.900969 | − | 0.566116i | −0.974928 | − | 0.222521i | 0.0573735 | + | 0.119137i | 0 | 0.752407 | + | 0.752407i | 0 | −0.103384 | + | 0.917554i | 0.900969 | + | 0.433884i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 965.1 | 0.623490 | − | 1.78183i | 0.433884 | − | 0.900969i | −2.00435 | − | 1.59842i | 0 | −1.33485 | − | 1.33485i | 0 | −2.49939 | + | 1.57047i | −0.623490 | − | 0.781831i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1034.1 | −0.900969 | + | 0.566116i | −0.974928 | + | 0.222521i | 0.0573735 | − | 0.119137i | 0 | 0.752407 | − | 0.752407i | 0 | −0.103384 | − | 0.917554i | 0.900969 | − | 0.433884i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1448.1 | −0.222521 | − | 0.0250721i | 0.781831 | + | 0.623490i | −0.926041 | − | 0.211363i | 0 | −0.158342 | − | 0.158342i | 0 | 0.412127 | + | 0.144209i | 0.222521 | + | 0.974928i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1655.1 | −0.222521 | + | 1.97493i | −0.781831 | − | 0.623490i | −2.87590 | − | 0.656405i | 0 | 1.40532 | − | 1.40532i | 0 | 1.27989 | − | 3.65773i | 0.222521 | + | 0.974928i | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
| 87.k | even | 28 | 1 | inner |
| 2001.bf | odd | 28 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2001.1.bf.a | ✓ | 12 |
| 3.b | odd | 2 | 1 | 2001.1.bf.b | yes | 12 | |
| 23.b | odd | 2 | 1 | CM | 2001.1.bf.a | ✓ | 12 |
| 29.f | odd | 28 | 1 | 2001.1.bf.b | yes | 12 | |
| 69.c | even | 2 | 1 | 2001.1.bf.b | yes | 12 | |
| 87.k | even | 28 | 1 | inner | 2001.1.bf.a | ✓ | 12 |
| 667.o | even | 28 | 1 | 2001.1.bf.b | yes | 12 | |
| 2001.bf | odd | 28 | 1 | inner | 2001.1.bf.a | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2001.1.bf.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 2001.1.bf.a | ✓ | 12 | 23.b | odd | 2 | 1 | CM |
| 2001.1.bf.a | ✓ | 12 | 87.k | even | 28 | 1 | inner |
| 2001.1.bf.a | ✓ | 12 | 2001.bf | odd | 28 | 1 | inner |
| 2001.1.bf.b | yes | 12 | 3.b | odd | 2 | 1 | |
| 2001.1.bf.b | yes | 12 | 29.f | odd | 28 | 1 | |
| 2001.1.bf.b | yes | 12 | 69.c | even | 2 | 1 | |
| 2001.1.bf.b | yes | 12 | 667.o | even | 28 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 2 T_{2}^{11} + 9 T_{2}^{10} + 14 T_{2}^{9} + 31 T_{2}^{8} + 34 T_{2}^{7} + 41 T_{2}^{6} + \cdots + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2001, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 2 T^{11} + \cdots + 1 \)
|
| $3$ |
\( T^{12} - T^{10} + \cdots + 1 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( T^{12} \)
|
| $13$ |
\( T^{12} - 4 T^{10} + \cdots + 1 \)
|
| $17$ |
\( T^{12} \)
|
| $19$ |
\( T^{12} \)
|
| $23$ |
\( T^{12} - T^{10} + \cdots + 1 \)
|
| $29$ |
\( T^{12} - T^{10} + \cdots + 1 \)
|
| $31$ |
\( T^{12} - 2 T^{11} + \cdots + 64 \)
|
| $37$ |
\( T^{12} \)
|
| $41$ |
\( T^{12} - 2 T^{11} + \cdots + 1 \)
|
| $43$ |
\( T^{12} \)
|
| $47$ |
\( T^{12} + 2 T^{11} + \cdots + 1 \)
|
| $53$ |
\( T^{12} \)
|
| $59$ |
\( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \)
|
| $61$ |
\( T^{12} \)
|
| $67$ |
\( T^{12} \)
|
| $71$ |
\( T^{12} + 14 T^{8} + \cdots + 49 \)
|
| $73$ |
\( T^{12} - 2 T^{11} + \cdots + 1 \)
|
| $79$ |
\( T^{12} \)
|
| $83$ |
\( T^{12} \)
|
| $89$ |
\( T^{12} \)
|
| $97$ |
\( T^{12} \)
|
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