Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2013,1,Mod(548,2013)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2013, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 6, 5]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2013.548");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2013 = 3 \cdot 11 \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2013.bm (of order \(10\), degree \(4\), 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: | \(1.00461787043\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | \(\Q(\zeta_{40})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - x^{12} + x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{20}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/2013\mathbb{Z}\right)^\times\).
| \(n\) | \(1222\) | \(1343\) | \(1465\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-\zeta_{40}^{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 548.1 |
|
−0.550672 | + | 1.69480i | 0.809017 | − | 0.587785i | −1.76007 | − | 1.27877i | 0 | 0.550672 | + | 1.69480i | 0 | 1.69480 | − | 1.23134i | 0.309017 | − | 0.951057i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 548.2 | −0.280582 | + | 0.863541i | 0.809017 | − | 0.587785i | 0.142040 | + | 0.103198i | 0 | 0.280582 | + | 0.863541i | 0 | −0.863541 | + | 0.627399i | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 548.3 | 0.280582 | − | 0.863541i | 0.809017 | − | 0.587785i | 0.142040 | + | 0.103198i | 0 | −0.280582 | − | 0.863541i | 0 | 0.863541 | − | 0.627399i | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 548.4 | 0.550672 | − | 1.69480i | 0.809017 | − | 0.587785i | −1.76007 | − | 1.27877i | 0 | −0.550672 | − | 1.69480i | 0 | −1.69480 | + | 1.23134i | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 731.1 | −0.550672 | − | 1.69480i | 0.809017 | + | 0.587785i | −1.76007 | + | 1.27877i | 0 | 0.550672 | − | 1.69480i | 0 | 1.69480 | + | 1.23134i | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 731.2 | −0.280582 | − | 0.863541i | 0.809017 | + | 0.587785i | 0.142040 | − | 0.103198i | 0 | 0.280582 | − | 0.863541i | 0 | −0.863541 | − | 0.627399i | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 731.3 | 0.280582 | + | 0.863541i | 0.809017 | + | 0.587785i | 0.142040 | − | 0.103198i | 0 | −0.280582 | + | 0.863541i | 0 | 0.863541 | + | 0.627399i | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 731.4 | 0.550672 | + | 1.69480i | 0.809017 | + | 0.587785i | −1.76007 | + | 1.27877i | 0 | −0.550672 | + | 1.69480i | 0 | −1.69480 | − | 1.23134i | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1280.1 | −1.59811 | − | 1.16110i | −0.309017 | + | 0.951057i | 0.896802 | + | 2.76007i | 0 | 1.59811 | − | 1.16110i | 0 | 1.16110 | − | 3.57349i | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1280.2 | −0.253116 | − | 0.183900i | −0.309017 | + | 0.951057i | −0.278768 | − | 0.857960i | 0 | 0.253116 | − | 0.183900i | 0 | −0.183900 | + | 0.565985i | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1280.3 | 0.253116 | + | 0.183900i | −0.309017 | + | 0.951057i | −0.278768 | − | 0.857960i | 0 | −0.253116 | + | 0.183900i | 0 | 0.183900 | − | 0.565985i | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1280.4 | 1.59811 | + | 1.16110i | −0.309017 | + | 0.951057i | 0.896802 | + | 2.76007i | 0 | −1.59811 | + | 1.16110i | 0 | −1.16110 | + | 3.57349i | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1829.1 | −1.59811 | + | 1.16110i | −0.309017 | − | 0.951057i | 0.896802 | − | 2.76007i | 0 | 1.59811 | + | 1.16110i | 0 | 1.16110 | + | 3.57349i | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1829.2 | −0.253116 | + | 0.183900i | −0.309017 | − | 0.951057i | −0.278768 | + | 0.857960i | 0 | 0.253116 | + | 0.183900i | 0 | −0.183900 | − | 0.565985i | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1829.3 | 0.253116 | − | 0.183900i | −0.309017 | − | 0.951057i | −0.278768 | + | 0.857960i | 0 | −0.253116 | − | 0.183900i | 0 | 0.183900 | + | 0.565985i | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1829.4 | 1.59811 | − | 1.16110i | −0.309017 | − | 0.951057i | 0.896802 | − | 2.76007i | 0 | −1.59811 | − | 1.16110i | 0 | −1.16110 | − | 3.57349i | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 183.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-183}) \) |
| 3.b | odd | 2 | 1 | inner |
| 11.c | even | 5 | 1 | inner |
| 33.h | odd | 10 | 1 | inner |
| 61.b | even | 2 | 1 | inner |
| 671.x | even | 10 | 1 | inner |
| 2013.bm | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2013.1.bm.d | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 2013.1.bm.d | ✓ | 16 |
| 11.c | even | 5 | 1 | inner | 2013.1.bm.d | ✓ | 16 |
| 33.h | odd | 10 | 1 | inner | 2013.1.bm.d | ✓ | 16 |
| 61.b | even | 2 | 1 | inner | 2013.1.bm.d | ✓ | 16 |
| 183.d | odd | 2 | 1 | CM | 2013.1.bm.d | ✓ | 16 |
| 671.x | even | 10 | 1 | inner | 2013.1.bm.d | ✓ | 16 |
| 2013.bm | odd | 10 | 1 | inner | 2013.1.bm.d | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2013.1.bm.d | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 2013.1.bm.d | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 2013.1.bm.d | ✓ | 16 | 11.c | even | 5 | 1 | inner |
| 2013.1.bm.d | ✓ | 16 | 33.h | odd | 10 | 1 | inner |
| 2013.1.bm.d | ✓ | 16 | 61.b | even | 2 | 1 | inner |
| 2013.1.bm.d | ✓ | 16 | 183.d | odd | 2 | 1 | CM |
| 2013.1.bm.d | ✓ | 16 | 671.x | even | 10 | 1 | inner |
| 2013.1.bm.d | ✓ | 16 | 2013.bm | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 4T_{2}^{14} + 17T_{2}^{12} + 72T_{2}^{10} + 230T_{2}^{8} + 228T_{2}^{6} + 92T_{2}^{4} - 4T_{2}^{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2013, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $3$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} \)
|
| $11$ |
\( T^{16} - T^{12} + \cdots + 1 \)
|
| $13$ |
\( (T^{8} + 5 T^{6} + 10 T^{4} + 25)^{2} \)
|
| $17$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $19$ |
\( (T^{8} + 5 T^{6} + 10 T^{4} + 25)^{2} \)
|
| $23$ |
\( (T^{8} - 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} \)
|
| $29$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $31$ |
\( T^{16} \)
|
| $37$ |
\( T^{16} \)
|
| $41$ |
\( T^{16} \)
|
| $43$ |
\( T^{16} \)
|
| $47$ |
\( T^{16} \)
|
| $53$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $59$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $61$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $67$ |
\( T^{16} \)
|
| $71$ |
\( (T^{8} + 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \)
|
| $73$ |
\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{4} \)
|
| $79$ |
\( T^{16} \)
|
| $83$ |
\( T^{16} \)
|
| $89$ |
\( (T^{8} - 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} \)
|
| $97$ |
\( T^{16} \)
|
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