Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [663,2,Mod(103,663)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(663, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("663.103");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 663 = 3 \cdot 13 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 663.b (of order \(2\), degree \(1\), 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: | \(5.29408165401\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 18x^{8} + 114x^{6} + 292x^{4} + 243x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{10} + 18x^{8} + 114x^{6} + 292x^{4} + 243x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{9} + 10\nu^{7} + 22\nu^{5} - 16\nu^{3} - 37\nu ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} + 10\nu^{4} + 25\nu^{2} + 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{9} + 4\nu^{8} + 10\nu^{7} + 48\nu^{6} + 18\nu^{5} + 168\nu^{4} - 44\nu^{3} + 152\nu^{2} - 69\nu - 4 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{9} + 4\nu^{8} - 10\nu^{7} + 48\nu^{6} - 18\nu^{5} + 168\nu^{4} + 44\nu^{3} + 152\nu^{2} + 69\nu - 4 ) / 8 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{9} - 14\nu^{7} - 66\nu^{5} - 120\nu^{3} - 67\nu ) / 4 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{9} + 4\nu^{8} + 10\nu^{7} + 48\nu^{6} + 18\nu^{5} + 176\nu^{4} - 44\nu^{3} + 208\nu^{2} - 69\nu + 44 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} - \beta_{6} - 7\beta_{2} + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{4} - 7\beta_{3} + 27\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -10\beta_{9} + 10\beta_{6} + \beta_{5} + 45\beta_{2} - 128 \)
|
| \(\nu^{7}\) | \(=\) |
\( -\beta_{8} - 11\beta_{7} + 11\beta_{6} - 12\beta_{4} + 43\beta_{3} - 153\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 78\beta_{9} + \beta_{7} - 77\beta_{6} - 12\beta_{5} - 284\beta_{2} + 765 \)
|
| \(\nu^{9}\) | \(=\) |
\( 10\beta_{8} + 88\beta_{7} - 88\beta_{6} + 102\beta_{4} - 260\beta_{3} + 893\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/663\mathbb{Z}\right)^\times\).
| \(n\) | \(443\) | \(547\) | \(613\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 103.1 |
|
− | 2.51305i | 1.00000 | −4.31544 | − | 3.69376i | − | 2.51305i | 3.30577i | 5.81882i | 1.00000 | −9.28262 | |||||||||||||||||||||||||||||||||||||||||||||
| 103.2 | − | 2.47415i | 1.00000 | −4.12140 | 1.00585i | − | 2.47415i | 2.77451i | 5.24865i | 1.00000 | 2.48861 | |||||||||||||||||||||||||||||||||||||||||||||||
| 103.3 | − | 2.01992i | 1.00000 | −2.08008 | − | 1.17254i | − | 2.01992i | − | 1.85817i | 0.161754i | 1.00000 | −2.36844 | |||||||||||||||||||||||||||||||||||||||||||||
| 103.4 | − | 1.18794i | 1.00000 | 0.588797 | 1.86665i | − | 1.18794i | − | 4.26328i | − | 3.07534i | 1.00000 | 2.21747 | |||||||||||||||||||||||||||||||||||||||||||||
| 103.5 | − | 0.268104i | 1.00000 | 1.92812 | − | 3.93512i | − | 0.268104i | − | 1.32125i | − | 1.05315i | 1.00000 | −1.05502 | ||||||||||||||||||||||||||||||||||||||||||||
| 103.6 | 0.268104i | 1.00000 | 1.92812 | 3.93512i | 0.268104i | 1.32125i | 1.05315i | 1.00000 | −1.05502 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 103.7 | 1.18794i | 1.00000 | 0.588797 | − | 1.86665i | 1.18794i | 4.26328i | 3.07534i | 1.00000 | 2.21747 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 103.8 | 2.01992i | 1.00000 | −2.08008 | 1.17254i | 2.01992i | 1.85817i | − | 0.161754i | 1.00000 | −2.36844 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 103.9 | 2.47415i | 1.00000 | −4.12140 | − | 1.00585i | 2.47415i | − | 2.77451i | − | 5.24865i | 1.00000 | 2.48861 | ||||||||||||||||||||||||||||||||||||||||||||||
| 103.10 | 2.51305i | 1.00000 | −4.31544 | 3.69376i | 2.51305i | − | 3.30577i | − | 5.81882i | 1.00000 | −9.28262 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 663.2.b.f | ✓ | 10 |
| 3.b | odd | 2 | 1 | 1989.2.b.i | 10 | ||
| 13.b | even | 2 | 1 | inner | 663.2.b.f | ✓ | 10 |
| 13.d | odd | 4 | 2 | 8619.2.a.bj | 10 | ||
| 39.d | odd | 2 | 1 | 1989.2.b.i | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 663.2.b.f | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 663.2.b.f | ✓ | 10 | 13.b | even | 2 | 1 | inner |
| 1989.2.b.i | 10 | 3.b | odd | 2 | 1 | ||
| 1989.2.b.i | 10 | 39.d | odd | 2 | 1 | ||
| 8619.2.a.bj | 10 | 13.d | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 18T_{2}^{8} + 114T_{2}^{6} + 292T_{2}^{4} + 243T_{2}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 18 T^{8} + \cdots + 16 \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( T^{10} + 35 T^{8} + \cdots + 1024 \)
|
| $7$ |
\( T^{10} + 42 T^{8} + \cdots + 9216 \)
|
| $11$ |
\( T^{10} + 95 T^{8} + \cdots + 118336 \)
|
| $13$ |
\( T^{10} + 5 T^{9} + \cdots + 371293 \)
|
| $17$ |
\( (T + 1)^{10} \)
|
| $19$ |
\( T^{10} + 119 T^{8} + \cdots + 82944 \)
|
| $23$ |
\( (T^{5} + 13 T^{4} + \cdots + 768)^{2} \)
|
| $29$ |
\( (T^{5} + 2 T^{4} - 44 T^{3} + \cdots + 48)^{2} \)
|
| $31$ |
\( T^{10} + 178 T^{8} + \cdots + 1218816 \)
|
| $37$ |
\( T^{10} + 102 T^{8} + \cdots + 746496 \)
|
| $41$ |
\( T^{10} + 35 T^{8} + \cdots + 1024 \)
|
| $43$ |
\( (T^{5} - 17 T^{4} + \cdots - 14992)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 364351744 \)
|
| $53$ |
\( (T^{5} - 20 T^{4} + \cdots - 9792)^{2} \)
|
| $59$ |
\( T^{10} + 350 T^{8} + \cdots + 8667136 \)
|
| $61$ |
\( (T^{5} - 10 T^{4} + \cdots - 8192)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 376049664 \)
|
| $71$ |
\( T^{10} + \cdots + 1350709504 \)
|
| $73$ |
\( T^{10} + 322 T^{8} + \cdots + 1498176 \)
|
| $79$ |
\( (T^{5} - 10 T^{4} + \cdots - 11776)^{2} \)
|
| $83$ |
\( T^{10} + 418 T^{8} + \cdots + 63744256 \)
|
| $89$ |
\( T^{10} + 126 T^{8} + \cdots + 12544 \)
|
| $97$ |
\( T^{10} + \cdots + 56430952704 \)
|
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