Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [639,1,Mod(70,639)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(639, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 3]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("639.70");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 639 = 3^{2} \cdot 71 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 639.g (of order \(6\), degree \(2\), 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: | \(0.318902543072\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{21})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{21}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/639\mathbb{Z}\right)^\times\).
| \(n\) | \(433\) | \(569\) |
| \(\chi(n)\) | \(-1\) | \(-\zeta_{42}^{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 70.1 |
|
−0.955573 | + | 1.65510i | 0.365341 | − | 0.930874i | −1.32624 | − | 2.29711i | 0.900969 | + | 1.56052i | 1.19158 | + | 1.49419i | 0 | 3.15813 | −0.733052 | − | 0.680173i | −3.44377 | ||||||||||||||||||||||||||||||||||||||||||
| 70.2 | −0.826239 | + | 1.43109i | −0.733052 | + | 0.680173i | −0.865341 | − | 1.49881i | −0.623490 | − | 1.07992i | −0.367711 | − | 1.61105i | 0 | 1.20744 | 0.0747301 | − | 0.997204i | 2.06061 | |||||||||||||||||||||||||||||||||||||||||||
| 70.3 | −0.365341 | + | 0.632789i | 0.0747301 | + | 0.997204i | 0.233052 | + | 0.403658i | 0.222521 | + | 0.385418i | −0.658322 | − | 0.317031i | 0 | −1.07126 | −0.988831 | + | 0.149042i | −0.325184 | |||||||||||||||||||||||||||||||||||||||||||
| 70.4 | −0.0747301 | + | 0.129436i | 0.955573 | − | 0.294755i | 0.488831 | + | 0.846680i | −0.623490 | − | 1.07992i | −0.0332580 | + | 0.145713i | 0 | −0.295582 | 0.826239 | − | 0.563320i | 0.186374 | |||||||||||||||||||||||||||||||||||||||||||
| 70.5 | 0.733052 | − | 1.26968i | −0.988831 | − | 0.149042i | −0.574730 | − | 0.995462i | 0.900969 | + | 1.56052i | −0.914101 | + | 1.14625i | 0 | −0.219124 | 0.955573 | + | 0.294755i | 2.64183 | |||||||||||||||||||||||||||||||||||||||||||
| 70.6 | 0.988831 | − | 1.71271i | 0.826239 | + | 0.563320i | −1.45557 | − | 2.52113i | 0.222521 | + | 0.385418i | 1.78181 | − | 0.858075i | 0 | −3.77960 | 0.365341 | + | 0.930874i | 0.880142 | |||||||||||||||||||||||||||||||||||||||||||
| 283.1 | −0.955573 | − | 1.65510i | 0.365341 | + | 0.930874i | −1.32624 | + | 2.29711i | 0.900969 | − | 1.56052i | 1.19158 | − | 1.49419i | 0 | 3.15813 | −0.733052 | + | 0.680173i | −3.44377 | |||||||||||||||||||||||||||||||||||||||||||
| 283.2 | −0.826239 | − | 1.43109i | −0.733052 | − | 0.680173i | −0.865341 | + | 1.49881i | −0.623490 | + | 1.07992i | −0.367711 | + | 1.61105i | 0 | 1.20744 | 0.0747301 | + | 0.997204i | 2.06061 | |||||||||||||||||||||||||||||||||||||||||||
| 283.3 | −0.365341 | − | 0.632789i | 0.0747301 | − | 0.997204i | 0.233052 | − | 0.403658i | 0.222521 | − | 0.385418i | −0.658322 | + | 0.317031i | 0 | −1.07126 | −0.988831 | − | 0.149042i | −0.325184 | |||||||||||||||||||||||||||||||||||||||||||
| 283.4 | −0.0747301 | − | 0.129436i | 0.955573 | + | 0.294755i | 0.488831 | − | 0.846680i | −0.623490 | + | 1.07992i | −0.0332580 | − | 0.145713i | 0 | −0.295582 | 0.826239 | + | 0.563320i | 0.186374 | |||||||||||||||||||||||||||||||||||||||||||
| 283.5 | 0.733052 | + | 1.26968i | −0.988831 | + | 0.149042i | −0.574730 | + | 0.995462i | 0.900969 | − | 1.56052i | −0.914101 | − | 1.14625i | 0 | −0.219124 | 0.955573 | − | 0.294755i | 2.64183 | |||||||||||||||||||||||||||||||||||||||||||
| 283.6 | 0.988831 | + | 1.71271i | 0.826239 | − | 0.563320i | −1.45557 | + | 2.52113i | 0.222521 | − | 0.385418i | 1.78181 | + | 0.858075i | 0 | −3.77960 | 0.365341 | − | 0.930874i | 0.880142 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 71.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-71}) \) |
| 9.c | even | 3 | 1 | inner |
| 639.g | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 639.1.g.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 1917.1.g.b | 12 | ||
| 9.c | even | 3 | 1 | inner | 639.1.g.b | ✓ | 12 |
| 9.d | odd | 6 | 1 | 1917.1.g.b | 12 | ||
| 71.b | odd | 2 | 1 | CM | 639.1.g.b | ✓ | 12 |
| 213.b | even | 2 | 1 | 1917.1.g.b | 12 | ||
| 639.g | odd | 6 | 1 | inner | 639.1.g.b | ✓ | 12 |
| 639.i | even | 6 | 1 | 1917.1.g.b | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 639.1.g.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 639.1.g.b | ✓ | 12 | 9.c | even | 3 | 1 | inner |
| 639.1.g.b | ✓ | 12 | 71.b | odd | 2 | 1 | CM |
| 639.1.g.b | ✓ | 12 | 639.g | odd | 6 | 1 | inner |
| 1917.1.g.b | 12 | 3.b | odd | 2 | 1 | ||
| 1917.1.g.b | 12 | 9.d | odd | 6 | 1 | ||
| 1917.1.g.b | 12 | 213.b | even | 2 | 1 | ||
| 1917.1.g.b | 12 | 639.i | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + T_{2}^{11} + 7 T_{2}^{10} + 6 T_{2}^{9} + 34 T_{2}^{8} + 28 T_{2}^{7} + 78 T_{2}^{6} + \cdots + 1 \)
acting on \(S_{1}^{\mathrm{new}}(639, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + T^{11} + \cdots + 1 \)
|
| $3$ |
\( T^{12} - T^{11} + \cdots + 1 \)
|
| $5$ |
\( (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( T^{12} \)
|
| $13$ |
\( T^{12} \)
|
| $17$ |
\( T^{12} \)
|
| $19$ |
\( (T^{6} - T^{5} - 6 T^{4} + \cdots + 1)^{2} \)
|
| $23$ |
\( T^{12} \)
|
| $29$ |
\( T^{12} + T^{11} + \cdots + 1 \)
|
| $31$ |
\( T^{12} \)
|
| $37$ |
\( (T^{3} + T^{2} - 2 T - 1)^{4} \)
|
| $41$ |
\( T^{12} \)
|
| $43$ |
\( T^{12} + T^{11} + \cdots + 1 \)
|
| $47$ |
\( T^{12} \)
|
| $53$ |
\( T^{12} \)
|
| $59$ |
\( T^{12} \)
|
| $61$ |
\( T^{12} \)
|
| $67$ |
\( T^{12} \)
|
| $71$ |
\( (T - 1)^{12} \)
|
| $73$ |
\( (T^{3} + T^{2} - 2 T - 1)^{4} \)
|
| $79$ |
\( (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} \)
|
| $83$ |
\( (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} \)
|
| $89$ |
\( (T^{6} - T^{5} - 6 T^{4} + \cdots + 1)^{2} \)
|
| $97$ |
\( T^{12} \)
|
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