Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [309,2,Mod(47,309)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(309, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("309.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 309 = 3 \cdot 103 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 309.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: | \(2.46737742246\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.1568160000.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 7x^{6} + 38x^{4} - 77x^{2} + 121 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 7x^{6} + 38x^{4} - 77x^{2} + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7\nu^{6} - 38\nu^{4} + 266\nu^{2} - 539 ) / 418 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{7} - 38\nu^{5} + 266\nu^{3} - 539\nu ) / 418 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 75\nu ) / 38 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{6} + 57\nu^{4} - 190\nu^{2} + 385 ) / 209 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} - 37 ) / 38 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -17\nu^{7} + 152\nu^{5} - 646\nu^{3} + 1309\nu ) / 418 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 3\beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{4} + 4\beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} + 10\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{7} + 17\beta_{3} \)
|
| \(\nu^{6}\) | \(=\) |
\( -38\beta_{6} - 37 \)
|
| \(\nu^{7}\) | \(=\) |
\( -38\beta_{4} - 75\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/309\mathbb{Z}\right)^\times\).
| \(n\) | \(104\) | \(211\) |
| \(\chi(n)\) | \(-1\) | \(1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 47.1 |
|
−1.86105 | − | 1.07448i | 1.73205i | 1.30902 | + | 2.26728i | 1.15020 | + | 1.99220i | 1.86105 | − | 3.22344i | 2.11803 | + | 3.66854i | − | 1.32813i | −3.00000 | − | 4.94345i | ||||||||||||||||||||||||||||||
| 47.2 | −1.33659 | − | 0.771681i | 1.73205i | 0.190983 | + | 0.330792i | −2.16265 | − | 3.74582i | 1.33659 | − | 2.31504i | −0.118034 | − | 0.204441i | 2.49721i | −3.00000 | 6.67550i | |||||||||||||||||||||||||||||||||
| 47.3 | 1.33659 | + | 0.771681i | 1.73205i | 0.190983 | + | 0.330792i | 2.16265 | + | 3.74582i | −1.33659 | + | 2.31504i | −0.118034 | − | 0.204441i | − | 2.49721i | −3.00000 | 6.67550i | ||||||||||||||||||||||||||||||||
| 47.4 | 1.86105 | + | 1.07448i | 1.73205i | 1.30902 | + | 2.26728i | −1.15020 | − | 1.99220i | −1.86105 | + | 3.22344i | 2.11803 | + | 3.66854i | 1.32813i | −3.00000 | − | 4.94345i | ||||||||||||||||||||||||||||||||
| 263.1 | −1.86105 | + | 1.07448i | − | 1.73205i | 1.30902 | − | 2.26728i | 1.15020 | − | 1.99220i | 1.86105 | + | 3.22344i | 2.11803 | − | 3.66854i | 1.32813i | −3.00000 | 4.94345i | ||||||||||||||||||||||||||||||||
| 263.2 | −1.33659 | + | 0.771681i | − | 1.73205i | 0.190983 | − | 0.330792i | −2.16265 | + | 3.74582i | 1.33659 | + | 2.31504i | −0.118034 | + | 0.204441i | − | 2.49721i | −3.00000 | − | 6.67550i | ||||||||||||||||||||||||||||||
| 263.3 | 1.33659 | − | 0.771681i | − | 1.73205i | 0.190983 | − | 0.330792i | 2.16265 | − | 3.74582i | −1.33659 | − | 2.31504i | −0.118034 | + | 0.204441i | 2.49721i | −3.00000 | − | 6.67550i | |||||||||||||||||||||||||||||||
| 263.4 | 1.86105 | − | 1.07448i | − | 1.73205i | 1.30902 | − | 2.26728i | −1.15020 | + | 1.99220i | −1.86105 | − | 3.22344i | 2.11803 | − | 3.66854i | − | 1.32813i | −3.00000 | 4.94345i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 103.d | odd | 6 | 1 | inner |
| 309.g | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 309.2.g.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 309.2.g.b | ✓ | 8 |
| 103.d | odd | 6 | 1 | inner | 309.2.g.b | ✓ | 8 |
| 309.g | even | 6 | 1 | inner | 309.2.g.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 309.2.g.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 309.2.g.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 309.2.g.b | ✓ | 8 | 103.d | odd | 6 | 1 | inner |
| 309.2.g.b | ✓ | 8 | 309.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 7T_{2}^{6} + 38T_{2}^{4} - 77T_{2}^{2} + 121 \)
acting on \(S_{2}^{\mathrm{new}}(309, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 7 T^{6} + \cdots + 121 \)
|
| $3$ |
\( (T^{2} + 3)^{4} \)
|
| $5$ |
\( T^{8} + 24 T^{6} + \cdots + 9801 \)
|
| $7$ |
\( (T^{4} - 4 T^{3} + 17 T^{2} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{8} + 24 T^{6} + \cdots + 9801 \)
|
| $13$ |
\( (T^{2} + 2 T - 4)^{4} \)
|
| $17$ |
\( T^{8} - 40 T^{6} + \cdots + 75625 \)
|
| $19$ |
\( (T^{4} + 2 T^{3} + \cdots + 361)^{2} \)
|
| $23$ |
\( (T^{4} + 28 T^{2} + 176)^{2} \)
|
| $29$ |
\( T^{8} - 40 T^{6} + \cdots + 75625 \)
|
| $31$ |
\( (T^{4} + 36 T^{2} + 144)^{2} \)
|
| $37$ |
\( (T^{4} + 36 T^{2} + 144)^{2} \)
|
| $41$ |
\( T^{8} - 160 T^{6} + \cdots + 75625 \)
|
| $43$ |
\( (T^{2} - 3 T + 3)^{4} \)
|
| $47$ |
\( T^{8} + 24 T^{6} + \cdots + 9801 \)
|
| $53$ |
\( T^{8} + 120 T^{6} + \cdots + 6125625 \)
|
| $59$ |
\( T^{8} - 40 T^{6} + \cdots + 75625 \)
|
| $61$ |
\( (T^{2} + 14 T + 44)^{4} \)
|
| $67$ |
\( (T^{4} - 6 T^{3} + \cdots + 3249)^{2} \)
|
| $71$ |
\( T^{8} + 264 T^{6} + \cdots + 143496441 \)
|
| $73$ |
\( (T^{4} + 276 T^{2} + 17424)^{2} \)
|
| $79$ |
\( (T^{2} + 8 T - 64)^{4} \)
|
| $83$ |
\( T^{8} - 208 T^{6} + \cdots + 111746041 \)
|
| $89$ |
\( (T^{4} - 96 T^{2} + 1584)^{2} \)
|
| $97$ |
\( (T^{4} + 10 T^{3} + \cdots + 25)^{2} \)
|
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