Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [310,2,Mod(191,310)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(310, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("310.191");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 310 = 2 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 310.e (of order \(3\), 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: | \(2.47536246266\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.309123.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(251\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 191.1 |
|
1.00000 | −1.23025 | + | 2.13086i | 1.00000 | 0.500000 | + | 0.866025i | −1.23025 | + | 2.13086i | −1.73025 | + | 2.99689i | 1.00000 | −1.52704 | − | 2.64491i | 0.500000 | + | 0.866025i | ||||||||||||||||||||||||
| 191.2 | 1.00000 | −0.119562 | + | 0.207087i | 1.00000 | 0.500000 | + | 0.866025i | −0.119562 | + | 0.207087i | −0.619562 | + | 1.07311i | 1.00000 | 1.47141 | + | 2.54856i | 0.500000 | + | 0.866025i | |||||||||||||||||||||||||
| 191.3 | 1.00000 | 0.849814 | − | 1.47192i | 1.00000 | 0.500000 | + | 0.866025i | 0.849814 | − | 1.47192i | 0.349814 | − | 0.605896i | 1.00000 | 0.0556321 | + | 0.0963576i | 0.500000 | + | 0.866025i | |||||||||||||||||||||||||
| 211.1 | 1.00000 | −1.23025 | − | 2.13086i | 1.00000 | 0.500000 | − | 0.866025i | −1.23025 | − | 2.13086i | −1.73025 | − | 2.99689i | 1.00000 | −1.52704 | + | 2.64491i | 0.500000 | − | 0.866025i | |||||||||||||||||||||||||
| 211.2 | 1.00000 | −0.119562 | − | 0.207087i | 1.00000 | 0.500000 | − | 0.866025i | −0.119562 | − | 0.207087i | −0.619562 | − | 1.07311i | 1.00000 | 1.47141 | − | 2.54856i | 0.500000 | − | 0.866025i | |||||||||||||||||||||||||
| 211.3 | 1.00000 | 0.849814 | + | 1.47192i | 1.00000 | 0.500000 | − | 0.866025i | 0.849814 | + | 1.47192i | 0.349814 | + | 0.605896i | 1.00000 | 0.0556321 | − | 0.0963576i | 0.500000 | − | 0.866025i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 310.2.e.c | ✓ | 6 |
| 5.b | even | 2 | 1 | 1550.2.e.l | 6 | ||
| 5.c | odd | 4 | 2 | 1550.2.p.g | 12 | ||
| 31.c | even | 3 | 1 | inner | 310.2.e.c | ✓ | 6 |
| 31.c | even | 3 | 1 | 9610.2.a.x | 3 | ||
| 31.e | odd | 6 | 1 | 9610.2.a.v | 3 | ||
| 155.j | even | 6 | 1 | 1550.2.e.l | 6 | ||
| 155.o | odd | 12 | 2 | 1550.2.p.g | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 310.2.e.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 310.2.e.c | ✓ | 6 | 31.c | even | 3 | 1 | inner |
| 1550.2.e.l | 6 | 5.b | even | 2 | 1 | ||
| 1550.2.e.l | 6 | 155.j | even | 6 | 1 | ||
| 1550.2.p.g | 12 | 5.c | odd | 4 | 2 | ||
| 1550.2.p.g | 12 | 155.o | odd | 12 | 2 | ||
| 9610.2.a.v | 3 | 31.e | odd | 6 | 1 | ||
| 9610.2.a.x | 3 | 31.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + T_{3}^{5} + 5T_{3}^{4} - 2T_{3}^{3} + 17T_{3}^{2} + 4T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(310, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} + T^{5} + 5 T^{4} + \cdots + 1 \)
|
| $5$ |
\( (T^{2} - T + 1)^{3} \)
|
| $7$ |
\( T^{6} + 4 T^{5} + \cdots + 9 \)
|
| $11$ |
\( T^{6} + 5 T^{5} + \cdots + 6561 \)
|
| $13$ |
\( T^{6} + T^{5} + \cdots + 1089 \)
|
| $17$ |
\( T^{6} - T^{5} + 7 T^{4} + \cdots + 9 \)
|
| $19$ |
\( T^{6} + 3 T^{5} + \cdots + 49 \)
|
| $23$ |
\( (T^{3} - 5 T^{2} + \cdots + 363)^{2} \)
|
| $29$ |
\( (T^{3} - 2 T^{2} + \cdots + 147)^{2} \)
|
| $31$ |
\( T^{6} + 16 T^{5} + \cdots + 29791 \)
|
| $37$ |
\( T^{6} + 5 T^{5} + \cdots + 38809 \)
|
| $41$ |
\( T^{6} + 3 T^{5} + \cdots + 6561 \)
|
| $43$ |
\( T^{6} - 9 T^{5} + \cdots + 434281 \)
|
| $47$ |
\( (T^{3} + 6 T^{2} + 3 T - 9)^{2} \)
|
| $53$ |
\( T^{6} + 11 T^{5} + \cdots + 335241 \)
|
| $59$ |
\( T^{6} - 10 T^{5} + \cdots + 1089 \)
|
| $61$ |
\( (T^{3} - 7 T^{2} + \cdots + 123)^{2} \)
|
| $67$ |
\( T^{6} - 25 T^{5} + \cdots + 185761 \)
|
| $71$ |
\( T^{6} + 9 T^{5} + \cdots + 6561 \)
|
| $73$ |
\( T^{6} + 28 T^{5} + \cdots + 53361 \)
|
| $79$ |
\( T^{6} - 9 T^{5} + \cdots + 78961 \)
|
| $83$ |
\( T^{6} - 3 T^{5} + \cdots + 227529 \)
|
| $89$ |
\( (T^{3} - 9 T^{2} + \cdots + 243)^{2} \)
|
| $97$ |
\( (T^{3} + 16 T^{2} + \cdots + 77)^{2} \)
|
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