Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [370,2,Mod(121,370)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(370, 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("370.121");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 370 = 2 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 370.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: | \(2.95446487479\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.2696112.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 18\nu^{2} + 8\nu - 40 ) / 82 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 10\nu^{4} + 9\nu^{3} - 36\nu^{2} + 16\nu - 121 ) / 41 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -10\nu^{5} + 9\nu^{4} - 45\nu^{3} - 25\nu^{2} - 162\nu + 72 ) / 82 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -26\nu^{5} + 7\nu^{4} - 117\nu^{3} - 65\nu^{2} - 454\nu - 26 ) / 82 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} - \beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 4\beta_{2} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{5} + 13\beta_{4} - 2\beta _1 - 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} + 11\beta_{4} + 7\beta_{3} - 18\beta_{2} - 18\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(297\) |
| \(\chi(n)\) | \(-1 + \beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 121.1 |
|
−0.500000 | + | 0.866025i | −1.38923 | − | 2.40621i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 2.77846 | −1.65389 | − | 2.86462i | 1.00000 | −2.35991 | + | 4.08749i | −1.00000 | ||||||||||||||||||||||||
| 121.2 | −0.500000 | + | 0.866025i | 0.144584 | + | 0.250427i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.289169 | −1.26222 | − | 2.18623i | 1.00000 | 1.45819 | − | 2.52566i | −1.00000 | |||||||||||||||||||||||||
| 121.3 | −0.500000 | + | 0.866025i | 1.24464 | + | 2.15579i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −2.48929 | 1.91611 | + | 3.31879i | 1.00000 | −1.59828 | + | 2.76830i | −1.00000 | |||||||||||||||||||||||||
| 211.1 | −0.500000 | − | 0.866025i | −1.38923 | + | 2.40621i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 2.77846 | −1.65389 | + | 2.86462i | 1.00000 | −2.35991 | − | 4.08749i | −1.00000 | |||||||||||||||||||||||||
| 211.2 | −0.500000 | − | 0.866025i | 0.144584 | − | 0.250427i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.289169 | −1.26222 | + | 2.18623i | 1.00000 | 1.45819 | + | 2.52566i | −1.00000 | |||||||||||||||||||||||||
| 211.3 | −0.500000 | − | 0.866025i | 1.24464 | − | 2.15579i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −2.48929 | 1.91611 | − | 3.31879i | 1.00000 | −1.59828 | − | 2.76830i | −1.00000 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 370.2.e.f | ✓ | 6 |
| 37.c | even | 3 | 1 | inner | 370.2.e.f | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 370.2.e.f | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 370.2.e.f | ✓ | 6 | 37.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\):
|
\( T_{3}^{6} + 7T_{3}^{4} - 4T_{3}^{3} + 49T_{3}^{2} - 14T_{3} + 4 \)
|
|
\( T_{7}^{6} + 2T_{7}^{5} + 18T_{7}^{4} + 36T_{7}^{3} + 260T_{7}^{2} + 448T_{7} + 1024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{3} \)
|
| $3$ |
\( T^{6} + 7 T^{4} + \cdots + 4 \)
|
| $5$ |
\( (T^{2} - T + 1)^{3} \)
|
| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 1024 \)
|
| $11$ |
\( (T^{3} - 5 T^{2} + 4 T + 2)^{2} \)
|
| $13$ |
\( T^{6} + 3 T^{5} + \cdots + 11881 \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots + 30976 \)
|
| $19$ |
\( T^{6} - 9 T^{5} + \cdots + 16 \)
|
| $23$ |
\( (T^{3} + 11 T^{2} + \cdots + 34)^{2} \)
|
| $29$ |
\( (T^{3} + 2 T^{2} - 28 T + 8)^{2} \)
|
| $31$ |
\( (T^{3} + 12 T^{2} + \cdots - 108)^{2} \)
|
| $37$ |
\( T^{6} - 9 T^{5} + \cdots + 50653 \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots + 1156 \)
|
| $43$ |
\( (T^{3} + 8 T^{2} - 11 T - 2)^{2} \)
|
| $47$ |
\( (T^{3} - 17 T^{2} + 722)^{2} \)
|
| $53$ |
\( T^{6} + 6 T^{5} + \cdots + 1156 \)
|
| $59$ |
\( T^{6} - 13 T^{5} + \cdots + 1156 \)
|
| $61$ |
\( T^{6} - 26 T^{5} + \cdots + 179776 \)
|
| $67$ |
\( T^{6} + 8 T^{5} + \cdots + 719104 \)
|
| $71$ |
\( T^{6} + 10 T^{5} + \cdots + 952576 \)
|
| $73$ |
\( (T^{3} - 2 T^{2} + \cdots + 176)^{2} \)
|
| $79$ |
\( T^{6} + 16 T^{5} + \cdots + 4096 \)
|
| $83$ |
\( T^{6} - 10 T^{5} + \cdots + 123904 \)
|
| $89$ |
\( T^{6} - 27 T^{5} + \cdots + 46656 \)
|
| $97$ |
\( (T^{3} - 6 T^{2} + \cdots + 872)^{2} \)
|
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