Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2240,2,Mod(1121,2240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2240.1121");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2240 = 2^{6} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2240.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: | \(17.8864900528\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.3534400.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -11\nu^{5} - 30\nu^{4} + 53\nu^{3} + 115\nu^{2} - 236\nu - 660 ) / 445 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -27\nu^{5} + 250\nu^{4} - 679\nu^{3} + 80\nu^{2} + 2293\nu - 2510 ) / 890 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -17\nu^{5} + 75\nu^{4} + \nu^{3} - 510\nu^{2} + 768\nu + 315 ) / 445 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -9\nu^{5} + 24\nu^{4} + 11\nu^{3} - 92\nu^{2} - 7\nu - 6 ) / 178 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -28\nu^{5} + 45\nu^{4} + 54\nu^{3} - 395\nu^{2} - 358\nu + 100 ) / 445 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{3} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{5} + 4\beta_{4} - \beta_{3} + \beta _1 + 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -8\beta_{5} + 11\beta_{4} + \beta_{3} - 3\beta_{2} - 7 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -13\beta_{5} + 30\beta_{4} - 7\beta_{3} - 2\beta_{2} - 15\beta _1 - 19 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -13\beta_{5} + 13\beta_{4} - 8\beta_{3} - 9\beta_{2} - 51\beta _1 - 92 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).
| \(n\) | \(897\) | \(1471\) | \(1541\) | \(1921\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1121.1 |
|
0 | − | 2.89511i | 0 | 1.00000i | 0 | 1.00000 | 0 | −5.38164 | 0 | |||||||||||||||||||||||||||||||||||
| 1121.2 | 0 | − | 2.29240i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | −2.25511 | 0 | |||||||||||||||||||||||||||||||||||
| 1121.3 | 0 | − | 0.602705i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | 2.63675 | 0 | |||||||||||||||||||||||||||||||||||
| 1121.4 | 0 | 0.602705i | 0 | 1.00000i | 0 | 1.00000 | 0 | 2.63675 | 0 | |||||||||||||||||||||||||||||||||||||
| 1121.5 | 0 | 2.29240i | 0 | 1.00000i | 0 | 1.00000 | 0 | −2.25511 | 0 | |||||||||||||||||||||||||||||||||||||
| 1121.6 | 0 | 2.89511i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | −5.38164 | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2240.2.b.f | yes | 6 |
| 4.b | odd | 2 | 1 | 2240.2.b.e | ✓ | 6 | |
| 8.b | even | 2 | 1 | inner | 2240.2.b.f | yes | 6 |
| 8.d | odd | 2 | 1 | 2240.2.b.e | ✓ | 6 | |
| 16.e | even | 4 | 1 | 8960.2.a.bh | 3 | ||
| 16.e | even | 4 | 1 | 8960.2.a.bn | 3 | ||
| 16.f | odd | 4 | 1 | 8960.2.a.bk | 3 | ||
| 16.f | odd | 4 | 1 | 8960.2.a.bq | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2240.2.b.e | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 2240.2.b.e | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 2240.2.b.f | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 2240.2.b.f | yes | 6 | 8.b | even | 2 | 1 | inner |
| 8960.2.a.bh | 3 | 16.e | even | 4 | 1 | ||
| 8960.2.a.bk | 3 | 16.f | odd | 4 | 1 | ||
| 8960.2.a.bn | 3 | 16.e | even | 4 | 1 | ||
| 8960.2.a.bq | 3 | 16.f | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2240, [\chi])\):
|
\( T_{3}^{6} + 14T_{3}^{4} + 49T_{3}^{2} + 16 \)
|
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\( T_{23}^{3} + 4T_{23}^{2} - 60T_{23} - 160 \)
|
|
\( T_{31}^{3} + 8T_{31}^{2} - 24T_{31} - 32 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 14 T^{4} + \cdots + 16 \)
|
| $5$ |
\( (T^{2} + 1)^{3} \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} + 38 T^{4} + \cdots + 100 \)
|
| $13$ |
\( T^{6} + 46 T^{4} + \cdots + 3364 \)
|
| $17$ |
\( (T^{3} + 2 T^{2} + \cdots - 106)^{2} \)
|
| $19$ |
\( T^{6} + 56 T^{4} + \cdots + 1024 \)
|
| $23$ |
\( (T^{3} + 4 T^{2} + \cdots - 160)^{2} \)
|
| $29$ |
\( T^{6} + 102 T^{4} + \cdots + 35344 \)
|
| $31$ |
\( (T^{3} + 8 T^{2} - 24 T - 32)^{2} \)
|
| $37$ |
\( T^{6} + 168 T^{4} + \cdots + 25600 \)
|
| $41$ |
\( (T^{3} + 18 T^{2} + \cdots + 80)^{2} \)
|
| $43$ |
\( (T^{2} + 4)^{3} \)
|
| $47$ |
\( (T^{3} - 10 T^{2} + \cdots + 64)^{2} \)
|
| $53$ |
\( T^{6} + 176 T^{4} + \cdots + 25600 \)
|
| $59$ |
\( T^{6} + 176 T^{4} + \cdots + 25600 \)
|
| $61$ |
\( T^{6} + 292 T^{4} + \cdots + 25600 \)
|
| $67$ |
\( T^{6} + 164 T^{4} + \cdots + 256 \)
|
| $71$ |
\( (T^{3} + 24 T^{2} + \cdots + 320)^{2} \)
|
| $73$ |
\( (T^{3} + 22 T^{2} + \cdots - 160)^{2} \)
|
| $79$ |
\( (T^{3} - 8 T^{2} + \cdots + 1472)^{2} \)
|
| $83$ |
\( T^{6} + 56 T^{4} + \cdots + 1024 \)
|
| $89$ |
\( (T^{3} + 26 T^{2} + \cdots + 160)^{2} \)
|
| $97$ |
\( (T^{3} - 14 T^{2} + \cdots + 230)^{2} \)
|
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