Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1040,2,Mod(209,1040)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1040, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1040.209");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1040 = 2^{4} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1040.d (of order \(2\), degree \(1\), not 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: | \(8.30444181021\) |
| 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: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 130) |
| 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
\( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{5} - 70\nu^{4} + 183\nu^{3} + 120\nu^{2} - 966\nu + 240 ) / 445 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\nu^{5} - 5\nu^{4} - 184\nu^{3} + 390\nu^{2} + 643\nu - 1000 ) / 445 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 8\nu^{4} - 26\nu^{3} - \nu^{2} + 57\nu - 180 ) / 89 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -9\nu^{5} + 24\nu^{4} + 11\nu^{3} - 92\nu^{2} - 7\nu - 6 ) / 178 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{5} - 4\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 14\beta_{5} - 14\beta_{4} + 9\beta_{3} - 3\beta_{2} - 10\beta _1 - 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - 32\beta_{4} + 6\beta_{3} - 17\beta_{2} - 25\beta _1 - 42 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(417\) | \(561\) | \(911\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 209.1 |
|
0 | − | 2.89511i | 0 | 1.44755 | + | 1.70429i | 0 | 4.38164i | 0 | −5.38164 | 0 | |||||||||||||||||||||||||||||||||
| 209.2 | 0 | − | 2.29240i | 0 | −1.14620 | + | 1.91995i | 0 | − | 1.25511i | 0 | −2.25511 | 0 | |||||||||||||||||||||||||||||||||
| 209.3 | 0 | − | 0.602705i | 0 | −0.301352 | − | 2.21567i | 0 | 3.63675i | 0 | 2.63675 | 0 | ||||||||||||||||||||||||||||||||||
| 209.4 | 0 | 0.602705i | 0 | −0.301352 | + | 2.21567i | 0 | − | 3.63675i | 0 | 2.63675 | 0 | ||||||||||||||||||||||||||||||||||
| 209.5 | 0 | 2.29240i | 0 | −1.14620 | − | 1.91995i | 0 | 1.25511i | 0 | −2.25511 | 0 | |||||||||||||||||||||||||||||||||||
| 209.6 | 0 | 2.89511i | 0 | 1.44755 | − | 1.70429i | 0 | − | 4.38164i | 0 | −5.38164 | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1040.2.d.b | 6 | |
| 4.b | odd | 2 | 1 | 130.2.b.a | ✓ | 6 | |
| 5.b | even | 2 | 1 | inner | 1040.2.d.b | 6 | |
| 5.c | odd | 4 | 1 | 5200.2.a.ce | 3 | ||
| 5.c | odd | 4 | 1 | 5200.2.a.cf | 3 | ||
| 12.b | even | 2 | 1 | 1170.2.e.f | 6 | ||
| 20.d | odd | 2 | 1 | 130.2.b.a | ✓ | 6 | |
| 20.e | even | 4 | 1 | 650.2.a.n | 3 | ||
| 20.e | even | 4 | 1 | 650.2.a.o | 3 | ||
| 52.b | odd | 2 | 1 | 1690.2.b.a | 6 | ||
| 52.f | even | 4 | 1 | 1690.2.c.a | 6 | ||
| 52.f | even | 4 | 1 | 1690.2.c.d | 6 | ||
| 60.h | even | 2 | 1 | 1170.2.e.f | 6 | ||
| 60.l | odd | 4 | 1 | 5850.2.a.cp | 3 | ||
| 60.l | odd | 4 | 1 | 5850.2.a.cs | 3 | ||
| 260.g | odd | 2 | 1 | 1690.2.b.a | 6 | ||
| 260.p | even | 4 | 1 | 8450.2.a.bs | 3 | ||
| 260.p | even | 4 | 1 | 8450.2.a.cc | 3 | ||
| 260.u | even | 4 | 1 | 1690.2.c.a | 6 | ||
| 260.u | even | 4 | 1 | 1690.2.c.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.2.b.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 130.2.b.a | ✓ | 6 | 20.d | odd | 2 | 1 | |
| 650.2.a.n | 3 | 20.e | even | 4 | 1 | ||
| 650.2.a.o | 3 | 20.e | even | 4 | 1 | ||
| 1040.2.d.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 1040.2.d.b | 6 | 5.b | even | 2 | 1 | inner | |
| 1170.2.e.f | 6 | 12.b | even | 2 | 1 | ||
| 1170.2.e.f | 6 | 60.h | even | 2 | 1 | ||
| 1690.2.b.a | 6 | 52.b | odd | 2 | 1 | ||
| 1690.2.b.a | 6 | 260.g | odd | 2 | 1 | ||
| 1690.2.c.a | 6 | 52.f | even | 4 | 1 | ||
| 1690.2.c.a | 6 | 260.u | even | 4 | 1 | ||
| 1690.2.c.d | 6 | 52.f | even | 4 | 1 | ||
| 1690.2.c.d | 6 | 260.u | even | 4 | 1 | ||
| 5200.2.a.ce | 3 | 5.c | odd | 4 | 1 | ||
| 5200.2.a.cf | 3 | 5.c | odd | 4 | 1 | ||
| 5850.2.a.cp | 3 | 60.l | odd | 4 | 1 | ||
| 5850.2.a.cs | 3 | 60.l | odd | 4 | 1 | ||
| 8450.2.a.bs | 3 | 260.p | even | 4 | 1 | ||
| 8450.2.a.cc | 3 | 260.p | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 14T_{3}^{4} + 49T_{3}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 14 T^{4} + \cdots + 16 \)
|
| $5$ |
\( T^{6} + 8 T^{4} + \cdots + 125 \)
|
| $7$ |
\( T^{6} + 34 T^{4} + \cdots + 400 \)
|
| $11$ |
\( (T + 2)^{6} \)
|
| $13$ |
\( (T^{2} + 1)^{3} \)
|
| $17$ |
\( T^{6} + 102 T^{4} + \cdots + 35344 \)
|
| $19$ |
\( (T^{3} - 2 T^{2} - 44 T - 40)^{2} \)
|
| $23$ |
\( T^{6} + 68 T^{4} + \cdots + 256 \)
|
| $29$ |
\( (T^{3} - 2 T^{2} - 44 T - 40)^{2} \)
|
| $31$ |
\( (T^{3} + 12 T^{2} + \cdots - 80)^{2} \)
|
| $37$ |
\( T^{6} + 62 T^{4} + \cdots + 4 \)
|
| $41$ |
\( (T^{3} - 2 T^{2} + \cdots + 320)^{2} \)
|
| $43$ |
\( T^{6} + 174 T^{4} + \cdots + 87616 \)
|
| $47$ |
\( T^{6} + 66 T^{4} + \cdots + 64 \)
|
| $53$ |
\( T^{6} + 68 T^{4} + \cdots + 256 \)
|
| $59$ |
\( (T^{3} - 2 T^{2} - 44 T - 40)^{2} \)
|
| $61$ |
\( (T^{3} - 10 T^{2} + \cdots - 32)^{2} \)
|
| $67$ |
\( T^{6} + 104 T^{4} + \cdots + 6400 \)
|
| $71$ |
\( (T^{3} - 8 T^{2} + \cdots + 200)^{2} \)
|
| $73$ |
\( (T^{2} + 36)^{3} \)
|
| $79$ |
\( (T^{3} - 28 T^{2} + \cdots - 320)^{2} \)
|
| $83$ |
\( T^{6} + 176 T^{4} + \cdots + 25600 \)
|
| $89$ |
\( (T^{3} - 2 T^{2} - 44 T - 40)^{2} \)
|
| $97$ |
\( T^{6} + 572 T^{4} + \cdots + 2534464 \)
|
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