Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1690,2,Mod(1689,1690)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1690, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1690.1689");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1690 = 2 \cdot 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1690.c (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: | \(13.4947179416\) |
| 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: | \( 2^{3} \) |
| 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
\( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -9\nu^{5} - 65\nu^{4} + 367\nu^{3} - 270\nu^{2} - 719\nu + 1240 ) / 445 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -11\nu^{5} - 30\nu^{4} + 53\nu^{3} + 115\nu^{2} - 681\nu - 660 ) / 445 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17\nu^{5} - 75\nu^{4} - \nu^{3} + 510\nu^{2} - 323\nu - 760 ) / 445 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -19\nu^{5} + 110\nu^{4} - 313\nu^{3} - 125\nu^{2} + 1251\nu - 1140 ) / 445 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -9\nu^{5} + 24\nu^{4} + 11\nu^{3} - 92\nu^{2} - 7\nu - 6 ) / 89 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} - \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{5} - \beta_{4} + 6\beta_{3} - 7\beta_{2} + 2\beta _1 - 8 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} - 6\beta_{4} + 4\beta_{3} - 8\beta_{2} - 5\beta _1 - 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{5} - 34\beta_{4} - 13\beta_{3} - 30\beta_{2} - 25\beta _1 - 84 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) | \(677\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1689.1 |
|
1.00000 | − | 2.89511i | 1.00000 | 1.70429 | + | 1.44755i | − | 2.89511i | 4.38164 | 1.00000 | −5.38164 | 1.70429 | + | 1.44755i | ||||||||||||||||||||||||||||||
| 1689.2 | 1.00000 | − | 2.29240i | 1.00000 | −1.91995 | + | 1.14620i | − | 2.29240i | 1.25511 | 1.00000 | −2.25511 | −1.91995 | + | 1.14620i | |||||||||||||||||||||||||||||||
| 1689.3 | 1.00000 | − | 0.602705i | 1.00000 | 2.21567 | + | 0.301352i | − | 0.602705i | −3.63675 | 1.00000 | 2.63675 | 2.21567 | + | 0.301352i | |||||||||||||||||||||||||||||||
| 1689.4 | 1.00000 | 0.602705i | 1.00000 | 2.21567 | − | 0.301352i | 0.602705i | −3.63675 | 1.00000 | 2.63675 | 2.21567 | − | 0.301352i | |||||||||||||||||||||||||||||||||
| 1689.5 | 1.00000 | 2.29240i | 1.00000 | −1.91995 | − | 1.14620i | 2.29240i | 1.25511 | 1.00000 | −2.25511 | −1.91995 | − | 1.14620i | |||||||||||||||||||||||||||||||||
| 1689.6 | 1.00000 | 2.89511i | 1.00000 | 1.70429 | − | 1.44755i | 2.89511i | 4.38164 | 1.00000 | −5.38164 | 1.70429 | − | 1.44755i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1690.2.c.d | 6 | |
| 5.b | even | 2 | 1 | 1690.2.c.a | 6 | ||
| 13.b | even | 2 | 1 | 1690.2.c.a | 6 | ||
| 13.d | odd | 4 | 1 | 130.2.b.a | ✓ | 6 | |
| 13.d | odd | 4 | 1 | 1690.2.b.a | 6 | ||
| 39.f | even | 4 | 1 | 1170.2.e.f | 6 | ||
| 52.f | even | 4 | 1 | 1040.2.d.b | 6 | ||
| 65.d | even | 2 | 1 | inner | 1690.2.c.d | 6 | |
| 65.f | even | 4 | 1 | 650.2.a.n | 3 | ||
| 65.f | even | 4 | 1 | 8450.2.a.bs | 3 | ||
| 65.g | odd | 4 | 1 | 130.2.b.a | ✓ | 6 | |
| 65.g | odd | 4 | 1 | 1690.2.b.a | 6 | ||
| 65.k | even | 4 | 1 | 650.2.a.o | 3 | ||
| 65.k | even | 4 | 1 | 8450.2.a.cc | 3 | ||
| 195.j | odd | 4 | 1 | 5850.2.a.cp | 3 | ||
| 195.n | even | 4 | 1 | 1170.2.e.f | 6 | ||
| 195.u | odd | 4 | 1 | 5850.2.a.cs | 3 | ||
| 260.l | odd | 4 | 1 | 5200.2.a.ce | 3 | ||
| 260.s | odd | 4 | 1 | 5200.2.a.cf | 3 | ||
| 260.u | even | 4 | 1 | 1040.2.d.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.2.b.a | ✓ | 6 | 13.d | odd | 4 | 1 | |
| 130.2.b.a | ✓ | 6 | 65.g | odd | 4 | 1 | |
| 650.2.a.n | 3 | 65.f | even | 4 | 1 | ||
| 650.2.a.o | 3 | 65.k | even | 4 | 1 | ||
| 1040.2.d.b | 6 | 52.f | even | 4 | 1 | ||
| 1040.2.d.b | 6 | 260.u | even | 4 | 1 | ||
| 1170.2.e.f | 6 | 39.f | even | 4 | 1 | ||
| 1170.2.e.f | 6 | 195.n | even | 4 | 1 | ||
| 1690.2.b.a | 6 | 13.d | odd | 4 | 1 | ||
| 1690.2.b.a | 6 | 65.g | odd | 4 | 1 | ||
| 1690.2.c.a | 6 | 5.b | even | 2 | 1 | ||
| 1690.2.c.a | 6 | 13.b | even | 2 | 1 | ||
| 1690.2.c.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 1690.2.c.d | 6 | 65.d | even | 2 | 1 | inner | |
| 5200.2.a.ce | 3 | 260.l | odd | 4 | 1 | ||
| 5200.2.a.cf | 3 | 260.s | odd | 4 | 1 | ||
| 5850.2.a.cp | 3 | 195.j | odd | 4 | 1 | ||
| 5850.2.a.cs | 3 | 195.u | odd | 4 | 1 | ||
| 8450.2.a.bs | 3 | 65.f | even | 4 | 1 | ||
| 8450.2.a.cc | 3 | 65.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1690, [\chi])\):
|
\( T_{3}^{6} + 14T_{3}^{4} + 49T_{3}^{2} + 16 \)
|
|
\( T_{7}^{3} - 2T_{7}^{2} - 15T_{7} + 20 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} + 14 T^{4} + \cdots + 16 \)
|
| $5$ |
\( T^{6} - 4 T^{5} + \cdots + 125 \)
|
| $7$ |
\( (T^{3} - 2 T^{2} - 15 T + 20)^{2} \)
|
| $11$ |
\( (T^{2} + 4)^{3} \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} + 102 T^{4} + \cdots + 35344 \)
|
| $19$ |
\( T^{6} + 92 T^{4} + \cdots + 1600 \)
|
| $23$ |
\( T^{6} + 68 T^{4} + \cdots + 256 \)
|
| $29$ |
\( (T^{3} - 2 T^{2} - 44 T - 40)^{2} \)
|
| $31$ |
\( T^{6} + 104 T^{4} + \cdots + 6400 \)
|
| $37$ |
\( (T^{3} - 8 T^{2} + T + 2)^{2} \)
|
| $41$ |
\( T^{6} + 164 T^{4} + \cdots + 102400 \)
|
| $43$ |
\( T^{6} + 174 T^{4} + \cdots + 87616 \)
|
| $47$ |
\( (T^{3} - 10 T^{2} + 17 T + 8)^{2} \)
|
| $53$ |
\( T^{6} + 68 T^{4} + \cdots + 256 \)
|
| $59$ |
\( T^{6} + 92 T^{4} + \cdots + 1600 \)
|
| $61$ |
\( (T^{3} - 10 T^{2} + \cdots - 32)^{2} \)
|
| $67$ |
\( (T^{3} + 12 T^{2} + \cdots - 80)^{2} \)
|
| $71$ |
\( T^{6} + 134 T^{4} + \cdots + 40000 \)
|
| $73$ |
\( (T + 6)^{6} \)
|
| $79$ |
\( (T^{3} + 28 T^{2} + \cdots + 320)^{2} \)
|
| $83$ |
\( (T^{3} - 16 T^{2} + \cdots + 160)^{2} \)
|
| $89$ |
\( T^{6} + 92 T^{4} + \cdots + 1600 \)
|
| $97$ |
\( (T^{3} + 26 T^{2} + \cdots - 1592)^{2} \)
|
| show more | |
| show less |