Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1690,2,Mod(1351,1690)]
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([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1690.1351");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1690 = 2 \cdot 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1690.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: | \(13.4947179416\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{12}^{2} - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( ( \beta_{2} + 1 ) / 2 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( \beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1351.1 |
|
− | 1.00000i | −2.73205 | −1.00000 | 1.00000i | 2.73205i | − | 3.00000i | 1.00000i | 4.46410 | 1.00000 | ||||||||||||||||||||||||||||
| 1351.2 | − | 1.00000i | 0.732051 | −1.00000 | 1.00000i | − | 0.732051i | − | 3.00000i | 1.00000i | −2.46410 | 1.00000 | ||||||||||||||||||||||||||||
| 1351.3 | 1.00000i | −2.73205 | −1.00000 | − | 1.00000i | − | 2.73205i | 3.00000i | − | 1.00000i | 4.46410 | 1.00000 | ||||||||||||||||||||||||||||
| 1351.4 | 1.00000i | 0.732051 | −1.00000 | − | 1.00000i | 0.732051i | 3.00000i | − | 1.00000i | −2.46410 | 1.00000 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | 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.d.f | 4 | |
| 13.b | even | 2 | 1 | inner | 1690.2.d.f | 4 | |
| 13.c | even | 3 | 1 | 130.2.l.a | ✓ | 4 | |
| 13.c | even | 3 | 1 | 1690.2.l.g | 4 | ||
| 13.d | odd | 4 | 1 | 1690.2.a.j | 2 | ||
| 13.d | odd | 4 | 1 | 1690.2.a.m | 2 | ||
| 13.e | even | 6 | 1 | 130.2.l.a | ✓ | 4 | |
| 13.e | even | 6 | 1 | 1690.2.l.g | 4 | ||
| 13.f | odd | 12 | 2 | 1690.2.e.l | 4 | ||
| 13.f | odd | 12 | 2 | 1690.2.e.n | 4 | ||
| 39.h | odd | 6 | 1 | 1170.2.bs.c | 4 | ||
| 39.i | odd | 6 | 1 | 1170.2.bs.c | 4 | ||
| 52.i | odd | 6 | 1 | 1040.2.da.a | 4 | ||
| 52.j | odd | 6 | 1 | 1040.2.da.a | 4 | ||
| 65.g | odd | 4 | 1 | 8450.2.a.bf | 2 | ||
| 65.g | odd | 4 | 1 | 8450.2.a.bm | 2 | ||
| 65.l | even | 6 | 1 | 650.2.m.a | 4 | ||
| 65.n | even | 6 | 1 | 650.2.m.a | 4 | ||
| 65.q | odd | 12 | 1 | 650.2.n.a | 4 | ||
| 65.q | odd | 12 | 1 | 650.2.n.b | 4 | ||
| 65.r | odd | 12 | 1 | 650.2.n.a | 4 | ||
| 65.r | odd | 12 | 1 | 650.2.n.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.2.l.a | ✓ | 4 | 13.c | even | 3 | 1 | |
| 130.2.l.a | ✓ | 4 | 13.e | even | 6 | 1 | |
| 650.2.m.a | 4 | 65.l | even | 6 | 1 | ||
| 650.2.m.a | 4 | 65.n | even | 6 | 1 | ||
| 650.2.n.a | 4 | 65.q | odd | 12 | 1 | ||
| 650.2.n.a | 4 | 65.r | odd | 12 | 1 | ||
| 650.2.n.b | 4 | 65.q | odd | 12 | 1 | ||
| 650.2.n.b | 4 | 65.r | odd | 12 | 1 | ||
| 1040.2.da.a | 4 | 52.i | odd | 6 | 1 | ||
| 1040.2.da.a | 4 | 52.j | odd | 6 | 1 | ||
| 1170.2.bs.c | 4 | 39.h | odd | 6 | 1 | ||
| 1170.2.bs.c | 4 | 39.i | odd | 6 | 1 | ||
| 1690.2.a.j | 2 | 13.d | odd | 4 | 1 | ||
| 1690.2.a.m | 2 | 13.d | odd | 4 | 1 | ||
| 1690.2.d.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 1690.2.d.f | 4 | 13.b | even | 2 | 1 | inner | |
| 1690.2.e.l | 4 | 13.f | odd | 12 | 2 | ||
| 1690.2.e.n | 4 | 13.f | odd | 12 | 2 | ||
| 1690.2.l.g | 4 | 13.c | even | 3 | 1 | ||
| 1690.2.l.g | 4 | 13.e | even | 6 | 1 | ||
| 8450.2.a.bf | 2 | 65.g | odd | 4 | 1 | ||
| 8450.2.a.bm | 2 | 65.g | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1690, [\chi])\):
|
\( T_{3}^{2} + 2T_{3} - 2 \)
|
|
\( T_{7}^{2} + 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
|
| $3$ |
\( (T^{2} + 2 T - 2)^{2} \)
|
| $5$ |
\( (T^{2} + 1)^{2} \)
|
| $7$ |
\( (T^{2} + 9)^{2} \)
|
| $11$ |
\( (T^{2} + 9)^{2} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} - 6 T - 18)^{2} \)
|
| $19$ |
\( T^{4} + 42T^{2} + 9 \)
|
| $23$ |
\( (T^{2} - 12 T + 24)^{2} \)
|
| $29$ |
\( (T^{2} + 12 T + 24)^{2} \)
|
| $31$ |
\( T^{4} + 24T^{2} + 36 \)
|
| $37$ |
\( T^{4} + 126T^{2} + 81 \)
|
| $41$ |
\( (T^{2} + 108)^{2} \)
|
| $43$ |
\( (T - 2)^{4} \)
|
| $47$ |
\( (T^{2} + 9)^{2} \)
|
| $53$ |
\( (T^{2} + 6 T - 3)^{2} \)
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| $59$ |
\( (T^{2} + 108)^{2} \)
|
| $61$ |
\( (T^{2} - 2 T - 26)^{2} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( (T^{2} + 36)^{2} \)
|
| $73$ |
\( T^{4} + 168T^{2} + 4356 \)
|
| $79$ |
\( (T^{2} - 2 T - 26)^{2} \)
|
| $83$ |
\( T^{4} + 72T^{2} + 324 \)
|
| $89$ |
\( T^{4} + 342 T^{2} + 13689 \)
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| $97$ |
\( T^{4} + 312 T^{2} + 19044 \)
|
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