Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1445,2,Mod(1,1445)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1445, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1445.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1445 = 5 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1445.a (trivial) |
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: | yes |
| Analytic conductor: | \(11.5383830921\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.7718912.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 7x^{4} + 4x^{3} + 9x^{2} - 2x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 85) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{6} - 2x^{5} - 7x^{4} + 4x^{3} + 9x^{2} - 2x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 3\nu^{2} + 5\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 3\nu^{3} + 5\nu^{2} - \nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + \nu^{2} + 6\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 7\nu^{2} + 4\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + 2\beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} - 2\beta_{4} + 3\beta_{3} + 2\beta_{2} + 9\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{5} - 9\beta_{4} + 12\beta_{3} + 10\beta_{2} + 28\beta _1 + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{5} - 28\beta_{4} + 41\beta_{3} + 31\beta_{2} + 102\beta _1 + 32 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.24891 | −0.198035 | 3.05761 | 1.00000 | 0.445364 | −1.89231 | −2.37848 | −2.96078 | −2.24891 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −0.783476 | 0.544977 | −1.38617 | 1.00000 | −0.426976 | 1.18848 | 2.65298 | −2.70300 | −0.783476 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.677603 | 2.35757 | −1.54085 | 1.00000 | −1.59750 | 4.27746 | 2.39929 | 2.55814 | −0.677603 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.12708 | −2.48265 | −0.729699 | 1.00000 | −2.79814 | 2.44256 | −3.07658 | 3.16356 | 1.12708 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.07061 | 2.52346 | 2.28744 | 1.00000 | 5.22511 | 0.368961 | 0.595174 | 3.36786 | 2.07061 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.51230 | 1.25468 | 4.31167 | 1.00000 | 3.15213 | 1.61485 | 5.80761 | −1.42578 | 2.51230 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(17\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1445.2.a.o | 6 | |
| 5.b | even | 2 | 1 | 7225.2.a.z | 6 | ||
| 17.b | even | 2 | 1 | 1445.2.a.n | 6 | ||
| 17.c | even | 4 | 2 | 1445.2.d.g | 12 | ||
| 17.d | even | 8 | 2 | 85.2.e.a | ✓ | 12 | |
| 51.g | odd | 8 | 2 | 765.2.k.b | 12 | ||
| 68.g | odd | 8 | 2 | 1360.2.bt.d | 12 | ||
| 85.c | even | 2 | 1 | 7225.2.a.bb | 6 | ||
| 85.k | odd | 8 | 2 | 425.2.j.b | 12 | ||
| 85.m | even | 8 | 2 | 425.2.e.f | 12 | ||
| 85.n | odd | 8 | 2 | 425.2.j.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.2.e.a | ✓ | 12 | 17.d | even | 8 | 2 | |
| 425.2.e.f | 12 | 85.m | even | 8 | 2 | ||
| 425.2.j.b | 12 | 85.k | odd | 8 | 2 | ||
| 425.2.j.c | 12 | 85.n | odd | 8 | 2 | ||
| 765.2.k.b | 12 | 51.g | odd | 8 | 2 | ||
| 1360.2.bt.d | 12 | 68.g | odd | 8 | 2 | ||
| 1445.2.a.n | 6 | 17.b | even | 2 | 1 | ||
| 1445.2.a.o | 6 | 1.a | even | 1 | 1 | trivial | |
| 1445.2.d.g | 12 | 17.c | even | 4 | 2 | ||
| 7225.2.a.z | 6 | 5.b | even | 2 | 1 | ||
| 7225.2.a.bb | 6 | 85.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1445))\):
|
\( T_{2}^{6} - 2T_{2}^{5} - 7T_{2}^{4} + 12T_{2}^{3} + 11T_{2}^{2} - 10T_{2} - 7 \)
|
|
\( T_{3}^{6} - 4T_{3}^{5} - 2T_{3}^{4} + 24T_{3}^{3} - 26T_{3}^{2} + 4T_{3} + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots - 7 \)
|
| $3$ |
\( T^{6} - 4 T^{5} + \cdots + 2 \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} - 8 T^{5} + \cdots - 14 \)
|
| $11$ |
\( T^{6} - 26 T^{4} + \cdots - 446 \)
|
| $13$ |
\( T^{6} - 58 T^{4} + \cdots - 316 \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( T^{6} + 12 T^{5} + \cdots - 1472 \)
|
| $23$ |
\( T^{6} - 42 T^{4} + \cdots - 14 \)
|
| $29$ |
\( T^{6} - 8 T^{5} + \cdots - 2312 \)
|
| $31$ |
\( T^{6} - 4 T^{5} + \cdots + 40562 \)
|
| $37$ |
\( T^{6} - 16 T^{5} + \cdots + 1912 \)
|
| $41$ |
\( T^{6} - 12 T^{5} + \cdots - 13888 \)
|
| $43$ |
\( T^{6} - 8 T^{5} + \cdots - 15004 \)
|
| $47$ |
\( T^{6} - 24 T^{5} + \cdots - 18076 \)
|
| $53$ |
\( T^{6} - 216 T^{4} + \cdots - 132112 \)
|
| $59$ |
\( T^{6} + 16 T^{5} + \cdots + 896 \)
|
| $61$ |
\( T^{6} + 4 T^{5} + \cdots - 992 \)
|
| $67$ |
\( T^{6} + 4 T^{5} + \cdots - 92 \)
|
| $71$ |
\( T^{6} - 32 T^{5} + \cdots - 170686 \)
|
| $73$ |
\( T^{6} + 8 T^{5} + \cdots - 736 \)
|
| $79$ |
\( T^{6} - 36 T^{5} + \cdots - 3902 \)
|
| $83$ |
\( T^{6} - 20 T^{5} + \cdots + 11204 \)
|
| $89$ |
\( T^{6} + 12 T^{5} + \cdots - 31292 \)
|
| $97$ |
\( T^{6} - 16 T^{5} + \cdots + 476536 \)
|
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