Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,3,Mod(145,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.145");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.be (of order \(6\), degree \(2\), 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: | \(24.8502001097\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 6.0.6967728.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 57) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{5} - 16\nu^{4} + 128\nu^{3} - 100\nu^{2} + 14\nu + 281 ) / 393 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -56\nu^{5} + 55\nu^{4} - 440\nu^{3} - 344\nu^{2} - 1964\nu - 8 ) / 393 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -56\nu^{5} + 55\nu^{4} - 440\nu^{3} - 344\nu^{2} - 2750\nu - 8 ) / 393 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 196\nu^{5} - 127\nu^{4} + 1540\nu^{3} + 1466\nu^{2} + 10018\nu + 1469 ) / 393 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -210\nu^{5} + 239\nu^{4} - 1650\nu^{3} - 766\nu^{2} - 10116\nu + 2459 ) / 393 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} - \beta_{4} - 11\beta_{3} - 9 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 7\beta _1 - 15 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{5} + 8\beta_{4} + 46\beta_{3} - 3\beta_{2} - 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -28\beta_{5} + 14\beta_{4} + 193\beta_{3} - 55\beta_{2} - 55\beta _1 + 165 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(229\) | \(305\) | \(799\) |
| \(\chi(n)\) | \(1 + \beta_{3}\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 145.1 |
|
0 | −1.50000 | + | 0.866025i | 0 | −3.13264 | − | 5.42589i | 0 | −8.36156 | 0 | 1.50000 | − | 2.59808i | 0 | ||||||||||||||||||||||||||||||
| 145.2 | 0 | −1.50000 | + | 0.866025i | 0 | −0.140435 | − | 0.243241i | 0 | 5.24143 | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||
| 145.3 | 0 | −1.50000 | + | 0.866025i | 0 | 2.27307 | + | 3.93708i | 0 | −9.87987 | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||
| 673.1 | 0 | −1.50000 | − | 0.866025i | 0 | −3.13264 | + | 5.42589i | 0 | −8.36156 | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||
| 673.2 | 0 | −1.50000 | − | 0.866025i | 0 | −0.140435 | + | 0.243241i | 0 | 5.24143 | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||
| 673.3 | 0 | −1.50000 | − | 0.866025i | 0 | 2.27307 | − | 3.93708i | 0 | −9.87987 | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 912.3.be.d | 6 | |
| 4.b | odd | 2 | 1 | 57.3.g.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 171.3.p.e | 6 | ||
| 19.d | odd | 6 | 1 | inner | 912.3.be.d | 6 | |
| 76.f | even | 6 | 1 | 57.3.g.a | ✓ | 6 | |
| 228.n | odd | 6 | 1 | 171.3.p.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.3.g.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 57.3.g.a | ✓ | 6 | 76.f | even | 6 | 1 | |
| 171.3.p.e | 6 | 12.b | even | 2 | 1 | ||
| 171.3.p.e | 6 | 228.n | odd | 6 | 1 | ||
| 912.3.be.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 912.3.be.d | 6 | 19.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\):
|
\( T_{5}^{6} + 2T_{5}^{5} + 32T_{5}^{4} - 40T_{5}^{3} + 800T_{5}^{2} + 224T_{5} + 64 \)
|
|
\( T_{7}^{3} + 13T_{7}^{2} - 13T_{7} - 433 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 3 T + 3)^{3} \)
|
| $5$ |
\( T^{6} + 2 T^{5} + \cdots + 64 \)
|
| $7$ |
\( (T^{3} + 13 T^{2} + \cdots - 433)^{2} \)
|
| $11$ |
\( (T^{3} - 264 T + 304)^{2} \)
|
| $13$ |
\( T^{6} + 15 T^{5} + \cdots + 3518667 \)
|
| $17$ |
\( T^{6} + 10 T^{5} + \cdots + 5184 \)
|
| $19$ |
\( T^{6} - 46 T^{5} + \cdots + 47045881 \)
|
| $23$ |
\( T^{6} - 24 T^{5} + \cdots + 1517824 \)
|
| $29$ |
\( T^{6} - 66 T^{5} + \cdots + 19293888 \)
|
| $31$ |
\( T^{6} + 2033 T^{4} + \cdots + 124768803 \)
|
| $37$ |
\( T^{6} + \cdots + 1689765867 \)
|
| $41$ |
\( T^{6} + \cdots + 1907539968 \)
|
| $43$ |
\( T^{6} + 11 T^{5} + \cdots + 2209 \)
|
| $47$ |
\( T^{6} + \cdots + 2266521664 \)
|
| $53$ |
\( T^{6} - 180 T^{5} + \cdots + 2495232 \)
|
| $59$ |
\( T^{6} + \cdots + 48350430912 \)
|
| $61$ |
\( T^{6} + \cdots + 558907255201 \)
|
| $67$ |
\( T^{6} + \cdots + 2349816507 \)
|
| $71$ |
\( T^{6} + \cdots + 100019515392 \)
|
| $73$ |
\( T^{6} + \cdots + 214090364601 \)
|
| $79$ |
\( T^{6} + \cdots + 2549342403 \)
|
| $83$ |
\( (T^{3} - 126 T^{2} + \cdots - 57704)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 3516172210368 \)
|
| $97$ |
\( T^{6} + \cdots + 3000768049152 \)
|
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