Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,15,Mod(17,48)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.17");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.e (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: | \(59.6779047129\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-35})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + 23x^{2} - 22x + 51 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{16}\cdot 3^{7} \) |
| Twist minimal: | no (minimal twist has level 6) |
| 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 in terms of a root \(\nu\) of
\( x^{4} - 2x^{3} + 23x^{2} - 22x + 51 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 256\nu^{3} - 384\nu^{2} + 3968\nu - 1920 ) / 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -280\nu^{3} - 876\nu^{2} - 7580\nu - 9888 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -160\nu^{3} + 2832\nu^{2} - 7664\nu + 31008 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( ( -16\beta_{3} - 32\beta_{2} - 45\beta _1 + 10368 ) / 20736 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 40\beta_{3} - 64\beta_{2} - 45\beta _1 - 217728 ) / 20736 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 308\beta_{3} + 400\beta_{2} + 1359\beta _1 - 331776 ) / 20736 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(31\) | \(37\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | −385.790 | − | 2152.70i | 0 | − | 105253.i | 0 | 1.21591e6 | 0 | −4.48530e6 | + | 1.66099e6i | 0 | |||||||||||||||||||||||||
| 17.2 | 0 | −385.790 | + | 2152.70i | 0 | 105253.i | 0 | 1.21591e6 | 0 | −4.48530e6 | − | 1.66099e6i | 0 | |||||||||||||||||||||||||||
| 17.3 | 0 | 2023.79 | − | 829.000i | 0 | − | 40425.0i | 0 | −388872. | 0 | 3.40849e6 | − | 3.35545e6i | 0 | ||||||||||||||||||||||||||
| 17.4 | 0 | 2023.79 | + | 829.000i | 0 | 40425.0i | 0 | −388872. | 0 | 3.40849e6 | + | 3.35545e6i | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 48.15.e.d | 4 | |
| 3.b | odd | 2 | 1 | inner | 48.15.e.d | 4 | |
| 4.b | odd | 2 | 1 | 6.15.b.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | 6.15.b.a | ✓ | 4 | |
| 20.d | odd | 2 | 1 | 150.15.d.a | 4 | ||
| 20.e | even | 4 | 2 | 150.15.b.a | 8 | ||
| 60.h | even | 2 | 1 | 150.15.d.a | 4 | ||
| 60.l | odd | 4 | 2 | 150.15.b.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.15.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 6.15.b.a | ✓ | 4 | 12.b | even | 2 | 1 | |
| 48.15.e.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 48.15.e.d | 4 | 3.b | odd | 2 | 1 | inner | |
| 150.15.b.a | 8 | 20.e | even | 4 | 2 | ||
| 150.15.b.a | 8 | 60.l | odd | 4 | 2 | ||
| 150.15.d.a | 4 | 20.d | odd | 2 | 1 | ||
| 150.15.d.a | 4 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 12712279680T_{5}^{2} + 18103614409876377600 \)
acting on \(S_{15}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 22876792454961 \)
|
| $5$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $7$ |
\( (T^{2} - 827036 T - 472833268796)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 30\!\cdots\!44 \)
|
| $13$ |
\( (T^{2} + \cdots + 38\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 15\!\cdots\!44 \)
|
| $19$ |
\( (T^{2} + \cdots - 11\!\cdots\!76)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 47\!\cdots\!64 \)
|
| $29$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + \cdots - 22\!\cdots\!16)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots - 37\!\cdots\!16)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 11\!\cdots\!64 \)
|
| $43$ |
\( (T^{2} + \cdots + 14\!\cdots\!56)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 88\!\cdots\!24 \)
|
| $59$ |
\( T^{4} + \cdots + 10\!\cdots\!64 \)
|
| $61$ |
\( (T^{2} + \cdots - 99\!\cdots\!04)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots - 34\!\cdots\!76)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 40\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots + 41\!\cdots\!80)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots + 39\!\cdots\!24)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 20\!\cdots\!04 \)
|
| $89$ |
\( T^{4} + \cdots + 19\!\cdots\!64 \)
|
| $97$ |
\( (T^{2} + \cdots + 13\!\cdots\!76)^{2} \)
|
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