Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,25,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, 25, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.17");
S:= CuspForms(chi, 25);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 25 \) |
| 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: | \(175.184233084\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + \cdots + 21\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{60}\cdot 3^{34}\cdot 17^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + \cdots + 21\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 74\!\cdots\!16 \nu^{7} + \cdots - 30\!\cdots\!50 ) / 71\!\cdots\!75 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 13\!\cdots\!96 \nu^{7} + \cdots - 83\!\cdots\!00 \nu ) / 86\!\cdots\!25 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 20\!\cdots\!84 \nu^{7} + \cdots + 18\!\cdots\!50 ) / 28\!\cdots\!75 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 40\!\cdots\!12 \nu^{7} + \cdots - 48\!\cdots\!50 ) / 71\!\cdots\!75 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 11\!\cdots\!52 \nu^{7} + \cdots + 19\!\cdots\!00 ) / 14\!\cdots\!75 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 88\!\cdots\!44 \nu^{7} + \cdots + 15\!\cdots\!50 ) / 71\!\cdots\!75 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 67\!\cdots\!16 \nu^{7} + \cdots + 69\!\cdots\!00 ) / 23\!\cdots\!25 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 145\beta_{2} + 155\beta_1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 22029313 \beta_{7} - 31312165 \beta_{6} + 30949615 \beta_{5} - 7494010 \beta_{4} + \cdots - 58\!\cdots\!72 ) / 144 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 192218719815945 \beta_{7} - 23049792898825 \beta_{6} - 153733842155035 \beta_{5} + \cdots - 21\!\cdots\!10 \beta_1 ) / 96 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 57\!\cdots\!71 \beta_{7} + \cdots + 12\!\cdots\!24 ) / 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 92\!\cdots\!45 \beta_{7} + \cdots + 93\!\cdots\!10 \beta_1 ) / 288 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 33\!\cdots\!93 \beta_{7} + \cdots - 72\!\cdots\!92 ) / 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 89\!\cdots\!60 \beta_{7} + \cdots - 90\!\cdots\!80 \beta_1 ) / 192 \)
|
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 | −473275. | − | 241744.i | 0 | 7.85820e7i | 0 | −2.19870e9 | 0 | 1.65549e11 | + | 2.28823e11i | 0 | ||||||||||||||||||||||||||||||||||||||
| 17.2 | 0 | −473275. | + | 241744.i | 0 | − | 7.85820e7i | 0 | −2.19870e9 | 0 | 1.65549e11 | − | 2.28823e11i | 0 | ||||||||||||||||||||||||||||||||||||||
| 17.3 | 0 | −269557. | − | 458005.i | 0 | 4.56581e8i | 0 | −1.81935e9 | 0 | −1.37107e11 | + | 2.46917e11i | 0 | |||||||||||||||||||||||||||||||||||||||
| 17.4 | 0 | −269557. | + | 458005.i | 0 | − | 4.56581e8i | 0 | −1.81935e9 | 0 | −1.37107e11 | − | 2.46917e11i | 0 | ||||||||||||||||||||||||||||||||||||||
| 17.5 | 0 | 317945. | − | 425841.i | 0 | − | 6.76938e7i | 0 | 2.41596e10 | 0 | −8.02515e10 | − | 2.70788e11i | 0 | ||||||||||||||||||||||||||||||||||||||
| 17.6 | 0 | 317945. | + | 425841.i | 0 | 6.76938e7i | 0 | 2.41596e10 | 0 | −8.02515e10 | + | 2.70788e11i | 0 | |||||||||||||||||||||||||||||||||||||||
| 17.7 | 0 | 490828. | − | 203759.i | 0 | − | 1.24013e8i | 0 | −1.50611e10 | 0 | 1.99394e11 | − | 2.00021e11i | 0 | ||||||||||||||||||||||||||||||||||||||
| 17.8 | 0 | 490828. | + | 203759.i | 0 | 1.24013e8i | 0 | −1.50611e10 | 0 | 1.99394e11 | + | 2.00021e11i | 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 | 48.25.e.d | 8 | |
| 3.b | odd | 2 | 1 | inner | 48.25.e.d | 8 | |
| 4.b | odd | 2 | 1 | 6.25.b.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 6.25.b.a | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.25.b.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 6.25.b.a | ✓ | 8 | 12.b | even | 2 | 1 | |
| 48.25.e.d | 8 | 1.a | even | 1 | 1 | trivial | |
| 48.25.e.d | 8 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + \cdots + 90\!\cdots\!00 \)
acting on \(S_{25}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 63\!\cdots\!21 \)
|
| $5$ |
\( T^{8} + \cdots + 90\!\cdots\!00 \)
|
| $7$ |
\( (T^{4} + \cdots - 14\!\cdots\!04)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 22\!\cdots\!36 \)
|
| $13$ |
\( (T^{4} + \cdots - 14\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 73\!\cdots\!56 \)
|
| $19$ |
\( (T^{4} + \cdots + 54\!\cdots\!36)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 36\!\cdots\!76 \)
|
| $29$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $31$ |
\( (T^{4} + \cdots - 23\!\cdots\!64)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots - 34\!\cdots\!64)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 27\!\cdots\!96 \)
|
| $43$ |
\( (T^{4} + \cdots - 18\!\cdots\!24)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 51\!\cdots\!36 \)
|
| $59$ |
\( T^{8} + \cdots + 14\!\cdots\!56 \)
|
| $61$ |
\( (T^{4} + \cdots + 20\!\cdots\!36)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots - 23\!\cdots\!24)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $73$ |
\( (T^{4} + \cdots + 15\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 50\!\cdots\!96)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 78\!\cdots\!76 \)
|
| $89$ |
\( T^{8} + \cdots + 13\!\cdots\!96 \)
|
| $97$ |
\( (T^{4} + \cdots - 78\!\cdots\!44)^{2} \)
|
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