Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,23,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, 23, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.17");
S:= CuspForms(chi, 23);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 23 \) |
| 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: | \(147.219568724\) |
| 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: |
\( x^{8} - 4 x^{7} - 12878974 x^{6} - 2567056924 x^{5} + 47744458496177 x^{4} + \cdots + 98\!\cdots\!84 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{63}\cdot 3^{31} \) |
| 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
\( x^{8} - 4 x^{7} - 12878974 x^{6} - 2567056924 x^{5} + 47744458496177 x^{4} + \cdots + 98\!\cdots\!84 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 51\!\cdots\!11 \nu^{7} + \cdots - 19\!\cdots\!04 ) / 56\!\cdots\!90 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 37\!\cdots\!36 \nu^{7} + \cdots + 53\!\cdots\!88 ) / 25\!\cdots\!79 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 13\!\cdots\!83 \nu^{7} + \cdots - 28\!\cdots\!32 ) / 17\!\cdots\!70 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 32\!\cdots\!25 \nu^{7} + \cdots + 10\!\cdots\!04 ) / 17\!\cdots\!70 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 16\!\cdots\!65 \nu^{7} + \cdots - 10\!\cdots\!68 ) / 56\!\cdots\!90 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 45\!\cdots\!41 \nu^{7} + \cdots - 85\!\cdots\!44 ) / 34\!\cdots\!34 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 27\!\cdots\!39 \nu^{7} + \cdots + 36\!\cdots\!72 ) / 17\!\cdots\!70 \)
|
| \(\nu\) | \(=\) |
\( ( 1536 \beta_{7} + 1764 \beta_{6} - 1612 \beta_{5} - 1944 \beta_{4} - 1140 \beta_{3} + \cdots + 1289945088 ) / 2579890176 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 15752 \beta_{7} - 15645 \beta_{6} - 15201 \beta_{5} - 34182 \beta_{4} - 859 \beta_{3} + \cdots + 43263488461824 ) / 13436928 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2012592192 \beta_{7} + 2305194156 \beta_{6} - 3337245028 \beta_{5} - 2575844712 \beta_{4} + \cdots + 63\!\cdots\!32 ) / 644972544 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 181721051392 \beta_{7} - 177656366256 \beta_{6} - 217468359696 \beta_{5} + \cdots + 47\!\cdots\!40 ) / 26873856 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 53\!\cdots\!96 \beta_{7} + \cdots + 33\!\cdots\!96 ) / 322486272 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 19\!\cdots\!32 \beta_{7} + \cdots + 56\!\cdots\!24 ) / 53747712 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 28\!\cdots\!08 \beta_{7} + \cdots + 28\!\cdots\!64 ) / 322486272 \)
|
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 | −152150. | − | 90727.5i | 0 | − | 3.00062e7i | 0 | −3.33386e9 | 0 | 1.49181e10 | + | 2.76083e10i | 0 | |||||||||||||||||||||||||||||||||||||
| 17.2 | 0 | −152150. | + | 90727.5i | 0 | 3.00062e7i | 0 | −3.33386e9 | 0 | 1.49181e10 | − | 2.76083e10i | 0 | |||||||||||||||||||||||||||||||||||||||
| 17.3 | 0 | −33954.9 | − | 173862.i | 0 | 7.14623e7i | 0 | 1.07711e9 | 0 | −2.90752e10 | + | 1.18070e10i | 0 | |||||||||||||||||||||||||||||||||||||||
| 17.4 | 0 | −33954.9 | + | 173862.i | 0 | − | 7.14623e7i | 0 | 1.07711e9 | 0 | −2.90752e10 | − | 1.18070e10i | 0 | ||||||||||||||||||||||||||||||||||||||
| 17.5 | 0 | 92471.5 | − | 151096.i | 0 | 4.00267e6i | 0 | 1.86396e9 | 0 | −1.42791e10 | − | 2.79442e10i | 0 | |||||||||||||||||||||||||||||||||||||||
| 17.6 | 0 | 92471.5 | + | 151096.i | 0 | − | 4.00267e6i | 0 | 1.86396e9 | 0 | −1.42791e10 | + | 2.79442e10i | 0 | ||||||||||||||||||||||||||||||||||||||
| 17.7 | 0 | 128493. | − | 121945.i | 0 | − | 8.15211e7i | 0 | −2.43000e9 | 0 | 1.64002e9 | − | 3.13382e10i | 0 | ||||||||||||||||||||||||||||||||||||||
| 17.8 | 0 | 128493. | + | 121945.i | 0 | 8.15211e7i | 0 | −2.43000e9 | 0 | 1.64002e9 | + | 3.13382e10i | 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 | 48.23.e.d | 8 | |
| 3.b | odd | 2 | 1 | inner | 48.23.e.d | 8 | |
| 4.b | odd | 2 | 1 | 6.23.b.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 6.23.b.a | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.23.b.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 6.23.b.a | ✓ | 8 | 12.b | even | 2 | 1 | |
| 48.23.e.d | 8 | 1.a | even | 1 | 1 | trivial | |
| 48.23.e.d | 8 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + \cdots + 48\!\cdots\!00 \)
acting on \(S_{23}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 96\!\cdots\!61 \)
|
| $5$ |
\( T^{8} + \cdots + 48\!\cdots\!00 \)
|
| $7$ |
\( (T^{4} + \cdots + 16\!\cdots\!76)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 96\!\cdots\!56 \)
|
| $13$ |
\( (T^{4} + \cdots + 13\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 58\!\cdots\!96 \)
|
| $19$ |
\( (T^{4} + \cdots - 13\!\cdots\!24)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 20\!\cdots\!56 \)
|
| $29$ |
\( T^{8} + \cdots + 98\!\cdots\!00 \)
|
| $31$ |
\( (T^{4} + \cdots - 29\!\cdots\!44)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots - 75\!\cdots\!24)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 10\!\cdots\!36 \)
|
| $43$ |
\( (T^{4} + \cdots + 50\!\cdots\!56)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 12\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 10\!\cdots\!76 \)
|
| $59$ |
\( T^{8} + \cdots + 56\!\cdots\!36 \)
|
| $61$ |
\( (T^{4} + \cdots + 38\!\cdots\!36)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots - 14\!\cdots\!64)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 39\!\cdots\!00 \)
|
| $73$ |
\( (T^{4} + \cdots - 30\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 21\!\cdots\!56)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 74\!\cdots\!76 \)
|
| $89$ |
\( T^{8} + \cdots + 46\!\cdots\!36 \)
|
| $97$ |
\( (T^{4} + \cdots + 41\!\cdots\!96)^{2} \)
|
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