Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,25,Mod(31,48)]
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([1, 0, 0]))
N = Newforms(chi, 25, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.31");
S:= CuspForms(chi, 25);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 25 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.g (of order \(2\), degree \(1\), 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: |
\( x^{8} - 2 x^{7} + 2573102906805 x^{6} + \cdots + 38\!\cdots\!76 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{52}\cdot 3^{15}\cdot 5^{2} \) |
| Twist minimal: | yes |
| 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
\( x^{8} - 2 x^{7} + 2573102906805 x^{6} + \cdots + 38\!\cdots\!76 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 91\!\cdots\!35 \nu^{7} + \cdots - 25\!\cdots\!28 ) / 67\!\cdots\!08 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 28\!\cdots\!25 \nu^{7} + \cdots - 17\!\cdots\!40 ) / 27\!\cdots\!42 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 23\!\cdots\!00 \nu^{7} + \cdots - 14\!\cdots\!80 ) / 22\!\cdots\!31 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 14\!\cdots\!16 \nu^{7} + \cdots - 22\!\cdots\!16 ) / 19\!\cdots\!75 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 24\!\cdots\!95 \nu^{7} + \cdots + 39\!\cdots\!16 ) / 56\!\cdots\!25 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 76\!\cdots\!04 \nu^{7} + \cdots + 30\!\cdots\!64 ) / 19\!\cdots\!75 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 21\!\cdots\!15 \nu^{7} + \cdots + 48\!\cdots\!04 ) / 11\!\cdots\!25 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - 120\beta _1 + 120 ) / 480 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 5425 \beta_{7} + 55550 \beta_{6} + 172075 \beta_{5} - 5425 \beta_{4} + 12453234 \beta_{3} + \cdots - 74\!\cdots\!40 ) / 11520 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2061617662825\beta_{6} - 67470135075\beta_{4} + 801844236499733\beta_{3} - 61642409353323043000320 ) / 92160 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 11\!\cdots\!75 \beta_{7} + \cdots - 12\!\cdots\!80 ) / 92160 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 50\!\cdots\!75 \beta_{7} + \cdots + 52\!\cdots\!76 ) / 36864 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 35\!\cdots\!25 \beta_{6} + \cdots + 30\!\cdots\!20 ) / 46080 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 80\!\cdots\!75 \beta_{7} + \cdots + 86\!\cdots\!20 ) / 184320 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | − | 306828.i | 0 | −3.49688e8 | 0 | 5.92123e9i | 0 | −9.41432e10 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 31.2 | 0 | − | 306828.i | 0 | −2.70145e7 | 0 | − | 9.36646e9i | 0 | −9.41432e10 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 31.3 | 0 | − | 306828.i | 0 | 2.72537e8 | 0 | 2.40937e10i | 0 | −9.41432e10 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 31.4 | 0 | − | 306828.i | 0 | 3.35414e8 | 0 | − | 6.89592e9i | 0 | −9.41432e10 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 31.5 | 0 | 306828.i | 0 | −3.49688e8 | 0 | − | 5.92123e9i | 0 | −9.41432e10 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 31.6 | 0 | 306828.i | 0 | −2.70145e7 | 0 | 9.36646e9i | 0 | −9.41432e10 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 31.7 | 0 | 306828.i | 0 | 2.72537e8 | 0 | − | 2.40937e10i | 0 | −9.41432e10 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 31.8 | 0 | 306828.i | 0 | 3.35414e8 | 0 | 6.89592e9i | 0 | −9.41432e10 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.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.g.c | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 48.25.g.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 48.25.g.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 48.25.g.c | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 231247800 T_{5}^{3} + \cdots + 86\!\cdots\!00 \)
acting on \(S_{25}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} + 94143178827)^{4} \)
|
| $5$ |
\( (T^{4} + \cdots + 86\!\cdots\!00)^{2} \)
|
| $7$ |
\( T^{8} + \cdots + 84\!\cdots\!16 \)
|
| $11$ |
\( T^{8} + \cdots + 77\!\cdots\!36 \)
|
| $13$ |
\( (T^{4} + \cdots + 49\!\cdots\!16)^{2} \)
|
| $17$ |
\( (T^{4} + \cdots + 19\!\cdots\!04)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 50\!\cdots\!56 \)
|
| $23$ |
\( T^{8} + \cdots + 16\!\cdots\!00 \)
|
| $29$ |
\( (T^{4} + \cdots - 25\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 96\!\cdots\!36 \)
|
| $37$ |
\( (T^{4} + \cdots - 11\!\cdots\!20)^{2} \)
|
| $41$ |
\( (T^{4} + \cdots + 69\!\cdots\!24)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 30\!\cdots\!56 \)
|
| $47$ |
\( T^{8} + \cdots + 19\!\cdots\!16 \)
|
| $53$ |
\( (T^{4} + \cdots - 11\!\cdots\!64)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 54\!\cdots\!96 \)
|
| $61$ |
\( (T^{4} + \cdots + 96\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 12\!\cdots\!36 \)
|
| $71$ |
\( T^{8} + \cdots + 38\!\cdots\!96 \)
|
| $73$ |
\( (T^{4} + \cdots + 61\!\cdots\!76)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 42\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 38\!\cdots\!56 \)
|
| $89$ |
\( (T^{4} + \cdots - 27\!\cdots\!64)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots - 20\!\cdots\!24)^{2} \)
|
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