Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1148,2,Mod(337,1148)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1148, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1148.337");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1148 = 2^{2} \cdot 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1148.k (of order \(4\), degree \(2\), 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: | \(9.16682615204\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.110166016.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} + 10\nu^{4} + 9\nu^{3} + 18\nu^{2} + 9\nu + 5 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} + 10\nu^{4} - 9\nu^{3} + 18\nu^{2} - 9\nu + 5 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + 9\nu^{4} + 11\nu^{2} + 4 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} + 10\nu^{5} + 9\nu^{4} + 18\nu^{3} + 9\nu^{2} + 5\nu - 4 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} + 10\nu^{5} - 9\nu^{4} + 18\nu^{3} - 9\nu^{2} + 5\nu ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{7} + 19\nu^{5} + 29\nu^{3} + 9\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta_{4} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 2\beta_{6} - 2\beta_{5} - \beta_{3} + \beta_{2} - 4\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{6} + 7\beta_{5} - 9\beta_{4} + 2\beta_{3} + 2\beta_{2} + 20 \)
|
| \(\nu^{5}\) | \(=\) |
\( -9\beta_{7} + 18\beta_{6} + 18\beta_{5} + 7\beta_{3} - 7\beta_{2} + 27\beta _1 + 18 \)
|
| \(\nu^{6}\) | \(=\) |
\( 52\beta_{6} - 52\beta_{5} + 72\beta_{4} - 18\beta_{3} - 18\beta_{2} - 151 \)
|
| \(\nu^{7}\) | \(=\) |
\( 72\beta_{7} - 142\beta_{6} - 142\beta_{5} - 52\beta_{3} + 52\beta_{2} - 203\beta _1 - 142 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).
| \(n\) | \(493\) | \(575\) | \(785\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
0 | −1.96195 | − | 1.96195i | 0 | 0.264927i | 0 | 0.707107 | + | 0.707107i | 0 | 4.69853i | 0 | ||||||||||||||||||||||||||||||||||||||
| 337.2 | 0 | −0.868559 | − | 0.868559i | 0 | − | 4.37966i | 0 | −0.707107 | − | 0.707107i | 0 | − | 1.49121i | 0 | |||||||||||||||||||||||||||||||||||||
| 337.3 | 0 | 0.254848 | + | 0.254848i | 0 | 1.56350i | 0 | 0.707107 | + | 0.707107i | 0 | − | 2.87011i | 0 | ||||||||||||||||||||||||||||||||||||||
| 337.4 | 0 | 0.575666 | + | 0.575666i | 0 | 0.551233i | 0 | −0.707107 | − | 0.707107i | 0 | − | 2.33722i | 0 | ||||||||||||||||||||||||||||||||||||||
| 729.1 | 0 | −1.96195 | + | 1.96195i | 0 | − | 0.264927i | 0 | 0.707107 | − | 0.707107i | 0 | − | 4.69853i | 0 | |||||||||||||||||||||||||||||||||||||
| 729.2 | 0 | −0.868559 | + | 0.868559i | 0 | 4.37966i | 0 | −0.707107 | + | 0.707107i | 0 | 1.49121i | 0 | |||||||||||||||||||||||||||||||||||||||
| 729.3 | 0 | 0.254848 | − | 0.254848i | 0 | − | 1.56350i | 0 | 0.707107 | − | 0.707107i | 0 | 2.87011i | 0 | ||||||||||||||||||||||||||||||||||||||
| 729.4 | 0 | 0.575666 | − | 0.575666i | 0 | − | 0.551233i | 0 | −0.707107 | + | 0.707107i | 0 | 2.33722i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.c | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1148.2.k.a | ✓ | 8 |
| 41.c | even | 4 | 1 | inner | 1148.2.k.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1148.2.k.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1148.2.k.a | ✓ | 8 | 41.c | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 4T_{3}^{7} + 8T_{3}^{6} - T_{3}^{4} + 8T_{3}^{2} - 4T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1148, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 4 T^{7} + \cdots + 1 \)
|
| $5$ |
\( T^{8} + 22 T^{6} + \cdots + 1 \)
|
| $7$ |
\( (T^{4} + 1)^{2} \)
|
| $11$ |
\( T^{8} - 12 T^{7} + \cdots + 625 \)
|
| $13$ |
\( T^{8} - 4 T^{7} + \cdots + 49 \)
|
| $17$ |
\( T^{8} - 8 T^{7} + \cdots + 6241 \)
|
| $19$ |
\( T^{8} - 8 T^{7} + \cdots + 5329 \)
|
| $23$ |
\( (T^{4} - 14 T^{3} + \cdots - 313)^{2} \)
|
| $29$ |
\( T^{8} + 16 T^{7} + \cdots + 54289 \)
|
| $31$ |
\( (T^{4} - 14 T^{3} + \cdots + 73)^{2} \)
|
| $37$ |
\( (T^{2} + 8 T + 14)^{4} \)
|
| $41$ |
\( T^{8} + 60 T^{6} + \cdots + 2825761 \)
|
| $43$ |
\( T^{8} + 124 T^{6} + \cdots + 253009 \)
|
| $47$ |
\( T^{8} - 20 T^{7} + \cdots + 16 \)
|
| $53$ |
\( T^{8} - 32 T^{7} + \cdots + 214369 \)
|
| $59$ |
\( (T^{4} + 10 T^{3} + \cdots + 3791)^{2} \)
|
| $61$ |
\( T^{8} + 500 T^{6} + \cdots + 178676689 \)
|
| $67$ |
\( T^{8} + 4 T^{7} + \cdots + 49 \)
|
| $71$ |
\( T^{8} - 8 T^{7} + \cdots + 49729 \)
|
| $73$ |
\( T^{8} + 76 T^{6} + \cdots + 57121 \)
|
| $79$ |
\( T^{8} + 12 T^{7} + \cdots + 583696 \)
|
| $83$ |
\( (T^{4} + 32 T^{3} + \cdots - 9623)^{2} \)
|
| $89$ |
\( T^{8} - 4 T^{7} + \cdots + 43256929 \)
|
| $97$ |
\( T^{8} - 56 T^{7} + \cdots + 18241441 \)
|
| show more | |
| show less |