Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1148,2,Mod(57,1148)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1148, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1148.57");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1148 = 2^{2} \cdot 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1148.n (of order \(5\), degree \(4\), 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: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.64000000.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{6} \)
|
| \(\nu^{7}\) | \(=\) |
\( 8\beta_{7} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).
| \(n\) | \(493\) | \(575\) | \(785\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 57.1 |
|
0 | −2.49207 | 0 | 0.335106 | + | 0.243469i | 0 | 0.309017 | + | 0.951057i | 0 | 3.21039 | 0 | ||||||||||||||||||||||||||||||||||||||
| 57.2 | 0 | −0.744002 | 0 | −1.95314 | − | 1.41904i | 0 | 0.309017 | + | 0.951057i | 0 | −2.44646 | 0 | |||||||||||||||||||||||||||||||||||||||
| 141.1 | 0 | −2.49207 | 0 | 0.335106 | − | 0.243469i | 0 | 0.309017 | − | 0.951057i | 0 | 3.21039 | 0 | |||||||||||||||||||||||||||||||||||||||
| 141.2 | 0 | −0.744002 | 0 | −1.95314 | + | 1.41904i | 0 | 0.309017 | − | 0.951057i | 0 | −2.44646 | 0 | |||||||||||||||||||||||||||||||||||||||
| 365.1 | 0 | −1.67021 | 0 | 0.746033 | − | 2.29605i | 0 | −0.809017 | − | 0.587785i | 0 | −0.210393 | 0 | |||||||||||||||||||||||||||||||||||||||
| 365.2 | 0 | 2.90628 | 0 | −0.127999 | + | 0.393941i | 0 | −0.809017 | − | 0.587785i | 0 | 5.44646 | 0 | |||||||||||||||||||||||||||||||||||||||
| 953.1 | 0 | −1.67021 | 0 | 0.746033 | + | 2.29605i | 0 | −0.809017 | + | 0.587785i | 0 | −0.210393 | 0 | |||||||||||||||||||||||||||||||||||||||
| 953.2 | 0 | 2.90628 | 0 | −0.127999 | − | 0.393941i | 0 | −0.809017 | + | 0.587785i | 0 | 5.44646 | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1148.2.n.a | ✓ | 8 |
| 41.d | even | 5 | 1 | inner | 1148.2.n.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1148.2.n.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1148.2.n.a | ✓ | 8 | 41.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2T_{3}^{3} - 7T_{3}^{2} - 18T_{3} - 9 \)
acting on \(S_{2}^{\mathrm{new}}(1148, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 2 T^{3} - 7 T^{2} + \cdots - 9)^{2} \)
|
| $5$ |
\( T^{8} + 2 T^{7} + \cdots + 1 \)
|
| $7$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{8} + 18 T^{6} + \cdots + 1681 \)
|
| $13$ |
\( T^{8} + 28 T^{6} + \cdots + 81 \)
|
| $17$ |
\( T^{8} + 14 T^{7} + \cdots + 81 \)
|
| $19$ |
\( T^{8} - 12 T^{7} + \cdots + 201601 \)
|
| $23$ |
\( T^{8} + 8 T^{7} + \cdots + 256 \)
|
| $29$ |
\( T^{8} - 2 T^{7} + \cdots + 1681 \)
|
| $31$ |
\( T^{8} - 18 T^{7} + \cdots + 531441 \)
|
| $37$ |
\( T^{8} - 14 T^{7} + \cdots + 1681 \)
|
| $41$ |
\( T^{8} + 22 T^{7} + \cdots + 2825761 \)
|
| $43$ |
\( T^{8} - 20 T^{7} + \cdots + 229441 \)
|
| $47$ |
\( T^{8} - 10 T^{7} + \cdots + 271441 \)
|
| $53$ |
\( T^{8} + 8 T^{7} + \cdots + 18800896 \)
|
| $59$ |
\( T^{8} + 16 T^{7} + \cdots + 1681 \)
|
| $61$ |
\( T^{8} - 14 T^{7} + \cdots + 2307361 \)
|
| $67$ |
\( T^{8} + 26 T^{7} + \cdots + 6241 \)
|
| $71$ |
\( T^{8} - 14 T^{7} + \cdots + 1185921 \)
|
| $73$ |
\( (T^{4} + 20 T^{3} + \cdots - 2416)^{2} \)
|
| $79$ |
\( (T^{4} + 36 T^{3} + \cdots - 18401)^{2} \)
|
| $83$ |
\( (T^{4} + 4 T^{3} + \cdots + 1359)^{2} \)
|
| $89$ |
\( T^{8} - 2 T^{7} + \cdots + 641601 \)
|
| $97$ |
\( T^{8} - 18 T^{7} + \cdots + 58081 \)
|
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