Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1150,2,Mod(643,1150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1150, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1150.643");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1150 = 2 \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1150.e (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: | \(9.18279623245\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{16})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 230) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{16}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{16}^{3} + \zeta_{16} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{16}^{4} \)
|
| \(\beta_{4}\) | \(=\) |
\( \zeta_{16}^{5} + \zeta_{16} \)
|
| \(\beta_{5}\) | \(=\) |
\( \zeta_{16}^{6} \)
|
| \(\beta_{6}\) | \(=\) |
\( \zeta_{16}^{7} + \zeta_{16} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{16}^{5} + \zeta_{16} \)
|
| \(\zeta_{16}\) | \(=\) |
\( ( \beta_{7} + \beta_{4} ) / 2 \)
|
| \(\zeta_{16}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{16}^{3}\) | \(=\) |
\( ( -\beta_{7} - \beta_{4} + 2\beta_{2} ) / 2 \)
|
| \(\zeta_{16}^{4}\) | \(=\) |
\( \beta_{3} \)
|
| \(\zeta_{16}^{5}\) | \(=\) |
\( ( -\beta_{7} + \beta_{4} ) / 2 \)
|
| \(\zeta_{16}^{6}\) | \(=\) |
\( \beta_{5} \)
|
| \(\zeta_{16}^{7}\) | \(=\) |
\( ( -\beta_{7} + 2\beta_{6} - \beta_{4} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).
| \(n\) | \(51\) | \(277\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 643.1 |
|
−0.707107 | + | 0.707107i | −2.00000 | − | 2.00000i | − | 1.00000i | 0 | 2.82843 | −2.61313 | − | 2.61313i | 0.707107 | + | 0.707107i | 5.00000i | 0 | |||||||||||||||||||||||||||||||||
| 643.2 | −0.707107 | + | 0.707107i | −2.00000 | − | 2.00000i | − | 1.00000i | 0 | 2.82843 | 2.61313 | + | 2.61313i | 0.707107 | + | 0.707107i | 5.00000i | 0 | ||||||||||||||||||||||||||||||||||
| 643.3 | 0.707107 | − | 0.707107i | −2.00000 | − | 2.00000i | − | 1.00000i | 0 | −2.82843 | −1.08239 | − | 1.08239i | −0.707107 | − | 0.707107i | 5.00000i | 0 | ||||||||||||||||||||||||||||||||||
| 643.4 | 0.707107 | − | 0.707107i | −2.00000 | − | 2.00000i | − | 1.00000i | 0 | −2.82843 | 1.08239 | + | 1.08239i | −0.707107 | − | 0.707107i | 5.00000i | 0 | ||||||||||||||||||||||||||||||||||
| 1057.1 | −0.707107 | − | 0.707107i | −2.00000 | + | 2.00000i | 1.00000i | 0 | 2.82843 | −2.61313 | + | 2.61313i | 0.707107 | − | 0.707107i | − | 5.00000i | 0 | ||||||||||||||||||||||||||||||||||
| 1057.2 | −0.707107 | − | 0.707107i | −2.00000 | + | 2.00000i | 1.00000i | 0 | 2.82843 | 2.61313 | − | 2.61313i | 0.707107 | − | 0.707107i | − | 5.00000i | 0 | ||||||||||||||||||||||||||||||||||
| 1057.3 | 0.707107 | + | 0.707107i | −2.00000 | + | 2.00000i | 1.00000i | 0 | −2.82843 | −1.08239 | + | 1.08239i | −0.707107 | + | 0.707107i | − | 5.00000i | 0 | ||||||||||||||||||||||||||||||||||
| 1057.4 | 0.707107 | + | 0.707107i | −2.00000 | + | 2.00000i | 1.00000i | 0 | −2.82843 | 1.08239 | − | 1.08239i | −0.707107 | + | 0.707107i | − | 5.00000i | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 23.b | odd | 2 | 1 | inner |
| 115.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1150.2.e.a | 8 | |
| 5.b | even | 2 | 1 | 230.2.e.c | ✓ | 8 | |
| 5.c | odd | 4 | 1 | 230.2.e.c | ✓ | 8 | |
| 5.c | odd | 4 | 1 | inner | 1150.2.e.a | 8 | |
| 23.b | odd | 2 | 1 | inner | 1150.2.e.a | 8 | |
| 115.c | odd | 2 | 1 | 230.2.e.c | ✓ | 8 | |
| 115.e | even | 4 | 1 | 230.2.e.c | ✓ | 8 | |
| 115.e | even | 4 | 1 | inner | 1150.2.e.a | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.2.e.c | ✓ | 8 | 5.b | even | 2 | 1 | |
| 230.2.e.c | ✓ | 8 | 5.c | odd | 4 | 1 | |
| 230.2.e.c | ✓ | 8 | 115.c | odd | 2 | 1 | |
| 230.2.e.c | ✓ | 8 | 115.e | even | 4 | 1 | |
| 1150.2.e.a | 8 | 1.a | even | 1 | 1 | trivial | |
| 1150.2.e.a | 8 | 5.c | odd | 4 | 1 | inner | |
| 1150.2.e.a | 8 | 23.b | odd | 2 | 1 | inner | |
| 1150.2.e.a | 8 | 115.e | even | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1150, [\chi])\):
|
\( T_{3}^{2} + 4T_{3} + 8 \)
|
|
\( T_{7}^{8} + 192T_{7}^{4} + 1024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{2} \)
|
| $3$ |
\( (T^{2} + 4 T + 8)^{4} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 192T^{4} + 1024 \)
|
| $11$ |
\( (T^{4} + 20 T^{2} + 2)^{2} \)
|
| $13$ |
\( (T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 16)^{2} \)
|
| $17$ |
\( T^{8} + 768 T^{4} + 16384 \)
|
| $19$ |
\( (T^{4} - 100 T^{2} + 1922)^{2} \)
|
| $23$ |
\( T^{8} - 8 T^{7} + \cdots + 279841 \)
|
| $29$ |
\( (T^{4} + 72 T^{2} + 784)^{2} \)
|
| $31$ |
\( (T^{2} + 4 T - 14)^{4} \)
|
| $37$ |
\( T^{8} + 204T^{4} + 9604 \)
|
| $41$ |
\( (T^{2} - 98)^{4} \)
|
| $43$ |
\( T^{8} + 4428 T^{4} + 9604 \)
|
| $47$ |
\( (T^{4} + 12 T^{3} + \cdots + 196)^{2} \)
|
| $53$ |
\( T^{8} + 2508 T^{4} + 9604 \)
|
| $59$ |
\( (T^{4} + 96 T^{2} + 256)^{2} \)
|
| $61$ |
\( (T^{4} + 148 T^{2} + 1058)^{2} \)
|
| $67$ |
\( T^{8} + 204T^{4} + 9604 \)
|
| $71$ |
\( (T^{2} + 4 T - 4)^{4} \)
|
| $73$ |
\( (T^{4} + 16 T^{3} + \cdots + 256)^{2} \)
|
| $79$ |
\( (T^{4} - 144 T^{2} + 2592)^{2} \)
|
| $83$ |
\( T^{8} + 19788 T^{4} + 1119364 \)
|
| $89$ |
\( (T^{4} - 256 T^{2} + 8192)^{2} \)
|
| $97$ |
\( T^{8} + 12288 T^{4} + 4194304 \)
|
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