Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1530,2,Mod(361,1530)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1530, 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("1530.361");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1530.q (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: | \(12.2171115093\) |
| 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: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 510) |
| 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
| \(\beta_{1}\) | \(=\) |
\( 2\zeta_{16} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{16}^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{16}^{4} \)
|
| \(\beta_{4}\) | \(=\) |
\( \zeta_{16}^{6} \)
|
| \(\beta_{5}\) | \(=\) |
\( \zeta_{16}^{7} + \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{3} - \zeta_{16} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{16}^{7} + \zeta_{16}^{5} - \zeta_{16}^{3} - \zeta_{16} \)
|
| \(\zeta_{16}\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\zeta_{16}^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{16}^{3}\) | \(=\) |
\( ( \beta_{6} + \beta_{5} ) / 2 \)
|
| \(\zeta_{16}^{4}\) | \(=\) |
\( \beta_{3} \)
|
| \(\zeta_{16}^{5}\) | \(=\) |
\( ( \beta_{7} + \beta_{5} ) / 2 \)
|
| \(\zeta_{16}^{6}\) | \(=\) |
\( \beta_{4} \)
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| \(\zeta_{16}^{7}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} - \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1530\mathbb{Z}\right)^\times\).
| \(n\) | \(307\) | \(1261\) | \(1361\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 361.1 |
|
1.00000i | 0 | −1.00000 | −0.707107 | + | 0.707107i | 0 | −1.84776 | − | 1.84776i | − | 1.00000i | 0 | −0.707107 | − | 0.707107i | |||||||||||||||||||||||||||||||||||
| 361.2 | 1.00000i | 0 | −1.00000 | −0.707107 | + | 0.707107i | 0 | 1.84776 | + | 1.84776i | − | 1.00000i | 0 | −0.707107 | − | 0.707107i | ||||||||||||||||||||||||||||||||||||
| 361.3 | 1.00000i | 0 | −1.00000 | 0.707107 | − | 0.707107i | 0 | −0.765367 | − | 0.765367i | − | 1.00000i | 0 | 0.707107 | + | 0.707107i | ||||||||||||||||||||||||||||||||||||
| 361.4 | 1.00000i | 0 | −1.00000 | 0.707107 | − | 0.707107i | 0 | 0.765367 | + | 0.765367i | − | 1.00000i | 0 | 0.707107 | + | 0.707107i | ||||||||||||||||||||||||||||||||||||
| 1441.1 | − | 1.00000i | 0 | −1.00000 | −0.707107 | − | 0.707107i | 0 | −1.84776 | + | 1.84776i | 1.00000i | 0 | −0.707107 | + | 0.707107i | ||||||||||||||||||||||||||||||||||||
| 1441.2 | − | 1.00000i | 0 | −1.00000 | −0.707107 | − | 0.707107i | 0 | 1.84776 | − | 1.84776i | 1.00000i | 0 | −0.707107 | + | 0.707107i | ||||||||||||||||||||||||||||||||||||
| 1441.3 | − | 1.00000i | 0 | −1.00000 | 0.707107 | + | 0.707107i | 0 | −0.765367 | + | 0.765367i | 1.00000i | 0 | 0.707107 | − | 0.707107i | ||||||||||||||||||||||||||||||||||||
| 1441.4 | − | 1.00000i | 0 | −1.00000 | 0.707107 | + | 0.707107i | 0 | 0.765367 | − | 0.765367i | 1.00000i | 0 | 0.707107 | − | 0.707107i | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.c | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1530.2.q.i | 8 | |
| 3.b | odd | 2 | 1 | 510.2.p.d | ✓ | 8 | |
| 17.c | even | 4 | 1 | inner | 1530.2.q.i | 8 | |
| 51.f | odd | 4 | 1 | 510.2.p.d | ✓ | 8 | |
| 51.g | odd | 8 | 1 | 8670.2.a.bt | 4 | ||
| 51.g | odd | 8 | 1 | 8670.2.a.bw | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 510.2.p.d | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 510.2.p.d | ✓ | 8 | 51.f | odd | 4 | 1 | |
| 1530.2.q.i | 8 | 1.a | even | 1 | 1 | trivial | |
| 1530.2.q.i | 8 | 17.c | even | 4 | 1 | inner | |
| 8670.2.a.bt | 4 | 51.g | odd | 8 | 1 | ||
| 8670.2.a.bw | 4 | 51.g | odd | 8 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1530, [\chi])\):
|
\( T_{7}^{8} + 48T_{7}^{4} + 64 \)
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\( T_{11}^{8} - 64T_{11}^{5} + 608T_{11}^{4} - 1536T_{11}^{3} + 2048T_{11}^{2} - 1024T_{11} + 256 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
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| $3$ |
\( T^{8} \)
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| $5$ |
\( (T^{4} + 1)^{2} \)
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| $7$ |
\( T^{8} + 48T^{4} + 64 \)
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| $11$ |
\( T^{8} - 64 T^{5} + \cdots + 256 \)
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| $13$ |
\( (T^{4} - 44 T^{2} + \cdots + 316)^{2} \)
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| $17$ |
\( T^{8} + 20 T^{6} + \cdots + 83521 \)
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| $19$ |
\( T^{8} + 96 T^{6} + \cdots + 4096 \)
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| $23$ |
\( T^{8} - 8 T^{7} + \cdots + 15376 \)
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| $29$ |
\( T^{8} - 32 T^{7} + \cdots + 295936 \)
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| $31$ |
\( T^{8} + 8 T^{7} + \cdots + 795664 \)
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| $37$ |
\( T^{8} - 16 T^{7} + \cdots + 246016 \)
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| $41$ |
\( T^{8} + 224 T^{5} + \cdots + 3655744 \)
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| $43$ |
\( T^{8} + 232 T^{6} + \cdots + 595984 \)
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| $47$ |
\( (T^{4} - 16 T^{3} + \cdots - 64)^{2} \)
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| $53$ |
\( T^{8} + 144 T^{6} + \cdots + 73984 \)
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| $59$ |
\( T^{8} + 296 T^{6} + \cdots + 204304 \)
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| $61$ |
\( T^{8} - 32 T^{7} + \cdots + 256 \)
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| $67$ |
\( (T^{4} - 40 T^{3} + \cdots + 5308)^{2} \)
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| $71$ |
\( T^{8} + 17712 T^{4} + 153664 \)
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| $73$ |
\( T^{8} + 800 T^{5} + \cdots + 17774656 \)
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| $79$ |
\( T^{8} + 40 T^{7} + \cdots + 4443664 \)
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| $83$ |
\( T^{8} + 192 T^{6} + \cdots + 65536 \)
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| $89$ |
\( (T^{4} - 8 T^{3} + \cdots + 3088)^{2} \)
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| $97$ |
\( T^{8} + 64 T^{7} + \cdots + 198021184 \)
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