Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [600,2,Mod(301,600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("600.301");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 600.k (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: | \(4.79102412128\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | 12.0.180227832610816.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | no (minimal twist has level 120) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{11} - \nu^{9} + 2\nu^{7} + 4\nu^{5} + 8\nu^{3} ) / 64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{10} + 3\nu^{8} - 2\nu^{6} + 12\nu^{4} - 8\nu^{2} + 32 ) / 32 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{10} - \nu^{8} - 6\nu^{6} - 4\nu^{4} + 8\nu^{2} + 32 ) / 32 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{11} + \nu^{9} - 2\nu^{7} + 12\nu^{5} + 8\nu^{3} ) / 32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{11} + 3\nu^{9} - 2\nu^{7} - 4\nu^{5} + 8\nu^{3} + 32\nu ) / 32 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{10} - 3\nu^{8} + 2\nu^{6} + 20\nu^{4} + 8\nu^{2} - 32 ) / 32 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -\nu^{11} + \nu^{9} + 6\nu^{7} + 4\nu^{5} - 8\nu^{3} - 32\nu ) / 32 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -\nu^{11} - 3\nu^{9} + 2\nu^{7} + 4\nu^{5} + 24\nu^{3} - 32\nu ) / 32 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 3\nu^{10} + \nu^{8} + 2\nu^{6} - 12\nu^{4} + 8\nu^{2} + 32 ) / 32 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} + \beta_{7} \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{10} - \beta_{7} + 2\beta_{6} + 4\beta_{3} \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{11} + \beta_{8} - 4\beta_{5} - \beta_{4} + 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{10} + 4\beta_{9} + \beta_{7} - 2\beta_{6} + 4\beta_{3} + 4\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -2\beta_{11} - 5\beta_{8} - 4\beta_{5} + 5\beta_{4} + 4\beta_{2} - 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( -\beta_{10} + 4\beta_{9} + 7\beta_{7} + 2\beta_{6} - 4\beta_{3} - 4\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 10\beta_{11} + 5\beta_{8} + 4\beta_{5} + 3\beta_{4} - 4\beta_{2} - 12 \)
|
| \(\nu^{11}\) | \(=\) |
\( -7\beta_{10} - 4\beta_{9} + \beta_{7} - 2\beta_{6} + 36\beta_{3} - 12\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).
| \(n\) | \(151\) | \(301\) | \(401\) | \(577\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 301.1 |
|
−1.37729 | − | 0.321037i | − | 1.00000i | 1.79387 | + | 0.884323i | 0 | −0.321037 | + | 1.37729i | −4.05705 | −2.18678 | − | 1.79387i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 301.2 | −1.37729 | + | 0.321037i | 1.00000i | 1.79387 | − | 0.884323i | 0 | −0.321037 | − | 1.37729i | −4.05705 | −2.18678 | + | 1.79387i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 301.3 | −0.806504 | − | 1.16170i | 1.00000i | −0.699104 | + | 1.87383i | 0 | 1.16170 | − | 0.806504i | −0.746175 | 2.74067 | − | 0.699104i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 301.4 | −0.806504 | + | 1.16170i | − | 1.00000i | −0.699104 | − | 1.87383i | 0 | 1.16170 | + | 0.806504i | −0.746175 | 2.74067 | + | 0.699104i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 301.5 | −0.450129 | − | 1.34067i | − | 1.00000i | −1.59477 | + | 1.20695i | 0 | −1.34067 | + | 0.450129i | 2.64265 | 2.33596 | + | 1.59477i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 301.6 | −0.450129 | + | 1.34067i | 1.00000i | −1.59477 | − | 1.20695i | 0 | −1.34067 | − | 0.450129i | 2.64265 | 2.33596 | − | 1.59477i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 301.7 | 0.450129 | − | 1.34067i | − | 1.00000i | −1.59477 | − | 1.20695i | 0 | −1.34067 | − | 0.450129i | −2.64265 | −2.33596 | + | 1.59477i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 301.8 | 0.450129 | + | 1.34067i | 1.00000i | −1.59477 | + | 1.20695i | 0 | −1.34067 | + | 0.450129i | −2.64265 | −2.33596 | − | 1.59477i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 301.9 | 0.806504 | − | 1.16170i | 1.00000i | −0.699104 | − | 1.87383i | 0 | 1.16170 | + | 0.806504i | 0.746175 | −2.74067 | − | 0.699104i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 301.10 | 0.806504 | + | 1.16170i | − | 1.00000i | −0.699104 | + | 1.87383i | 0 | 1.16170 | − | 0.806504i | 0.746175 | −2.74067 | + | 0.699104i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 301.11 | 1.37729 | − | 0.321037i | − | 1.00000i | 1.79387 | − | 0.884323i | 0 | −0.321037 | − | 1.37729i | 4.05705 | 2.18678 | − | 1.79387i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 301.12 | 1.37729 | + | 0.321037i | 1.00000i | 1.79387 | + | 0.884323i | 0 | −0.321037 | + | 1.37729i | 4.05705 | 2.18678 | + | 1.79387i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 40.f | even | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} - 24T_{7}^{4} + 128T_{7}^{2} - 64 \)
acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + T^{10} + \cdots + 64 \)
|
| $3$ |
\( (T^{2} + 1)^{6} \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( (T^{6} - 24 T^{4} + \cdots - 64)^{2} \)
|
| $11$ |
\( (T^{6} + 32 T^{4} + \cdots + 64)^{2} \)
|
| $13$ |
\( (T^{6} + 48 T^{4} + \cdots + 3136)^{2} \)
|
| $17$ |
\( (T^{6} - 36 T^{4} + \cdots - 1024)^{2} \)
|
| $19$ |
\( (T^{6} + 60 T^{4} + \cdots + 1024)^{2} \)
|
| $23$ |
\( (T^{6} - 92 T^{4} + \cdots - 16384)^{2} \)
|
| $29$ |
\( (T^{6} + 108 T^{4} + \cdots + 12544)^{2} \)
|
| $31$ |
\( (T^{3} + 8 T^{2} - 4 T - 64)^{4} \)
|
| $37$ |
\( (T^{6} + 64 T^{4} + \cdots + 64)^{2} \)
|
| $41$ |
\( (T^{3} + 2 T^{2} - 100 T + 56)^{4} \)
|
| $43$ |
\( (T^{6} + 128 T^{4} + \cdots + 4096)^{2} \)
|
| $47$ |
\( (T^{6} - 60 T^{4} + \cdots - 1024)^{2} \)
|
| $53$ |
\( (T^{6} + 80 T^{4} + \cdots + 64)^{2} \)
|
| $59$ |
\( (T^{6} + 176 T^{4} + \cdots + 179776)^{2} \)
|
| $61$ |
\( (T^{6} + 176 T^{4} + \cdots + 65536)^{2} \)
|
| $67$ |
\( (T^{6} + 128 T^{4} + \cdots + 4096)^{2} \)
|
| $71$ |
\( (T^{3} - 8 T^{2} + \cdots + 128)^{4} \)
|
| $73$ |
\( (T^{6} - 384 T^{4} + \cdots - 16384)^{2} \)
|
| $79$ |
\( (T^{3} + 8 T^{2} - 4 T - 64)^{4} \)
|
| $83$ |
\( (T^{6} + 192 T^{4} + \cdots + 200704)^{2} \)
|
| $89$ |
\( (T^{3} - 10 T^{2} + \cdots + 1384)^{4} \)
|
| $97$ |
\( (T^{6} - 336 T^{4} + \cdots - 262144)^{2} \)
|
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