Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [600,2,Mod(349,600)]
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, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("600.349");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 600.d (of order \(2\), degree \(1\), not 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: | \(6\) |
| Coefficient field: | 6.0.399424.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + \nu^{3} - 4 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - \nu^{3} + 3\nu^{2} + 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 3\nu^{3} + 3\nu^{2} - 2\nu + 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} + 2\nu^{3} - 3\nu^{2} + 3\nu - 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{5} - 3\beta_{3} + \beta_{2} + 3 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -3\beta_{4} + 3\beta_{3} + \beta_{2} + \beta _1 + 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( \beta_{5} + 3\beta_{3} - \beta_{2} + 4\beta _1 + 5 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 349.1 |
|
−1.24464 | − | 0.671462i | −1.00000 | 1.09828 | + | 1.67146i | 0 | 1.24464 | + | 0.671462i | 4.68585i | −0.244644 | − | 2.81783i | 1.00000 | 0 | ||||||||||||||||||||||||||||
| 349.2 | −1.24464 | + | 0.671462i | −1.00000 | 1.09828 | − | 1.67146i | 0 | 1.24464 | − | 0.671462i | − | 4.68585i | −0.244644 | + | 2.81783i | 1.00000 | 0 | ||||||||||||||||||||||||||||
| 349.3 | −0.144584 | − | 1.40680i | −1.00000 | −1.95819 | + | 0.406803i | 0 | 0.144584 | + | 1.40680i | 3.62721i | 0.855416 | + | 2.69597i | 1.00000 | 0 | |||||||||||||||||||||||||||||
| 349.4 | −0.144584 | + | 1.40680i | −1.00000 | −1.95819 | − | 0.406803i | 0 | 0.144584 | − | 1.40680i | − | 3.62721i | 0.855416 | − | 2.69597i | 1.00000 | 0 | ||||||||||||||||||||||||||||
| 349.5 | 1.38923 | − | 0.264658i | −1.00000 | 1.85991 | − | 0.735342i | 0 | −1.38923 | + | 0.264658i | − | 0.941367i | 2.38923 | − | 1.51380i | 1.00000 | 0 | ||||||||||||||||||||||||||||
| 349.6 | 1.38923 | + | 0.264658i | −1.00000 | 1.85991 | + | 0.735342i | 0 | −1.38923 | − | 0.264658i | 0.941367i | 2.38923 | + | 1.51380i | 1.00000 | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 40.f | even | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):
|
\( T_{7}^{6} + 36T_{7}^{4} + 320T_{7}^{2} + 256 \)
|
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\( T_{11}^{6} + 64T_{11}^{4} + 1088T_{11}^{2} + 4096 \)
|
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\( T_{13}^{3} - 28T_{13} - 16 \)
|
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\( T_{37}^{3} + 4T_{37}^{2} - 60T_{37} - 256 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{4} - 2 T^{3} + \cdots + 8 \)
|
| $3$ |
\( (T + 1)^{6} \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + 36 T^{4} + \cdots + 256 \)
|
| $11$ |
\( T^{6} + 64 T^{4} + \cdots + 4096 \)
|
| $13$ |
\( (T^{3} - 28 T - 16)^{2} \)
|
| $17$ |
\( T^{6} + 68 T^{4} + \cdots + 1024 \)
|
| $19$ |
\( T^{6} + 40 T^{4} + \cdots + 256 \)
|
| $23$ |
\( T^{6} + 40 T^{4} + \cdots + 256 \)
|
| $29$ |
\( (T^{2} + 4)^{3} \)
|
| $31$ |
\( (T^{3} + 6 T^{2} - 16 T - 64)^{2} \)
|
| $37$ |
\( (T^{3} + 4 T^{2} + \cdots - 256)^{2} \)
|
| $41$ |
\( (T^{3} + 10 T^{2} + \cdots - 232)^{2} \)
|
| $43$ |
\( (T^{3} - 4 T^{2} + \cdots + 128)^{2} \)
|
| $47$ |
\( T^{6} + 200 T^{4} + \cdots + 246016 \)
|
| $53$ |
\( (T + 2)^{6} \)
|
| $59$ |
\( T^{6} + 80 T^{4} + \cdots + 1024 \)
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| $61$ |
\( T^{6} + 256 T^{4} + \cdots + 262144 \)
|
| $67$ |
\( (T + 4)^{6} \)
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| $71$ |
\( (T^{3} + 4 T^{2} - 112 T + 64)^{2} \)
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| $73$ |
\( (T^{2} + 36)^{3} \)
|
| $79$ |
\( (T^{3} + 18 T^{2} + \cdots + 64)^{2} \)
|
| $83$ |
\( (T^{3} + 16 T^{2} + \cdots - 256)^{2} \)
|
| $89$ |
\( (T^{3} - 14 T^{2} + \cdots + 184)^{2} \)
|
| $97$ |
\( T^{6} + 332 T^{4} + \cdots + 107584 \)
|
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