Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3800,2,Mod(3649,3800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3800.3649");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3800.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: | \(30.3431527681\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.4227136.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 9x^{4} + 22x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} + 9x^{4} + 22x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 6\nu^{3} + 7\nu ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} + 5\nu^{2} + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 5\beta_{2} + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{4} - 6\beta_{3} + 17\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).
| \(n\) | \(401\) | \(951\) | \(1901\) | \(1977\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3649.1 |
|
0 | − | 2.49086i | 0 | 0 | 0 | − | 2.49086i | 0 | −3.20440 | 0 | ||||||||||||||||||||||||||||||||||
| 3649.2 | 0 | − | 1.83424i | 0 | 0 | 0 | − | 1.83424i | 0 | −0.364448 | 0 | |||||||||||||||||||||||||||||||||||
| 3649.3 | 0 | − | 0.656620i | 0 | 0 | 0 | − | 0.656620i | 0 | 2.56885 | 0 | |||||||||||||||||||||||||||||||||||
| 3649.4 | 0 | 0.656620i | 0 | 0 | 0 | 0.656620i | 0 | 2.56885 | 0 | |||||||||||||||||||||||||||||||||||||
| 3649.5 | 0 | 1.83424i | 0 | 0 | 0 | 1.83424i | 0 | −0.364448 | 0 | |||||||||||||||||||||||||||||||||||||
| 3649.6 | 0 | 2.49086i | 0 | 0 | 0 | 2.49086i | 0 | −3.20440 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3800.2.d.m | 6 | |
| 5.b | even | 2 | 1 | inner | 3800.2.d.m | 6 | |
| 5.c | odd | 4 | 1 | 3800.2.a.u | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 3800.2.a.v | yes | 3 | |
| 20.e | even | 4 | 1 | 7600.2.a.bt | 3 | ||
| 20.e | even | 4 | 1 | 7600.2.a.bu | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3800.2.a.u | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 3800.2.a.v | yes | 3 | 5.c | odd | 4 | 1 | |
| 3800.2.d.m | 6 | 1.a | even | 1 | 1 | trivial | |
| 3800.2.d.m | 6 | 5.b | even | 2 | 1 | inner | |
| 7600.2.a.bt | 3 | 20.e | even | 4 | 1 | ||
| 7600.2.a.bu | 3 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3800, [\chi])\):
|
\( T_{3}^{6} + 10T_{3}^{4} + 25T_{3}^{2} + 9 \)
|
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\( T_{7}^{6} + 10T_{7}^{4} + 25T_{7}^{2} + 9 \)
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\( T_{11}^{3} + 3T_{11}^{2} - 2T_{11} - 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} + 10 T^{4} + \cdots + 9 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + 10 T^{4} + \cdots + 9 \)
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| $11$ |
\( (T^{3} + 3 T^{2} - 2 T - 1)^{2} \)
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| $13$ |
\( T^{6} + 9 T^{4} + \cdots + 9 \)
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| $17$ |
\( T^{6} + 42 T^{4} + \cdots + 2025 \)
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| $19$ |
\( (T - 1)^{6} \)
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| $23$ |
\( T^{6} + 18 T^{4} + \cdots + 25 \)
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| $29$ |
\( (T^{3} - T^{2} - 78 T - 49)^{2} \)
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| $31$ |
\( (T^{3} + 3 T^{2} - 36 T - 81)^{2} \)
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| $37$ |
\( T^{6} + 49 T^{4} + \cdots + 2025 \)
|
| $41$ |
\( (T^{3} + 9 T^{2} + \cdots - 175)^{2} \)
|
| $43$ |
\( T^{6} + 83 T^{4} + \cdots + 12769 \)
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| $47$ |
\( T^{6} + 109 T^{4} + \cdots + 81 \)
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| $53$ |
\( T^{6} + 67 T^{4} + \cdots + 81 \)
|
| $59$ |
\( (T^{3} - 10 T^{2} + 16 T + 8)^{2} \)
|
| $61$ |
\( (T^{3} + 21 T^{2} + \cdots - 305)^{2} \)
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| $67$ |
\( T^{6} + 97 T^{4} + \cdots + 625 \)
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| $71$ |
\( (T^{3} - T^{2} - 81 T + 281)^{2} \)
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| $73$ |
\( T^{6} + 373 T^{4} + \cdots + 1155625 \)
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| $79$ |
\( (T^{3} - 20 T^{2} + \cdots + 301)^{2} \)
|
| $83$ |
\( T^{6} + 237 T^{4} + \cdots + 11881 \)
|
| $89$ |
\( (T^{3} + 4 T^{2} + \cdots + 355)^{2} \)
|
| $97$ |
\( T^{6} + 153 T^{4} + \cdots + 12769 \)
|
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