Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3900,2,Mod(1249,3900)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3900, 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("3900.1249");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3900.h (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: | \(31.1416567883\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.60217600.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 16x^{4} + 64x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{6} + 16x^{4} + 64x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 8\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 8\nu^{2} ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{2} + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 13\nu^{3} + 40\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} - 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{4} + 2\beta_{3} + 40 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - 26\beta_{2} + 64\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times\).
| \(n\) | \(301\) | \(1301\) | \(1951\) | \(3277\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1249.1 |
|
0 | − | 1.00000i | 0 | 0 | 0 | − | 4.68450i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1249.2 | 0 | − | 1.00000i | 0 | 0 | 0 | 0.732893i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1249.3 | 0 | − | 1.00000i | 0 | 0 | 0 | 4.95160i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1249.4 | 0 | 1.00000i | 0 | 0 | 0 | − | 4.95160i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1249.5 | 0 | 1.00000i | 0 | 0 | 0 | − | 0.732893i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1249.6 | 0 | 1.00000i | 0 | 0 | 0 | 4.68450i | 0 | −1.00000 | 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 | 3900.2.h.k | 6 | |
| 5.b | even | 2 | 1 | inner | 3900.2.h.k | 6 | |
| 5.c | odd | 4 | 1 | 3900.2.a.v | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 3900.2.a.w | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3900.2.a.v | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 3900.2.a.w | yes | 3 | 5.c | odd | 4 | 1 | |
| 3900.2.h.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 3900.2.h.k | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3900, [\chi])\):
|
\( T_{7}^{6} + 47T_{7}^{4} + 563T_{7}^{2} + 289 \)
|
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\( T_{11}^{3} + 3T_{11}^{2} - 5T_{11} - 9 \)
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|
\( T_{19}^{3} - 2T_{19}^{2} - 20T_{19} - 20 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 1)^{3} \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + 47 T^{4} + \cdots + 289 \)
|
| $11$ |
\( (T^{3} + 3 T^{2} - 5 T - 9)^{2} \)
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| $13$ |
\( (T^{2} + 1)^{3} \)
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| $17$ |
\( T^{6} + 67 T^{4} + \cdots + 2209 \)
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| $19$ |
\( (T^{3} - 2 T^{2} - 20 T - 20)^{2} \)
|
| $23$ |
\( T^{6} + 148 T^{4} + \cdots + 32400 \)
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| $29$ |
\( (T^{3} + 5 T^{2} + \cdots - 409)^{2} \)
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| $31$ |
\( (T^{3} - 3 T^{2} + \cdots + 125)^{2} \)
|
| $37$ |
\( T^{6} + 152 T^{4} + \cdots + 115600 \)
|
| $41$ |
\( (T^{3} - 2 T^{2} + \cdots + 228)^{2} \)
|
| $43$ |
\( T^{6} + 108 T^{4} + \cdots + 1296 \)
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| $47$ |
\( T^{6} + 71 T^{4} + \cdots + 225 \)
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| $53$ |
\( T^{6} + 307 T^{4} + \cdots + 55225 \)
|
| $59$ |
\( (T^{3} + 5 T^{2} + \cdots - 633)^{2} \)
|
| $61$ |
\( (T^{3} + T^{2} - 141 T - 261)^{2} \)
|
| $67$ |
\( T^{6} + 251 T^{4} + \cdots + 2809 \)
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| $71$ |
\( (T^{3} + 10 T^{2} + \cdots + 76)^{2} \)
|
| $73$ |
\( T^{6} + 432 T^{4} + \cdots + 2624400 \)
|
| $79$ |
\( (T^{3} + 14 T^{2} + \cdots - 940)^{2} \)
|
| $83$ |
\( T^{6} + 455 T^{4} + \cdots + 1755625 \)
|
| $89$ |
\( (T^{3} + 8 T^{2} + \cdots - 304)^{2} \)
|
| $97$ |
\( T^{6} + 76 T^{4} + \cdots + 5184 \)
|
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