Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7744,2,Mod(1,7744)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7744, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("7744.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7744 = 2^{6} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7744.a (trivial) |
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: | yes |
| Analytic conductor: | \(61.8361513253\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 3872) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $N(\mathrm{U}(1))$ |
$q$-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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 0 | 0 | −4.46410 | 0 | 0 | 0 | −3.00000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 2.46410 | 0 | 0 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(11\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7744.2.a.cf | 2 | |
| 4.b | odd | 2 | 1 | CM | 7744.2.a.cf | 2 | |
| 8.b | even | 2 | 1 | 3872.2.a.v | ✓ | 2 | |
| 8.d | odd | 2 | 1 | 3872.2.a.v | ✓ | 2 | |
| 11.b | odd | 2 | 1 | 7744.2.a.ce | 2 | ||
| 44.c | even | 2 | 1 | 7744.2.a.ce | 2 | ||
| 88.b | odd | 2 | 1 | 3872.2.a.w | yes | 2 | |
| 88.g | even | 2 | 1 | 3872.2.a.w | yes | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3872.2.a.v | ✓ | 2 | 8.b | even | 2 | 1 | |
| 3872.2.a.v | ✓ | 2 | 8.d | odd | 2 | 1 | |
| 3872.2.a.w | yes | 2 | 88.b | odd | 2 | 1 | |
| 3872.2.a.w | yes | 2 | 88.g | even | 2 | 1 | |
| 7744.2.a.ce | 2 | 11.b | odd | 2 | 1 | ||
| 7744.2.a.ce | 2 | 44.c | even | 2 | 1 | ||
| 7744.2.a.cf | 2 | 1.a | even | 1 | 1 | trivial | |
| 7744.2.a.cf | 2 | 4.b | odd | 2 | 1 | CM | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7744))\):
|
\( T_{3} \)
|
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\( T_{5}^{2} + 2T_{5} - 11 \)
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|
\( T_{7} \)
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\( T_{13}^{2} - 6T_{13} - 3 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
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| $5$ |
\( T^{2} + 2T - 11 \)
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| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} \)
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| $13$ |
\( T^{2} - 6T - 3 \)
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| $17$ |
\( T^{2} + 2T - 47 \)
|
| $19$ |
\( T^{2} \)
|
| $23$ |
\( T^{2} \)
|
| $29$ |
\( T^{2} + 10T + 13 \)
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| $31$ |
\( T^{2} \)
|
| $37$ |
\( T^{2} + 2T - 107 \)
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| $41$ |
\( T^{2} + 10T - 23 \)
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| $43$ |
\( T^{2} \)
|
| $47$ |
\( T^{2} \)
|
| $53$ |
\( T^{2} - 14T + 37 \)
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| $59$ |
\( T^{2} \)
|
| $61$ |
\( (T - 10)^{2} \)
|
| $67$ |
\( T^{2} \)
|
| $71$ |
\( T^{2} \)
|
| $73$ |
\( (T + 6)^{2} \)
|
| $79$ |
\( T^{2} \)
|
| $83$ |
\( T^{2} \)
|
| $89$ |
\( T^{2} + 10T - 167 \)
|
| $97$ |
\( T^{2} + 18T + 33 \)
|
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