Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6600,2,Mod(1,6600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6600, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("6600.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6600.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: | \(52.7012653340\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.2389280.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 11x^{3} + 7x^{2} + 26x - 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 1320) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 11x^{3} + 7x^{2} + 26x - 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 5\nu + 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 2\nu^{2} + 7\nu - 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 8\nu^{2} + 2\nu + 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 2\beta _1 + 8 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{3} + 9\beta_{2} + 4\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{4} + 13\beta_{3} + 15\beta_{2} + 20\beta _1 + 48 ) / 2 \)
|
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 | 1.00000 | 0 | 0 | 0 | −4.20542 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | 0 | 0 | −2.99623 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.00000 | 0 | 0 | 0 | 2.56437 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.00000 | 0 | 0 | 0 | 2.71268 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.00000 | 0 | 0 | 0 | 3.92460 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(11\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6600.2.a.bz | 5 | |
| 5.b | even | 2 | 1 | 6600.2.a.bx | 5 | ||
| 5.c | odd | 4 | 2 | 1320.2.d.c | ✓ | 10 | |
| 15.e | even | 4 | 2 | 3960.2.d.h | 10 | ||
| 20.e | even | 4 | 2 | 2640.2.d.k | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1320.2.d.c | ✓ | 10 | 5.c | odd | 4 | 2 | |
| 2640.2.d.k | 10 | 20.e | even | 4 | 2 | ||
| 3960.2.d.h | 10 | 15.e | even | 4 | 2 | ||
| 6600.2.a.bx | 5 | 5.b | even | 2 | 1 | ||
| 6600.2.a.bz | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6600))\):
|
\( T_{7}^{5} - 2T_{7}^{4} - 26T_{7}^{3} + 56T_{7}^{2} + 152T_{7} - 344 \)
|
|
\( T_{13}^{5} - 46T_{13}^{3} + 84T_{13}^{2} + 296T_{13} - 648 \)
|
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\( T_{17}^{5} + 6T_{17}^{4} - 16T_{17}^{3} - 108T_{17}^{2} - 64T_{17} + 128 \)
|
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\( T_{19}^{5} - 2T_{19}^{4} - 52T_{19}^{3} + 72T_{19}^{2} + 640T_{19} - 512 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T - 1)^{5} \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - 2 T^{4} + \cdots - 344 \)
|
| $11$ |
\( (T + 1)^{5} \)
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| $13$ |
\( T^{5} - 46 T^{3} + \cdots - 648 \)
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| $17$ |
\( T^{5} + 6 T^{4} + \cdots + 128 \)
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| $19$ |
\( T^{5} - 2 T^{4} + \cdots - 512 \)
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| $23$ |
\( T^{5} - 4 T^{4} + \cdots - 1536 \)
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| $29$ |
\( T^{5} - 4 T^{4} + \cdots - 1472 \)
|
| $31$ |
\( T^{5} - 4 T^{4} + \cdots - 1792 \)
|
| $37$ |
\( T^{5} - 8 T^{4} + \cdots - 128 \)
|
| $41$ |
\( T^{5} - 8 T^{4} + \cdots - 3840 \)
|
| $43$ |
\( T^{5} - 22 T^{4} + \cdots + 96 \)
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| $47$ |
\( T^{5} - 166 T^{3} + \cdots + 448 \)
|
| $53$ |
\( T^{5} + 4 T^{4} + \cdots + 12000 \)
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| $59$ |
\( T^{5} - 12 T^{4} + \cdots + 11392 \)
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| $61$ |
\( T^{5} - 10 T^{4} + \cdots - 7184 \)
|
| $67$ |
\( T^{5} - 20 T^{4} + \cdots + 40960 \)
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| $71$ |
\( T^{5} - 4 T^{4} + \cdots + 1952 \)
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| $73$ |
\( T^{5} - 4 T^{4} + \cdots + 2192 \)
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| $79$ |
\( T^{5} - 18 T^{4} + \cdots - 16 \)
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| $83$ |
\( T^{5} - 24 T^{4} + \cdots + 3392 \)
|
| $89$ |
\( T^{5} - 34 T^{4} + \cdots + 6752 \)
|
| $97$ |
\( T^{5} - 252 T^{3} + \cdots - 35584 \)
|
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