Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8925,2,Mod(1,8925)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8925, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8925.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8925 = 3 \cdot 5^{2} \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8925.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: | \(71.2664838040\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.5869904.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 6x^{4} + 4x^{3} + 9x^{2} - 3x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1785) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 6x^{4} + 4x^{3} + 9x^{2} - 3x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 5\nu^{3} + 3\nu^{2} + 5\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + \beta _1 + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 6\beta_{3} + 7\beta_{2} + 11\beta _1 + 1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.23321 | −1.00000 | 2.98722 | 0 | 2.23321 | 1.00000 | −2.20468 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.56429 | −1.00000 | 0.447013 | 0 | 1.56429 | 1.00000 | 2.42933 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.672910 | −1.00000 | −1.54719 | 0 | 0.672910 | 1.00000 | 2.38694 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.364731 | −1.00000 | −1.86697 | 0 | −0.364731 | 1.00000 | −1.41040 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.27240 | −1.00000 | −0.381001 | 0 | −1.27240 | 1.00000 | −3.02958 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.83328 | −1.00000 | 1.36093 | 0 | −1.83328 | 1.00000 | −1.17160 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 8925.2.a.cd | 6 | |
| 5.b | even | 2 | 1 | 8925.2.a.ce | 6 | ||
| 5.c | odd | 4 | 2 | 1785.2.g.d | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1785.2.g.d | ✓ | 12 | 5.c | odd | 4 | 2 | |
| 8925.2.a.cd | 6 | 1.a | even | 1 | 1 | trivial | |
| 8925.2.a.ce | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8925))\):
|
\( T_{2}^{6} + T_{2}^{5} - 6T_{2}^{4} - 4T_{2}^{3} + 9T_{2}^{2} + 3T_{2} - 2 \)
|
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\( T_{11}^{6} - 4T_{11}^{5} - 32T_{11}^{4} + 92T_{11}^{3} + 248T_{11}^{2} - 188T_{11} + 32 \)
|
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\( T_{13}^{6} + 7T_{13}^{5} - 22T_{13}^{4} - 226T_{13}^{3} - 220T_{13}^{2} + 684T_{13} + 668 \)
|
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\( T_{23}^{6} + 17T_{23}^{5} + 98T_{23}^{4} + 208T_{23}^{3} + 80T_{23}^{2} - 80T_{23} - 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 6 T^{4} + \cdots - 2 \)
|
| $3$ |
\( (T + 1)^{6} \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} - 4 T^{5} + \cdots + 32 \)
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| $13$ |
\( T^{6} + 7 T^{5} + \cdots + 668 \)
|
| $17$ |
\( (T + 1)^{6} \)
|
| $19$ |
\( T^{6} - 48 T^{4} + \cdots + 256 \)
|
| $23$ |
\( T^{6} + 17 T^{5} + \cdots - 32 \)
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| $29$ |
\( T^{6} - 8 T^{5} + \cdots - 1816 \)
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| $31$ |
\( T^{6} - 7 T^{5} + \cdots + 292 \)
|
| $37$ |
\( T^{6} - 7 T^{5} + \cdots + 32 \)
|
| $41$ |
\( T^{6} - 7 T^{5} + \cdots - 32 \)
|
| $43$ |
\( T^{6} + 20 T^{5} + \cdots - 2272 \)
|
| $47$ |
\( T^{6} - 15 T^{5} + \cdots - 1832 \)
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| $53$ |
\( T^{6} - 8 T^{5} + \cdots + 752 \)
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| $59$ |
\( T^{6} - 8 T^{5} + \cdots + 8000 \)
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| $61$ |
\( T^{6} - 25 T^{5} + \cdots + 284 \)
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| $67$ |
\( T^{6} + 36 T^{5} + \cdots - 7984 \)
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| $71$ |
\( T^{6} + 14 T^{5} + \cdots + 119528 \)
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| $73$ |
\( T^{6} + 10 T^{5} + \cdots + 1216 \)
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| $79$ |
\( T^{6} + 30 T^{5} + \cdots + 684032 \)
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| $83$ |
\( T^{6} + 37 T^{5} + \cdots - 67768 \)
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| $89$ |
\( T^{6} - 26 T^{5} + \cdots + 150272 \)
|
| $97$ |
\( T^{6} + 38 T^{5} + \cdots - 109456 \)
|
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