Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1925,2,Mod(1,1925)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1925, 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("1925.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1925 = 5^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1925.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: | \(15.3712023891\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.9921856.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 7x^{4} + 11x^{2} - 2x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 385) |
| 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
\( x^{6} - 7x^{4} + 11x^{2} - 2x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 6\nu^{3} - \nu^{2} + 7\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 7\nu^{3} - 11\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{2} + 4\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + 5\beta_{2} + \beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{5} + 7\beta_{4} + 7\beta_{2} + 17\beta _1 - 5 \)
|
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 |
|
−2.05338 | 0.555739 | 2.21636 | 0 | −1.14114 | −1.00000 | −0.444261 | −2.69115 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.68955 | 2.93524 | 0.854580 | 0 | −4.95924 | −1.00000 | 1.93524 | 5.61566 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.226645 | 1.89494 | −1.94863 | 0 | −0.429477 | −1.00000 | 0.894936 | 0.590784 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.441021 | −0.678305 | −1.80550 | 0 | −0.299147 | −1.00000 | −1.67831 | −2.53990 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.28581 | −2.01740 | −0.346698 | 0 | −2.59399 | −1.00000 | −3.01740 | 1.06991 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.24274 | 3.30979 | 3.02989 | 0 | 7.42300 | −1.00000 | 2.30979 | 7.95470 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(1\) |
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 | 1925.2.a.z | 6 | |
| 5.b | even | 2 | 1 | 1925.2.a.y | 6 | ||
| 5.c | odd | 4 | 2 | 385.2.b.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 385.2.b.c | ✓ | 12 | 5.c | odd | 4 | 2 | |
| 1925.2.a.y | 6 | 5.b | even | 2 | 1 | ||
| 1925.2.a.z | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1925))\):
|
\( T_{2}^{6} - 7T_{2}^{4} + 11T_{2}^{2} - 2T_{2} - 1 \)
|
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\( T_{3}^{6} - 6T_{3}^{5} + 4T_{3}^{4} + 28T_{3}^{3} - 36T_{3}^{2} - 14T_{3} + 14 \)
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\( T_{13}^{6} - 6T_{13}^{5} - 22T_{13}^{4} + 102T_{13}^{3} + 56T_{13}^{2} - 442T_{13} + 338 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 7 T^{4} + \cdots - 1 \)
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| $3$ |
\( T^{6} - 6 T^{5} + \cdots + 14 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( (T + 1)^{6} \)
|
| $11$ |
\( (T + 1)^{6} \)
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| $13$ |
\( T^{6} - 6 T^{5} + \cdots + 338 \)
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| $17$ |
\( T^{6} + 2 T^{5} + \cdots - 38 \)
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| $19$ |
\( T^{6} + 2 T^{5} + \cdots + 72 \)
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| $23$ |
\( T^{6} - 8 T^{5} + \cdots + 12 \)
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| $29$ |
\( T^{6} + 8 T^{5} + \cdots + 16 \)
|
| $31$ |
\( T^{6} - 4 T^{5} + \cdots + 794 \)
|
| $37$ |
\( T^{6} - 22 T^{5} + \cdots + 38044 \)
|
| $41$ |
\( T^{6} - 4 T^{5} + \cdots + 38 \)
|
| $43$ |
\( T^{6} - 30 T^{5} + \cdots - 20396 \)
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| $47$ |
\( T^{6} - 16 T^{5} + \cdots + 93774 \)
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| $53$ |
\( T^{6} - 6 T^{5} + \cdots + 876 \)
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| $59$ |
\( T^{6} - 14 T^{5} + \cdots - 69814 \)
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| $61$ |
\( T^{6} - 12 T^{5} + \cdots + 1966 \)
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| $67$ |
\( T^{6} - 46 T^{5} + \cdots - 66644 \)
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| $71$ |
\( T^{6} - 4 T^{5} + \cdots - 11728 \)
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| $73$ |
\( T^{6} + 4 T^{5} + \cdots - 1578774 \)
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| $79$ |
\( T^{6} + 8 T^{5} + \cdots + 552316 \)
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| $83$ |
\( T^{6} - 14 T^{5} + \cdots + 1256 \)
|
| $89$ |
\( T^{6} - 8 T^{5} + \cdots - 1097208 \)
|
| $97$ |
\( T^{6} - 42 T^{5} + \cdots - 235096 \)
|
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