Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9025,2,Mod(1,9025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("9025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9025 = 5^{2} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9025.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: | \(72.0649878242\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.4227136.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 6x^{4} + 7x^{2} - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 95) |
| 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
\( x^{6} - 6x^{4} + 7x^{2} - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 3\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 6\nu^{3} + 6\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 5\beta_{2} + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( -5\beta_{5} + 6\beta_{4} + 12\beta_1 \)
|
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 |
|
−2.45620 | −1.56104 | 4.03293 | 0 | 3.83424 | 4.50527 | −4.99330 | −0.563139 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −0.862781 | −3.07914 | −1.25561 | 0 | 2.65662 | 0.566520 | 2.80888 | 6.48108 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.471884 | 1.04022 | −1.77733 | 0 | −0.490864 | −1.17540 | 1.78246 | −1.91794 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.471884 | −1.04022 | −1.77733 | 0 | −0.490864 | 1.17540 | −1.78246 | −1.91794 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0.862781 | 3.07914 | −1.25561 | 0 | 2.65662 | −0.566520 | −2.80888 | 6.48108 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.45620 | 1.56104 | 4.03293 | 0 | 3.83424 | −4.50527 | 4.99330 | −0.563139 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 9025.2.a.bt | 6 | |
| 5.b | even | 2 | 1 | inner | 9025.2.a.bt | 6 | |
| 5.c | odd | 4 | 2 | 1805.2.b.g | 6 | ||
| 19.b | odd | 2 | 1 | 9025.2.a.bu | 6 | ||
| 19.d | odd | 6 | 2 | 475.2.e.g | 12 | ||
| 95.d | odd | 2 | 1 | 9025.2.a.bu | 6 | ||
| 95.g | even | 4 | 2 | 1805.2.b.f | 6 | ||
| 95.h | odd | 6 | 2 | 475.2.e.g | 12 | ||
| 95.l | even | 12 | 4 | 95.2.i.b | ✓ | 12 | |
| 285.w | odd | 12 | 4 | 855.2.be.d | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.2.i.b | ✓ | 12 | 95.l | even | 12 | 4 | |
| 475.2.e.g | 12 | 19.d | odd | 6 | 2 | ||
| 475.2.e.g | 12 | 95.h | odd | 6 | 2 | ||
| 855.2.be.d | 12 | 285.w | odd | 12 | 4 | ||
| 1805.2.b.f | 6 | 95.g | even | 4 | 2 | ||
| 1805.2.b.g | 6 | 5.c | odd | 4 | 2 | ||
| 9025.2.a.bt | 6 | 1.a | even | 1 | 1 | trivial | |
| 9025.2.a.bt | 6 | 5.b | even | 2 | 1 | inner | |
| 9025.2.a.bu | 6 | 19.b | odd | 2 | 1 | ||
| 9025.2.a.bu | 6 | 95.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):
|
\( T_{2}^{6} - 7T_{2}^{4} + 6T_{2}^{2} - 1 \)
|
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\( T_{3}^{6} - 13T_{3}^{4} + 36T_{3}^{2} - 25 \)
|
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\( T_{7}^{6} - 22T_{7}^{4} + 35T_{7}^{2} - 9 \)
|
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\( T_{11}^{3} - T_{11}^{2} - 4T_{11} + 3 \)
|
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\( T_{29}^{3} - 6T_{29}^{2} - 27T_{29} + 27 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 7 T^{4} + \cdots - 1 \)
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| $3$ |
\( T^{6} - 13 T^{4} + \cdots - 25 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} - 22 T^{4} + \cdots - 9 \)
|
| $11$ |
\( (T^{3} - T^{2} - 4 T + 3)^{2} \)
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| $13$ |
\( T^{6} - 31 T^{4} + \cdots - 9 \)
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| $17$ |
\( T^{6} - 35 T^{4} + \cdots - 81 \)
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| $19$ |
\( T^{6} \)
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| $23$ |
\( T^{6} - 12 T^{4} + \cdots - 1 \)
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| $29$ |
\( (T^{3} - 6 T^{2} - 27 T + 27)^{2} \)
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| $31$ |
\( (T^{3} + 15 T^{2} + \cdots + 97)^{2} \)
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| $37$ |
\( T^{6} - 98 T^{4} + \cdots - 729 \)
|
| $41$ |
\( (T^{3} + 6 T^{2} - 41 T - 3)^{2} \)
|
| $43$ |
\( T^{6} - 127 T^{4} + \cdots - 441 \)
|
| $47$ |
\( T^{6} - 214 T^{4} + \cdots - 218089 \)
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| $53$ |
\( T^{6} - 99 T^{4} + \cdots - 11449 \)
|
| $59$ |
\( (T^{3} + 10 T^{2} + \cdots - 291)^{2} \)
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| $61$ |
\( (T^{3} + T^{2} - 41 T - 113)^{2} \)
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| $67$ |
\( T^{6} - 76 T^{4} + \cdots - 14400 \)
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| $71$ |
\( (T^{3} - T^{2} - 124 T - 477)^{2} \)
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| $73$ |
\( T^{6} - 236 T^{4} + \cdots - 99225 \)
|
| $79$ |
\( (T^{3} + 12 T^{2} + \cdots - 448)^{2} \)
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| $83$ |
\( T^{6} - 459 T^{4} + \cdots - 966289 \)
|
| $89$ |
\( (T^{3} + 18 T^{2} + \cdots - 72)^{2} \)
|
| $97$ |
\( T^{6} - 529 T^{4} + \cdots - 1238769 \)
|
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