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: | 6.6.71593280.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 8x^{4} + 13x^{2} - 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - 8x^{4} + 13x^{2} - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 7\nu^{3} + 6\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 7\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 7\beta_{3} + 29\beta_1 \)
|
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 |
|
−2.44118 | −1.94894 | 3.95934 | 0 | 4.75770 | −4.16098 | −4.78310 | 0.798360 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.21237 | 1.36806 | −0.530168 | 0 | −1.65859 | −1.59825 | 3.06749 | −1.12842 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.755530 | −2.51596 | −1.42917 | 0 | 1.90089 | 3.75923 | 2.59085 | 3.33006 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.755530 | 2.51596 | −1.42917 | 0 | 1.90089 | 3.75923 | −2.59085 | 3.33006 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.21237 | −1.36806 | −0.530168 | 0 | −1.65859 | −1.59825 | −3.06749 | −1.12842 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.44118 | 1.94894 | 3.95934 | 0 | 4.75770 | −4.16098 | 4.78310 | 0.798360 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(19\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.b | odd | 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.bv | ✓ | 6 |
| 5.b | even | 2 | 1 | 9025.2.a.bw | yes | 6 | |
| 19.b | odd | 2 | 1 | inner | 9025.2.a.bv | ✓ | 6 |
| 95.d | odd | 2 | 1 | 9025.2.a.bw | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9025.2.a.bv | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 9025.2.a.bv | ✓ | 6 | 19.b | odd | 2 | 1 | inner |
| 9025.2.a.bw | yes | 6 | 5.b | even | 2 | 1 | |
| 9025.2.a.bw | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):
|
\( T_{2}^{6} - 8T_{2}^{4} + 13T_{2}^{2} - 5 \)
|
|
\( T_{3}^{6} - 12T_{3}^{4} + 43T_{3}^{2} - 45 \)
|
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\( T_{7}^{3} + 2T_{7}^{2} - 15T_{7} - 25 \)
|
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\( T_{11}^{3} + 5T_{11}^{2} - 2T_{11} - 15 \)
|
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\( T_{29}^{6} - 115T_{29}^{4} + 1250T_{29}^{2} - 3125 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 8 T^{4} + \cdots - 5 \)
|
| $3$ |
\( T^{6} - 12 T^{4} + \cdots - 45 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( (T^{3} + 2 T^{2} - 15 T - 25)^{2} \)
|
| $11$ |
\( (T^{3} + 5 T^{2} - 2 T - 15)^{2} \)
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| $13$ |
\( T^{6} - 37 T^{4} + \cdots - 605 \)
|
| $17$ |
\( (T^{3} + 2 T^{2} - 15 T - 25)^{2} \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( (T^{3} + 4 T^{2} + \cdots - 225)^{2} \)
|
| $29$ |
\( T^{6} - 115 T^{4} + \cdots - 3125 \)
|
| $31$ |
\( T^{6} - 65 T^{4} + \cdots - 3125 \)
|
| $37$ |
\( T^{6} - 123 T^{4} + \cdots - 53045 \)
|
| $41$ |
\( T^{6} - 215 T^{4} + \cdots - 253125 \)
|
| $43$ |
\( (T^{3} + 11 T^{2} + \cdots - 75)^{2} \)
|
| $47$ |
\( (T^{3} + 17 T^{2} + \cdots - 425)^{2} \)
|
| $53$ |
\( T^{6} - 107 T^{4} + \cdots - 21125 \)
|
| $59$ |
\( T^{6} - 260 T^{4} + \cdots - 200000 \)
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| $61$ |
\( (T^{3} - 5 T^{2} + \cdots + 505)^{2} \)
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| $67$ |
\( T^{6} - 43 T^{4} + \cdots - 45 \)
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| $71$ |
\( T^{6} - 315 T^{4} + \cdots - 3125 \)
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| $73$ |
\( (T^{3} + 11 T^{2} + \cdots - 25)^{2} \)
|
| $79$ |
\( T^{6} - 260 T^{4} + \cdots - 3125 \)
|
| $83$ |
\( (T^{3} + 11 T^{2} + \cdots - 25)^{2} \)
|
| $89$ |
\( T^{6} - 200 T^{4} + \cdots - 78125 \)
|
| $97$ |
\( T^{6} - 473 T^{4} + \cdots - 658845 \)
|
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