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: | 6.6.66064384.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 9x^{4} + 13x^{2} - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{6} - 9x^{4} + 13x^{2} - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 8\nu^{2} + 5 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 8\nu^{3} + 7\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 9\nu^{3} - 13\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{3} + 8\beta_{2} + 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{5} + 18\beta_{4} + 41\beta_1 \)
|
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 |
|
−2.41987 | 0.537080 | 3.85577 | 0 | −1.29966 | 3.18676 | −4.49073 | −2.71155 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.82254 | −2.31446 | 1.32164 | 0 | 4.21819 | 1.45033 | 1.23634 | 2.35673 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.906968 | 3.21789 | −1.17741 | 0 | −2.91852 | −2.59637 | 2.88181 | 7.35482 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.906968 | −3.21789 | −1.17741 | 0 | −2.91852 | 2.59637 | −2.88181 | 7.35482 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.82254 | 2.31446 | 1.32164 | 0 | 4.21819 | −1.45033 | −1.23634 | 2.35673 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.41987 | −0.537080 | 3.85577 | 0 | −1.29966 | −3.18676 | 4.49073 | −2.71155 | 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.bx | 6 | |
| 5.b | even | 2 | 1 | inner | 9025.2.a.bx | 6 | |
| 5.c | odd | 4 | 2 | 1805.2.b.e | 6 | ||
| 19.b | odd | 2 | 1 | 475.2.a.j | 6 | ||
| 57.d | even | 2 | 1 | 4275.2.a.br | 6 | ||
| 76.d | even | 2 | 1 | 7600.2.a.ck | 6 | ||
| 95.d | odd | 2 | 1 | 475.2.a.j | 6 | ||
| 95.g | even | 4 | 2 | 95.2.b.b | ✓ | 6 | |
| 285.b | even | 2 | 1 | 4275.2.a.br | 6 | ||
| 285.j | odd | 4 | 2 | 855.2.c.d | 6 | ||
| 380.d | even | 2 | 1 | 7600.2.a.ck | 6 | ||
| 380.j | odd | 4 | 2 | 1520.2.d.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.2.b.b | ✓ | 6 | 95.g | even | 4 | 2 | |
| 475.2.a.j | 6 | 19.b | odd | 2 | 1 | ||
| 475.2.a.j | 6 | 95.d | odd | 2 | 1 | ||
| 855.2.c.d | 6 | 285.j | odd | 4 | 2 | ||
| 1520.2.d.h | 6 | 380.j | odd | 4 | 2 | ||
| 1805.2.b.e | 6 | 5.c | odd | 4 | 2 | ||
| 4275.2.a.br | 6 | 57.d | even | 2 | 1 | ||
| 4275.2.a.br | 6 | 285.b | even | 2 | 1 | ||
| 7600.2.a.ck | 6 | 76.d | even | 2 | 1 | ||
| 7600.2.a.ck | 6 | 380.d | even | 2 | 1 | ||
| 9025.2.a.bx | 6 | 1.a | even | 1 | 1 | trivial | |
| 9025.2.a.bx | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):
|
\( T_{2}^{6} - 10T_{2}^{4} + 27T_{2}^{2} - 16 \)
|
|
\( T_{3}^{6} - 16T_{3}^{4} + 60T_{3}^{2} - 16 \)
|
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\( T_{7}^{6} - 19T_{7}^{4} + 104T_{7}^{2} - 144 \)
|
|
\( T_{11}^{3} - T_{11}^{2} - 16T_{11} + 12 \)
|
|
\( T_{29} + 6 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 10 T^{4} + \cdots - 16 \)
|
| $3$ |
\( T^{6} - 16 T^{4} + \cdots - 16 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 19 T^{4} + \cdots - 144 \)
|
| $11$ |
\( (T^{3} - T^{2} - 16 T + 12)^{2} \)
|
| $13$ |
\( T^{6} - 28 T^{4} + \cdots - 576 \)
|
| $17$ |
\( T^{6} - 59 T^{4} + \cdots - 5184 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - 36 T^{4} + \cdots - 64 \)
|
| $29$ |
\( (T + 6)^{6} \)
|
| $31$ |
\( (T^{3} - 56 T - 128)^{2} \)
|
| $37$ |
\( T^{6} - 56 T^{4} + \cdots - 1296 \)
|
| $41$ |
\( (T^{3} + 6 T^{2} - 44 T + 24)^{2} \)
|
| $43$ |
\( T^{6} - 19 T^{4} + \cdots - 144 \)
|
| $47$ |
\( T^{6} - 187 T^{4} + \cdots - 85264 \)
|
| $53$ |
\( T^{6} - 156 T^{4} + \cdots - 64 \)
|
| $59$ |
\( (T^{3} + 10 T^{2} + \cdots - 48)^{2} \)
|
| $61$ |
\( (T^{3} + 7 T^{2} + \cdots - 776)^{2} \)
|
| $67$ |
\( T^{6} - 340 T^{4} + \cdots - 484416 \)
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| $71$ |
\( (T^{3} + 26 T^{2} + \cdots + 432)^{2} \)
|
| $73$ |
\( T^{6} - 131 T^{4} + \cdots - 5184 \)
|
| $79$ |
\( (T^{3} - 12 T^{2} + \cdots + 32)^{2} \)
|
| $83$ |
\( T^{6} - 228 T^{4} + \cdots - 141376 \)
|
| $89$ |
\( (T^{3} + 12 T^{2} + \cdots - 3456)^{2} \)
|
| $97$ |
\( T^{6} - 28 T^{4} + \cdots - 576 \)
|
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