Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3025,2,Mod(1,3025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3025, 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("3025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3025 = 5^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3025.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: | \(24.1547466114\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.1480160000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 9x^{6} + 27x^{4} - 31x^{2} + 11 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 55) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 9x^{6} + 27x^{4} - 31x^{2} + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{7} + 7\nu^{5} - 13\nu^{3} + 5\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 7\nu^{4} + 14\nu^{2} - 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 8\nu \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{7} + 8\nu^{5} - 19\nu^{3} + 12\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{4} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 5\beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 6\beta_{6} + 5\beta_{4} + 11\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{5} + 7\beta_{3} + 21\beta_{2} + 22 \)
|
| \(\nu^{7}\) | \(=\) |
\( 7\beta_{7} + 29\beta_{6} + 21\beta_{4} + 43\beta_1 \)
|
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 |
|
−2.02368 | −2.62059 | 2.09529 | 0 | 5.30325 | 0.965823 | −0.192845 | 3.86752 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.65458 | 1.97479 | 0.737640 | 0 | −3.26745 | 2.24307 | 2.08868 | 0.899788 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.23399 | 0.363982 | −0.477260 | 0 | −0.449152 | −2.58558 | 3.05692 | −2.86752 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.802699 | −1.76074 | −1.35567 | 0 | 1.41335 | −0.592103 | 2.69360 | 0.100212 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.802699 | 1.76074 | −1.35567 | 0 | 1.41335 | 0.592103 | −2.69360 | 0.100212 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.23399 | −0.363982 | −0.477260 | 0 | −0.449152 | 2.58558 | −3.05692 | −2.86752 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.65458 | −1.97479 | 0.737640 | 0 | −3.26745 | −2.24307 | −2.08868 | 0.899788 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.02368 | 2.62059 | 2.09529 | 0 | 5.30325 | −0.965823 | 0.192845 | 3.86752 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(11\) | \(1\) |
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 | 3025.2.a.bl | 8 | |
| 5.b | even | 2 | 1 | inner | 3025.2.a.bl | 8 | |
| 5.c | odd | 4 | 2 | 605.2.b.g | 8 | ||
| 11.b | odd | 2 | 1 | 3025.2.a.bk | 8 | ||
| 11.c | even | 5 | 2 | 275.2.h.d | 16 | ||
| 55.d | odd | 2 | 1 | 3025.2.a.bk | 8 | ||
| 55.e | even | 4 | 2 | 605.2.b.f | 8 | ||
| 55.j | even | 10 | 2 | 275.2.h.d | 16 | ||
| 55.k | odd | 20 | 4 | 55.2.j.a | ✓ | 16 | |
| 55.k | odd | 20 | 4 | 605.2.j.h | 16 | ||
| 55.l | even | 20 | 4 | 605.2.j.d | 16 | ||
| 55.l | even | 20 | 4 | 605.2.j.g | 16 | ||
| 165.v | even | 20 | 4 | 495.2.ba.a | 16 | ||
| 220.v | even | 20 | 4 | 880.2.cd.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.2.j.a | ✓ | 16 | 55.k | odd | 20 | 4 | |
| 275.2.h.d | 16 | 11.c | even | 5 | 2 | ||
| 275.2.h.d | 16 | 55.j | even | 10 | 2 | ||
| 495.2.ba.a | 16 | 165.v | even | 20 | 4 | ||
| 605.2.b.f | 8 | 55.e | even | 4 | 2 | ||
| 605.2.b.g | 8 | 5.c | odd | 4 | 2 | ||
| 605.2.j.d | 16 | 55.l | even | 20 | 4 | ||
| 605.2.j.g | 16 | 55.l | even | 20 | 4 | ||
| 605.2.j.h | 16 | 55.k | odd | 20 | 4 | ||
| 880.2.cd.c | 16 | 220.v | even | 20 | 4 | ||
| 3025.2.a.bk | 8 | 11.b | odd | 2 | 1 | ||
| 3025.2.a.bk | 8 | 55.d | odd | 2 | 1 | ||
| 3025.2.a.bl | 8 | 1.a | even | 1 | 1 | trivial | |
| 3025.2.a.bl | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):
|
\( T_{2}^{8} - 9T_{2}^{6} + 27T_{2}^{4} - 31T_{2}^{2} + 11 \)
|
|
\( T_{3}^{8} - 14T_{3}^{6} + 62T_{3}^{4} - 91T_{3}^{2} + 11 \)
|
|
\( T_{19}^{4} - 6T_{19}^{3} - 4T_{19}^{2} + 39T_{19} + 11 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 9 T^{6} + \cdots + 11 \)
|
| $3$ |
\( T^{8} - 14 T^{6} + \cdots + 11 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} - 13 T^{6} + \cdots + 11 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} - 45 T^{6} + \cdots + 6875 \)
|
| $17$ |
\( T^{8} - 81 T^{6} + \cdots + 40931 \)
|
| $19$ |
\( (T^{4} - 6 T^{3} - 4 T^{2} + \cdots + 11)^{2} \)
|
| $23$ |
\( T^{8} - 99 T^{6} + \cdots + 26411 \)
|
| $29$ |
\( (T^{4} - 12 T^{3} + \cdots + 11)^{2} \)
|
| $31$ |
\( (T^{4} - 7 T^{3} + 5 T^{2} + \cdots - 1)^{2} \)
|
| $37$ |
\( T^{8} - 101 T^{6} + \cdots + 3971 \)
|
| $41$ |
\( (T^{4} - 17 T^{3} + \cdots - 1969)^{2} \)
|
| $43$ |
\( T^{8} - 173 T^{6} + \cdots + 212531 \)
|
| $47$ |
\( T^{8} - 191 T^{6} + \cdots + 244211 \)
|
| $53$ |
\( T^{8} - 249 T^{6} + \cdots + 489731 \)
|
| $59$ |
\( (T^{4} + 3 T^{3} - 75 T^{2} + \cdots - 1)^{2} \)
|
| $61$ |
\( (T^{4} - 10 T^{3} + \cdots + 209)^{2} \)
|
| $67$ |
\( T^{8} - 149 T^{6} + \cdots + 18491 \)
|
| $71$ |
\( (T^{4} + 21 T^{3} + \cdots - 2511)^{2} \)
|
| $73$ |
\( T^{8} - 297 T^{6} + \cdots + 1279091 \)
|
| $79$ |
\( (T^{4} - 8 T^{3} + \cdots - 319)^{2} \)
|
| $83$ |
\( T^{8} - 111 T^{6} + \cdots + 212531 \)
|
| $89$ |
\( (T^{4} + 6 T^{3} + \cdots + 1871)^{2} \)
|
| $97$ |
\( T^{8} - 162 T^{6} + \cdots + 161051 \)
|
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