Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1875,2,Mod(1,1875)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1875, 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("1875.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1875 = 3 \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1875.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: | \(14.9719503790\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.5444000000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 75) |
| 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
\( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - \nu^{3} + 7\nu^{2} - 3\nu - 1 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 3\nu^{4} + 11\nu^{3} + 3\nu^{2} - 9\nu - 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 4\nu^{6} - \nu^{5} + 16\nu^{4} - 5\nu^{3} - 16\nu^{2} + 5\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 5\beta_{2} + 7\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} + 7\beta_{3} + 9\beta_{2} + 21\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 3\beta_{5} + 12\beta_{4} + 16\beta_{3} + 28\beta_{2} + 46\beta _1 + 33 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{7} + 4\beta_{6} + 13\beta_{5} + 35\beta_{4} + 44\beta_{3} + 62\beta_{2} + 124\beta _1 + 64 \)
|
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 |
|
−2.53767 | 1.00000 | 4.43979 | 0 | −2.53767 | −1.04054 | −6.19138 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.35083 | 1.00000 | 3.52640 | 0 | −2.35083 | −3.48189 | −3.58831 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.53655 | 1.00000 | 0.360976 | 0 | −1.53655 | −1.49550 | 2.51844 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.08982 | 1.00000 | −0.812294 | 0 | −1.08982 | 3.08724 | 3.06489 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.0898194 | 1.00000 | −1.99193 | 0 | 0.0898194 | −4.36070 | −0.358553 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.536547 | 1.00000 | −1.71212 | 0 | 0.536547 | 2.57318 | −1.99173 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.35083 | 1.00000 | −0.175259 | 0 | 1.35083 | −1.59580 | −2.93840 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.53767 | 1.00000 | 0.364440 | 0 | 1.53767 | −1.68601 | −2.51496 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 1875.2.a.m | 8 | |
| 3.b | odd | 2 | 1 | 5625.2.a.bd | 8 | ||
| 5.b | even | 2 | 1 | 1875.2.a.p | 8 | ||
| 5.c | odd | 4 | 2 | 1875.2.b.h | 16 | ||
| 15.d | odd | 2 | 1 | 5625.2.a.t | 8 | ||
| 25.d | even | 5 | 2 | 375.2.g.e | 16 | ||
| 25.e | even | 10 | 2 | 375.2.g.d | 16 | ||
| 25.f | odd | 20 | 2 | 75.2.i.a | ✓ | 16 | |
| 25.f | odd | 20 | 2 | 375.2.i.c | 16 | ||
| 75.l | even | 20 | 2 | 225.2.m.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.2.i.a | ✓ | 16 | 25.f | odd | 20 | 2 | |
| 225.2.m.b | 16 | 75.l | even | 20 | 2 | ||
| 375.2.g.d | 16 | 25.e | even | 10 | 2 | ||
| 375.2.g.e | 16 | 25.d | even | 5 | 2 | ||
| 375.2.i.c | 16 | 25.f | odd | 20 | 2 | ||
| 1875.2.a.m | 8 | 1.a | even | 1 | 1 | trivial | |
| 1875.2.a.p | 8 | 5.b | even | 2 | 1 | ||
| 1875.2.b.h | 16 | 5.c | odd | 4 | 2 | ||
| 5625.2.a.t | 8 | 15.d | odd | 2 | 1 | ||
| 5625.2.a.bd | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 4T_{2}^{7} - 2T_{2}^{6} - 20T_{2}^{5} - 4T_{2}^{4} + 30T_{2}^{3} + 7T_{2}^{2} - 12T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 4 T^{7} + \cdots + 1 \)
|
| $3$ |
\( (T - 1)^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 8 T^{7} + \cdots + 505 \)
|
| $11$ |
\( T^{8} - 2 T^{7} + \cdots + 5281 \)
|
| $13$ |
\( T^{8} + 16 T^{7} + \cdots + 281 \)
|
| $17$ |
\( T^{8} + 16 T^{7} + \cdots - 7339 \)
|
| $19$ |
\( T^{8} + 14 T^{7} + \cdots + 2525 \)
|
| $23$ |
\( T^{8} + 14 T^{7} + \cdots - 2095 \)
|
| $29$ |
\( T^{8} - 2 T^{7} + \cdots - 395 \)
|
| $31$ |
\( T^{8} + 22 T^{7} + \cdots + 125 \)
|
| $37$ |
\( T^{8} + 28 T^{7} + \cdots + 93025 \)
|
| $41$ |
\( T^{8} - 8 T^{7} + \cdots + 4705 \)
|
| $43$ |
\( T^{8} + 20 T^{7} + \cdots + 22961 \)
|
| $47$ |
\( T^{8} + 10 T^{7} + \cdots - 6057019 \)
|
| $53$ |
\( T^{8} + 44 T^{7} + \cdots - 200995 \)
|
| $59$ |
\( T^{8} - 14 T^{7} + \cdots - 3595 \)
|
| $61$ |
\( T^{8} + 20 T^{7} + \cdots + 16604261 \)
|
| $67$ |
\( T^{8} + 16 T^{7} + \cdots - 3739 \)
|
| $71$ |
\( T^{8} - 16 T^{7} + \cdots - 159779 \)
|
| $73$ |
\( T^{8} + 24 T^{7} + \cdots - 870295 \)
|
| $79$ |
\( T^{8} + 30 T^{7} + \cdots - 1984975 \)
|
| $83$ |
\( T^{8} + 12 T^{7} + \cdots + 48541 \)
|
| $89$ |
\( T^{8} - 16 T^{7} + \cdots + 5 \)
|
| $97$ |
\( T^{8} + 16 T^{7} + \cdots - 14719 \)
|
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