Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1875,2,Mod(1249,1875)]
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, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1875.1249");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1875 = 3 \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1875.b (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(14.9719503790\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.6724000000.12 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 16x^{6} + 86x^{4} + 181x^{2} + 121 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 75) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} + 16x^{6} + 86x^{4} + 181x^{2} + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 8\nu^{4} + 4\nu^{2} - 22 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{7} - 58\nu^{5} - 155\nu^{3} - 25\nu ) / 99 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{7} - 43\nu^{5} - 260\nu^{3} - 406\nu ) / 99 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{6} - 25\nu^{4} - 80\nu^{2} - 46 ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{6} + 25\nu^{4} + 89\nu^{2} + 82 ) / 9 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{7} + 25\nu^{5} + 89\nu^{3} + 91\nu ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{4} + 4\beta_{3} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{6} - 9\beta_{5} - 2\beta_{2} + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{7} - 15\beta_{4} - 38\beta_{3} + 30\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 60\beta_{6} + 68\beta_{5} + 25\beta_{2} - 138 \)
|
| \(\nu^{7}\) | \(=\) |
\( 85\beta_{7} + 143\beta_{4} + 297\beta_{3} - 198\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).
| \(n\) | \(626\) | \(1252\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1249.1 |
|
− | 2.70636i | 1.00000i | −5.32440 | 0 | 2.70636 | − | 0.470294i | 8.99702i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1249.2 | − | 2.12233i | 1.00000i | −2.50430 | 0 | 2.12233 | − | 4.35840i | 1.07029i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1249.3 | − | 1.70636i | − | 1.00000i | −0.911672 | 0 | −1.70636 | − | 3.94243i | − | 1.85708i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1249.4 | − | 1.12233i | − | 1.00000i | 0.740367 | 0 | −1.12233 | 1.11373i | − | 3.07561i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1249.5 | 1.12233i | 1.00000i | 0.740367 | 0 | −1.12233 | − | 1.11373i | 3.07561i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1249.6 | 1.70636i | 1.00000i | −0.911672 | 0 | −1.70636 | 3.94243i | 1.85708i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1249.7 | 2.12233i | − | 1.00000i | −2.50430 | 0 | 2.12233 | 4.35840i | − | 1.07029i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1249.8 | 2.70636i | − | 1.00000i | −5.32440 | 0 | 2.70636 | 0.470294i | − | 8.99702i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
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 | 1875.2.b.c | 8 | |
| 5.b | even | 2 | 1 | inner | 1875.2.b.c | 8 | |
| 5.c | odd | 4 | 1 | 1875.2.a.e | 4 | ||
| 5.c | odd | 4 | 1 | 1875.2.a.h | 4 | ||
| 15.e | even | 4 | 1 | 5625.2.a.i | 4 | ||
| 15.e | even | 4 | 1 | 5625.2.a.n | 4 | ||
| 25.d | even | 5 | 2 | 375.2.i.b | 16 | ||
| 25.e | even | 10 | 2 | 375.2.i.b | 16 | ||
| 25.f | odd | 20 | 2 | 75.2.g.b | ✓ | 8 | |
| 25.f | odd | 20 | 2 | 375.2.g.b | 8 | ||
| 75.l | even | 20 | 2 | 225.2.h.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.2.g.b | ✓ | 8 | 25.f | odd | 20 | 2 | |
| 225.2.h.c | 8 | 75.l | even | 20 | 2 | ||
| 375.2.g.b | 8 | 25.f | odd | 20 | 2 | ||
| 375.2.i.b | 16 | 25.d | even | 5 | 2 | ||
| 375.2.i.b | 16 | 25.e | even | 10 | 2 | ||
| 1875.2.a.e | 4 | 5.c | odd | 4 | 1 | ||
| 1875.2.a.h | 4 | 5.c | odd | 4 | 1 | ||
| 1875.2.b.c | 8 | 1.a | even | 1 | 1 | trivial | |
| 1875.2.b.c | 8 | 5.b | even | 2 | 1 | inner | |
| 5625.2.a.i | 4 | 15.e | even | 4 | 1 | ||
| 5625.2.a.n | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 16T_{2}^{6} + 86T_{2}^{4} + 181T_{2}^{2} + 121 \)
acting on \(S_{2}^{\mathrm{new}}(1875, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 16 T^{6} + \cdots + 121 \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 36 T^{6} + \cdots + 81 \)
|
| $11$ |
\( (T^{4} + 7 T^{3} + \cdots - 109)^{2} \)
|
| $13$ |
\( T^{8} + 29 T^{6} + \cdots + 81 \)
|
| $17$ |
\( T^{8} + 16 T^{6} + \cdots + 121 \)
|
| $19$ |
\( (T^{4} + 5 T^{3} + \cdots + 275)^{2} \)
|
| $23$ |
\( T^{8} + 49 T^{6} + \cdots + 361 \)
|
| $29$ |
\( (T^{4} - 20 T^{3} + \cdots + 405)^{2} \)
|
| $31$ |
\( (T^{4} - 23 T^{3} + \cdots + 711)^{2} \)
|
| $37$ |
\( T^{8} + 96 T^{6} + \cdots + 1 \)
|
| $41$ |
\( (T^{2} + 6 T - 11)^{4} \)
|
| $43$ |
\( T^{8} + 134 T^{6} + \cdots + 9801 \)
|
| $47$ |
\( T^{8} + 76 T^{6} + \cdots + 96721 \)
|
| $53$ |
\( T^{8} + 84 T^{6} + \cdots + 68121 \)
|
| $59$ |
\( (T^{4} - 15 T^{3} + \cdots - 3645)^{2} \)
|
| $61$ |
\( (T^{4} + 2 T^{3} - 56 T^{2} + \cdots - 9)^{2} \)
|
| $67$ |
\( T^{8} + 66 T^{6} + \cdots + 29241 \)
|
| $71$ |
\( (T^{4} + 2 T^{3} + \cdots + 911)^{2} \)
|
| $73$ |
\( T^{8} + 354 T^{6} + \cdots + 23707161 \)
|
| $79$ |
\( (T^{4} + 35 T^{3} + \cdots - 9845)^{2} \)
|
| $83$ |
\( T^{8} + 364 T^{6} + \cdots + 958441 \)
|
| $89$ |
\( (T^{4} - 35 T^{3} + \cdots + 3305)^{2} \)
|
| $97$ |
\( T^{8} + 316 T^{6} + \cdots + 4414201 \)
|
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