Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1375,2,Mod(1,1375)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1375, 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("1375.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1375 = 5^{3} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1375.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: | \(10.9794302779\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.6988960000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 9x^{6} + 22x^{4} - 11x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 9x^{6} + 22x^{4} - 11x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 8\nu^{4} + 16\nu^{2} - 5 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 10\nu^{4} - 26\nu^{2} + 9 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{7} + 9\nu^{5} - 21\nu^{3} + 6\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 9\nu^{5} + 22\nu^{3} - 11\nu \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{7} + 26\nu^{5} - 58\nu^{3} + 17\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{7} + 5\beta_{6} + 8\beta_{5} + 24\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{4} + 10\beta_{3} + 24\beta_{2} + 37 \)
|
| \(\nu^{7}\) | \(=\) |
\( -18\beta_{7} + 24\beta_{6} + 50\beta_{5} + 117\beta_1 \)
|
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 |
|
−2.25947 | 0.716111 | 3.10522 | 0 | −1.61803 | 3.36025 | −2.49721 | −2.48718 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.80692 | −0.342036 | 1.26498 | 0 | 0.618034 | 3.40246 | 1.32813 | −2.88301 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.716111 | 2.25947 | −1.48718 | 0 | −1.61803 | −2.22368 | 2.49721 | 2.10522 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.342036 | −1.80692 | −1.88301 | 0 | 0.618034 | −0.432668 | 1.32813 | 0.264977 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.342036 | 1.80692 | −1.88301 | 0 | 0.618034 | 0.432668 | −1.32813 | 0.264977 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.716111 | −2.25947 | −1.48718 | 0 | −1.61803 | 2.22368 | −2.49721 | 2.10522 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.80692 | 0.342036 | 1.26498 | 0 | 0.618034 | −3.40246 | −1.32813 | −2.88301 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.25947 | −0.716111 | 3.10522 | 0 | −1.61803 | −3.36025 | 2.49721 | −2.48718 | 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 | 1375.2.a.c | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 1375.2.a.c | ✓ | 8 |
| 5.c | odd | 4 | 2 | 1375.2.b.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1375.2.a.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1375.2.a.c | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 1375.2.b.b | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 9T_{2}^{6} + 22T_{2}^{4} - 11T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1375))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 9 T^{6} + \cdots + 1 \)
|
| $3$ |
\( T^{8} - 9 T^{6} + \cdots + 1 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} - 28 T^{6} + \cdots + 121 \)
|
| $11$ |
\( (T + 1)^{8} \)
|
| $13$ |
\( T^{8} - 60 T^{6} + \cdots + 10000 \)
|
| $17$ |
\( T^{8} - 56 T^{6} + \cdots + 15376 \)
|
| $19$ |
\( (T^{4} + 9 T^{3} + \cdots + 176)^{2} \)
|
| $23$ |
\( T^{8} - 59 T^{6} + \cdots + 10201 \)
|
| $29$ |
\( (T^{4} + 18 T^{3} + \cdots + 251)^{2} \)
|
| $31$ |
\( (T^{4} + 8 T^{3} + \cdots + 484)^{2} \)
|
| $37$ |
\( T^{8} - 121 T^{6} + \cdots + 364816 \)
|
| $41$ |
\( (T^{4} + 23 T^{3} + \cdots + 461)^{2} \)
|
| $43$ |
\( T^{8} - 93 T^{6} + \cdots + 128881 \)
|
| $47$ |
\( T^{8} - 181 T^{6} + \cdots + 109561 \)
|
| $53$ |
\( T^{8} - 344 T^{6} + \cdots + 16128256 \)
|
| $59$ |
\( (T^{4} - 7 T^{3} + \cdots + 844)^{2} \)
|
| $61$ |
\( (T^{4} + 10 T^{3} + \cdots + 359)^{2} \)
|
| $67$ |
\( T^{8} - 404 T^{6} + \cdots + 15335056 \)
|
| $71$ |
\( (T^{4} - 9 T^{3} + \cdots - 1936)^{2} \)
|
| $73$ |
\( T^{8} - 327 T^{6} + \cdots + 55696 \)
|
| $79$ |
\( (T^{4} + 17 T^{3} + \cdots - 3764)^{2} \)
|
| $83$ |
\( T^{8} - 331 T^{6} + \cdots + 274576 \)
|
| $89$ |
\( (T^{4} + 36 T^{3} + \cdots - 1159)^{2} \)
|
| $97$ |
\( T^{8} - 732 T^{6} + \cdots + 939790336 \)
|
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