Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6975,2,Mod(1,6975)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6975, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6975.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6975.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: | \(55.6956554098\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.116450197504.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 14x^{6} + 58x^{4} - 62x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| 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
\( x^{8} - 14x^{6} + 58x^{4} - 62x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - 6\nu^{2} + 1 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 8\nu^{2} - 9 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 9\nu^{4} + 17\nu^{2} - 5 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 13\nu^{5} - 49\nu^{3} + 37\nu ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 15\nu^{5} + 65\nu^{3} - 67\nu ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 13\nu^{5} + 49\nu^{3} - 47\nu ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 14\nu^{5} + 59\nu^{3} - 68\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} - 2\beta_{4} ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{7} - 8\beta_{6} - 5\beta_{5} - 11\beta_{4} ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{2} + 8\beta _1 + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 40\beta_{7} - 49\beta_{6} - 50\beta_{5} - 68\beta_{4} ) / 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4\beta_{3} + 37\beta_{2} + 55\beta _1 + 144 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 275\beta_{7} - 282\beta_{6} - 405\beta_{5} - 439\beta_{4} ) / 5 \)
|
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 |
|
−2.72741 | 0 | 5.43874 | 0 | 0 | 3.85713 | −9.37885 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.10974 | 0 | 2.45100 | 0 | 0 | −2.98362 | −0.951486 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.77541 | 0 | 1.15208 | 0 | 0 | 2.51081 | 1.50540 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.978865 | 0 | −1.04182 | 0 | 0 | −1.38432 | 2.97753 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.978865 | 0 | −1.04182 | 0 | 0 | −1.38432 | −2.97753 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.77541 | 0 | 1.15208 | 0 | 0 | 2.51081 | −1.50540 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.10974 | 0 | 2.45100 | 0 | 0 | −2.98362 | 0.951486 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.72741 | 0 | 5.43874 | 0 | 0 | 3.85713 | 9.37885 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(31\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6975.2.a.cg | yes | 8 |
| 3.b | odd | 2 | 1 | inner | 6975.2.a.cg | yes | 8 |
| 5.b | even | 2 | 1 | 6975.2.a.cf | ✓ | 8 | |
| 15.d | odd | 2 | 1 | 6975.2.a.cf | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6975.2.a.cf | ✓ | 8 | 5.b | even | 2 | 1 | |
| 6975.2.a.cf | ✓ | 8 | 15.d | odd | 2 | 1 | |
| 6975.2.a.cg | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 6975.2.a.cg | yes | 8 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6975))\):
|
\( T_{2}^{8} - 16T_{2}^{6} + 85T_{2}^{4} - 172T_{2}^{2} + 100 \)
|
|
\( T_{7}^{4} - 2T_{7}^{3} - 14T_{7}^{2} + 16T_{7} + 40 \)
|
|
\( T_{11}^{8} - 50T_{11}^{6} + 813T_{11}^{4} - 5308T_{11}^{2} + 12100 \)
|
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\( T_{13}^{4} - 4T_{13}^{3} - 22T_{13}^{2} + 64T_{13} - 40 \)
|
|
\( T_{17}^{8} - 42T_{17}^{6} + 610T_{17}^{4} - 3578T_{17}^{2} + 7225 \)
|
|
\( T_{29}^{8} - 98T_{29}^{6} + 3082T_{29}^{4} - 34082T_{29}^{2} + 87025 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 16 T^{6} + \cdots + 100 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} - 2 T^{3} - 14 T^{2} + \cdots + 40)^{2} \)
|
| $11$ |
\( T^{8} - 50 T^{6} + \cdots + 12100 \)
|
| $13$ |
\( (T^{4} - 4 T^{3} - 22 T^{2} + \cdots - 40)^{2} \)
|
| $17$ |
\( T^{8} - 42 T^{6} + \cdots + 7225 \)
|
| $19$ |
\( (T^{4} - 45 T^{2} + \cdots + 206)^{2} \)
|
| $23$ |
\( T^{8} - 110 T^{6} + \cdots + 270400 \)
|
| $29$ |
\( T^{8} - 98 T^{6} + \cdots + 87025 \)
|
| $31$ |
\( (T + 1)^{8} \)
|
| $37$ |
\( (T^{4} - 6 T^{3} + \cdots - 340)^{2} \)
|
| $41$ |
\( T^{8} - 72 T^{6} + \cdots + 10000 \)
|
| $43$ |
\( (T^{4} - 26 T^{3} + \cdots + 1180)^{2} \)
|
| $47$ |
\( T^{8} - 156 T^{6} + \cdots + 462400 \)
|
| $53$ |
\( T^{8} - 218 T^{6} + \cdots + 16900 \)
|
| $59$ |
\( (T^{2} - 50)^{4} \)
|
| $61$ |
\( (T^{4} + 14 T^{3} + \cdots - 8048)^{2} \)
|
| $67$ |
\( (T^{4} - 12 T^{3} + \cdots - 810)^{2} \)
|
| $71$ |
\( T^{8} - 276 T^{6} + \cdots + 4000000 \)
|
| $73$ |
\( (T^{4} - 28 T^{3} + \cdots + 640)^{2} \)
|
| $79$ |
\( (T^{4} + 2 T^{3} - 53 T^{2} + \cdots - 2)^{2} \)
|
| $83$ |
\( T^{8} - 702 T^{6} + \cdots + 150798400 \)
|
| $89$ |
\( T^{8} - 314 T^{6} + \cdots + 133225 \)
|
| $97$ |
\( (T^{4} - 22 T^{3} + \cdots - 37895)^{2} \)
|
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