Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7497,2,Mod(1,7497)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7497, 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("7497.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7497 = 3^{2} \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7497.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: | \(59.8638463954\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.17314349056.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 12x^{6} + 40x^{4} - 32x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{8} - 12x^{6} + 40x^{4} - 32x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 10\nu^{3} + 22\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + 10\nu^{4} - 22\nu^{2} + 4 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 10\nu^{4} + 26\nu^{2} - 16 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 12\nu^{4} + 38\nu^{2} - 20 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 11\nu^{5} + 32\nu^{3} - 18\nu ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 12\nu^{5} + 38\nu^{3} - 20\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} - \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{5} + 8\beta_{4} + 6\beta_{3} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{7} + 10\beta_{6} - 6\beta_{2} + 28\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -20\beta_{5} + 58\beta_{4} + 34\beta_{3} + 98 \)
|
| \(\nu^{7}\) | \(=\) |
\( -78\beta_{7} + 82\beta_{6} - 34\beta_{2} + 166\beta_1 \)
|
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 |
|
−2.60593 | 0 | 4.79087 | −1.35178 | 0 | 0 | −7.27281 | 0 | 3.52265 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.03016 | 0 | 2.12154 | 1.29601 | 0 | 0 | −0.246742 | 0 | −2.63111 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.966796 | 0 | −1.06531 | −4.16083 | 0 | 0 | 2.96352 | 0 | 4.02268 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.391024 | 0 | −1.84710 | −1.78339 | 0 | 0 | 1.50431 | 0 | 0.697349 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.391024 | 0 | −1.84710 | −1.78339 | 0 | 0 | −1.50431 | 0 | −0.697349 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.966796 | 0 | −1.06531 | −4.16083 | 0 | 0 | −2.96352 | 0 | −4.02268 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.03016 | 0 | 2.12154 | 1.29601 | 0 | 0 | 0.246742 | 0 | 2.63111 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.60593 | 0 | 4.79087 | −1.35178 | 0 | 0 | 7.27281 | 0 | −3.52265 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(1\) |
| \(17\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7497.2.a.cd | ✓ | 8 |
| 3.b | odd | 2 | 1 | 7497.2.a.ce | yes | 8 | |
| 7.b | odd | 2 | 1 | 7497.2.a.ce | yes | 8 | |
| 21.c | even | 2 | 1 | inner | 7497.2.a.cd | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7497.2.a.cd | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 7497.2.a.cd | ✓ | 8 | 21.c | even | 2 | 1 | inner |
| 7497.2.a.ce | yes | 8 | 3.b | odd | 2 | 1 | |
| 7497.2.a.ce | yes | 8 | 7.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7497))\):
|
\( T_{2}^{8} - 12T_{2}^{6} + 40T_{2}^{4} - 32T_{2}^{2} + 4 \)
|
|
\( T_{5}^{4} + 6T_{5}^{3} + 6T_{5}^{2} - 10T_{5} - 13 \)
|
|
\( T_{11}^{8} - 32T_{11}^{6} + 306T_{11}^{4} - 1000T_{11}^{2} + 625 \)
|
|
\( T_{19}^{8} - 88T_{19}^{6} + 1990T_{19}^{4} - 14352T_{19}^{2} + 32041 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 12 T^{6} + \cdots + 4 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 6 T^{3} + 6 T^{2} + \cdots - 13)^{2} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} - 32 T^{6} + \cdots + 625 \)
|
| $13$ |
\( T^{8} - 48 T^{6} + \cdots + 529 \)
|
| $17$ |
\( (T + 1)^{8} \)
|
| $19$ |
\( T^{8} - 88 T^{6} + \cdots + 32041 \)
|
| $23$ |
\( T^{8} - 56 T^{6} + \cdots + 3025 \)
|
| $29$ |
\( T^{8} - 64 T^{6} + \cdots + 64 \)
|
| $31$ |
\( T^{8} - 212 T^{6} + \cdots + 3587236 \)
|
| $37$ |
\( (T^{4} - 6 T^{3} + \cdots - 806)^{2} \)
|
| $41$ |
\( (T^{4} + 14 T^{3} + \cdots - 1289)^{2} \)
|
| $43$ |
\( (T^{4} + 12 T^{3} + \cdots - 731)^{2} \)
|
| $47$ |
\( (T^{4} + 10 T^{3} + \cdots - 1234)^{2} \)
|
| $53$ |
\( T^{8} - 172 T^{6} + \cdots + 289444 \)
|
| $59$ |
\( (T^{4} + 6 T^{3} + \cdots - 1090)^{2} \)
|
| $61$ |
\( T^{8} - 20 T^{6} + \cdots + 100 \)
|
| $67$ |
\( (T^{4} - 24 T^{3} + \cdots - 1516)^{2} \)
|
| $71$ |
\( T^{8} - 336 T^{6} + \cdots + 38440000 \)
|
| $73$ |
\( T^{8} - 208 T^{6} + \cdots + 64 \)
|
| $79$ |
\( (T^{4} - 30 T^{3} + \cdots - 358)^{2} \)
|
| $83$ |
\( (T^{4} + 20 T^{3} + \cdots - 1040)^{2} \)
|
| $89$ |
\( (T^{4} + 8 T^{3} - 124 T^{2} + \cdots + 40)^{2} \)
|
| $97$ |
\( T^{8} - 352 T^{6} + \cdots + 1763584 \)
|
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