Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,16,Mod(1,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.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: | \(107.020128825\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 206884 x^{6} + 5065964 x^{5} + 12902022496 x^{4} - 638065050800 x^{3} + \cdots + 96\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{8}\cdot 5^{14} \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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
\( x^{8} - x^{7} - 206884 x^{6} + 5065964 x^{5} + 12902022496 x^{4} - 638065050800 x^{3} + \cdots + 96\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 35\nu - 51726 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2581 \nu^{7} + 31693 \nu^{6} - 683774226 \nu^{5} + 2890812920 \nu^{4} + \cdots - 20\!\cdots\!12 ) / 11388268099584 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2581 \nu^{7} - 31693 \nu^{6} + 683774226 \nu^{5} - 2890812920 \nu^{4} + \cdots + 25\!\cdots\!60 ) / 11388268099584 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 56903 \nu^{7} - 58895923 \nu^{6} + 13639097274 \nu^{5} + 9939087037640 \nu^{4} + \cdots + 40\!\cdots\!68 ) / 45553072398336 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 473989 \nu^{7} - 55931569 \nu^{6} + 87901444566 \nu^{5} + 8587020066616 \nu^{4} + \cdots + 29\!\cdots\!52 ) / 22776536199168 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1430365 \nu^{7} + 235564033 \nu^{6} - 269464422942 \nu^{5} - 34056074566936 \nu^{4} + \cdots - 84\!\cdots\!28 ) / 45553072398336 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 35\beta _1 + 51726 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 4\beta_{2} + 82116\beta _1 - 1832418 \)
|
| \(\nu^{4}\) | \(=\) |
\( 14 \beta_{7} + 20 \beta_{6} + 18 \beta_{5} + 55 \beta_{4} + 51 \beta_{3} + 114528 \beta_{2} + \cdots + 4247434154 \)
|
| \(\nu^{5}\) | \(=\) |
\( 194 \beta_{7} - 348 \beta_{6} + 1326 \beta_{5} + 132143 \beta_{4} + 80619 \beta_{3} + \cdots - 238535247846 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2401506 \beta_{7} + 3446148 \beta_{6} + 2278350 \beta_{5} + 8733991 \beta_{4} + 5003587 \beta_{3} + \cdots + 400771550176714 \)
|
| \(\nu^{7}\) | \(=\) |
\( 6226226 \beta_{7} - 156911452 \beta_{6} + 303154686 \beta_{5} + 14768675407 \beta_{4} + \cdots - 24\!\cdots\!26 \)
|
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 |
|
−288.290 | 2187.00 | 50343.1 | 0 | −630490. | 1.20275e6 | −5.06672e6 | 4.78297e6 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −269.266 | 2187.00 | 39736.0 | 0 | −588884. | −2.21965e6 | −1.87623e6 | 4.78297e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −116.927 | 2187.00 | −19096.0 | 0 | −255720. | −38467.9 | 6.06431e6 | 4.78297e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −12.8400 | 2187.00 | −32603.1 | 0 | −28081.0 | 3.37654e6 | 839364. | 4.78297e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 165.545 | 2187.00 | −5362.86 | 0 | 362047. | −3.09920e6 | −6.31237e6 | 4.78297e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 182.436 | 2187.00 | 514.866 | 0 | 398987. | −1.87870e6 | −5.88413e6 | 4.78297e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 253.903 | 2187.00 | 31698.5 | 0 | 555285. | 3.96221e6 | −271536. | 4.78297e6 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 358.439 | 2187.00 | 95710.6 | 0 | 783906. | −156354. | 2.25611e7 | 4.78297e6 | 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 | 75.16.a.l | 8 | |
| 5.b | even | 2 | 1 | 75.16.a.k | 8 | ||
| 5.c | odd | 4 | 2 | 15.16.b.a | ✓ | 16 | |
| 15.e | even | 4 | 2 | 45.16.b.d | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.16.b.a | ✓ | 16 | 5.c | odd | 4 | 2 | |
| 45.16.b.d | 16 | 15.e | even | 4 | 2 | ||
| 75.16.a.k | 8 | 5.b | even | 2 | 1 | ||
| 75.16.a.l | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 273 T_{2}^{7} - 174278 T_{2}^{6} + 45045000 T_{2}^{5} + 8548359216 T_{2}^{4} + \cdots + 32\!\cdots\!16 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 32\!\cdots\!16 \)
|
| $3$ |
\( (T - 2187)^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots - 12\!\cdots\!00 \)
|
| $11$ |
\( T^{8} + \cdots - 27\!\cdots\!24 \)
|
| $13$ |
\( T^{8} + \cdots - 29\!\cdots\!04 \)
|
| $17$ |
\( T^{8} + \cdots + 71\!\cdots\!96 \)
|
| $19$ |
\( T^{8} + \cdots + 50\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots - 73\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 52\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots - 13\!\cdots\!00 \)
|
| $41$ |
\( T^{8} + \cdots + 45\!\cdots\!00 \)
|
| $43$ |
\( T^{8} + \cdots - 10\!\cdots\!24 \)
|
| $47$ |
\( T^{8} + \cdots - 99\!\cdots\!44 \)
|
| $53$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $59$ |
\( T^{8} + \cdots - 11\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 41\!\cdots\!36 \)
|
| $67$ |
\( T^{8} + \cdots - 47\!\cdots\!84 \)
|
| $71$ |
\( T^{8} + \cdots - 18\!\cdots\!04 \)
|
| $73$ |
\( T^{8} + \cdots + 69\!\cdots\!00 \)
|
| $79$ |
\( T^{8} + \cdots + 38\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 93\!\cdots\!76 \)
|
| $89$ |
\( T^{8} + \cdots - 11\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots - 65\!\cdots\!24 \)
|
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