Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [26,5,Mod(7,26)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([11]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26.7");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 26 = 2 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 26.f (of order \(12\), degree \(4\), 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: | \(2.68761904018\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 494x^{6} + 90747x^{4} - 7343258x^{2} + 221087161 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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
\( x^{8} - 494x^{6} + 90747x^{4} - 7343258x^{2} + 221087161 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - 247\nu^{2} + 26\nu + 14869 ) / 52 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 247\nu^{2} + 26\nu - 14869 ) / 52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 494\nu^{5} - 75878\nu^{3} + 3670615\nu + 386594 ) / 773188 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 370\nu^{4} + 45498\nu^{2} - 1858625 ) / 1262 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1625 \nu^{7} - 14869 \nu^{6} - 609453 \nu^{5} + 5501530 \nu^{4} + 75943985 \nu^{3} + \cdots + 25309075053 ) / 487881628 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1625 \nu^{7} - 14869 \nu^{6} + 609453 \nu^{5} + 5501530 \nu^{4} - 75943985 \nu^{3} + \cdots + 25309075053 ) / 487881628 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 13\nu^{6} - 370\nu^{5} + 4810\nu^{4} + 44236\nu^{3} - 591474\nu^{2} - 1702137\nu + 24162125 ) / 32812 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 13\beta_{6} + 13\beta_{5} + \beta_{4} + 124 \)
|
| \(\nu^{3}\) | \(=\) |
\( -26\beta_{7} - 124\beta_{6} + 124\beta_{5} - 13\beta_{4} + 26\beta_{3} + 123\beta_{2} + 123\beta _1 - 13 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3211\beta_{6} + 3211\beta_{5} + 247\beta_{4} - 26\beta_{2} + 26\beta _1 + 15759 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 6370 \beta_{7} - 31642 \beta_{6} + 31642 \beta_{5} - 3185 \beta_{4} + 12870 \beta_{3} + 15512 \beta_{2} + \cdots - 6435 \)
|
| \(\nu^{6}\) | \(=\) |
\( 596596\beta_{6} + 596596\beta_{5} + 47154\beta_{4} - 9620\beta_{2} + 9620\beta _1 + 2047703 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1173952 \beta_{7} - 6222276 \beta_{6} + 6222276 \beta_{5} - 586976 \beta_{4} + 3611764 \beta_{3} + \cdots - 1805882 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) |
| \(\chi(n)\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
2.73205 | − | 0.732051i | −6.32978 | + | 10.9635i | 6.92820 | − | 4.00000i | 24.0516 | + | 24.0516i | −9.26744 | + | 34.5865i | −4.96375 | − | 1.33003i | 16.0000 | − | 16.0000i | −39.6322 | − | 68.6449i | 83.3172 | + | 48.1032i | ||||||||||||||||||||||||
| 7.2 | 2.73205 | − | 0.732051i | 3.73170 | − | 6.46350i | 6.92820 | − | 4.00000i | −3.43687 | − | 3.43687i | 5.46359 | − | 20.3904i | 5.09773 | + | 1.36593i | 16.0000 | − | 16.0000i | 12.6488 | + | 21.9084i | −11.9057 | − | 6.87373i | |||||||||||||||||||||||||
| 11.1 | −0.732051 | + | 2.73205i | −4.74800 | − | 8.22378i | −6.92820 | − | 4.00000i | −19.2341 | − | 19.2341i | 25.9436 | − | 6.95155i | −5.11403 | − | 19.0858i | 16.0000 | − | 16.0000i | −4.58700 | + | 7.94492i | 66.6289 | − | 38.4682i | |||||||||||||||||||||||||
| 11.2 | −0.732051 | + | 2.73205i | 7.34608 | + | 12.7238i | −6.92820 | − | 4.00000i | −10.3806 | − | 10.3806i | −40.1397 | + | 10.7554i | 6.98005 | + | 26.0499i | 16.0000 | − | 16.0000i | −67.4297 | + | 116.792i | 35.9596 | − | 20.7613i | |||||||||||||||||||||||||
| 15.1 | 2.73205 | + | 0.732051i | −6.32978 | − | 10.9635i | 6.92820 | + | 4.00000i | 24.0516 | − | 24.0516i | −9.26744 | − | 34.5865i | −4.96375 | + | 1.33003i | 16.0000 | + | 16.0000i | −39.6322 | + | 68.6449i | 83.3172 | − | 48.1032i | |||||||||||||||||||||||||
| 15.2 | 2.73205 | + | 0.732051i | 3.73170 | + | 6.46350i | 6.92820 | + | 4.00000i | −3.43687 | + | 3.43687i | 5.46359 | + | 20.3904i | 5.09773 | − | 1.36593i | 16.0000 | + | 16.0000i | 12.6488 | − | 21.9084i | −11.9057 | + | 6.87373i | |||||||||||||||||||||||||
| 19.1 | −0.732051 | − | 2.73205i | −4.74800 | + | 8.22378i | −6.92820 | + | 4.00000i | −19.2341 | + | 19.2341i | 25.9436 | + | 6.95155i | −5.11403 | + | 19.0858i | 16.0000 | + | 16.0000i | −4.58700 | − | 7.94492i | 66.6289 | + | 38.4682i | |||||||||||||||||||||||||
| 19.2 | −0.732051 | − | 2.73205i | 7.34608 | − | 12.7238i | −6.92820 | + | 4.00000i | −10.3806 | + | 10.3806i | −40.1397 | − | 10.7554i | 6.98005 | − | 26.0499i | 16.0000 | + | 16.0000i | −67.4297 | − | 116.792i | 35.9596 | + | 20.7613i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.f | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 26.5.f.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 234.5.bb.a | 8 | ||
| 13.c | even | 3 | 1 | 338.5.d.f | 8 | ||
| 13.e | even | 6 | 1 | 338.5.d.g | 8 | ||
| 13.f | odd | 12 | 1 | inner | 26.5.f.b | ✓ | 8 |
| 13.f | odd | 12 | 1 | 338.5.d.f | 8 | ||
| 13.f | odd | 12 | 1 | 338.5.d.g | 8 | ||
| 39.k | even | 12 | 1 | 234.5.bb.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.5.f.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 26.5.f.b | ✓ | 8 | 13.f | odd | 12 | 1 | inner |
| 234.5.bb.a | 8 | 3.b | odd | 2 | 1 | ||
| 234.5.bb.a | 8 | 39.k | even | 12 | 1 | ||
| 338.5.d.f | 8 | 13.c | even | 3 | 1 | ||
| 338.5.d.f | 8 | 13.f | odd | 12 | 1 | ||
| 338.5.d.g | 8 | 13.e | even | 6 | 1 | ||
| 338.5.d.g | 8 | 13.f | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 261T_{3}^{6} + 468T_{3}^{5} + 54939T_{3}^{4} + 61074T_{3}^{3} + 3495258T_{3}^{2} - 3084588T_{3} + 173765124 \)
acting on \(S_{5}^{\mathrm{new}}(26, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 64)^{2} \)
|
| $3$ |
\( T^{8} + 261 T^{6} + \cdots + 173765124 \)
|
| $5$ |
\( T^{8} + \cdots + 4358376324 \)
|
| $7$ |
\( T^{8} - 4 T^{7} + \cdots + 208860304 \)
|
| $11$ |
\( T^{8} + \cdots + 58\!\cdots\!44 \)
|
| $13$ |
\( T^{8} + \cdots + 66\!\cdots\!41 \)
|
| $17$ |
\( T^{8} + \cdots + 46\!\cdots\!29 \)
|
| $19$ |
\( T^{8} + \cdots + 71\!\cdots\!16 \)
|
| $23$ |
\( T^{8} + \cdots + 80\!\cdots\!96 \)
|
| $29$ |
\( T^{8} + \cdots + 19\!\cdots\!81 \)
|
| $31$ |
\( T^{8} + \cdots + 99\!\cdots\!84 \)
|
| $37$ |
\( T^{8} + \cdots + 12\!\cdots\!21 \)
|
| $41$ |
\( T^{8} + \cdots + 43\!\cdots\!69 \)
|
| $43$ |
\( T^{8} + \cdots + 33\!\cdots\!04 \)
|
| $47$ |
\( T^{8} + \cdots + 11\!\cdots\!04 \)
|
| $53$ |
\( (T^{4} + \cdots + 43469639270772)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 56\!\cdots\!64 \)
|
| $61$ |
\( T^{8} + \cdots + 11\!\cdots\!69 \)
|
| $67$ |
\( T^{8} + \cdots + 25\!\cdots\!64 \)
|
| $71$ |
\( T^{8} + \cdots + 11\!\cdots\!84 \)
|
| $73$ |
\( T^{8} + \cdots + 18\!\cdots\!16 \)
|
| $79$ |
\( (T^{4} + \cdots + 393593299652832)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 38\!\cdots\!76 \)
|
| $89$ |
\( T^{8} + \cdots + 69\!\cdots\!36 \)
|
| $97$ |
\( T^{8} + \cdots + 71\!\cdots\!64 \)
|
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