Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [26,7,Mod(5,26)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26.5");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 26 = 2 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 26.d (of order \(4\), degree \(2\), 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: | \(5.98140617412\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 1578x^{4} + 622521x^{2} + 72284004 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 1578x^{4} + 622521x^{2} + 72284004 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 789\nu ) / 8502 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 789\nu^{2} ) / 8502 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 17\nu^{4} + 1315\nu^{3} + 21915\nu^{2} + 270480\nu + 4472052 ) / 17004 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 17\nu^{4} - 1315\nu^{3} + 21915\nu^{2} - 270480\nu + 4472052 ) / 17004 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 17\beta_{3} - 526 \)
|
| \(\nu^{3}\) | \(=\) |
\( 8502\beta_{2} - 789\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -789\beta_{5} - 789\beta_{4} + 21915\beta_{3} + 415014 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8502\beta_{5} + 8502\beta_{4} - 11180130\beta_{2} + 767055\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
4.00000 | − | 4.00000i | −16.9739 | − | 32.0000i | −151.676 | + | 151.676i | −67.8957 | + | 67.8957i | 319.655 | + | 319.655i | −128.000 | − | 128.000i | −440.886 | 1213.41i | |||||||||||||||||||||||||
| 5.2 | 4.00000 | − | 4.00000i | −15.4487 | − | 32.0000i | 36.7769 | − | 36.7769i | −61.7947 | + | 61.7947i | −379.688 | − | 379.688i | −128.000 | − | 128.000i | −490.338 | − | 294.216i | |||||||||||||||||||||||||
| 5.3 | 4.00000 | − | 4.00000i | 32.4226 | − | 32.0000i | 39.8988 | − | 39.8988i | 129.690 | − | 129.690i | 135.033 | + | 135.033i | −128.000 | − | 128.000i | 322.225 | − | 319.191i | |||||||||||||||||||||||||
| 21.1 | 4.00000 | + | 4.00000i | −16.9739 | 32.0000i | −151.676 | − | 151.676i | −67.8957 | − | 67.8957i | 319.655 | − | 319.655i | −128.000 | + | 128.000i | −440.886 | − | 1213.41i | ||||||||||||||||||||||||||
| 21.2 | 4.00000 | + | 4.00000i | −15.4487 | 32.0000i | 36.7769 | + | 36.7769i | −61.7947 | − | 61.7947i | −379.688 | + | 379.688i | −128.000 | + | 128.000i | −490.338 | 294.216i | |||||||||||||||||||||||||||
| 21.3 | 4.00000 | + | 4.00000i | 32.4226 | 32.0000i | 39.8988 | + | 39.8988i | 129.690 | + | 129.690i | 135.033 | − | 135.033i | −128.000 | + | 128.000i | 322.225 | 319.191i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 26.7.d.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 234.7.i.a | 6 | ||
| 4.b | odd | 2 | 1 | 208.7.t.a | 6 | ||
| 13.d | odd | 4 | 1 | inner | 26.7.d.a | ✓ | 6 |
| 39.f | even | 4 | 1 | 234.7.i.a | 6 | ||
| 52.f | even | 4 | 1 | 208.7.t.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.7.d.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 26.7.d.a | ✓ | 6 | 13.d | odd | 4 | 1 | inner |
| 208.7.t.a | 6 | 4.b | odd | 2 | 1 | ||
| 208.7.t.a | 6 | 52.f | even | 4 | 1 | ||
| 234.7.i.a | 6 | 3.b | odd | 2 | 1 | ||
| 234.7.i.a | 6 | 39.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 789T_{3} - 8502 \)
acting on \(S_{7}^{\mathrm{new}}(26, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8 T + 32)^{3} \)
|
| $3$ |
\( (T^{3} - 789 T - 8502)^{2} \)
|
| $5$ |
\( T^{6} + \cdots + 396272531250 \)
|
| $7$ |
\( T^{6} + \cdots + 21\!\cdots\!02 \)
|
| $11$ |
\( T^{6} + \cdots + 90667601418888 \)
|
| $13$ |
\( T^{6} + \cdots + 11\!\cdots\!29 \)
|
| $17$ |
\( T^{6} + \cdots + 43\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots + 88\!\cdots\!32 \)
|
| $23$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} - 42798 T^{2} + \cdots + 863541600636)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 60\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 14\!\cdots\!62 \)
|
| $41$ |
\( T^{6} + \cdots + 11\!\cdots\!28 \)
|
| $43$ |
\( T^{6} + \cdots + 38\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 18\!\cdots\!62 \)
|
| $53$ |
\( (T^{3} + \cdots + 16\!\cdots\!32)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 26\!\cdots\!88 \)
|
| $61$ |
\( (T^{3} + \cdots - 470197434055512)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 62\!\cdots\!48 \)
|
| $71$ |
\( T^{6} + \cdots + 58\!\cdots\!22 \)
|
| $73$ |
\( T^{6} + \cdots + 88\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} + \cdots - 12\!\cdots\!56)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 88\!\cdots\!28 \)
|
| $89$ |
\( T^{6} + \cdots + 80\!\cdots\!92 \)
|
| $97$ |
\( T^{6} + \cdots + 37\!\cdots\!08 \)
|
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