Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [28,11,Mod(13,28)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("28.13");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 28 = 2^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 28.b (of order \(2\), degree \(1\), 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: | \(17.7900030749\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 4500x^{4} + 5859828x^{2} + 2030476896 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{19}\cdot 3^{2}\cdot 7^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 4500x^{4} + 5859828x^{2} + 2030476896 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 3405\nu^{3} - 2108358\nu ) / 64386 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{5} + 27756\nu^{3} + 36489348\nu ) / 96579 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\nu^{5} + 38133\nu^{3} + 386316\nu^{2} + 24943590\nu + 579474000 ) / 27594 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13\nu^{5} + 38133\nu^{3} - 386316\nu^{2} + 24943590\nu - 579474000 ) / 27594 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4\nu^{4} + 11940\nu^{2} + 6536208 ) / 21 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 4\beta_{2} + 74\beta_1 ) / 896 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} - 42000 ) / 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -1209\beta_{4} - 1209\beta_{3} - 804\beta_{2} - 76026\beta_1 ) / 448 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 147\beta_{5} + 2985\beta_{4} - 2985\beta_{3} + 79616544 ) / 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 1531233\beta_{4} + 1531233\beta_{3} - 739548\beta_{2} + 76007178\beta_1 ) / 224 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
0 | − | 454.668i | 0 | 4676.96i | 0 | −2465.27 | − | 16625.2i | 0 | −147674. | 0 | |||||||||||||||||||||||||||||||||
| 13.2 | 0 | − | 195.108i | 0 | − | 4544.88i | 0 | −15892.1 | − | 5469.55i | 0 | 20981.7 | 0 | |||||||||||||||||||||||||||||||||
| 13.3 | 0 | − | 190.833i | 0 | 207.001i | 0 | 10524.4 | + | 13103.9i | 0 | 22631.6 | 0 | ||||||||||||||||||||||||||||||||||
| 13.4 | 0 | 190.833i | 0 | − | 207.001i | 0 | 10524.4 | − | 13103.9i | 0 | 22631.6 | 0 | ||||||||||||||||||||||||||||||||||
| 13.5 | 0 | 195.108i | 0 | 4544.88i | 0 | −15892.1 | + | 5469.55i | 0 | 20981.7 | 0 | |||||||||||||||||||||||||||||||||||
| 13.6 | 0 | 454.668i | 0 | − | 4676.96i | 0 | −2465.27 | + | 16625.2i | 0 | −147674. | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 28.11.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 252.11.d.b | 6 | ||
| 4.b | odd | 2 | 1 | 112.11.c.c | 6 | ||
| 7.b | odd | 2 | 1 | inner | 28.11.b.a | ✓ | 6 |
| 7.c | even | 3 | 2 | 196.11.h.a | 12 | ||
| 7.d | odd | 6 | 2 | 196.11.h.a | 12 | ||
| 21.c | even | 2 | 1 | 252.11.d.b | 6 | ||
| 28.d | even | 2 | 1 | 112.11.c.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.11.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 28.11.b.a | ✓ | 6 | 7.b | odd | 2 | 1 | inner |
| 112.11.c.c | 6 | 4.b | odd | 2 | 1 | ||
| 112.11.c.c | 6 | 28.d | even | 2 | 1 | ||
| 196.11.h.a | 12 | 7.c | even | 3 | 2 | ||
| 196.11.h.a | 12 | 7.d | odd | 6 | 2 | ||
| 252.11.d.b | 6 | 3.b | odd | 2 | 1 | ||
| 252.11.d.b | 6 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(28, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 286582804862976 \)
|
| $5$ |
\( T^{6} + \cdots + 19\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 22\!\cdots\!49 \)
|
| $11$ |
\( (T^{3} + \cdots - 72013139033256)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 19\!\cdots\!76 \)
|
| $17$ |
\( T^{6} + \cdots + 39\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots + 23\!\cdots\!00 \)
|
| $23$ |
\( (T^{3} + \cdots + 23\!\cdots\!00)^{2} \)
|
| $29$ |
\( (T^{3} + \cdots + 13\!\cdots\!56)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $37$ |
\( (T^{3} + \cdots + 42\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 48\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + \cdots - 92\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 74\!\cdots\!96 \)
|
| $53$ |
\( (T^{3} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 31\!\cdots\!00 \)
|
| $67$ |
\( (T^{3} + \cdots - 27\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{3} + \cdots - 79\!\cdots\!20)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 48\!\cdots\!84 \)
|
| $79$ |
\( (T^{3} + \cdots + 12\!\cdots\!44)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 16\!\cdots\!44 \)
|
| $89$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 78\!\cdots\!96 \)
|
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