Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [28,5,Mod(5,28)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("28.5");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 28 = 2^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 28.h (of order \(6\), 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: | \(2.89435896635\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 6.0.11337408.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 18x^{4} + 81x^{2} + 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{3}\cdot 7^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{6} + 18x^{4} + 81x^{2} + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 9\nu + 2 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 9\nu^{4} - 15\nu^{3} - 87\nu^{2} - 36\nu - 36 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 9\nu^{4} + 15\nu^{3} - 87\nu^{2} + 36\nu - 36 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9\nu^{5} - 3\nu^{4} + 135\nu^{3} + 27\nu^{2} + 492\nu + 324 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 9\nu^{5} + 3\nu^{4} + 135\nu^{3} - 27\nu^{2} + 492\nu - 324 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 9\beta_{3} + 9\beta_{2} ) / 42 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{5} + 3\beta_{4} - \beta_{3} - \beta_{2} - 252 ) / 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{5} - 3\beta_{4} + 27\beta_{3} - 27\beta_{2} + 56\beta _1 - 28 ) / 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 29\beta_{5} - 29\beta_{4} - 9\beta_{3} - 9\beta_{2} + 2268 ) / 42 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 33\beta_{5} + 33\beta_{4} - 241\beta_{3} + 241\beta_{2} - 840\beta _1 + 420 ) / 14 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) |
| \(\chi(n)\) | \(1\) | \(1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
0 | −7.81152 | + | 4.50998i | 0 | −26.9260 | − | 15.5457i | 0 | −48.6720 | + | 5.65972i | 0 | 0.179888 | − | 0.311574i | 0 | ||||||||||||||||||||||||||||
| 5.2 | 0 | −1.35901 | + | 0.784623i | 0 | 38.3327 | + | 22.1314i | 0 | 42.3967 | − | 24.5667i | 0 | −39.2687 | + | 68.0154i | 0 | |||||||||||||||||||||||||||||
| 5.3 | 0 | 13.6705 | − | 7.89268i | 0 | −24.9067 | − | 14.3799i | 0 | 39.2754 | + | 29.2993i | 0 | 84.0888 | − | 145.646i | 0 | |||||||||||||||||||||||||||||
| 17.1 | 0 | −7.81152 | − | 4.50998i | 0 | −26.9260 | + | 15.5457i | 0 | −48.6720 | − | 5.65972i | 0 | 0.179888 | + | 0.311574i | 0 | |||||||||||||||||||||||||||||
| 17.2 | 0 | −1.35901 | − | 0.784623i | 0 | 38.3327 | − | 22.1314i | 0 | 42.3967 | + | 24.5667i | 0 | −39.2687 | − | 68.0154i | 0 | |||||||||||||||||||||||||||||
| 17.3 | 0 | 13.6705 | + | 7.89268i | 0 | −24.9067 | + | 14.3799i | 0 | 39.2754 | − | 29.2993i | 0 | 84.0888 | + | 145.646i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 28.5.h.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 252.5.z.f | 6 | ||
| 4.b | odd | 2 | 1 | 112.5.s.c | 6 | ||
| 5.b | even | 2 | 1 | 700.5.s.a | 6 | ||
| 5.c | odd | 4 | 2 | 700.5.o.a | 12 | ||
| 7.b | odd | 2 | 1 | 196.5.h.c | 6 | ||
| 7.c | even | 3 | 1 | 196.5.b.a | 6 | ||
| 7.c | even | 3 | 1 | 196.5.h.c | 6 | ||
| 7.d | odd | 6 | 1 | inner | 28.5.h.a | ✓ | 6 |
| 7.d | odd | 6 | 1 | 196.5.b.a | 6 | ||
| 21.g | even | 6 | 1 | 252.5.z.f | 6 | ||
| 28.f | even | 6 | 1 | 112.5.s.c | 6 | ||
| 28.f | even | 6 | 1 | 784.5.c.e | 6 | ||
| 28.g | odd | 6 | 1 | 784.5.c.e | 6 | ||
| 35.i | odd | 6 | 1 | 700.5.s.a | 6 | ||
| 35.k | even | 12 | 2 | 700.5.o.a | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.5.h.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 28.5.h.a | ✓ | 6 | 7.d | odd | 6 | 1 | inner |
| 112.5.s.c | 6 | 4.b | odd | 2 | 1 | ||
| 112.5.s.c | 6 | 28.f | even | 6 | 1 | ||
| 196.5.b.a | 6 | 7.c | even | 3 | 1 | ||
| 196.5.b.a | 6 | 7.d | odd | 6 | 1 | ||
| 196.5.h.c | 6 | 7.b | odd | 2 | 1 | ||
| 196.5.h.c | 6 | 7.c | even | 3 | 1 | ||
| 252.5.z.f | 6 | 3.b | odd | 2 | 1 | ||
| 252.5.z.f | 6 | 21.g | even | 6 | 1 | ||
| 700.5.o.a | 12 | 5.c | odd | 4 | 2 | ||
| 700.5.o.a | 12 | 35.k | even | 12 | 2 | ||
| 700.5.s.a | 6 | 5.b | even | 2 | 1 | ||
| 700.5.s.a | 6 | 35.i | odd | 6 | 1 | ||
| 784.5.c.e | 6 | 28.f | even | 6 | 1 | ||
| 784.5.c.e | 6 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(28, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 9 T^{5} + \cdots + 49923 \)
|
| $5$ |
\( T^{6} + \cdots + 1566504603 \)
|
| $7$ |
\( T^{6} + \cdots + 13841287201 \)
|
| $11$ |
\( T^{6} + \cdots + 1920280041 \)
|
| $13$ |
\( T^{6} + \cdots + 65933832204288 \)
|
| $17$ |
\( T^{6} + \cdots + 86645980184427 \)
|
| $19$ |
\( T^{6} + \cdots + 163848759776883 \)
|
| $23$ |
\( T^{6} + \cdots + 18\!\cdots\!21 \)
|
| $29$ |
\( (T^{3} + 270 T^{2} + \cdots + 401089968)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 81\!\cdots\!63 \)
|
| $37$ |
\( T^{6} + \cdots + 19\!\cdots\!81 \)
|
| $41$ |
\( T^{6} + \cdots + 30\!\cdots\!92 \)
|
| $43$ |
\( (T^{3} + 474 T^{2} + \cdots + 2925826856)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 12\!\cdots\!27 \)
|
| $53$ |
\( T^{6} + \cdots + 34\!\cdots\!01 \)
|
| $59$ |
\( T^{6} + \cdots + 40\!\cdots\!87 \)
|
| $61$ |
\( T^{6} + \cdots + 15\!\cdots\!47 \)
|
| $67$ |
\( T^{6} + \cdots + 30\!\cdots\!41 \)
|
| $71$ |
\( (T^{3} - 1134 T^{2} + \cdots + 66080643048)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 51\!\cdots\!67 \)
|
| $79$ |
\( T^{6} + \cdots + 32\!\cdots\!81 \)
|
| $83$ |
\( T^{6} + \cdots + 21\!\cdots\!72 \)
|
| $89$ |
\( T^{6} + \cdots + 12\!\cdots\!07 \)
|
| $97$ |
\( T^{6} + \cdots + 22\!\cdots\!32 \)
|
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