Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [28,7,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, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("28.13");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 28 = 2^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| 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: | \(6.44151434136\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.903168.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 24x^{2} + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3\cdot 7^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 24x^{2} + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} + 156\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( -2\nu^{3} - 44\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 28\nu^{2} + 336 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta_1 ) / 112 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 336 ) / 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -39\beta_{2} - 33\beta_1 ) / 56 \)
|
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 | − | 37.3711i | 0 | − | 45.3237i | 0 | −321.299 | + | 120.066i | 0 | −667.598 | 0 | ||||||||||||||||||||||||||
| 13.2 | 0 | − | 11.8068i | 0 | 175.982i | 0 | 307.299 | − | 152.369i | 0 | 589.598 | 0 | ||||||||||||||||||||||||||||
| 13.3 | 0 | 11.8068i | 0 | − | 175.982i | 0 | 307.299 | + | 152.369i | 0 | 589.598 | 0 | ||||||||||||||||||||||||||||
| 13.4 | 0 | 37.3711i | 0 | 45.3237i | 0 | −321.299 | − | 120.066i | 0 | −667.598 | 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 | 28.7.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 252.7.d.a | 4 | ||
| 4.b | odd | 2 | 1 | 112.7.c.d | 4 | ||
| 5.b | even | 2 | 1 | 700.7.d.b | 4 | ||
| 5.c | odd | 4 | 2 | 700.7.h.a | 8 | ||
| 7.b | odd | 2 | 1 | inner | 28.7.b.a | ✓ | 4 |
| 7.c | even | 3 | 2 | 196.7.h.b | 8 | ||
| 7.d | odd | 6 | 2 | 196.7.h.b | 8 | ||
| 8.b | even | 2 | 1 | 448.7.c.f | 4 | ||
| 8.d | odd | 2 | 1 | 448.7.c.g | 4 | ||
| 21.c | even | 2 | 1 | 252.7.d.a | 4 | ||
| 28.d | even | 2 | 1 | 112.7.c.d | 4 | ||
| 35.c | odd | 2 | 1 | 700.7.d.b | 4 | ||
| 35.f | even | 4 | 2 | 700.7.h.a | 8 | ||
| 56.e | even | 2 | 1 | 448.7.c.g | 4 | ||
| 56.h | odd | 2 | 1 | 448.7.c.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.7.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 28.7.b.a | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 112.7.c.d | 4 | 4.b | odd | 2 | 1 | ||
| 112.7.c.d | 4 | 28.d | even | 2 | 1 | ||
| 196.7.h.b | 8 | 7.c | even | 3 | 2 | ||
| 196.7.h.b | 8 | 7.d | odd | 6 | 2 | ||
| 252.7.d.a | 4 | 3.b | odd | 2 | 1 | ||
| 252.7.d.a | 4 | 21.c | even | 2 | 1 | ||
| 448.7.c.f | 4 | 8.b | even | 2 | 1 | ||
| 448.7.c.f | 4 | 56.h | odd | 2 | 1 | ||
| 448.7.c.g | 4 | 8.d | odd | 2 | 1 | ||
| 448.7.c.g | 4 | 56.e | even | 2 | 1 | ||
| 700.7.d.b | 4 | 5.b | even | 2 | 1 | ||
| 700.7.d.b | 4 | 35.c | odd | 2 | 1 | ||
| 700.7.h.a | 8 | 5.c | odd | 4 | 2 | ||
| 700.7.h.a | 8 | 35.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(28, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 1536 T^{2} + 194688 \)
|
| $5$ |
\( T^{4} + 33024 T^{2} + 63619200 \)
|
| $7$ |
\( T^{4} + \cdots + 13841287201 \)
|
| $11$ |
\( (T^{2} + 108 T - 1577628)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 74197833257088 \)
|
| $17$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $23$ |
\( (T^{2} + 6876 T - 13468860)^{2} \)
|
| $29$ |
\( (T^{2} - 17460 T - 380564316)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 28\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} + 79468 T + 1564565860)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 84\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} - 107540 T - 4853452700)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 91\!\cdots\!72 \)
|
| $53$ |
\( (T^{2} + 79020 T - 11241366300)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 29\!\cdots\!00 \)
|
| $67$ |
\( (T^{2} + 255500 T + 15687844900)^{2} \)
|
| $71$ |
\( (T^{2} - 824292 T + 115477411140)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 48\!\cdots\!52 \)
|
| $79$ |
\( (T^{2} + 1413884 T + 454357567108)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 83\!\cdots\!08 \)
|
| $89$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 73\!\cdots\!68 \)
|
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