Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [980,3,Mod(197,980)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(980, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("980.197");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 980.l (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: | \(26.7030659073\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{14})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 49 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 140) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 7\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) | \(491\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 197.1 |
|
0 | −2.87083 | + | 2.87083i | 0 | 1.12917 | + | 4.87083i | 0 | 0 | 0 | − | 7.48331i | 0 | |||||||||||||||||||||||||
| 197.2 | 0 | 0.870829 | − | 0.870829i | 0 | 4.87083 | + | 1.12917i | 0 | 0 | 0 | 7.48331i | 0 | |||||||||||||||||||||||||||
| 393.1 | 0 | −2.87083 | − | 2.87083i | 0 | 1.12917 | − | 4.87083i | 0 | 0 | 0 | 7.48331i | 0 | |||||||||||||||||||||||||||
| 393.2 | 0 | 0.870829 | + | 0.870829i | 0 | 4.87083 | − | 1.12917i | 0 | 0 | 0 | − | 7.48331i | 0 | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 980.3.l.b | 4 | |
| 5.c | odd | 4 | 1 | inner | 980.3.l.b | 4 | |
| 7.b | odd | 2 | 1 | 140.3.l.a | ✓ | 4 | |
| 21.c | even | 2 | 1 | 1260.3.y.a | 4 | ||
| 28.d | even | 2 | 1 | 560.3.bh.a | 4 | ||
| 35.c | odd | 2 | 1 | 700.3.l.a | 4 | ||
| 35.f | even | 4 | 1 | 140.3.l.a | ✓ | 4 | |
| 35.f | even | 4 | 1 | 700.3.l.a | 4 | ||
| 105.k | odd | 4 | 1 | 1260.3.y.a | 4 | ||
| 140.j | odd | 4 | 1 | 560.3.bh.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.3.l.a | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 140.3.l.a | ✓ | 4 | 35.f | even | 4 | 1 | |
| 560.3.bh.a | 4 | 28.d | even | 2 | 1 | ||
| 560.3.bh.a | 4 | 140.j | odd | 4 | 1 | ||
| 700.3.l.a | 4 | 35.c | odd | 2 | 1 | ||
| 700.3.l.a | 4 | 35.f | even | 4 | 1 | ||
| 980.3.l.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 980.3.l.b | 4 | 5.c | odd | 4 | 1 | inner | |
| 1260.3.y.a | 4 | 21.c | even | 2 | 1 | ||
| 1260.3.y.a | 4 | 105.k | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 4T_{3}^{3} + 8T_{3}^{2} - 20T_{3} + 25 \)
acting on \(S_{3}^{\mathrm{new}}(980, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 4 T^{3} + \cdots + 25 \)
|
| $5$ |
\( T^{4} - 12 T^{3} + \cdots + 625 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 22 T + 65)^{2} \)
|
| $13$ |
\( T^{4} - 20 T^{3} + \cdots + 169 \)
|
| $17$ |
\( T^{4} + 4 T^{3} + \cdots + 116281 \)
|
| $19$ |
\( T^{4} + 100T^{2} + 484 \)
|
| $23$ |
\( T^{4} - 104 T^{3} + \cdots + 1752976 \)
|
| $29$ |
\( T^{4} + 2130 T^{2} + 528529 \)
|
| $31$ |
\( (T^{2} - 1134)^{2} \)
|
| $37$ |
\( T^{4} - 104 T^{3} + \cdots + 1210000 \)
|
| $41$ |
\( (T^{2} + 56 T - 350)^{2} \)
|
| $43$ |
\( T^{4} + 4 T^{3} + \cdots + 7828804 \)
|
| $47$ |
\( T^{4} + 12 T^{3} + \cdots + 18983449 \)
|
| $53$ |
\( T^{4} + 140 T^{3} + \cdots + 3062500 \)
|
| $59$ |
\( T^{4} + 1820 T^{2} + 432964 \)
|
| $61$ |
\( (T^{2} - 168 T + 6832)^{2} \)
|
| $67$ |
\( T^{4} + 140 T^{3} + \cdots + 5866084 \)
|
| $71$ |
\( (T^{2} - 112 T - 3640)^{2} \)
|
| $73$ |
\( T^{4} - 168 T^{3} + \cdots + 529984 \)
|
| $79$ |
\( T^{4} + 40594 T^{2} + 405418225 \)
|
| $83$ |
\( T^{4} + 112 T^{3} + \cdots + 1254400 \)
|
| $89$ |
\( T^{4} + 17696 T^{2} + 52998400 \)
|
| $97$ |
\( T^{4} + 132 T^{3} + \cdots + 4012009 \)
|
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