Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,3,Mod(197,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("441.197");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 441.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: | \(12.0163796583\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.959512576.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 7x^{4} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 7^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 7x^{4} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{7} + 9\nu^{5} + 13\nu^{3} + 9\nu ) / 135 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 7\nu^{3} - 27\nu ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{2} ) / 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{6} - 18\nu^{4} + 80\nu^{2} - 63 ) / 45 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4\nu^{6} + 36\nu^{4} + 64\nu^{2} + 126 ) / 45 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{7} + 54\nu^{5} - 148\nu^{3} + 324\nu ) / 135 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -4\nu^{7} - 3\nu^{5} - 19\nu^{3} + 87\nu ) / 45 \)
|
| \(\nu\) | \(=\) |
\( ( 6\beta_{7} + \beta_{6} - 14\beta_{2} ) / 28 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{5} + 10\beta_{4} - 14\beta_{3} ) / 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} - 6\beta_{6} - 14\beta_{2} + 35\beta_1 ) / 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 25\beta_{5} - 20\beta_{4} - 98 ) / 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -36\beta_{7} + 29\beta_{6} + 14\beta_{2} + 210\beta_1 ) / 28 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 5\beta_{5} + 10\beta_{4} + 112\beta_{3} ) / 14 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -148\beta_{7} + 57\beta_{6} - 182\beta_{2} - 490\beta_1 ) / 28 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).
| \(n\) | \(199\) | \(344\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 197.1 |
|
− | 3.05231i | 0 | −5.31662 | − | 8.31662i | 0 | 0 | 4.01875i | 0 | −25.3850 | ||||||||||||||||||||||||||||||||||||||||
| 197.2 | − | 3.05231i | 0 | −5.31662 | 8.31662i | 0 | 0 | 4.01875i | 0 | 25.3850 | ||||||||||||||||||||||||||||||||||||||||||
| 197.3 | − | 1.63810i | 0 | 1.31662 | − | 1.68338i | 0 | 0 | − | 8.70917i | 0 | −2.75754 | ||||||||||||||||||||||||||||||||||||||||
| 197.4 | − | 1.63810i | 0 | 1.31662 | 1.68338i | 0 | 0 | − | 8.70917i | 0 | 2.75754 | |||||||||||||||||||||||||||||||||||||||||
| 197.5 | 1.63810i | 0 | 1.31662 | − | 1.68338i | 0 | 0 | 8.70917i | 0 | 2.75754 | ||||||||||||||||||||||||||||||||||||||||||
| 197.6 | 1.63810i | 0 | 1.31662 | 1.68338i | 0 | 0 | 8.70917i | 0 | −2.75754 | |||||||||||||||||||||||||||||||||||||||||||
| 197.7 | 3.05231i | 0 | −5.31662 | − | 8.31662i | 0 | 0 | − | 4.01875i | 0 | 25.3850 | |||||||||||||||||||||||||||||||||||||||||
| 197.8 | 3.05231i | 0 | −5.31662 | 8.31662i | 0 | 0 | − | 4.01875i | 0 | −25.3850 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 441.3.b.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 441.3.b.e | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 441.3.b.e | ✓ | 8 |
| 7.c | even | 3 | 2 | 441.3.q.e | 16 | ||
| 7.d | odd | 6 | 2 | 441.3.q.e | 16 | ||
| 21.c | even | 2 | 1 | inner | 441.3.b.e | ✓ | 8 |
| 21.g | even | 6 | 2 | 441.3.q.e | 16 | ||
| 21.h | odd | 6 | 2 | 441.3.q.e | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 441.3.b.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 441.3.b.e | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 441.3.b.e | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 441.3.b.e | ✓ | 8 | 21.c | even | 2 | 1 | inner |
| 441.3.q.e | 16 | 7.c | even | 3 | 2 | ||
| 441.3.q.e | 16 | 7.d | odd | 6 | 2 | ||
| 441.3.q.e | 16 | 21.g | even | 6 | 2 | ||
| 441.3.q.e | 16 | 21.h | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\):
|
\( T_{2}^{4} + 12T_{2}^{2} + 25 \)
|
|
\( T_{13}^{4} - 708T_{13}^{2} + 122500 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 12 T^{2} + 25)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 72 T^{2} + 196)^{2} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} + 460 T^{2} + 27556)^{2} \)
|
| $13$ |
\( (T^{4} - 708 T^{2} + 122500)^{2} \)
|
| $17$ |
\( (T^{4} + 184 T^{2} + 4900)^{2} \)
|
| $19$ |
\( (T^{4} - 1072 T^{2} + 19600)^{2} \)
|
| $23$ |
\( (T^{4} + 828 T^{2} + 136900)^{2} \)
|
| $29$ |
\( (T^{4} + 2124 T^{2} + 1444)^{2} \)
|
| $31$ |
\( (T^{4} - 1584 T^{2} + 94864)^{2} \)
|
| $37$ |
\( (T^{2} + 32 T - 140)^{4} \)
|
| $41$ |
\( (T^{4} + 5224 T^{2} + 6708100)^{2} \)
|
| $43$ |
\( (T^{2} + 36 T + 280)^{4} \)
|
| $47$ |
\( (T^{4} + 6056 T^{2} + 19600)^{2} \)
|
| $53$ |
\( (T^{2} + 1250)^{4} \)
|
| $59$ |
\( (T^{4} + 8808 T^{2} + 19324816)^{2} \)
|
| $61$ |
\( (T^{4} - 3644 T^{2} + 3161284)^{2} \)
|
| $67$ |
\( (T^{2} - 1100)^{4} \)
|
| $71$ |
\( (T^{4} + 2124 T^{2} + 1444)^{2} \)
|
| $73$ |
\( (T^{4} - 15948 T^{2} + 62568100)^{2} \)
|
| $79$ |
\( (T^{2} - 52 T + 500)^{4} \)
|
| $83$ |
\( (T^{4} + 26784 T^{2} + 144961600)^{2} \)
|
| $89$ |
\( (T^{4} + 10984 T^{2} + 6002500)^{2} \)
|
| $97$ |
\( (T^{4} - 2748 T^{2} + 1768900)^{2} \)
|
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