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: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{7})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 8x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 63) |
| 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
\( x^{4} + 8x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 6\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 6\beta_1 \)
|
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 |
|
− | 2.57794i | 0 | −2.64575 | 5.15587i | 0 | 0 | − | 3.49117i | 0 | 13.2915 | ||||||||||||||||||||||||||||
| 197.2 | − | 1.16372i | 0 | 2.64575 | 2.32744i | 0 | 0 | − | 7.73381i | 0 | 2.70850 | |||||||||||||||||||||||||||||
| 197.3 | 1.16372i | 0 | 2.64575 | − | 2.32744i | 0 | 0 | 7.73381i | 0 | 2.70850 | ||||||||||||||||||||||||||||||
| 197.4 | 2.57794i | 0 | −2.64575 | − | 5.15587i | 0 | 0 | 3.49117i | 0 | 13.2915 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 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.b | 4 | |
| 3.b | odd | 2 | 1 | inner | 441.3.b.b | 4 | |
| 7.b | odd | 2 | 1 | 63.3.b.a | ✓ | 4 | |
| 7.c | even | 3 | 2 | 441.3.q.a | 8 | ||
| 7.d | odd | 6 | 2 | 441.3.q.b | 8 | ||
| 21.c | even | 2 | 1 | 63.3.b.a | ✓ | 4 | |
| 21.g | even | 6 | 2 | 441.3.q.b | 8 | ||
| 21.h | odd | 6 | 2 | 441.3.q.a | 8 | ||
| 28.d | even | 2 | 1 | 1008.3.d.d | 4 | ||
| 35.c | odd | 2 | 1 | 1575.3.c.a | 4 | ||
| 35.f | even | 4 | 2 | 1575.3.f.a | 8 | ||
| 56.e | even | 2 | 1 | 4032.3.d.c | 4 | ||
| 56.h | odd | 2 | 1 | 4032.3.d.b | 4 | ||
| 63.l | odd | 6 | 2 | 567.3.r.a | 8 | ||
| 63.o | even | 6 | 2 | 567.3.r.a | 8 | ||
| 84.h | odd | 2 | 1 | 1008.3.d.d | 4 | ||
| 105.g | even | 2 | 1 | 1575.3.c.a | 4 | ||
| 105.k | odd | 4 | 2 | 1575.3.f.a | 8 | ||
| 168.e | odd | 2 | 1 | 4032.3.d.c | 4 | ||
| 168.i | even | 2 | 1 | 4032.3.d.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 63.3.b.a | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 63.3.b.a | ✓ | 4 | 21.c | even | 2 | 1 | |
| 441.3.b.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 441.3.b.b | 4 | 3.b | odd | 2 | 1 | inner | |
| 441.3.q.a | 8 | 7.c | even | 3 | 2 | ||
| 441.3.q.a | 8 | 21.h | odd | 6 | 2 | ||
| 441.3.q.b | 8 | 7.d | odd | 6 | 2 | ||
| 441.3.q.b | 8 | 21.g | even | 6 | 2 | ||
| 567.3.r.a | 8 | 63.l | odd | 6 | 2 | ||
| 567.3.r.a | 8 | 63.o | even | 6 | 2 | ||
| 1008.3.d.d | 4 | 28.d | even | 2 | 1 | ||
| 1008.3.d.d | 4 | 84.h | odd | 2 | 1 | ||
| 1575.3.c.a | 4 | 35.c | odd | 2 | 1 | ||
| 1575.3.c.a | 4 | 105.g | even | 2 | 1 | ||
| 1575.3.f.a | 8 | 35.f | even | 4 | 2 | ||
| 1575.3.f.a | 8 | 105.k | odd | 4 | 2 | ||
| 4032.3.d.b | 4 | 56.h | odd | 2 | 1 | ||
| 4032.3.d.b | 4 | 168.i | even | 2 | 1 | ||
| 4032.3.d.c | 4 | 56.e | even | 2 | 1 | ||
| 4032.3.d.c | 4 | 168.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\):
|
\( T_{2}^{4} + 8T_{2}^{2} + 9 \)
|
|
\( T_{13}^{2} + 12T_{13} - 76 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 8T^{2} + 9 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 32T^{2} + 144 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 212T^{2} + 36 \)
|
| $13$ |
\( (T^{2} + 12 T - 76)^{2} \)
|
| $17$ |
\( T^{4} + 1152 T^{2} + 104976 \)
|
| $19$ |
\( (T^{2} + 12 T - 664)^{2} \)
|
| $23$ |
\( T^{4} + 1332 T^{2} + 116964 \)
|
| $29$ |
\( T^{4} + 2228 T^{2} + 1004004 \)
|
| $31$ |
\( (T^{2} - 76 T + 1416)^{2} \)
|
| $37$ |
\( (T^{2} + 64 T + 912)^{2} \)
|
| $41$ |
\( T^{4} + 4064 T^{2} + 1838736 \)
|
| $43$ |
\( (T^{2} - 40 T - 608)^{2} \)
|
| $47$ |
\( T^{4} + 1584 T^{2} + 46656 \)
|
| $53$ |
\( T^{4} + 4068 T^{2} + 3992004 \)
|
| $59$ |
\( T^{4} + 6192 T^{2} + 4359744 \)
|
| $61$ |
\( (T^{2} - 24 T - 9964)^{2} \)
|
| $67$ |
\( (T^{2} + 12 T - 5452)^{2} \)
|
| $71$ |
\( T^{4} + 5364 T^{2} + 2802276 \)
|
| $73$ |
\( (T^{2} + 8 T - 1356)^{2} \)
|
| $79$ |
\( (T + 22)^{4} \)
|
| $83$ |
\( T^{4} + 1152 T^{2} + 186624 \)
|
| $89$ |
\( T^{4} + 17600 T^{2} + 65610000 \)
|
| $97$ |
\( (T^{2} + 80 T + 1572)^{2} \)
|
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