Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,2,Mod(11,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 3, 6, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.bv (of order \(12\), degree \(4\), 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: | \(7.28235666434\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 8.0.303595776.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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
\( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 45 ) / 144 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + 13\nu ) / 48 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} - 13 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{7} + 16\nu^{5} + 80\nu^{3} + 225\nu ) / 144 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - 2\beta_{4} - \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} - 3\beta_{5} - 3\beta_{3} - 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + 15\beta_{3} \)
|
| \(\nu^{6}\) | \(=\) |
\( -16\beta_{6} - 13 \)
|
| \(\nu^{7}\) | \(=\) |
\( 48\beta_{5} - 13\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(229\) | \(305\) | \(799\) |
| \(\chi(n)\) | \(-1 - \beta_{4}\) | \(\beta_{3} + \beta_{5}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−0.366025 | − | 1.36603i | −1.68614 | − | 0.396143i | −1.73205 | + | 1.00000i | −0.732051 | − | 2.73205i | 0.0760282 | + | 2.44831i | −2.00000 | 2.00000 | + | 2.00000i | 2.68614 | + | 1.33591i | −3.46410 | + | 2.00000i | ||||||||||||||||||||||||||
| 11.2 | −0.366025 | − | 1.36603i | 1.18614 | + | 1.26217i | −1.73205 | + | 1.00000i | −0.732051 | − | 2.73205i | 1.29000 | − | 2.08228i | −2.00000 | 2.00000 | + | 2.00000i | −0.186141 | + | 2.99422i | −3.46410 | + | 2.00000i | |||||||||||||||||||||||||||
| 83.1 | −0.366025 | + | 1.36603i | −1.68614 | + | 0.396143i | −1.73205 | − | 1.00000i | −0.732051 | + | 2.73205i | 0.0760282 | − | 2.44831i | −2.00000 | 2.00000 | − | 2.00000i | 2.68614 | − | 1.33591i | −3.46410 | − | 2.00000i | |||||||||||||||||||||||||||
| 83.2 | −0.366025 | + | 1.36603i | 1.18614 | − | 1.26217i | −1.73205 | − | 1.00000i | −0.732051 | + | 2.73205i | 1.29000 | + | 2.08228i | −2.00000 | 2.00000 | − | 2.00000i | −0.186141 | − | 2.99422i | −3.46410 | − | 2.00000i | |||||||||||||||||||||||||||
| 467.1 | 1.36603 | − | 0.366025i | −1.68614 | − | 0.396143i | 1.73205 | − | 1.00000i | 2.73205 | − | 0.732051i | −2.44831 | + | 0.0760282i | −2.00000 | 2.00000 | − | 2.00000i | 2.68614 | + | 1.33591i | 3.46410 | − | 2.00000i | |||||||||||||||||||||||||||
| 467.2 | 1.36603 | − | 0.366025i | 1.18614 | + | 1.26217i | 1.73205 | − | 1.00000i | 2.73205 | − | 0.732051i | 2.08228 | + | 1.29000i | −2.00000 | 2.00000 | − | 2.00000i | −0.186141 | + | 2.99422i | 3.46410 | − | 2.00000i | |||||||||||||||||||||||||||
| 539.1 | 1.36603 | + | 0.366025i | −1.68614 | + | 0.396143i | 1.73205 | + | 1.00000i | 2.73205 | + | 0.732051i | −2.44831 | − | 0.0760282i | −2.00000 | 2.00000 | + | 2.00000i | 2.68614 | − | 1.33591i | 3.46410 | + | 2.00000i | |||||||||||||||||||||||||||
| 539.2 | 1.36603 | + | 0.366025i | 1.18614 | − | 1.26217i | 1.73205 | + | 1.00000i | 2.73205 | + | 0.732051i | 2.08228 | − | 1.29000i | −2.00000 | 2.00000 | + | 2.00000i | −0.186141 | − | 2.99422i | 3.46410 | + | 2.00000i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
| 48.k | even | 4 | 1 | inner |
| 912.bv | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 912.2.bv.b | yes | 8 |
| 3.b | odd | 2 | 1 | 912.2.bv.a | ✓ | 8 | |
| 16.f | odd | 4 | 1 | 912.2.bv.a | ✓ | 8 | |
| 19.c | even | 3 | 1 | inner | 912.2.bv.b | yes | 8 |
| 48.k | even | 4 | 1 | inner | 912.2.bv.b | yes | 8 |
| 57.h | odd | 6 | 1 | 912.2.bv.a | ✓ | 8 | |
| 304.y | odd | 12 | 1 | 912.2.bv.a | ✓ | 8 | |
| 912.bv | even | 12 | 1 | inner | 912.2.bv.b | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 912.2.bv.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 912.2.bv.a | ✓ | 8 | 16.f | odd | 4 | 1 | |
| 912.2.bv.a | ✓ | 8 | 57.h | odd | 6 | 1 | |
| 912.2.bv.a | ✓ | 8 | 304.y | odd | 12 | 1 | |
| 912.2.bv.b | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 912.2.bv.b | yes | 8 | 19.c | even | 3 | 1 | inner |
| 912.2.bv.b | yes | 8 | 48.k | even | 4 | 1 | inner |
| 912.2.bv.b | yes | 8 | 912.bv | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 4T_{5}^{3} + 8T_{5}^{2} - 32T_{5} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \)
|
| $3$ |
\( (T^{4} + T^{3} - 2 T^{2} + \cdots + 9)^{2} \)
|
| $5$ |
\( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 64)^{2} \)
|
| $7$ |
\( (T + 2)^{8} \)
|
| $11$ |
\( (T^{4} - 6 T^{3} + 18 T^{2} + \cdots + 1)^{2} \)
|
| $13$ |
\( T^{8} - 484 T^{4} + 234256 \)
|
| $17$ |
\( (T^{4} - 16 T^{2} + 256)^{2} \)
|
| $19$ |
\( T^{8} + 18 T^{7} + \cdots + 130321 \)
|
| $23$ |
\( T^{8} - 24 T^{6} + \cdots + 10000 \)
|
| $29$ |
\( T^{8} - 12 T^{7} + \cdots + 256 \)
|
| $31$ |
\( (T^{4} + 120 T^{2} + 1444)^{2} \)
|
| $37$ |
\( (T^{4} + 16 T^{3} + \cdots + 100)^{2} \)
|
| $41$ |
\( (T^{4} - 12 T^{3} + \cdots + 625)^{2} \)
|
| $43$ |
\( T^{8} + 4 T^{7} + \cdots + 54700816 \)
|
| $47$ |
\( (T^{2} - 2 T + 4)^{4} \)
|
| $53$ |
\( T^{8} - 4 T^{7} + \cdots + 54700816 \)
|
| $59$ |
\( T^{8} - 2 T^{7} + \cdots + 625 \)
|
| $61$ |
\( T^{8} - 8 T^{7} + \cdots + 40960000 \)
|
| $67$ |
\( T^{8} + 6 T^{7} + \cdots + 4100625 \)
|
| $71$ |
\( T^{8} - 24 T^{6} + \cdots + 10000 \)
|
| $73$ |
\( T^{8} - 222 T^{6} + \cdots + 62742241 \)
|
| $79$ |
\( (T^{4} - 44 T^{2} + 1936)^{2} \)
|
| $83$ |
\( (T^{4} - 10 T^{3} + \cdots + 49)^{2} \)
|
| $89$ |
\( (T^{4} + 176 T^{2} + 30976)^{2} \)
|
| $97$ |
\( (T^{4} + 99 T^{2} + 9801)^{2} \)
|
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