Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [900,2,Mod(301,900)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(900, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("900.301");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 900.i (of order \(3\), 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: | \(7.18653618192\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.142635249.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} + 3x^{6} + 3x^{5} - 11x^{4} + 6x^{3} + 12x^{2} - 24x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - 3x^{7} + 3x^{6} + 3x^{5} - 11x^{4} + 6x^{3} + 12x^{2} - 24x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - \nu^{5} + 7\nu^{4} - 7\nu^{3} - 6\nu^{2} + 16\nu - 8 ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} - \nu^{5} - 5\nu^{4} + 9\nu^{3} + 4\nu^{2} - 12\nu + 16 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + 3\nu^{5} - \nu^{4} - 3\nu^{3} + 16\nu^{2} + 8\nu - 16 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} + \nu^{5} + 5\nu^{4} - 5\nu^{3} + 4\nu - 12 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} + 5\nu^{6} + 3\nu^{5} - 13\nu^{4} + 5\nu^{3} + 18\nu^{2} - 28\nu ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + \nu^{5} - 3\nu^{4} + 3\nu^{3} + 4\nu^{2} - 8\nu + 4 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{7} + 15\nu^{6} - 15\nu^{5} - 27\nu^{4} + 55\nu^{3} - 8\nu^{2} - 92\nu + 96 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 3\beta_{2} + \beta _1 + 2 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{6} - 3\beta_{5} + \beta_{3} + \beta_{2} - 3\beta _1 - 5 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{7} + 3\beta_{6} - 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - 3\beta_{2} + 4\beta _1 + 2 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -\beta_{7} + 3\beta_{6} - 5\beta_{5} - \beta_{3} - 4\beta_{2} - 11\beta _1 + 1 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2\beta_{7} + 2\beta_{6} - 3\beta_{5} - \beta_{4} + 5\beta_{3} - 15\beta_{2} - 3\beta _1 + 6 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -2\beta_{7} - 11\beta_{6} + \beta_{5} + 5\beta_{4} - 3\beta_{3} + 13\beta_{2} - 23\beta _1 + 6 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(451\) | \(577\) |
| \(\chi(n)\) | \(-1 - \beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 301.1 |
|
0 | −1.36091 | + | 1.07141i | 0 | 0 | 0 | −0.0432397 | − | 0.0748933i | 0 | 0.704170 | − | 2.91619i | 0 | ||||||||||||||||||||||||||||||||||||
| 301.2 | 0 | −1.02936 | − | 1.39299i | 0 | 0 | 0 | 0.340213 | + | 0.589266i | 0 | −0.880830 | + | 2.86778i | 0 | |||||||||||||||||||||||||||||||||||||
| 301.3 | 0 | 1.16098 | + | 1.28535i | 0 | 0 | 0 | −2.49787 | − | 4.32643i | 0 | −0.304233 | + | 2.98453i | 0 | |||||||||||||||||||||||||||||||||||||
| 301.4 | 0 | 1.72929 | − | 0.0977414i | 0 | 0 | 0 | 1.70089 | + | 2.94604i | 0 | 2.98089 | − | 0.338047i | 0 | |||||||||||||||||||||||||||||||||||||
| 601.1 | 0 | −1.36091 | − | 1.07141i | 0 | 0 | 0 | −0.0432397 | + | 0.0748933i | 0 | 0.704170 | + | 2.91619i | 0 | |||||||||||||||||||||||||||||||||||||
| 601.2 | 0 | −1.02936 | + | 1.39299i | 0 | 0 | 0 | 0.340213 | − | 0.589266i | 0 | −0.880830 | − | 2.86778i | 0 | |||||||||||||||||||||||||||||||||||||
| 601.3 | 0 | 1.16098 | − | 1.28535i | 0 | 0 | 0 | −2.49787 | + | 4.32643i | 0 | −0.304233 | − | 2.98453i | 0 | |||||||||||||||||||||||||||||||||||||
| 601.4 | 0 | 1.72929 | + | 0.0977414i | 0 | 0 | 0 | 1.70089 | − | 2.94604i | 0 | 2.98089 | + | 0.338047i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 900.2.i.e | yes | 8 |
| 3.b | odd | 2 | 1 | 2700.2.i.d | 8 | ||
| 5.b | even | 2 | 1 | 900.2.i.d | ✓ | 8 | |
| 5.c | odd | 4 | 2 | 900.2.s.d | 16 | ||
| 9.c | even | 3 | 1 | inner | 900.2.i.e | yes | 8 |
| 9.c | even | 3 | 1 | 8100.2.a.z | 4 | ||
| 9.d | odd | 6 | 1 | 2700.2.i.d | 8 | ||
| 9.d | odd | 6 | 1 | 8100.2.a.ba | 4 | ||
| 15.d | odd | 2 | 1 | 2700.2.i.e | 8 | ||
| 15.e | even | 4 | 2 | 2700.2.s.d | 16 | ||
| 45.h | odd | 6 | 1 | 2700.2.i.e | 8 | ||
| 45.h | odd | 6 | 1 | 8100.2.a.y | 4 | ||
| 45.j | even | 6 | 1 | 900.2.i.d | ✓ | 8 | |
| 45.j | even | 6 | 1 | 8100.2.a.x | 4 | ||
| 45.k | odd | 12 | 2 | 900.2.s.d | 16 | ||
| 45.k | odd | 12 | 2 | 8100.2.d.q | 8 | ||
| 45.l | even | 12 | 2 | 2700.2.s.d | 16 | ||
| 45.l | even | 12 | 2 | 8100.2.d.s | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 900.2.i.d | ✓ | 8 | 5.b | even | 2 | 1 | |
| 900.2.i.d | ✓ | 8 | 45.j | even | 6 | 1 | |
| 900.2.i.e | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 900.2.i.e | yes | 8 | 9.c | even | 3 | 1 | inner |
| 900.2.s.d | 16 | 5.c | odd | 4 | 2 | ||
| 900.2.s.d | 16 | 45.k | odd | 12 | 2 | ||
| 2700.2.i.d | 8 | 3.b | odd | 2 | 1 | ||
| 2700.2.i.d | 8 | 9.d | odd | 6 | 1 | ||
| 2700.2.i.e | 8 | 15.d | odd | 2 | 1 | ||
| 2700.2.i.e | 8 | 45.h | odd | 6 | 1 | ||
| 2700.2.s.d | 16 | 15.e | even | 4 | 2 | ||
| 2700.2.s.d | 16 | 45.l | even | 12 | 2 | ||
| 8100.2.a.x | 4 | 45.j | even | 6 | 1 | ||
| 8100.2.a.y | 4 | 45.h | odd | 6 | 1 | ||
| 8100.2.a.z | 4 | 9.c | even | 3 | 1 | ||
| 8100.2.a.ba | 4 | 9.d | odd | 6 | 1 | ||
| 8100.2.d.q | 8 | 45.k | odd | 12 | 2 | ||
| 8100.2.d.s | 8 | 45.l | even | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):
|
\( T_{7}^{8} + T_{7}^{7} + 19T_{7}^{6} - 38T_{7}^{5} + 313T_{7}^{4} - 182T_{7}^{3} + 118T_{7}^{2} + 10T_{7} + 1 \)
|
|
\( T_{11}^{8} - 3T_{11}^{7} + 24T_{11}^{6} - 45T_{11}^{5} + 387T_{11}^{4} - 837T_{11}^{3} + 1620T_{11}^{2} - 1215T_{11} + 729 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - T^{7} + \cdots + 81 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + T^{7} + 19 T^{6} + \cdots + 1 \)
|
| $11$ |
\( T^{8} - 3 T^{7} + \cdots + 729 \)
|
| $13$ |
\( T^{8} - 2 T^{7} + \cdots + 1849 \)
|
| $17$ |
\( (T^{4} - 9 T^{3} + 12 T^{2} + \cdots - 27)^{2} \)
|
| $19$ |
\( (T^{4} + 4 T^{3} - 27 T^{2} + \cdots - 23)^{2} \)
|
| $23$ |
\( T^{8} - 3 T^{7} + \cdots + 729 \)
|
| $29$ |
\( T^{8} + 9 T^{7} + \cdots + 1347921 \)
|
| $31$ |
\( T^{8} + 2 T^{7} + \cdots + 625 \)
|
| $37$ |
\( (T^{4} - T^{3} - 39 T^{2} + \cdots + 97)^{2} \)
|
| $41$ |
\( T^{8} - 9 T^{7} + \cdots + 6561 \)
|
| $43$ |
\( T^{8} - 8 T^{7} + \cdots + 1369 \)
|
| $47$ |
\( T^{8} + 12 T^{7} + \cdots + 998001 \)
|
| $53$ |
\( (T^{4} - 12 T^{3} + \cdots + 2781)^{2} \)
|
| $59$ |
\( T^{8} + 15 T^{7} + \cdots + 101425041 \)
|
| $61$ |
\( T^{8} - T^{7} + \cdots + 100489 \)
|
| $67$ |
\( T^{8} - 11 T^{7} + \cdots + 8162449 \)
|
| $71$ |
\( (T^{4} + 12 T^{3} + \cdots - 729)^{2} \)
|
| $73$ |
\( (T^{4} - 10 T^{3} + \cdots - 515)^{2} \)
|
| $79$ |
\( T^{8} - 7 T^{7} + \cdots + 240033049 \)
|
| $83$ |
\( T^{8} + 12 T^{7} + \cdots + 13682601 \)
|
| $89$ |
\( (T^{4} - 3 T^{3} + \cdots + 5913)^{2} \)
|
| $97$ |
\( T^{8} - 5 T^{7} + \cdots + 4044121 \)
|
| show more | |
| show less |