Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1089,3,Mod(604,1089)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1089.604");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1089 = 3^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1089.c (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: | \(29.6731007888\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.79010463744.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 6x^{6} - 44x^{5} + 108x^{3} + 538x^{2} + 360x + 825 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{8} - 6x^{6} - 44x^{5} + 108x^{3} + 538x^{2} + 360x + 825 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -23\nu^{7} + 181\nu^{6} + 550\nu^{5} - 610\nu^{4} - 5638\nu^{3} - 5185\nu^{2} + 1947\nu - 3885 ) / 33345 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{7} + 29\nu^{6} - 58\nu^{5} - 2\nu^{4} - 242\nu^{3} + 1579\nu^{2} - 48\nu - 1605 ) / 2565 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 88\nu^{7} + 274\nu^{6} - 1460\nu^{5} - 4915\nu^{4} + 308\nu^{3} + 23060\nu^{2} + 9948\nu - 600 ) / 33345 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{7} + 2\nu^{6} - 4\nu^{5} - 141\nu^{4} - 56\nu^{3} - 68\nu^{2} + 2031\nu + 420 ) / 855 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 148\nu^{7} + 124\nu^{6} - 1445\nu^{5} - 5740\nu^{4} - 3187\nu^{3} + 32720\nu^{2} + 55128\nu + 74775 ) / 33345 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 22\nu^{7} - 17\nu^{6} - 251\nu^{5} + 11\nu^{4} + 761\nu^{3} + 2573\nu^{2} - 8286\nu + 3270 ) / 2565 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -362\nu^{7} + 1399\nu^{6} + 280\nu^{5} + 6830\nu^{4} - 34612\nu^{3} - 1225\nu^{2} - 62142\nu + 139890 ) / 33345 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{7} + 2\beta_{5} - 2\beta_{4} - 2\beta_{3} + 3\beta_{2} + 4\beta _1 + 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{7} + \beta_{6} - 7\beta_{5} + 4\beta_{4} - \beta_{3} + 12\beta_{2} + 4\beta _1 + 33 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} + 4\beta_{6} - 2\beta_{5} + 8\beta_{4} - 15\beta_{3} + 15\beta_{2} + 2\beta _1 + 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -31\beta_{7} + 9\beta_{6} - \beta_{5} - 8\beta_{4} - 53\beta_{3} + 84\beta_{2} + 160\beta _1 + 195 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -57\beta_{7} + 76\beta_{6} - 190\beta_{5} + 202\beta_{4} - 118\beta_{3} + 405\beta_{2} + 280\beta _1 + 753 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -176\beta_{7} + 356\beta_{6} - 148\beta_{5} + 529\beta_{4} - 789\beta_{3} + 867\beta_{2} + 1057\beta _1 + 1008 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).
| \(n\) | \(244\) | \(848\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 604.1 |
|
− | 3.11962i | 0 | −5.73205 | −6.02665 | 0 | − | 7.07107i | 5.40335i | 0 | 18.8009i | ||||||||||||||||||||||||||||||||||||||||
| 604.2 | − | 3.11962i | 0 | −5.73205 | 6.02665 | 0 | 7.07107i | 5.40335i | 0 | − | 18.8009i | |||||||||||||||||||||||||||||||||||||||||
| 604.3 | − | 2.50359i | 0 | −2.26795 | −1.29595 | 0 | 7.07107i | − | 4.33634i | 0 | 3.24453i | |||||||||||||||||||||||||||||||||||||||||
| 604.4 | − | 2.50359i | 0 | −2.26795 | 1.29595 | 0 | − | 7.07107i | − | 4.33634i | 0 | − | 3.24453i | |||||||||||||||||||||||||||||||||||||||
| 604.5 | 2.50359i | 0 | −2.26795 | −1.29595 | 0 | − | 7.07107i | 4.33634i | 0 | − | 3.24453i | |||||||||||||||||||||||||||||||||||||||||
| 604.6 | 2.50359i | 0 | −2.26795 | 1.29595 | 0 | 7.07107i | 4.33634i | 0 | 3.24453i | |||||||||||||||||||||||||||||||||||||||||||
| 604.7 | 3.11962i | 0 | −5.73205 | −6.02665 | 0 | 7.07107i | − | 5.40335i | 0 | − | 18.8009i | |||||||||||||||||||||||||||||||||||||||||
| 604.8 | 3.11962i | 0 | −5.73205 | 6.02665 | 0 | − | 7.07107i | − | 5.40335i | 0 | 18.8009i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 33.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1089.3.c.i | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 1089.3.c.i | ✓ | 8 |
| 11.b | odd | 2 | 1 | inner | 1089.3.c.i | ✓ | 8 |
| 33.d | even | 2 | 1 | inner | 1089.3.c.i | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1089.3.c.i | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1089.3.c.i | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 1089.3.c.i | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
| 1089.3.c.i | ✓ | 8 | 33.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 16T_{2}^{2} + 61 \)
acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 16 T^{2} + 61)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} - 38 T^{2} + 61)^{2} \)
|
| $7$ |
\( (T^{2} + 50)^{4} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( (T^{4} + 300 T^{2} + 5625)^{2} \)
|
| $17$ |
\( (T^{4} + 1056 T^{2} + 92781)^{2} \)
|
| $19$ |
\( (T^{4} + 1344 T^{2} + 278784)^{2} \)
|
| $23$ |
\( (T^{4} - 1584 T^{2} + 19764)^{2} \)
|
| $29$ |
\( (T^{4} + 2200 T^{2} + 38125)^{2} \)
|
| $31$ |
\( (T - 34)^{8} \)
|
| $37$ |
\( (T^{2} - 40 T + 325)^{4} \)
|
| $41$ |
\( (T^{4} + 10800 T^{2} + 27793125)^{2} \)
|
| $43$ |
\( (T^{2} + 1350)^{4} \)
|
| $47$ |
\( (T^{4} - 8888 T^{2} + 16749136)^{2} \)
|
| $53$ |
\( (T^{4} - 1898 T^{2} + 893101)^{2} \)
|
| $59$ |
\( (T^{4} - 4400 T^{2} + 152500)^{2} \)
|
| $61$ |
\( (T^{4} + 10684 T^{2} + 11600836)^{2} \)
|
| $67$ |
\( (T^{2} - 130 T + 3550)^{4} \)
|
| $71$ |
\( (T^{4} - 11600 T^{2} + 18452500)^{2} \)
|
| $73$ |
\( (T^{4} + 7900 T^{2} + 12602500)^{2} \)
|
| $79$ |
\( (T^{4} + 4464 T^{2} + 2742336)^{2} \)
|
| $83$ |
\( (T^{4} + 3856 T^{2} + 472384)^{2} \)
|
| $89$ |
\( (T^{4} - 14550 T^{2} + 41518125)^{2} \)
|
| $97$ |
\( (T^{2} + 100 T - 1175)^{4} \)
|
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