Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1098,2,Mod(487,1098)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1098, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1098.487");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1098 = 2 \cdot 3^{2} \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1098.d (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: | \(8.76757414194\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.7718676736.7 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 8x^{6} + 2x^{5} + 5x^{4} - 22x^{3} + 50x^{2} + 20x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{8} - 4x^{7} + 8x^{6} + 2x^{5} + 5x^{4} - 22x^{3} + 50x^{2} + 20x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -8\nu^{7} + 52\nu^{6} - 194\nu^{5} + 469\nu^{4} - 566\nu^{3} + 298\nu^{2} + 148\nu + 3349 ) / 1293 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -203\nu^{7} + 673\nu^{6} - 1367\nu^{5} - 221\nu^{4} - 4988\nu^{3} + 4006\nu^{2} + 1816\nu - 13772 ) / 10344 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 727\nu^{7} - 1493\nu^{6} - 149\nu^{5} + 12817\nu^{4} + 4564\nu^{3} - 9302\nu^{2} - 3752\nu + 49780 ) / 10344 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 857\nu^{7} - 3631\nu^{6} + 7529\nu^{5} + 347\nu^{4} + 4064\nu^{3} - 23842\nu^{2} + 46856\nu + 8612 ) / 10344 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 265\nu^{7} - 1076\nu^{6} + 2224\nu^{5} + 142\nu^{4} + 2263\nu^{3} - 6962\nu^{2} + 16432\nu + 3010 ) / 2586 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 592\nu^{7} - 2555\nu^{6} + 5305\nu^{5} + 205\nu^{4} + 1801\nu^{3} - 14294\nu^{2} + 30424\nu + 5602 ) / 2586 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 542\nu^{7} - 2230\nu^{6} + 4739\nu^{5} + 227\nu^{4} + 2789\nu^{3} - 10492\nu^{2} + 30056\nu + 5522 ) / 1293 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} - 4\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} + 3\beta_{6} + 6\beta_{5} - 11\beta_{4} - \beta_{3} - 6\beta_{2} + 3\beta _1 - 11 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{3} - 10\beta_{2} + 9\beta _1 - 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 11\beta_{7} - 35\beta_{6} - 46\beta_{5} + 99\beta_{4} - 11\beta_{3} - 46\beta_{2} + 35\beta _1 - 99 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 23\beta_{7} - 80\beta_{6} - 90\beta_{5} + 216\beta_{4} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 103\beta_{7} - 329\beta_{6} - 386\beta_{5} + 869\beta_{4} + 103\beta_{3} + 386\beta_{2} - 329\beta _1 + 869 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1098\mathbb{Z}\right)^\times\).
| \(n\) | \(245\) | \(307\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 487.1 |
|
− | 1.00000i | 0 | −1.00000 | −3.18168 | 0 | 1.78668i | 1.00000i | 0 | 3.18168i | |||||||||||||||||||||||||||||||||||||||||
| 487.2 | − | 1.00000i | 0 | −1.00000 | −1.37000 | 0 | − | 4.87932i | 1.00000i | 0 | 1.37000i | |||||||||||||||||||||||||||||||||||||||||
| 487.3 | − | 1.00000i | 0 | −1.00000 | 1.37000 | 0 | 4.87932i | 1.00000i | 0 | − | 1.37000i | |||||||||||||||||||||||||||||||||||||||||
| 487.4 | − | 1.00000i | 0 | −1.00000 | 3.18168 | 0 | − | 1.78668i | 1.00000i | 0 | − | 3.18168i | ||||||||||||||||||||||||||||||||||||||||
| 487.5 | 1.00000i | 0 | −1.00000 | −3.18168 | 0 | − | 1.78668i | − | 1.00000i | 0 | − | 3.18168i | ||||||||||||||||||||||||||||||||||||||||
| 487.6 | 1.00000i | 0 | −1.00000 | −1.37000 | 0 | 4.87932i | − | 1.00000i | 0 | − | 1.37000i | |||||||||||||||||||||||||||||||||||||||||
| 487.7 | 1.00000i | 0 | −1.00000 | 1.37000 | 0 | − | 4.87932i | − | 1.00000i | 0 | 1.37000i | |||||||||||||||||||||||||||||||||||||||||
| 487.8 | 1.00000i | 0 | −1.00000 | 3.18168 | 0 | 1.78668i | − | 1.00000i | 0 | 3.18168i | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 61.b | even | 2 | 1 | inner |
| 183.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1098.2.d.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 1098.2.d.e | ✓ | 8 |
| 61.b | even | 2 | 1 | inner | 1098.2.d.e | ✓ | 8 |
| 183.d | odd | 2 | 1 | inner | 1098.2.d.e | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1098.2.d.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1098.2.d.e | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 1098.2.d.e | ✓ | 8 | 61.b | even | 2 | 1 | inner |
| 1098.2.d.e | ✓ | 8 | 183.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 12T_{5}^{2} + 19 \)
acting on \(S_{2}^{\mathrm{new}}(1098, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} - 12 T^{2} + 19)^{2} \)
|
| $7$ |
\( (T^{4} + 27 T^{2} + 76)^{2} \)
|
| $11$ |
\( (T^{4} + 13 T^{2} + 4)^{2} \)
|
| $13$ |
\( (T^{2} - T - 4)^{4} \)
|
| $17$ |
\( (T^{4} + 9 T^{2} + 16)^{2} \)
|
| $19$ |
\( (T - 2)^{8} \)
|
| $23$ |
\( (T^{2} + 17)^{4} \)
|
| $29$ |
\( (T^{4} + 36 T^{2} + 256)^{2} \)
|
| $31$ |
\( (T^{4} + 48 T^{2} + 304)^{2} \)
|
| $37$ |
\( (T^{4} + 41 T^{2} + 76)^{2} \)
|
| $41$ |
\( (T^{4} - 79 T^{2} + 1216)^{2} \)
|
| $43$ |
\( (T^{4} + 41 T^{2} + 76)^{2} \)
|
| $47$ |
\( (T^{4} - 108 T^{2} + 1216)^{2} \)
|
| $53$ |
\( (T^{4} + 168 T^{2} + 2704)^{2} \)
|
| $59$ |
\( (T^{4} + 93 T^{2} + 1444)^{2} \)
|
| $61$ |
\( (T^{4} - 6 T^{3} + \cdots + 3721)^{2} \)
|
| $67$ |
\( (T^{4} + 71 T^{2} + 304)^{2} \)
|
| $71$ |
\( (T^{4} + 49 T^{2} + 256)^{2} \)
|
| $73$ |
\( (T^{2} + 4 T - 13)^{4} \)
|
| $79$ |
\( (T^{4} + 71 T^{2} + 304)^{2} \)
|
| $83$ |
\( (T^{4} - 129 T^{2} + 76)^{2} \)
|
| $89$ |
\( (T^{4} + 121 T^{2} + 2704)^{2} \)
|
| $97$ |
\( (T^{2} + T - 106)^{4} \)
|
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