Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,8,Mod(7,18)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.7");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 18.c (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: | \(5.62293045871\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.14601465675.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 52x^{4} - 99x^{3} + 709x^{2} - 660x + 1872 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} + 52x^{4} - 99x^{3} + 709x^{2} - 660x + 1872 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 6\nu^{3} + 4\nu^{2} + 1067\nu + 612 ) / 2292 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 93\nu^{4} - 188\nu^{3} + 2481\nu^{2} + 517\nu + 3132 ) / 382 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 98\nu^{4} + 194\nu^{3} - 2485\nu^{2} + 5292\nu - 6036 ) / 382 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 184\nu^{5} + 113\nu^{4} + 6282\nu^{3} + 14530\nu^{2} + 25031\nu + 172896 ) / 2292 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -187\nu^{5} + 181\nu^{4} - 6864\nu^{3} + 3239\nu^{2} - 51221\nu + 13674 ) / 1146 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 6\beta _1 + 6 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} + 4\beta_{4} + 3\beta_{3} + \beta_{2} + 6\beta _1 - 288 ) / 18 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{4} - 17\beta_{3} - 23\beta_{2} + 432\beta _1 - 648 ) / 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -52\beta_{5} - 92\beta_{4} - 123\beta_{3} - 11\beta_{2} + 858\beta _1 + 5478 ) / 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -126\beta_{5} - 240\beta_{4} + 283\beta_{3} + 577\beta_{2} - 16566\beta _1 + 23772 ) / 18 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−4.00000 | + | 6.92820i | −43.9518 | + | 15.9763i | −32.0000 | − | 55.4256i | −1.70819 | − | 2.95867i | 65.1201 | − | 368.412i | 407.819 | − | 706.363i | 512.000 | 1676.52 | − | 1404.37i | 27.3311 | ||||||||||||||||||||||
| 7.2 | −4.00000 | + | 6.92820i | −11.7819 | − | 45.2569i | −32.0000 | − | 55.4256i | 187.789 | + | 325.261i | 360.677 | + | 99.4005i | −528.867 | + | 916.024i | 512.000 | −1909.38 | + | 1066.42i | −3004.63 | |||||||||||||||||||||||
| 7.3 | −4.00000 | + | 6.92820i | 42.2336 | − | 20.0828i | −32.0000 | − | 55.4256i | −159.081 | − | 275.536i | −29.7966 | + | 372.935i | 226.048 | − | 391.526i | 512.000 | 1380.36 | − | 1696.34i | 2545.30 | |||||||||||||||||||||||
| 13.1 | −4.00000 | − | 6.92820i | −43.9518 | − | 15.9763i | −32.0000 | + | 55.4256i | −1.70819 | + | 2.95867i | 65.1201 | + | 368.412i | 407.819 | + | 706.363i | 512.000 | 1676.52 | + | 1404.37i | 27.3311 | |||||||||||||||||||||||
| 13.2 | −4.00000 | − | 6.92820i | −11.7819 | + | 45.2569i | −32.0000 | + | 55.4256i | 187.789 | − | 325.261i | 360.677 | − | 99.4005i | −528.867 | − | 916.024i | 512.000 | −1909.38 | − | 1066.42i | −3004.63 | |||||||||||||||||||||||
| 13.3 | −4.00000 | − | 6.92820i | 42.2336 | + | 20.0828i | −32.0000 | + | 55.4256i | −159.081 | + | 275.536i | −29.7966 | − | 372.935i | 226.048 | + | 391.526i | 512.000 | 1380.36 | + | 1696.34i | 2545.30 | |||||||||||||||||||||||
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 | 18.8.c.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 54.8.c.a | 6 | ||
| 4.b | odd | 2 | 1 | 144.8.i.a | 6 | ||
| 9.c | even | 3 | 1 | inner | 18.8.c.a | ✓ | 6 |
| 9.c | even | 3 | 1 | 162.8.a.f | 3 | ||
| 9.d | odd | 6 | 1 | 54.8.c.a | 6 | ||
| 9.d | odd | 6 | 1 | 162.8.a.e | 3 | ||
| 12.b | even | 2 | 1 | 432.8.i.a | 6 | ||
| 36.f | odd | 6 | 1 | 144.8.i.a | 6 | ||
| 36.h | even | 6 | 1 | 432.8.i.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.8.c.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 18.8.c.a | ✓ | 6 | 9.c | even | 3 | 1 | inner |
| 54.8.c.a | 6 | 3.b | odd | 2 | 1 | ||
| 54.8.c.a | 6 | 9.d | odd | 6 | 1 | ||
| 144.8.i.a | 6 | 4.b | odd | 2 | 1 | ||
| 144.8.i.a | 6 | 36.f | odd | 6 | 1 | ||
| 162.8.a.e | 3 | 9.d | odd | 6 | 1 | ||
| 162.8.a.f | 3 | 9.c | even | 3 | 1 | ||
| 432.8.i.a | 6 | 12.b | even | 2 | 1 | ||
| 432.8.i.a | 6 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 54T_{5}^{5} + 122607T_{5}^{4} + 7279794T_{5}^{3} + 14303890521T_{5}^{2} + 48862653840T_{5} + 166659897600 \)
acting on \(S_{8}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8 T + 64)^{3} \)
|
| $3$ |
\( T^{6} + \cdots + 10460353203 \)
|
| $5$ |
\( T^{6} + \cdots + 166659897600 \)
|
| $7$ |
\( T^{6} + \cdots + 15\!\cdots\!56 \)
|
| $11$ |
\( T^{6} + \cdots + 35\!\cdots\!25 \)
|
| $13$ |
\( T^{6} + \cdots + 18\!\cdots\!64 \)
|
| $17$ |
\( (T^{3} + \cdots + 2214690411708)^{2} \)
|
| $19$ |
\( (T^{3} + \cdots - 22074972070832)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 22\!\cdots\!36 \)
|
| $31$ |
\( T^{6} + \cdots + 81\!\cdots\!00 \)
|
| $37$ |
\( (T^{3} + \cdots - 81\!\cdots\!16)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 16\!\cdots\!61 \)
|
| $43$ |
\( T^{6} + \cdots + 14\!\cdots\!69 \)
|
| $47$ |
\( T^{6} + \cdots + 17\!\cdots\!04 \)
|
| $53$ |
\( (T^{3} + \cdots + 33\!\cdots\!40)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 94\!\cdots\!81 \)
|
| $61$ |
\( T^{6} + \cdots + 65\!\cdots\!16 \)
|
| $67$ |
\( T^{6} + \cdots + 53\!\cdots\!61 \)
|
| $71$ |
\( (T^{3} + \cdots + 47\!\cdots\!44)^{2} \)
|
| $73$ |
\( (T^{3} + \cdots + 84\!\cdots\!08)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 87\!\cdots\!84 \)
|
| $83$ |
\( T^{6} + \cdots + 35\!\cdots\!84 \)
|
| $89$ |
\( (T^{3} + \cdots + 28\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 88\!\cdots\!61 \)
|
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