Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.7");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(2.88690875663\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.47347183152.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{4} \) |
| 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
\( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} + 176\nu^{3} - 269\nu^{2} + 16626\nu + 22035 ) / 60606 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 331\nu^{5} - 2875\nu^{4} + 35573\nu^{3} - 258101\nu^{2} + 670179\nu - 4096833 ) / 60606 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 331\nu^{5} - 418\nu^{4} + 30659\nu^{3} - 22229\nu^{2} + 527673\nu - 168909 ) / 30303 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 991\nu^{5} + 3665\nu^{4} + 82325\nu^{3} + 495697\nu^{2} + 769179\nu + 11037819 ) / 60606 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1655\nu^{5} - 9461\nu^{4} + 168037\nu^{3} - 636943\nu^{2} + 3065883\nu - 5840445 ) / 60606 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 2\beta_{4} + 4\beta_{3} - 7\beta_{2} + 2\beta _1 + 9 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - \beta_{3} - 3\beta_{2} - 112 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -44\beta_{5} + 109\beta_{4} - 221\beta_{3} + 353\beta_{2} + 2870\beta _1 - 2505 ) / 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -101\beta_{5} + 17\beta_{4} + 98\beta_{3} + 264\beta_{2} + 976\beta _1 + 5208 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2119\beta_{5} - 6779\beta_{4} + 16081\beta_{3} - 20746\beta_{2} - 261628\beta _1 + 221196 ) / 18 \)
|
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 |
|
−2.00000 | + | 3.46410i | −8.58066 | + | 13.0143i | −8.00000 | − | 13.8564i | −39.2420 | − | 67.9691i | −27.9216 | − | 55.7529i | −110.566 | + | 191.505i | 64.0000 | −95.7446 | − | 223.343i | 313.936 | ||||||||||||||||||||||
| 7.2 | −2.00000 | + | 3.46410i | −2.43381 | − | 15.3973i | −8.00000 | − | 13.8564i | −20.8014 | − | 36.0292i | 58.2054 | + | 22.3636i | 101.661 | − | 176.082i | 64.0000 | −231.153 | + | 74.9483i | 166.412 | |||||||||||||||||||||||
| 7.3 | −2.00000 | + | 3.46410i | 15.5145 | + | 1.51695i | −8.00000 | − | 13.8564i | 33.0434 | + | 57.2329i | −36.2838 | + | 50.7098i | −57.0952 | + | 98.8918i | 64.0000 | 238.398 | + | 47.0695i | −264.347 | |||||||||||||||||||||||
| 13.1 | −2.00000 | − | 3.46410i | −8.58066 | − | 13.0143i | −8.00000 | + | 13.8564i | −39.2420 | + | 67.9691i | −27.9216 | + | 55.7529i | −110.566 | − | 191.505i | 64.0000 | −95.7446 | + | 223.343i | 313.936 | |||||||||||||||||||||||
| 13.2 | −2.00000 | − | 3.46410i | −2.43381 | + | 15.3973i | −8.00000 | + | 13.8564i | −20.8014 | + | 36.0292i | 58.2054 | − | 22.3636i | 101.661 | + | 176.082i | 64.0000 | −231.153 | − | 74.9483i | 166.412 | |||||||||||||||||||||||
| 13.3 | −2.00000 | − | 3.46410i | 15.5145 | − | 1.51695i | −8.00000 | + | 13.8564i | 33.0434 | − | 57.2329i | −36.2838 | − | 50.7098i | −57.0952 | − | 98.8918i | 64.0000 | 238.398 | − | 47.0695i | −264.347 | |||||||||||||||||||||||
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 | 18.6.c.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 54.6.c.b | 6 | ||
| 4.b | odd | 2 | 1 | 144.6.i.b | 6 | ||
| 9.c | even | 3 | 1 | inner | 18.6.c.b | ✓ | 6 |
| 9.c | even | 3 | 1 | 162.6.a.j | 3 | ||
| 9.d | odd | 6 | 1 | 54.6.c.b | 6 | ||
| 9.d | odd | 6 | 1 | 162.6.a.i | 3 | ||
| 12.b | even | 2 | 1 | 432.6.i.b | 6 | ||
| 36.f | odd | 6 | 1 | 144.6.i.b | 6 | ||
| 36.h | even | 6 | 1 | 432.6.i.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.6.c.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 18.6.c.b | ✓ | 6 | 9.c | even | 3 | 1 | inner |
| 54.6.c.b | 6 | 3.b | odd | 2 | 1 | ||
| 54.6.c.b | 6 | 9.d | odd | 6 | 1 | ||
| 144.6.i.b | 6 | 4.b | odd | 2 | 1 | ||
| 144.6.i.b | 6 | 36.f | odd | 6 | 1 | ||
| 162.6.a.i | 3 | 9.d | odd | 6 | 1 | ||
| 162.6.a.j | 3 | 9.c | even | 3 | 1 | ||
| 432.6.i.b | 6 | 12.b | even | 2 | 1 | ||
| 432.6.i.b | 6 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 54T_{5}^{5} + 7587T_{5}^{4} + 179334T_{5}^{3} + 33470577T_{5}^{2} + 1007927064T_{5} + 46562734656 \)
acting on \(S_{6}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4 T + 16)^{3} \)
|
| $3$ |
\( T^{6} - 9 T^{5} + \cdots + 14348907 \)
|
| $5$ |
\( T^{6} + \cdots + 46562734656 \)
|
| $7$ |
\( T^{6} + \cdots + 26358756910084 \)
|
| $11$ |
\( T^{6} + \cdots + 17\!\cdots\!69 \)
|
| $13$ |
\( T^{6} + \cdots + 14\!\cdots\!84 \)
|
| $17$ |
\( (T^{3} - 1449 T^{2} + \cdots + 930192444)^{2} \)
|
| $19$ |
\( (T^{3} - 1131 T^{2} + \cdots - 352455920)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 18\!\cdots\!96 \)
|
| $29$ |
\( T^{6} + \cdots + 42\!\cdots\!16 \)
|
| $31$ |
\( T^{6} + \cdots + 35\!\cdots\!16 \)
|
| $37$ |
\( (T^{3} - 19968 T^{2} + \cdots - 188019064016)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 46\!\cdots\!69 \)
|
| $43$ |
\( T^{6} + \cdots + 26\!\cdots\!21 \)
|
| $47$ |
\( T^{6} + \cdots + 71\!\cdots\!00 \)
|
| $53$ |
\( (T^{3} + \cdots + 1764512817552)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 33\!\cdots\!49 \)
|
| $61$ |
\( T^{6} + \cdots + 28\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 92\!\cdots\!25 \)
|
| $71$ |
\( (T^{3} + 64836 T^{2} + \cdots - 139951336896)^{2} \)
|
| $73$ |
\( (T^{3} + \cdots + 14322358753732)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 13\!\cdots\!16 \)
|
| $83$ |
\( T^{6} + \cdots + 10\!\cdots\!16 \)
|
| $89$ |
\( (T^{3} + \cdots + 9104584153608)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 85\!\cdots\!09 \)
|
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