Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [201,3,Mod(17,201)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(201, base_ring=CyclotomicField(66))
chi = DirichletCharacter(H, H._module([33, 64]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("201.17");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 201 = 3 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 201.o (of order \(66\), degree \(20\), 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: | \(5.47685331364\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\Q(\zeta_{33})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{66}]$ |
$q$-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 primitive root of unity \(\zeta_{33}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).
| \(n\) | \(68\) | \(136\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{33}^{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | −2.52376 | + | 1.62192i | 3.92771 | − | 0.757005i | 0 | 0 | −6.83405 | − | 9.59709i | 0 | 3.73874 | − | 8.18669i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.1 | 0 | 0.426945 | − | 2.96946i | −3.55534 | + | 1.83291i | 0 | 0 | 6.63942 | − | 6.33068i | 0 | −8.63544 | − | 2.53559i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.1 | 0 | −2.52376 | + | 1.62192i | −1.30827 | + | 3.78000i | 0 | 0 | −9.12076 | + | 0.870927i | 0 | 3.73874 | − | 8.18669i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 35.1 | 0 | 0.426945 | + | 2.96946i | −3.55534 | − | 1.83291i | 0 | 0 | 6.63942 | + | 6.33068i | 0 | −8.63544 | + | 2.53559i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.1 | 0 | 0.426945 | − | 2.96946i | 0.190328 | − | 3.99547i | 0 | 0 | −1.36948 | − | 5.64509i | 0 | −8.63544 | − | 2.53559i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 56.1 | 0 | −1.24625 | − | 2.72890i | −3.14421 | + | 2.47264i | 0 | 0 | 13.1880 | + | 2.54178i | 0 | −5.89375 | + | 6.80175i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.1 | 0 | 2.87848 | + | 0.845198i | −3.98189 | − | 0.380224i | 0 | 0 | −11.9416 | + | 6.15630i | 0 | 7.57128 | + | 4.86577i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 71.1 | 0 | −2.52376 | − | 1.62192i | 3.92771 | + | 0.757005i | 0 | 0 | −6.83405 | + | 9.59709i | 0 | 3.73874 | + | 8.18669i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 77.1 | 0 | 0.426945 | + | 2.96946i | 0.190328 | + | 3.99547i | 0 | 0 | −1.36948 | + | 5.64509i | 0 | −8.63544 | + | 2.53559i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.1 | 0 | −1.24625 | − | 2.72890i | 3.71347 | + | 1.48665i | 0 | 0 | 4.57895 | + | 13.2300i | 0 | −5.89375 | + | 6.80175i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 86.1 | 0 | 2.87848 | − | 0.845198i | 2.32023 | + | 3.25830i | 0 | 0 | −0.560201 | − | 11.7601i | 0 | 7.57128 | − | 4.86577i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 116.1 | 0 | −2.52376 | − | 1.62192i | −1.30827 | − | 3.78000i | 0 | 0 | −9.12076 | − | 0.870927i | 0 | 3.73874 | + | 8.18669i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 122.1 | 0 | 1.96458 | − | 2.26725i | 2.89494 | + | 2.76032i | 0 | 0 | 4.57702 | − | 3.59941i | 0 | −1.28083 | − | 8.90839i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 140.1 | 0 | −1.24625 | + | 2.72890i | −3.14421 | − | 2.47264i | 0 | 0 | 13.1880 | − | 2.54178i | 0 | −5.89375 | − | 6.80175i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 155.1 | 0 | −1.24625 | + | 2.72890i | 3.71347 | − | 1.48665i | 0 | 0 | 4.57895 | − | 13.2300i | 0 | −5.89375 | − | 6.80175i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 167.1 | 0 | 2.87848 | − | 0.845198i | −3.98189 | + | 0.380224i | 0 | 0 | −11.9416 | − | 6.15630i | 0 | 7.57128 | − | 4.86577i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 170.1 | 0 | 1.96458 | + | 2.26725i | 0.943036 | + | 3.88725i | 0 | 0 | 1.84264 | − | 0.737681i | 0 | −1.28083 | + | 8.90839i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 173.1 | 0 | 1.96458 | + | 2.26725i | 2.89494 | − | 2.76032i | 0 | 0 | 4.57702 | + | 3.59941i | 0 | −1.28083 | + | 8.90839i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 188.1 | 0 | 1.96458 | − | 2.26725i | 0.943036 | − | 3.88725i | 0 | 0 | 1.84264 | + | 0.737681i | 0 | −1.28083 | − | 8.90839i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 194.1 | 0 | 2.87848 | + | 0.845198i | 2.32023 | − | 3.25830i | 0 | 0 | −0.560201 | + | 11.7601i | 0 | 7.57128 | + | 4.86577i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
| 67.g | even | 33 | 1 | inner |
| 201.o | odd | 66 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 201.3.o.a | ✓ | 20 |
| 3.b | odd | 2 | 1 | CM | 201.3.o.a | ✓ | 20 |
| 67.g | even | 33 | 1 | inner | 201.3.o.a | ✓ | 20 |
| 201.o | odd | 66 | 1 | inner | 201.3.o.a | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 201.3.o.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 201.3.o.a | ✓ | 20 | 3.b | odd | 2 | 1 | CM |
| 201.3.o.a | ✓ | 20 | 67.g | even | 33 | 1 | inner |
| 201.3.o.a | ✓ | 20 | 201.o | odd | 66 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{3}^{\mathrm{new}}(201, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( (T^{10} - 3 T^{9} + \cdots + 59049)^{2} \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( T^{20} + \cdots + 39\!\cdots\!61 \)
|
| $11$ |
\( T^{20} \)
|
| $13$ |
\( T^{20} + \cdots + 82\!\cdots\!21 \)
|
| $17$ |
\( T^{20} \)
|
| $19$ |
\( T^{20} + \cdots + 66\!\cdots\!21 \)
|
| $23$ |
\( T^{20} \)
|
| $29$ |
\( T^{20} \)
|
| $31$ |
\( T^{20} + \cdots + 81\!\cdots\!01 \)
|
| $37$ |
\( T^{20} + \cdots + 97\!\cdots\!01 \)
|
| $41$ |
\( T^{20} \)
|
| $43$ |
\( T^{20} + \cdots + 43\!\cdots\!01 \)
|
| $47$ |
\( T^{20} \)
|
| $53$ |
\( T^{20} \)
|
| $59$ |
\( T^{20} \)
|
| $61$ |
\( T^{20} + \cdots + 29\!\cdots\!01 \)
|
| $67$ |
\( T^{20} + \cdots + 33\!\cdots\!01 \)
|
| $71$ |
\( T^{20} \)
|
| $73$ |
\( T^{20} + \cdots + 72\!\cdots\!41 \)
|
| $79$ |
\( T^{20} + \cdots + 12\!\cdots\!61 \)
|
| $83$ |
\( T^{20} \)
|
| $89$ |
\( T^{20} \)
|
| $97$ |
\( T^{20} + \cdots + 61\!\cdots\!61 \)
|
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