Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [201,2,Mod(2,201)]
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, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("201.2");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 201 = 3 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 201.p (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: | \(1.60499308063\) |
| 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 \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).
| \(n\) | \(68\) | \(136\) |
| \(\chi(n)\) | \(-1\) | \(-\zeta_{33}^{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
0 | 0.487975 | − | 1.66189i | 1.99094 | + | 0.190112i | 0 | 0 | 2.32668 | + | 4.51312i | 0 | −2.52376 | − | 1.62192i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.1 | 0 | 1.57553 | − | 0.719520i | 1.57211 | − | 1.23632i | 0 | 0 | −0.656000 | + | 3.40365i | 0 | 1.96458 | − | 2.26725i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 20.1 | 0 | 1.71442 | + | 0.246497i | −0.0951638 | + | 1.99773i | 0 | 0 | −0.730437 | + | 0.177202i | 0 | 2.87848 | + | 0.845198i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 32.1 | 0 | −1.71442 | + | 0.246497i | 1.77767 | + | 0.916453i | 0 | 0 | −3.50336 | + | 3.67422i | 0 | 2.87848 | − | 0.845198i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.1 | 0 | −0.936417 | − | 1.45709i | 0.654136 | − | 1.89000i | 0 | 0 | −0.0717192 | − | 0.751078i | 0 | −1.24625 | + | 2.72890i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 44.1 | 0 | −1.71442 | − | 0.246497i | 1.77767 | − | 0.916453i | 0 | 0 | −3.50336 | − | 3.67422i | 0 | 2.87848 | + | 0.845198i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 50.1 | 0 | 0.936417 | + | 1.45709i | −1.96386 | + | 0.378502i | 0 | 0 | −4.31033 | + | 3.06937i | 0 | −1.24625 | + | 2.72890i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 74.1 | 0 | −0.487975 | + | 1.66189i | −1.16011 | + | 1.62915i | 0 | 0 | −2.19700 | − | 0.104656i | 0 | −2.52376 | − | 1.62192i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.1 | 0 | −1.30900 | − | 1.13425i | −0.471518 | + | 1.94362i | 0 | 0 | −0.816487 | + | 2.03949i | 0 | 0.426945 | + | 2.96946i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 95.1 | 0 | 1.30900 | − | 1.13425i | −1.44747 | + | 1.38016i | 0 | 0 | 2.75125 | − | 3.49850i | 0 | 0.426945 | − | 2.96946i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 98.1 | 0 | −1.30900 | + | 1.13425i | −0.471518 | − | 1.94362i | 0 | 0 | −0.816487 | − | 2.03949i | 0 | 0.426945 | − | 2.96946i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.1 | 0 | 0.487975 | + | 1.66189i | 1.99094 | − | 0.190112i | 0 | 0 | 2.32668 | − | 4.51312i | 0 | −2.52376 | + | 1.62192i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.1 | 0 | −1.57553 | − | 0.719520i | −1.85674 | + | 0.743325i | 0 | 0 | 4.20741 | + | 1.45620i | 0 | 1.96458 | + | 2.26725i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 128.1 | 0 | 1.57553 | + | 0.719520i | 1.57211 | + | 1.23632i | 0 | 0 | −0.656000 | − | 3.40365i | 0 | 1.96458 | + | 2.26725i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 146.1 | 0 | 1.30900 | + | 1.13425i | −1.44747 | − | 1.38016i | 0 | 0 | 2.75125 | + | 3.49850i | 0 | 0.426945 | + | 2.96946i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 152.1 | 0 | −0.936417 | + | 1.45709i | 0.654136 | + | 1.89000i | 0 | 0 | −0.0717192 | + | 0.751078i | 0 | −1.24625 | − | 2.72890i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 182.1 | 0 | −0.487975 | − | 1.66189i | −1.16011 | − | 1.62915i | 0 | 0 | −2.19700 | + | 0.104656i | 0 | −2.52376 | + | 1.62192i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 185.1 | 0 | −1.57553 | + | 0.719520i | −1.85674 | − | 0.743325i | 0 | 0 | 4.20741 | − | 1.45620i | 0 | 1.96458 | − | 2.26725i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.1 | 0 | 1.71442 | − | 0.246497i | −0.0951638 | − | 1.99773i | 0 | 0 | −0.730437 | − | 0.177202i | 0 | 2.87848 | − | 0.845198i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 197.1 | 0 | 0.936417 | − | 1.45709i | −1.96386 | − | 0.378502i | 0 | 0 | −4.31033 | − | 3.06937i | 0 | −1.24625 | − | 2.72890i | 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.h | odd | 66 | 1 | inner |
| 201.p | even | 66 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 201.2.p.a | ✓ | 20 |
| 3.b | odd | 2 | 1 | CM | 201.2.p.a | ✓ | 20 |
| 67.h | odd | 66 | 1 | inner | 201.2.p.a | ✓ | 20 |
| 201.p | even | 66 | 1 | inner | 201.2.p.a | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 201.2.p.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 201.2.p.a | ✓ | 20 | 3.b | odd | 2 | 1 | CM |
| 201.2.p.a | ✓ | 20 | 67.h | odd | 66 | 1 | inner |
| 201.2.p.a | ✓ | 20 | 201.p | even | 66 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(201, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} - 3 T^{18} + \cdots + 59049 \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( T^{20} + \cdots + 659102929 \)
|
| $11$ |
\( T^{20} \)
|
| $13$ |
\( T^{20} + \cdots + 496668334009 \)
|
| $17$ |
\( T^{20} \)
|
| $19$ |
\( T^{20} + \cdots + 338632050241 \)
|
| $23$ |
\( T^{20} \)
|
| $29$ |
\( T^{20} \)
|
| $31$ |
\( T^{20} + \cdots + 183137731399921 \)
|
| $37$ |
\( T^{20} + \cdots + 61\!\cdots\!29 \)
|
| $41$ |
\( T^{20} \)
|
| $43$ |
\( T^{20} + \cdots + 40\!\cdots\!49 \)
|
| $47$ |
\( T^{20} \)
|
| $53$ |
\( T^{20} \)
|
| $59$ |
\( T^{20} \)
|
| $61$ |
\( T^{20} + \cdots + 75\!\cdots\!21 \)
|
| $67$ |
\( T^{20} + \cdots + 18\!\cdots\!49 \)
|
| $71$ |
\( T^{20} \)
|
| $73$ |
\( T^{20} + \cdots + 83\!\cdots\!09 \)
|
| $79$ |
\( T^{20} + \cdots + 16\!\cdots\!61 \)
|
| $83$ |
\( T^{20} \)
|
| $89$ |
\( T^{20} \)
|
| $97$ |
\( T^{20} + \cdots + 43\!\cdots\!29 \)
|
| show more | |
| show less |