Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [273,2,Mod(22,273)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(273, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("273.22");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 273.k (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: | \(2.17991597518\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.771147.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 5x^{4} + 6x^{3} + 15x^{2} + 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{6} - x^{5} + 5x^{4} + 6x^{3} + 15x^{2} + 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 5\nu^{4} - 25\nu^{3} + 15\nu^{2} + 4\nu - 20 ) / 79 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{5} + 25\nu^{4} - 46\nu^{3} + 75\nu^{2} + 20\nu + 216 ) / 79 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 20\nu^{5} - 21\nu^{4} + 105\nu^{3} + 95\nu^{2} + 315\nu + 5 ) / 79 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 38\nu^{5} - 32\nu^{4} + 160\nu^{3} + 299\nu^{2} + 480\nu + 128 ) / 79 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 5\beta_{2} - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{5} + 11\beta_{4} + 5\beta_{3} + 5\beta_{2} - 15\beta _1 - 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{5} + 25\beta_{4} + 41\beta_{2} - 41\beta _1 + 25 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).
| \(n\) | \(92\) | \(106\) | \(157\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 22.1 |
|
−0.825547 | + | 1.42989i | 0.500000 | − | 0.866025i | −0.363055 | − | 0.628829i | 2.92498 | 0.825547 | + | 1.42989i | 0.500000 | + | 0.866025i | −2.10331 | −0.500000 | − | 0.866025i | −2.41471 | + | 4.18240i | ||||||||||||||||||||||
| 22.2 | 0.636945 | − | 1.10322i | 0.500000 | − | 0.866025i | 0.188601 | + | 0.326667i | 1.10331 | −0.636945 | − | 1.10322i | 0.500000 | + | 0.866025i | 3.02830 | −0.500000 | − | 0.866025i | 0.702750 | − | 1.21720i | |||||||||||||||||||||||
| 22.3 | 1.18860 | − | 2.05872i | 0.500000 | − | 0.866025i | −1.82555 | − | 3.16194i | −4.02830 | −1.18860 | − | 2.05872i | 0.500000 | + | 0.866025i | −3.92498 | −0.500000 | − | 0.866025i | −4.78804 | + | 8.29313i | |||||||||||||||||||||||
| 211.1 | −0.825547 | − | 1.42989i | 0.500000 | + | 0.866025i | −0.363055 | + | 0.628829i | 2.92498 | 0.825547 | − | 1.42989i | 0.500000 | − | 0.866025i | −2.10331 | −0.500000 | + | 0.866025i | −2.41471 | − | 4.18240i | |||||||||||||||||||||||
| 211.2 | 0.636945 | + | 1.10322i | 0.500000 | + | 0.866025i | 0.188601 | − | 0.326667i | 1.10331 | −0.636945 | + | 1.10322i | 0.500000 | − | 0.866025i | 3.02830 | −0.500000 | + | 0.866025i | 0.702750 | + | 1.21720i | |||||||||||||||||||||||
| 211.3 | 1.18860 | + | 2.05872i | 0.500000 | + | 0.866025i | −1.82555 | + | 3.16194i | −4.02830 | −1.18860 | + | 2.05872i | 0.500000 | − | 0.866025i | −3.92498 | −0.500000 | + | 0.866025i | −4.78804 | − | 8.29313i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 273.2.k.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | 819.2.o.d | 6 | ||
| 13.c | even | 3 | 1 | inner | 273.2.k.d | ✓ | 6 |
| 13.c | even | 3 | 1 | 3549.2.a.h | 3 | ||
| 13.e | even | 6 | 1 | 3549.2.a.s | 3 | ||
| 39.i | odd | 6 | 1 | 819.2.o.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.2.k.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 273.2.k.d | ✓ | 6 | 13.c | even | 3 | 1 | inner |
| 819.2.o.d | 6 | 3.b | odd | 2 | 1 | ||
| 819.2.o.d | 6 | 39.i | odd | 6 | 1 | ||
| 3549.2.a.h | 3 | 13.c | even | 3 | 1 | ||
| 3549.2.a.s | 3 | 13.e | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 2T_{2}^{5} + 7T_{2}^{4} - 4T_{2}^{3} + 19T_{2}^{2} - 15T_{2} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots + 25 \)
|
| $3$ |
\( (T^{2} - T + 1)^{3} \)
|
| $5$ |
\( (T^{3} - 13 T + 13)^{2} \)
|
| $7$ |
\( (T^{2} - T + 1)^{3} \)
|
| $11$ |
\( T^{6} - 8 T^{5} + \cdots + 25 \)
|
| $13$ |
\( T^{6} + 65T^{3} + 2197 \)
|
| $17$ |
\( T^{6} + 4 T^{5} + \cdots + 1 \)
|
| $19$ |
\( T^{6} - 7 T^{5} + \cdots + 2209 \)
|
| $23$ |
\( T^{6} + 9 T^{5} + \cdots + 1 \)
|
| $29$ |
\( T^{6} - 7 T^{5} + \cdots + 25 \)
|
| $31$ |
\( (T^{3} + 7 T^{2} + \cdots - 281)^{2} \)
|
| $37$ |
\( T^{6} + 13 T^{4} + \cdots + 169 \)
|
| $41$ |
\( T^{6} + 2 T^{5} + \cdots + 40000 \)
|
| $43$ |
\( T^{6} - 19 T^{5} + \cdots + 52441 \)
|
| $47$ |
\( (T^{3} + 17 T^{2} + \cdots - 547)^{2} \)
|
| $53$ |
\( (T^{3} - 13 T^{2} + \cdots + 13)^{2} \)
|
| $59$ |
\( T^{6} - 3 T^{5} + \cdots + 625 \)
|
| $61$ |
\( T^{6} + 13 T^{5} + \cdots + 169 \)
|
| $67$ |
\( T^{6} + 5 T^{5} + \cdots + 156025 \)
|
| $71$ |
\( T^{6} + 8 T^{5} + \cdots + 265225 \)
|
| $73$ |
\( (T^{3} + 2 T^{2} - 81 T + 73)^{2} \)
|
| $79$ |
\( (T^{3} + T^{2} - 290 T - 337)^{2} \)
|
| $83$ |
\( (T^{3} - 2 T^{2} + \cdots - 229)^{2} \)
|
| $89$ |
\( T^{6} - 19 T^{5} + \cdots + 1038361 \)
|
| $97$ |
\( T^{6} + 27 T^{5} + \cdots + 358801 \)
|
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