Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2010,2,Mod(841,2010)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2010, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2010.841");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2010 = 2 \cdot 3 \cdot 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2010.i (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: | \(16.0499308063\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 5x^{9} + 39x^{8} - 126x^{7} + 479x^{6} - 1017x^{5} + 2214x^{4} - 2870x^{3} + 3178x^{2} - 1893x + 439 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 5x^{9} + 39x^{8} - 126x^{7} + 479x^{6} - 1017x^{5} + 2214x^{4} - 2870x^{3} + 3178x^{2} - 1893x + 439 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{8} + 8\nu^{7} - 47\nu^{6} + 113\nu^{5} - 280\nu^{4} + 381\nu^{3} - 386\nu^{2} + 213\nu - 87 ) / 49 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2 \nu^{9} - 8 \nu^{8} + 3 \nu^{7} - 214 \nu^{6} + 295 \nu^{5} - 1159 \nu^{4} + 611 \nu^{3} + \cdots + 1110 ) / 833 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -11\nu^{8} + 44\nu^{7} - 283\nu^{6} + 695\nu^{5} - 1883\nu^{4} + 2659\nu^{3} - 2564\nu^{2} + 1343\nu + 1163 ) / 245 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{8} + 44\nu^{7} - 283\nu^{6} + 695\nu^{5} - 1883\nu^{4} + 2659\nu^{3} - 2809\nu^{2} + 1588\nu - 307 ) / 245 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 4 \nu^{9} + 18 \nu^{8} - 130 \nu^{7} + 371 \nu^{6} - 1331 \nu^{5} + 2442 \nu^{4} - 5255 \nu^{3} + \cdots + 2866 ) / 833 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 23 \nu^{9} - 299 \nu^{8} + 1232 \nu^{7} - 6311 \nu^{6} + 12749 \nu^{5} - 35283 \nu^{4} + 32830 \nu^{3} + \cdots - 559 ) / 4165 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 23\nu^{8} - 92\nu^{7} + 614\nu^{6} - 1520\nu^{5} + 4494\nu^{4} - 6562\nu^{3} + 8947\nu^{2} - 5904\nu + 1711 ) / 245 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 101 \nu^{9} - 548 \nu^{8} + 3359 \nu^{7} - 10732 \nu^{6} + 30858 \nu^{5} - 58626 \nu^{4} + \cdots - 8693 ) / 4165 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 131 \nu^{9} + 683 \nu^{8} - 4929 \nu^{7} + 15597 \nu^{6} - 54823 \nu^{5} + 106691 \nu^{4} + \cdots + 65293 ) / 4165 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 2\beta_{2} + \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 3\beta_{4} + 3\beta_{3} - 2\beta_{2} + \beta _1 - 17 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{9} - 3 \beta_{8} + 2 \beta_{7} + 4 \beta_{6} - 11 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + \cdots - 20 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 2 \beta_{9} - 6 \beta_{8} + 7 \beta_{7} + 8 \beta_{6} - 23 \beta_{5} + 33 \beta_{4} - 22 \beta_{3} + \cdots + 124 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 14 \beta_{9} + 27 \beta_{8} - 13 \beta_{7} - 41 \beta_{6} + 44 \beta_{5} + 69 \beta_{4} - 53 \beta_{3} + \cdots + 302 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 47 \beta_{9} + 96 \beta_{8} - 91 \beta_{7} - 143 \beta_{6} + 190 \beta_{5} - 270 \beta_{4} + 157 \beta_{3} + \cdots - 776 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 139 \beta_{9} - 180 \beta_{8} + 44 \beta_{7} + 277 \beta_{6} + 71 \beta_{5} - 981 \beta_{4} + \cdots - 3583 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 260 \beta_{9} - 394 \beta_{8} + 327 \beta_{7} + 598 \beta_{6} - 219 \beta_{5} + 569 \beta_{4} + \cdots + 1018 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 801 \beta_{9} + 729 \beta_{8} + 516 \beta_{7} - 921 \beta_{6} - 2750 \beta_{5} + 11439 \beta_{4} + \cdots + 36337 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2010\mathbb{Z}\right)^\times\).
| \(n\) | \(671\) | \(1141\) | \(1207\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 841.1 |
|
0.500000 | + | 0.866025i | 1.00000 | −0.500000 | + | 0.866025i | 1.00000 | 0.500000 | + | 0.866025i | −2.21042 | + | 3.82857i | −1.00000 | 1.00000 | 0.500000 | + | 0.866025i | ||||||||||||||||||||||||||||||||||||||
| 841.2 | 0.500000 | + | 0.866025i | 1.00000 | −0.500000 | + | 0.866025i | 1.00000 | 0.500000 | + | 0.866025i | −0.806500 | + | 1.39690i | −1.00000 | 1.00000 | 0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||||||
| 841.3 | 0.500000 | + | 0.866025i | 1.00000 | −0.500000 | + | 0.866025i | 1.00000 | 0.500000 | + | 0.866025i | −0.100287 | + | 0.173703i | −1.00000 | 1.00000 | 0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||||||
| 841.4 | 0.500000 | + | 0.866025i | 1.00000 | −0.500000 | + | 0.866025i | 1.00000 | 0.500000 | + | 0.866025i | 0.935361 | − | 1.62009i | −1.00000 | 1.00000 | 0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||||||
| 841.5 | 0.500000 | + | 0.866025i | 1.00000 | −0.500000 | + | 0.866025i | 1.00000 | 0.500000 | + | 0.866025i | 1.68185 | − | 2.91305i | −1.00000 | 1.00000 | 0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||||||
| 1771.1 | 0.500000 | − | 0.866025i | 1.00000 | −0.500000 | − | 0.866025i | 1.00000 | 0.500000 | − | 0.866025i | −2.21042 | − | 3.82857i | −1.00000 | 1.00000 | 0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||||||
| 1771.2 | 0.500000 | − | 0.866025i | 1.00000 | −0.500000 | − | 0.866025i | 1.00000 | 0.500000 | − | 0.866025i | −0.806500 | − | 1.39690i | −1.00000 | 1.00000 | 0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||||||
| 1771.3 | 0.500000 | − | 0.866025i | 1.00000 | −0.500000 | − | 0.866025i | 1.00000 | 0.500000 | − | 0.866025i | −0.100287 | − | 0.173703i | −1.00000 | 1.00000 | 0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||||||
| 1771.4 | 0.500000 | − | 0.866025i | 1.00000 | −0.500000 | − | 0.866025i | 1.00000 | 0.500000 | − | 0.866025i | 0.935361 | + | 1.62009i | −1.00000 | 1.00000 | 0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||||||
| 1771.5 | 0.500000 | − | 0.866025i | 1.00000 | −0.500000 | − | 0.866025i | 1.00000 | 0.500000 | − | 0.866025i | 1.68185 | + | 2.91305i | −1.00000 | 1.00000 | 0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 67.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2010.2.i.f | ✓ | 10 |
| 67.c | even | 3 | 1 | inner | 2010.2.i.f | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2010.2.i.f | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 2010.2.i.f | ✓ | 10 | 67.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{10} + T_{7}^{9} + 19 T_{7}^{8} - 12 T_{7}^{7} + 282 T_{7}^{6} - 27 T_{7}^{5} + 810 T_{7}^{4} + \cdots + 81 \)
acting on \(S_{2}^{\mathrm{new}}(2010, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{5} \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( (T - 1)^{10} \)
|
| $7$ |
\( T^{10} + T^{9} + \cdots + 81 \)
|
| $11$ |
\( T^{10} - 3 T^{9} + \cdots + 625 \)
|
| $13$ |
\( T^{10} + 6 T^{9} + \cdots + 5625 \)
|
| $17$ |
\( T^{10} - 12 T^{9} + \cdots + 1530169 \)
|
| $19$ |
\( (T^{2} - T + 1)^{5} \)
|
| $23$ |
\( T^{10} + T^{9} + \cdots + 7263025 \)
|
| $29$ |
\( T^{10} + 2 T^{9} + \cdots + 49 \)
|
| $31$ |
\( T^{10} + 55 T^{8} + \cdots + 225 \)
|
| $37$ |
\( T^{10} - 16 T^{9} + \cdots + 19140625 \)
|
| $41$ |
\( T^{10} - 8 T^{9} + \cdots + 41024025 \)
|
| $43$ |
\( (T^{5} + 3 T^{4} + \cdots - 5261)^{2} \)
|
| $47$ |
\( T^{10} + 3 T^{9} + \cdots + 273529 \)
|
| $53$ |
\( (T^{5} + 12 T^{4} + \cdots + 2187)^{2} \)
|
| $59$ |
\( (T^{5} + 8 T^{4} + \cdots - 5625)^{2} \)
|
| $61$ |
\( T^{10} - 5 T^{9} + \cdots + 625 \)
|
| $67$ |
\( T^{10} + \cdots + 1350125107 \)
|
| $71$ |
\( T^{10} - 6 T^{9} + \cdots + 5625 \)
|
| $73$ |
\( T^{10} - 8 T^{9} + \cdots + 79121025 \)
|
| $79$ |
\( T^{10} + 11 T^{9} + \cdots + 32661225 \)
|
| $83$ |
\( T^{10} + 4 T^{9} + \cdots + 33489 \)
|
| $89$ |
\( (T^{5} + T^{4} + \cdots - 79471)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 9729257769 \)
|
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