Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [494,2,Mod(191,494)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(494, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("494.191");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 494 = 2 \cdot 13 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 494.g (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: | \(3.94460985985\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 9x^{6} + 8x^{5} + 25x^{4} + 3x^{3} + 11x^{2} + 2x + 4 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} + 9x^{6} + 8x^{5} + 25x^{4} + 3x^{3} + 11x^{2} + 2x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -257\nu^{7} + 924\nu^{6} - 3003\nu^{5} + 1306\nu^{4} - 2079\nu^{3} + 10857\nu^{2} - 859\nu + 142 ) / 4478 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 313\nu^{7} - 1892\nu^{6} + 6149\nu^{5} - 10564\nu^{4} + 4257\nu^{3} - 22231\nu^{2} + 9009\nu - 9460 ) / 4478 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 164\nu^{7} - 276\nu^{6} + 897\nu^{5} + 2634\nu^{4} + 621\nu^{3} - 3243\nu^{2} - 7798\nu - 1380 ) / 2239 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -205\nu^{7} + 345\nu^{6} - 1681\nu^{5} - 2173\nu^{4} - 5814\nu^{3} - 984\nu^{2} - 328\nu - 2753 ) / 2239 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 605\nu^{7} - 1182\nu^{6} + 4961\nu^{5} + 6413\nu^{4} + 10496\nu^{3} + 2904\nu^{2} + 968\nu + 5285 ) / 2239 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -786\nu^{7} + 1432\nu^{6} - 6893\nu^{5} - 7436\nu^{4} - 21134\nu^{3} - 7803\nu^{2} - 9318\nu - 1796 ) / 2239 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{3} + 3\beta_{2} + 2\beta _1 - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{7} + 2\beta_{6} - 8\beta_{5} - 3\beta_{4} - 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( -11\beta_{4} - 9\beta_{3} - 25\beta_{2} - 25\beta _1 - 25 \)
|
| \(\nu^{5}\) | \(=\) |
\( -36\beta_{7} - 27\beta_{6} + 86\beta_{5} - 27\beta_{3} - 77\beta_{2} - 86\beta _1 + 86 \)
|
| \(\nu^{6}\) | \(=\) |
\( -122\beta_{7} - 95\beta_{6} + 285\beta_{5} + 122\beta_{4} + 552 \)
|
| \(\nu^{7}\) | \(=\) |
\( 407\beta_{4} + 312\beta_{3} + 882\beta_{2} + 959\beta _1 + 882 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/494\mathbb{Z}\right)^\times\).
| \(n\) | \(287\) | \(457\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 191.1 |
|
0.500000 | − | 0.866025i | −0.691332 | + | 1.19742i | −0.500000 | − | 0.866025i | −1.23060 | 0.691332 | + | 1.19742i | −0.468090 | − | 0.810755i | −1.00000 | 0.544121 | + | 0.942445i | −0.615300 | + | 1.06573i | ||||||||||||||||||||||||||||
| 191.2 | 0.500000 | − | 0.866025i | −0.320522 | + | 0.555160i | −0.500000 | − | 0.866025i | 3.42689 | 0.320522 | + | 0.555160i | 0.739436 | + | 1.28074i | −1.00000 | 1.29453 | + | 2.24219i | 1.71345 | − | 2.96778i | |||||||||||||||||||||||||||||
| 191.3 | 0.500000 | − | 0.866025i | 0.336767 | − | 0.583297i | −0.500000 | − | 0.866025i | −0.0759953 | −0.336767 | − | 0.583297i | −1.64794 | − | 2.85432i | −1.00000 | 1.27318 | + | 2.20521i | −0.0379976 | + | 0.0658138i | |||||||||||||||||||||||||||||
| 191.4 | 0.500000 | − | 0.866025i | 1.67509 | − | 2.90133i | −0.500000 | − | 0.866025i | −3.12030 | −1.67509 | − | 2.90133i | 0.876594 | + | 1.51831i | −1.00000 | −4.11183 | − | 7.12190i | −1.56015 | + | 2.70226i | |||||||||||||||||||||||||||||
| 419.1 | 0.500000 | + | 0.866025i | −0.691332 | − | 1.19742i | −0.500000 | + | 0.866025i | −1.23060 | 0.691332 | − | 1.19742i | −0.468090 | + | 0.810755i | −1.00000 | 0.544121 | − | 0.942445i | −0.615300 | − | 1.06573i | |||||||||||||||||||||||||||||
| 419.2 | 0.500000 | + | 0.866025i | −0.320522 | − | 0.555160i | −0.500000 | + | 0.866025i | 3.42689 | 0.320522 | − | 0.555160i | 0.739436 | − | 1.28074i | −1.00000 | 1.29453 | − | 2.24219i | 1.71345 | + | 2.96778i | |||||||||||||||||||||||||||||
| 419.3 | 0.500000 | + | 0.866025i | 0.336767 | + | 0.583297i | −0.500000 | + | 0.866025i | −0.0759953 | −0.336767 | + | 0.583297i | −1.64794 | + | 2.85432i | −1.00000 | 1.27318 | − | 2.20521i | −0.0379976 | − | 0.0658138i | |||||||||||||||||||||||||||||
| 419.4 | 0.500000 | + | 0.866025i | 1.67509 | + | 2.90133i | −0.500000 | + | 0.866025i | −3.12030 | −1.67509 | + | 2.90133i | 0.876594 | − | 1.51831i | −1.00000 | −4.11183 | + | 7.12190i | −1.56015 | − | 2.70226i | |||||||||||||||||||||||||||||
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 | 494.2.g.e | ✓ | 8 |
| 13.c | even | 3 | 1 | inner | 494.2.g.e | ✓ | 8 |
| 13.c | even | 3 | 1 | 6422.2.a.z | 4 | ||
| 13.e | even | 6 | 1 | 6422.2.a.bb | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 494.2.g.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 494.2.g.e | ✓ | 8 | 13.c | even | 3 | 1 | inner |
| 6422.2.a.z | 4 | 13.c | even | 3 | 1 | ||
| 6422.2.a.bb | 4 | 13.e | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(494, [\chi])\):
|
\( T_{3}^{8} - 2T_{3}^{7} + 9T_{3}^{6} + 8T_{3}^{5} + 25T_{3}^{4} + 3T_{3}^{3} + 11T_{3}^{2} + 2T_{3} + 4 \)
|
|
\( T_{5}^{4} + T_{5}^{3} - 11T_{5}^{2} - 14T_{5} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{4} \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 4 \)
|
| $5$ |
\( (T^{4} + T^{3} - 11 T^{2} + \cdots - 1)^{2} \)
|
| $7$ |
\( T^{8} + T^{7} + \cdots + 64 \)
|
| $11$ |
\( T^{8} + 2 T^{7} + \cdots + 36 \)
|
| $13$ |
\( T^{8} - T^{7} + \cdots + 28561 \)
|
| $17$ |
\( T^{8} + T^{7} + \cdots + 5184 \)
|
| $19$ |
\( (T^{2} - T + 1)^{4} \)
|
| $23$ |
\( T^{8} - 10 T^{7} + \cdots + 144 \)
|
| $29$ |
\( T^{8} - T^{7} + \cdots + 1274641 \)
|
| $31$ |
\( (T^{4} - 11 T^{3} + 38 T^{2} + \cdots + 8)^{2} \)
|
| $37$ |
\( T^{8} - 9 T^{7} + \cdots + 153664 \)
|
| $41$ |
\( T^{8} + 4 T^{7} + \cdots + 2209 \)
|
| $43$ |
\( T^{8} - 15 T^{7} + \cdots + 16384 \)
|
| $47$ |
\( (T^{4} - 25 T^{3} + \cdots - 11166)^{2} \)
|
| $53$ |
\( (T^{4} + 8 T^{3} + 5 T^{2} + \cdots - 1)^{2} \)
|
| $59$ |
\( T^{8} + 28 T^{7} + \cdots + 8213956 \)
|
| $61$ |
\( T^{8} - 15 T^{7} + \cdots + 5329 \)
|
| $67$ |
\( T^{8} + 164 T^{6} + \cdots + 1364224 \)
|
| $71$ |
\( T^{8} - T^{7} + \cdots + 675684 \)
|
| $73$ |
\( (T^{4} - 6 T^{3} + \cdots - 2117)^{2} \)
|
| $79$ |
\( (T^{4} + 12 T^{3} + \cdots - 5294)^{2} \)
|
| $83$ |
\( (T^{4} + 17 T^{3} + \cdots - 538)^{2} \)
|
| $89$ |
\( T^{8} - 15 T^{7} + \cdots + 1562500 \)
|
| $97$ |
\( T^{8} + 2 T^{7} + \cdots + 76597504 \)
|
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