Newspace parameters
| Level: | \( N \) | \(=\) | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1014.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(8.09683076496\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.153664.1 |
| Defining polynomial: |
\( x^{6} + 5x^{4} + 6x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
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
\( x^{6} + 5x^{4} + 6x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 3\nu^{2} + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 4\nu^{3} + 3\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 3\beta_{2} + 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 4\beta_{3} + 9\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).
| \(n\) | \(677\) | \(847\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
− | 1.00000i | 1.00000 | −1.00000 | − | 0.692021i | − | 1.00000i | − | 0.356896i | 1.00000i | 1.00000 | −0.692021 | ||||||||||||||||||||||||||||||||
| 337.2 | − | 1.00000i | 1.00000 | −1.00000 | − | 0.356896i | − | 1.00000i | 4.04892i | 1.00000i | 1.00000 | −0.356896 | ||||||||||||||||||||||||||||||||||
| 337.3 | − | 1.00000i | 1.00000 | −1.00000 | 4.04892i | − | 1.00000i | − | 0.692021i | 1.00000i | 1.00000 | 4.04892 | ||||||||||||||||||||||||||||||||||
| 337.4 | 1.00000i | 1.00000 | −1.00000 | − | 4.04892i | 1.00000i | 0.692021i | − | 1.00000i | 1.00000 | 4.04892 | |||||||||||||||||||||||||||||||||||
| 337.5 | 1.00000i | 1.00000 | −1.00000 | 0.356896i | 1.00000i | − | 4.04892i | − | 1.00000i | 1.00000 | −0.356896 | |||||||||||||||||||||||||||||||||||
| 337.6 | 1.00000i | 1.00000 | −1.00000 | 0.692021i | 1.00000i | 0.356896i | − | 1.00000i | 1.00000 | −0.692021 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1014.2.b.g | 6 | |
| 3.b | odd | 2 | 1 | 3042.2.b.r | 6 | ||
| 13.b | even | 2 | 1 | inner | 1014.2.b.g | 6 | |
| 13.c | even | 3 | 2 | 1014.2.i.g | 12 | ||
| 13.d | odd | 4 | 1 | 1014.2.a.m | ✓ | 3 | |
| 13.d | odd | 4 | 1 | 1014.2.a.o | yes | 3 | |
| 13.e | even | 6 | 2 | 1014.2.i.g | 12 | ||
| 13.f | odd | 12 | 2 | 1014.2.e.k | 6 | ||
| 13.f | odd | 12 | 2 | 1014.2.e.m | 6 | ||
| 39.d | odd | 2 | 1 | 3042.2.b.r | 6 | ||
| 39.f | even | 4 | 1 | 3042.2.a.bd | 3 | ||
| 39.f | even | 4 | 1 | 3042.2.a.be | 3 | ||
| 52.f | even | 4 | 1 | 8112.2.a.bz | 3 | ||
| 52.f | even | 4 | 1 | 8112.2.a.ce | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1014.2.a.m | ✓ | 3 | 13.d | odd | 4 | 1 | |
| 1014.2.a.o | yes | 3 | 13.d | odd | 4 | 1 | |
| 1014.2.b.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 1014.2.b.g | 6 | 13.b | even | 2 | 1 | inner | |
| 1014.2.e.k | 6 | 13.f | odd | 12 | 2 | ||
| 1014.2.e.m | 6 | 13.f | odd | 12 | 2 | ||
| 1014.2.i.g | 12 | 13.c | even | 3 | 2 | ||
| 1014.2.i.g | 12 | 13.e | even | 6 | 2 | ||
| 3042.2.a.bd | 3 | 39.f | even | 4 | 1 | ||
| 3042.2.a.be | 3 | 39.f | even | 4 | 1 | ||
| 3042.2.b.r | 6 | 3.b | odd | 2 | 1 | ||
| 3042.2.b.r | 6 | 39.d | odd | 2 | 1 | ||
| 8112.2.a.bz | 3 | 52.f | even | 4 | 1 | ||
| 8112.2.a.ce | 3 | 52.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 17T_{5}^{4} + 10T_{5}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1014, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{3} \)
$3$
\( (T - 1)^{6} \)
$5$
\( T^{6} + 17 T^{4} + 10 T^{2} + 1 \)
$7$
\( T^{6} + 17 T^{4} + 10 T^{2} + 1 \)
$11$
\( T^{6} + 33 T^{4} + 230 T^{2} + \cdots + 169 \)
$13$
\( T^{6} \)
$17$
\( (T^{3} + 12 T^{2} + 20 T - 104)^{2} \)
$19$
\( T^{6} + 80 T^{4} + 1536 T^{2} + \cdots + 4096 \)
$23$
\( (T^{3} + 16 T^{2} + 76 T + 104)^{2} \)
$29$
\( (T^{3} - 13 T^{2} + 12 T + 223)^{2} \)
$31$
\( T^{6} + 125 T^{4} + 1006 T^{2} + \cdots + 841 \)
$37$
\( T^{6} + 104 T^{4} + 208 T^{2} + \cdots + 64 \)
$41$
\( T^{6} + 84 T^{4} + 1568 T^{2} + \cdots + 3136 \)
$43$
\( (T^{3} - 8 T^{2} - 44 T + 344)^{2} \)
$47$
\( T^{6} + 80 T^{4} + 1536 T^{2} + \cdots + 4096 \)
$53$
\( (T^{3} - 15 T^{2} - 72 T + 1247)^{2} \)
$59$
\( T^{6} + 41 T^{4} + 166 T^{2} + \cdots + 169 \)
$61$
\( (T^{3} + 10 T^{2} + 24 T + 8)^{2} \)
$67$
\( T^{6} + 404 T^{4} + 47200 T^{2} + \cdots + 1236544 \)
$71$
\( T^{6} + 180 T^{4} + 6432 T^{2} + \cdots + 10816 \)
$73$
\( T^{6} + 69 T^{4} + 614 T^{2} + \cdots + 169 \)
$79$
\( (T^{3} + 5 T^{2} - 204 T - 1469)^{2} \)
$83$
\( T^{6} + 497 T^{4} + 70854 T^{2} + \cdots + 2181529 \)
$89$
\( T^{6} + 52 T^{4} + 416 T^{2} + \cdots + 64 \)
$97$
\( T^{6} + 77 T^{4} + 294 T^{2} + \cdots + 49 \)
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