Newspace parameters
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.g (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(7.43424066072\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{193})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 49\nu^{2} - 49\nu + 2304 ) / 2352 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 97 ) / 49 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 48\beta_{2} + \beta _1 - 49 \)
|
| \(\nu^{3}\) | \(=\) |
\( 49\beta_{3} - 97 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −1.72311 | + | 2.98452i | 0 | 15.3924 | − | 10.2992i | 8.00000 | 0 | −3.44622 | − | 5.96903i | ||||||||||||||||||||
| 37.2 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 5.22311 | − | 9.04669i | 0 | −12.3924 | + | 13.7633i | 8.00000 | 0 | 10.4462 | + | 18.0934i | |||||||||||||||||||||
| 109.1 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −1.72311 | − | 2.98452i | 0 | 15.3924 | + | 10.2992i | 8.00000 | 0 | −3.44622 | + | 5.96903i | |||||||||||||||||||||
| 109.2 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 5.22311 | + | 9.04669i | 0 | −12.3924 | − | 13.7633i | 8.00000 | 0 | 10.4462 | − | 18.0934i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 126.4.g.e | ✓ | 4 |
| 3.b | odd | 2 | 1 | 126.4.g.f | yes | 4 | |
| 7.b | odd | 2 | 1 | 882.4.g.z | 4 | ||
| 7.c | even | 3 | 1 | inner | 126.4.g.e | ✓ | 4 |
| 7.c | even | 3 | 1 | 882.4.a.bd | 2 | ||
| 7.d | odd | 6 | 1 | 882.4.a.bh | 2 | ||
| 7.d | odd | 6 | 1 | 882.4.g.z | 4 | ||
| 21.c | even | 2 | 1 | 882.4.g.bj | 4 | ||
| 21.g | even | 6 | 1 | 882.4.a.u | 2 | ||
| 21.g | even | 6 | 1 | 882.4.g.bj | 4 | ||
| 21.h | odd | 6 | 1 | 126.4.g.f | yes | 4 | |
| 21.h | odd | 6 | 1 | 882.4.a.ba | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.4.g.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 126.4.g.e | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 126.4.g.f | yes | 4 | 3.b | odd | 2 | 1 | |
| 126.4.g.f | yes | 4 | 21.h | odd | 6 | 1 | |
| 882.4.a.u | 2 | 21.g | even | 6 | 1 | ||
| 882.4.a.ba | 2 | 21.h | odd | 6 | 1 | ||
| 882.4.a.bd | 2 | 7.c | even | 3 | 1 | ||
| 882.4.a.bh | 2 | 7.d | odd | 6 | 1 | ||
| 882.4.g.z | 4 | 7.b | odd | 2 | 1 | ||
| 882.4.g.z | 4 | 7.d | odd | 6 | 1 | ||
| 882.4.g.bj | 4 | 21.c | even | 2 | 1 | ||
| 882.4.g.bj | 4 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 7T_{5}^{3} + 85T_{5}^{2} + 252T_{5} + 1296 \)
acting on \(S_{4}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2 T + 4)^{2} \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 7 T^{3} + 85 T^{2} + \cdots + 1296 \)
$7$
\( T^{4} - 6 T^{3} - 77 T^{2} + \cdots + 117649 \)
$11$
\( T^{4} + 25 T^{3} + 2833 T^{2} + \cdots + 4875264 \)
$13$
\( (T^{2} + 49 T - 606)^{2} \)
$17$
\( T^{4} - 98 T^{3} + 12028 T^{2} + \cdots + 5875776 \)
$19$
\( T^{4} - 119 T^{3} + 11055 T^{2} + \cdots + 9647236 \)
$23$
\( T^{4} - 122 T^{3} + \cdots + 32901696 \)
$29$
\( (T^{2} + 73 T - 1032)^{2} \)
$31$
\( T^{4} + 98 T^{3} + \cdots + 1255072329 \)
$37$
\( T^{4} + 289 T^{3} + 83919 T^{2} + \cdots + 158404 \)
$41$
\( (T + 168)^{4} \)
$43$
\( (T^{2} - 307 T + 2284)^{2} \)
$47$
\( T^{4} - 672 T^{3} + \cdots + 7242690816 \)
$53$
\( T^{4} + 375 T^{3} + \cdots + 24444697104 \)
$59$
\( T^{4} - 763 T^{3} + \cdots + 1155728016 \)
$61$
\( T^{4} - 406 T^{3} + \cdots + 212226624 \)
$67$
\( T^{4} - 1041 T^{3} + \cdots + 44865170596 \)
$71$
\( (T^{2} + 1652 T + 644448)^{2} \)
$73$
\( T^{4} - 189 T^{3} + \cdots + 198073062916 \)
$79$
\( T^{4} + 524 T^{3} + \cdots + 155826773001 \)
$83$
\( (T^{2} + 287 T - 361596)^{2} \)
$89$
\( T^{4} - 2394 T^{3} + \cdots + 1669553420544 \)
$97$
\( (T^{2} + 63 T - 1158214)^{2} \)
show more
show less