Newspace parameters
| Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 42.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(13.1201710703\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{2881})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 721x^{2} + 720x + 518400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{4} - x^{3} + 721x^{2} + 720x + 518400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 721\nu^{2} - 721\nu + 518400 ) / 519120 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} + 1441\nu + 720 ) / 720 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 721\nu + 1441 ) / 721 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 2162\beta _1 - 2163 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1442\beta_{3} + 721\beta_{2} + 721\beta _1 - 4323 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
−4.00000 | − | 6.92820i | 13.5000 | − | 23.3827i | −32.0000 | + | 55.4256i | −198.019 | − | 342.978i | −216.000 | −346.587 | + | 838.702i | 512.000 | −364.500 | − | 631.333i | −1584.15 | + | 2743.83i | ||||||||||||||||
| 25.2 | −4.00000 | − | 6.92820i | 13.5000 | − | 23.3827i | −32.0000 | + | 55.4256i | 43.5186 | + | 75.3765i | −216.000 | 780.587 | − | 462.847i | 512.000 | −364.500 | − | 631.333i | 348.149 | − | 603.012i | |||||||||||||||||
| 37.1 | −4.00000 | + | 6.92820i | 13.5000 | + | 23.3827i | −32.0000 | − | 55.4256i | −198.019 | + | 342.978i | −216.000 | −346.587 | − | 838.702i | 512.000 | −364.500 | + | 631.333i | −1584.15 | − | 2743.83i | |||||||||||||||||
| 37.2 | −4.00000 | + | 6.92820i | 13.5000 | + | 23.3827i | −32.0000 | − | 55.4256i | 43.5186 | − | 75.3765i | −216.000 | 780.587 | + | 462.847i | 512.000 | −364.500 | + | 631.333i | 348.149 | + | 603.012i | |||||||||||||||||
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 | 42.8.e.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 126.8.g.f | 4 | ||
| 7.b | odd | 2 | 1 | 294.8.e.t | 4 | ||
| 7.c | even | 3 | 1 | inner | 42.8.e.c | ✓ | 4 |
| 7.c | even | 3 | 1 | 294.8.a.s | 2 | ||
| 7.d | odd | 6 | 1 | 294.8.a.t | 2 | ||
| 7.d | odd | 6 | 1 | 294.8.e.t | 4 | ||
| 21.h | odd | 6 | 1 | 126.8.g.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.8.e.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 42.8.e.c | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 126.8.g.f | 4 | 3.b | odd | 2 | 1 | ||
| 126.8.g.f | 4 | 21.h | odd | 6 | 1 | ||
| 294.8.a.s | 2 | 7.c | even | 3 | 1 | ||
| 294.8.a.t | 2 | 7.d | odd | 6 | 1 | ||
| 294.8.e.t | 4 | 7.b | odd | 2 | 1 | ||
| 294.8.e.t | 4 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 309T_{5}^{3} + 129951T_{5}^{2} - 10651230T_{5} + 1188180900 \)
acting on \(S_{8}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8 T + 64)^{2} \)
|
| $3$ |
\( (T^{2} - 27 T + 729)^{2} \)
|
| $5$ |
\( T^{4} + \cdots + 1188180900 \)
|
| $7$ |
\( T^{4} + \cdots + 678223072849 \)
|
| $11$ |
\( T^{4} + \cdots + 81729012968100 \)
|
| $13$ |
\( (T^{2} + 7139 T - 6867476)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 30\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots + 83\!\cdots\!84 \)
|
| $23$ |
\( T^{4} + \cdots + 57\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + 394359 T + 38382378660)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 82\!\cdots\!21 \)
|
| $37$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} - 567006 T + 80065888560)^{2} \)
|
| $43$ |
\( (T^{2} + 265505 T - 568015812050)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 19\!\cdots\!84 \)
|
| $53$ |
\( T^{4} + \cdots + 61\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots + 39\!\cdots\!04 \)
|
| $61$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $71$ |
\( (T^{2} + \cdots + 8314342748532)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 20\!\cdots\!24 \)
|
| $79$ |
\( T^{4} + \cdots + 47\!\cdots\!61 \)
|
| $83$ |
\( (T^{2} + \cdots - 13321578284910)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots + 68727373321558)^{2} \)
|
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