Newspace parameters
| Level: | \( N \) | \(=\) | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 294.f (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.34760181943\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 42) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{4} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{24}^{5} + \zeta_{24} \)
|
| \(\beta_{4}\) | \(=\) |
\( \zeta_{24}^{6} \)
|
| \(\beta_{5}\) | \(=\) |
\( \zeta_{24}^{7} + \zeta_{24}^{3} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{24}^{5} + 2\zeta_{24} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{24}^{7} + 2\zeta_{24}^{3} \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{6} + \beta_{3} ) / 3 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( ( \beta_{7} + \beta_{5} ) / 3 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( -\beta_{6} + 2\beta_{3} ) / 3 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( \beta_{4} \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( -\beta_{7} + 2\beta_{5} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(199\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 215.1 |
|
−0.866025 | − | 0.500000i | −1.67303 | − | 0.448288i | 0.500000 | + | 0.866025i | 1.22474 | − | 2.12132i | 1.22474 | + | 1.22474i | 0 | − | 1.00000i | 2.59808 | + | 1.50000i | −2.12132 | + | 1.22474i | |||||||||||||||||||||||||||
| 215.2 | −0.866025 | − | 0.500000i | 1.67303 | + | 0.448288i | 0.500000 | + | 0.866025i | −1.22474 | + | 2.12132i | −1.22474 | − | 1.22474i | 0 | − | 1.00000i | 2.59808 | + | 1.50000i | 2.12132 | − | 1.22474i | ||||||||||||||||||||||||||||
| 215.3 | 0.866025 | + | 0.500000i | −0.448288 | + | 1.67303i | 0.500000 | + | 0.866025i | −1.22474 | + | 2.12132i | −1.22474 | + | 1.22474i | 0 | 1.00000i | −2.59808 | − | 1.50000i | −2.12132 | + | 1.22474i | |||||||||||||||||||||||||||||
| 215.4 | 0.866025 | + | 0.500000i | 0.448288 | − | 1.67303i | 0.500000 | + | 0.866025i | 1.22474 | − | 2.12132i | 1.22474 | − | 1.22474i | 0 | 1.00000i | −2.59808 | − | 1.50000i | 2.12132 | − | 1.22474i | |||||||||||||||||||||||||||||
| 227.1 | −0.866025 | + | 0.500000i | −1.67303 | + | 0.448288i | 0.500000 | − | 0.866025i | 1.22474 | + | 2.12132i | 1.22474 | − | 1.22474i | 0 | 1.00000i | 2.59808 | − | 1.50000i | −2.12132 | − | 1.22474i | |||||||||||||||||||||||||||||
| 227.2 | −0.866025 | + | 0.500000i | 1.67303 | − | 0.448288i | 0.500000 | − | 0.866025i | −1.22474 | − | 2.12132i | −1.22474 | + | 1.22474i | 0 | 1.00000i | 2.59808 | − | 1.50000i | 2.12132 | + | 1.22474i | |||||||||||||||||||||||||||||
| 227.3 | 0.866025 | − | 0.500000i | −0.448288 | − | 1.67303i | 0.500000 | − | 0.866025i | −1.22474 | − | 2.12132i | −1.22474 | − | 1.22474i | 0 | − | 1.00000i | −2.59808 | + | 1.50000i | −2.12132 | − | 1.22474i | ||||||||||||||||||||||||||||
| 227.4 | 0.866025 | − | 0.500000i | 0.448288 | + | 1.67303i | 0.500000 | − | 0.866025i | 1.22474 | + | 2.12132i | 1.22474 | + | 1.22474i | 0 | − | 1.00000i | −2.59808 | + | 1.50000i | 2.12132 | + | 1.22474i | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
| 21.g | even | 6 | 1 | inner |
| 21.h | odd | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 6T_{5}^{2} + 36 \)
acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{2} + 1)^{2} \)
|
| $3$ |
\( T^{8} - 9T^{4} + 81 \)
|
| $5$ |
\( (T^{4} + 6 T^{2} + 36)^{2} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( (T^{2} + 6)^{4} \)
|
| $17$ |
\( (T^{4} + 24 T^{2} + 576)^{2} \)
|
| $19$ |
\( (T^{4} - 6 T^{2} + 36)^{2} \)
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| $23$ |
\( (T^{4} - 36 T^{2} + 1296)^{2} \)
|
| $29$ |
\( (T^{2} + 36)^{4} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( (T^{2} - 2 T + 4)^{4} \)
|
| $41$ |
\( (T^{2} - 24)^{4} \)
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| $43$ |
\( (T - 4)^{8} \)
|
| $47$ |
\( (T^{4} + 24 T^{2} + 576)^{2} \)
|
| $53$ |
\( (T^{4} - 36 T^{2} + 1296)^{2} \)
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| $59$ |
\( (T^{4} + 150 T^{2} + 22500)^{2} \)
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| $61$ |
\( (T^{4} - 150 T^{2} + 22500)^{2} \)
|
| $67$ |
\( (T^{2} + 8 T + 64)^{4} \)
|
| $71$ |
\( T^{8} \)
|
| $73$ |
\( (T^{4} - 96 T^{2} + 9216)^{2} \)
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| $79$ |
\( (T^{2} - 10 T + 100)^{4} \)
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| $83$ |
\( (T^{2} - 6)^{4} \)
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| $89$ |
\( T^{8} \)
|
| $97$ |
\( (T^{2} + 24)^{4} \)
|
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