Newspace parameters
| Level: | \( N \) | \(=\) | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 294.e (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(47.1528430250\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{4705})\) |
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|
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| Defining polynomial: |
\( x^{4} - x^{3} + 1177x^{2} + 1176x + 1382976 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} - x^{3} + 1177x^{2} + 1176x + 1382976 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 1177\nu^{2} - 1177\nu + 1382976 ) / 1384152 \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 1177\nu^{2} + 2769481\nu - 1382976 ) / 1384152 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + 3529 ) / 1177 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 2353\beta _1 - 2353 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1177\beta_{3} - 3529 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(199\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
−2.00000 | − | 3.46410i | −4.50000 | + | 7.79423i | −8.00000 | + | 13.8564i | −38.7965 | − | 67.1975i | 36.0000 | 0 | 64.0000 | −40.5000 | − | 70.1481i | −155.186 | + | 268.790i | ||||||||||||||||||
| 67.2 | −2.00000 | − | 3.46410i | −4.50000 | + | 7.79423i | −8.00000 | + | 13.8564i | 29.7965 | + | 51.6091i | 36.0000 | 0 | 64.0000 | −40.5000 | − | 70.1481i | 119.186 | − | 206.436i | |||||||||||||||||||
| 79.1 | −2.00000 | + | 3.46410i | −4.50000 | − | 7.79423i | −8.00000 | − | 13.8564i | −38.7965 | + | 67.1975i | 36.0000 | 0 | 64.0000 | −40.5000 | + | 70.1481i | −155.186 | − | 268.790i | |||||||||||||||||||
| 79.2 | −2.00000 | + | 3.46410i | −4.50000 | − | 7.79423i | −8.00000 | − | 13.8564i | 29.7965 | − | 51.6091i | 36.0000 | 0 | 64.0000 | −40.5000 | + | 70.1481i | 119.186 | + | 206.436i | |||||||||||||||||||
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 | 294.6.e.t | 4 | |
| 7.b | odd | 2 | 1 | 294.6.e.w | 4 | ||
| 7.c | even | 3 | 1 | 294.6.a.v | yes | 2 | |
| 7.c | even | 3 | 1 | inner | 294.6.e.t | 4 | |
| 7.d | odd | 6 | 1 | 294.6.a.s | ✓ | 2 | |
| 7.d | odd | 6 | 1 | 294.6.e.w | 4 | ||
| 21.g | even | 6 | 1 | 882.6.a.bg | 2 | ||
| 21.h | odd | 6 | 1 | 882.6.a.bc | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 294.6.a.s | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 294.6.a.v | yes | 2 | 7.c | even | 3 | 1 | |
| 294.6.e.t | 4 | 1.a | even | 1 | 1 | trivial | |
| 294.6.e.t | 4 | 7.c | even | 3 | 1 | inner | |
| 294.6.e.w | 4 | 7.b | odd | 2 | 1 | ||
| 294.6.e.w | 4 | 7.d | odd | 6 | 1 | ||
| 882.6.a.bc | 2 | 21.h | odd | 6 | 1 | ||
| 882.6.a.bg | 2 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(294, [\chi])\):
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\( T_{5}^{4} + 18T_{5}^{3} + 4948T_{5}^{2} - 83232T_{5} + 21381376 \)
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\( T_{11}^{4} + 2T_{11}^{3} + 381108T_{11}^{2} - 762208T_{11} + 145240258816 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4 T + 16)^{2} \)
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| $3$ |
\( (T^{2} + 9 T + 81)^{2} \)
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| $5$ |
\( T^{4} + 18 T^{3} + \cdots + 21381376 \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( T^{4} + \cdots + 145240258816 \)
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| $13$ |
\( (T^{2} - 288 T - 54544)^{2} \)
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| $17$ |
\( T^{4} + \cdots + 224107560000 \)
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| $19$ |
\( T^{4} + \cdots + 13844816896 \)
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| $23$ |
\( T^{4} + \cdots + 44283701160000 \)
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| $29$ |
\( (T^{2} + 3976 T - 9767636)^{2} \)
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| $31$ |
\( T^{4} + \cdots + 19972175512576 \)
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| $37$ |
\( T^{4} + \cdots + 141930147941136 \)
|
| $41$ |
\( (T^{2} - 36630 T + 328282920)^{2} \)
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| $43$ |
\( (T^{2} + 23032 T + 35055376)^{2} \)
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| $47$ |
\( T^{4} + \cdots + 16\!\cdots\!36 \)
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| $53$ |
\( T^{4} + \cdots + 21\!\cdots\!00 \)
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| $59$ |
\( T^{4} + \cdots + 53\!\cdots\!96 \)
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| $61$ |
\( T^{4} + \cdots + 21\!\cdots\!36 \)
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| $67$ |
\( T^{4} + \cdots + 22\!\cdots\!56 \)
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| $71$ |
\( (T^{2} - 73706 T - 252025016)^{2} \)
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| $73$ |
\( T^{4} + \cdots + 23\!\cdots\!56 \)
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| $79$ |
\( T^{4} + \cdots + 91\!\cdots\!96 \)
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| $83$ |
\( (T^{2} - 147816 T + 5438001744)^{2} \)
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| $89$ |
\( T^{4} + \cdots + 45\!\cdots\!76 \)
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| $97$ |
\( (T^{2} - 162036 T + 3353807744)^{2} \)
|
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