Newspace parameters
| Level: | \( N \) | \(=\) | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 294.g (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(30.3907691467\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−1.41421 | + | 2.44949i | −4.50000 | + | 2.59808i | −4.00000 | − | 6.92820i | 14.4853 | + | 8.36308i | − | 14.6969i | 0 | 22.6274 | 13.5000 | − | 23.3827i | −40.9706 | + | 23.6544i | |||||||||||||||||
| 19.2 | 1.41421 | − | 2.44949i | −4.50000 | + | 2.59808i | −4.00000 | − | 6.92820i | −2.48528 | − | 1.43488i | 14.6969i | 0 | −22.6274 | 13.5000 | − | 23.3827i | −7.02944 | + | 4.05845i | |||||||||||||||||||
| 31.1 | −1.41421 | − | 2.44949i | −4.50000 | − | 2.59808i | −4.00000 | + | 6.92820i | 14.4853 | − | 8.36308i | 14.6969i | 0 | 22.6274 | 13.5000 | + | 23.3827i | −40.9706 | − | 23.6544i | |||||||||||||||||||
| 31.2 | 1.41421 | + | 2.44949i | −4.50000 | − | 2.59808i | −4.00000 | + | 6.92820i | −2.48528 | + | 1.43488i | − | 14.6969i | 0 | −22.6274 | 13.5000 | + | 23.3827i | −7.02944 | − | 4.05845i | ||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 294.5.g.a | 4 | |
| 7.b | odd | 2 | 1 | 294.5.g.b | 4 | ||
| 7.c | even | 3 | 1 | 42.5.c.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | 294.5.g.b | 4 | ||
| 7.d | odd | 6 | 1 | 42.5.c.a | ✓ | 4 | |
| 7.d | odd | 6 | 1 | inner | 294.5.g.a | 4 | |
| 21.g | even | 6 | 1 | 126.5.c.b | 4 | ||
| 21.h | odd | 6 | 1 | 126.5.c.b | 4 | ||
| 28.f | even | 6 | 1 | 336.5.f.a | 4 | ||
| 28.g | odd | 6 | 1 | 336.5.f.a | 4 | ||
| 35.i | odd | 6 | 1 | 1050.5.f.a | 4 | ||
| 35.j | even | 6 | 1 | 1050.5.f.a | 4 | ||
| 35.k | even | 12 | 2 | 1050.5.h.a | 8 | ||
| 35.l | odd | 12 | 2 | 1050.5.h.a | 8 | ||
| 84.j | odd | 6 | 1 | 1008.5.f.f | 4 | ||
| 84.n | even | 6 | 1 | 1008.5.f.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.5.c.a | ✓ | 4 | 7.c | even | 3 | 1 | |
| 42.5.c.a | ✓ | 4 | 7.d | odd | 6 | 1 | |
| 126.5.c.b | 4 | 21.g | even | 6 | 1 | ||
| 126.5.c.b | 4 | 21.h | odd | 6 | 1 | ||
| 294.5.g.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 294.5.g.a | 4 | 7.d | odd | 6 | 1 | inner | |
| 294.5.g.b | 4 | 7.b | odd | 2 | 1 | ||
| 294.5.g.b | 4 | 7.c | even | 3 | 1 | ||
| 336.5.f.a | 4 | 28.f | even | 6 | 1 | ||
| 336.5.f.a | 4 | 28.g | odd | 6 | 1 | ||
| 1008.5.f.f | 4 | 84.j | odd | 6 | 1 | ||
| 1008.5.f.f | 4 | 84.n | even | 6 | 1 | ||
| 1050.5.f.a | 4 | 35.i | odd | 6 | 1 | ||
| 1050.5.f.a | 4 | 35.j | even | 6 | 1 | ||
| 1050.5.h.a | 8 | 35.k | even | 12 | 2 | ||
| 1050.5.h.a | 8 | 35.l | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 24T_{5}^{3} + 144T_{5}^{2} + 1152T_{5} + 2304 \)
acting on \(S_{5}^{\mathrm{new}}(294, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 8T^{2} + 64 \)
|
| $3$ |
\( (T^{2} + 9 T + 27)^{2} \)
|
| $5$ |
\( T^{4} - 24 T^{3} + \cdots + 2304 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 204 T^{3} + \cdots + 13749264 \)
|
| $13$ |
\( T^{4} + 103968 T^{2} + 300259584 \)
|
| $17$ |
\( T^{4} - 480 T^{3} + \cdots + 282240000 \)
|
| $19$ |
\( T^{4} + \cdots + 11903682816 \)
|
| $23$ |
\( T^{4} + \cdots + 5014339344 \)
|
| $29$ |
\( (T^{2} + 924 T - 100188)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 3537890141184 \)
|
| $37$ |
\( T^{4} + \cdots + 683208046096 \)
|
| $41$ |
\( T^{4} + \cdots + 25205540742144 \)
|
| $43$ |
\( (T^{2} - 308 T - 1553372)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 25374512185344 \)
|
| $53$ |
\( T^{4} + \cdots + 15522622895376 \)
|
| $59$ |
\( T^{4} + \cdots + 997521767610624 \)
|
| $61$ |
\( T^{4} + \cdots + 363850436618496 \)
|
| $67$ |
\( T^{4} + \cdots + 4223337365776 \)
|
| $71$ |
\( (T^{2} + 8076 T + 16079652)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 18235222953984 \)
|
| $79$ |
\( T^{4} + \cdots + 4830834743056 \)
|
| $83$ |
\( T^{4} + \cdots + 28\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots + 36\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots + 425280742096896 \)
|
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