Newspace parameters
| Level: | \( N \) | \(=\) | \( 494 = 2 \cdot 13 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 494.x (of order \(9\), degree \(6\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(3.94460985985\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{18})\) |
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| Defining polynomial: |
\( x^{6} - x^{3} + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/494\mathbb{Z}\right)^\times\).
| \(n\) | \(287\) | \(457\) |
| \(\chi(n)\) | \(-\zeta_{18}^{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 131.1 |
|
0.173648 | − | 0.984808i | 1.17365 | + | 0.984808i | −0.939693 | − | 0.342020i | 2.20574 | − | 0.802823i | 1.17365 | − | 0.984808i | −1.26604 | − | 2.19285i | −0.500000 | + | 0.866025i | −0.113341 | − | 0.642788i | −0.407604 | − | 2.31164i | ||||||||||||||||||
| 157.1 | −0.939693 | + | 0.342020i | 0.0603074 | − | 0.342020i | 0.766044 | − | 0.642788i | −0.0923963 | − | 0.0775297i | 0.0603074 | + | 0.342020i | −0.673648 | − | 1.16679i | −0.500000 | + | 0.866025i | 2.70574 | + | 0.984808i | 0.113341 | + | 0.0412527i | |||||||||||||||||||
| 313.1 | 0.173648 | + | 0.984808i | 1.17365 | − | 0.984808i | −0.939693 | + | 0.342020i | 2.20574 | + | 0.802823i | 1.17365 | + | 0.984808i | −1.26604 | + | 2.19285i | −0.500000 | − | 0.866025i | −0.113341 | + | 0.642788i | −0.407604 | + | 2.31164i | |||||||||||||||||||
| 339.1 | 0.766044 | + | 0.642788i | 1.76604 | − | 0.642788i | 0.173648 | + | 0.984808i | −0.613341 | + | 3.47843i | 1.76604 | + | 0.642788i | 0.439693 | + | 0.761570i | −0.500000 | + | 0.866025i | 0.407604 | − | 0.342020i | −2.70574 | + | 2.27038i | |||||||||||||||||||
| 365.1 | −0.939693 | − | 0.342020i | 0.0603074 | + | 0.342020i | 0.766044 | + | 0.642788i | −0.0923963 | + | 0.0775297i | 0.0603074 | − | 0.342020i | −0.673648 | + | 1.16679i | −0.500000 | − | 0.866025i | 2.70574 | − | 0.984808i | 0.113341 | − | 0.0412527i | |||||||||||||||||||
| 443.1 | 0.766044 | − | 0.642788i | 1.76604 | + | 0.642788i | 0.173648 | − | 0.984808i | −0.613341 | − | 3.47843i | 1.76604 | − | 0.642788i | 0.439693 | − | 0.761570i | −0.500000 | − | 0.866025i | 0.407604 | + | 0.342020i | −2.70574 | − | 2.27038i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.e | even | 9 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 494.2.x.a | ✓ | 6 |
| 19.e | even | 9 | 1 | inner | 494.2.x.a | ✓ | 6 |
| 19.e | even | 9 | 1 | 9386.2.a.ba | 3 | ||
| 19.f | odd | 18 | 1 | 9386.2.a.x | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 494.2.x.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 494.2.x.a | ✓ | 6 | 19.e | even | 9 | 1 | inner |
| 9386.2.a.x | 3 | 19.f | odd | 18 | 1 | ||
| 9386.2.a.ba | 3 | 19.e | even | 9 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 6T_{3}^{5} + 15T_{3}^{4} - 19T_{3}^{3} + 12T_{3}^{2} - 3T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(494, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{3} + 1 \)
|
| $3$ |
\( T^{6} - 6 T^{5} + \cdots + 1 \)
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| $5$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
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| $7$ |
\( T^{6} + 3 T^{5} + \cdots + 9 \)
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| $11$ |
\( T^{6} + 3 T^{5} + \cdots + 289 \)
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| $13$ |
\( T^{6} + T^{3} + 1 \)
|
| $17$ |
\( T^{6} - 9 T^{5} + \cdots + 6561 \)
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| $19$ |
\( T^{6} - 9 T^{5} + \cdots + 6859 \)
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| $23$ |
\( T^{6} - 9 T^{5} + \cdots + 1369 \)
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| $29$ |
\( T^{6} - 27 T^{5} + \cdots + 2809 \)
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| $31$ |
\( T^{6} - 15 T^{5} + \cdots + 1369 \)
|
| $37$ |
\( (T^{3} - 48 T - 64)^{2} \)
|
| $41$ |
\( T^{6} - 12 T^{5} + \cdots + 64 \)
|
| $43$ |
\( T^{6} - 3 T^{5} + \cdots + 361 \)
|
| $47$ |
\( T^{6} - 42 T^{5} + \cdots + 23104 \)
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| $53$ |
\( T^{6} + 18 T^{5} + \cdots + 1369 \)
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| $59$ |
\( T^{6} + 6 T^{5} + \cdots + 9 \)
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| $61$ |
\( T^{6} + 3 T^{5} + \cdots + 361 \)
|
| $67$ |
\( T^{6} + 21 T^{5} + \cdots + 239121 \)
|
| $71$ |
\( T^{6} + 24 T^{5} + \cdots + 104329 \)
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| $73$ |
\( T^{6} + 39 T^{5} + \cdots + 848241 \)
|
| $79$ |
\( T^{6} + 30 T^{5} + \cdots + 597529 \)
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| $83$ |
\( T^{6} - 6 T^{5} + \cdots + 2601 \)
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| $89$ |
\( T^{6} + 27 T^{5} + \cdots + 110889 \)
|
| $97$ |
\( T^{6} + 42 T^{5} + \cdots + 5041 \)
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