Newspace parameters
| Level: | \( N \) | \(=\) | \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1368.ba (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.682720937282\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\zeta_{6})\) |
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|
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| Defining polynomial: |
\( x^{2} - x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{3}\) |
| Projective field: | Galois closure of 3.1.233928.2 |
| Artin image: | $C_3\times S_3$ |
| Artin field: | Galois closure of 6.0.14971392.2 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).
| \(n\) | \(343\) | \(685\) | \(1009\) | \(1217\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-\zeta_{6}\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 619.1 |
|
−0.500000 | − | 0.866025i | 1.00000 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 0 | 1.00000 | 1.00000 | 0 | ||||||||||||||||||
| 1147.1 | −0.500000 | + | 0.866025i | 1.00000 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0 | 1.00000 | 1.00000 | 0 | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
| 171.g | even | 3 | 1 | inner |
| 1368.ba | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1368.1.ba.b | ✓ | 2 |
| 8.d | odd | 2 | 1 | CM | 1368.1.ba.b | ✓ | 2 |
| 9.c | even | 3 | 1 | 1368.1.cj.a | yes | 2 | |
| 19.c | even | 3 | 1 | 1368.1.cj.a | yes | 2 | |
| 72.p | odd | 6 | 1 | 1368.1.cj.a | yes | 2 | |
| 152.k | odd | 6 | 1 | 1368.1.cj.a | yes | 2 | |
| 171.g | even | 3 | 1 | inner | 1368.1.ba.b | ✓ | 2 |
| 1368.ba | odd | 6 | 1 | inner | 1368.1.ba.b | ✓ | 2 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1368.1.ba.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 1368.1.ba.b | ✓ | 2 | 8.d | odd | 2 | 1 | CM |
| 1368.1.ba.b | ✓ | 2 | 171.g | even | 3 | 1 | inner |
| 1368.1.ba.b | ✓ | 2 | 1368.ba | odd | 6 | 1 | inner |
| 1368.1.cj.a | yes | 2 | 9.c | even | 3 | 1 | |
| 1368.1.cj.a | yes | 2 | 19.c | even | 3 | 1 | |
| 1368.1.cj.a | yes | 2 | 72.p | odd | 6 | 1 | |
| 1368.1.cj.a | yes | 2 | 152.k | odd | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1368, [\chi])\):
|
\( T_{5} \)
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\( T_{11}^{2} + 2T_{11} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + T + 1 \)
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| $3$ |
\( (T - 1)^{2} \)
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| $5$ |
\( T^{2} \)
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| $7$ |
\( T^{2} \)
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| $11$ |
\( T^{2} + 2T + 4 \)
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| $13$ |
\( T^{2} \)
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| $17$ |
\( T^{2} - T + 1 \)
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| $19$ |
\( (T - 1)^{2} \)
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| $23$ |
\( T^{2} \)
|
| $29$ |
\( T^{2} \)
|
| $31$ |
\( T^{2} \)
|
| $37$ |
\( T^{2} \)
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| $41$ |
\( (T + 1)^{2} \)
|
| $43$ |
\( T^{2} - T + 1 \)
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| $47$ |
\( T^{2} \)
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| $53$ |
\( T^{2} \)
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| $59$ |
\( (T + 1)^{2} \)
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| $61$ |
\( T^{2} \)
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| $67$ |
\( T^{2} - T + 1 \)
|
| $71$ |
\( T^{2} \)
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| $73$ |
\( T^{2} + 2T + 4 \)
|
| $79$ |
\( T^{2} \)
|
| $83$ |
\( T^{2} - T + 1 \)
|
| $89$ |
\( T^{2} + 2T + 4 \)
|
| $97$ |
\( T^{2} + 2T + 4 \)
|
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