Newspace parameters
| Level: | \( N \) | \(=\) | \( 3636 = 2^{2} \cdot 3^{2} \cdot 101 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3636.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.81460038593\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{14})^+\) |
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| Defining polynomial: |
\( x^{3} - x^{2} - 2x + 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 404) |
| Projective image: | \(D_{7}\) |
| Projective field: | Galois closure of 7.1.65939264.1 |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3636\mathbb{Z}\right)^\times\).
| \(n\) | \(1819\) | \(3133\) | \(3233\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2827.1 |
|
−1.00000 | 0 | 1.00000 | −1.24698 | 0 | −1.80194 | −1.00000 | 0 | 1.24698 | |||||||||||||||||||||||||||
| 2827.2 | −1.00000 | 0 | 1.00000 | 0.445042 | 0 | 1.24698 | −1.00000 | 0 | −0.445042 | ||||||||||||||||||||||||||||
| 2827.3 | −1.00000 | 0 | 1.00000 | 1.80194 | 0 | −0.445042 | −1.00000 | 0 | −1.80194 | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 404.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-101}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3636.1.h.b | 3 | |
| 3.b | odd | 2 | 1 | 404.1.d.b | yes | 3 | |
| 4.b | odd | 2 | 1 | 3636.1.h.c | 3 | ||
| 12.b | even | 2 | 1 | 404.1.d.a | ✓ | 3 | |
| 101.b | even | 2 | 1 | 3636.1.h.c | 3 | ||
| 303.d | odd | 2 | 1 | 404.1.d.a | ✓ | 3 | |
| 404.d | odd | 2 | 1 | CM | 3636.1.h.b | 3 | |
| 1212.d | even | 2 | 1 | 404.1.d.b | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 404.1.d.a | ✓ | 3 | 12.b | even | 2 | 1 | |
| 404.1.d.a | ✓ | 3 | 303.d | odd | 2 | 1 | |
| 404.1.d.b | yes | 3 | 3.b | odd | 2 | 1 | |
| 404.1.d.b | yes | 3 | 1212.d | even | 2 | 1 | |
| 3636.1.h.b | 3 | 1.a | even | 1 | 1 | trivial | |
| 3636.1.h.b | 3 | 404.d | odd | 2 | 1 | CM | |
| 3636.1.h.c | 3 | 4.b | odd | 2 | 1 | ||
| 3636.1.h.c | 3 | 101.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3636, [\chi])\):
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\( T_{5}^{3} - T_{5}^{2} - 2T_{5} + 1 \)
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\( T_{7}^{3} + T_{7}^{2} - 2T_{7} - 1 \)
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\( T_{11}^{3} - T_{11}^{2} - 2T_{11} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{3} \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} - T^{2} - 2T + 1 \)
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| $7$ |
\( T^{3} + T^{2} - 2T - 1 \)
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| $11$ |
\( T^{3} - T^{2} - 2T + 1 \)
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| $13$ |
\( T^{3} + T^{2} - 2T - 1 \)
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| $17$ |
\( T^{3} - T^{2} - 2T + 1 \)
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| $19$ |
\( T^{3} \)
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| $23$ |
\( T^{3} \)
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| $29$ |
\( T^{3} \)
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| $31$ |
\( T^{3} \)
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| $37$ |
\( T^{3} + T^{2} - 2T - 1 \)
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| $41$ |
\( T^{3} \)
|
| $43$ |
\( T^{3} \)
|
| $47$ |
\( T^{3} \)
|
| $53$ |
\( T^{3} \)
|
| $59$ |
\( T^{3} - T^{2} - 2T + 1 \)
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| $61$ |
\( T^{3} \)
|
| $67$ |
\( T^{3} + T^{2} - 2T - 1 \)
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| $71$ |
\( T^{3} \)
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| $73$ |
\( T^{3} \)
|
| $79$ |
\( T^{3} \)
|
| $83$ |
\( T^{3} - T^{2} - 2T + 1 \)
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| $89$ |
\( T^{3} \)
|
| $97$ |
\( T^{3} + T^{2} - 2T - 1 \)
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