Newspace parameters
| Level: | \( N \) | \(=\) | \( 2034 = 2 \cdot 3^{2} \cdot 113 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2034.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(16.2415717711\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{13})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - x^{2} + 2x + 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 226) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} - 2x^{3} - x^{2} + 2x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 9\nu^{2} - 7\nu - 12 ) / 21 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} - 7\nu + 3 ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{3} + 6\nu^{2} + 28\nu - 15 ) / 21 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 4\beta _1 + 3 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} - 7\beta_{2} + 3\beta _1 + 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2034\mathbb{Z}\right)^\times\).
| \(n\) | \(227\) | \(1585\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 451.1 |
|
−1.00000 | 0 | 1.00000 | − | 1.41421i | 0 | −2.60555 | −1.00000 | 0 | 1.41421i | |||||||||||||||||||||||||||||
| 451.2 | −1.00000 | 0 | 1.00000 | − | 1.41421i | 0 | 4.60555 | −1.00000 | 0 | 1.41421i | ||||||||||||||||||||||||||||||
| 451.3 | −1.00000 | 0 | 1.00000 | 1.41421i | 0 | −2.60555 | −1.00000 | 0 | − | 1.41421i | ||||||||||||||||||||||||||||||
| 451.4 | −1.00000 | 0 | 1.00000 | 1.41421i | 0 | 4.60555 | −1.00000 | 0 | − | 1.41421i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 113.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2034.2.c.e | 4 | |
| 3.b | odd | 2 | 1 | 226.2.b.d | ✓ | 4 | |
| 12.b | even | 2 | 1 | 1808.2.c.d | 4 | ||
| 113.b | even | 2 | 1 | inner | 2034.2.c.e | 4 | |
| 339.c | odd | 2 | 1 | 226.2.b.d | ✓ | 4 | |
| 1356.g | even | 2 | 1 | 1808.2.c.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 226.2.b.d | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 226.2.b.d | ✓ | 4 | 339.c | odd | 2 | 1 | |
| 1808.2.c.d | 4 | 12.b | even | 2 | 1 | ||
| 1808.2.c.d | 4 | 1356.g | even | 2 | 1 | ||
| 2034.2.c.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 2034.2.c.e | 4 | 113.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2034, [\chi])\):
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\( T_{5}^{2} + 2 \)
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\( T_{7}^{2} - 2T_{7} - 12 \)
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\( T_{11}^{2} + 2T_{11} - 12 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
|
| $3$ |
\( T^{4} \)
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| $5$ |
\( (T^{2} + 2)^{2} \)
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| $7$ |
\( (T^{2} - 2 T - 12)^{2} \)
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| $11$ |
\( (T^{2} + 2 T - 12)^{2} \)
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| $13$ |
\( (T + 4)^{4} \)
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| $17$ |
\( (T^{2} + 32)^{2} \)
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| $19$ |
\( T^{4} + 62T^{2} + 324 \)
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| $23$ |
\( T^{4} + 38T^{2} + 36 \)
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| $29$ |
\( (T^{2} + 26)^{2} \)
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| $31$ |
\( (T^{2} - 6 T - 4)^{2} \)
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| $37$ |
\( T^{4} + 68T^{2} + 324 \)
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| $41$ |
\( (T^{2} - 2 T - 12)^{2} \)
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| $43$ |
\( T^{4} + 38T^{2} + 36 \)
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| $47$ |
\( T^{4} + 166T^{2} + 1156 \)
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| $53$ |
\( T^{4} \)
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| $59$ |
\( T^{4} + 94T^{2} + 1156 \)
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| $61$ |
\( (T + 10)^{4} \)
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| $67$ |
\( T^{4} + 302 T^{2} + 19044 \)
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| $71$ |
\( T^{4} + 22T^{2} + 4 \)
|
| $73$ |
\( T^{4} + 272T^{2} + 5184 \)
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| $79$ |
\( T^{4} + 14T^{2} + 36 \)
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| $83$ |
\( (T^{2} - 26 T + 156)^{2} \)
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| $89$ |
\( (T^{2} + 32)^{2} \)
|
| $97$ |
\( (T^{2} - 208)^{2} \)
|
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