Newspace parameters
| Level: | \( N \) | \(=\) | \( 2034 = 2 \cdot 3^{2} \cdot 113 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2034.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(16.2415717711\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.77976.1 |
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|
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| Defining polynomial: |
\( x^{4} - x^{3} - 12x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 678) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - x^{3} - 12x^{2} - x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 10\nu + 5 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 12\nu - 1 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 2\beta_{2} + 2\beta _1 + 6 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} - 2\beta_{2} + 14\beta _1 + 7 \)
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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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | 0 | 1.00000 | −4.06815 | 0 | −1.93410 | 1.00000 | 0 | −4.06815 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | −1.20423 | 0 | 5.02657 | 1.00000 | 0 | −1.20423 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | 1.20423 | 0 | 1.24835 | 1.00000 | 0 | 1.20423 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | 4.06815 | 0 | 0.659180 | 1.00000 | 0 | 4.06815 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(113\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2034.2.a.s | 4 | |
| 3.b | odd | 2 | 1 | 678.2.a.k | ✓ | 4 | |
| 12.b | even | 2 | 1 | 5424.2.a.bm | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 678.2.a.k | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 2034.2.a.s | 4 | 1.a | even | 1 | 1 | trivial | |
| 5424.2.a.bm | 4 | 12.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2034))\):
|
\( T_{5}^{4} - 18T_{5}^{2} + 24 \)
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\( T_{7}^{4} - 5T_{7}^{3} - 3T_{7}^{2} + 16T_{7} - 8 \)
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\( T_{11}^{4} + 3T_{11}^{3} - 9T_{11}^{2} - 24T_{11} - 12 \)
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\( T_{13} - 2 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} - 18T^{2} + 24 \)
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| $7$ |
\( T^{4} - 5 T^{3} + \cdots - 8 \)
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| $11$ |
\( T^{4} + 3 T^{3} + \cdots - 12 \)
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| $13$ |
\( (T - 2)^{4} \)
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| $17$ |
\( T^{4} - 3 T^{3} + \cdots + 252 \)
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| $19$ |
\( T^{4} - 5 T^{3} + \cdots - 8 \)
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| $23$ |
\( T^{4} + 3 T^{3} + \cdots - 18 \)
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| $29$ |
\( T^{4} - 18T^{2} + 24 \)
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| $31$ |
\( T^{4} - 11 T^{3} + \cdots - 224 \)
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| $37$ |
\( T^{4} - 11 T^{3} + \cdots - 998 \)
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| $41$ |
\( (T^{2} - 6 T - 48)^{2} \)
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| $43$ |
\( T^{4} - 11 T^{3} + \cdots - 200 \)
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| $47$ |
\( T^{4} + 3 T^{3} + \cdots + 1050 \)
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| $53$ |
\( T^{4} + 9 T^{3} + \cdots - 972 \)
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| $59$ |
\( T^{4} - 276 T^{2} + 18816 \)
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| $61$ |
\( T^{4} - 2 T^{3} + \cdots - 1712 \)
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| $67$ |
\( T^{4} - 11 T^{3} + \cdots - 200 \)
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| $71$ |
\( T^{4} + 9 T^{3} + \cdots + 114 \)
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| $73$ |
\( (T^{2} - 10 T - 32)^{2} \)
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| $79$ |
\( T^{4} + 4 T^{3} + \cdots + 2344 \)
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| $83$ |
\( T^{4} - 3 T^{3} + \cdots + 252 \)
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| $89$ |
\( T^{4} + 3 T^{3} + \cdots - 588 \)
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| $97$ |
\( T^{4} + T^{3} + \cdots - 1568 \)
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