Newspace parameters
| Level: | \( N \) | \(=\) | \( 2254 = 2 \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2254.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(17.9982806156\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1957.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 4x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 322) |
| 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
\( x^{4} - 4x^{2} - x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 4\nu - 1 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + \nu^{2} + 3\nu - 1 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 4\beta _1 + 1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −2.57641 | 1.00000 | −1.32664 | −2.57641 | 0 | 1.00000 | 3.63791 | −1.32664 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.74747 | 1.00000 | −4.26608 | −1.74747 | 0 | 1.00000 | 0.0536460 | −4.26608 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.197126 | 1.00000 | 0.308875 | 0.197126 | 0 | 1.00000 | −2.96114 | 0.308875 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.12676 | 1.00000 | −1.71616 | 1.12676 | 0 | 1.00000 | −1.73042 | −1.71616 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 2254.2.a.x | 4 | |
| 7.b | odd | 2 | 1 | 2254.2.a.z | 4 | ||
| 7.d | odd | 6 | 2 | 322.2.e.a | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.2.e.a | ✓ | 8 | 7.d | odd | 6 | 2 | |
| 2254.2.a.x | 4 | 1.a | even | 1 | 1 | trivial | |
| 2254.2.a.z | 4 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2254))\):
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\( T_{3}^{4} + 3T_{3}^{3} - T_{3}^{2} - 5T_{3} + 1 \)
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\( T_{5}^{4} + 7T_{5}^{3} + 13T_{5}^{2} + 5T_{5} - 3 \)
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\( T_{11}^{4} - 2T_{11}^{3} - 12T_{11}^{2} + 29T_{11} - 9 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
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| $3$ |
\( T^{4} + 3 T^{3} + \cdots + 1 \)
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| $5$ |
\( T^{4} + 7 T^{3} + \cdots - 3 \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( T^{4} - 2 T^{3} + \cdots - 9 \)
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| $13$ |
\( T^{4} + T^{3} + \cdots + 343 \)
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| $17$ |
\( T^{4} + 5 T^{3} + \cdots - 423 \)
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| $19$ |
\( T^{4} + 11 T^{3} + \cdots + 101 \)
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| $23$ |
\( (T + 1)^{4} \)
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| $29$ |
\( T^{4} - 2 T^{3} + \cdots - 21 \)
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| $31$ |
\( T^{4} + 6 T^{3} + \cdots - 707 \)
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| $37$ |
\( T^{4} + 8 T^{3} + \cdots - 1367 \)
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| $41$ |
\( T^{4} + 9 T^{3} + \cdots + 417 \)
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| $43$ |
\( T^{4} - 4 T^{3} + \cdots + 1021 \)
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| $47$ |
\( T^{4} + 11 T^{3} + \cdots + 687 \)
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| $53$ |
\( T^{4} - T^{3} + \cdots - 933 \)
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| $59$ |
\( T^{4} + 12 T^{3} + \cdots - 81 \)
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| $61$ |
\( T^{4} + 21 T^{3} + \cdots - 12473 \)
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| $67$ |
\( T^{4} + 3 T^{3} + \cdots + 5827 \)
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| $71$ |
\( T^{4} - 11 T^{3} + \cdots - 189 \)
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| $73$ |
\( T^{4} - 16 T^{3} + \cdots - 1493 \)
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| $79$ |
\( T^{4} + 21 T^{3} + \cdots - 53 \)
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| $83$ |
\( T^{4} + 4 T^{3} + \cdots + 5913 \)
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| $89$ |
\( T^{4} + 27 T^{3} + \cdots - 3999 \)
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| $97$ |
\( T^{4} + 6 T^{3} + \cdots - 23 \)
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