Newspace parameters
| Level: | \( N \) | \(=\) | \( 217 = 7 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 217.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.73275372386\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.6809.1 |
|
|
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| Defining polynomial: |
\( x^{4} - 5x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - 5x^{2} - x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta _1 + 1 \)
|
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 |
|
−2.29041 | 0.563397 | 3.24598 | 3.29041 | −1.29041 | 1.00000 | −2.85381 | −2.68258 | −7.53639 | ||||||||||||||||||||||||||||||
| 1.2 | −0.361989 | −1.76251 | −1.86896 | 1.36199 | 0.638011 | 1.00000 | 1.40052 | 0.106451 | −0.493026 | |||||||||||||||||||||||||||||||
| 1.3 | 0.582772 | 2.71594 | −1.66038 | 0.417228 | 1.58277 | 1.00000 | −2.13317 | 4.37631 | 0.243149 | |||||||||||||||||||||||||||||||
| 1.4 | 2.06963 | 1.48318 | 2.28336 | −1.06963 | 3.06963 | 1.00000 | 0.586449 | −0.800181 | −2.21373 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
| \(31\) | \( +1 \) |
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 | 217.2.a.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1953.2.a.t | 4 | ||
| 4.b | odd | 2 | 1 | 3472.2.a.v | 4 | ||
| 5.b | even | 2 | 1 | 5425.2.a.u | 4 | ||
| 7.b | odd | 2 | 1 | 1519.2.a.d | 4 | ||
| 31.b | odd | 2 | 1 | 6727.2.a.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 217.2.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1519.2.a.d | 4 | 7.b | odd | 2 | 1 | ||
| 1953.2.a.t | 4 | 3.b | odd | 2 | 1 | ||
| 3472.2.a.v | 4 | 4.b | odd | 2 | 1 | ||
| 5425.2.a.u | 4 | 5.b | even | 2 | 1 | ||
| 6727.2.a.g | 4 | 31.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(217))\):
|
\( T_{2}^{4} - 5T_{2}^{2} + T_{2} + 1 \)
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\( T_{3}^{4} - 3T_{3}^{3} - 2T_{3}^{2} + 9T_{3} - 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 5T^{2} + T + 1 \)
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| $3$ |
\( T^{4} - 3 T^{3} + \cdots - 4 \)
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| $5$ |
\( T^{4} - 4 T^{3} + \cdots - 2 \)
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| $7$ |
\( (T - 1)^{4} \)
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| $11$ |
\( T^{4} - 2 T^{3} + \cdots - 68 \)
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| $13$ |
\( T^{4} + T^{3} - 18 T^{2} + \cdots - 2 \)
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| $17$ |
\( T^{4} - 8 T^{3} + \cdots + 214 \)
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| $19$ |
\( T^{4} - 5 T^{3} + \cdots + 4 \)
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| $23$ |
\( T^{4} - 20 T^{3} + \cdots - 160 \)
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| $29$ |
\( T^{4} + 7 T^{3} + \cdots - 110 \)
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| $31$ |
\( (T + 1)^{4} \)
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| $37$ |
\( T^{4} - 179 T^{2} + \cdots + 7058 \)
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| $41$ |
\( T^{4} - 5 T^{3} + \cdots - 254 \)
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| $43$ |
\( T^{4} + 15 T^{3} + \cdots + 116 \)
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| $47$ |
\( T^{4} - 19 T^{3} + \cdots + 32 \)
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| $53$ |
\( T^{4} - 3 T^{3} + \cdots + 2390 \)
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| $59$ |
\( T^{4} - 5 T^{3} + \cdots - 556 \)
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| $61$ |
\( T^{4} + 5 T^{3} + \cdots + 22 \)
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| $67$ |
\( T^{4} + 14 T^{3} + \cdots - 10076 \)
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| $71$ |
\( T^{4} + 5 T^{3} + \cdots + 1720 \)
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| $73$ |
\( T^{4} + 9 T^{3} + \cdots - 1766 \)
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| $79$ |
\( T^{4} + 4 T^{3} + \cdots - 1088 \)
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| $83$ |
\( T^{4} - 25 T^{3} + \cdots - 5732 \)
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| $89$ |
\( T^{4} - 21 T^{3} + \cdots - 118 \)
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| $97$ |
\( T^{4} + 15 T^{3} + \cdots - 1298 \)
|
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