Newspace parameters
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 50 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(30.4132410198\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 104434803447206332x + 4289992005756109702361620 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 104434803447206332x + 4289992005756109702361620 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 160\nu^{2} + 17410920480\nu - 11139712373505648960 ) / 12670249 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -11698720\nu^{2} + 6966187410422880\nu + 814502346867205307288640 ) / 12670249 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 73117\beta _1 + 216760320 ) / 650280960 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -12090917\beta_{2} + 4837630146127\beta _1 + 5030515869856343941939200 ) / 72253440 \)
|
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 |
|
1.67772e7 | −6.77152e11 | 2.81475e14 | −2.33026e17 | −1.13607e19 | −5.15430e20 | 4.72237e21 | 2.19235e23 | −3.90952e24 | |||||||||||||||||||||||||||
| 1.2 | 1.67772e7 | −1.33407e11 | 2.81475e14 | 1.87739e17 | −2.23819e18 | 8.28497e20 | 4.72237e21 | −2.21502e23 | 3.14974e24 | ||||||||||||||||||||||||||||
| 1.3 | 1.67772e7 | 7.94355e11 | 2.81475e14 | −5.65262e16 | 1.33271e19 | −4.38546e20 | 4.72237e21 | 3.91700e23 | −9.48353e23 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
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 | 2.50.a.b | ✓ | 3 |
| 4.b | odd | 2 | 1 | 16.50.a.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.50.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 16.50.a.b | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 16203614388T_{3}^{2} - 553534181046157255103952T_{3} - 71759294724891880256790697012966464 \)
acting on \(S_{50}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16777216)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 71\!\cdots\!64 \)
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| $5$ |
\( T^{3} + \cdots - 24\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots - 18\!\cdots\!32 \)
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| $11$ |
\( T^{3} + \cdots - 90\!\cdots\!68 \)
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| $13$ |
\( T^{3} + \cdots + 10\!\cdots\!56 \)
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| $17$ |
\( T^{3} + \cdots + 59\!\cdots\!68 \)
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| $19$ |
\( T^{3} + \cdots + 90\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 31\!\cdots\!56 \)
|
| $29$ |
\( T^{3} + \cdots - 14\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 13\!\cdots\!08 \)
|
| $37$ |
\( T^{3} + \cdots + 24\!\cdots\!88 \)
|
| $41$ |
\( T^{3} + \cdots + 66\!\cdots\!32 \)
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| $43$ |
\( T^{3} + \cdots - 17\!\cdots\!04 \)
|
| $47$ |
\( T^{3} + \cdots - 29\!\cdots\!92 \)
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| $53$ |
\( T^{3} + \cdots - 10\!\cdots\!04 \)
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| $59$ |
\( T^{3} + \cdots + 52\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 31\!\cdots\!92 \)
|
| $67$ |
\( T^{3} + \cdots - 49\!\cdots\!12 \)
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| $71$ |
\( T^{3} + \cdots - 87\!\cdots\!88 \)
|
| $73$ |
\( T^{3} + \cdots - 11\!\cdots\!84 \)
|
| $79$ |
\( T^{3} + \cdots - 16\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 12\!\cdots\!56 \)
|
| $89$ |
\( T^{3} + \cdots + 13\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 56\!\cdots\!28 \)
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