Newspace parameters
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 38 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(86.7140381246\) |
| 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: |
\( x^{3} - x^{2} - 354229391290x - 56510296564182800 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{5}\cdot 5^{3}\cdot 7 \) |
| 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
\( x^{3} - x^{2} - 354229391290x - 56510296564182800 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 18\nu^{2} + 197982\nu - 4250752761480 ) / 5083 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -90\nu^{2} + 43563690\nu + 21253748956200 ) / 221 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 115\beta _1 + 67200 ) / 201600 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -10999\beta_{2} + 55664715\beta _1 + 47608430189443200 ) / 201600 \)
|
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 |
|
−262144. | −9.18650e8 | 6.87195e10 | 3.81470e12 | 2.40818e14 | −4.46427e15 | −1.80144e16 | 3.93633e17 | −1.00000e18 | |||||||||||||||||||||||||||
| 1.2 | −262144. | −1.94103e8 | 6.87195e10 | 3.81470e12 | 5.08830e13 | 1.33426e15 | −1.80144e16 | −4.12608e17 | −1.00000e18 | ||||||||||||||||||||||||||||
| 1.3 | −262144. | 5.62358e8 | 6.87195e10 | 3.81470e12 | −1.47419e14 | −1.40493e15 | −1.80144e16 | −1.34037e17 | −1.00000e18 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 10.38.a.b | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.38.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 550394388T_{3}^{2} - 447453021323175552T_{3} - 100275628892080438522808064 \)
acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(10))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 262144)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 10\!\cdots\!64 \)
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| $5$ |
\( (T - 3814697265625)^{3} \)
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| $7$ |
\( T^{3} + \cdots - 83\!\cdots\!52 \)
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| $11$ |
\( T^{3} + \cdots - 22\!\cdots\!08 \)
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| $13$ |
\( T^{3} + \cdots - 16\!\cdots\!44 \)
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| $17$ |
\( T^{3} + \cdots + 30\!\cdots\!68 \)
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| $19$ |
\( T^{3} + \cdots + 11\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 37\!\cdots\!36 \)
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| $29$ |
\( T^{3} + \cdots - 24\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 25\!\cdots\!48 \)
|
| $37$ |
\( T^{3} + \cdots - 12\!\cdots\!52 \)
|
| $41$ |
\( T^{3} + \cdots - 94\!\cdots\!48 \)
|
| $43$ |
\( T^{3} + \cdots - 46\!\cdots\!24 \)
|
| $47$ |
\( T^{3} + \cdots - 13\!\cdots\!92 \)
|
| $53$ |
\( T^{3} + \cdots - 54\!\cdots\!44 \)
|
| $59$ |
\( T^{3} + \cdots - 76\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 15\!\cdots\!12 \)
|
| $67$ |
\( T^{3} + \cdots - 13\!\cdots\!92 \)
|
| $71$ |
\( T^{3} + \cdots + 10\!\cdots\!72 \)
|
| $73$ |
\( T^{3} + \cdots + 64\!\cdots\!56 \)
|
| $79$ |
\( T^{3} + \cdots + 27\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 16\!\cdots\!16 \)
|
| $89$ |
\( T^{3} + \cdots + 22\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 47\!\cdots\!52 \)
|
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