Newspace parameters
| Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 78.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(83.6401245825\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 686684708x^{2} - 387038512644x + 108192419023985712 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 686684708x^{2} - 387038512644x + 108192419023985712 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11\nu^{3} - 330317\nu^{2} - 3838789306\nu + 110214008358642 ) / 69778650 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -488\nu^{3} + 1967036\nu^{2} + 179583213298\nu - 533504297040861 ) / 104667975 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -33\beta_{3} - 976\beta_{2} + 1463\beta _1 + 1373368181 ) / 4 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -990951\beta_{3} - 3934072\beta_{2} + 741893853\beta _1 + 1162802182619 ) / 4 \)
|
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 |
|
64.0000 | 729.000 | 4096.00 | −32873.9 | 46656.0 | −220937. | 262144. | 531441. | −2.10393e6 | ||||||||||||||||||||||||||||||
| 1.2 | 64.0000 | 729.000 | 4096.00 | −18749.3 | 46656.0 | 256855. | 262144. | 531441. | −1.19995e6 | |||||||||||||||||||||||||||||||
| 1.3 | 64.0000 | 729.000 | 4096.00 | 44766.3 | 46656.0 | 401020. | 262144. | 531441. | 2.86504e6 | |||||||||||||||||||||||||||||||
| 1.4 | 64.0000 | 729.000 | 4096.00 | 50286.9 | 46656.0 | −365904. | 262144. | 531441. | 3.21836e6 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 78.14.a.h | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.14.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 43430T_{5}^{3} - 2039426996T_{5}^{2} + 57624729759000T_{5} + 1387525964471640000 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(78))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 64)^{4} \)
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| $3$ |
\( (T - 729)^{4} \)
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| $5$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
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| $7$ |
\( T^{4} + \cdots + 83\!\cdots\!80 \)
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| $11$ |
\( T^{4} + \cdots + 11\!\cdots\!40 \)
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| $13$ |
\( (T - 4826809)^{4} \)
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| $17$ |
\( T^{4} + \cdots + 16\!\cdots\!88 \)
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| $19$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
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| $23$ |
\( T^{4} + \cdots + 46\!\cdots\!92 \)
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| $29$ |
\( T^{4} + \cdots - 32\!\cdots\!00 \)
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| $31$ |
\( T^{4} + \cdots - 50\!\cdots\!20 \)
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| $37$ |
\( T^{4} + \cdots + 86\!\cdots\!48 \)
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| $41$ |
\( T^{4} + \cdots + 96\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
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| $47$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
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| $53$ |
\( T^{4} + \cdots - 11\!\cdots\!56 \)
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| $59$ |
\( T^{4} + \cdots + 76\!\cdots\!00 \)
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| $61$ |
\( T^{4} + \cdots + 22\!\cdots\!84 \)
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| $67$ |
\( T^{4} + \cdots + 21\!\cdots\!12 \)
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| $71$ |
\( T^{4} + \cdots - 34\!\cdots\!00 \)
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| $73$ |
\( T^{4} + \cdots + 36\!\cdots\!60 \)
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| $79$ |
\( T^{4} + \cdots + 35\!\cdots\!00 \)
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| $83$ |
\( T^{4} + \cdots - 10\!\cdots\!12 \)
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| $89$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
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| $97$ |
\( T^{4} + \cdots - 60\!\cdots\!84 \)
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