Newspace parameters
| Level: | \( N \) | \(=\) | \( 114 = 2 \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 114.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(162.670595814\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - 28164679x^{2} - 71978627334x - 44502120040968 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2}\cdot 5 \) |
| 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
\( x^{4} - 28164679x^{2} - 71978627334x - 44502120040968 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 73\nu^{3} - 539691\nu^{2} - 173173645\nu + 3659282040558 ) / 52840425 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -413\nu^{3} + 881796\nu^{2} + 9483425045\nu + 9877602754752 ) / 52840425 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 6 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -73\beta_{3} - 413\beta_{2} + 1958\beta _1 + 42246982 ) / 3 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -1079382\beta_{3} - 1763592\beta_{2} + 31323337\beta _1 + 323903283312 ) / 6 \)
|
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 |
|
128.000 | 2187.00 | 16384.0 | −347847. | 279936. | 1.25343e6 | 2.09715e6 | 4.78297e6 | −4.45244e7 | ||||||||||||||||||||||||||||||
| 1.2 | 128.000 | 2187.00 | 16384.0 | −64118.7 | 279936. | −257801. | 2.09715e6 | 4.78297e6 | −8.20720e6 | |||||||||||||||||||||||||||||||
| 1.3 | 128.000 | 2187.00 | 16384.0 | −42961.1 | 279936. | −402377. | 2.09715e6 | 4.78297e6 | −5.49902e6 | |||||||||||||||||||||||||||||||
| 1.4 | 128.000 | 2187.00 | 16384.0 | 151327. | 279936. | −2.39468e6 | 2.09715e6 | 4.78297e6 | 1.93698e7 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 114.16.a.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.16.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 303600T_{5}^{3} - 28840588875T_{5}^{2} - 5095189157208750T_{5} - 144998694866793285000 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(114))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 128)^{4} \)
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| $3$ |
\( (T - 2187)^{4} \)
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| $5$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
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| $7$ |
\( T^{4} + \cdots - 31\!\cdots\!36 \)
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| $11$ |
\( T^{4} + \cdots - 14\!\cdots\!68 \)
|
| $13$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
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| $17$ |
\( T^{4} + \cdots + 19\!\cdots\!88 \)
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| $19$ |
\( (T - 893871739)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 55\!\cdots\!32 \)
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| $29$ |
\( T^{4} + \cdots + 53\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots - 88\!\cdots\!40 \)
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| $41$ |
\( T^{4} + \cdots - 88\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots - 24\!\cdots\!68 \)
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| $47$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
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| $53$ |
\( T^{4} + \cdots + 62\!\cdots\!00 \)
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| $59$ |
\( T^{4} + \cdots - 18\!\cdots\!80 \)
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| $61$ |
\( T^{4} + \cdots + 69\!\cdots\!00 \)
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| $67$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots - 67\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots + 36\!\cdots\!96 \)
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| $79$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
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| $83$ |
\( T^{4} + \cdots - 13\!\cdots\!72 \)
|
| $89$ |
\( T^{4} + \cdots + 13\!\cdots\!28 \)
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| $97$ |
\( T^{4} + \cdots - 20\!\cdots\!00 \)
|
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