Newspace parameters
| Level: | \( N \) | \(=\) | \( 114 = 2 \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 114.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(122.243259005\) |
| 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: |
\( x^{4} - x^{3} - 661858x^{2} + 249481534x - 25069361604 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{3} \) |
| 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
\( x^{4} - x^{3} - 661858x^{2} + 249481534x - 25069361604 \)
:
| \(\beta_{1}\) | \(=\) |
\( 36\nu - 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 315\nu^{2} + 563536\nu - 82513536 ) / 1218 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{3} - 1053\nu^{2} + 3110000\nu - 585382218 ) / 1131 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 9 ) / 36 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 39\beta_{3} - 210\beta_{2} - 280\beta _1 + 5956602 ) / 18 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -12285\beta_{3} + 44226\beta_{2} + 369968\beta _1 - 3359037366 ) / 18 \)
|
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 | −51833.9 | −46656.0 | −331088. | 262144. | 531441. | −3.31737e6 | ||||||||||||||||||||||||||||||
| 1.2 | 64.0000 | −729.000 | 4096.00 | −23578.4 | −46656.0 | 45014.5 | 262144. | 531441. | −1.50902e6 | |||||||||||||||||||||||||||||||
| 1.3 | 64.0000 | −729.000 | 4096.00 | −9993.06 | −46656.0 | 435862. | 262144. | 531441. | −639556. | |||||||||||||||||||||||||||||||
| 1.4 | 64.0000 | −729.000 | 4096.00 | 27961.4 | −46656.0 | 5354.27 | 262144. | 531441. | 1.78953e6 | |||||||||||||||||||||||||||||||
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 | 114.14.a.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.14.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 57444T_{5}^{3} - 412291755T_{5}^{2} - 43031934049050T_{5} - 341496547497630000 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(114))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 64)^{4} \)
|
| $3$ |
\( (T + 729)^{4} \)
|
| $5$ |
\( T^{4} + \cdots - 34\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots - 34\!\cdots\!28 \)
|
| $11$ |
\( T^{4} + \cdots - 10\!\cdots\!60 \)
|
| $13$ |
\( T^{4} + \cdots + 14\!\cdots\!84 \)
|
| $17$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
| $19$ |
\( (T - 47045881)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 98\!\cdots\!40 \)
|
| $29$ |
\( T^{4} + \cdots + 12\!\cdots\!08 \)
|
| $31$ |
\( T^{4} + \cdots - 24\!\cdots\!32 \)
|
| $37$ |
\( T^{4} + \cdots - 45\!\cdots\!36 \)
|
| $41$ |
\( T^{4} + \cdots - 80\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 14\!\cdots\!80 \)
|
| $47$ |
\( T^{4} + \cdots + 79\!\cdots\!40 \)
|
| $53$ |
\( T^{4} + \cdots + 16\!\cdots\!56 \)
|
| $59$ |
\( T^{4} + \cdots - 26\!\cdots\!72 \)
|
| $61$ |
\( T^{4} + \cdots - 22\!\cdots\!24 \)
|
| $67$ |
\( T^{4} + \cdots + 87\!\cdots\!60 \)
|
| $71$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 12\!\cdots\!56 \)
|
| $79$ |
\( T^{4} + \cdots - 44\!\cdots\!48 \)
|
| $83$ |
\( T^{4} + \cdots - 69\!\cdots\!40 \)
|
| $89$ |
\( T^{4} + \cdots - 12\!\cdots\!60 \)
|
| $97$ |
\( T^{4} + \cdots - 48\!\cdots\!80 \)
|
| show more | |
| show less |