Newspace parameters
| Level: | \( N \) | \(=\) | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| Character orbit: | \([\chi]\) | \(=\) | 84.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(192.206025107\) |
| 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: |
\( x^{4} - x^{3} - 134334816755x^{2} + 25646529546394449x - 1331716811709552720606 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{4}\cdot 5\cdot 7 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 134334816755x^{2} + 25646529546394449x - 1331716811709552720606 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 337\nu^{3} + 56190504\nu^{2} - 36945170063563\nu + 2707965134184629004 ) / 74570891850 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5818\nu^{3} - 900657456\nu^{2} + 631701667694782\nu - 51413376275518788006 ) / 10966307625 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -88072\nu^{3} - 10849400424\nu^{2} + 10444498755891328\nu - 965323491497699948124 ) / 186427229625 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{3} - 13\beta_{2} - 1108\beta _1 + 6602 ) / 24192 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 352820\beta_{3} + 2674915\beta_{2} + 350906476\beta _1 + 1624913768021290 ) / 24192 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 379690049356\beta_{3} - 1871192975767\beta_{2} - 174625641418684\beta _1 - 465328107406044964546 ) / 24192 \)
|
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 |
|
0 | −19683.0 | 0 | −7.42148e6 | 0 | −4.03536e7 | 0 | 3.87420e8 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −19683.0 | 0 | −1.67225e6 | 0 | −4.03536e7 | 0 | 3.87420e8 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −19683.0 | 0 | 1.39252e6 | 0 | −4.03536e7 | 0 | 3.87420e8 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −19683.0 | 0 | 5.19906e6 | 0 | −4.03536e7 | 0 | 3.87420e8 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(7\) | \(1\) |
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 | 84.20.a.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.20.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 2502150T_{5}^{3} - 40291665747800T_{5}^{2} - 15968736728988108000T_{5} + 89849733189702751930000000 \)
acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(84))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 19683)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 89\!\cdots\!00 \)
|
| $7$ |
\( (T + 40353607)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 18\!\cdots\!12 \)
|
| $13$ |
\( T^{4} + \cdots + 20\!\cdots\!72 \)
|
| $17$ |
\( T^{4} + \cdots - 67\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots - 42\!\cdots\!64 \)
|
| $23$ |
\( T^{4} + \cdots + 20\!\cdots\!08 \)
|
| $29$ |
\( T^{4} + \cdots - 28\!\cdots\!08 \)
|
| $31$ |
\( T^{4} + \cdots - 23\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 13\!\cdots\!84 \)
|
| $41$ |
\( T^{4} + \cdots - 32\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 17\!\cdots\!84 \)
|
| $47$ |
\( T^{4} + \cdots - 12\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots + 86\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 26\!\cdots\!80 \)
|
| $61$ |
\( T^{4} + \cdots + 41\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 72\!\cdots\!76 \)
|
| $71$ |
\( T^{4} + \cdots - 21\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots - 52\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 17\!\cdots\!52 \)
|
| $83$ |
\( T^{4} + \cdots - 13\!\cdots\!88 \)
|
| $89$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 37\!\cdots\!80 \)
|
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