Newspace parameters
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(146.270043669\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2922585.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 360x + 60 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 114) |
| 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
\( x^{3} - x^{2} - 360x + 60 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 21\nu - 250 ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 3\nu + 238 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 3 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{2} + 2\beta _1 + 479 ) / 2 \)
|
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 | 9.00000 | 0 | −69.4272 | 0 | 167.663 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | 24.6902 | 0 | −239.263 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 9.00000 | 0 | 107.737 | 0 | −53.4003 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||
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 | 912.6.a.o | 3 | |
| 4.b | odd | 2 | 1 | 114.6.a.h | ✓ | 3 | |
| 12.b | even | 2 | 1 | 342.6.a.o | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.6.a.h | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 342.6.a.o | 3 | 12.b | even | 2 | 1 | ||
| 912.6.a.o | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 63T_{5}^{2} - 6534T_{5} + 184680 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 9)^{3} \)
|
| $5$ |
\( T^{3} - 63 T^{2} + \cdots + 184680 \)
|
| $7$ |
\( T^{3} + 125 T^{2} + \cdots - 2142176 \)
|
| $11$ |
\( T^{3} - 473 T^{2} + \cdots + 158381280 \)
|
| $13$ |
\( T^{3} + 812 T^{2} + \cdots + 571824 \)
|
| $17$ |
\( T^{3} + \cdots - 1400814036 \)
|
| $19$ |
\( (T + 361)^{3} \)
|
| $23$ |
\( T^{3} + \cdots + 18455076000 \)
|
| $29$ |
\( T^{3} + \cdots + 19566842760 \)
|
| $31$ |
\( T^{3} + \cdots + 256187655168 \)
|
| $37$ |
\( T^{3} + \cdots + 250544964416 \)
|
| $41$ |
\( T^{3} + \cdots + 16913974176 \)
|
| $43$ |
\( T^{3} + \cdots + 1232020979664 \)
|
| $47$ |
\( T^{3} + \cdots - 497752133400 \)
|
| $53$ |
\( T^{3} + \cdots + 484621940664 \)
|
| $59$ |
\( T^{3} + \cdots + 10803978277632 \)
|
| $61$ |
\( T^{3} + \cdots + 35958120853924 \)
|
| $67$ |
\( T^{3} + \cdots - 18394458377472 \)
|
| $71$ |
\( T^{3} + \cdots + 35051765760000 \)
|
| $73$ |
\( T^{3} + \cdots + 53634950800100 \)
|
| $79$ |
\( T^{3} + \cdots + 30939023653120 \)
|
| $83$ |
\( T^{3} + \cdots - 365118946951680 \)
|
| $89$ |
\( T^{3} + \cdots - 196132218012936 \)
|
| $97$ |
\( T^{3} + \cdots - 156744792431720 \)
|
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