Newspace parameters
| Level: | \( N \) | \(=\) | \( 8954 = 2 \cdot 11^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8954.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(71.4980499699\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.249645904.1 |
|
|
|
| Defining polynomial: |
\( x^{6} - 12x^{4} - 6x^{3} + 34x^{2} + 34x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 12x^{4} - 6x^{3} + 34x^{2} + 34x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 12\nu^{3} - 4\nu^{2} + 34\nu + 20 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 11\nu^{3} - 4\nu^{2} - 30\nu - 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + \beta_{3} + 8\beta_{2} + 10\beta _1 + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{4} + 12\beta_{3} + 4\beta_{2} + 42\beta _1 + 32 \)
|
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 |
|
−1.00000 | −2.43488 | 1.00000 | 1.86346 | 2.43488 | 3.63624 | −1.00000 | 2.92863 | −1.86346 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.79965 | 1.00000 | 3.33826 | 1.79965 | −4.67488 | −1.00000 | 0.238734 | −3.33826 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −0.676613 | 1.00000 | 1.30701 | 0.676613 | 0.545661 | −1.00000 | −2.54219 | −1.30701 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −0.389103 | 1.00000 | −3.04234 | 0.389103 | −5.00817 | −1.00000 | −2.84860 | 3.04234 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.35251 | 1.00000 | 0.813501 | −2.35251 | −3.74156 | −1.00000 | 2.53430 | −0.813501 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.94773 | 1.00000 | −3.27989 | −2.94773 | 1.24272 | −1.00000 | 5.68913 | 3.27989 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(11\) | \( +1 \) |
| \(37\) | \( +1 \) |
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 | 8954.2.a.be | ✓ | 6 |
| 11.b | odd | 2 | 1 | 8954.2.a.bf | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8954.2.a.be | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 8954.2.a.bf | yes | 6 | 11.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8954))\):
|
\( T_{3}^{6} - 12T_{3}^{4} - 6T_{3}^{3} + 34T_{3}^{2} + 34T_{3} + 8 \)
|
|
\( T_{5}^{6} - T_{5}^{5} - 18T_{5}^{4} + 24T_{5}^{3} + 71T_{5}^{2} - 145T_{5} + 66 \)
|
|
\( T_{7}^{6} + 8T_{7}^{5} - 6T_{7}^{4} - 142T_{7}^{3} - 80T_{7}^{2} + 482T_{7} - 216 \)
|
|
\( T_{17}^{6} + 5T_{17}^{5} - 28T_{17}^{4} - 114T_{17}^{3} + 153T_{17}^{2} + 629T_{17} + 402 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} - 12 T^{4} + \cdots + 8 \)
|
| $5$ |
\( T^{6} - T^{5} + \cdots + 66 \)
|
| $7$ |
\( T^{6} + 8 T^{5} + \cdots - 216 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} + 12 T^{5} + \cdots + 108 \)
|
| $17$ |
\( T^{6} + 5 T^{5} + \cdots + 402 \)
|
| $19$ |
\( T^{6} + 2 T^{5} + \cdots - 24 \)
|
| $23$ |
\( T^{6} - 14 T^{5} + \cdots - 768 \)
|
| $29$ |
\( T^{6} + 4 T^{5} + \cdots + 12 \)
|
| $31$ |
\( T^{6} - 4 T^{5} + \cdots + 16 \)
|
| $37$ |
\( (T + 1)^{6} \)
|
| $41$ |
\( T^{6} - 98 T^{4} + \cdots - 13824 \)
|
| $43$ |
\( T^{6} + 4 T^{5} + \cdots - 1752 \)
|
| $47$ |
\( T^{6} - 15 T^{5} + \cdots + 16248 \)
|
| $53$ |
\( T^{6} - 10 T^{5} + \cdots + 4224 \)
|
| $59$ |
\( T^{6} + 18 T^{5} + \cdots + 161064 \)
|
| $61$ |
\( T^{6} + 12 T^{5} + \cdots + 93732 \)
|
| $67$ |
\( T^{6} - 10 T^{5} + \cdots + 32 \)
|
| $71$ |
\( T^{6} - 35 T^{5} + \cdots + 253044 \)
|
| $73$ |
\( T^{6} + 4 T^{5} + \cdots - 369312 \)
|
| $79$ |
\( T^{6} - T^{5} + \cdots + 1800 \)
|
| $83$ |
\( T^{6} + 3 T^{5} + \cdots - 9132 \)
|
| $89$ |
\( T^{6} + 4 T^{5} + \cdots - 1334832 \)
|
| $97$ |
\( T^{6} - 12 T^{5} + \cdots - 810484 \)
|
| show more | |
| show less |