Newspace parameters
| Level: | \( N \) | \(=\) | \( 11 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 11.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.43623528033\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{4} - x^{3} - 341x^{2} + 1417x - 1412 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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
\( x^{4} - x^{3} - 341x^{2} + 1417x - 1412 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 4\nu^{2} - 321\nu + 204 ) / 28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -9\nu^{3} - 8\nu^{2} + 3029\nu - 6652 ) / 28 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + 27\beta_{2} - 5\beta _1 + 511 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -12\beta_{3} - 24\beta_{2} + 341\beta _1 - 2335 ) / 3 \)
|
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 |
|
−21.2011 | −77.4315 | 321.485 | −138.468 | 1641.63 | −91.3115 | −4102.09 | 3808.63 | 2935.68 | ||||||||||||||||||||||||||||||
| 1.2 | −10.6261 | 12.2971 | −15.0863 | 512.130 | −130.670 | 973.904 | 1520.45 | −2035.78 | −5441.94 | |||||||||||||||||||||||||||||||
| 1.3 | 11.0598 | 59.6211 | −5.68000 | −60.1766 | 659.400 | 698.069 | −1478.48 | 1367.68 | −665.544 | |||||||||||||||||||||||||||||||
| 1.4 | 20.7673 | −29.4867 | 303.281 | 223.515 | −612.360 | −1410.66 | 3640.12 | −1317.53 | 4641.80 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
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 | 11.8.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 99.8.a.g | 4 | ||
| 4.b | odd | 2 | 1 | 176.8.a.j | 4 | ||
| 5.b | even | 2 | 1 | 275.8.a.b | 4 | ||
| 7.b | odd | 2 | 1 | 539.8.a.b | 4 | ||
| 11.b | odd | 2 | 1 | 121.8.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.8.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 99.8.a.g | 4 | 3.b | odd | 2 | 1 | ||
| 121.8.a.c | 4 | 11.b | odd | 2 | 1 | ||
| 176.8.a.j | 4 | 4.b | odd | 2 | 1 | ||
| 275.8.a.b | 4 | 5.b | even | 2 | 1 | ||
| 539.8.a.b | 4 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 558T_{2}^{2} + 140T_{2} + 51744 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(11))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 558 T^{2} + 140 T + 51744 \)
$3$
\( T^{4} + 35 T^{3} - 4673 T^{2} + \cdots + 1673964 \)
$5$
\( T^{4} - 537 T^{3} + \cdots + 953818350 \)
$7$
\( T^{4} - 170 T^{3} + \cdots + 87571440704 \)
$11$
\( (T + 1331)^{4} \)
$13$
\( T^{4} + \cdots - 518474088880000 \)
$17$
\( T^{4} - 54300 T^{3} + \cdots - 58\!\cdots\!56 \)
$19$
\( T^{4} - 67844 T^{3} + \cdots - 31\!\cdots\!84 \)
$23$
\( T^{4} + 9015 T^{3} + \cdots + 10\!\cdots\!44 \)
$29$
\( T^{4} - 234078 T^{3} + \cdots - 41\!\cdots\!64 \)
$31$
\( T^{4} - 189857 T^{3} + \cdots - 61\!\cdots\!36 \)
$37$
\( T^{4} - 127895 T^{3} + \cdots + 41\!\cdots\!94 \)
$41$
\( T^{4} - 289842 T^{3} + \cdots - 66\!\cdots\!84 \)
$43$
\( T^{4} - 704930 T^{3} + \cdots + 28\!\cdots\!64 \)
$47$
\( T^{4} + 1729080 T^{3} + \cdots - 74\!\cdots\!04 \)
$53$
\( T^{4} - 1098660 T^{3} + \cdots + 18\!\cdots\!84 \)
$59$
\( T^{4} + 4665777 T^{3} + \cdots + 29\!\cdots\!64 \)
$61$
\( T^{4} - 310610 T^{3} + \cdots + 10\!\cdots\!84 \)
$67$
\( T^{4} + 3368245 T^{3} + \cdots - 84\!\cdots\!64 \)
$71$
\( T^{4} + 3416541 T^{3} + \cdots + 59\!\cdots\!44 \)
$73$
\( T^{4} - 11466230 T^{3} + \cdots - 60\!\cdots\!76 \)
$79$
\( T^{4} - 566282 T^{3} + \cdots + 25\!\cdots\!16 \)
$83$
\( T^{4} + 4220790 T^{3} + \cdots + 70\!\cdots\!96 \)
$89$
\( T^{4} - 18265191 T^{3} + \cdots - 24\!\cdots\!34 \)
$97$
\( T^{4} - 11425325 T^{3} + \cdots + 46\!\cdots\!46 \)
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