Newspace parameters
| Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 121.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(37.7985880836\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 395x^{3} + 660x^{2} + 21600x - 76032 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 395x^{3} + 660x^{2} + 21600x - 76032 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{4} + 14\nu^{3} - 1879\nu^{2} - 5892\nu + 72864 ) / 144 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 10\nu^{3} - 419\nu^{2} - 3216\nu + 21672 ) / 36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{4} - 14\nu^{3} + 2023\nu^{2} + 5892\nu - 95760 ) / 144 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 159 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{4} + 5\beta_{3} + 2\beta_{2} + 283\beta _1 - 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( 359\beta_{4} - 14\beta_{3} + 399\beta_{2} + 386\beta _1 + 45269 \)
|
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 |
|
−19.7320 | 35.1324 | 261.350 | −485.746 | −693.232 | 1057.93 | −2631.26 | −952.713 | 9584.72 | |||||||||||||||||||||||||||||||||
| 1.2 | −11.0249 | −21.6215 | −6.45150 | 243.519 | 238.374 | −646.506 | 1482.31 | −1719.51 | −2684.78 | ||||||||||||||||||||||||||||||||||
| 1.3 | 2.59754 | 60.4351 | −121.253 | −21.1080 | 156.982 | 295.244 | −647.443 | 1465.40 | −54.8289 | ||||||||||||||||||||||||||||||||||
| 1.4 | 3.55452 | −81.6995 | −115.365 | −319.514 | −290.403 | 1172.95 | −865.048 | 4487.81 | −1135.72 | ||||||||||||||||||||||||||||||||||
| 1.5 | 16.6048 | −6.24652 | 147.719 | 67.8486 | −103.722 | −593.613 | 327.438 | −2147.98 | 1126.61 | ||||||||||||||||||||||||||||||||||
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 | 121.8.a.d | ✓ | 5 |
| 11.b | odd | 2 | 1 | 121.8.a.e | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 121.8.a.d | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 121.8.a.e | yes | 5 | 11.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 8T_{2}^{4} - 371T_{2}^{3} - 1678T_{2}^{2} + 19516T_{2} - 33352 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(121))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 8 T^{4} + \cdots - 33352 \)
|
| $3$ |
\( T^{5} + 14 T^{4} + \cdots + 23428224 \)
|
| $5$ |
\( T^{5} + \cdots + 54127925375 \)
|
| $7$ |
\( T^{5} + \cdots - 140602327290304 \)
|
| $11$ |
\( T^{5} \)
|
| $13$ |
\( T^{5} + \cdots + 13\!\cdots\!03 \)
|
| $17$ |
\( T^{5} + \cdots + 48\!\cdots\!65 \)
|
| $19$ |
\( T^{5} + \cdots + 46\!\cdots\!16 \)
|
| $23$ |
\( T^{5} + \cdots + 23\!\cdots\!76 \)
|
| $29$ |
\( T^{5} + \cdots + 13\!\cdots\!11 \)
|
| $31$ |
\( T^{5} + \cdots + 89\!\cdots\!92 \)
|
| $37$ |
\( T^{5} + \cdots + 52\!\cdots\!59 \)
|
| $41$ |
\( T^{5} + \cdots - 65\!\cdots\!15 \)
|
| $43$ |
\( T^{5} + \cdots - 60\!\cdots\!00 \)
|
| $47$ |
\( T^{5} + \cdots - 23\!\cdots\!16 \)
|
| $53$ |
\( T^{5} + \cdots - 56\!\cdots\!77 \)
|
| $59$ |
\( T^{5} + \cdots - 46\!\cdots\!32 \)
|
| $61$ |
\( T^{5} + \cdots + 52\!\cdots\!36 \)
|
| $67$ |
\( T^{5} + \cdots + 32\!\cdots\!44 \)
|
| $71$ |
\( T^{5} + \cdots - 17\!\cdots\!08 \)
|
| $73$ |
\( T^{5} + \cdots + 92\!\cdots\!96 \)
|
| $79$ |
\( T^{5} + \cdots - 20\!\cdots\!60 \)
|
| $83$ |
\( T^{5} + \cdots - 33\!\cdots\!12 \)
|
| $89$ |
\( T^{5} + \cdots - 20\!\cdots\!51 \)
|
| $97$ |
\( T^{5} + \cdots - 36\!\cdots\!79 \)
|
| show more | |
| show less |