Newspace parameters
| Level: | \( N \) | \(=\) | \( 220 = 2^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 220.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(35.2844403589\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.399324.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 159x - 680 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 159x - 680 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} - 4\nu - 107 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{2} + 20\nu + 211 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5\beta _1 + 108 \)
|
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 | −13.9603 | 0 | 25.0000 | 0 | −9.35268 | 0 | −48.1099 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −5.13342 | 0 | 25.0000 | 0 | 82.4100 | 0 | −216.648 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 20.0937 | 0 | 25.0000 | 0 | −228.057 | 0 | 160.758 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
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 | 220.6.a.b | ✓ | 3 |
| 4.b | odd | 2 | 1 | 880.6.a.j | 3 | ||
| 5.b | even | 2 | 1 | 1100.6.a.d | 3 | ||
| 5.c | odd | 4 | 2 | 1100.6.b.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 220.6.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 880.6.a.j | 3 | 4.b | odd | 2 | 1 | ||
| 1100.6.a.d | 3 | 5.b | even | 2 | 1 | ||
| 1100.6.b.d | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - T_{3}^{2} - 312T_{3} - 1440 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(220))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - T^{2} + \cdots - 1440 \)
|
| $5$ |
\( (T - 25)^{3} \)
|
| $7$ |
\( T^{3} + 155 T^{2} + \cdots - 175776 \)
|
| $11$ |
\( (T + 121)^{3} \)
|
| $13$ |
\( T^{3} - 1094 T^{2} + \cdots + 31937144 \)
|
| $17$ |
\( T^{3} + 19 T^{2} + \cdots - 493345380 \)
|
| $19$ |
\( T^{3} + \cdots - 13139307248 \)
|
| $23$ |
\( T^{3} + \cdots - 25323871968 \)
|
| $29$ |
\( T^{3} + \cdots + 57124516620 \)
|
| $31$ |
\( T^{3} + \cdots - 63522375360 \)
|
| $37$ |
\( T^{3} + \cdots - 156308522956 \)
|
| $41$ |
\( T^{3} + \cdots + 25081114200 \)
|
| $43$ |
\( T^{3} + \cdots - 1547993112000 \)
|
| $47$ |
\( T^{3} + \cdots - 382738023840 \)
|
| $53$ |
\( T^{3} + \cdots + 208612572228 \)
|
| $59$ |
\( T^{3} + \cdots - 27595270294560 \)
|
| $61$ |
\( T^{3} + \cdots - 230105423484 \)
|
| $67$ |
\( T^{3} + \cdots - 96069431484160 \)
|
| $71$ |
\( T^{3} + \cdots - 259369439040 \)
|
| $73$ |
\( T^{3} + \cdots - 38281908027960 \)
|
| $79$ |
\( T^{3} + \cdots - 159067707906944 \)
|
| $83$ |
\( T^{3} + \cdots + 2693394216480 \)
|
| $89$ |
\( T^{3} + \cdots + 854013774895068 \)
|
| $97$ |
\( T^{3} + \cdots - 700110118756688 \)
|
| show more | |
| show less |