Newspace parameters
| Level: | \( N \) | \(=\) | \( 220 = 2^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 220.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.99456581593\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 5 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} + 16\nu - 9 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 4\nu^{2} - 2\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{3} - 5\nu^{2} + 5\nu + 21 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{3} + \beta_{2} + 14\beta _1 + 7 ) / 40 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -6\beta_{3} - 23\beta_{2} - 2\beta _1 + 39 ) / 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{3} + 3\beta_{2} + 2\beta _1 + 17 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/220\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(111\) | \(177\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 109.1 |
|
0 | − | 5.98844i | 0 | −4.55842 | − | 2.05446i | 0 | 0 | 0 | −26.8614 | 0 | |||||||||||||||||||||||||||
| 109.2 | 0 | − | 2.67181i | 0 | 4.05842 | + | 2.92048i | 0 | 0 | 0 | 1.86141 | 0 | ||||||||||||||||||||||||||||
| 109.3 | 0 | 2.67181i | 0 | 4.05842 | − | 2.92048i | 0 | 0 | 0 | 1.86141 | 0 | |||||||||||||||||||||||||||||
| 109.4 | 0 | 5.98844i | 0 | −4.55842 | + | 2.05446i | 0 | 0 | 0 | −26.8614 | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
| 5.b | even | 2 | 1 | inner |
| 55.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 220.3.e.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1980.3.p.a | 4 | ||
| 4.b | odd | 2 | 1 | 880.3.i.e | 4 | ||
| 5.b | even | 2 | 1 | inner | 220.3.e.a | ✓ | 4 |
| 5.c | odd | 4 | 2 | 1100.3.f.b | 4 | ||
| 11.b | odd | 2 | 1 | CM | 220.3.e.a | ✓ | 4 |
| 15.d | odd | 2 | 1 | 1980.3.p.a | 4 | ||
| 20.d | odd | 2 | 1 | 880.3.i.e | 4 | ||
| 33.d | even | 2 | 1 | 1980.3.p.a | 4 | ||
| 44.c | even | 2 | 1 | 880.3.i.e | 4 | ||
| 55.d | odd | 2 | 1 | inner | 220.3.e.a | ✓ | 4 |
| 55.e | even | 4 | 2 | 1100.3.f.b | 4 | ||
| 165.d | even | 2 | 1 | 1980.3.p.a | 4 | ||
| 220.g | even | 2 | 1 | 880.3.i.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 220.3.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 220.3.e.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 220.3.e.a | ✓ | 4 | 11.b | odd | 2 | 1 | CM |
| 220.3.e.a | ✓ | 4 | 55.d | odd | 2 | 1 | inner |
| 880.3.i.e | 4 | 4.b | odd | 2 | 1 | ||
| 880.3.i.e | 4 | 20.d | odd | 2 | 1 | ||
| 880.3.i.e | 4 | 44.c | even | 2 | 1 | ||
| 880.3.i.e | 4 | 220.g | even | 2 | 1 | ||
| 1100.3.f.b | 4 | 5.c | odd | 4 | 2 | ||
| 1100.3.f.b | 4 | 55.e | even | 4 | 2 | ||
| 1980.3.p.a | 4 | 3.b | odd | 2 | 1 | ||
| 1980.3.p.a | 4 | 15.d | odd | 2 | 1 | ||
| 1980.3.p.a | 4 | 33.d | even | 2 | 1 | ||
| 1980.3.p.a | 4 | 165.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 43T_{3}^{2} + 256 \)
acting on \(S_{3}^{\mathrm{new}}(220, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 43T^{2} + 256 \)
|
| $5$ |
\( T^{4} + T^{3} + \cdots + 625 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T - 11)^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + 2283 T^{2} + 484416 \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( (T^{2} + 37 T - 1514)^{2} \)
|
| $37$ |
\( T^{4} + 3363 T^{2} + 553536 \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( (T^{2} + 6336)^{2} \)
|
| $53$ |
\( (T^{2} + 6336)^{2} \)
|
| $59$ |
\( (T^{2} + 107 T + 1006)^{2} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} + 10203 T^{2} + 10653696 \)
|
| $71$ |
\( (T^{2} - 133 T + 2566)^{2} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( (T^{2} + 97 T - 14354)^{2} \)
|
| $97$ |
\( T^{4} + 27843 T^{2} + 147456 \)
|
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