Newspace parameters
| Level: | \( N \) | \(=\) | \( 110 = 2 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 110.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(17.6422201794\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.709185.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 166x - 704 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| 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
\( x^{3} - x^{2} - 166x - 704 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{2} + 27\nu + 102 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 7\nu - 108 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 3 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 27\beta_{2} + 7\beta _1 + 1101 ) / 10 \)
|
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 |
|
4.00000 | −13.2599 | 16.0000 | 25.0000 | −53.0397 | 18.0952 | 64.0000 | −67.1746 | 100.000 | |||||||||||||||||||||||||||
| 1.2 | 4.00000 | 15.0517 | 16.0000 | 25.0000 | 60.2069 | 193.859 | 64.0000 | −16.4453 | 100.000 | ||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 23.2082 | 16.0000 | 25.0000 | 92.8327 | −52.9544 | 64.0000 | 295.620 | 100.000 | ||||||||||||||||||||||||||||
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 | 110.6.a.h | ✓ | 3 |
| 3.b | odd | 2 | 1 | 990.6.a.v | 3 | ||
| 4.b | odd | 2 | 1 | 880.6.a.i | 3 | ||
| 5.b | even | 2 | 1 | 550.6.a.k | 3 | ||
| 5.c | odd | 4 | 2 | 550.6.b.l | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.6.a.h | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 550.6.a.k | 3 | 5.b | even | 2 | 1 | ||
| 550.6.b.l | 6 | 5.c | odd | 4 | 2 | ||
| 880.6.a.i | 3 | 4.b | odd | 2 | 1 | ||
| 990.6.a.v | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 25T_{3}^{2} - 158T_{3} + 4632 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(110))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{3} \)
|
| $3$ |
\( T^{3} - 25 T^{2} + \cdots + 4632 \)
|
| $5$ |
\( (T - 25)^{3} \)
|
| $7$ |
\( T^{3} - 159 T^{2} + \cdots + 185760 \)
|
| $11$ |
\( (T - 121)^{3} \)
|
| $13$ |
\( T^{3} - 444 T^{2} + \cdots + 398396448 \)
|
| $17$ |
\( T^{3} + \cdots + 1244827144 \)
|
| $19$ |
\( T^{3} + \cdots + 3670188400 \)
|
| $23$ |
\( T^{3} + \cdots + 15084641088 \)
|
| $29$ |
\( T^{3} - 1305 T^{2} + \cdots + 28124500 \)
|
| $31$ |
\( T^{3} + \cdots - 28863295488 \)
|
| $37$ |
\( T^{3} + \cdots + 558761410332 \)
|
| $41$ |
\( T^{3} + \cdots - 376445579672 \)
|
| $43$ |
\( T^{3} + \cdots + 1021689891200 \)
|
| $47$ |
\( T^{3} + \cdots - 1149845848672 \)
|
| $53$ |
\( T^{3} + \cdots + 4161422398780 \)
|
| $59$ |
\( T^{3} + \cdots - 12735694359200 \)
|
| $61$ |
\( T^{3} + \cdots - 10752273603940 \)
|
| $67$ |
\( T^{3} + \cdots + 36912328239616 \)
|
| $71$ |
\( T^{3} + \cdots - 137777074006720 \)
|
| $73$ |
\( T^{3} + \cdots + 5146054668016 \)
|
| $79$ |
\( T^{3} + \cdots - 38185482179200 \)
|
| $83$ |
\( T^{3} + \cdots - 293039494873344 \)
|
| $89$ |
\( T^{3} + \cdots - 421429288376700 \)
|
| $97$ |
\( T^{3} + \cdots + 277986854215152 \)
|
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