Newspace parameters
| Level: | \( N \) | \(=\) | \( 55 = 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 55.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.24510505032\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.568.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 6x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{3} - x^{2} - 6x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 4 \)
|
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 |
|
−3.38776 | −9.87859 | 3.47689 | −5.00000 | 33.4663 | −19.5308 | 15.3232 | 70.5866 | 16.9388 | |||||||||||||||||||||||||||
| 1.2 | 3.77801 | 7.42401 | 6.27334 | −5.00000 | 28.0480 | −28.7933 | −6.52332 | 28.1159 | −18.8900 | ||||||||||||||||||||||||||||
| 1.3 | 4.60975 | −0.545414 | 13.2498 | −5.00000 | −2.51422 | 33.3241 | 24.2001 | −26.7025 | −23.0487 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(11\) | \( +1 \) |
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 | 55.4.a.c | ✓ | 3 |
| 3.b | odd | 2 | 1 | 495.4.a.f | 3 | ||
| 4.b | odd | 2 | 1 | 880.4.a.x | 3 | ||
| 5.b | even | 2 | 1 | 275.4.a.d | 3 | ||
| 5.c | odd | 4 | 2 | 275.4.b.d | 6 | ||
| 11.b | odd | 2 | 1 | 605.4.a.h | 3 | ||
| 15.d | odd | 2 | 1 | 2475.4.a.ba | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.4.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 275.4.a.d | 3 | 5.b | even | 2 | 1 | ||
| 275.4.b.d | 6 | 5.c | odd | 4 | 2 | ||
| 495.4.a.f | 3 | 3.b | odd | 2 | 1 | ||
| 605.4.a.h | 3 | 11.b | odd | 2 | 1 | ||
| 880.4.a.x | 3 | 4.b | odd | 2 | 1 | ||
| 2475.4.a.ba | 3 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 5T_{2}^{2} - 11T_{2} + 59 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(55))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 5 T^{2} + \cdots + 59 \)
|
| $3$ |
\( T^{3} + 3 T^{2} + \cdots - 40 \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( T^{3} + 15 T^{2} + \cdots - 18740 \)
|
| $11$ |
\( (T + 11)^{3} \)
|
| $13$ |
\( T^{3} + 14 T^{2} + \cdots + 50800 \)
|
| $17$ |
\( T^{3} - 91 T^{2} + \cdots + 1256 \)
|
| $19$ |
\( T^{3} + 19 T^{2} + \cdots - 549824 \)
|
| $23$ |
\( T^{3} - 330 T^{2} + \cdots + 1346224 \)
|
| $29$ |
\( T^{3} - 413 T^{2} + \cdots - 296164 \)
|
| $31$ |
\( T^{3} + 127 T^{2} + \cdots - 7744 \)
|
| $37$ |
\( T^{3} + 543 T^{2} + \cdots + 4866188 \)
|
| $41$ |
\( T^{3} - 58 T^{2} + \cdots + 8259848 \)
|
| $43$ |
\( T^{3} - 532 T^{2} + \cdots + 26232736 \)
|
| $47$ |
\( T^{3} - 478 T^{2} + \cdots + 907024 \)
|
| $53$ |
\( T^{3} - 261 T^{2} + \cdots - 581092 \)
|
| $59$ |
\( T^{3} + 630 T^{2} + \cdots - 215392 \)
|
| $61$ |
\( T^{3} - 1095 T^{2} + \cdots - 20829740 \)
|
| $67$ |
\( T^{3} + 976 T^{2} + \cdots + 20007488 \)
|
| $71$ |
\( T^{3} + 445 T^{2} + \cdots + 69739136 \)
|
| $73$ |
\( T^{3} + 1838 T^{2} + \cdots + 163030384 \)
|
| $79$ |
\( T^{3} - 318 T^{2} + \cdots + 502905248 \)
|
| $83$ |
\( T^{3} + 330 T^{2} + \cdots + 37348312 \)
|
| $89$ |
\( T^{3} + \cdots + 1186825660 \)
|
| $97$ |
\( T^{3} + 2744 T^{2} + \cdots - 146434160 \)
|
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