Newspace parameters
| Level: | \( N \) | \(=\) | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 495.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(13.4877730858\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{10}, \sqrt{22})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 16x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{4} - 16x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 15\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 15\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).
| \(n\) | \(46\) | \(56\) | \(397\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 |
|
−3.92635 | 0 | 11.4162 | −5.00000 | 0 | −6.21855 | −29.1186 | 0 | 19.6317 | ||||||||||||||||||||||||||||||
| 109.2 | −0.764069 | 0 | −3.41620 | −5.00000 | 0 | −12.5431 | 5.66649 | 0 | 3.82035 | |||||||||||||||||||||||||||||||
| 109.3 | 0.764069 | 0 | −3.41620 | −5.00000 | 0 | 12.5431 | −5.66649 | 0 | −3.82035 | |||||||||||||||||||||||||||||||
| 109.4 | 3.92635 | 0 | 11.4162 | −5.00000 | 0 | 6.21855 | 29.1186 | 0 | −19.6317 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 55.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-55}) \) |
| 5.b | even | 2 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 495.3.h.e | ✓ | 4 |
| 3.b | odd | 2 | 1 | 495.3.h.f | yes | 4 | |
| 5.b | even | 2 | 1 | inner | 495.3.h.e | ✓ | 4 |
| 11.b | odd | 2 | 1 | inner | 495.3.h.e | ✓ | 4 |
| 15.d | odd | 2 | 1 | 495.3.h.f | yes | 4 | |
| 33.d | even | 2 | 1 | 495.3.h.f | yes | 4 | |
| 55.d | odd | 2 | 1 | CM | 495.3.h.e | ✓ | 4 |
| 165.d | even | 2 | 1 | 495.3.h.f | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 495.3.h.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 495.3.h.e | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 495.3.h.e | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
| 495.3.h.e | ✓ | 4 | 55.d | odd | 2 | 1 | CM |
| 495.3.h.f | yes | 4 | 3.b | odd | 2 | 1 | |
| 495.3.h.f | yes | 4 | 15.d | odd | 2 | 1 | |
| 495.3.h.f | yes | 4 | 33.d | even | 2 | 1 | |
| 495.3.h.f | yes | 4 | 165.d | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(495, [\chi])\):
|
\( T_{2}^{4} - 16T_{2}^{2} + 9 \)
|
|
\( T_{7}^{4} - 196T_{7}^{2} + 6084 \)
|
|
\( T_{89} - 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 16T^{2} + 9 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 5)^{4} \)
|
| $7$ |
\( T^{4} - 196T^{2} + 6084 \)
|
| $11$ |
\( (T + 11)^{4} \)
|
| $13$ |
\( T^{4} - 676 T^{2} + 26244 \)
|
| $17$ |
\( T^{4} - 1156 T^{2} + 161604 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( (T^{2} - 3520)^{2} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} - 7396 T^{2} + 12404484 \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( (T^{2} - 3520)^{2} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( (T^{2} - 14080)^{2} \)
|
| $73$ |
\( T^{4} - 21316 T^{2} + 113167044 \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} - 27556 T^{2} + 185014404 \)
|
| $89$ |
\( (T - 2)^{4} \)
|
| $97$ |
\( T^{4} \)
|
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