Newspace parameters
| Level: | \( N \) | \(=\) | \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1452.f (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(39.5641343851\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{3})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 4x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 4x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 4\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 4\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times\).
| \(n\) | \(485\) | \(727\) | \(1333\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 241.1 |
|
0 | −1.73205 | 0 | 3.73205 | 0 | − | 2.44949i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||
| 241.2 | 0 | −1.73205 | 0 | 3.73205 | 0 | 2.44949i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||
| 241.3 | 0 | 1.73205 | 0 | 0.267949 | 0 | − | 2.44949i | 0 | 3.00000 | 0 | ||||||||||||||||||||||||||||||
| 241.4 | 0 | 1.73205 | 0 | 0.267949 | 0 | 2.44949i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 1452.3.f.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 4356.3.f.d | 4 | ||
| 11.b | odd | 2 | 1 | inner | 1452.3.f.a | ✓ | 4 |
| 33.d | even | 2 | 1 | 4356.3.f.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1452.3.f.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1452.3.f.a | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
| 4356.3.f.d | 4 | 3.b | odd | 2 | 1 | ||
| 4356.3.f.d | 4 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 4T_{5} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(1452, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 3)^{2} \)
|
| $5$ |
\( (T^{2} - 4 T + 1)^{2} \)
|
| $7$ |
\( (T^{2} + 6)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 268 T^{2} + 11881 \)
|
| $17$ |
\( T^{4} + 412 T^{2} + 24649 \)
|
| $19$ |
\( T^{4} + 64T^{2} + 256 \)
|
| $23$ |
\( (T^{2} + 2 T - 506)^{2} \)
|
| $29$ |
\( T^{4} + 412 T^{2} + 24649 \)
|
| $31$ |
\( (T^{2} + 24 T + 132)^{2} \)
|
| $37$ |
\( (T^{2} + 60 T + 393)^{2} \)
|
| $41$ |
\( T^{4} + 4548 T^{2} + 62001 \)
|
| $43$ |
\( (T^{2} + 18)^{2} \)
|
| $47$ |
\( (T^{2} + 56 T + 772)^{2} \)
|
| $53$ |
\( (T + 17)^{4} \)
|
| $59$ |
\( (T^{2} + 34 T + 262)^{2} \)
|
| $61$ |
\( T^{4} + 10524 T^{2} + 3218436 \)
|
| $67$ |
\( (T^{2} + 2 T - 4106)^{2} \)
|
| $71$ |
\( (T^{2} - 162 T + 6558)^{2} \)
|
| $73$ |
\( T^{4} + 13132 T^{2} + 42276004 \)
|
| $79$ |
\( T^{4} + 2368 T^{2} + 1364224 \)
|
| $83$ |
\( T^{4} + 30784 T^{2} + 201412864 \)
|
| $89$ |
\( (T^{2} + 200 T + 9853)^{2} \)
|
| $97$ |
\( (T^{2} + 216 T + 11589)^{2} \)
|
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