Newspace parameters
| Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 252.t (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.01223013094\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
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|
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| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 2\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 4\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{3} + 4\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | 0 | 0 | −1.22474 | − | 2.12132i | 0 | 0.500000 | − | 2.59808i | 0 | 0 | 0 | ||||||||||||||||||||||||||
| 17.2 | 0 | 0 | 0 | 1.22474 | + | 2.12132i | 0 | 0.500000 | − | 2.59808i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 89.1 | 0 | 0 | 0 | −1.22474 | + | 2.12132i | 0 | 0.500000 | + | 2.59808i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 89.2 | 0 | 0 | 0 | 1.22474 | − | 2.12132i | 0 | 0.500000 | + | 2.59808i | 0 | 0 | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
| 21.g | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(252, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 6T^{2} + 36 \)
|
| $7$ |
\( (T^{2} - T + 7)^{2} \)
|
| $11$ |
\( T^{4} - 18T^{2} + 324 \)
|
| $13$ |
\( (T^{2} + 3)^{2} \)
|
| $17$ |
\( T^{4} + 24T^{2} + 576 \)
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| $19$ |
\( (T^{2} - 9 T + 27)^{2} \)
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| $23$ |
\( T^{4} \)
|
| $29$ |
\( (T^{2} + 72)^{2} \)
|
| $31$ |
\( (T^{2} + 3 T + 3)^{2} \)
|
| $37$ |
\( (T^{2} - 5 T + 25)^{2} \)
|
| $41$ |
\( (T^{2} - 150)^{2} \)
|
| $43$ |
\( (T + 11)^{4} \)
|
| $47$ |
\( T^{4} + 6T^{2} + 36 \)
|
| $53$ |
\( T^{4} - 72T^{2} + 5184 \)
|
| $59$ |
\( T^{4} + 24T^{2} + 576 \)
|
| $61$ |
\( (T^{2} + 6 T + 12)^{2} \)
|
| $67$ |
\( (T^{2} - 7 T + 49)^{2} \)
|
| $71$ |
\( (T^{2} + 162)^{2} \)
|
| $73$ |
\( (T^{2} - 27 T + 243)^{2} \)
|
| $79$ |
\( (T^{2} + 11 T + 121)^{2} \)
|
| $83$ |
\( (T^{2} - 150)^{2} \)
|
| $89$ |
\( T^{4} + 216 T^{2} + 46656 \)
|
| $97$ |
\( (T^{2} + 12)^{2} \)
|
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