Newspace parameters
| Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 144.e (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(3.92371580679\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 18) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(65\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | 0 | 0 | − | 4.24264i | 0 | 4.00000 | 0 | 0 | 0 | |||||||||||||||||||||||
| 17.2 | 0 | 0 | 0 | 4.24264i | 0 | 4.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 18 \)
acting on \(S_{3}^{\mathrm{new}}(144, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} + 18 \)
$7$
\( (T - 4)^{2} \)
$11$
\( T^{2} + 288 \)
$13$
\( (T - 8)^{2} \)
$17$
\( T^{2} + 162 \)
$19$
\( (T - 16)^{2} \)
$23$
\( T^{2} + 288 \)
$29$
\( T^{2} + 18 \)
$31$
\( (T + 44)^{2} \)
$37$
\( (T + 34)^{2} \)
$41$
\( T^{2} + 2178 \)
$43$
\( (T - 40)^{2} \)
$47$
\( T^{2} + 7200 \)
$53$
\( T^{2} + 1458 \)
$59$
\( T^{2} + 1152 \)
$61$
\( (T - 50)^{2} \)
$67$
\( (T + 8)^{2} \)
$71$
\( T^{2} + 2592 \)
$73$
\( (T + 16)^{2} \)
$79$
\( (T - 76)^{2} \)
$83$
\( T^{2} + 14112 \)
$89$
\( T^{2} + 162 \)
$97$
\( (T - 176)^{2} \)
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