Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(8.38432032861\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{649}) \) |
| Defining polynomial: |
\( x^{2} - x - 162 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 3^{2}\cdot 7 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 63\sqrt{649}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−1271.96 | 59049.0 | −479282. | −2.12776e7 | −7.51077e7 | 6.32076e8 | 3.27711e9 | 3.48678e9 | 2.70641e10 | ||||||||||||||||||||||||
| 1.2 | 1937.96 | 59049.0 | 1.65852e6 | 2.22745e7 | 1.14434e8 | 4.78205e7 | −8.50053e8 | 3.48678e9 | 4.31669e10 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
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 | 3.22.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 9.22.a.e | 2 | ||
| 4.b | odd | 2 | 1 | 48.22.a.g | 2 | ||
| 5.b | even | 2 | 1 | 75.22.a.d | 2 | ||
| 5.c | odd | 4 | 2 | 75.22.b.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.22.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 9.22.a.e | 2 | 3.b | odd | 2 | 1 | ||
| 48.22.a.g | 2 | 4.b | odd | 2 | 1 | ||
| 75.22.a.d | 2 | 5.b | even | 2 | 1 | ||
| 75.22.b.d | 4 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 666T_{2} - 2464992 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 666 T - 2464992 \)
$3$
\( (T - 59049)^{2} \)
$5$
\( T^{2} + \cdots - 473947100199900 \)
$7$
\( T^{2} - 679896112 T + 30\!\cdots\!00 \)
$11$
\( T^{2} - 219869122968 T + 95\!\cdots\!12 \)
$13$
\( T^{2} + 48468909956 T - 58\!\cdots\!92 \)
$17$
\( T^{2} + 11333529041436 T + 24\!\cdots\!24 \)
$19$
\( T^{2} - 11960585011624 T - 12\!\cdots\!20 \)
$23$
\( T^{2} + 146508390063504 T - 85\!\cdots\!00 \)
$29$
\( T^{2} + \cdots - 26\!\cdots\!00 \)
$31$
\( T^{2} + \cdots + 23\!\cdots\!00 \)
$37$
\( T^{2} + \cdots - 54\!\cdots\!20 \)
$41$
\( T^{2} + \cdots - 32\!\cdots\!80 \)
$43$
\( T^{2} + \cdots + 20\!\cdots\!44 \)
$47$
\( T^{2} + \cdots + 17\!\cdots\!16 \)
$53$
\( T^{2} + \cdots - 20\!\cdots\!60 \)
$59$
\( T^{2} + \cdots - 83\!\cdots\!20 \)
$61$
\( T^{2} + \cdots - 74\!\cdots\!84 \)
$67$
\( T^{2} + \cdots + 38\!\cdots\!04 \)
$71$
\( T^{2} + \cdots + 73\!\cdots\!84 \)
$73$
\( T^{2} + \cdots - 11\!\cdots\!40 \)
$79$
\( T^{2} + \cdots + 69\!\cdots\!00 \)
$83$
\( T^{2} + \cdots + 33\!\cdots\!12 \)
$89$
\( T^{2} + \cdots - 17\!\cdots\!80 \)
$97$
\( T^{2} + \cdots - 27\!\cdots\!76 \)
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