Newspace parameters
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 28 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(41.5670017354\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{30001}) \) |
| Defining polynomial: |
\( x^{2} - x - 7500 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 3^{2}\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 3) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 63\sqrt{30001}\). 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 |
|
−21703.1 | 0 | 3.36807e8 | 6.93920e8 | 0 | 3.81056e11 | −4.39681e12 | 0 | −1.50602e13 | ||||||||||||||||||||||||
| 1.2 | 121.102 | 0 | −1.34203e8 | 1.07803e9 | 0 | −1.13904e10 | −3.25063e10 | 0 | 1.30551e11 | |||||||||||||||||||||||||
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 | 9.28.a.b | 2 | |
| 3.b | odd | 2 | 1 | 3.28.a.b | ✓ | 2 | |
| 12.b | even | 2 | 1 | 48.28.a.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.28.a.b | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 9.28.a.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 48.28.a.e | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 21582T_{2} - 2628288 \)
acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 21582 T - 2628288 \)
$3$
\( T^{2} \)
$5$
\( T^{2} - 1771946100 T + 74\!\cdots\!00 \)
$7$
\( T^{2} - 369665199904 T - 43\!\cdots\!80 \)
$11$
\( T^{2} + 75762335668248 T - 63\!\cdots\!48 \)
$13$
\( T^{2} + 103021079177588 T - 15\!\cdots\!08 \)
$17$
\( T^{2} + \cdots - 24\!\cdots\!84 \)
$19$
\( T^{2} + \cdots + 21\!\cdots\!80 \)
$23$
\( T^{2} + \cdots - 17\!\cdots\!40 \)
$29$
\( T^{2} + \cdots - 15\!\cdots\!40 \)
$31$
\( T^{2} + \cdots - 10\!\cdots\!00 \)
$37$
\( T^{2} + \cdots + 34\!\cdots\!20 \)
$41$
\( T^{2} + \cdots - 34\!\cdots\!20 \)
$43$
\( T^{2} + \cdots + 90\!\cdots\!76 \)
$47$
\( T^{2} + \cdots + 16\!\cdots\!64 \)
$53$
\( T^{2} + \cdots + 45\!\cdots\!20 \)
$59$
\( T^{2} + \cdots - 75\!\cdots\!00 \)
$61$
\( T^{2} + \cdots + 88\!\cdots\!16 \)
$67$
\( T^{2} + \cdots + 41\!\cdots\!56 \)
$71$
\( T^{2} + \cdots - 41\!\cdots\!36 \)
$73$
\( T^{2} + \cdots + 15\!\cdots\!20 \)
$79$
\( T^{2} + \cdots + 43\!\cdots\!00 \)
$83$
\( T^{2} + \cdots + 36\!\cdots\!28 \)
$89$
\( T^{2} + \cdots - 45\!\cdots\!20 \)
$97$
\( T^{2} + \cdots + 44\!\cdots\!96 \)
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