Newspace parameters
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.81146522936\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{10}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} - 10 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{10}\). 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 |
|
−18.9737 | 0 | 232.000 | 303.579 | 0 | 260.000 | −1973.26 | 0 | −5760.00 | ||||||||||||||||||||||||
| 1.2 | 18.9737 | 0 | 232.000 | −303.579 | 0 | 260.000 | 1973.26 | 0 | −5760.00 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 9.8.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | inner | 9.8.a.b | ✓ | 2 |
| 4.b | odd | 2 | 1 | 144.8.a.m | 2 | ||
| 5.b | even | 2 | 1 | 225.8.a.q | 2 | ||
| 5.c | odd | 4 | 2 | 225.8.b.k | 4 | ||
| 7.b | odd | 2 | 1 | 441.8.a.k | 2 | ||
| 8.b | even | 2 | 1 | 576.8.a.bj | 2 | ||
| 8.d | odd | 2 | 1 | 576.8.a.bi | 2 | ||
| 9.c | even | 3 | 2 | 81.8.c.f | 4 | ||
| 9.d | odd | 6 | 2 | 81.8.c.f | 4 | ||
| 12.b | even | 2 | 1 | 144.8.a.m | 2 | ||
| 15.d | odd | 2 | 1 | 225.8.a.q | 2 | ||
| 15.e | even | 4 | 2 | 225.8.b.k | 4 | ||
| 21.c | even | 2 | 1 | 441.8.a.k | 2 | ||
| 24.f | even | 2 | 1 | 576.8.a.bi | 2 | ||
| 24.h | odd | 2 | 1 | 576.8.a.bj | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.8.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 9.8.a.b | ✓ | 2 | 3.b | odd | 2 | 1 | inner |
| 81.8.c.f | 4 | 9.c | even | 3 | 2 | ||
| 81.8.c.f | 4 | 9.d | odd | 6 | 2 | ||
| 144.8.a.m | 2 | 4.b | odd | 2 | 1 | ||
| 144.8.a.m | 2 | 12.b | even | 2 | 1 | ||
| 225.8.a.q | 2 | 5.b | even | 2 | 1 | ||
| 225.8.a.q | 2 | 15.d | odd | 2 | 1 | ||
| 225.8.b.k | 4 | 5.c | odd | 4 | 2 | ||
| 225.8.b.k | 4 | 15.e | even | 4 | 2 | ||
| 441.8.a.k | 2 | 7.b | odd | 2 | 1 | ||
| 441.8.a.k | 2 | 21.c | even | 2 | 1 | ||
| 576.8.a.bi | 2 | 8.d | odd | 2 | 1 | ||
| 576.8.a.bi | 2 | 24.f | even | 2 | 1 | ||
| 576.8.a.bj | 2 | 8.b | even | 2 | 1 | ||
| 576.8.a.bj | 2 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 360 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 360 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 92160 \)
|
| $7$ |
\( (T - 260)^{2} \)
|
| $11$ |
\( T^{2} - 36864000 \)
|
| $13$ |
\( (T - 6890)^{2} \)
|
| $17$ |
\( T^{2} - 560701440 \)
|
| $19$ |
\( (T - 33176)^{2} \)
|
| $23$ |
\( T^{2} - 996802560 \)
|
| $29$ |
\( T^{2} - 19079424000 \)
|
| $31$ |
\( (T - 1508)^{2} \)
|
| $37$ |
\( (T + 380770)^{2} \)
|
| $41$ |
\( T^{2} - 7750656000 \)
|
| $43$ |
\( (T - 7640)^{2} \)
|
| $47$ |
\( T^{2} - 320209551360 \)
|
| $53$ |
\( T^{2} - 1060987299840 \)
|
| $59$ |
\( T^{2} - 7332839424000 \)
|
| $61$ |
\( (T + 988858)^{2} \)
|
| $67$ |
\( (T - 3857360)^{2} \)
|
| $71$ |
\( T^{2} - 17857511424000 \)
|
| $73$ |
\( (T + 2004730)^{2} \)
|
| $79$ |
\( (T - 2699684)^{2} \)
|
| $83$ |
\( T^{2} - 7352582307840 \)
|
| $89$ |
\( T^{2} - 59927040000000 \)
|
| $97$ |
\( (T + 12957490)^{2} \)
|
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