Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.22213583018\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-14}) \) |
| Defining polynomial: | \(x^{2} + 14\) |
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{-14}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
− | 22.4499i | 45.0000 | + | 67.3498i | −248.000 | 224.499i | 1512.00 | − | 1010.25i | −1750.00 | − | 179.600i | −2511.00 | + | 6061.48i | 5040.00 | ||||||||||||||||
| 2.2 | 22.4499i | 45.0000 | − | 67.3498i | −248.000 | − | 224.499i | 1512.00 | + | 1010.25i | −1750.00 | 179.600i | −2511.00 | − | 6061.48i | 5040.00 | ||||||||||||||||||
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 | 3.9.b.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | inner | 3.9.b.a | ✓ | 2 |
| 4.b | odd | 2 | 1 | 48.9.e.b | 2 | ||
| 5.b | even | 2 | 1 | 75.9.c.c | 2 | ||
| 5.c | odd | 4 | 2 | 75.9.d.b | 4 | ||
| 8.b | even | 2 | 1 | 192.9.e.e | 2 | ||
| 8.d | odd | 2 | 1 | 192.9.e.f | 2 | ||
| 9.c | even | 3 | 2 | 81.9.d.d | 4 | ||
| 9.d | odd | 6 | 2 | 81.9.d.d | 4 | ||
| 12.b | even | 2 | 1 | 48.9.e.b | 2 | ||
| 15.d | odd | 2 | 1 | 75.9.c.c | 2 | ||
| 15.e | even | 4 | 2 | 75.9.d.b | 4 | ||
| 24.f | even | 2 | 1 | 192.9.e.f | 2 | ||
| 24.h | odd | 2 | 1 | 192.9.e.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.9.b.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 3.9.b.a | ✓ | 2 | 3.b | odd | 2 | 1 | inner |
| 48.9.e.b | 2 | 4.b | odd | 2 | 1 | ||
| 48.9.e.b | 2 | 12.b | even | 2 | 1 | ||
| 75.9.c.c | 2 | 5.b | even | 2 | 1 | ||
| 75.9.c.c | 2 | 15.d | odd | 2 | 1 | ||
| 75.9.d.b | 4 | 5.c | odd | 4 | 2 | ||
| 75.9.d.b | 4 | 15.e | even | 4 | 2 | ||
| 81.9.d.d | 4 | 9.c | even | 3 | 2 | ||
| 81.9.d.d | 4 | 9.d | odd | 6 | 2 | ||
| 192.9.e.e | 2 | 8.b | even | 2 | 1 | ||
| 192.9.e.e | 2 | 24.h | odd | 2 | 1 | ||
| 192.9.e.f | 2 | 8.d | odd | 2 | 1 | ||
| 192.9.e.f | 2 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(3, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( 504 + T^{2} \) |
| $3$ | \( 6561 - 90 T + T^{2} \) |
| $5$ | \( 50400 + T^{2} \) |
| $7$ | \( ( 1750 + T )^{2} \) |
| $11$ | \( 48434400 + T^{2} \) |
| $13$ | \( ( -25730 + T )^{2} \) |
| $17$ | \( 5608963584 + T^{2} \) |
| $19$ | \( ( -18938 + T )^{2} \) |
| $23$ | \( 221333583744 + T^{2} \) |
| $29$ | \( 212426373600 + T^{2} \) |
| $31$ | \( ( 351478 + T )^{2} \) |
| $37$ | \( ( -1335170 + T )^{2} \) |
| $41$ | \( 3517381526400 + T^{2} \) |
| $43$ | \( ( 3526150 + T )^{2} \) |
| $47$ | \( 16654893018624 + T^{2} \) |
| $53$ | \( 43583305427424 + T^{2} \) |
| $59$ | \( 188098358162400 + T^{2} \) |
| $61$ | \( ( -753602 + T )^{2} \) |
| $67$ | \( ( -2268890 + T )^{2} \) |
| $71$ | \( 289748374473600 + T^{2} \) |
| $73$ | \( ( -27672770 + T )^{2} \) |
| $79$ | \( ( 22980982 + T )^{2} \) |
| $83$ | \( 2152513621054944 + T^{2} \) |
| $89$ | \( 5272973501846400 + T^{2} \) |
| $97$ | \( ( -147271010 + T )^{2} \) |
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