Newspace parameters
| Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 99.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.84118909057\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{3}) \) |
| Defining polynomial: | \(x^{2} - 3\) |
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 11) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). 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 |
|
−2.73205 | 0 | −0.535898 | −14.8564 | 0 | 3.07180 | 23.3205 | 0 | 40.5885 | ||||||||||||||||||||||||
| 1.2 | 0.732051 | 0 | −7.46410 | 12.8564 | 0 | 16.9282 | −11.3205 | 0 | 9.41154 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 99.4.a.c | 2 | |
| 3.b | odd | 2 | 1 | 11.4.a.a | ✓ | 2 | |
| 4.b | odd | 2 | 1 | 1584.4.a.bc | 2 | ||
| 5.b | even | 2 | 1 | 2475.4.a.q | 2 | ||
| 11.b | odd | 2 | 1 | 1089.4.a.v | 2 | ||
| 12.b | even | 2 | 1 | 176.4.a.i | 2 | ||
| 15.d | odd | 2 | 1 | 275.4.a.b | 2 | ||
| 15.e | even | 4 | 2 | 275.4.b.c | 4 | ||
| 21.c | even | 2 | 1 | 539.4.a.e | 2 | ||
| 24.f | even | 2 | 1 | 704.4.a.n | 2 | ||
| 24.h | odd | 2 | 1 | 704.4.a.p | 2 | ||
| 33.d | even | 2 | 1 | 121.4.a.c | 2 | ||
| 33.f | even | 10 | 4 | 121.4.c.f | 8 | ||
| 33.h | odd | 10 | 4 | 121.4.c.c | 8 | ||
| 39.d | odd | 2 | 1 | 1859.4.a.a | 2 | ||
| 132.d | odd | 2 | 1 | 1936.4.a.w | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.4.a.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 99.4.a.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 121.4.a.c | 2 | 33.d | even | 2 | 1 | ||
| 121.4.c.c | 8 | 33.h | odd | 10 | 4 | ||
| 121.4.c.f | 8 | 33.f | even | 10 | 4 | ||
| 176.4.a.i | 2 | 12.b | even | 2 | 1 | ||
| 275.4.a.b | 2 | 15.d | odd | 2 | 1 | ||
| 275.4.b.c | 4 | 15.e | even | 4 | 2 | ||
| 539.4.a.e | 2 | 21.c | even | 2 | 1 | ||
| 704.4.a.n | 2 | 24.f | even | 2 | 1 | ||
| 704.4.a.p | 2 | 24.h | odd | 2 | 1 | ||
| 1089.4.a.v | 2 | 11.b | odd | 2 | 1 | ||
| 1584.4.a.bc | 2 | 4.b | odd | 2 | 1 | ||
| 1859.4.a.a | 2 | 39.d | odd | 2 | 1 | ||
| 1936.4.a.w | 2 | 132.d | odd | 2 | 1 | ||
| 2475.4.a.q | 2 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 T_{2} - 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(99))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( -2 + 2 T + T^{2} \) |
| $3$ | \( T^{2} \) |
| $5$ | \( -191 + 2 T + T^{2} \) |
| $7$ | \( 52 - 20 T + T^{2} \) |
| $11$ | \( ( -11 + T )^{2} \) |
| $13$ | \( 400 - 80 T + T^{2} \) |
| $17$ | \( 3412 - 124 T + T^{2} \) |
| $19$ | \( -9504 - 72 T + T^{2} \) |
| $23$ | \( -1487 - 98 T + T^{2} \) |
| $29$ | \( -4224 + 144 T + T^{2} \) |
| $31$ | \( -2063 + 34 T + T^{2} \) |
| $37$ | \( 537 - 54 T + T^{2} \) |
| $41$ | \( 71776 + 536 T + T^{2} \) |
| $43$ | \( 132 + 60 T + T^{2} \) |
| $47$ | \( -24704 - 272 T + T^{2} \) |
| $53$ | \( 51108 - 492 T + T^{2} \) |
| $59$ | \( 48217 + 634 T + T^{2} \) |
| $61$ | \( 74832 - 840 T + T^{2} \) |
| $67$ | \( 140929 - 754 T + T^{2} \) |
| $71$ | \( 97593 - 678 T + T^{2} \) |
| $73$ | \( -617072 + 400 T + T^{2} \) |
| $79$ | \( -1266044 - 316 T + T^{2} \) |
| $83$ | \( 11556 + 468 T + T^{2} \) |
| $89$ | \( 525489 - 1842 T + T^{2} \) |
| $97$ | \( 1141201 - 2194 T + T^{2} \) |
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