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: | \(\Q(\sqrt{3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
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
\( T_{2}^{2} + 2T_{2} - 2 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(99))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 2T - 2 \)
$3$
\( T^{2} \)
$5$
\( T^{2} + 2T - 191 \)
$7$
\( T^{2} - 20T + 52 \)
$11$
\( (T - 11)^{2} \)
$13$
\( T^{2} - 80T + 400 \)
$17$
\( T^{2} - 124T + 3412 \)
$19$
\( T^{2} - 72T - 9504 \)
$23$
\( T^{2} - 98T - 1487 \)
$29$
\( T^{2} + 144T - 4224 \)
$31$
\( T^{2} + 34T - 2063 \)
$37$
\( T^{2} - 54T + 537 \)
$41$
\( T^{2} + 536T + 71776 \)
$43$
\( T^{2} + 60T + 132 \)
$47$
\( T^{2} - 272T - 24704 \)
$53$
\( T^{2} - 492T + 51108 \)
$59$
\( T^{2} + 634T + 48217 \)
$61$
\( T^{2} - 840T + 74832 \)
$67$
\( T^{2} - 754T + 140929 \)
$71$
\( T^{2} - 678T + 97593 \)
$73$
\( T^{2} + 400T - 617072 \)
$79$
\( T^{2} - 316 T - 1266044 \)
$83$
\( T^{2} + 468T + 11556 \)
$89$
\( T^{2} - 1842 T + 525489 \)
$97$
\( T^{2} - 2194 T + 1141201 \)
show more
show less