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: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). 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 |
|
−2.46410 | 0 | −1.92820 | 13.4641 | 0 | −25.3205 | 24.4641 | 0 | −33.1769 | ||||||||||||||||||||||||
| 1.2 | 4.46410 | 0 | 11.9282 | 6.53590 | 0 | 9.32051 | 17.5359 | 0 | 29.1769 | |||||||||||||||||||||||||
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.g | yes | 2 |
| 3.b | odd | 2 | 1 | 99.4.a.d | ✓ | 2 | |
| 4.b | odd | 2 | 1 | 1584.4.a.bk | 2 | ||
| 5.b | even | 2 | 1 | 2475.4.a.m | 2 | ||
| 11.b | odd | 2 | 1 | 1089.4.a.l | 2 | ||
| 12.b | even | 2 | 1 | 1584.4.a.w | 2 | ||
| 15.d | odd | 2 | 1 | 2475.4.a.r | 2 | ||
| 33.d | even | 2 | 1 | 1089.4.a.w | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 99.4.a.d | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 99.4.a.g | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 1089.4.a.l | 2 | 11.b | odd | 2 | 1 | ||
| 1089.4.a.w | 2 | 33.d | even | 2 | 1 | ||
| 1584.4.a.w | 2 | 12.b | even | 2 | 1 | ||
| 1584.4.a.bk | 2 | 4.b | odd | 2 | 1 | ||
| 2475.4.a.m | 2 | 5.b | even | 2 | 1 | ||
| 2475.4.a.r | 2 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 2T_{2} - 11 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(99))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 2T - 11 \)
$3$
\( T^{2} \)
$5$
\( T^{2} - 20T + 88 \)
$7$
\( T^{2} + 16T - 236 \)
$11$
\( (T + 11)^{2} \)
$13$
\( T^{2} - 80T + 148 \)
$17$
\( T^{2} - 164T + 6292 \)
$19$
\( T^{2} + 36T - 10476 \)
$23$
\( T^{2} - 172T + 6424 \)
$29$
\( T^{2} - 108T - 34716 \)
$31$
\( T^{2} + 448T + 42064 \)
$37$
\( T^{2} - 108T + 1716 \)
$41$
\( T^{2} - 212T + 11044 \)
$43$
\( T^{2} - 156T - 95484 \)
$47$
\( T^{2} + 20T - 5192 \)
$53$
\( T^{2} + 132T - 415272 \)
$59$
\( T^{2} - 688T - 6512 \)
$61$
\( T^{2} + 96T - 51564 \)
$67$
\( T^{2} - 448T - 146432 \)
$71$
\( T^{2} - 132T - 903144 \)
$73$
\( T^{2} - 428T + 30244 \)
$79$
\( T^{2} - 424T + 44932 \)
$83$
\( T^{2} - 720T + 118800 \)
$89$
\( T^{2} - 1056T - 92928 \)
$97$
\( T^{2} - 52T - 2238812 \)
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