Newspace parameters
| Level: | \( N \) | \(=\) | \( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9900.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(79.0518980011\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{3}, \sqrt{19})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 11x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 220) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 11x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 3\nu ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 11\nu ) / 4 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 6 \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 6 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{2} + 11\beta_1 ) / 2 \)
|
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 |
|
0 | 0 | 0 | 0 | 0 | −3.46410 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 0 | 0 | −3.46410 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 0 | 0 | 3.46410 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 0 | 0 | 3.46410 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 9900.2.a.cb | 4 | |
| 3.b | odd | 2 | 1 | 1100.2.a.j | 4 | ||
| 5.b | even | 2 | 1 | inner | 9900.2.a.cb | 4 | |
| 5.c | odd | 4 | 2 | 1980.2.c.g | 4 | ||
| 12.b | even | 2 | 1 | 4400.2.a.cd | 4 | ||
| 15.d | odd | 2 | 1 | 1100.2.a.j | 4 | ||
| 15.e | even | 4 | 2 | 220.2.b.b | ✓ | 4 | |
| 60.h | even | 2 | 1 | 4400.2.a.cd | 4 | ||
| 60.l | odd | 4 | 2 | 880.2.b.i | 4 | ||
| 165.l | odd | 4 | 2 | 2420.2.b.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 220.2.b.b | ✓ | 4 | 15.e | even | 4 | 2 | |
| 880.2.b.i | 4 | 60.l | odd | 4 | 2 | ||
| 1100.2.a.j | 4 | 3.b | odd | 2 | 1 | ||
| 1100.2.a.j | 4 | 15.d | odd | 2 | 1 | ||
| 1980.2.c.g | 4 | 5.c | odd | 4 | 2 | ||
| 2420.2.b.e | 4 | 165.l | odd | 4 | 2 | ||
| 4400.2.a.cd | 4 | 12.b | even | 2 | 1 | ||
| 4400.2.a.cd | 4 | 60.h | even | 2 | 1 | ||
| 9900.2.a.cb | 4 | 1.a | even | 1 | 1 | trivial | |
| 9900.2.a.cb | 4 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9900))\):
|
\( T_{7}^{2} - 12 \)
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\( T_{13}^{4} - 44T_{13}^{2} + 256 \)
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\( T_{17}^{2} - 12 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( (T^{2} - 12)^{2} \)
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| $11$ |
\( (T + 1)^{4} \)
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| $13$ |
\( T^{4} - 44T^{2} + 256 \)
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| $17$ |
\( (T^{2} - 12)^{2} \)
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| $19$ |
\( (T - 4)^{4} \)
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| $23$ |
\( T^{4} - 83T^{2} + 1024 \)
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| $29$ |
\( (T^{2} - 2 T - 56)^{2} \)
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| $31$ |
\( (T^{2} - 11 T + 16)^{2} \)
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| $37$ |
\( T^{4} - 83T^{2} + 1024 \)
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| $41$ |
\( (T^{2} + 2 T - 56)^{2} \)
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| $43$ |
\( (T^{2} - 12)^{2} \)
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| $47$ |
\( T^{4} - 44T^{2} + 256 \)
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| $53$ |
\( T^{4} \)
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| $59$ |
\( (T^{2} + 21 T + 96)^{2} \)
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| $61$ |
\( (T^{2} + 2 T - 56)^{2} \)
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| $67$ |
\( T^{4} - 123T^{2} + 576 \)
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| $71$ |
\( (T^{2} + 3 T - 12)^{2} \)
|
| $73$ |
\( T^{4} - 348 T^{2} + 28224 \)
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| $79$ |
\( (T^{2} + 6 T - 48)^{2} \)
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| $83$ |
\( T^{4} - 248T^{2} + 784 \)
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| $89$ |
\( (T^{2} + 9 T + 6)^{2} \)
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| $97$ |
\( T^{4} - 99T^{2} + 1296 \)
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