Newspace parameters
| Level: | \( N \) | \(=\) | \( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1980.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(116.823781811\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.9192.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 18x + 30 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 18x + 30 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{2} + 12 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2\nu - 13 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta _1 + 12 \)
|
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 | 5.00000 | 0 | −22.8386 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 5.00000 | 0 | 2.58315 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 5.00000 | 0 | 15.2554 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
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 | 1980.4.a.l | 3 | |
| 3.b | odd | 2 | 1 | 220.4.a.f | ✓ | 3 | |
| 12.b | even | 2 | 1 | 880.4.a.w | 3 | ||
| 15.d | odd | 2 | 1 | 1100.4.a.i | 3 | ||
| 15.e | even | 4 | 2 | 1100.4.b.h | 6 | ||
| 33.d | even | 2 | 1 | 2420.4.a.i | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 220.4.a.f | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 880.4.a.w | 3 | 12.b | even | 2 | 1 | ||
| 1100.4.a.i | 3 | 15.d | odd | 2 | 1 | ||
| 1100.4.b.h | 6 | 15.e | even | 4 | 2 | ||
| 1980.4.a.l | 3 | 1.a | even | 1 | 1 | trivial | |
| 2420.4.a.i | 3 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1980))\):
|
\( T_{7}^{3} + 5T_{7}^{2} - 368T_{7} + 900 \)
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|
\( T_{17}^{3} + 77T_{17}^{2} - 5928T_{17} - 455496 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( (T - 5)^{3} \)
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| $7$ |
\( T^{3} + 5 T^{2} + \cdots + 900 \)
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| $11$ |
\( (T + 11)^{3} \)
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| $13$ |
\( T^{3} - 2 T^{2} + \cdots + 65360 \)
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| $17$ |
\( T^{3} + 77 T^{2} + \cdots - 455496 \)
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| $19$ |
\( T^{3} - 171 T^{2} + \cdots + 46336 \)
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| $23$ |
\( T^{3} + 222 T^{2} + \cdots - 1048464 \)
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| $29$ |
\( T^{3} + 55 T^{2} + \cdots - 1502868 \)
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| $31$ |
\( T^{3} - 181 T^{2} + \cdots + 10494720 \)
|
| $37$ |
\( T^{3} - 317 T^{2} + \cdots + 45828764 \)
|
| $41$ |
\( T^{3} + 302 T^{2} + \cdots - 4589880 \)
|
| $43$ |
\( T^{3} + 188 T^{2} + \cdots - 51956064 \)
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| $47$ |
\( T^{3} + 662 T^{2} + \cdots - 3431760 \)
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| $53$ |
\( T^{3} + 81 T^{2} + \cdots - 47414700 \)
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| $59$ |
\( T^{3} - 42 T^{2} + \cdots - 46277856 \)
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| $61$ |
\( T^{3} - 349 T^{2} + \cdots + 44132796 \)
|
| $67$ |
\( T^{3} - 152 T^{2} + \cdots - 30647104 \)
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| $71$ |
\( T^{3} + 927 T^{2} + \cdots - 103206528 \)
|
| $73$ |
\( T^{3} + 2378 T^{2} + \cdots + 374556144 \)
|
| $79$ |
\( T^{3} + 1146 T^{2} + \cdots + 14051104 \)
|
| $83$ |
\( T^{3} - 458 T^{2} + \cdots - 95875608 \)
|
| $89$ |
\( T^{3} - 875 T^{2} + \cdots - 3833436 \)
|
| $97$ |
\( T^{3} + \cdots - 1770497888 \)
|
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