Newspace parameters
| Level: | \( N \) | \(=\) | \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2178.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(128.506159993\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.5157648.2 |
|
|
|
| Defining polynomial: |
\( x^{4} - 123x^{2} - 132x + 2148 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 726) |
| 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
\( x^{4} - 123x^{2} - 132x + 2148 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 22\nu^{2} - 63\nu + 1254 ) / 424 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11\nu^{3} - 30\nu^{2} - 693\nu + 650 ) / 212 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -21\nu^{3} + 38\nu^{2} + 2595\nu - 46 ) / 424 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta_1 ) / 3 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 22\beta _1 + 62 \)
|
| \(\nu^{3}\) | \(=\) |
\( 21\beta_{3} + 43\beta_{2} - 81\beta _1 + 110 \)
|
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 |
|
−2.00000 | 0 | 4.00000 | −12.2570 | 0 | 11.9445 | −8.00000 | 0 | 24.5139 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | −10.2063 | 0 | 29.4304 | −8.00000 | 0 | 20.4127 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | 6.27815 | 0 | −32.0906 | −8.00000 | 0 | −12.5563 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 0 | 4.00000 | 22.1852 | 0 | 2.71575 | −8.00000 | 0 | −44.3703 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(11\) | \( +1 \) |
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 | 2178.4.a.bx | 4 | |
| 3.b | odd | 2 | 1 | 726.4.a.y | yes | 4 | |
| 11.b | odd | 2 | 1 | 2178.4.a.cb | 4 | ||
| 33.d | even | 2 | 1 | 726.4.a.v | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 726.4.a.v | ✓ | 4 | 33.d | even | 2 | 1 | |
| 726.4.a.y | yes | 4 | 3.b | odd | 2 | 1 | |
| 2178.4.a.bx | 4 | 1.a | even | 1 | 1 | trivial | |
| 2178.4.a.cb | 4 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2178))\):
|
\( T_{5}^{4} - 6T_{5}^{3} - 375T_{5}^{2} - 432T_{5} + 17424 \)
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\( T_{7}^{4} - 12T_{7}^{3} - 951T_{7}^{2} + 13932T_{7} - 30636 \)
|
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\( T_{17}^{4} + 108T_{17}^{3} - 4671T_{17}^{2} - 625968T_{17} - 11710656 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 6 T^{3} + \cdots + 17424 \)
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| $7$ |
\( T^{4} - 12 T^{3} + \cdots - 30636 \)
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| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 12 T^{3} + \cdots - 2567808 \)
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| $17$ |
\( T^{4} + 108 T^{3} + \cdots - 11710656 \)
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| $19$ |
\( T^{4} - 192 T^{3} + \cdots - 3118896 \)
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| $23$ |
\( T^{4} + 156 T^{3} + \cdots + 177076944 \)
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| $29$ |
\( T^{4} + 408 T^{3} + \cdots + 56524068 \)
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| $31$ |
\( T^{4} + 26 T^{3} + \cdots + 549207376 \)
|
| $37$ |
\( T^{4} + 224 T^{3} + \cdots - 691955147 \)
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| $41$ |
\( T^{4} + \cdots - 3970969596 \)
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| $43$ |
\( T^{4} + \cdots - 15489329328 \)
|
| $47$ |
\( T^{4} + \cdots + 34460360064 \)
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| $53$ |
\( T^{4} + \cdots - 6826674492 \)
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| $59$ |
\( T^{4} + \cdots + 13387922064 \)
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| $61$ |
\( T^{4} + \cdots + 1167035616 \)
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| $67$ |
\( T^{4} + \cdots - 39899770364 \)
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| $71$ |
\( T^{4} + \cdots + 4103639424 \)
|
| $73$ |
\( T^{4} + \cdots - 11096170416 \)
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| $79$ |
\( T^{4} + \cdots - 80077424544 \)
|
| $83$ |
\( T^{4} + \cdots + 152428283136 \)
|
| $89$ |
\( T^{4} + \cdots + 37459324512 \)
|
| $97$ |
\( T^{4} + \cdots + 411402264937 \)
|
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