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: | \(\Q(\sqrt{3}, \sqrt{67})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 35x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of 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
\( x^{4} - 35x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 19\nu ) / 16 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 67\nu ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{2} - 53 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 53 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 19\beta_{2} + 67\beta_1 ) / 3 \)
|
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 | −16.6082 | 0 | 7.94790 | 8.00000 | 0 | −33.2163 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | 0 | 4.00000 | −7.94790 | 0 | 16.6082 | 8.00000 | 0 | −15.8958 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0 | 4.00000 | 7.94790 | 0 | −16.6082 | 8.00000 | 0 | 15.8958 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 0 | 4.00000 | 16.6082 | 0 | −7.94790 | 8.00000 | 0 | 33.2163 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(11\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 33.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2178.4.a.ca | yes | 4 |
| 3.b | odd | 2 | 1 | 2178.4.a.bv | ✓ | 4 | |
| 11.b | odd | 2 | 1 | 2178.4.a.bv | ✓ | 4 | |
| 33.d | even | 2 | 1 | inner | 2178.4.a.ca | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2178.4.a.bv | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 2178.4.a.bv | ✓ | 4 | 11.b | odd | 2 | 1 | |
| 2178.4.a.ca | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 2178.4.a.ca | yes | 4 | 33.d | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2178))\):
|
\( T_{5}^{4} - 339T_{5}^{2} + 17424 \)
|
|
\( T_{7}^{4} - 339T_{7}^{2} + 17424 \)
|
|
\( T_{17}^{2} + 117T_{17} - 648 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 339 T^{2} + 17424 \)
|
| $7$ |
\( T^{4} - 339 T^{2} + 17424 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 963 T^{2} + 32400 \)
|
| $17$ |
\( (T^{2} + 117 T - 648)^{2} \)
|
| $19$ |
\( T^{4} - 23667 T^{2} + 8643600 \)
|
| $23$ |
\( T^{4} - 21696 T^{2} + 71368704 \)
|
| $29$ |
\( (T^{2} + 111 T - 19080)^{2} \)
|
| $31$ |
\( (T^{2} + 121 T - 18500)^{2} \)
|
| $37$ |
\( (T^{2} + 184 T - 7817)^{2} \)
|
| $41$ |
\( (T^{2} - 21 T - 11196)^{2} \)
|
| $43$ |
\( T^{4} - 143100 T^{2} + 683090496 \)
|
| $47$ |
\( T^{4} - 48816 T^{2} + 361304064 \)
|
| $53$ |
\( T^{4} - 189675 T^{2} + 729000000 \)
|
| $59$ |
\( T^{4} - 49200 T^{2} + 391090176 \)
|
| $61$ |
\( T^{4} + \cdots + 178249462416 \)
|
| $67$ |
\( (T^{2} + 517 T - 552308)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 4305459456 \)
|
| $73$ |
\( T^{4} + \cdots + 13493145600 \)
|
| $79$ |
\( T^{4} + \cdots + 246857935104 \)
|
| $83$ |
\( (T^{2} + 528 T + 40752)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 870197928336 \)
|
| $97$ |
\( (T^{2} + 1544 T + 73183)^{2} \)
|
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