Newspace parameters
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(38.4766383424\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 102516x^{2} - 1667140x + 2022689200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{2}\cdot 5\cdot 7 \) |
| 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
\( x^{4} - x^{3} - 102516x^{2} - 1667140x + 2022689200 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} - 102\nu - 205008 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 135\nu^{2} - 59622\nu + 5607524 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 51\beta _1 + 205008 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 24\beta_{3} + 135\beta_{2} + 126129\beta _1 + 5245984 ) / 4 \)
|
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 |
|
−585.858 | −6561.00 | 212158. | −559169. | 3.84381e6 | 5.76480e6 | −4.75046e7 | 4.30467e7 | 3.27593e8 | ||||||||||||||||||||||||||||||
| 1.2 | −300.731 | −6561.00 | −40632.6 | 685014. | 1.97310e6 | 5.76480e6 | 5.16370e7 | 4.30467e7 | −2.06005e8 | |||||||||||||||||||||||||||||||
| 1.3 | 366.034 | −6561.00 | 2908.63 | 1.67136e6 | −2.40155e6 | 5.76480e6 | −4.69121e7 | 4.30467e7 | 6.11775e8 | |||||||||||||||||||||||||||||||
| 1.4 | 502.556 | −6561.00 | 121490. | −945700. | −3.29727e6 | 5.76480e6 | −4.81531e6 | 4.30467e7 | −4.75267e8 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 21.18.a.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 63.18.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.18.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 63.18.a.c | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 18T_{2}^{3} - 409944T_{2}^{2} + 10056960T_{2} + 32409815040 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(21))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 32409815040 \)
|
| $3$ |
\( (T + 6561)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 60\!\cdots\!00 \)
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| $7$ |
\( (T - 5764801)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 38\!\cdots\!44 \)
|
| $13$ |
\( T^{4} + \cdots + 93\!\cdots\!40 \)
|
| $17$ |
\( T^{4} + \cdots - 42\!\cdots\!76 \)
|
| $19$ |
\( T^{4} + \cdots + 37\!\cdots\!44 \)
|
| $23$ |
\( T^{4} + \cdots + 51\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 85\!\cdots\!36 \)
|
| $31$ |
\( T^{4} + \cdots - 58\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 16\!\cdots\!84 \)
|
| $41$ |
\( T^{4} + \cdots - 47\!\cdots\!04 \)
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| $43$ |
\( T^{4} + \cdots - 10\!\cdots\!60 \)
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| $47$ |
\( T^{4} + \cdots - 92\!\cdots\!00 \)
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| $53$ |
\( T^{4} + \cdots + 18\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 13\!\cdots\!60 \)
|
| $61$ |
\( T^{4} + \cdots - 47\!\cdots\!84 \)
|
| $67$ |
\( T^{4} + \cdots - 52\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots - 25\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots + 87\!\cdots\!84 \)
|
| $79$ |
\( T^{4} + \cdots + 17\!\cdots\!96 \)
|
| $83$ |
\( T^{4} + \cdots - 73\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots + 18\!\cdots\!84 \)
|
| $97$ |
\( T^{4} + \cdots - 47\!\cdots\!56 \)
|
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