Newspace parameters
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 21 \) |
| Character orbit: | \([\chi]\) | \(=\) | 18.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(45.6324777185\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 8995x^{2} + 8996x + 20268006 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{10}\cdot 5^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} - 2x^{3} - 8995x^{2} + 8996x + 20268006 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 3\nu^{2} - 8987\nu + 4494 ) / 18009 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2880\nu^{3} + 4320\nu^{2} + 38874240\nu - 19437840 ) / 667 \)
|
| \(\beta_{3}\) | \(=\) |
\( 9720\nu^{2} - 9720\nu - 43720560 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + 38880\beta _1 + 19440 ) / 38880 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + \beta_{2} + 38880\beta _1 + 174901680 ) / 38880 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{3} + 4495\beta_{2} + 524860560\beta _1 + 262342800 ) / 38880 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
− | 724.077i | 0 | −524288. | − | 1.17877e7i | 0 | 5.17614e7 | 3.79625e8i | 0 | −8.53519e9 | ||||||||||||||||||||||||||||
| 17.2 | − | 724.077i | 0 | −524288. | 1.40324e7i | 0 | 4.65449e7 | 3.79625e8i | 0 | 1.01606e10 | ||||||||||||||||||||||||||||||
| 17.3 | 724.077i | 0 | −524288. | − | 1.40324e7i | 0 | 4.65449e7 | − | 3.79625e8i | 0 | 1.01606e10 | |||||||||||||||||||||||||||||
| 17.4 | 724.077i | 0 | −524288. | 1.17877e7i | 0 | 5.17614e7 | − | 3.79625e8i | 0 | −8.53519e9 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 18.21.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 18.21.b.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.21.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 18.21.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 335857739019300T_{5}^{2} + 27360283984671775389120562500 \)
acting on \(S_{21}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 524288)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $7$ |
\( (T^{2} + \cdots + 24\!\cdots\!84)^{2} \)
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| $11$ |
\( T^{4} + \cdots + 21\!\cdots\!84 \)
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| $13$ |
\( (T^{2} + \cdots + 44\!\cdots\!56)^{2} \)
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| $17$ |
\( T^{4} + \cdots + 39\!\cdots\!64 \)
|
| $19$ |
\( (T^{2} + \cdots - 28\!\cdots\!04)^{2} \)
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| $23$ |
\( T^{4} + \cdots + 56\!\cdots\!84 \)
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| $29$ |
\( T^{4} + \cdots + 30\!\cdots\!04 \)
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| $31$ |
\( (T^{2} + \cdots + 63\!\cdots\!76)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots - 17\!\cdots\!76)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 17\!\cdots\!44 \)
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| $43$ |
\( (T^{2} + \cdots - 39\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 67\!\cdots\!44 \)
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| $59$ |
\( T^{4} + \cdots + 39\!\cdots\!64 \)
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| $61$ |
\( (T^{2} + \cdots + 67\!\cdots\!00)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots - 76\!\cdots\!24)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 23\!\cdots\!44 \)
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| $73$ |
\( (T^{2} + \cdots - 43\!\cdots\!36)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots + 38\!\cdots\!36)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 15\!\cdots\!64 \)
|
| $89$ |
\( T^{4} + \cdots + 30\!\cdots\!84 \)
|
| $97$ |
\( (T^{2} + \cdots - 93\!\cdots\!36)^{2} \)
|
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