Newspace parameters
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.e (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(101.212748257\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{1289})\) |
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|
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| Defining polynomial: |
\( x^{4} - x^{3} + 323x^{2} + 322x + 103684 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 108) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 323x^{2} + 322x + 103684 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 323\nu^{2} - 323\nu + 103684 ) / 104006 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 9\nu^{3} - 2907\nu^{2} + 1875015\nu - 933156 ) / 104006 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 18\nu^{3} + 8703 ) / 323 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + 9\beta_1 ) / 18 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 5805\beta _1 - 5805 ) / 18 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 323\beta_{3} - 8703 ) / 18 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).
| \(n\) | \(163\) | \(245\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 109.1 |
|
0 | 0 | 0 | −80.5619 | + | 139.537i | 0 | −239.686 | − | 415.148i | 0 | 0 | 0 | ||||||||||||||||||||||||||
| 109.2 | 0 | 0 | 0 | 242.562 | − | 420.130i | 0 | 729.686 | + | 1263.85i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 217.1 | 0 | 0 | 0 | −80.5619 | − | 139.537i | 0 | −239.686 | + | 415.148i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 217.2 | 0 | 0 | 0 | 242.562 | + | 420.130i | 0 | 729.686 | − | 1263.85i | 0 | 0 | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 324.8.e.j | 4 | |
| 3.b | odd | 2 | 1 | 324.8.e.g | 4 | ||
| 9.c | even | 3 | 1 | 108.8.a.b | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 324.8.e.j | 4 | |
| 9.d | odd | 6 | 1 | 108.8.a.e | yes | 2 | |
| 9.d | odd | 6 | 1 | 324.8.e.g | 4 | ||
| 36.f | odd | 6 | 1 | 432.8.a.i | 2 | ||
| 36.h | even | 6 | 1 | 432.8.a.r | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 108.8.a.b | ✓ | 2 | 9.c | even | 3 | 1 | |
| 108.8.a.e | yes | 2 | 9.d | odd | 6 | 1 | |
| 324.8.e.g | 4 | 3.b | odd | 2 | 1 | ||
| 324.8.e.g | 4 | 9.d | odd | 6 | 1 | ||
| 324.8.e.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 324.8.e.j | 4 | 9.c | even | 3 | 1 | inner | |
| 432.8.a.i | 2 | 36.f | odd | 6 | 1 | ||
| 432.8.a.r | 2 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(324, [\chi])\):
|
\( T_{5}^{4} - 324T_{5}^{3} + 183141T_{5}^{2} + 25325460T_{5} + 6109767225 \)
|
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\( T_{7}^{4} - 980T_{7}^{3} + 1659981T_{7}^{2} + 685589380T_{7} + 489413575561 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} + \cdots + 6109767225 \)
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| $7$ |
\( T^{4} + \cdots + 489413575561 \)
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| $11$ |
\( T^{4} + \cdots + 2269599497361 \)
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| $13$ |
\( T^{4} + \cdots + 19\!\cdots\!56 \)
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| $17$ |
\( (T^{2} - 20736 T - 59558976)^{2} \)
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| $19$ |
\( (T^{2} - 10456 T - 66636116)^{2} \)
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| $23$ |
\( T^{4} + \cdots + 56\!\cdots\!96 \)
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| $29$ |
\( T^{4} + \cdots + 16\!\cdots\!36 \)
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| $31$ |
\( T^{4} + \cdots + 13\!\cdots\!21 \)
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| $37$ |
\( (T^{2} + 10460 T - 3821580476)^{2} \)
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| $41$ |
\( T^{4} + \cdots + 76\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots + 57\!\cdots\!56 \)
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| $47$ |
\( T^{4} + \cdots + 23\!\cdots\!56 \)
|
| $53$ |
\( (T^{2} - 2079756 T + 883947989259)^{2} \)
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| $59$ |
\( T^{4} + \cdots + 37\!\cdots\!00 \)
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| $61$ |
\( T^{4} + \cdots + 32\!\cdots\!00 \)
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| $67$ |
\( T^{4} + \cdots + 80\!\cdots\!76 \)
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| $71$ |
\( (T^{2} - 3201120 T + 473739274944)^{2} \)
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| $73$ |
\( (T^{2} + \cdots - 4430654478791)^{2} \)
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| $79$ |
\( T^{4} + \cdots + 44\!\cdots\!00 \)
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| $83$ |
\( T^{4} + \cdots + 14\!\cdots\!61 \)
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| $89$ |
\( (T^{2} + \cdots + 3197429581356)^{2} \)
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| $97$ |
\( T^{4} + \cdots + 11\!\cdots\!21 \)
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