Newspace parameters
| Level: | \( N \) | \(=\) | \( 234 = 2 \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 234.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(73.0980959633\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 643x + 6370 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 643x + 6370 \)
:
| \(\beta_{1}\) | \(=\) |
\( 12\nu - 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( 12\nu^{2} + 168\nu - 5204 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 4 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 14\beta _1 + 5148 ) / 12 \)
|
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 |
|
8.00000 | 0 | 64.0000 | −396.674 | 0 | 334.498 | 512.000 | 0 | −3173.40 | |||||||||||||||||||||||||||
| 1.2 | 8.00000 | 0 | 64.0000 | 109.075 | 0 | −997.288 | 512.000 | 0 | 872.600 | ||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 0 | 64.0000 | 149.600 | 0 | 740.791 | 512.000 | 0 | 1196.80 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 234.8.a.o | yes | 3 |
| 3.b | odd | 2 | 1 | 234.8.a.n | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 234.8.a.n | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 234.8.a.o | yes | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 138T_{5}^{2} - 86292T_{5} + 6472760 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(234))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 138 T^{2} + \cdots + 6472760 \)
|
| $7$ |
\( T^{3} - 78 T^{2} + \cdots + 247120952 \)
|
| $11$ |
\( T^{3} + \cdots + 31374773248 \)
|
| $13$ |
\( (T + 2197)^{3} \)
|
| $17$ |
\( T^{3} + \cdots - 7940542814208 \)
|
| $19$ |
\( T^{3} + \cdots - 10727715751416 \)
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| $23$ |
\( T^{3} + \cdots + 35856853293248 \)
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| $29$ |
\( T^{3} + \cdots + 254844683706368 \)
|
| $31$ |
\( T^{3} + \cdots + 33209987379320 \)
|
| $37$ |
\( T^{3} + \cdots - 21\!\cdots\!72 \)
|
| $41$ |
\( T^{3} + \cdots - 52\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots - 61\!\cdots\!48 \)
|
| $47$ |
\( T^{3} + \cdots - 21\!\cdots\!44 \)
|
| $53$ |
\( T^{3} + \cdots + 41\!\cdots\!28 \)
|
| $59$ |
\( T^{3} + \cdots + 21\!\cdots\!32 \)
|
| $61$ |
\( T^{3} + \cdots + 72\!\cdots\!00 \)
|
| $67$ |
\( T^{3} + \cdots - 12\!\cdots\!04 \)
|
| $71$ |
\( T^{3} + \cdots + 55\!\cdots\!00 \)
|
| $73$ |
\( T^{3} + \cdots + 62\!\cdots\!24 \)
|
| $79$ |
\( T^{3} + \cdots - 16\!\cdots\!88 \)
|
| $83$ |
\( T^{3} + \cdots - 56\!\cdots\!40 \)
|
| $89$ |
\( T^{3} + \cdots - 30\!\cdots\!40 \)
|
| $97$ |
\( T^{3} + \cdots - 14\!\cdots\!92 \)
|
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