Newspace parameters
| Level: | \( N \) | \(=\) | \( 234 = 2 \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 234.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.37603818603\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
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|
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| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| 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
| \(\beta_{1}\) | \(=\) |
\( 6\zeta_{8}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( -\zeta_{8}^{3} + \zeta_{8} \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\zeta_{8}^{3} + 3\zeta_{8} \)
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| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} ) / 6 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( ( \beta_1 ) / 6 \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( \beta_{3} - 3\beta_{2} ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(209\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 233.1 |
|
−1.41421 | 0 | 2.00000 | 8.48528 | 0 | − | 12.0000i | −2.82843 | 0 | −12.0000 | |||||||||||||||||||||||||||||
| 233.2 | −1.41421 | 0 | 2.00000 | 8.48528 | 0 | 12.0000i | −2.82843 | 0 | −12.0000 | |||||||||||||||||||||||||||||||
| 233.3 | 1.41421 | 0 | 2.00000 | −8.48528 | 0 | − | 12.0000i | 2.82843 | 0 | −12.0000 | ||||||||||||||||||||||||||||||
| 233.4 | 1.41421 | 0 | 2.00000 | −8.48528 | 0 | 12.0000i | 2.82843 | 0 | −12.0000 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
| 39.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 234.3.d.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 234.3.d.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1872.3.l.a | 4 | ||
| 12.b | even | 2 | 1 | 1872.3.l.a | 4 | ||
| 13.b | even | 2 | 1 | inner | 234.3.d.a | ✓ | 4 |
| 13.d | odd | 4 | 1 | 3042.3.c.a | 2 | ||
| 13.d | odd | 4 | 1 | 3042.3.c.i | 2 | ||
| 39.d | odd | 2 | 1 | inner | 234.3.d.a | ✓ | 4 |
| 39.f | even | 4 | 1 | 3042.3.c.a | 2 | ||
| 39.f | even | 4 | 1 | 3042.3.c.i | 2 | ||
| 52.b | odd | 2 | 1 | 1872.3.l.a | 4 | ||
| 156.h | even | 2 | 1 | 1872.3.l.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 234.3.d.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 234.3.d.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 234.3.d.a | ✓ | 4 | 13.b | even | 2 | 1 | inner |
| 234.3.d.a | ✓ | 4 | 39.d | odd | 2 | 1 | inner |
| 1872.3.l.a | 4 | 4.b | odd | 2 | 1 | ||
| 1872.3.l.a | 4 | 12.b | even | 2 | 1 | ||
| 1872.3.l.a | 4 | 52.b | odd | 2 | 1 | ||
| 1872.3.l.a | 4 | 156.h | even | 2 | 1 | ||
| 3042.3.c.a | 2 | 13.d | odd | 4 | 1 | ||
| 3042.3.c.a | 2 | 39.f | even | 4 | 1 | ||
| 3042.3.c.i | 2 | 13.d | odd | 4 | 1 | ||
| 3042.3.c.i | 2 | 39.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(234, [\chi])\):
|
\( T_{5}^{2} - 72 \)
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\( T_{7}^{2} + 144 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{2} \)
|
| $3$ |
\( T^{4} \)
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| $5$ |
\( (T^{2} - 72)^{2} \)
|
| $7$ |
\( (T^{2} + 144)^{2} \)
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| $11$ |
\( (T^{2} - 18)^{2} \)
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| $13$ |
\( (T + 13)^{4} \)
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| $17$ |
\( (T^{2} + 18)^{2} \)
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| $19$ |
\( (T^{2} + 324)^{2} \)
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| $23$ |
\( (T^{2} + 288)^{2} \)
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| $29$ |
\( (T^{2} + 1458)^{2} \)
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| $31$ |
\( (T^{2} + 1764)^{2} \)
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| $37$ |
\( (T^{2} + 3600)^{2} \)
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| $41$ |
\( (T^{2} - 72)^{2} \)
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| $43$ |
\( (T - 68)^{4} \)
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| $47$ |
\( (T^{2} - 1458)^{2} \)
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| $53$ |
\( (T^{2} + 6498)^{2} \)
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| $59$ |
\( (T^{2} - 450)^{2} \)
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| $61$ |
\( (T - 40)^{4} \)
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| $67$ |
\( (T^{2} + 3600)^{2} \)
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| $71$ |
\( (T^{2} - 3042)^{2} \)
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| $73$ |
\( (T^{2} + 144)^{2} \)
|
| $79$ |
\( (T + 68)^{4} \)
|
| $83$ |
\( (T^{2} - 13122)^{2} \)
|
| $89$ |
\( (T^{2} - 648)^{2} \)
|
| $97$ |
\( (T^{2} + 11664)^{2} \)
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