Newspace parameters
| Level: | \( N \) | \(=\) | \( 234 = 2 \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 234.h (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.86849940730\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{10})\) |
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|
|
| Defining polynomial: |
\( x^{4} + 10x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} + 10x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 10\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 10\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(209\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −4.16228 | 0 | −1.58114 | + | 2.73861i | −1.00000 | 0 | −2.08114 | − | 3.60464i | ||||||||||||||||||||||
| 55.2 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 2.16228 | 0 | 1.58114 | − | 2.73861i | −1.00000 | 0 | 1.08114 | + | 1.87259i | |||||||||||||||||||||||
| 217.1 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | −4.16228 | 0 | −1.58114 | − | 2.73861i | −1.00000 | 0 | −2.08114 | + | 3.60464i | |||||||||||||||||||||||
| 217.2 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | 2.16228 | 0 | 1.58114 | + | 2.73861i | −1.00000 | 0 | 1.08114 | − | 1.87259i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 234.2.h.e | yes | 4 |
| 3.b | odd | 2 | 1 | 234.2.h.d | ✓ | 4 | |
| 4.b | odd | 2 | 1 | 1872.2.t.n | 4 | ||
| 12.b | even | 2 | 1 | 1872.2.t.p | 4 | ||
| 13.c | even | 3 | 1 | inner | 234.2.h.e | yes | 4 |
| 13.c | even | 3 | 1 | 3042.2.a.r | 2 | ||
| 13.e | even | 6 | 1 | 3042.2.a.x | 2 | ||
| 13.f | odd | 12 | 2 | 3042.2.b.j | 4 | ||
| 39.h | odd | 6 | 1 | 3042.2.a.q | 2 | ||
| 39.i | odd | 6 | 1 | 234.2.h.d | ✓ | 4 | |
| 39.i | odd | 6 | 1 | 3042.2.a.w | 2 | ||
| 39.k | even | 12 | 2 | 3042.2.b.k | 4 | ||
| 52.j | odd | 6 | 1 | 1872.2.t.n | 4 | ||
| 156.p | even | 6 | 1 | 1872.2.t.p | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 234.2.h.d | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 234.2.h.d | ✓ | 4 | 39.i | odd | 6 | 1 | |
| 234.2.h.e | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 234.2.h.e | yes | 4 | 13.c | even | 3 | 1 | inner |
| 1872.2.t.n | 4 | 4.b | odd | 2 | 1 | ||
| 1872.2.t.n | 4 | 52.j | odd | 6 | 1 | ||
| 1872.2.t.p | 4 | 12.b | even | 2 | 1 | ||
| 1872.2.t.p | 4 | 156.p | even | 6 | 1 | ||
| 3042.2.a.q | 2 | 39.h | odd | 6 | 1 | ||
| 3042.2.a.r | 2 | 13.c | even | 3 | 1 | ||
| 3042.2.a.w | 2 | 39.i | odd | 6 | 1 | ||
| 3042.2.a.x | 2 | 13.e | even | 6 | 1 | ||
| 3042.2.b.j | 4 | 13.f | odd | 12 | 2 | ||
| 3042.2.b.k | 4 | 39.k | even | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(234, [\chi])\):
|
\( T_{5}^{2} + 2T_{5} - 9 \)
|
|
\( T_{7}^{4} + 10T_{7}^{2} + 100 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{2} \)
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| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 2 T - 9)^{2} \)
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| $7$ |
\( T^{4} + 10T^{2} + 100 \)
|
| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 36 \)
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| $13$ |
\( T^{4} + 6 T^{3} + \cdots + 169 \)
|
| $17$ |
\( (T^{2} + 3 T + 9)^{2} \)
|
| $19$ |
\( T^{4} - 4 T^{3} + \cdots + 36 \)
|
| $23$ |
\( T^{4} - 8 T^{3} + \cdots + 36 \)
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| $29$ |
\( T^{4} + 10 T^{3} + \cdots + 225 \)
|
| $31$ |
\( (T^{2} - 40)^{2} \)
|
| $37$ |
\( T^{4} + 14 T^{3} + \cdots + 1521 \)
|
| $41$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $43$ |
\( T^{4} + 12 T^{3} + \cdots + 676 \)
|
| $47$ |
\( (T^{2} + 16 T + 54)^{2} \)
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| $53$ |
\( (T^{2} - 6 T - 81)^{2} \)
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| $59$ |
\( T^{4} - 8 T^{3} + \cdots + 576 \)
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| $61$ |
\( T^{4} - 6 T^{3} + \cdots + 1 \)
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| $67$ |
\( T^{4} - 12 T^{3} + \cdots + 676 \)
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| $71$ |
\( T^{4} + 8 T^{3} + \cdots + 36 \)
|
| $73$ |
\( (T + 1)^{4} \)
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| $79$ |
\( (T + 4)^{4} \)
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| $83$ |
\( (T^{2} + 12 T - 54)^{2} \)
|
| $89$ |
\( (T^{2} - 12 T + 144)^{2} \)
|
| $97$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
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