Newspace parameters
| Level: | \( N \) | \(=\) | \( 702 = 2 \cdot 3^{3} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 702.h (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.60549822189\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{37})\) |
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|
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| Defining polynomial: |
\( x^{4} - x^{3} + 10x^{2} + 9x + 81 \)
|
| 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
\( x^{4} - x^{3} + 10x^{2} + 9x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 10\nu^{2} - 10\nu + 81 ) / 90 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 19 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 9\beta_{2} + \beta _1 - 10 \)
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| \(\nu^{3}\) | \(=\) |
\( 10\beta_{3} - 19 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/702\mathbb{Z}\right)^\times\).
| \(n\) | \(379\) | \(677\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) |
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 | −2.54138 | 0 | −0.770691 | + | 1.33488i | 1.00000 | 0 | 1.27069 | + | 2.20090i | ||||||||||||||||||||||
| 55.2 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | 3.54138 | 0 | 2.27069 | − | 3.93295i | 1.00000 | 0 | −1.77069 | − | 3.06693i | |||||||||||||||||||||||
| 217.1 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −2.54138 | 0 | −0.770691 | − | 1.33488i | 1.00000 | 0 | 1.27069 | − | 2.20090i | |||||||||||||||||||||||
| 217.2 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 3.54138 | 0 | 2.27069 | + | 3.93295i | 1.00000 | 0 | −1.77069 | + | 3.06693i | |||||||||||||||||||||||
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 | 702.2.h.i | ✓ | 4 |
| 3.b | odd | 2 | 1 | 702.2.h.j | yes | 4 | |
| 13.c | even | 3 | 1 | inner | 702.2.h.i | ✓ | 4 |
| 13.c | even | 3 | 1 | 9126.2.a.bz | 2 | ||
| 13.e | even | 6 | 1 | 9126.2.a.bq | 2 | ||
| 39.h | odd | 6 | 1 | 9126.2.a.ca | 2 | ||
| 39.i | odd | 6 | 1 | 702.2.h.j | yes | 4 | |
| 39.i | odd | 6 | 1 | 9126.2.a.bp | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 702.2.h.i | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 702.2.h.i | ✓ | 4 | 13.c | even | 3 | 1 | inner |
| 702.2.h.j | yes | 4 | 3.b | odd | 2 | 1 | |
| 702.2.h.j | yes | 4 | 39.i | odd | 6 | 1 | |
| 9126.2.a.bp | 2 | 39.i | odd | 6 | 1 | ||
| 9126.2.a.bq | 2 | 13.e | even | 6 | 1 | ||
| 9126.2.a.bz | 2 | 13.c | even | 3 | 1 | ||
| 9126.2.a.ca | 2 | 39.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(702, [\chi])\):
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\( T_{5}^{2} - T_{5} - 9 \)
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\( T_{7}^{4} - 3T_{7}^{3} + 16T_{7}^{2} + 21T_{7} + 49 \)
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\( T_{11}^{4} + 7T_{11}^{3} + 46T_{11}^{2} + 21T_{11} + 9 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{2} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( (T^{2} - T - 9)^{2} \)
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| $7$ |
\( T^{4} - 3 T^{3} + \cdots + 49 \)
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| $11$ |
\( T^{4} + 7 T^{3} + \cdots + 9 \)
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| $13$ |
\( (T^{2} - 5 T + 13)^{2} \)
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| $17$ |
\( T^{4} - 5 T^{3} + \cdots + 9 \)
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| $19$ |
\( T^{4} + 37T^{2} + 1369 \)
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| $23$ |
\( (T^{2} + 3 T + 9)^{2} \)
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| $29$ |
\( (T^{2} + 3 T + 9)^{2} \)
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| $31$ |
\( (T - 2)^{4} \)
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| $37$ |
\( T^{4} - 3 T^{3} + \cdots + 49 \)
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| $41$ |
\( T^{4} + 4 T^{3} + \cdots + 1089 \)
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| $43$ |
\( T^{4} - 6 T^{3} + \cdots + 784 \)
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| $47$ |
\( (T^{2} + 2 T - 36)^{2} \)
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| $53$ |
\( (T^{2} - 16 T + 27)^{2} \)
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| $59$ |
\( T^{4} + 11 T^{3} + \cdots + 441 \)
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| $61$ |
\( T^{4} - 9 T^{3} + \cdots + 121 \)
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| $67$ |
\( T^{4} + T^{3} + \cdots + 6889 \)
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| $71$ |
\( T^{4} - 7 T^{3} + \cdots + 9 \)
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| $73$ |
\( (T^{2} - T - 83)^{2} \)
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| $79$ |
\( (T^{2} - 7 T - 71)^{2} \)
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| $83$ |
\( T^{4} \)
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| $89$ |
\( T^{4} + 4 T^{3} + \cdots + 1089 \)
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| $97$ |
\( T^{4} - 6 T^{3} + \cdots + 784 \)
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