Newspace parameters
| Level: | \( N \) | \(=\) | \( 702 = 2 \cdot 3^{3} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 702.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.60549822189\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{10})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 in terms of a root \(\nu\) of
\( x^{4} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/702\mathbb{Z}\right)^\times\).
| \(n\) | \(379\) | \(677\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 649.1 |
|
− | 1.00000i | 0 | −1.00000 | − | 4.16228i | 0 | 4.16228i | 1.00000i | 0 | −4.16228 | ||||||||||||||||||||||||||||
| 649.2 | − | 1.00000i | 0 | −1.00000 | 2.16228i | 0 | − | 2.16228i | 1.00000i | 0 | 2.16228 | |||||||||||||||||||||||||||||
| 649.3 | 1.00000i | 0 | −1.00000 | − | 2.16228i | 0 | 2.16228i | − | 1.00000i | 0 | 2.16228 | |||||||||||||||||||||||||||||
| 649.4 | 1.00000i | 0 | −1.00000 | 4.16228i | 0 | − | 4.16228i | − | 1.00000i | 0 | −4.16228 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 702.2.b.g | yes | 4 |
| 3.b | odd | 2 | 1 | 702.2.b.f | ✓ | 4 | |
| 13.b | even | 2 | 1 | inner | 702.2.b.g | yes | 4 |
| 13.d | odd | 4 | 1 | 9126.2.a.bn | 2 | ||
| 13.d | odd | 4 | 1 | 9126.2.a.cc | 2 | ||
| 39.d | odd | 2 | 1 | 702.2.b.f | ✓ | 4 | |
| 39.f | even | 4 | 1 | 9126.2.a.bo | 2 | ||
| 39.f | even | 4 | 1 | 9126.2.a.cb | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 702.2.b.f | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 702.2.b.f | ✓ | 4 | 39.d | odd | 2 | 1 | |
| 702.2.b.g | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 702.2.b.g | yes | 4 | 13.b | even | 2 | 1 | inner |
| 9126.2.a.bn | 2 | 13.d | odd | 4 | 1 | ||
| 9126.2.a.bo | 2 | 39.f | even | 4 | 1 | ||
| 9126.2.a.cb | 2 | 39.f | even | 4 | 1 | ||
| 9126.2.a.cc | 2 | 13.d | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(702, [\chi])\):
|
\( T_{5}^{4} + 22T_{5}^{2} + 81 \)
|
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\( T_{7}^{4} + 22T_{7}^{2} + 81 \)
|
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\( T_{17}^{2} - 4T_{17} - 6 \)
|
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\( T_{23}^{2} - 4T_{23} - 6 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 22T^{2} + 81 \)
|
| $7$ |
\( T^{4} + 22T^{2} + 81 \)
|
| $11$ |
\( (T^{2} + 9)^{2} \)
|
| $13$ |
\( T^{4} + 8 T^{3} + \cdots + 169 \)
|
| $17$ |
\( (T^{2} - 4 T - 6)^{2} \)
|
| $19$ |
\( (T^{2} + 36)^{2} \)
|
| $23$ |
\( (T^{2} - 4 T - 6)^{2} \)
|
| $29$ |
\( (T^{2} + 4 T - 36)^{2} \)
|
| $31$ |
\( T^{4} + 70T^{2} + 225 \)
|
| $37$ |
\( T^{4} + 88T^{2} + 1296 \)
|
| $41$ |
\( T^{4} + 52T^{2} + 36 \)
|
| $43$ |
\( (T^{2} - 10)^{2} \)
|
| $47$ |
\( (T^{2} + 36)^{2} \)
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| $53$ |
\( (T^{2} + 6 T - 81)^{2} \)
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| $59$ |
\( T^{4} + 88T^{2} + 1296 \)
|
| $61$ |
\( (T^{2} - 10)^{2} \)
|
| $67$ |
\( T^{4} + 52T^{2} + 36 \)
|
| $71$ |
\( (T^{2} + 90)^{2} \)
|
| $73$ |
\( T^{4} + 322 T^{2} + 25281 \)
|
| $79$ |
\( (T^{2} + 8 T - 144)^{2} \)
|
| $83$ |
\( T^{4} + 82T^{2} + 1521 \)
|
| $89$ |
\( (T^{2} + 90)^{2} \)
|
| $97$ |
\( (T^{2} + 9)^{2} \)
|
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