Newspace parameters
| Level: | \( N \) | \(=\) | \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 700.m (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.58952814149\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.7965941760000.3 |
|
|
|
| Defining polynomial: |
\( x^{8} + 113x^{4} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 113x^{4} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 117\nu^{2} ) / 44 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 117\nu^{3} ) / 44 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 62 ) / 11 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 117\nu ) / 22 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 113\nu^{3} ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{6} - 329\nu^{2} ) / 22 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + 6\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{6} + 11\beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( 11\beta_{4} - 62 \)
|
| \(\nu^{5}\) | \(=\) |
\( 22\beta_{5} - 117\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -117\beta_{7} - 658\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( 234\beta_{6} - 1243\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(351\) | \(477\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 293.1 |
|
0 | −2.30472 | + | 2.30472i | 0 | 0 | 0 | 1.87083 | + | 1.87083i | 0 | − | 7.62348i | 0 | |||||||||||||||||||||||||||||||||||||
| 293.2 | 0 | −0.433892 | + | 0.433892i | 0 | 0 | 0 | −1.87083 | − | 1.87083i | 0 | 2.62348i | 0 | |||||||||||||||||||||||||||||||||||||||
| 293.3 | 0 | 0.433892 | − | 0.433892i | 0 | 0 | 0 | 1.87083 | + | 1.87083i | 0 | 2.62348i | 0 | |||||||||||||||||||||||||||||||||||||||
| 293.4 | 0 | 2.30472 | − | 2.30472i | 0 | 0 | 0 | −1.87083 | − | 1.87083i | 0 | − | 7.62348i | 0 | ||||||||||||||||||||||||||||||||||||||
| 657.1 | 0 | −2.30472 | − | 2.30472i | 0 | 0 | 0 | 1.87083 | − | 1.87083i | 0 | 7.62348i | 0 | |||||||||||||||||||||||||||||||||||||||
| 657.2 | 0 | −0.433892 | − | 0.433892i | 0 | 0 | 0 | −1.87083 | + | 1.87083i | 0 | − | 2.62348i | 0 | ||||||||||||||||||||||||||||||||||||||
| 657.3 | 0 | 0.433892 | + | 0.433892i | 0 | 0 | 0 | 1.87083 | − | 1.87083i | 0 | − | 2.62348i | 0 | ||||||||||||||||||||||||||||||||||||||
| 657.4 | 0 | 2.30472 | + | 2.30472i | 0 | 0 | 0 | −1.87083 | + | 1.87083i | 0 | 7.62348i | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 35.c | odd | 2 | 1 | CM by \(\Q(\sqrt{-35}) \) |
| 5.b | even | 2 | 1 | inner |
| 5.c | odd | 4 | 2 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.f | even | 4 | 2 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 700.2.m.a | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 700.2.m.a | ✓ | 8 |
| 5.c | odd | 4 | 2 | inner | 700.2.m.a | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 700.2.m.a | ✓ | 8 |
| 35.c | odd | 2 | 1 | CM | 700.2.m.a | ✓ | 8 |
| 35.f | even | 4 | 2 | inner | 700.2.m.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 700.2.m.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 700.2.m.a | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 700.2.m.a | ✓ | 8 | 5.c | odd | 4 | 2 | inner |
| 700.2.m.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 700.2.m.a | ✓ | 8 | 35.c | odd | 2 | 1 | CM |
| 700.2.m.a | ✓ | 8 | 35.f | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 113T_{3}^{4} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 113T^{4} + 16 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} + 49)^{2} \)
|
| $11$ |
\( (T^{2} - 3 T - 24)^{4} \)
|
| $13$ |
\( T^{8} + 2993 T^{4} + 1048576 \)
|
| $17$ |
\( T^{8} + 1233 T^{4} + 20736 \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( (T^{4} + 93 T^{2} + 36)^{2} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} \)
|
| $41$ |
\( T^{8} \)
|
| $43$ |
\( T^{8} \)
|
| $47$ |
\( T^{8} + 35793 T^{4} + 37015056 \)
|
| $53$ |
\( T^{8} \)
|
| $59$ |
\( T^{8} \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( T^{8} \)
|
| $71$ |
\( (T - 12)^{8} \)
|
| $73$ |
\( (T^{4} + 12544)^{2} \)
|
| $79$ |
\( (T^{4} + 473 T^{2} + 55696)^{2} \)
|
| $83$ |
\( (T^{4} + 63504)^{2} \)
|
| $89$ |
\( T^{8} \)
|
| $97$ |
\( T^{8} + 51713 T^{4} + 7311616 \)
|
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