Newspace parameters
| Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 200.e (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.44960528721\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} + 6\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 6\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{3} + 3\beta_{2} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(151\) | \(177\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
− | 2.00000i | − | 5.89898i | −4.00000 | 0 | −11.7980 | 0 | 8.00000i | −25.7980 | 0 | ||||||||||||||||||||||||||||
| 99.2 | − | 2.00000i | 3.89898i | −4.00000 | 0 | 7.79796 | 0 | 8.00000i | −6.20204 | 0 | ||||||||||||||||||||||||||||||
| 99.3 | 2.00000i | − | 3.89898i | −4.00000 | 0 | 7.79796 | 0 | − | 8.00000i | −6.20204 | 0 | |||||||||||||||||||||||||||||
| 99.4 | 2.00000i | 5.89898i | −4.00000 | 0 | −11.7980 | 0 | − | 8.00000i | −25.7980 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
| 5.b | even | 2 | 1 | inner |
| 40.e | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 200.3.e.b | 4 | |
| 4.b | odd | 2 | 1 | 800.3.e.b | 4 | ||
| 5.b | even | 2 | 1 | inner | 200.3.e.b | 4 | |
| 5.c | odd | 4 | 1 | 200.3.g.b | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 200.3.g.d | yes | 2 | |
| 8.b | even | 2 | 1 | 800.3.e.b | 4 | ||
| 8.d | odd | 2 | 1 | CM | 200.3.e.b | 4 | |
| 20.d | odd | 2 | 1 | 800.3.e.b | 4 | ||
| 20.e | even | 4 | 1 | 800.3.g.b | 2 | ||
| 20.e | even | 4 | 1 | 800.3.g.d | 2 | ||
| 40.e | odd | 2 | 1 | inner | 200.3.e.b | 4 | |
| 40.f | even | 2 | 1 | 800.3.e.b | 4 | ||
| 40.i | odd | 4 | 1 | 800.3.g.b | 2 | ||
| 40.i | odd | 4 | 1 | 800.3.g.d | 2 | ||
| 40.k | even | 4 | 1 | 200.3.g.b | ✓ | 2 | |
| 40.k | even | 4 | 1 | 200.3.g.d | yes | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 200.3.e.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 200.3.e.b | 4 | 5.b | even | 2 | 1 | inner | |
| 200.3.e.b | 4 | 8.d | odd | 2 | 1 | CM | |
| 200.3.e.b | 4 | 40.e | odd | 2 | 1 | inner | |
| 200.3.g.b | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 200.3.g.b | ✓ | 2 | 40.k | even | 4 | 1 | |
| 200.3.g.d | yes | 2 | 5.c | odd | 4 | 1 | |
| 200.3.g.d | yes | 2 | 40.k | even | 4 | 1 | |
| 800.3.e.b | 4 | 4.b | odd | 2 | 1 | ||
| 800.3.e.b | 4 | 8.b | even | 2 | 1 | ||
| 800.3.e.b | 4 | 20.d | odd | 2 | 1 | ||
| 800.3.e.b | 4 | 40.f | even | 2 | 1 | ||
| 800.3.g.b | 2 | 20.e | even | 4 | 1 | ||
| 800.3.g.b | 2 | 40.i | odd | 4 | 1 | ||
| 800.3.g.d | 2 | 20.e | even | 4 | 1 | ||
| 800.3.g.d | 2 | 40.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 50T_{3}^{2} + 529 \)
acting on \(S_{3}^{\mathrm{new}}(200, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{2} \)
|
| $3$ |
\( T^{4} + 50T^{2} + 529 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 14 T - 167)^{2} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 1730 T^{2} + 744769 \)
|
| $19$ |
\( (T^{2} + 34 T + 73)^{2} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( (T^{2} - 46 T - 2927)^{2} \)
|
| $43$ |
\( (T^{2} + 196)^{2} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( (T - 82)^{4} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} + 23090 T^{2} + 92602129 \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} + 11810 T^{2} + 17447329 \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} + 16370 T^{2} + 18464209 \)
|
| $89$ |
\( (T^{2} - 146 T - 2447)^{2} \)
|
| $97$ |
\( (T^{2} + 8836)^{2} \)
|
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