Newspace parameters
| Level: | \( N \) | \(=\) | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 650.k (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(17.7112171834\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 151.1 |
|
1.00000 | + | 1.00000i | −4.44949 | 2.00000i | 0 | −4.44949 | − | 4.44949i | −0.674235 | + | 0.674235i | −2.00000 | + | 2.00000i | 10.7980 | 0 | ||||||||||||||||||||||
| 151.2 | 1.00000 | + | 1.00000i | 0.449490 | 2.00000i | 0 | 0.449490 | + | 0.449490i | 6.67423 | − | 6.67423i | −2.00000 | + | 2.00000i | −8.79796 | 0 | |||||||||||||||||||||||
| 551.1 | 1.00000 | − | 1.00000i | −4.44949 | − | 2.00000i | 0 | −4.44949 | + | 4.44949i | −0.674235 | − | 0.674235i | −2.00000 | − | 2.00000i | 10.7980 | 0 | ||||||||||||||||||||||
| 551.2 | 1.00000 | − | 1.00000i | 0.449490 | − | 2.00000i | 0 | 0.449490 | − | 0.449490i | 6.67423 | + | 6.67423i | −2.00000 | − | 2.00000i | −8.79796 | 0 | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 650.3.k.g | yes | 4 |
| 5.b | even | 2 | 1 | 650.3.k.f | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 650.3.f.g | 4 | ||
| 5.c | odd | 4 | 1 | 650.3.f.h | 4 | ||
| 13.d | odd | 4 | 1 | inner | 650.3.k.g | yes | 4 |
| 65.f | even | 4 | 1 | 650.3.f.g | 4 | ||
| 65.g | odd | 4 | 1 | 650.3.k.f | ✓ | 4 | |
| 65.k | even | 4 | 1 | 650.3.f.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 650.3.f.g | 4 | 5.c | odd | 4 | 1 | ||
| 650.3.f.g | 4 | 65.f | even | 4 | 1 | ||
| 650.3.f.h | 4 | 5.c | odd | 4 | 1 | ||
| 650.3.f.h | 4 | 65.k | even | 4 | 1 | ||
| 650.3.k.f | ✓ | 4 | 5.b | even | 2 | 1 | |
| 650.3.k.f | ✓ | 4 | 65.g | odd | 4 | 1 | |
| 650.3.k.g | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 650.3.k.g | yes | 4 | 13.d | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 4T_{3} - 2 \)
acting on \(S_{3}^{\mathrm{new}}(650, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $3$ |
\( (T^{2} + 4 T - 2)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 12 T^{3} + \cdots + 81 \)
|
| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 58081 \)
|
| $13$ |
\( (T^{2} + 169)^{2} \)
|
| $17$ |
\( T^{4} + 450 T^{2} + 42849 \)
|
| $19$ |
\( (T^{2} - 12 T + 72)^{2} \)
|
| $23$ |
\( T^{4} + 1460 T^{2} + 386884 \)
|
| $29$ |
\( (T^{2} - 38 T - 503)^{2} \)
|
| $31$ |
\( T^{4} - 60 T^{3} + \cdots + 178929 \)
|
| $37$ |
\( T^{4} + 68 T^{3} + \cdots + 21316 \)
|
| $41$ |
\( T^{4} - 64 T^{3} + \cdots + 1478656 \)
|
| $43$ |
\( T^{4} + 3860 T^{2} + 917764 \)
|
| $47$ |
\( T^{4} + 12 T^{3} + \cdots + 60606225 \)
|
| $53$ |
\( (T^{2} + 130 T + 4009)^{2} \)
|
| $59$ |
\( T^{4} - 52 T^{3} + \cdots + 8579041 \)
|
| $61$ |
\( (T^{2} + 50 T - 1319)^{2} \)
|
| $67$ |
\( T^{4} + 44 T^{3} + \cdots + 18671041 \)
|
| $71$ |
\( T^{4} - 56 T^{3} + \cdots + 42510400 \)
|
| $73$ |
\( T^{4} + 36 T^{3} + \cdots + 45562500 \)
|
| $79$ |
\( (T^{2} + 32 T - 3200)^{2} \)
|
| $83$ |
\( T^{4} + 140 T^{3} + \cdots + 667489 \)
|
| $89$ |
\( T^{4} - 84 T^{3} + \cdots + 202500 \)
|
| $97$ |
\( T^{4} - 128 T^{3} + \cdots + 102400 \)
|
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