Newspace parameters
| Level: | \( N \) | \(=\) | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 650.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.19027613138\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
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|
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| Defining polynomial: |
\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - \nu - 4 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 3\nu + 2 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} + \nu^{2} + \nu - 8 ) / 2 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 5\beta _1 + 4 ) / 3 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} - \beta_{2} - \beta _1 + 7 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(1\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 451.1 |
|
0.500000 | − | 0.866025i | −0.895644 | + | 1.55130i | −0.500000 | − | 0.866025i | 0 | 0.895644 | + | 1.55130i | −1.39564 | − | 2.41733i | −1.00000 | −0.104356 | − | 0.180750i | 0 | ||||||||||||||||||
| 451.2 | 0.500000 | − | 0.866025i | 1.39564 | − | 2.41733i | −0.500000 | − | 0.866025i | 0 | −1.39564 | − | 2.41733i | 0.895644 | + | 1.55130i | −1.00000 | −2.39564 | − | 4.14938i | 0 | |||||||||||||||||||
| 601.1 | 0.500000 | + | 0.866025i | −0.895644 | − | 1.55130i | −0.500000 | + | 0.866025i | 0 | 0.895644 | − | 1.55130i | −1.39564 | + | 2.41733i | −1.00000 | −0.104356 | + | 0.180750i | 0 | |||||||||||||||||||
| 601.2 | 0.500000 | + | 0.866025i | 1.39564 | + | 2.41733i | −0.500000 | + | 0.866025i | 0 | −1.39564 | + | 2.41733i | 0.895644 | − | 1.55130i | −1.00000 | −2.39564 | + | 4.14938i | 0 | |||||||||||||||||||
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 | 650.2.e.i | yes | 4 |
| 5.b | even | 2 | 1 | 650.2.e.d | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 650.2.o.f | 8 | ||
| 13.c | even | 3 | 1 | inner | 650.2.e.i | yes | 4 |
| 13.c | even | 3 | 1 | 8450.2.a.bb | 2 | ||
| 13.e | even | 6 | 1 | 8450.2.a.bh | 2 | ||
| 65.l | even | 6 | 1 | 8450.2.a.be | 2 | ||
| 65.n | even | 6 | 1 | 650.2.e.d | ✓ | 4 | |
| 65.n | even | 6 | 1 | 8450.2.a.bk | 2 | ||
| 65.q | odd | 12 | 2 | 650.2.o.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 650.2.e.d | ✓ | 4 | 5.b | even | 2 | 1 | |
| 650.2.e.d | ✓ | 4 | 65.n | even | 6 | 1 | |
| 650.2.e.i | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 650.2.e.i | yes | 4 | 13.c | even | 3 | 1 | inner |
| 650.2.o.f | 8 | 5.c | odd | 4 | 2 | ||
| 650.2.o.f | 8 | 65.q | odd | 12 | 2 | ||
| 8450.2.a.bb | 2 | 13.c | even | 3 | 1 | ||
| 8450.2.a.be | 2 | 65.l | even | 6 | 1 | ||
| 8450.2.a.bh | 2 | 13.e | even | 6 | 1 | ||
| 8450.2.a.bk | 2 | 65.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\):
|
\( T_{3}^{4} - T_{3}^{3} + 6T_{3}^{2} + 5T_{3} + 25 \)
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\( T_{7}^{4} + T_{7}^{3} + 6T_{7}^{2} - 5T_{7} + 25 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{2} \)
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| $3$ |
\( T^{4} - T^{3} + \cdots + 25 \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + T^{3} + \cdots + 25 \)
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| $11$ |
\( T^{4} - 3 T^{3} + \cdots + 9 \)
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| $13$ |
\( (T^{2} + 7 T + 13)^{2} \)
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| $17$ |
\( T^{4} + 3 T^{3} + \cdots + 9 \)
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| $19$ |
\( (T^{2} + 5 T + 25)^{2} \)
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| $23$ |
\( T^{4} + 21T^{2} + 441 \)
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| $29$ |
\( T^{4} - 3 T^{3} + \cdots + 9 \)
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| $31$ |
\( (T^{2} - 10 T + 4)^{2} \)
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| $37$ |
\( T^{4} + T^{3} + \cdots + 2209 \)
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| $41$ |
\( T^{4} + 6 T^{3} + \cdots + 5625 \)
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| $43$ |
\( T^{4} + T^{3} + \cdots + 17161 \)
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| $47$ |
\( (T^{2} + 3 T - 3)^{2} \)
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| $53$ |
\( (T^{2} - 21)^{2} \)
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| $59$ |
\( T^{4} + 21T^{2} + 441 \)
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| $61$ |
\( T^{4} - 8 T^{3} + \cdots + 25 \)
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| $67$ |
\( (T^{2} + 11 T + 121)^{2} \)
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| $71$ |
\( T^{4} + 6 T^{3} + \cdots + 144 \)
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| $73$ |
\( (T^{2} - 4 T - 80)^{2} \)
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| $79$ |
\( (T^{2} - 7 T - 119)^{2} \)
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| $83$ |
\( (T^{2} - 6 T - 75)^{2} \)
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| $89$ |
\( T^{4} + 15 T^{3} + \cdots + 81 \)
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| $97$ |
\( T^{4} - 23 T^{3} + \cdots + 16129 \)
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