Newspace parameters
| Level: | \( N \) | \(=\) | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 650.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(38.3512415037\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{19})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 9x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 130) |
| 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
\( x^{4} - 9x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 4\nu ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 14\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 9 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} + 7\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 599.1 |
|
− | 2.00000i | − | 5.35890i | −4.00000 | 0 | −10.7178 | − | 26.1534i | 8.00000i | −1.71780 | 0 | |||||||||||||||||||||||||||
| 599.2 | − | 2.00000i | 3.35890i | −4.00000 | 0 | 6.71780 | 26.1534i | 8.00000i | 15.7178 | 0 | ||||||||||||||||||||||||||||||
| 599.3 | 2.00000i | − | 3.35890i | −4.00000 | 0 | 6.71780 | − | 26.1534i | − | 8.00000i | 15.7178 | 0 | ||||||||||||||||||||||||||||
| 599.4 | 2.00000i | 5.35890i | −4.00000 | 0 | −10.7178 | 26.1534i | − | 8.00000i | −1.71780 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 650.4.b.l | 4 | |
| 5.b | even | 2 | 1 | inner | 650.4.b.l | 4 | |
| 5.c | odd | 4 | 1 | 130.4.a.d | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 650.4.a.r | 2 | ||
| 15.e | even | 4 | 1 | 1170.4.a.ba | 2 | ||
| 20.e | even | 4 | 1 | 1040.4.a.i | 2 | ||
| 65.h | odd | 4 | 1 | 1690.4.a.s | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.4.a.d | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 650.4.a.r | 2 | 5.c | odd | 4 | 1 | ||
| 650.4.b.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 650.4.b.l | 4 | 5.b | even | 2 | 1 | inner | |
| 1040.4.a.i | 2 | 20.e | even | 4 | 1 | ||
| 1170.4.a.ba | 2 | 15.e | even | 4 | 1 | ||
| 1690.4.a.s | 2 | 65.h | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(650, [\chi])\):
|
\( T_{3}^{4} + 40T_{3}^{2} + 324 \)
|
|
\( T_{7}^{2} + 684 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{2} \)
|
| $3$ |
\( T^{4} + 40T^{2} + 324 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 684)^{2} \)
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| $11$ |
\( (T^{2} + 50 T - 306)^{2} \)
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| $13$ |
\( (T^{2} + 169)^{2} \)
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| $17$ |
\( T^{4} + 11320 T^{2} + 3196944 \)
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| $19$ |
\( (T^{2} - 94 T - 2066)^{2} \)
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| $23$ |
\( T^{4} + 7784 T^{2} + 12602500 \)
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| $29$ |
\( (T^{2} - 260 T + 7704)^{2} \)
|
| $31$ |
\( (T^{2} + 402 T + 40382)^{2} \)
|
| $37$ |
\( T^{4} + 90112 T^{2} + 37748736 \)
|
| $41$ |
\( (T^{2} + 536 T + 49860)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 22361613444 \)
|
| $47$ |
\( T^{4} + \cdots + 19453554576 \)
|
| $53$ |
\( T^{4} + \cdots + 71513456400 \)
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| $59$ |
\( (T^{2} - 330 T + 24014)^{2} \)
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| $61$ |
\( (T^{2} - 620 T + 13336)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 57237691536 \)
|
| $71$ |
\( (T^{2} + 734 T + 131478)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 210717721600 \)
|
| $79$ |
\( (T^{2} - 924 T - 51112)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 1759390016400 \)
|
| $89$ |
\( (T^{2} - 748 T + 71476)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 79625552400 \)
|
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