Newspace parameters
| Level: | \( N \) | \(=\) | \( 35 = 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 35.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.953680925261\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{10})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| 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
\( x^{4} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{2} ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 10\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 5\nu ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -10\beta_{3} + 5\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).
| \(n\) | \(22\) | \(31\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 34.1 |
|
− | 3.00000i | −3.16228 | −5.00000 | 1.58114 | − | 4.74342i | 9.48683i | 6.32456 | + | 3.00000i | 3.00000i | 1.00000 | −14.2302 | − | 4.74342i | |||||||||||||||||||||||
| 34.2 | − | 3.00000i | 3.16228 | −5.00000 | −1.58114 | + | 4.74342i | − | 9.48683i | −6.32456 | + | 3.00000i | 3.00000i | 1.00000 | 14.2302 | + | 4.74342i | |||||||||||||||||||||||
| 34.3 | 3.00000i | −3.16228 | −5.00000 | 1.58114 | + | 4.74342i | − | 9.48683i | 6.32456 | − | 3.00000i | − | 3.00000i | 1.00000 | −14.2302 | + | 4.74342i | |||||||||||||||||||||||
| 34.4 | 3.00000i | 3.16228 | −5.00000 | −1.58114 | − | 4.74342i | 9.48683i | −6.32456 | − | 3.00000i | − | 3.00000i | 1.00000 | 14.2302 | − | 4.74342i | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 35.3.c.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 315.3.e.c | 4 | ||
| 4.b | odd | 2 | 1 | 560.3.p.f | 4 | ||
| 5.b | even | 2 | 1 | inner | 35.3.c.c | ✓ | 4 |
| 5.c | odd | 4 | 1 | 175.3.d.b | 2 | ||
| 5.c | odd | 4 | 1 | 175.3.d.h | 2 | ||
| 7.b | odd | 2 | 1 | inner | 35.3.c.c | ✓ | 4 |
| 7.c | even | 3 | 2 | 245.3.i.c | 8 | ||
| 7.d | odd | 6 | 2 | 245.3.i.c | 8 | ||
| 15.d | odd | 2 | 1 | 315.3.e.c | 4 | ||
| 20.d | odd | 2 | 1 | 560.3.p.f | 4 | ||
| 21.c | even | 2 | 1 | 315.3.e.c | 4 | ||
| 28.d | even | 2 | 1 | 560.3.p.f | 4 | ||
| 35.c | odd | 2 | 1 | inner | 35.3.c.c | ✓ | 4 |
| 35.f | even | 4 | 1 | 175.3.d.b | 2 | ||
| 35.f | even | 4 | 1 | 175.3.d.h | 2 | ||
| 35.i | odd | 6 | 2 | 245.3.i.c | 8 | ||
| 35.j | even | 6 | 2 | 245.3.i.c | 8 | ||
| 105.g | even | 2 | 1 | 315.3.e.c | 4 | ||
| 140.c | even | 2 | 1 | 560.3.p.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.3.c.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 35.3.c.c | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 35.3.c.c | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 35.3.c.c | ✓ | 4 | 35.c | odd | 2 | 1 | inner |
| 175.3.d.b | 2 | 5.c | odd | 4 | 1 | ||
| 175.3.d.b | 2 | 35.f | even | 4 | 1 | ||
| 175.3.d.h | 2 | 5.c | odd | 4 | 1 | ||
| 175.3.d.h | 2 | 35.f | even | 4 | 1 | ||
| 245.3.i.c | 8 | 7.c | even | 3 | 2 | ||
| 245.3.i.c | 8 | 7.d | odd | 6 | 2 | ||
| 245.3.i.c | 8 | 35.i | odd | 6 | 2 | ||
| 245.3.i.c | 8 | 35.j | even | 6 | 2 | ||
| 315.3.e.c | 4 | 3.b | odd | 2 | 1 | ||
| 315.3.e.c | 4 | 15.d | odd | 2 | 1 | ||
| 315.3.e.c | 4 | 21.c | even | 2 | 1 | ||
| 315.3.e.c | 4 | 105.g | even | 2 | 1 | ||
| 560.3.p.f | 4 | 4.b | odd | 2 | 1 | ||
| 560.3.p.f | 4 | 20.d | odd | 2 | 1 | ||
| 560.3.p.f | 4 | 28.d | even | 2 | 1 | ||
| 560.3.p.f | 4 | 140.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(35, [\chi])\):
|
\( T_{2}^{2} + 9 \)
|
|
\( T_{3}^{2} - 10 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 9)^{2} \)
|
| $3$ |
\( (T^{2} - 10)^{2} \)
|
| $5$ |
\( T^{4} + 40T^{2} + 625 \)
|
| $7$ |
\( T^{4} - 62T^{2} + 2401 \)
|
| $11$ |
\( (T - 14)^{4} \)
|
| $13$ |
\( (T^{2} - 10)^{2} \)
|
| $17$ |
\( (T^{2} - 40)^{2} \)
|
| $19$ |
\( (T^{2} + 810)^{2} \)
|
| $23$ |
\( (T^{2} + 144)^{2} \)
|
| $29$ |
\( (T - 14)^{4} \)
|
| $31$ |
\( (T^{2} + 1440)^{2} \)
|
| $37$ |
\( (T^{2} + 324)^{2} \)
|
| $41$ |
\( (T^{2} + 360)^{2} \)
|
| $43$ |
\( (T^{2} + 1764)^{2} \)
|
| $47$ |
\( (T^{2} - 1960)^{2} \)
|
| $53$ |
\( (T^{2} + 2916)^{2} \)
|
| $59$ |
\( (T^{2} + 90)^{2} \)
|
| $61$ |
\( (T^{2} + 4410)^{2} \)
|
| $67$ |
\( (T^{2} + 10404)^{2} \)
|
| $71$ |
\( (T + 16)^{4} \)
|
| $73$ |
\( (T^{2} - 4000)^{2} \)
|
| $79$ |
\( (T + 76)^{4} \)
|
| $83$ |
\( (T^{2} - 5290)^{2} \)
|
| $89$ |
\( (T^{2} + 3240)^{2} \)
|
| $97$ |
\( (T^{2} - 4840)^{2} \)
|
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