Newspace parameters
| Level: | \( N \) | \(=\) | \( 290 = 2 \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 290.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.31566165862\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{29})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 15x^{2} + 49 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} + 15x^{2} + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 8\nu ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{2} - 8\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/290\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(117\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 231.1 |
|
− | 1.00000i | − | 3.19258i | −1.00000 | −1.00000 | −3.19258 | 0.192582 | 1.00000i | −7.19258 | 1.00000i | ||||||||||||||||||||||||||||
| 231.2 | − | 1.00000i | 2.19258i | −1.00000 | −1.00000 | 2.19258 | −5.19258 | 1.00000i | −1.80742 | 1.00000i | ||||||||||||||||||||||||||||||
| 231.3 | 1.00000i | − | 2.19258i | −1.00000 | −1.00000 | 2.19258 | −5.19258 | − | 1.00000i | −1.80742 | − | 1.00000i | ||||||||||||||||||||||||||||
| 231.4 | 1.00000i | 3.19258i | −1.00000 | −1.00000 | −3.19258 | 0.192582 | − | 1.00000i | −7.19258 | − | 1.00000i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 290.2.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 2610.2.f.e | 4 | ||
| 4.b | odd | 2 | 1 | 2320.2.g.f | 4 | ||
| 5.b | even | 2 | 1 | 1450.2.c.c | 4 | ||
| 5.c | odd | 4 | 1 | 1450.2.d.f | 4 | ||
| 5.c | odd | 4 | 1 | 1450.2.d.g | 4 | ||
| 29.b | even | 2 | 1 | inner | 290.2.c.b | ✓ | 4 |
| 29.c | odd | 4 | 1 | 8410.2.a.o | 2 | ||
| 29.c | odd | 4 | 1 | 8410.2.a.t | 2 | ||
| 87.d | odd | 2 | 1 | 2610.2.f.e | 4 | ||
| 116.d | odd | 2 | 1 | 2320.2.g.f | 4 | ||
| 145.d | even | 2 | 1 | 1450.2.c.c | 4 | ||
| 145.h | odd | 4 | 1 | 1450.2.d.f | 4 | ||
| 145.h | odd | 4 | 1 | 1450.2.d.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 290.2.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 290.2.c.b | ✓ | 4 | 29.b | even | 2 | 1 | inner |
| 1450.2.c.c | 4 | 5.b | even | 2 | 1 | ||
| 1450.2.c.c | 4 | 145.d | even | 2 | 1 | ||
| 1450.2.d.f | 4 | 5.c | odd | 4 | 1 | ||
| 1450.2.d.f | 4 | 145.h | odd | 4 | 1 | ||
| 1450.2.d.g | 4 | 5.c | odd | 4 | 1 | ||
| 1450.2.d.g | 4 | 145.h | odd | 4 | 1 | ||
| 2320.2.g.f | 4 | 4.b | odd | 2 | 1 | ||
| 2320.2.g.f | 4 | 116.d | odd | 2 | 1 | ||
| 2610.2.f.e | 4 | 3.b | odd | 2 | 1 | ||
| 2610.2.f.e | 4 | 87.d | odd | 2 | 1 | ||
| 8410.2.a.o | 2 | 29.c | odd | 4 | 1 | ||
| 8410.2.a.t | 2 | 29.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 15T_{3}^{2} + 49 \)
acting on \(S_{2}^{\mathrm{new}}(290, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
|
| $3$ |
\( T^{4} + 15T^{2} + 49 \)
|
| $5$ |
\( (T + 1)^{4} \)
|
| $7$ |
\( (T^{2} + 5 T - 1)^{2} \)
|
| $11$ |
\( (T^{2} + 4)^{2} \)
|
| $13$ |
\( (T^{2} + 3 T - 5)^{2} \)
|
| $17$ |
\( T^{4} + 19T^{2} + 25 \)
|
| $19$ |
\( T^{4} + 76T^{2} + 400 \)
|
| $23$ |
\( (T^{2} - T - 7)^{2} \)
|
| $29$ |
\( (T^{2} + 4 T + 29)^{2} \)
|
| $31$ |
\( T^{4} + 27T^{2} + 1 \)
|
| $37$ |
\( (T^{2} + 64)^{2} \)
|
| $41$ |
\( T^{4} + 76T^{2} + 400 \)
|
| $43$ |
\( T^{4} + 27T^{2} + 1 \)
|
| $47$ |
\( T^{4} + 108T^{2} + 16 \)
|
| $53$ |
\( (T^{2} - 11 T + 23)^{2} \)
|
| $59$ |
\( (T^{2} + 23 T + 125)^{2} \)
|
| $61$ |
\( T^{4} + 235 T^{2} + 10609 \)
|
| $67$ |
\( (T^{2} - 14 T + 20)^{2} \)
|
| $71$ |
\( (T - 6)^{4} \)
|
| $73$ |
\( T^{4} + 155T^{2} + 2809 \)
|
| $79$ |
\( T^{4} + 367 T^{2} + 32041 \)
|
| $83$ |
\( (T^{2} + 4 T - 112)^{2} \)
|
| $89$ |
\( T^{4} + 60T^{2} + 784 \)
|
| $97$ |
\( T^{4} + 155T^{2} + 2809 \)
|
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