Newspace parameters
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.95633484952\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 4\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
− | 2.44949i | − | 1.73205i | −4.00000 | −1.41421 | − | 1.73205i | −4.24264 | 0 | 4.89898i | 0 | −4.24264 | + | 3.46410i | ||||||||||||||||||||||||
| 99.2 | − | 2.44949i | 1.73205i | −4.00000 | 1.41421 | + | 1.73205i | 4.24264 | 0 | 4.89898i | 0 | 4.24264 | − | 3.46410i | ||||||||||||||||||||||||||
| 99.3 | 2.44949i | − | 1.73205i | −4.00000 | 1.41421 | − | 1.73205i | 4.24264 | 0 | − | 4.89898i | 0 | 4.24264 | + | 3.46410i | |||||||||||||||||||||||||
| 99.4 | 2.44949i | 1.73205i | −4.00000 | −1.41421 | + | 1.73205i | −4.24264 | 0 | − | 4.89898i | 0 | −4.24264 | − | 3.46410i | ||||||||||||||||||||||||||
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 | 245.2.b.e | ✓ | 4 |
| 3.b | odd | 2 | 1 | 2205.2.d.i | 4 | ||
| 5.b | even | 2 | 1 | inner | 245.2.b.e | ✓ | 4 |
| 5.c | odd | 4 | 2 | 1225.2.a.bb | 4 | ||
| 7.b | odd | 2 | 1 | inner | 245.2.b.e | ✓ | 4 |
| 7.c | even | 3 | 1 | 245.2.j.a | 4 | ||
| 7.c | even | 3 | 1 | 245.2.j.f | 4 | ||
| 7.d | odd | 6 | 1 | 245.2.j.a | 4 | ||
| 7.d | odd | 6 | 1 | 245.2.j.f | 4 | ||
| 15.d | odd | 2 | 1 | 2205.2.d.i | 4 | ||
| 21.c | even | 2 | 1 | 2205.2.d.i | 4 | ||
| 35.c | odd | 2 | 1 | inner | 245.2.b.e | ✓ | 4 |
| 35.f | even | 4 | 2 | 1225.2.a.bb | 4 | ||
| 35.i | odd | 6 | 1 | 245.2.j.a | 4 | ||
| 35.i | odd | 6 | 1 | 245.2.j.f | 4 | ||
| 35.j | even | 6 | 1 | 245.2.j.a | 4 | ||
| 35.j | even | 6 | 1 | 245.2.j.f | 4 | ||
| 105.g | even | 2 | 1 | 2205.2.d.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 245.2.b.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 245.2.b.e | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 245.2.b.e | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 245.2.b.e | ✓ | 4 | 35.c | odd | 2 | 1 | inner |
| 245.2.j.a | 4 | 7.c | even | 3 | 1 | ||
| 245.2.j.a | 4 | 7.d | odd | 6 | 1 | ||
| 245.2.j.a | 4 | 35.i | odd | 6 | 1 | ||
| 245.2.j.a | 4 | 35.j | even | 6 | 1 | ||
| 245.2.j.f | 4 | 7.c | even | 3 | 1 | ||
| 245.2.j.f | 4 | 7.d | odd | 6 | 1 | ||
| 245.2.j.f | 4 | 35.i | odd | 6 | 1 | ||
| 245.2.j.f | 4 | 35.j | even | 6 | 1 | ||
| 1225.2.a.bb | 4 | 5.c | odd | 4 | 2 | ||
| 1225.2.a.bb | 4 | 35.f | even | 4 | 2 | ||
| 2205.2.d.i | 4 | 3.b | odd | 2 | 1 | ||
| 2205.2.d.i | 4 | 15.d | odd | 2 | 1 | ||
| 2205.2.d.i | 4 | 21.c | even | 2 | 1 | ||
| 2205.2.d.i | 4 | 105.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\):
|
\( T_{2}^{2} + 6 \)
|
|
\( T_{3}^{2} + 3 \)
|
|
\( T_{19}^{2} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 6)^{2} \)
|
| $3$ |
\( (T^{2} + 3)^{2} \)
|
| $5$ |
\( T^{4} + 2T^{2} + 25 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T - 5)^{4} \)
|
| $13$ |
\( (T^{2} + 3)^{2} \)
|
| $17$ |
\( (T^{2} + 3)^{2} \)
|
| $19$ |
\( (T^{2} - 8)^{2} \)
|
| $23$ |
\( (T^{2} + 6)^{2} \)
|
| $29$ |
\( (T + 5)^{4} \)
|
| $31$ |
\( (T^{2} - 2)^{2} \)
|
| $37$ |
\( (T^{2} + 6)^{2} \)
|
| $41$ |
\( (T^{2} - 98)^{2} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( (T^{2} + 75)^{2} \)
|
| $53$ |
\( (T^{2} + 54)^{2} \)
|
| $59$ |
\( (T^{2} - 2)^{2} \)
|
| $61$ |
\( (T^{2} - 8)^{2} \)
|
| $67$ |
\( (T^{2} + 54)^{2} \)
|
| $71$ |
\( (T - 2)^{4} \)
|
| $73$ |
\( (T^{2} + 192)^{2} \)
|
| $79$ |
\( (T - 3)^{4} \)
|
| $83$ |
\( (T^{2} + 192)^{2} \)
|
| $89$ |
\( (T^{2} - 288)^{2} \)
|
| $97$ |
\( (T^{2} + 3)^{2} \)
|
| show more | |
| show less |