Newspace parameters
| Level: | \( N \) | \(=\) | \( 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 61.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.487087452330\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.29952.1 |
|
|
|
| Defining polynomial: |
\( x^{4} + 8x^{2} + 13 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{4} + 8x^{2} + 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 5\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/61\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 60.1 |
|
− | 2.39417i | −2.73205 | −3.73205 | 1.73205 | 6.54099i | − | 4.14682i | 4.14682i | 4.46410 | − | 4.14682i | |||||||||||||||||||||||||||
| 60.2 | − | 1.50597i | 0.732051 | −0.267949 | −1.73205 | − | 1.10245i | 2.60842i | − | 2.60842i | −2.46410 | 2.60842i | ||||||||||||||||||||||||||||
| 60.3 | 1.50597i | 0.732051 | −0.267949 | −1.73205 | 1.10245i | − | 2.60842i | 2.60842i | −2.46410 | − | 2.60842i | |||||||||||||||||||||||||||||
| 60.4 | 2.39417i | −2.73205 | −3.73205 | 1.73205 | − | 6.54099i | 4.14682i | − | 4.14682i | 4.46410 | 4.14682i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 61.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 61.2.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 549.2.c.c | 4 | ||
| 4.b | odd | 2 | 1 | 976.2.h.d | 4 | ||
| 5.b | even | 2 | 1 | 1525.2.c.b | 4 | ||
| 5.c | odd | 4 | 2 | 1525.2.d.c | 8 | ||
| 61.b | even | 2 | 1 | inner | 61.2.b.a | ✓ | 4 |
| 61.d | odd | 4 | 2 | 3721.2.a.f | 4 | ||
| 183.d | odd | 2 | 1 | 549.2.c.c | 4 | ||
| 244.c | odd | 2 | 1 | 976.2.h.d | 4 | ||
| 305.d | even | 2 | 1 | 1525.2.c.b | 4 | ||
| 305.h | odd | 4 | 2 | 1525.2.d.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 61.2.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 61.2.b.a | ✓ | 4 | 61.b | even | 2 | 1 | inner |
| 549.2.c.c | 4 | 3.b | odd | 2 | 1 | ||
| 549.2.c.c | 4 | 183.d | odd | 2 | 1 | ||
| 976.2.h.d | 4 | 4.b | odd | 2 | 1 | ||
| 976.2.h.d | 4 | 244.c | odd | 2 | 1 | ||
| 1525.2.c.b | 4 | 5.b | even | 2 | 1 | ||
| 1525.2.c.b | 4 | 305.d | even | 2 | 1 | ||
| 1525.2.d.c | 8 | 5.c | odd | 4 | 2 | ||
| 1525.2.d.c | 8 | 305.h | odd | 4 | 2 | ||
| 3721.2.a.f | 4 | 61.d | odd | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(61, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 8T^{2} + 13 \)
|
| $3$ |
\( (T^{2} + 2 T - 2)^{2} \)
|
| $5$ |
\( (T^{2} - 3)^{2} \)
|
| $7$ |
\( T^{4} + 24T^{2} + 117 \)
|
| $11$ |
\( T^{4} + 8T^{2} + 13 \)
|
| $13$ |
\( (T - 3)^{4} \)
|
| $17$ |
\( T^{4} + 20T^{2} + 52 \)
|
| $19$ |
\( (T^{2} - 6 T + 6)^{2} \)
|
| $23$ |
\( T^{4} + 80T^{2} + 1573 \)
|
| $29$ |
\( T^{4} + 20T^{2} + 52 \)
|
| $31$ |
\( T^{4} + 60T^{2} + 468 \)
|
| $37$ |
\( T^{4} + 60T^{2} + 468 \)
|
| $41$ |
\( (T^{2} + 6 T - 3)^{2} \)
|
| $43$ |
\( T^{4} + 132T^{2} + 468 \)
|
| $47$ |
\( (T^{2} - 12 T - 12)^{2} \)
|
| $53$ |
\( T^{4} + 128T^{2} + 208 \)
|
| $59$ |
\( T^{4} + 80T^{2} + 13 \)
|
| $61$ |
\( T^{4} - 8 T^{3} + \cdots + 3721 \)
|
| $67$ |
\( T^{4} + 240 T^{2} + 14157 \)
|
| $71$ |
\( T^{4} + 44T^{2} + 52 \)
|
| $73$ |
\( (T^{2} - 8 T - 11)^{2} \)
|
| $79$ |
\( T^{4} + 312 T^{2} + 19773 \)
|
| $83$ |
\( (T^{2} - 12 T + 24)^{2} \)
|
| $89$ |
\( T^{4} + 320 T^{2} + 25168 \)
|
| $97$ |
\( (T^{2} - 12)^{2} \)
|
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