Newspace parameters
| Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 775.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(6.18840615665\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{24})^+\) |
| Defining polynomial: |
\( x^{4} - 4x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 155) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.41421 | −0.517638 | 0 | 0 | 0.732051 | 2.44949 | 2.82843 | −2.73205 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −1.41421 | 1.93185 | 0 | 0 | −2.73205 | −2.44949 | 2.82843 | 0.732051 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 1.41421 | −1.93185 | 0 | 0 | −2.73205 | 2.44949 | −2.82843 | 0.732051 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 1.41421 | 0.517638 | 0 | 0 | 0.732051 | −2.44949 | −2.82843 | −2.73205 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(31\) | \(-1\) |
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 | 775.2.a.f | 4 | |
| 3.b | odd | 2 | 1 | 6975.2.a.bk | 4 | ||
| 5.b | even | 2 | 1 | inner | 775.2.a.f | 4 | |
| 5.c | odd | 4 | 2 | 155.2.b.a | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 6975.2.a.bk | 4 | ||
| 15.e | even | 4 | 2 | 1395.2.c.c | 4 | ||
| 20.e | even | 4 | 2 | 2480.2.d.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 155.2.b.a | ✓ | 4 | 5.c | odd | 4 | 2 | |
| 775.2.a.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 775.2.a.f | 4 | 5.b | even | 2 | 1 | inner | |
| 1395.2.c.c | 4 | 15.e | even | 4 | 2 | ||
| 2480.2.d.b | 4 | 20.e | even | 4 | 2 | ||
| 6975.2.a.bk | 4 | 3.b | odd | 2 | 1 | ||
| 6975.2.a.bk | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(775))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 2)^{2} \)
$3$
\( T^{4} - 4T^{2} + 1 \)
$5$
\( T^{4} \)
$7$
\( (T^{2} - 6)^{2} \)
$11$
\( (T^{2} + 6 T + 6)^{2} \)
$13$
\( (T^{2} - 6)^{2} \)
$17$
\( T^{4} - 28T^{2} + 169 \)
$19$
\( (T^{2} + 4 T - 23)^{2} \)
$23$
\( (T^{2} - 2)^{2} \)
$29$
\( (T^{2} + 6 T - 18)^{2} \)
$31$
\( (T - 1)^{4} \)
$37$
\( T^{4} - 12T^{2} + 9 \)
$41$
\( (T^{2} + 18 T + 69)^{2} \)
$43$
\( T^{4} - 156T^{2} + 9 \)
$47$
\( T^{4} - 112T^{2} + 2704 \)
$53$
\( T^{4} - 76T^{2} + 121 \)
$59$
\( (T^{2} + 6 T - 39)^{2} \)
$61$
\( (T^{2} + 8 T - 92)^{2} \)
$67$
\( (T^{2} - 6)^{2} \)
$71$
\( (T^{2} + 12 T + 33)^{2} \)
$73$
\( T^{4} - 84T^{2} + 1089 \)
$79$
\( (T^{2} - 2 T - 26)^{2} \)
$83$
\( T^{4} - 268 T^{2} + 11881 \)
$89$
\( (T^{2} + 18 T + 54)^{2} \)
$97$
\( T^{4} - 144T^{2} + 1296 \)
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