Newspace parameters
| Level: | \( N \) | \(=\) | \( 95 = 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 95.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(9.82014649297\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{10}, \sqrt{38})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 24x^{2} + 49 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{4} - 24x^{2} + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 17\nu ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{2} + 17\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) | \(77\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 94.1 |
|
−7.82562 | −9.16655 | 45.2404 | −25.0000 | 71.7340 | 0 | −228.824 | 3.02565 | 195.641 | ||||||||||||||||||||||||||||||
| 94.2 | −1.66121 | 15.4911 | −13.2404 | −25.0000 | −25.7340 | 0 | 48.5744 | 158.974 | 41.5302 | |||||||||||||||||||||||||||||||
| 94.3 | 1.66121 | −15.4911 | −13.2404 | −25.0000 | −25.7340 | 0 | −48.5744 | 158.974 | −41.5302 | |||||||||||||||||||||||||||||||
| 94.4 | 7.82562 | 9.16655 | 45.2404 | −25.0000 | 71.7340 | 0 | 228.824 | 3.02565 | −195.641 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 95.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-95}) \) |
| 5.b | even | 2 | 1 | inner |
| 19.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 95.5.d.d | ✓ | 4 |
| 5.b | even | 2 | 1 | inner | 95.5.d.d | ✓ | 4 |
| 19.b | odd | 2 | 1 | inner | 95.5.d.d | ✓ | 4 |
| 95.d | odd | 2 | 1 | CM | 95.5.d.d | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.5.d.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 95.5.d.d | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 95.5.d.d | ✓ | 4 | 19.b | odd | 2 | 1 | inner |
| 95.5.d.d | ✓ | 4 | 95.d | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 64T_{2}^{2} + 169 \)
acting on \(S_{5}^{\mathrm{new}}(95, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 64T^{2} + 169 \)
$3$
\( T^{4} - 324 T^{2} + 20164 \)
$5$
\( (T + 25)^{4} \)
$7$
\( T^{4} \)
$11$
\( (T^{2} - 54720)^{2} \)
$13$
\( T^{4} - 114244 T^{2} + \cdots + 2769074884 \)
$17$
\( T^{4} \)
$19$
\( (T + 361)^{4} \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( T^{4} \)
$37$
\( T^{4} - 7496644 T^{2} + \cdots + 10480098340804 \)
$41$
\( T^{4} \)
$43$
\( T^{4} \)
$47$
\( T^{4} \)
$53$
\( T^{4} + \cdots + 229655293801924 \)
$59$
\( T^{4} \)
$61$
\( (T^{2} - 4432320)^{2} \)
$67$
\( T^{4} - 80604484 T^{2} + \cdots + 11320681202884 \)
$71$
\( T^{4} \)
$73$
\( T^{4} \)
$79$
\( T^{4} \)
$83$
\( T^{4} \)
$89$
\( T^{4} \)
$97$
\( T^{4} - 354117124 T^{2} + \cdots + 36\!\cdots\!24 \)
show more
show less