Newspace parameters
| Level: | \( N \) | \(=\) | \( 95 = 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 95.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(38.7009679558\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{95}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 95 \)
|
| 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 \(\beta = \sqrt{95}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−29.2404 | 77.9744 | 599.000 | 625.000 | −2280.00 | 0 | −10029.5 | −481.000 | −18275.2 | ||||||||||||||||||||||||
| 94.2 | 29.2404 | −77.9744 | 599.000 | 625.000 | −2280.00 | 0 | 10029.5 | −481.000 | 18275.2 | |||||||||||||||||||||||||
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 | 95.9.d.d | ✓ | 2 |
| 5.b | even | 2 | 1 | inner | 95.9.d.d | ✓ | 2 |
| 19.b | odd | 2 | 1 | inner | 95.9.d.d | ✓ | 2 |
| 95.d | odd | 2 | 1 | CM | 95.9.d.d | ✓ | 2 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.9.d.d | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 95.9.d.d | ✓ | 2 | 5.b | even | 2 | 1 | inner |
| 95.9.d.d | ✓ | 2 | 19.b | odd | 2 | 1 | inner |
| 95.9.d.d | ✓ | 2 | 95.d | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 855 \)
acting on \(S_{9}^{\mathrm{new}}(95, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 855 \)
$3$
\( T^{2} - 6080 \)
$5$
\( (T - 625)^{2} \)
$7$
\( T^{2} \)
$11$
\( (T - 25438)^{2} \)
$13$
\( T^{2} - 493848000 \)
$17$
\( T^{2} \)
$19$
\( (T - 130321)^{2} \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} - 3569819474880 \)
$41$
\( T^{2} \)
$43$
\( T^{2} \)
$47$
\( T^{2} \)
$53$
\( T^{2} - 19383467843520 \)
$59$
\( T^{2} \)
$61$
\( (T + 23259362)^{2} \)
$67$
\( T^{2} - 16\!\cdots\!80 \)
$71$
\( T^{2} \)
$73$
\( T^{2} \)
$79$
\( T^{2} \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} - 27\!\cdots\!20 \)
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