Newspace parameters
| Level: | \( N \) | \(=\) | \( 95 = 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 95.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(60.3589390040\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.7600.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 9x^{2} + 19 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5^{2}\cdot 11 \) |
| 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
\( x^{4} - 9x^{2} + 19 \)
:
| \(\beta_{1}\) | \(=\) |
\( -5\nu^{3} + 40\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -6\nu^{3} + 26\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 10\nu^{2} - 45 \)
|
| \(\nu\) | \(=\) |
\( ( -5\beta_{2} + 6\beta_1 ) / 110 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 45 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -20\beta_{2} + 13\beta_1 ) / 55 \)
|
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 |
|
−63.4580 | 76.3219 | 3002.92 | 3125.00 | −4843.23 | 0 | −125578. | −53224.0 | −198306. | ||||||||||||||||||||||||||||||
| 94.2 | −8.31143 | −479.970 | −954.920 | 3125.00 | 3989.23 | 0 | 16447.7 | 171322. | −25973.2 | |||||||||||||||||||||||||||||||
| 94.3 | 8.31143 | 479.970 | −954.920 | 3125.00 | 3989.23 | 0 | −16447.7 | 171322. | 25973.2 | |||||||||||||||||||||||||||||||
| 94.4 | 63.4580 | −76.3219 | 3002.92 | 3125.00 | −4843.23 | 0 | 125578. | −53224.0 | 198306. | |||||||||||||||||||||||||||||||
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 | 95.11.d.c | ✓ | 4 |
| 5.b | even | 2 | 1 | inner | 95.11.d.c | ✓ | 4 |
| 19.b | odd | 2 | 1 | inner | 95.11.d.c | ✓ | 4 |
| 95.d | odd | 2 | 1 | CM | 95.11.d.c | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.11.d.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 95.11.d.c | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 95.11.d.c | ✓ | 4 | 19.b | odd | 2 | 1 | inner |
| 95.11.d.c | ✓ | 4 | 95.d | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 4096T_{2}^{2} + 278179 \)
acting on \(S_{11}^{\mathrm{new}}(95, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 4096 T^{2} + 278179 \)
$3$
\( T^{4} - 236196 T^{2} + \cdots + 1341917104 \)
$5$
\( (T - 3125)^{4} \)
$7$
\( T^{4} \)
$11$
\( (T^{2} - 1954264500)^{2} \)
$13$
\( T^{4} - 551433967396 T^{2} + \cdots + 22\!\cdots\!04 \)
$17$
\( T^{4} \)
$19$
\( (T + 2476099)^{4} \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( T^{4} \)
$37$
\( T^{4} + \cdots + 86\!\cdots\!04 \)
$41$
\( T^{4} \)
$43$
\( T^{4} \)
$47$
\( T^{4} \)
$53$
\( T^{4} + \cdots + 70\!\cdots\!04 \)
$59$
\( T^{4} \)
$61$
\( (T^{2} - 16\!\cdots\!00)^{2} \)
$67$
\( T^{4} + \cdots + 93\!\cdots\!04 \)
$71$
\( T^{4} \)
$73$
\( T^{4} \)
$79$
\( T^{4} \)
$83$
\( T^{4} \)
$89$
\( T^{4} \)
$97$
\( T^{4} + \cdots + 16\!\cdots\!04 \)
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