Newspace parameters
| Level: | \( N \) | \(=\) | \( 190 = 2 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 190.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.51715763840\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.5161984.1 |
| Defining polynomial: |
\( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} - 5\beta_{2} + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/190\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) | \(77\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 39.1 |
|
− | 1.00000i | − | 2.12489i | −1.00000 | 1.80487 | + | 1.32001i | −2.12489 | − | 4.12489i | 1.00000i | −1.51514 | 1.32001 | − | 1.80487i | |||||||||||||||||||||||||||||
| 39.2 | − | 1.00000i | 1.36333i | −1.00000 | 1.38900 | − | 1.75233i | 1.36333 | − | 0.636672i | 1.00000i | 1.14134 | −1.75233 | − | 1.38900i | |||||||||||||||||||||||||||||||
| 39.3 | − | 1.00000i | 2.76156i | −1.00000 | −2.19388 | + | 0.432320i | 2.76156 | 0.761557i | 1.00000i | −4.62620 | 0.432320 | + | 2.19388i | ||||||||||||||||||||||||||||||||
| 39.4 | 1.00000i | − | 2.76156i | −1.00000 | −2.19388 | − | 0.432320i | 2.76156 | − | 0.761557i | − | 1.00000i | −4.62620 | 0.432320 | − | 2.19388i | ||||||||||||||||||||||||||||||
| 39.5 | 1.00000i | − | 1.36333i | −1.00000 | 1.38900 | + | 1.75233i | 1.36333 | 0.636672i | − | 1.00000i | 1.14134 | −1.75233 | + | 1.38900i | |||||||||||||||||||||||||||||||
| 39.6 | 1.00000i | 2.12489i | −1.00000 | 1.80487 | − | 1.32001i | −2.12489 | 4.12489i | − | 1.00000i | −1.51514 | 1.32001 | + | 1.80487i | ||||||||||||||||||||||||||||||||
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 | 190.2.b.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1710.2.d.d | 6 | ||
| 4.b | odd | 2 | 1 | 1520.2.d.j | 6 | ||
| 5.b | even | 2 | 1 | inner | 190.2.b.b | ✓ | 6 |
| 5.c | odd | 4 | 1 | 950.2.a.i | 3 | ||
| 5.c | odd | 4 | 1 | 950.2.a.n | 3 | ||
| 15.d | odd | 2 | 1 | 1710.2.d.d | 6 | ||
| 15.e | even | 4 | 1 | 8550.2.a.ck | 3 | ||
| 15.e | even | 4 | 1 | 8550.2.a.cl | 3 | ||
| 20.d | odd | 2 | 1 | 1520.2.d.j | 6 | ||
| 20.e | even | 4 | 1 | 7600.2.a.bi | 3 | ||
| 20.e | even | 4 | 1 | 7600.2.a.cd | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 190.2.b.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 190.2.b.b | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 950.2.a.i | 3 | 5.c | odd | 4 | 1 | ||
| 950.2.a.n | 3 | 5.c | odd | 4 | 1 | ||
| 1520.2.d.j | 6 | 4.b | odd | 2 | 1 | ||
| 1520.2.d.j | 6 | 20.d | odd | 2 | 1 | ||
| 1710.2.d.d | 6 | 3.b | odd | 2 | 1 | ||
| 1710.2.d.d | 6 | 15.d | odd | 2 | 1 | ||
| 7600.2.a.bi | 3 | 20.e | even | 4 | 1 | ||
| 7600.2.a.cd | 3 | 20.e | even | 4 | 1 | ||
| 8550.2.a.ck | 3 | 15.e | even | 4 | 1 | ||
| 8550.2.a.cl | 3 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 14T_{3}^{4} + 57T_{3}^{2} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(190, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{3} \)
$3$
\( T^{6} + 14 T^{4} + 57 T^{2} + 64 \)
$5$
\( T^{6} - 2 T^{5} - 3 T^{4} + 24 T^{3} + \cdots + 125 \)
$7$
\( T^{6} + 18 T^{4} + 17 T^{2} + 4 \)
$11$
\( (T^{3} - 10 T - 8)^{2} \)
$13$
\( T^{6} + 38 T^{4} + 201 T^{2} + 4 \)
$17$
\( T^{6} + 18 T^{4} + 65 T^{2} + 16 \)
$19$
\( (T + 1)^{6} \)
$23$
\( T^{6} + 98 T^{4} + 2401 T^{2} + \cdots + 14884 \)
$29$
\( (T^{3} - 8 T^{2} - 51 T + 410)^{2} \)
$31$
\( (T^{3} - 4 T^{2} - 62 T + 232)^{2} \)
$37$
\( T^{6} + 116 T^{4} + 1152 T^{2} + \cdots + 256 \)
$41$
\( (T^{3} - 2 T^{2} - 50 T - 100)^{2} \)
$43$
\( T^{6} + 128 T^{4} + 4276 T^{2} + \cdots + 21904 \)
$47$
\( T^{6} + 132 T^{4} + 2816 T^{2} + \cdots + 4096 \)
$53$
\( T^{6} + 102 T^{4} + 2537 T^{2} + \cdots + 11236 \)
$59$
\( (T^{3} + 2 T^{2} - 29 T - 80)^{2} \)
$61$
\( (T^{3} + 30 T^{2} + 290 T + 892)^{2} \)
$67$
\( T^{6} + 126 T^{4} + 3977 T^{2} + \cdots + 4096 \)
$71$
\( (T^{3} + 8 T^{2} - 122 T - 1016)^{2} \)
$73$
\( T^{6} + 290 T^{4} + 5745 T^{2} + \cdots + 26896 \)
$79$
\( (T^{3} - 228 T + 880)^{2} \)
$83$
\( T^{6} + 92 T^{4} + 880 T^{2} + \cdots + 64 \)
$89$
\( (T^{3} - 14 T^{2} + 46 T + 20)^{2} \)
$97$
\( T^{6} + 300 T^{4} + 19760 T^{2} + \cdots + 238144 \)
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