Newspace parameters
| Level: | \( N \) | \(=\) | \( 145 = 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 145.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.15783082931\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.84345856.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 13x^{4} + 41x^{2} + 1 \)
|
| 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
\( x^{6} + 13x^{4} + 41x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 14\nu^{3} + 6\nu^{2} + 47\nu - 5 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} + 14\nu^{3} - 6\nu^{2} + 47\nu + 5 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 14\nu^{3} + 10\nu^{2} - 47\nu + 15 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 12\nu^{3} + 35\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{3} + \beta_{2} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{4} - 8\beta_{3} + 2\beta_{2} + 35 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14\beta_{5} - 12\beta_{3} - 12\beta_{2} + 37\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/145\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(117\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 59.1 |
|
− | 2.77035i | 0.269894i | −5.67486 | −1.96358 | − | 1.06975i | 0.747703 | − | 1.86960i | 10.1807i | 2.92716 | −2.96358 | + | 5.43981i | ||||||||||||||||||||||||||||||
| 59.2 | − | 2.30229i | 2.89028i | −3.30056 | 2.17686 | + | 0.511167i | 6.65427 | 3.91261i | 2.99427i | −5.35371 | 1.17686 | − | 5.01177i | ||||||||||||||||||||||||||||||||
| 59.3 | − | 0.156785i | − | 2.56387i | 1.97542 | 1.28672 | + | 1.82876i | −0.401976 | 1.09364i | − | 0.623285i | −3.57344 | 0.286721 | − | 0.201738i | ||||||||||||||||||||||||||||||
| 59.4 | 0.156785i | 2.56387i | 1.97542 | 1.28672 | − | 1.82876i | −0.401976 | − | 1.09364i | 0.623285i | −3.57344 | 0.286721 | + | 0.201738i | ||||||||||||||||||||||||||||||||
| 59.5 | 2.30229i | − | 2.89028i | −3.30056 | 2.17686 | − | 0.511167i | 6.65427 | − | 3.91261i | − | 2.99427i | −5.35371 | 1.17686 | + | 5.01177i | ||||||||||||||||||||||||||||||
| 59.6 | 2.77035i | − | 0.269894i | −5.67486 | −1.96358 | + | 1.06975i | 0.747703 | 1.86960i | − | 10.1807i | 2.92716 | −2.96358 | − | 5.43981i | |||||||||||||||||||||||||||||||
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 | 145.2.b.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1305.2.c.h | 6 | ||
| 4.b | odd | 2 | 1 | 2320.2.d.g | 6 | ||
| 5.b | even | 2 | 1 | inner | 145.2.b.c | ✓ | 6 |
| 5.c | odd | 4 | 2 | 725.2.a.l | 6 | ||
| 15.d | odd | 2 | 1 | 1305.2.c.h | 6 | ||
| 15.e | even | 4 | 2 | 6525.2.a.bt | 6 | ||
| 20.d | odd | 2 | 1 | 2320.2.d.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 145.2.b.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 145.2.b.c | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 725.2.a.l | 6 | 5.c | odd | 4 | 2 | ||
| 1305.2.c.h | 6 | 3.b | odd | 2 | 1 | ||
| 1305.2.c.h | 6 | 15.d | odd | 2 | 1 | ||
| 2320.2.d.g | 6 | 4.b | odd | 2 | 1 | ||
| 2320.2.d.g | 6 | 20.d | odd | 2 | 1 | ||
| 6525.2.a.bt | 6 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 13T_{2}^{4} + 41T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(145, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 13 T^{4} + 41 T^{2} + 1 \)
$3$
\( T^{6} + 15 T^{4} + 56 T^{2} + 4 \)
$5$
\( T^{6} - 3 T^{5} - T^{4} + 14 T^{3} + \cdots + 125 \)
$7$
\( T^{6} + 20 T^{4} + 76 T^{2} + 64 \)
$11$
\( (T^{3} + 5 T^{2} - 6 T - 38)^{2} \)
$13$
\( T^{6} + 59 T^{4} + 1048 T^{2} + \cdots + 5776 \)
$17$
\( T^{6} + 48 T^{4} + 380 T^{2} + \cdots + 784 \)
$19$
\( (T^{3} + 8 T^{2} + 6 T - 8)^{2} \)
$23$
\( T^{6} + 56 T^{4} + 60 T^{2} + 16 \)
$29$
\( (T - 1)^{6} \)
$31$
\( (T^{3} - 11 T^{2} + 26 T + 22)^{2} \)
$37$
\( T^{6} + 20 T^{4} + 76 T^{2} + 64 \)
$41$
\( T^{6} \)
$43$
\( T^{6} + 127 T^{4} + 1400 T^{2} + \cdots + 196 \)
$47$
\( T^{6} + 63 T^{4} + 1304 T^{2} + \cdots + 8836 \)
$53$
\( T^{6} + 187 T^{4} + 7640 T^{2} + \cdots + 5776 \)
$59$
\( (T + 10)^{6} \)
$61$
\( (T^{3} - 6 T^{2} - 64 T - 64)^{2} \)
$67$
\( T^{6} + 140 T^{4} + 6316 T^{2} + \cdots + 92416 \)
$71$
\( (T - 2)^{6} \)
$73$
\( T^{6} + 188 T^{4} + 8556 T^{2} + \cdots + 3136 \)
$79$
\( (T^{3} + T^{2} - 54 T + 98)^{2} \)
$83$
\( T^{6} + 48 T^{4} + 380 T^{2} + \cdots + 784 \)
$89$
\( (T^{3} - 16 T^{2} + 28 T + 304)^{2} \)
$97$
\( T^{6} + 476 T^{4} + 52396 T^{2} + \cdots + 7744 \)
show more
show less