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: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 2\nu - 3 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + \beta_{2} + 2\beta _1 + 4 \)
|
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 |
|
− | 1.73205i | − | 2.52434i | −1.00000 | −2.18614 | + | 0.469882i | −4.37228 | 1.58457i | − | 1.73205i | −3.37228 | 0.813859 | + | 3.78651i | |||||||||||||||||||||||
| 59.2 | − | 1.73205i | 0.792287i | −1.00000 | 0.686141 | + | 2.12819i | 1.37228 | − | 5.04868i | − | 1.73205i | 2.37228 | 3.68614 | − | 1.18843i | ||||||||||||||||||||||||
| 59.3 | 1.73205i | − | 0.792287i | −1.00000 | 0.686141 | − | 2.12819i | 1.37228 | 5.04868i | 1.73205i | 2.37228 | 3.68614 | + | 1.18843i | ||||||||||||||||||||||||||
| 59.4 | 1.73205i | 2.52434i | −1.00000 | −2.18614 | − | 0.469882i | −4.37228 | − | 1.58457i | 1.73205i | −3.37228 | 0.813859 | − | 3.78651i | ||||||||||||||||||||||||||
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.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1305.2.c.e | 4 | ||
| 4.b | odd | 2 | 1 | 2320.2.d.c | 4 | ||
| 5.b | even | 2 | 1 | inner | 145.2.b.a | ✓ | 4 |
| 5.c | odd | 4 | 2 | 725.2.a.g | 4 | ||
| 15.d | odd | 2 | 1 | 1305.2.c.e | 4 | ||
| 15.e | even | 4 | 2 | 6525.2.a.bk | 4 | ||
| 20.d | odd | 2 | 1 | 2320.2.d.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 145.2.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 145.2.b.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 725.2.a.g | 4 | 5.c | odd | 4 | 2 | ||
| 1305.2.c.e | 4 | 3.b | odd | 2 | 1 | ||
| 1305.2.c.e | 4 | 15.d | odd | 2 | 1 | ||
| 2320.2.d.c | 4 | 4.b | odd | 2 | 1 | ||
| 2320.2.d.c | 4 | 20.d | odd | 2 | 1 | ||
| 6525.2.a.bk | 4 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(145, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 3)^{2} \)
$3$
\( T^{4} + 7T^{2} + 4 \)
$5$
\( T^{4} + 3 T^{3} + 4 T^{2} + 15 T + 25 \)
$7$
\( T^{4} + 28T^{2} + 64 \)
$11$
\( (T^{2} - 7 T + 4)^{2} \)
$13$
\( T^{4} + 19T^{2} + 16 \)
$17$
\( T^{4} + 28T^{2} + 64 \)
$19$
\( (T + 4)^{4} \)
$23$
\( (T^{2} + 12)^{2} \)
$29$
\( (T - 1)^{4} \)
$31$
\( (T^{2} + T - 8)^{2} \)
$37$
\( T^{4} + 112T^{2} + 1024 \)
$41$
\( (T^{2} - 2 T - 32)^{2} \)
$43$
\( T^{4} + 151T^{2} + 3844 \)
$47$
\( T^{4} + 151T^{2} + 3844 \)
$53$
\( T^{4} + 19T^{2} + 16 \)
$59$
\( (T^{2} - 10 T - 8)^{2} \)
$61$
\( (T - 6)^{4} \)
$67$
\( T^{4} + 76T^{2} + 256 \)
$71$
\( (T^{2} - 2 T - 32)^{2} \)
$73$
\( (T^{2} + 48)^{2} \)
$79$
\( (T^{2} + 17 T + 64)^{2} \)
$83$
\( T^{4} + 376 T^{2} + 26896 \)
$89$
\( (T^{2} + 10 T - 8)^{2} \)
$97$
\( (T^{2} + 48)^{2} \)
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