Newspace parameters
Level: | \( N \) | \(=\) | \( 460 = 2^{2} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 460.i (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.67311849298\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | 8.0.11574317056.3 |
Defining polynomial: |
\( x^{8} + 45x^{4} + 4 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 45x^{4} + 4 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 47\nu^{3} ) / 14 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 47\nu^{2} ) / 14 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 3\nu^{6} + 2\nu^{5} + 2\nu^{4} + 127\nu^{2} + 94\nu + 38 ) / 28 \)
|
\(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} - 2\nu^{5} + 2\nu^{4} - 127\nu^{2} - 94\nu + 38 ) / 28 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 7\nu^{7} + 2\nu^{5} + 315\nu^{3} + 94\nu ) / 28 \)
|
\(\beta_{7}\) | \(=\) |
\( ( -7\nu^{7} + 2\nu^{5} - 315\nu^{3} + 94\nu ) / 28 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 3\beta_{3} \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} + 7\beta_{2} \)
|
\(\nu^{4}\) | \(=\) |
\( 7\beta_{5} + 7\beta_{4} - 19 \)
|
\(\nu^{5}\) | \(=\) |
\( 7\beta_{7} + 7\beta_{6} - 47\beta_1 \)
|
\(\nu^{6}\) | \(=\) |
\( -47\beta_{7} - 47\beta_{6} - 47\beta_{5} + 47\beta_{4} + 127\beta_{3} \)
|
\(\nu^{7}\) | \(=\) |
\( -47\beta_{7} + 47\beta_{6} - 315\beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).
\(n\) | \(231\) | \(277\) | \(281\) |
\(\chi(n)\) | \(1\) | \(-\beta_{3}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
137.1 |
|
0 | −1.00000 | + | 1.00000i | 0 | −1.83051 | − | 1.28422i | 0 | 1.83051 | − | 1.83051i | 0 | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||
137.2 | 0 | −1.00000 | + | 1.00000i | 0 | −0.386289 | + | 2.20245i | 0 | 0.386289 | − | 0.386289i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
137.3 | 0 | −1.00000 | + | 1.00000i | 0 | 0.386289 | − | 2.20245i | 0 | −0.386289 | + | 0.386289i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
137.4 | 0 | −1.00000 | + | 1.00000i | 0 | 1.83051 | + | 1.28422i | 0 | −1.83051 | + | 1.83051i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
413.1 | 0 | −1.00000 | − | 1.00000i | 0 | −1.83051 | + | 1.28422i | 0 | 1.83051 | + | 1.83051i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||
413.2 | 0 | −1.00000 | − | 1.00000i | 0 | −0.386289 | − | 2.20245i | 0 | 0.386289 | + | 0.386289i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||
413.3 | 0 | −1.00000 | − | 1.00000i | 0 | 0.386289 | + | 2.20245i | 0 | −0.386289 | − | 0.386289i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||
413.4 | 0 | −1.00000 | − | 1.00000i | 0 | 1.83051 | − | 1.28422i | 0 | −1.83051 | − | 1.83051i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
23.b | odd | 2 | 1 | inner |
115.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 460.2.i.a | ✓ | 8 |
5.b | even | 2 | 1 | 2300.2.i.c | 8 | ||
5.c | odd | 4 | 1 | inner | 460.2.i.a | ✓ | 8 |
5.c | odd | 4 | 1 | 2300.2.i.c | 8 | ||
23.b | odd | 2 | 1 | inner | 460.2.i.a | ✓ | 8 |
115.c | odd | 2 | 1 | 2300.2.i.c | 8 | ||
115.e | even | 4 | 1 | inner | 460.2.i.a | ✓ | 8 |
115.e | even | 4 | 1 | 2300.2.i.c | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
460.2.i.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
460.2.i.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
460.2.i.a | ✓ | 8 | 23.b | odd | 2 | 1 | inner |
460.2.i.a | ✓ | 8 | 115.e | even | 4 | 1 | inner |
2300.2.i.c | 8 | 5.b | even | 2 | 1 | ||
2300.2.i.c | 8 | 5.c | odd | 4 | 1 | ||
2300.2.i.c | 8 | 115.c | odd | 2 | 1 | ||
2300.2.i.c | 8 | 115.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 2T_{3} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(460, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{2} + 2 T + 2)^{4} \)
$5$
\( T^{8} + 6 T^{6} + 18 T^{4} + 150 T^{2} + \cdots + 625 \)
$7$
\( T^{8} + 45T^{4} + 4 \)
$11$
\( (T^{4} + 14 T^{2} + 8)^{2} \)
$13$
\( (T^{4} + 2 T^{3} + 2 T^{2} - 40 T + 400)^{2} \)
$17$
\( T^{8} + 2109 T^{4} + 2500 \)
$19$
\( (T^{4} - 58 T^{2} + 800)^{2} \)
$23$
\( T^{8} + 14 T^{7} + 98 T^{6} + \cdots + 279841 \)
$29$
\( (T^{4} + 81 T^{2} + 400)^{2} \)
$31$
\( (T^{2} + T - 92)^{4} \)
$37$
\( T^{8} + 2109 T^{4} + 2500 \)
$41$
\( (T^{2} - 9 T + 10)^{4} \)
$43$
\( T^{8} + 16500 T^{4} + \cdots + 17909824 \)
$47$
\( (T^{4} + 6724)^{2} \)
$53$
\( T^{8} + 4125 T^{4} + \cdots + 1119364 \)
$59$
\( (T^{4} + 21 T^{2} + 100)^{2} \)
$61$
\( (T^{4} + 202 T^{2} + 8192)^{2} \)
$67$
\( T^{8} + 13965 T^{4} + \cdots + 48469444 \)
$71$
\( (T^{2} - 11 T + 20)^{4} \)
$73$
\( (T^{4} - 16 T^{3} + 128 T^{2} + 800 T + 2500)^{2} \)
$79$
\( (T^{4} - 234 T^{2} + 10368)^{2} \)
$83$
\( T^{8} + 45T^{4} + 4 \)
$89$
\( (T^{4} - 478 T^{2} + 55112)^{2} \)
$97$
\( T^{8} + 69540 T^{4} + \cdots + 945685504 \)
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