Newspace parameters
Level: | \( N \) | \(=\) | \( 280 = 2^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 280.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.23581125660\) |
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 \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).
\(n\) | \(57\) | \(71\) | \(141\) | \(241\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
169.1 |
|
0 | − | 3.12489i | 0 | 1.32001 | − | 1.80487i | 0 | − | 1.00000i | 0 | −6.76491 | 0 | ||||||||||||||||||||||||||||||||
169.2 | 0 | − | 1.76156i | 0 | 0.432320 | − | 2.19388i | 0 | 1.00000i | 0 | −0.103084 | 0 | ||||||||||||||||||||||||||||||||||
169.3 | 0 | − | 0.363328i | 0 | −1.75233 | + | 1.38900i | 0 | 1.00000i | 0 | 2.86799 | 0 | ||||||||||||||||||||||||||||||||||
169.4 | 0 | 0.363328i | 0 | −1.75233 | − | 1.38900i | 0 | − | 1.00000i | 0 | 2.86799 | 0 | ||||||||||||||||||||||||||||||||||
169.5 | 0 | 1.76156i | 0 | 0.432320 | + | 2.19388i | 0 | − | 1.00000i | 0 | −0.103084 | 0 | ||||||||||||||||||||||||||||||||||
169.6 | 0 | 3.12489i | 0 | 1.32001 | + | 1.80487i | 0 | 1.00000i | 0 | −6.76491 | 0 | |||||||||||||||||||||||||||||||||||
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 | 280.2.g.b | ✓ | 6 |
3.b | odd | 2 | 1 | 2520.2.t.g | 6 | ||
4.b | odd | 2 | 1 | 560.2.g.f | 6 | ||
5.b | even | 2 | 1 | inner | 280.2.g.b | ✓ | 6 |
5.c | odd | 4 | 1 | 1400.2.a.s | 3 | ||
5.c | odd | 4 | 1 | 1400.2.a.t | 3 | ||
7.b | odd | 2 | 1 | 1960.2.g.c | 6 | ||
8.b | even | 2 | 1 | 2240.2.g.l | 6 | ||
8.d | odd | 2 | 1 | 2240.2.g.m | 6 | ||
12.b | even | 2 | 1 | 5040.2.t.y | 6 | ||
15.d | odd | 2 | 1 | 2520.2.t.g | 6 | ||
20.d | odd | 2 | 1 | 560.2.g.f | 6 | ||
20.e | even | 4 | 1 | 2800.2.a.bq | 3 | ||
20.e | even | 4 | 1 | 2800.2.a.br | 3 | ||
35.c | odd | 2 | 1 | 1960.2.g.c | 6 | ||
35.f | even | 4 | 1 | 9800.2.a.cd | 3 | ||
35.f | even | 4 | 1 | 9800.2.a.cg | 3 | ||
40.e | odd | 2 | 1 | 2240.2.g.m | 6 | ||
40.f | even | 2 | 1 | 2240.2.g.l | 6 | ||
60.h | even | 2 | 1 | 5040.2.t.y | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
280.2.g.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
280.2.g.b | ✓ | 6 | 5.b | even | 2 | 1 | inner |
560.2.g.f | 6 | 4.b | odd | 2 | 1 | ||
560.2.g.f | 6 | 20.d | odd | 2 | 1 | ||
1400.2.a.s | 3 | 5.c | odd | 4 | 1 | ||
1400.2.a.t | 3 | 5.c | odd | 4 | 1 | ||
1960.2.g.c | 6 | 7.b | odd | 2 | 1 | ||
1960.2.g.c | 6 | 35.c | odd | 2 | 1 | ||
2240.2.g.l | 6 | 8.b | even | 2 | 1 | ||
2240.2.g.l | 6 | 40.f | even | 2 | 1 | ||
2240.2.g.m | 6 | 8.d | odd | 2 | 1 | ||
2240.2.g.m | 6 | 40.e | odd | 2 | 1 | ||
2520.2.t.g | 6 | 3.b | odd | 2 | 1 | ||
2520.2.t.g | 6 | 15.d | odd | 2 | 1 | ||
2800.2.a.bq | 3 | 20.e | even | 4 | 1 | ||
2800.2.a.br | 3 | 20.e | even | 4 | 1 | ||
5040.2.t.y | 6 | 12.b | even | 2 | 1 | ||
5040.2.t.y | 6 | 60.h | even | 2 | 1 | ||
9800.2.a.cd | 3 | 35.f | even | 4 | 1 | ||
9800.2.a.cg | 3 | 35.f | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 13T_{3}^{4} + 32T_{3}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(280, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( T^{6} + 13 T^{4} + 32 T^{2} + 4 \)
$5$
\( T^{6} + 5 T^{4} + 8 T^{3} + 25 T^{2} + \cdots + 125 \)
$7$
\( (T^{2} + 1)^{3} \)
$11$
\( (T^{3} - 7 T^{2} + 8 T + 8)^{2} \)
$13$
\( T^{6} + 69 T^{4} + 1544 T^{2} + \cdots + 11236 \)
$17$
\( T^{6} + 49 T^{4} + 536 T^{2} + \cdots + 400 \)
$19$
\( (T^{3} + 4 T^{2} - 14 T + 8)^{2} \)
$23$
\( T^{6} + 92 T^{4} + 2416 T^{2} + \cdots + 18496 \)
$29$
\( (T^{3} + 3 T^{2} - 72 T - 108)^{2} \)
$31$
\( (T^{3} - 10 T^{2} + 8 T + 80)^{2} \)
$37$
\( (T^{2} + 36)^{3} \)
$41$
\( (T^{3} - 18 T^{2} + 68 T + 88)^{2} \)
$43$
\( T^{6} + 80 T^{4} + 1600 T^{2} + \cdots + 4096 \)
$47$
\( T^{6} + 177 T^{4} + 6480 T^{2} + \cdots + 53824 \)
$53$
\( T^{6} + 188 T^{4} + 11376 T^{2} + \cdots + 222784 \)
$59$
\( (T^{3} + 6 T^{2} - 78 T + 44)^{2} \)
$61$
\( (T^{3} - 24 T^{2} + 182 T - 440)^{2} \)
$67$
\( T^{6} + 228 T^{4} + 14336 T^{2} + \cdots + 262144 \)
$71$
\( (T^{3} - 4 T^{2} - 20 T + 64)^{2} \)
$73$
\( T^{6} + 92 T^{4} + 2416 T^{2} + \cdots + 18496 \)
$79$
\( (T^{3} + 17 T^{2} - 32 T - 548)^{2} \)
$83$
\( T^{6} + 428 T^{4} + 39940 T^{2} + \cdots + 678976 \)
$89$
\( (T^{3} - 172 T + 464)^{2} \)
$97$
\( T^{6} + 113 T^{4} + 1048 T^{2} + \cdots + 1936 \)
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