Newspace parameters
Level: | \( N \) | \(=\) | \( 280 = 2^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 280.ba (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.23581125660\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.3317760000.3 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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 \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{6} + 14\nu^{4} + 7\nu^{2} - 36 ) / 63 \) |
\(\beta_{2}\) | \(=\) | \( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \) |
\(\beta_{3}\) | \(=\) | \( ( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63 \) |
\(\beta_{4}\) | \(=\) | \( ( -5\nu^{7} - 7\nu^{5} - 35\nu^{3} + 180\nu ) / 189 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{7} + 13\nu ) / 21 \) |
\(\beta_{6}\) | \(=\) | \( ( -8\nu^{6} + 14\nu^{4} + 7\nu^{2} + 162 ) / 63 \) |
\(\beta_{7}\) | \(=\) | \( ( 19\nu^{7} - 49\nu^{5} + 133\nu^{3} - 684\nu ) / 189 \) |
\(\nu\) | \(=\) | \( ( \beta_{5} + \beta_{4} + \beta_{2} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{6} - 2\beta_{3} + 2 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{7} + \beta_{5} + 7\beta_{2} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{3} + 4\beta_1 \) |
\(\nu^{5}\) | \(=\) | \( ( -5\beta_{7} - 19\beta_{4} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( -7\beta_{6} + 7\beta _1 + 22 \) |
\(\nu^{7}\) | \(=\) | \( ( 29\beta_{5} - 13\beta_{4} - 13\beta_{2} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).
\(n\) | \(57\) | \(71\) | \(141\) | \(241\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(1 - \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
|
−0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | −1.93649 | − | 1.11803i | 0 | 1.41421 | + | 2.23607i | 2.82843 | 1.50000 | − | 2.59808i | 2.73861 | − | 1.58114i | ||||||||||||||||||||||||||||||
19.2 | −0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | 1.93649 | + | 1.11803i | 0 | 1.41421 | − | 2.23607i | 2.82843 | 1.50000 | − | 2.59808i | −2.73861 | + | 1.58114i | |||||||||||||||||||||||||||||||
19.3 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | −1.93649 | − | 1.11803i | 0 | −1.41421 | + | 2.23607i | −2.82843 | 1.50000 | − | 2.59808i | −2.73861 | + | 1.58114i | |||||||||||||||||||||||||||||||
19.4 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | 1.93649 | + | 1.11803i | 0 | −1.41421 | − | 2.23607i | −2.82843 | 1.50000 | − | 2.59808i | 2.73861 | − | 1.58114i | |||||||||||||||||||||||||||||||
59.1 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | −1.93649 | + | 1.11803i | 0 | 1.41421 | − | 2.23607i | 2.82843 | 1.50000 | + | 2.59808i | 2.73861 | + | 1.58114i | |||||||||||||||||||||||||||||||
59.2 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | 1.93649 | − | 1.11803i | 0 | 1.41421 | + | 2.23607i | 2.82843 | 1.50000 | + | 2.59808i | −2.73861 | − | 1.58114i | |||||||||||||||||||||||||||||||
59.3 | 0.707107 | + | 1.22474i | 0 | −1.00000 | + | 1.73205i | −1.93649 | + | 1.11803i | 0 | −1.41421 | − | 2.23607i | −2.82843 | 1.50000 | + | 2.59808i | −2.73861 | − | 1.58114i | |||||||||||||||||||||||||||||||
59.4 | 0.707107 | + | 1.22474i | 0 | −1.00000 | + | 1.73205i | 1.93649 | − | 1.11803i | 0 | −1.41421 | + | 2.23607i | −2.82843 | 1.50000 | + | 2.59808i | 2.73861 | + | 1.58114i | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
40.e | odd | 2 | 1 | CM by \(\Q(\sqrt{-10}) \) |
5.b | even | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
8.d | odd | 2 | 1 | inner |
35.i | odd | 6 | 1 | inner |
56.m | even | 6 | 1 | inner |
280.ba | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 280.2.ba.a | ✓ | 8 |
4.b | odd | 2 | 1 | 1120.2.bq.a | 8 | ||
5.b | even | 2 | 1 | inner | 280.2.ba.a | ✓ | 8 |
7.d | odd | 6 | 1 | inner | 280.2.ba.a | ✓ | 8 |
8.b | even | 2 | 1 | 1120.2.bq.a | 8 | ||
8.d | odd | 2 | 1 | inner | 280.2.ba.a | ✓ | 8 |
20.d | odd | 2 | 1 | 1120.2.bq.a | 8 | ||
28.f | even | 6 | 1 | 1120.2.bq.a | 8 | ||
35.i | odd | 6 | 1 | inner | 280.2.ba.a | ✓ | 8 |
40.e | odd | 2 | 1 | CM | 280.2.ba.a | ✓ | 8 |
40.f | even | 2 | 1 | 1120.2.bq.a | 8 | ||
56.j | odd | 6 | 1 | 1120.2.bq.a | 8 | ||
56.m | even | 6 | 1 | inner | 280.2.ba.a | ✓ | 8 |
140.s | even | 6 | 1 | 1120.2.bq.a | 8 | ||
280.ba | even | 6 | 1 | inner | 280.2.ba.a | ✓ | 8 |
280.bk | odd | 6 | 1 | 1120.2.bq.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
280.2.ba.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
280.2.ba.a | ✓ | 8 | 5.b | even | 2 | 1 | inner |
280.2.ba.a | ✓ | 8 | 7.d | odd | 6 | 1 | inner |
280.2.ba.a | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
280.2.ba.a | ✓ | 8 | 35.i | odd | 6 | 1 | inner |
280.2.ba.a | ✓ | 8 | 40.e | odd | 2 | 1 | CM |
280.2.ba.a | ✓ | 8 | 56.m | even | 6 | 1 | inner |
280.2.ba.a | ✓ | 8 | 280.ba | even | 6 | 1 | inner |
1120.2.bq.a | 8 | 4.b | odd | 2 | 1 | ||
1120.2.bq.a | 8 | 8.b | even | 2 | 1 | ||
1120.2.bq.a | 8 | 20.d | odd | 2 | 1 | ||
1120.2.bq.a | 8 | 28.f | even | 6 | 1 | ||
1120.2.bq.a | 8 | 40.f | even | 2 | 1 | ||
1120.2.bq.a | 8 | 56.j | odd | 6 | 1 | ||
1120.2.bq.a | 8 | 140.s | even | 6 | 1 | ||
1120.2.bq.a | 8 | 280.bk | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{2}^{\mathrm{new}}(280, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + 2 T^{2} + 4)^{2} \)
$3$
\( T^{8} \)
$5$
\( (T^{4} - 5 T^{2} + 25)^{2} \)
$7$
\( (T^{4} + 6 T^{2} + 49)^{2} \)
$11$
\( (T^{4} + 2 T^{3} + 33 T^{2} - 58 T + 841)^{2} \)
$13$
\( (T^{4} + 58 T^{2} + 361)^{2} \)
$17$
\( T^{8} \)
$19$
\( (T^{4} - 18 T^{3} + 125 T^{2} - 306 T + 289)^{2} \)
$23$
\( T^{8} + 66 T^{6} + 4347 T^{4} + \cdots + 81 \)
$29$
\( T^{8} \)
$31$
\( T^{8} \)
$37$
\( T^{8} + 94 T^{6} + 8547 T^{4} + \cdots + 83521 \)
$41$
\( (T^{4} + 86 T^{2} + 1369)^{2} \)
$43$
\( T^{8} \)
$47$
\( T^{8} - 102 T^{6} + 8883 T^{4} + \cdots + 2313441 \)
$53$
\( T^{8} + 286 T^{6} + \cdots + 260144641 \)
$59$
\( (T^{4} - 40 T^{2} + 1600)^{2} \)
$61$
\( T^{8} \)
$67$
\( T^{8} \)
$71$
\( T^{8} \)
$73$
\( T^{8} \)
$79$
\( T^{8} \)
$83$
\( T^{8} \)
$89$
\( (T^{4} - 160 T^{2} + 25600)^{2} \)
$97$
\( T^{8} \)
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