Newspace parameters
Level: | \( N \) | \(=\) | \( 1120 = 2^{5} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1120.w (of order \(4\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.94324502638\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | 8.0.40282095616.8 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{6} + 8x^{4} - 36x^{2} + 81 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 5 \) |
Twist minimal: | no (minimal twist has level 280) |
Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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 \( x^{8} - 4x^{6} + 8x^{4} - 36x^{2} + 81 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{7} + 23\nu^{5} - 46\nu^{3} + 72\nu ) / 135 \) |
\(\beta_{2}\) | \(=\) | \( ( -4\nu^{7} - 2\nu^{5} - 41\nu^{3} + 297\nu ) / 135 \) |
\(\beta_{3}\) | \(=\) | \( ( 2\nu^{6} + \nu^{4} + 43\nu^{2} - 81 ) / 45 \) |
\(\beta_{4}\) | \(=\) | \( ( -2\nu^{6} - \nu^{4} + 2\nu^{2} + 36 ) / 45 \) |
\(\beta_{5}\) | \(=\) | \( ( 7\nu^{7} - 19\nu^{5} - 7\nu^{3} - 126\nu ) / 135 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{7} + 2\nu^{5} - 9\nu^{3} + 23\nu ) / 15 \) |
\(\beta_{7}\) | \(=\) | \( ( -7\nu^{6} + 19\nu^{4} - 38\nu^{2} + 171 ) / 45 \) |
\(\nu\) | \(=\) | \( ( -2\beta_{6} - \beta_{5} + 3\beta_{2} + \beta_1 ) / 5 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{4} + \beta_{3} + 1 \) |
\(\nu^{3}\) | \(=\) | \( ( -8\beta_{6} - 9\beta_{5} + 2\beta_{2} - \beta_1 ) / 5 \) |
\(\nu^{4}\) | \(=\) | \( 2\beta_{7} - 5\beta_{4} + 2\beta_{3} \) |
\(\nu^{5}\) | \(=\) | \( ( -7\beta_{6} - 16\beta_{5} - 7\beta_{2} + 21\beta_1 ) / 5 \) |
\(\nu^{6}\) | \(=\) | \( -\beta_{7} - 19\beta_{4} + 19 \) |
\(\nu^{7}\) | \(=\) | \( ( -63\beta_{6} + 26\beta_{5} + 37\beta_{2} + 74\beta_1 ) / 5 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times\).
\(n\) | \(351\) | \(421\) | \(801\) | \(897\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
433.1 |
|
0 | −2.42298 | − | 2.42298i | 0 | −1.75060 | + | 1.39119i | 0 | −1.87083 | + | 1.87083i | 0 | 8.74166i | 0 | ||||||||||||||||||||||||||||||||||||
433.2 | 0 | −1.45917 | − | 1.45917i | 0 | 2.22158 | − | 0.254137i | 0 | 1.87083 | − | 1.87083i | 0 | 1.25834i | 0 | |||||||||||||||||||||||||||||||||||||
433.3 | 0 | 1.45917 | + | 1.45917i | 0 | −2.22158 | + | 0.254137i | 0 | 1.87083 | − | 1.87083i | 0 | 1.25834i | 0 | |||||||||||||||||||||||||||||||||||||
433.4 | 0 | 2.42298 | + | 2.42298i | 0 | 1.75060 | − | 1.39119i | 0 | −1.87083 | + | 1.87083i | 0 | 8.74166i | 0 | |||||||||||||||||||||||||||||||||||||
657.1 | 0 | −2.42298 | + | 2.42298i | 0 | −1.75060 | − | 1.39119i | 0 | −1.87083 | − | 1.87083i | 0 | − | 8.74166i | 0 | ||||||||||||||||||||||||||||||||||||
657.2 | 0 | −1.45917 | + | 1.45917i | 0 | 2.22158 | + | 0.254137i | 0 | 1.87083 | + | 1.87083i | 0 | − | 1.25834i | 0 | ||||||||||||||||||||||||||||||||||||
657.3 | 0 | 1.45917 | − | 1.45917i | 0 | −2.22158 | − | 0.254137i | 0 | 1.87083 | + | 1.87083i | 0 | − | 1.25834i | 0 | ||||||||||||||||||||||||||||||||||||
657.4 | 0 | 2.42298 | − | 2.42298i | 0 | 1.75060 | + | 1.39119i | 0 | −1.87083 | − | 1.87083i | 0 | − | 8.74166i | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
56.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-14}) \) |
5.c | odd | 4 | 1 | inner |
7.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
35.f | even | 4 | 1 | inner |
40.i | odd | 4 | 1 | inner |
280.s | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1120.2.w.b | 8 | |
4.b | odd | 2 | 1 | 280.2.s.b | ✓ | 8 | |
5.c | odd | 4 | 1 | inner | 1120.2.w.b | 8 | |
7.b | odd | 2 | 1 | inner | 1120.2.w.b | 8 | |
8.b | even | 2 | 1 | inner | 1120.2.w.b | 8 | |
8.d | odd | 2 | 1 | 280.2.s.b | ✓ | 8 | |
20.e | even | 4 | 1 | 280.2.s.b | ✓ | 8 | |
28.d | even | 2 | 1 | 280.2.s.b | ✓ | 8 | |
35.f | even | 4 | 1 | inner | 1120.2.w.b | 8 | |
40.i | odd | 4 | 1 | inner | 1120.2.w.b | 8 | |
40.k | even | 4 | 1 | 280.2.s.b | ✓ | 8 | |
56.e | even | 2 | 1 | 280.2.s.b | ✓ | 8 | |
56.h | odd | 2 | 1 | CM | 1120.2.w.b | 8 | |
140.j | odd | 4 | 1 | 280.2.s.b | ✓ | 8 | |
280.s | even | 4 | 1 | inner | 1120.2.w.b | 8 | |
280.y | odd | 4 | 1 | 280.2.s.b | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
280.2.s.b | ✓ | 8 | 4.b | odd | 2 | 1 | |
280.2.s.b | ✓ | 8 | 8.d | odd | 2 | 1 | |
280.2.s.b | ✓ | 8 | 20.e | even | 4 | 1 | |
280.2.s.b | ✓ | 8 | 28.d | even | 2 | 1 | |
280.2.s.b | ✓ | 8 | 40.k | even | 4 | 1 | |
280.2.s.b | ✓ | 8 | 56.e | even | 2 | 1 | |
280.2.s.b | ✓ | 8 | 140.j | odd | 4 | 1 | |
280.2.s.b | ✓ | 8 | 280.y | odd | 4 | 1 | |
1120.2.w.b | 8 | 1.a | even | 1 | 1 | trivial | |
1120.2.w.b | 8 | 5.c | odd | 4 | 1 | inner | |
1120.2.w.b | 8 | 7.b | odd | 2 | 1 | inner | |
1120.2.w.b | 8 | 8.b | even | 2 | 1 | inner | |
1120.2.w.b | 8 | 35.f | even | 4 | 1 | inner | |
1120.2.w.b | 8 | 40.i | odd | 4 | 1 | inner | |
1120.2.w.b | 8 | 56.h | odd | 2 | 1 | CM | |
1120.2.w.b | 8 | 280.s | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 156T_{3}^{4} + 2500 \)
acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} + 156T^{4} + 2500 \)
$5$
\( T^{8} - 12 T^{6} + 72 T^{4} + \cdots + 625 \)
$7$
\( (T^{4} + 49)^{2} \)
$11$
\( T^{8} \)
$13$
\( T^{8} + 156T^{4} + 2500 \)
$17$
\( T^{8} \)
$19$
\( (T^{4} + 88 T^{2} + 1250)^{2} \)
$23$
\( (T^{4} - 12 T^{3} + 72 T^{2} + 120 T + 100)^{2} \)
$29$
\( T^{8} \)
$31$
\( T^{8} \)
$37$
\( T^{8} \)
$41$
\( T^{8} \)
$43$
\( T^{8} \)
$47$
\( T^{8} \)
$53$
\( T^{8} \)
$59$
\( (T^{4} + 128 T^{2} + 50)^{2} \)
$61$
\( (T^{4} - 232 T^{2} + 6050)^{2} \)
$67$
\( T^{8} \)
$71$
\( (T^{2} + 8 T - 110)^{4} \)
$73$
\( T^{8} \)
$79$
\( (T^{4} + 316 T^{2} + 16900)^{2} \)
$83$
\( T^{8} + 70716 T^{4} + \cdots + 71402500 \)
$89$
\( T^{8} \)
$97$
\( T^{8} \)
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