Newspace parameters
Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 448.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.57729801055\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.629407744.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{10} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{7} + 4\nu^{5} - 10\nu^{3} + 28\nu ) / 12 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{6} + 2\nu^{4} - 2\nu^{2} - 4 ) / 6 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{7} + 2\nu^{3} - 4\nu ) / 4 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{6} - \nu^{4} + 4\nu^{2} - 7 ) / 3 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{7} - 4\nu^{5} + 2\nu^{3} + 4\nu ) / 4 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{7} + \nu^{5} + 2\nu^{3} + 10\nu ) / 3 \) |
\(\beta_{7}\) | \(=\) | \( ( -\nu^{6} + 2\nu^{4} + 2\nu^{2} + 4 ) / 2 \) |
\(\nu\) | \(=\) | \( ( \beta_{6} + \beta_{5} + \beta_{3} + 2\beta_1 ) / 8 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{7} + 2\beta_{4} - \beta_{2} + 2 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{6} + \beta_{3} - \beta_1 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( \beta_{7} + 3\beta_{2} ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( \beta_{6} - 3\beta_{5} + 5\beta_{3} + 2\beta_1 ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( -\beta_{7} + 2\beta_{4} + 5\beta_{2} + 10 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( -\beta_{6} + \beta_{5} + 7\beta_{3} + 4\beta_1 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(129\) | \(197\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
223.1 |
|
0 | − | 3.36028i | 0 | −2.16991 | 0 | 2.64575i | 0 | −8.29150 | 0 | |||||||||||||||||||||||||||||||||||||||||
223.2 | 0 | − | 3.36028i | 0 | 2.16991 | 0 | − | 2.64575i | 0 | −8.29150 | 0 | |||||||||||||||||||||||||||||||||||||||||
223.3 | 0 | − | 0.841723i | 0 | −3.91044 | 0 | 2.64575i | 0 | 2.29150 | 0 | ||||||||||||||||||||||||||||||||||||||||||
223.4 | 0 | − | 0.841723i | 0 | 3.91044 | 0 | − | 2.64575i | 0 | 2.29150 | 0 | |||||||||||||||||||||||||||||||||||||||||
223.5 | 0 | 0.841723i | 0 | −3.91044 | 0 | − | 2.64575i | 0 | 2.29150 | 0 | ||||||||||||||||||||||||||||||||||||||||||
223.6 | 0 | 0.841723i | 0 | 3.91044 | 0 | 2.64575i | 0 | 2.29150 | 0 | |||||||||||||||||||||||||||||||||||||||||||
223.7 | 0 | 3.36028i | 0 | −2.16991 | 0 | − | 2.64575i | 0 | −8.29150 | 0 | ||||||||||||||||||||||||||||||||||||||||||
223.8 | 0 | 3.36028i | 0 | 2.16991 | 0 | 2.64575i | 0 | −8.29150 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
56.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-14}) \) |
4.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
28.d | even | 2 | 1 | inner |
56.e | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 448.2.e.a | ✓ | 8 |
3.b | odd | 2 | 1 | 4032.2.p.h | 8 | ||
4.b | odd | 2 | 1 | inner | 448.2.e.a | ✓ | 8 |
7.b | odd | 2 | 1 | inner | 448.2.e.a | ✓ | 8 |
8.b | even | 2 | 1 | inner | 448.2.e.a | ✓ | 8 |
8.d | odd | 2 | 1 | inner | 448.2.e.a | ✓ | 8 |
12.b | even | 2 | 1 | 4032.2.p.h | 8 | ||
16.e | even | 4 | 2 | 1792.2.f.k | 8 | ||
16.f | odd | 4 | 2 | 1792.2.f.k | 8 | ||
21.c | even | 2 | 1 | 4032.2.p.h | 8 | ||
24.f | even | 2 | 1 | 4032.2.p.h | 8 | ||
24.h | odd | 2 | 1 | 4032.2.p.h | 8 | ||
28.d | even | 2 | 1 | inner | 448.2.e.a | ✓ | 8 |
56.e | even | 2 | 1 | inner | 448.2.e.a | ✓ | 8 |
56.h | odd | 2 | 1 | CM | 448.2.e.a | ✓ | 8 |
84.h | odd | 2 | 1 | 4032.2.p.h | 8 | ||
112.j | even | 4 | 2 | 1792.2.f.k | 8 | ||
112.l | odd | 4 | 2 | 1792.2.f.k | 8 | ||
168.e | odd | 2 | 1 | 4032.2.p.h | 8 | ||
168.i | even | 2 | 1 | 4032.2.p.h | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
448.2.e.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
448.2.e.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
448.2.e.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
448.2.e.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
448.2.e.a | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
448.2.e.a | ✓ | 8 | 28.d | even | 2 | 1 | inner |
448.2.e.a | ✓ | 8 | 56.e | even | 2 | 1 | inner |
448.2.e.a | ✓ | 8 | 56.h | odd | 2 | 1 | CM |
1792.2.f.k | 8 | 16.e | even | 4 | 2 | ||
1792.2.f.k | 8 | 16.f | odd | 4 | 2 | ||
1792.2.f.k | 8 | 112.j | even | 4 | 2 | ||
1792.2.f.k | 8 | 112.l | odd | 4 | 2 | ||
4032.2.p.h | 8 | 3.b | odd | 2 | 1 | ||
4032.2.p.h | 8 | 12.b | even | 2 | 1 | ||
4032.2.p.h | 8 | 21.c | even | 2 | 1 | ||
4032.2.p.h | 8 | 24.f | even | 2 | 1 | ||
4032.2.p.h | 8 | 24.h | odd | 2 | 1 | ||
4032.2.p.h | 8 | 84.h | odd | 2 | 1 | ||
4032.2.p.h | 8 | 168.e | odd | 2 | 1 | ||
4032.2.p.h | 8 | 168.i | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 12T_{3}^{2} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} + 12 T^{2} + 8)^{2} \)
$5$
\( (T^{4} - 20 T^{2} + 72)^{2} \)
$7$
\( (T^{2} + 7)^{4} \)
$11$
\( T^{8} \)
$13$
\( (T^{4} - 52 T^{2} + 648)^{2} \)
$17$
\( T^{8} \)
$19$
\( (T^{4} + 76 T^{2} + 72)^{2} \)
$23$
\( (T^{2} + 36)^{4} \)
$29$
\( T^{8} \)
$31$
\( T^{8} \)
$37$
\( T^{8} \)
$41$
\( T^{8} \)
$43$
\( T^{8} \)
$47$
\( T^{8} \)
$53$
\( T^{8} \)
$59$
\( (T^{4} + 236 T^{2} + 5832)^{2} \)
$61$
\( (T^{4} - 244 T^{2} + 72)^{2} \)
$67$
\( T^{8} \)
$71$
\( (T^{2} + 252)^{4} \)
$73$
\( T^{8} \)
$79$
\( (T^{2} + 28)^{4} \)
$83$
\( (T^{4} + 332 T^{2} + 648)^{2} \)
$89$
\( T^{8} \)
$97$
\( T^{8} \)
show more
show less