Newspace parameters
Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 224.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.10355792167\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.4694952902656.3 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 20x^{6} + 56x^{4} + 20x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{9} \) |
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 \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} + 20x^{6} + 56x^{4} + 20x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( 10\nu^{6} + 196\nu^{4} + 490\nu^{2} + 88 ) / 21 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{7} - \nu^{6} + 21\nu^{5} - 21\nu^{4} + 77\nu^{3} - 77\nu^{2} + 97\nu - 48 ) / 7 \) |
\(\beta_{3}\) | \(=\) | \( ( -3\nu^{7} + 7\nu^{6} - 63\nu^{5} + 133\nu^{4} - 231\nu^{3} + 259\nu^{2} - 291\nu - 56 ) / 21 \) |
\(\beta_{4}\) | \(=\) | \( ( 2\nu^{7} + 38\nu^{5} + 74\nu^{3} - 34\nu ) / 3 \) |
\(\beta_{5}\) | \(=\) | \( 2\nu^{6} + 40\nu^{4} + 110\nu^{2} + 20 \) |
\(\beta_{6}\) | \(=\) | \( ( -13\nu^{7} - 259\nu^{5} - 707\nu^{3} - 197\nu ) / 7 \) |
\(\beta_{7}\) | \(=\) | \( ( -7\nu^{7} - 139\nu^{5} - 373\nu^{3} - 97\nu ) / 3 \) |
\(\nu\) | \(=\) | \( ( 2\beta_{7} - 2\beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 8 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{5} - 2\beta_{3} - 2\beta_{2} + 5\beta _1 - 20 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( -38\beta_{7} + 42\beta_{6} - 13\beta_{4} + 7\beta_{3} - 7\beta_{2} - 7\beta_1 ) / 8 \) |
\(\nu^{4}\) | \(=\) | \( ( 10\beta_{5} + 15\beta_{3} + 15\beta_{2} - 48\beta _1 + 144 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 646\beta_{7} - 726\beta_{6} + 197\beta_{4} - 97\beta_{3} + 97\beta_{2} + 97\beta_1 ) / 8 \) |
\(\nu^{6}\) | \(=\) | \( ( -343\beta_{5} - 490\beta_{3} - 490\beta_{2} + 1645\beta _1 - 4700 ) / 4 \) |
\(\nu^{7}\) | \(=\) | \( ( -10834\beta_{7} + 12206\beta_{6} - 3233\beta_{4} + 1567\beta_{3} - 1567\beta_{2} - 1567\beta_1 ) / 8 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(129\) | \(197\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
97.1 |
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0 | − | 5.43734i | 0 | − | 6.12900i | 0 | 6.21005 | − | 3.23038i | 0 | −20.5647 | 0 | ||||||||||||||||||||||||||||||||||||||
97.2 | 0 | − | 5.43734i | 0 | 6.12900i | 0 | −6.21005 | − | 3.23038i | 0 | −20.5647 | 0 | ||||||||||||||||||||||||||||||||||||||||
97.3 | 0 | − | 1.56056i | 0 | − | 3.23038i | 0 | 3.38162 | + | 6.12900i | 0 | 6.56466 | 0 | |||||||||||||||||||||||||||||||||||||||
97.4 | 0 | − | 1.56056i | 0 | 3.23038i | 0 | −3.38162 | + | 6.12900i | 0 | 6.56466 | 0 | ||||||||||||||||||||||||||||||||||||||||
97.5 | 0 | 1.56056i | 0 | − | 3.23038i | 0 | −3.38162 | − | 6.12900i | 0 | 6.56466 | 0 | ||||||||||||||||||||||||||||||||||||||||
97.6 | 0 | 1.56056i | 0 | 3.23038i | 0 | 3.38162 | − | 6.12900i | 0 | 6.56466 | 0 | |||||||||||||||||||||||||||||||||||||||||
97.7 | 0 | 5.43734i | 0 | − | 6.12900i | 0 | −6.21005 | + | 3.23038i | 0 | −20.5647 | 0 | ||||||||||||||||||||||||||||||||||||||||
97.8 | 0 | 5.43734i | 0 | 6.12900i | 0 | 6.21005 | + | 3.23038i | 0 | −20.5647 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
28.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 224.3.c.a | ✓ | 8 |
3.b | odd | 2 | 1 | 2016.3.f.c | 8 | ||
4.b | odd | 2 | 1 | inner | 224.3.c.a | ✓ | 8 |
7.b | odd | 2 | 1 | inner | 224.3.c.a | ✓ | 8 |
8.b | even | 2 | 1 | 448.3.c.g | 8 | ||
8.d | odd | 2 | 1 | 448.3.c.g | 8 | ||
12.b | even | 2 | 1 | 2016.3.f.c | 8 | ||
21.c | even | 2 | 1 | 2016.3.f.c | 8 | ||
28.d | even | 2 | 1 | inner | 224.3.c.a | ✓ | 8 |
56.e | even | 2 | 1 | 448.3.c.g | 8 | ||
56.h | odd | 2 | 1 | 448.3.c.g | 8 | ||
84.h | odd | 2 | 1 | 2016.3.f.c | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
224.3.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
224.3.c.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
224.3.c.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
224.3.c.a | ✓ | 8 | 28.d | even | 2 | 1 | inner |
448.3.c.g | 8 | 8.b | even | 2 | 1 | ||
448.3.c.g | 8 | 8.d | odd | 2 | 1 | ||
448.3.c.g | 8 | 56.e | even | 2 | 1 | ||
448.3.c.g | 8 | 56.h | odd | 2 | 1 | ||
2016.3.f.c | 8 | 3.b | odd | 2 | 1 | ||
2016.3.f.c | 8 | 12.b | even | 2 | 1 | ||
2016.3.f.c | 8 | 21.c | even | 2 | 1 | ||
2016.3.f.c | 8 | 84.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 32T_{3}^{2} + 72 \)
acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} + 32 T^{2} + 72)^{2} \)
$5$
\( (T^{4} + 48 T^{2} + 392)^{2} \)
$7$
\( T^{8} - 4 T^{6} + 1862 T^{4} + \cdots + 5764801 \)
$11$
\( (T^{4} - 248 T^{2} + 3600)^{2} \)
$13$
\( (T^{4} + 208 T^{2} + 1800)^{2} \)
$17$
\( (T^{4} + 1152 T^{2} + 320000)^{2} \)
$19$
\( (T^{4} + 256 T^{2} + 16200)^{2} \)
$23$
\( (T^{4} - 440 T^{2} + 1296)^{2} \)
$29$
\( (T^{2} + 28 T - 540)^{4} \)
$31$
\( (T^{4} + 1664 T^{2} + 115200)^{2} \)
$37$
\( (T^{2} - 52 T - 60)^{4} \)
$41$
\( (T^{4} + 1856 T^{2} + 10368)^{2} \)
$43$
\( (T^{4} - 4856 T^{2} + 4717584)^{2} \)
$47$
\( (T^{4} + 2048 T^{2} + 1036800)^{2} \)
$53$
\( (T + 30)^{8} \)
$59$
\( (T^{4} + 9344 T^{2} + 21385800)^{2} \)
$61$
\( (T^{4} + 10672 T^{2} + 342792)^{2} \)
$67$
\( (T^{2} - 828)^{4} \)
$71$
\( (T^{4} - 8200 T^{2} + 250000)^{2} \)
$73$
\( (T^{4} + 12928 T^{2} + 35280000)^{2} \)
$79$
\( (T^{4} - 25992 T^{2} + \cdots + 128595600)^{2} \)
$83$
\( (T^{4} + 19424 T^{2} + 89191368)^{2} \)
$89$
\( (T^{4} + 768 T^{2} + 3200)^{2} \)
$97$
\( (T^{4} + 18432 T^{2} + 83980800)^{2} \)
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