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: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
Defining polynomial: |
\( x^{8} + 20x^{6} + 116x^{4} + 180x^{2} + 81 \)
|
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
\( x^{8} + 20x^{6} + 116x^{4} + 180x^{2} + 81 \)
:
\(\beta_{1}\) | \(=\) |
\( ( 4\nu^{7} + 80\nu^{5} + 464\nu^{3} + 828\nu ) / 27 \)
|
\(\beta_{2}\) | \(=\) |
\( ( 7\nu^{7} + 131\nu^{5} + 659\nu^{3} + 567\nu ) / 27 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -5\nu^{7} - 9\nu^{6} - 91\nu^{5} - 153\nu^{4} - 427\nu^{3} - 585\nu^{2} - 261\nu - 54 ) / 27 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 2\nu^{6} + 40\nu^{4} + 214\nu^{2} + 180 ) / 9 \)
|
\(\beta_{5}\) | \(=\) |
\( ( -13\nu^{7} - 251\nu^{5} - 1301\nu^{3} - 1107\nu ) / 27 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 2\nu^{7} + 15\nu^{6} + 40\nu^{5} + 273\nu^{4} + 232\nu^{3} + 1335\nu^{2} + 306\nu + 1134 ) / 27 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 5\nu^{7} + 15\nu^{6} + 91\nu^{5} + 273\nu^{4} + 427\nu^{3} + 1335\nu^{2} + 261\nu + 1134 ) / 27 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 ) / 8 \)
|
\(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{4} + \beta_{3} - 20 ) / 4 \)
|
\(\nu^{3}\) | \(=\) |
\( ( -13\beta_{7} + 13\beta_{6} + 4\beta_{5} + 17\beta_{2} - 7\beta_1 ) / 8 \)
|
\(\nu^{4}\) | \(=\) |
\( ( -6\beta_{7} - \beta_{6} + 10\beta_{4} - 5\beta_{3} + \beta_{2} + 84 ) / 2 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 123\beta_{7} - 123\beta_{6} - 68\beta_{5} - 215\beta_{2} + 63\beta_1 ) / 8 \)
|
\(\nu^{6}\) | \(=\) |
\( ( 133\beta_{7} + 40\beta_{6} - 275\beta_{4} + 93\beta_{3} - 40\beta_{2} - 1580 ) / 4 \)
|
\(\nu^{7}\) | \(=\) |
\( ( -1159\beta_{7} + 1159\beta_{6} + 896\beta_{5} + 2535\beta_{2} - 601\beta_1 ) / 8 \)
|
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 |
|
0 | − | 3.29066i | 0 | − | 5.90156i | 0 | −6.86601 | + | 1.36303i | 0 | −1.82843 | 0 | ||||||||||||||||||||||||||||||||||||||
97.2 | 0 | − | 3.29066i | 0 | 5.90156i | 0 | 6.86601 | + | 1.36303i | 0 | −1.82843 | 0 | ||||||||||||||||||||||||||||||||||||||||
97.3 | 0 | − | 2.27411i | 0 | − | 5.40107i | 0 | 4.34256 | − | 5.49019i | 0 | 3.82843 | 0 | |||||||||||||||||||||||||||||||||||||||
97.4 | 0 | − | 2.27411i | 0 | 5.40107i | 0 | −4.34256 | − | 5.49019i | 0 | 3.82843 | 0 | ||||||||||||||||||||||||||||||||||||||||
97.5 | 0 | 2.27411i | 0 | − | 5.40107i | 0 | −4.34256 | + | 5.49019i | 0 | 3.82843 | 0 | ||||||||||||||||||||||||||||||||||||||||
97.6 | 0 | 2.27411i | 0 | 5.40107i | 0 | 4.34256 | + | 5.49019i | 0 | 3.82843 | 0 | |||||||||||||||||||||||||||||||||||||||||
97.7 | 0 | 3.29066i | 0 | − | 5.90156i | 0 | 6.86601 | − | 1.36303i | 0 | −1.82843 | 0 | ||||||||||||||||||||||||||||||||||||||||
97.8 | 0 | 3.29066i | 0 | 5.90156i | 0 | −6.86601 | − | 1.36303i | 0 | −1.82843 | 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.b | ✓ | 8 |
3.b | odd | 2 | 1 | 2016.3.f.b | 8 | ||
4.b | odd | 2 | 1 | inner | 224.3.c.b | ✓ | 8 |
7.b | odd | 2 | 1 | inner | 224.3.c.b | ✓ | 8 |
8.b | even | 2 | 1 | 448.3.c.h | 8 | ||
8.d | odd | 2 | 1 | 448.3.c.h | 8 | ||
12.b | even | 2 | 1 | 2016.3.f.b | 8 | ||
21.c | even | 2 | 1 | 2016.3.f.b | 8 | ||
28.d | even | 2 | 1 | inner | 224.3.c.b | ✓ | 8 |
56.e | even | 2 | 1 | 448.3.c.h | 8 | ||
56.h | odd | 2 | 1 | 448.3.c.h | 8 | ||
84.h | odd | 2 | 1 | 2016.3.f.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
224.3.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
224.3.c.b | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
224.3.c.b | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
224.3.c.b | ✓ | 8 | 28.d | even | 2 | 1 | inner |
448.3.c.h | 8 | 8.b | even | 2 | 1 | ||
448.3.c.h | 8 | 8.d | odd | 2 | 1 | ||
448.3.c.h | 8 | 56.e | even | 2 | 1 | ||
448.3.c.h | 8 | 56.h | odd | 2 | 1 | ||
2016.3.f.b | 8 | 3.b | odd | 2 | 1 | ||
2016.3.f.b | 8 | 12.b | even | 2 | 1 | ||
2016.3.f.b | 8 | 21.c | even | 2 | 1 | ||
2016.3.f.b | 8 | 84.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 16T_{3}^{2} + 56 \)
acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} + 16 T^{2} + 56)^{2} \)
$5$
\( (T^{4} + 64 T^{2} + 1016)^{2} \)
$7$
\( T^{8} - 68 T^{6} + 2758 T^{4} + \cdots + 5764801 \)
$11$
\( (T^{4} - 472 T^{2} + 14224)^{2} \)
$13$
\( (T^{4} + 544 T^{2} + 49784)^{2} \)
$17$
\( (T^{4} + 512 T^{2} + 65024)^{2} \)
$19$
\( (T^{4} + 1072 T^{2} + 123704)^{2} \)
$23$
\( (T^{4} - 1112 T^{2} + 14224)^{2} \)
$29$
\( (T^{2} - 4 T - 284)^{4} \)
$31$
\( (T^{4} + 4096 T^{2} + 1895936)^{2} \)
$37$
\( (T^{2} + 12 T - 1532)^{4} \)
$41$
\( (T^{4} + 4736 T^{2} + 16256)^{2} \)
$43$
\( (T^{4} - 1112 T^{2} + 14224)^{2} \)
$47$
\( (T^{4} + 2944 T^{2} + 3584)^{2} \)
$53$
\( (T^{2} - 68 T + 644)^{4} \)
$59$
\( (T^{4} + 6064 T^{2} + 2784824)^{2} \)
$61$
\( (T^{4} + 7744 T^{2} + 14875256)^{2} \)
$67$
\( (T^{4} - 15064 T^{2} + 34151824)^{2} \)
$71$
\( (T^{4} - 8648 T^{2} + 696976)^{2} \)
$73$
\( (T^{4} + 6336 T^{2} + 1316736)^{2} \)
$79$
\( (T^{4} - 17352 T^{2} + 71703184)^{2} \)
$83$
\( (T^{4} + 8912 T^{2} + 9367736)^{2} \)
$89$
\( (T^{4} + 13376 T^{2} + 39030656)^{2} \)
$97$
\( (T^{4} + 27008 T^{2} + \cdots + 143638016)^{2} \)
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