Newspace parameters
Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 224.x (of order \(8\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.78864900528\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{8})\) |
Coefficient field: | 8.0.157351936.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + x^{4} + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{8}]$ |
$q$-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 \( x^{8} + x^{4} + 16 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{4} + 2 ) / 3 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{5} + 5\nu ) / 6 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{6} + 5\nu^{2} ) / 12 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{5} + \nu ) / 6 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{6} + 7\nu^{2} ) / 12 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{7} + 7\nu^{3} ) / 24 \) |
\(\beta_{7}\) | \(=\) | \( ( \nu^{7} + 5\nu^{3} ) / 12 \) |
\(\nu\) | \(=\) | \( \beta_{4} + \beta_{2} \) |
\(\nu^{2}\) | \(=\) | \( \beta_{5} + \beta_{3} \) |
\(\nu^{3}\) | \(=\) | \( \beta_{7} + 2\beta_{6} \) |
\(\nu^{4}\) | \(=\) | \( 3\beta _1 - 2 \) |
\(\nu^{5}\) | \(=\) | \( -5\beta_{4} + \beta_{2} \) |
\(\nu^{6}\) | \(=\) | \( -5\beta_{5} + 7\beta_{3} \) |
\(\nu^{7}\) | \(=\) | \( 7\beta_{7} - 10\beta_{6} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(129\) | \(197\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
27.1 |
|
−1.28897 | + | 0.581861i | 0 | 1.32288 | − | 1.50000i | 0 | 0 | −1.87083 | + | 1.87083i | −0.832353 | + | 2.70318i | 2.12132 | − | 2.12132i | 0 | ||||||||||||||||||||||||||||||||
27.2 | 0.581861 | − | 1.28897i | 0 | −1.32288 | − | 1.50000i | 0 | 0 | 1.87083 | − | 1.87083i | −2.70318 | + | 0.832353i | 2.12132 | − | 2.12132i | 0 | |||||||||||||||||||||||||||||||||
83.1 | −1.28897 | − | 0.581861i | 0 | 1.32288 | + | 1.50000i | 0 | 0 | −1.87083 | − | 1.87083i | −0.832353 | − | 2.70318i | 2.12132 | + | 2.12132i | 0 | |||||||||||||||||||||||||||||||||
83.2 | 0.581861 | + | 1.28897i | 0 | −1.32288 | + | 1.50000i | 0 | 0 | 1.87083 | + | 1.87083i | −2.70318 | − | 0.832353i | 2.12132 | + | 2.12132i | 0 | |||||||||||||||||||||||||||||||||
139.1 | −0.581861 | + | 1.28897i | 0 | −1.32288 | − | 1.50000i | 0 | 0 | −1.87083 | + | 1.87083i | 2.70318 | − | 0.832353i | −2.12132 | + | 2.12132i | 0 | |||||||||||||||||||||||||||||||||
139.2 | 1.28897 | − | 0.581861i | 0 | 1.32288 | − | 1.50000i | 0 | 0 | 1.87083 | − | 1.87083i | 0.832353 | − | 2.70318i | −2.12132 | + | 2.12132i | 0 | |||||||||||||||||||||||||||||||||
195.1 | −0.581861 | − | 1.28897i | 0 | −1.32288 | + | 1.50000i | 0 | 0 | −1.87083 | − | 1.87083i | 2.70318 | + | 0.832353i | −2.12132 | − | 2.12132i | 0 | |||||||||||||||||||||||||||||||||
195.2 | 1.28897 | + | 0.581861i | 0 | 1.32288 | + | 1.50000i | 0 | 0 | 1.87083 | + | 1.87083i | 0.832353 | + | 2.70318i | −2.12132 | − | 2.12132i | 0 | |||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
32.h | odd | 8 | 1 | inner |
224.x | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 224.2.x.a | ✓ | 8 |
4.b | odd | 2 | 1 | 896.2.x.a | 8 | ||
7.b | odd | 2 | 1 | CM | 224.2.x.a | ✓ | 8 |
28.d | even | 2 | 1 | 896.2.x.a | 8 | ||
32.g | even | 8 | 1 | 896.2.x.a | 8 | ||
32.h | odd | 8 | 1 | inner | 224.2.x.a | ✓ | 8 |
224.v | odd | 8 | 1 | 896.2.x.a | 8 | ||
224.x | even | 8 | 1 | inner | 224.2.x.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
224.2.x.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
224.2.x.a | ✓ | 8 | 7.b | odd | 2 | 1 | CM |
224.2.x.a | ✓ | 8 | 32.h | odd | 8 | 1 | inner |
224.2.x.a | ✓ | 8 | 224.x | even | 8 | 1 | inner |
896.2.x.a | 8 | 4.b | odd | 2 | 1 | ||
896.2.x.a | 8 | 28.d | even | 2 | 1 | ||
896.2.x.a | 8 | 32.g | even | 8 | 1 | ||
896.2.x.a | 8 | 224.v | odd | 8 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + T^{4} + 16 \)
$3$
\( T^{8} \)
$5$
\( T^{8} \)
$7$
\( (T^{4} + 49)^{2} \)
$11$
\( T^{8} - 12 T^{6} - 176 T^{5} + \cdots + 42436 \)
$13$
\( T^{8} \)
$17$
\( T^{8} \)
$19$
\( T^{8} \)
$23$
\( (T^{4} - 16 T^{3} + 128 T^{2} - 288 T + 324)^{2} \)
$29$
\( T^{8} - 108 T^{6} + 232 T^{5} + \cdots + 1522756 \)
$31$
\( T^{8} \)
$37$
\( T^{8} - 76 T^{6} - 888 T^{5} + \cdots + 1674436 \)
$41$
\( T^{8} \)
$43$
\( T^{8} + 48 T^{7} + 1036 T^{6} + \cdots + 111556 \)
$47$
\( T^{8} \)
$53$
\( T^{8} - 40 T^{7} + 812 T^{6} + \cdots + 31158724 \)
$59$
\( T^{8} \)
$61$
\( T^{8} \)
$67$
\( T^{8} - 16 T^{7} + 364 T^{6} + \cdots + 24462916 \)
$71$
\( (T^{4} + 65536)^{2} \)
$73$
\( T^{8} \)
$79$
\( (T^{4} - 316 T^{2} + 8836)^{2} \)
$83$
\( T^{8} \)
$89$
\( T^{8} \)
$97$
\( T^{8} \)
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