Newspace parameters
Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 448.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(12.2071158433\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.98344960000.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 17x^{4} + 81 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{10} \) |
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 \( x^{8} - 17x^{4} + 81 \) :
\(\beta_{1}\) | \(=\) | \( ( 2\nu^{7} - 9\nu^{5} - 7\nu^{3} + 99\nu ) / 27 \) |
\(\beta_{2}\) | \(=\) | \( ( -2\nu^{6} + 16\nu^{2} ) / 9 \) |
\(\beta_{3}\) | \(=\) | \( ( 4\nu^{7} - 9\nu^{5} - 41\nu^{3} + 45\nu ) / 27 \) |
\(\beta_{4}\) | \(=\) | \( ( -4\nu^{7} + 3\nu^{6} - 9\nu^{5} + 41\nu^{3} - 78\nu^{2} + 45\nu ) / 27 \) |
\(\beta_{5}\) | \(=\) | \( ( 4\nu^{7} + 18\nu^{5} - 14\nu^{3} - 198\nu ) / 27 \) |
\(\beta_{6}\) | \(=\) | \( ( -4\nu^{7} - 3\nu^{6} - 9\nu^{5} + 41\nu^{3} + 78\nu^{2} + 45\nu ) / 27 \) |
\(\beta_{7}\) | \(=\) | \( 4\nu^{4} - 34 \) |
\(\nu\) | \(=\) | \( ( -\beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{3} + 2\beta_1 ) / 8 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{6} - \beta_{4} - \beta_{2} ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{6} + 2\beta_{5} + \beta_{4} - 2\beta_{3} + 4\beta_1 ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( \beta_{7} + 34 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( -11\beta_{6} - 5\beta_{5} - 11\beta_{4} - 22\beta_{3} + 10\beta_1 ) / 8 \) |
\(\nu^{6}\) | \(=\) | \( ( 4\beta_{6} - 4\beta_{4} - 13\beta_{2} ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( 7\beta_{6} + 41\beta_{5} + 7\beta_{4} - 14\beta_{3} + 82\beta_1 ) / 8 \) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
97.1 |
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0 | −3.16228 | 0 | −3.74166 | 0 | 5.91608 | − | 3.74166i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
97.2 | 0 | −3.16228 | 0 | −3.74166 | 0 | 5.91608 | + | 3.74166i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
97.3 | 0 | −3.16228 | 0 | 3.74166 | 0 | −5.91608 | − | 3.74166i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
97.4 | 0 | −3.16228 | 0 | 3.74166 | 0 | −5.91608 | + | 3.74166i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
97.5 | 0 | 3.16228 | 0 | −3.74166 | 0 | −5.91608 | − | 3.74166i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
97.6 | 0 | 3.16228 | 0 | −3.74166 | 0 | −5.91608 | + | 3.74166i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
97.7 | 0 | 3.16228 | 0 | 3.74166 | 0 | 5.91608 | − | 3.74166i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
97.8 | 0 | 3.16228 | 0 | 3.74166 | 0 | 5.91608 | + | 3.74166i | 0 | 1.00000 | 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 |
8.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
28.d | even | 2 | 1 | inner |
56.e | even | 2 | 1 | inner |
56.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 448.3.h.a | ✓ | 8 |
4.b | odd | 2 | 1 | inner | 448.3.h.a | ✓ | 8 |
7.b | odd | 2 | 1 | inner | 448.3.h.a | ✓ | 8 |
8.b | even | 2 | 1 | inner | 448.3.h.a | ✓ | 8 |
8.d | odd | 2 | 1 | inner | 448.3.h.a | ✓ | 8 |
28.d | even | 2 | 1 | inner | 448.3.h.a | ✓ | 8 |
56.e | even | 2 | 1 | inner | 448.3.h.a | ✓ | 8 |
56.h | odd | 2 | 1 | inner | 448.3.h.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
448.3.h.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
448.3.h.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
448.3.h.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
448.3.h.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
448.3.h.a | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
448.3.h.a | ✓ | 8 | 28.d | even | 2 | 1 | inner |
448.3.h.a | ✓ | 8 | 56.e | even | 2 | 1 | inner |
448.3.h.a | ✓ | 8 | 56.h | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 10 \)
acting on \(S_{3}^{\mathrm{new}}(448, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{2} - 10)^{4} \)
$5$
\( (T^{2} - 14)^{4} \)
$7$
\( (T^{4} - 42 T^{2} + 2401)^{2} \)
$11$
\( (T^{2} + 4)^{4} \)
$13$
\( (T^{2} - 126)^{4} \)
$17$
\( (T^{2} + 40)^{4} \)
$19$
\( (T^{2} - 10)^{4} \)
$23$
\( (T^{2} - 560)^{4} \)
$29$
\( (T^{2} + 560)^{4} \)
$31$
\( (T^{2} + 1400)^{4} \)
$37$
\( (T^{2} + 2240)^{4} \)
$41$
\( (T^{2} + 2560)^{4} \)
$43$
\( (T^{2} + 100)^{4} \)
$47$
\( (T^{2} + 6776)^{4} \)
$53$
\( (T^{2} + 5040)^{4} \)
$59$
\( (T^{2} - 3610)^{4} \)
$61$
\( (T^{2} - 3150)^{4} \)
$67$
\( (T^{2} + 4900)^{4} \)
$71$
\( (T^{2} - 1260)^{4} \)
$73$
\( (T^{2} + 11560)^{4} \)
$79$
\( (T^{2} - 140)^{4} \)
$83$
\( (T^{2} - 15210)^{4} \)
$89$
\( (T^{2} + 3240)^{4} \)
$97$
\( (T^{2} + 21160)^{4} \)
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