Newspace parameters
Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 256.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(104.288924176\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(i, \sqrt{39})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 19x^{2} + 100 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{12} \) |
Twist minimal: | no (minimal twist has level 4) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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 \( x^{4} - 19x^{2} + 100 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} - 9\nu ) / 5 \) |
\(\beta_{2}\) | \(=\) | \( ( -8\nu^{3} + 232\nu ) / 5 \) |
\(\beta_{3}\) | \(=\) | \( 64\nu^{2} - 608 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} + 8\beta_1 ) / 32 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} + 608 ) / 64 \) |
\(\nu^{3}\) | \(=\) | \( ( 9\beta_{2} + 232\beta_1 ) / 32 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).
\(n\) | \(5\) | \(255\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
127.1 |
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0 | −99.9200 | 0 | − | 610.000i | 0 | − | 1398.88i | 0 | 3423.00 | 0 | ||||||||||||||||||||||||||||
127.2 | 0 | −99.9200 | 0 | 610.000i | 0 | 1398.88i | 0 | 3423.00 | 0 | |||||||||||||||||||||||||||||||
127.3 | 0 | 99.9200 | 0 | − | 610.000i | 0 | 1398.88i | 0 | 3423.00 | 0 | ||||||||||||||||||||||||||||||
127.4 | 0 | 99.9200 | 0 | 610.000i | 0 | − | 1398.88i | 0 | 3423.00 | 0 | ||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 256.9.d.e | 4 | |
4.b | odd | 2 | 1 | inner | 256.9.d.e | 4 | |
8.b | even | 2 | 1 | inner | 256.9.d.e | 4 | |
8.d | odd | 2 | 1 | inner | 256.9.d.e | 4 | |
16.e | even | 4 | 1 | 4.9.b.b | ✓ | 2 | |
16.e | even | 4 | 1 | 64.9.c.b | 2 | ||
16.f | odd | 4 | 1 | 4.9.b.b | ✓ | 2 | |
16.f | odd | 4 | 1 | 64.9.c.b | 2 | ||
48.i | odd | 4 | 1 | 36.9.d.b | 2 | ||
48.k | even | 4 | 1 | 36.9.d.b | 2 | ||
80.i | odd | 4 | 1 | 100.9.d.b | 4 | ||
80.j | even | 4 | 1 | 100.9.d.b | 4 | ||
80.k | odd | 4 | 1 | 100.9.b.c | 2 | ||
80.q | even | 4 | 1 | 100.9.b.c | 2 | ||
80.s | even | 4 | 1 | 100.9.d.b | 4 | ||
80.t | odd | 4 | 1 | 100.9.d.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4.9.b.b | ✓ | 2 | 16.e | even | 4 | 1 | |
4.9.b.b | ✓ | 2 | 16.f | odd | 4 | 1 | |
36.9.d.b | 2 | 48.i | odd | 4 | 1 | ||
36.9.d.b | 2 | 48.k | even | 4 | 1 | ||
64.9.c.b | 2 | 16.e | even | 4 | 1 | ||
64.9.c.b | 2 | 16.f | odd | 4 | 1 | ||
100.9.b.c | 2 | 80.k | odd | 4 | 1 | ||
100.9.b.c | 2 | 80.q | even | 4 | 1 | ||
100.9.d.b | 4 | 80.i | odd | 4 | 1 | ||
100.9.d.b | 4 | 80.j | even | 4 | 1 | ||
100.9.d.b | 4 | 80.s | even | 4 | 1 | ||
100.9.d.b | 4 | 80.t | odd | 4 | 1 | ||
256.9.d.e | 4 | 1.a | even | 1 | 1 | trivial | |
256.9.d.e | 4 | 4.b | odd | 2 | 1 | inner | |
256.9.d.e | 4 | 8.b | even | 2 | 1 | inner | |
256.9.d.e | 4 | 8.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 9984 \)
acting on \(S_{9}^{\mathrm{new}}(256, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} - 9984)^{2} \)
$5$
\( (T^{2} + 372100)^{2} \)
$7$
\( (T^{2} + 1956864)^{2} \)
$11$
\( (T^{2} - 341702400)^{2} \)
$13$
\( (T^{2} + 29920900)^{2} \)
$17$
\( (T - 73090)^{4} \)
$19$
\( (T^{2} - 379641600)^{2} \)
$23$
\( (T^{2} + 56268585984)^{2} \)
$29$
\( (T^{2} + 16440881284)^{2} \)
$31$
\( (T^{2} + 4616601600)^{2} \)
$37$
\( (T^{2} + 12054992320900)^{2} \)
$41$
\( (T + 2146882)^{4} \)
$43$
\( (T^{2} - 35142983526144)^{2} \)
$47$
\( (T^{2} + 58160324112384)^{2} \)
$53$
\( (T^{2} + 679454004100)^{2} \)
$59$
\( (T^{2} - 13879469510400)^{2} \)
$61$
\( (T^{2} + 217446816382084)^{2} \)
$67$
\( (T^{2} - 232766284318464)^{2} \)
$71$
\( (T^{2} + 1430516505600)^{2} \)
$73$
\( (T - 5725630)^{4} \)
$79$
\( (T^{2} + 12\!\cdots\!00)^{2} \)
$83$
\( (T^{2} - 26\!\cdots\!24)^{2} \)
$89$
\( (T - 83324222)^{4} \)
$97$
\( (T - 120619010)^{4} \)
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