Newspace parameters
Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 256.g (of order \(8\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.04417029174\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{8})\) |
Coefficient field: | 8.0.18939904.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{3} \) |
Twist minimal: | no (minimal twist has level 32) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{7} - 4\nu^{6} + 14\nu^{5} - 27\nu^{4} + 41\nu^{3} - 37\nu^{2} + 24\nu - 5 \) |
\(\beta_{2}\) | \(=\) | \( \nu^{7} - 4\nu^{6} + 14\nu^{5} - 28\nu^{4} + 43\nu^{3} - 44\nu^{2} + 30\nu - 10 \) |
\(\beta_{3}\) | \(=\) | \( -5\nu^{7} + 17\nu^{6} - 59\nu^{5} + 102\nu^{4} - 146\nu^{3} + 121\nu^{2} - 66\nu + 15 \) |
\(\beta_{4}\) | \(=\) | \( 5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19 \) |
\(\beta_{5}\) | \(=\) | \( -5\nu^{7} + 18\nu^{6} - 62\nu^{5} + 113\nu^{4} - 163\nu^{3} + 145\nu^{2} - 82\nu + 20 \) |
\(\beta_{6}\) | \(=\) | \( -5\nu^{7} + 18\nu^{6} - 63\nu^{5} + 115\nu^{4} - 170\nu^{3} + 152\nu^{2} - 89\nu + 23 \) |
\(\beta_{7}\) | \(=\) | \( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \) |
\(\nu\) | \(=\) | \( ( -\beta_{7} + 2\beta_{6} + \beta_{2} + \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{7} + 3\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2\beta _1 - 4 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( 5\beta_{7} - 5\beta_{6} - 2\beta_{5} + 3\beta_{4} + \beta_{3} - 4\beta_{2} - \beta _1 - 3 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 11\beta_{7} - 19\beta_{6} + 3\beta_{5} - \beta_{4} - 5\beta_{3} - 4\beta_{2} - 8\beta _1 + 12 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( -13\beta_{7} + 2\beta_{6} + 15\beta_{5} - 16\beta_{4} - 10\beta_{3} + 13\beta_{2} - 2\beta _1 + 23 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( -67\beta_{7} + 90\beta_{6} + 4\beta_{5} - 10\beta_{4} + 16\beta_{3} + 31\beta_{2} + 33\beta _1 - 28 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( -7\beta_{7} + 87\beta_{6} - 68\beta_{5} + 71\beta_{4} + 65\beta_{3} - 26\beta_{2} + 37\beta _1 - 125 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).
\(n\) | \(5\) | \(255\) |
\(\chi(n)\) | \(-\beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
33.1 |
|
0 | −0.0794708 | + | 0.191860i | 0 | −0.707107 | + | 0.292893i | 0 | −2.27133 | + | 2.27133i | 0 | 2.09083 | + | 2.09083i | 0 | ||||||||||||||||||||||||||||||||||
33.2 | 0 | 1.07947 | − | 2.60607i | 0 | −0.707107 | + | 0.292893i | 0 | 1.68554 | − | 1.68554i | 0 | −3.50504 | − | 3.50504i | 0 | |||||||||||||||||||||||||||||||||||
97.1 | 0 | −1.27882 | + | 0.529706i | 0 | 0.707107 | − | 1.70711i | 0 | −2.74912 | − | 2.74912i | 0 | −0.766519 | + | 0.766519i | 0 | |||||||||||||||||||||||||||||||||||
97.2 | 0 | 2.27882 | − | 0.943920i | 0 | 0.707107 | − | 1.70711i | 0 | −0.665096 | − | 0.665096i | 0 | 2.18073 | − | 2.18073i | 0 | |||||||||||||||||||||||||||||||||||
161.1 | 0 | −1.27882 | − | 0.529706i | 0 | 0.707107 | + | 1.70711i | 0 | −2.74912 | + | 2.74912i | 0 | −0.766519 | − | 0.766519i | 0 | |||||||||||||||||||||||||||||||||||
161.2 | 0 | 2.27882 | + | 0.943920i | 0 | 0.707107 | + | 1.70711i | 0 | −0.665096 | + | 0.665096i | 0 | 2.18073 | + | 2.18073i | 0 | |||||||||||||||||||||||||||||||||||
225.1 | 0 | −0.0794708 | − | 0.191860i | 0 | −0.707107 | − | 0.292893i | 0 | −2.27133 | − | 2.27133i | 0 | 2.09083 | − | 2.09083i | 0 | |||||||||||||||||||||||||||||||||||
225.2 | 0 | 1.07947 | + | 2.60607i | 0 | −0.707107 | − | 0.292893i | 0 | 1.68554 | + | 1.68554i | 0 | −3.50504 | + | 3.50504i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
32.g | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 256.2.g.d | 8 | |
4.b | odd | 2 | 1 | 256.2.g.c | 8 | ||
8.b | even | 2 | 1 | 32.2.g.b | ✓ | 8 | |
8.d | odd | 2 | 1 | 128.2.g.b | 8 | ||
16.e | even | 4 | 1 | 512.2.g.e | 8 | ||
16.e | even | 4 | 1 | 512.2.g.h | 8 | ||
16.f | odd | 4 | 1 | 512.2.g.f | 8 | ||
16.f | odd | 4 | 1 | 512.2.g.g | 8 | ||
24.f | even | 2 | 1 | 1152.2.v.b | 8 | ||
24.h | odd | 2 | 1 | 288.2.v.b | 8 | ||
32.g | even | 8 | 1 | 32.2.g.b | ✓ | 8 | |
32.g | even | 8 | 1 | inner | 256.2.g.d | 8 | |
32.g | even | 8 | 1 | 512.2.g.e | 8 | ||
32.g | even | 8 | 1 | 512.2.g.h | 8 | ||
32.h | odd | 8 | 1 | 128.2.g.b | 8 | ||
32.h | odd | 8 | 1 | 256.2.g.c | 8 | ||
32.h | odd | 8 | 1 | 512.2.g.f | 8 | ||
32.h | odd | 8 | 1 | 512.2.g.g | 8 | ||
40.f | even | 2 | 1 | 800.2.y.b | 8 | ||
40.i | odd | 4 | 1 | 800.2.ba.c | 8 | ||
40.i | odd | 4 | 1 | 800.2.ba.d | 8 | ||
64.i | even | 16 | 2 | 4096.2.a.k | 8 | ||
64.j | odd | 16 | 2 | 4096.2.a.q | 8 | ||
96.o | even | 8 | 1 | 1152.2.v.b | 8 | ||
96.p | odd | 8 | 1 | 288.2.v.b | 8 | ||
160.v | odd | 8 | 1 | 800.2.ba.c | 8 | ||
160.z | even | 8 | 1 | 800.2.y.b | 8 | ||
160.bb | odd | 8 | 1 | 800.2.ba.d | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
32.2.g.b | ✓ | 8 | 8.b | even | 2 | 1 | |
32.2.g.b | ✓ | 8 | 32.g | even | 8 | 1 | |
128.2.g.b | 8 | 8.d | odd | 2 | 1 | ||
128.2.g.b | 8 | 32.h | odd | 8 | 1 | ||
256.2.g.c | 8 | 4.b | odd | 2 | 1 | ||
256.2.g.c | 8 | 32.h | odd | 8 | 1 | ||
256.2.g.d | 8 | 1.a | even | 1 | 1 | trivial | |
256.2.g.d | 8 | 32.g | even | 8 | 1 | inner | |
288.2.v.b | 8 | 24.h | odd | 2 | 1 | ||
288.2.v.b | 8 | 96.p | odd | 8 | 1 | ||
512.2.g.e | 8 | 16.e | even | 4 | 1 | ||
512.2.g.e | 8 | 32.g | even | 8 | 1 | ||
512.2.g.f | 8 | 16.f | odd | 4 | 1 | ||
512.2.g.f | 8 | 32.h | odd | 8 | 1 | ||
512.2.g.g | 8 | 16.f | odd | 4 | 1 | ||
512.2.g.g | 8 | 32.h | odd | 8 | 1 | ||
512.2.g.h | 8 | 16.e | even | 4 | 1 | ||
512.2.g.h | 8 | 32.g | even | 8 | 1 | ||
800.2.y.b | 8 | 40.f | even | 2 | 1 | ||
800.2.y.b | 8 | 160.z | even | 8 | 1 | ||
800.2.ba.c | 8 | 40.i | odd | 4 | 1 | ||
800.2.ba.c | 8 | 160.v | odd | 8 | 1 | ||
800.2.ba.d | 8 | 40.i | odd | 4 | 1 | ||
800.2.ba.d | 8 | 160.bb | odd | 8 | 1 | ||
1152.2.v.b | 8 | 24.f | even | 2 | 1 | ||
1152.2.v.b | 8 | 96.o | even | 8 | 1 | ||
4096.2.a.k | 8 | 64.i | even | 16 | 2 | ||
4096.2.a.q | 8 | 64.j | odd | 16 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 4T_{3}^{7} + 8T_{3}^{6} - 32T_{3}^{4} + 24T_{3}^{3} + 96T_{3}^{2} + 16T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(256, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} - 4 T^{7} + 8 T^{6} - 32 T^{4} + \cdots + 4 \)
$5$
\( (T^{4} + 2 T^{2} + 4 T + 2)^{2} \)
$7$
\( T^{8} + 8 T^{7} + 32 T^{6} + 48 T^{5} + \cdots + 784 \)
$11$
\( T^{8} + 4 T^{7} + 8 T^{6} + 64 T^{5} + \cdots + 4 \)
$13$
\( T^{8} - 8 T^{7} + 36 T^{6} + \cdots + 6724 \)
$17$
\( T^{8} + 64 T^{6} + 1056 T^{4} + \cdots + 256 \)
$19$
\( T^{8} + 4 T^{7} - 8 T^{6} - 48 T^{5} + \cdots + 196 \)
$23$
\( T^{8} + 8 T^{7} + 32 T^{6} + 16 T^{5} + \cdots + 16 \)
$29$
\( T^{8} - 12 T^{6} - 168 T^{5} + \cdots + 188356 \)
$31$
\( (T^{2} - 8 T + 8)^{4} \)
$37$
\( T^{8} - 8 T^{7} - 44 T^{6} + \cdots + 64516 \)
$41$
\( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 26896 \)
$43$
\( T^{8} - 12 T^{7} + 56 T^{6} + \cdots + 31684 \)
$47$
\( T^{8} + 64 T^{6} + 544 T^{4} + \cdots + 256 \)
$53$
\( T^{8} + 8 T^{7} + 100 T^{6} + \cdots + 158404 \)
$59$
\( T^{8} - 20 T^{7} + 136 T^{6} + \cdots + 643204 \)
$61$
\( T^{8} + 24 T^{7} + 132 T^{6} + \cdots + 42436 \)
$67$
\( T^{8} - 36 T^{7} + 504 T^{6} + \cdots + 1285956 \)
$71$
\( T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 21196816 \)
$73$
\( T^{8} + 32 T^{7} + 512 T^{6} + \cdots + 38416 \)
$79$
\( T^{8} + 512 T^{6} + \cdots + 99361024 \)
$83$
\( T^{8} + 20 T^{7} + \cdots + 138250564 \)
$89$
\( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 17007376 \)
$97$
\( (T^{4} - 16 T^{3} + 40 T^{2} + 288 T - 992)^{2} \)
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