Newspace parameters
Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 128.f (of order \(4\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.48774738381\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Relative dimension: | \(3\) over \(\Q(i)\) |
Coefficient field: | 6.0.399424.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 2^{6} \) |
Twist minimal: | no (minimal twist has level 16) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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 \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{5} + 4\nu^{4} - 5\nu^{3} + 8\nu^{2} - 14\nu + 4 ) / 4 \) |
\(\beta_{3}\) | \(=\) | \( ( -3\nu^{5} + 4\nu^{4} - 9\nu^{3} + 8\nu^{2} + 2\nu + 12 ) / 4 \) |
\(\beta_{4}\) | \(=\) | \( ( 3\nu^{5} - 4\nu^{4} + 9\nu^{3} - 8\nu^{2} + 14\nu - 20 ) / 4 \) |
\(\beta_{5}\) | \(=\) | \( ( -5\nu^{5} + 4\nu^{4} - 7\nu^{3} + 16\nu^{2} - 10\nu + 20 ) / 4 \) |
\(\nu\) | \(=\) | \( ( \beta_{4} + \beta_{3} + 2 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{5} + 2\beta_{4} + \beta_{2} + 2\beta_1 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( 2\beta_{5} + \beta_{4} - \beta_{3} - 4\beta _1 + 6 ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( \beta_{5} + 2\beta_{3} + \beta_{2} - 10\beta _1 + 8 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( -2\beta_{5} + 3\beta_{4} + \beta_{3} + 4\beta_{2} + 4\beta _1 + 10 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).
\(n\) | \(5\) | \(127\) |
\(\chi(n)\) | \(-\beta_{1}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 |
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0 | −3.24914 | − | 3.24914i | 0 | 0.0586332 | + | 0.0586332i | 0 | −4.61555 | 0 | 12.1138i | 0 | ||||||||||||||||||||||||||||||||
31.2 | 0 | 0.146365 | + | 0.146365i | 0 | −3.68585 | − | 3.68585i | 0 | 9.66442 | 0 | − | 8.95715i | 0 | ||||||||||||||||||||||||||||||||
31.3 | 0 | 2.10278 | + | 2.10278i | 0 | 4.62721 | + | 4.62721i | 0 | −3.04888 | 0 | − | 0.156674i | 0 | ||||||||||||||||||||||||||||||||
95.1 | 0 | −3.24914 | + | 3.24914i | 0 | 0.0586332 | − | 0.0586332i | 0 | −4.61555 | 0 | − | 12.1138i | 0 | ||||||||||||||||||||||||||||||||
95.2 | 0 | 0.146365 | − | 0.146365i | 0 | −3.68585 | + | 3.68585i | 0 | 9.66442 | 0 | 8.95715i | 0 | |||||||||||||||||||||||||||||||||
95.3 | 0 | 2.10278 | − | 2.10278i | 0 | 4.62721 | − | 4.62721i | 0 | −3.04888 | 0 | 0.156674i | 0 | |||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.f | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 128.3.f.a | 6 | |
3.b | odd | 2 | 1 | 1152.3.m.b | 6 | ||
4.b | odd | 2 | 1 | 128.3.f.b | 6 | ||
8.b | even | 2 | 1 | 64.3.f.a | 6 | ||
8.d | odd | 2 | 1 | 16.3.f.a | ✓ | 6 | |
12.b | even | 2 | 1 | 1152.3.m.a | 6 | ||
16.e | even | 4 | 1 | 16.3.f.a | ✓ | 6 | |
16.e | even | 4 | 1 | 128.3.f.b | 6 | ||
16.f | odd | 4 | 1 | 64.3.f.a | 6 | ||
16.f | odd | 4 | 1 | inner | 128.3.f.a | 6 | |
24.f | even | 2 | 1 | 144.3.m.a | 6 | ||
24.h | odd | 2 | 1 | 576.3.m.a | 6 | ||
32.g | even | 8 | 2 | 1024.3.c.j | 12 | ||
32.g | even | 8 | 2 | 1024.3.d.k | 12 | ||
32.h | odd | 8 | 2 | 1024.3.c.j | 12 | ||
32.h | odd | 8 | 2 | 1024.3.d.k | 12 | ||
40.e | odd | 2 | 1 | 400.3.r.c | 6 | ||
40.k | even | 4 | 1 | 400.3.k.c | 6 | ||
40.k | even | 4 | 1 | 400.3.k.d | 6 | ||
48.i | odd | 4 | 1 | 144.3.m.a | 6 | ||
48.i | odd | 4 | 1 | 1152.3.m.a | 6 | ||
48.k | even | 4 | 1 | 576.3.m.a | 6 | ||
48.k | even | 4 | 1 | 1152.3.m.b | 6 | ||
80.i | odd | 4 | 1 | 400.3.k.d | 6 | ||
80.q | even | 4 | 1 | 400.3.r.c | 6 | ||
80.t | odd | 4 | 1 | 400.3.k.c | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
16.3.f.a | ✓ | 6 | 8.d | odd | 2 | 1 | |
16.3.f.a | ✓ | 6 | 16.e | even | 4 | 1 | |
64.3.f.a | 6 | 8.b | even | 2 | 1 | ||
64.3.f.a | 6 | 16.f | odd | 4 | 1 | ||
128.3.f.a | 6 | 1.a | even | 1 | 1 | trivial | |
128.3.f.a | 6 | 16.f | odd | 4 | 1 | inner | |
128.3.f.b | 6 | 4.b | odd | 2 | 1 | ||
128.3.f.b | 6 | 16.e | even | 4 | 1 | ||
144.3.m.a | 6 | 24.f | even | 2 | 1 | ||
144.3.m.a | 6 | 48.i | odd | 4 | 1 | ||
400.3.k.c | 6 | 40.k | even | 4 | 1 | ||
400.3.k.c | 6 | 80.t | odd | 4 | 1 | ||
400.3.k.d | 6 | 40.k | even | 4 | 1 | ||
400.3.k.d | 6 | 80.i | odd | 4 | 1 | ||
400.3.r.c | 6 | 40.e | odd | 2 | 1 | ||
400.3.r.c | 6 | 80.q | even | 4 | 1 | ||
576.3.m.a | 6 | 24.h | odd | 2 | 1 | ||
576.3.m.a | 6 | 48.k | even | 4 | 1 | ||
1024.3.c.j | 12 | 32.g | even | 8 | 2 | ||
1024.3.c.j | 12 | 32.h | odd | 8 | 2 | ||
1024.3.d.k | 12 | 32.g | even | 8 | 2 | ||
1024.3.d.k | 12 | 32.h | odd | 8 | 2 | ||
1152.3.m.a | 6 | 12.b | even | 2 | 1 | ||
1152.3.m.a | 6 | 48.i | odd | 4 | 1 | ||
1152.3.m.b | 6 | 3.b | odd | 2 | 1 | ||
1152.3.m.b | 6 | 48.k | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 2T_{3}^{5} + 2T_{3}^{4} - 32T_{3}^{3} + 196T_{3}^{2} - 56T_{3} + 8 \)
acting on \(S_{3}^{\mathrm{new}}(128, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( T^{6} + 2 T^{5} + 2 T^{4} - 32 T^{3} + \cdots + 8 \)
$5$
\( T^{6} - 2 T^{5} + 2 T^{4} + 64 T^{3} + \cdots + 8 \)
$7$
\( (T^{3} - 2 T^{2} - 60 T - 136)^{2} \)
$11$
\( T^{6} + 18 T^{5} + 162 T^{4} + \cdots + 587528 \)
$13$
\( T^{6} - 2 T^{5} + 2 T^{4} + \cdots + 1286408 \)
$17$
\( (T^{3} + 2 T^{2} - 260 T - 1544)^{2} \)
$19$
\( T^{6} - 30 T^{5} + 450 T^{4} + \cdots + 13448 \)
$23$
\( (T^{3} + 30 T^{2} + 164 T - 968)^{2} \)
$29$
\( T^{6} - 18 T^{5} + 162 T^{4} + \cdots + 19046792 \)
$31$
\( T^{6} + 1920 T^{4} + \cdots + 16777216 \)
$37$
\( T^{6} + 46 T^{5} + 1058 T^{4} + \cdots + 42632 \)
$41$
\( T^{6} + 4992 T^{4} + \cdots + 67108864 \)
$43$
\( T^{6} + 114 T^{5} + 6498 T^{4} + \cdots + 42632 \)
$47$
\( T^{6} + 8576 T^{4} + \cdots + 6056574976 \)
$53$
\( T^{6} + 78 T^{5} + 3042 T^{4} + \cdots + 783752 \)
$59$
\( T^{6} - 206 T^{5} + \cdots + 8410007432 \)
$61$
\( T^{6} + 30 T^{5} + \cdots + 151449608 \)
$67$
\( T^{6} + 226 T^{5} + \cdots + 87233303432 \)
$71$
\( (T^{3} - 130 T^{2} - 3548 T + 391864)^{2} \)
$73$
\( T^{6} + 18848 T^{4} + \cdots + 7310934016 \)
$79$
\( T^{6} + 37376 T^{4} + \cdots + 1550483193856 \)
$83$
\( T^{6} - 318 T^{5} + \cdots + 105636303368 \)
$89$
\( T^{6} + 16288 T^{4} + \cdots + 25681985536 \)
$97$
\( (T^{3} + 2 T^{2} - 17540 T + 519928)^{2} \)
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