Newspace parameters
Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 128.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.48774738381\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{7} \) |
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) | \( 2\zeta_{8}^{3} + 2\zeta_{8}^{2} + 2\zeta_{8} \) |
\(\beta_{2}\) | \(=\) | \( -4\zeta_{8}^{3} + 4\zeta_{8} \) |
\(\beta_{3}\) | \(=\) | \( 2\zeta_{8}^{3} - 6\zeta_{8}^{2} + 2\zeta_{8} \) |
\(\zeta_{8}\) | \(=\) | \( ( \beta_{3} + 2\beta_{2} + 3\beta_1 ) / 16 \) |
\(\zeta_{8}^{2}\) | \(=\) | \( ( -\beta_{3} + \beta_1 ) / 8 \) |
\(\zeta_{8}^{3}\) | \(=\) | \( ( \beta_{3} - 2\beta_{2} + 3\beta_1 ) / 16 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).
\(n\) | \(5\) | \(127\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | − | 4.82843i | 0 | −7.65685 | 0 | 1.65685i | 0 | −14.3137 | 0 | |||||||||||||||||||||||||||||
127.2 | 0 | − | 0.828427i | 0 | 3.65685 | 0 | 9.65685i | 0 | 8.31371 | 0 | ||||||||||||||||||||||||||||||
127.3 | 0 | 0.828427i | 0 | 3.65685 | 0 | − | 9.65685i | 0 | 8.31371 | 0 | ||||||||||||||||||||||||||||||
127.4 | 0 | 4.82843i | 0 | −7.65685 | 0 | − | 1.65685i | 0 | −14.3137 | 0 | ||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 128.3.c.a | ✓ | 4 |
3.b | odd | 2 | 1 | 1152.3.g.b | 4 | ||
4.b | odd | 2 | 1 | inner | 128.3.c.a | ✓ | 4 |
8.b | even | 2 | 1 | 128.3.c.b | yes | 4 | |
8.d | odd | 2 | 1 | 128.3.c.b | yes | 4 | |
12.b | even | 2 | 1 | 1152.3.g.b | 4 | ||
16.e | even | 4 | 1 | 256.3.d.d | 4 | ||
16.e | even | 4 | 1 | 256.3.d.e | 4 | ||
16.f | odd | 4 | 1 | 256.3.d.d | 4 | ||
16.f | odd | 4 | 1 | 256.3.d.e | 4 | ||
24.f | even | 2 | 1 | 1152.3.g.a | 4 | ||
24.h | odd | 2 | 1 | 1152.3.g.a | 4 | ||
48.i | odd | 4 | 1 | 2304.3.b.j | 4 | ||
48.i | odd | 4 | 1 | 2304.3.b.p | 4 | ||
48.k | even | 4 | 1 | 2304.3.b.j | 4 | ||
48.k | even | 4 | 1 | 2304.3.b.p | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
128.3.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
128.3.c.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
128.3.c.b | yes | 4 | 8.b | even | 2 | 1 | |
128.3.c.b | yes | 4 | 8.d | odd | 2 | 1 | |
256.3.d.d | 4 | 16.e | even | 4 | 1 | ||
256.3.d.d | 4 | 16.f | odd | 4 | 1 | ||
256.3.d.e | 4 | 16.e | even | 4 | 1 | ||
256.3.d.e | 4 | 16.f | odd | 4 | 1 | ||
1152.3.g.a | 4 | 24.f | even | 2 | 1 | ||
1152.3.g.a | 4 | 24.h | odd | 2 | 1 | ||
1152.3.g.b | 4 | 3.b | odd | 2 | 1 | ||
1152.3.g.b | 4 | 12.b | even | 2 | 1 | ||
2304.3.b.j | 4 | 48.i | odd | 4 | 1 | ||
2304.3.b.j | 4 | 48.k | even | 4 | 1 | ||
2304.3.b.p | 4 | 48.i | odd | 4 | 1 | ||
2304.3.b.p | 4 | 48.k | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 4T_{5} - 28 \)
acting on \(S_{3}^{\mathrm{new}}(128, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 24T^{2} + 16 \)
$5$
\( (T^{2} + 4 T - 28)^{2} \)
$7$
\( T^{4} + 96T^{2} + 256 \)
$11$
\( T^{4} + 344T^{2} + 784 \)
$13$
\( (T^{2} - 12 T + 4)^{2} \)
$17$
\( (T^{2} + 4 T - 124)^{2} \)
$19$
\( T^{4} + 664 T^{2} + 99856 \)
$23$
\( T^{4} + 1632 T^{2} + 565504 \)
$29$
\( (T^{2} + 68 T + 1124)^{2} \)
$31$
\( (T^{2} + 2048)^{2} \)
$37$
\( (T^{2} + 20 T - 1468)^{2} \)
$41$
\( (T^{2} - 4 T - 508)^{2} \)
$43$
\( T^{4} + 4632 T^{2} + \cdots + 5326864 \)
$47$
\( T^{4} + 1408 T^{2} + 200704 \)
$53$
\( (T^{2} - 44 T + 452)^{2} \)
$59$
\( T^{4} + 2904 T^{2} + 234256 \)
$61$
\( (T^{2} + 148 T + 3908)^{2} \)
$67$
\( T^{4} + 5976 T^{2} + \cdots + 8088336 \)
$71$
\( T^{4} + 1888 T^{2} + 430336 \)
$73$
\( (T^{2} - 44 T - 668)^{2} \)
$79$
\( T^{4} + 23936 T^{2} + \cdots + 93392896 \)
$83$
\( T^{4} + 22104 T^{2} + \cdots + 117765904 \)
$89$
\( (T^{2} - 108 T - 284)^{2} \)
$97$
\( (T^{2} + 164 T + 3524)^{2} \)
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