Newspace parameters
Level: | \( N \) | \(=\) | \( 768 = 2^{8} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 768.i (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.383281929702\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{4} + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{4}\) |
Projective field: | Galois closure of 4.2.18432.2 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).
\(n\) | \(257\) | \(511\) | \(517\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-\zeta_{8}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
65.1 |
|
0 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | − | 1.41421i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||
65.2 | 0 | 0.707107 | + | 0.707107i | 0 | 0 | 0 | 1.41421i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||
449.1 | 0 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | 1.41421i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||
449.2 | 0 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | − | 1.41421i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
16.e | even | 4 | 1 | inner |
16.f | odd | 4 | 1 | inner |
48.i | odd | 4 | 1 | inner |
48.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 768.1.i.a | ✓ | 4 |
3.b | odd | 2 | 1 | CM | 768.1.i.a | ✓ | 4 |
4.b | odd | 2 | 1 | inner | 768.1.i.a | ✓ | 4 |
8.b | even | 2 | 1 | 768.1.i.b | yes | 4 | |
8.d | odd | 2 | 1 | 768.1.i.b | yes | 4 | |
12.b | even | 2 | 1 | inner | 768.1.i.a | ✓ | 4 |
16.e | even | 4 | 1 | inner | 768.1.i.a | ✓ | 4 |
16.e | even | 4 | 1 | 768.1.i.b | yes | 4 | |
16.f | odd | 4 | 1 | inner | 768.1.i.a | ✓ | 4 |
16.f | odd | 4 | 1 | 768.1.i.b | yes | 4 | |
24.f | even | 2 | 1 | 768.1.i.b | yes | 4 | |
24.h | odd | 2 | 1 | 768.1.i.b | yes | 4 | |
32.g | even | 8 | 1 | 3072.1.e.a | 2 | ||
32.g | even | 8 | 1 | 3072.1.e.b | 2 | ||
32.g | even | 8 | 2 | 3072.1.h.a | 4 | ||
32.h | odd | 8 | 1 | 3072.1.e.a | 2 | ||
32.h | odd | 8 | 1 | 3072.1.e.b | 2 | ||
32.h | odd | 8 | 2 | 3072.1.h.a | 4 | ||
48.i | odd | 4 | 1 | inner | 768.1.i.a | ✓ | 4 |
48.i | odd | 4 | 1 | 768.1.i.b | yes | 4 | |
48.k | even | 4 | 1 | inner | 768.1.i.a | ✓ | 4 |
48.k | even | 4 | 1 | 768.1.i.b | yes | 4 | |
96.o | even | 8 | 1 | 3072.1.e.a | 2 | ||
96.o | even | 8 | 1 | 3072.1.e.b | 2 | ||
96.o | even | 8 | 2 | 3072.1.h.a | 4 | ||
96.p | odd | 8 | 1 | 3072.1.e.a | 2 | ||
96.p | odd | 8 | 1 | 3072.1.e.b | 2 | ||
96.p | odd | 8 | 2 | 3072.1.h.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
768.1.i.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
768.1.i.a | ✓ | 4 | 3.b | odd | 2 | 1 | CM |
768.1.i.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
768.1.i.a | ✓ | 4 | 12.b | even | 2 | 1 | inner |
768.1.i.a | ✓ | 4 | 16.e | even | 4 | 1 | inner |
768.1.i.a | ✓ | 4 | 16.f | odd | 4 | 1 | inner |
768.1.i.a | ✓ | 4 | 48.i | odd | 4 | 1 | inner |
768.1.i.a | ✓ | 4 | 48.k | even | 4 | 1 | inner |
768.1.i.b | yes | 4 | 8.b | even | 2 | 1 | |
768.1.i.b | yes | 4 | 8.d | odd | 2 | 1 | |
768.1.i.b | yes | 4 | 16.e | even | 4 | 1 | |
768.1.i.b | yes | 4 | 16.f | odd | 4 | 1 | |
768.1.i.b | yes | 4 | 24.f | even | 2 | 1 | |
768.1.i.b | yes | 4 | 24.h | odd | 2 | 1 | |
768.1.i.b | yes | 4 | 48.i | odd | 4 | 1 | |
768.1.i.b | yes | 4 | 48.k | even | 4 | 1 | |
3072.1.e.a | 2 | 32.g | even | 8 | 1 | ||
3072.1.e.a | 2 | 32.h | odd | 8 | 1 | ||
3072.1.e.a | 2 | 96.o | even | 8 | 1 | ||
3072.1.e.a | 2 | 96.p | odd | 8 | 1 | ||
3072.1.e.b | 2 | 32.g | even | 8 | 1 | ||
3072.1.e.b | 2 | 32.h | odd | 8 | 1 | ||
3072.1.e.b | 2 | 96.o | even | 8 | 1 | ||
3072.1.e.b | 2 | 96.p | odd | 8 | 1 | ||
3072.1.h.a | 4 | 32.g | even | 8 | 2 | ||
3072.1.h.a | 4 | 32.h | odd | 8 | 2 | ||
3072.1.h.a | 4 | 96.o | even | 8 | 2 | ||
3072.1.h.a | 4 | 96.p | odd | 8 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{13}^{2} + 2T_{13} + 2 \)
acting on \(S_{1}^{\mathrm{new}}(768, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 1 \)
$5$
\( T^{4} \)
$7$
\( (T^{2} + 2)^{2} \)
$11$
\( T^{4} \)
$13$
\( (T^{2} + 2 T + 2)^{2} \)
$17$
\( T^{4} \)
$19$
\( T^{4} \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( (T^{2} - 2)^{2} \)
$37$
\( (T^{2} - 2 T + 2)^{2} \)
$41$
\( T^{4} \)
$43$
\( T^{4} \)
$47$
\( T^{4} \)
$53$
\( T^{4} \)
$59$
\( T^{4} \)
$61$
\( (T^{2} - 2 T + 2)^{2} \)
$67$
\( T^{4} + 16 \)
$71$
\( T^{4} \)
$73$
\( T^{4} \)
$79$
\( (T^{2} - 2)^{2} \)
$83$
\( T^{4} \)
$89$
\( T^{4} \)
$97$
\( T^{4} \)
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