Newspace parameters
Level: | \( N \) | \(=\) | \( 2023 = 7 \cdot 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2023.l (of order \(8\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.00960852056\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{8})\) |
Coefficient field: | \(\Q(\zeta_{16})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{3}\) |
Projective field: | Galois closure of 3.1.2023.1 |
Artin image: | $S_3\times C_{16}$ |
Artin field: | Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2023\mathbb{Z}\right)^\times\).
\(n\) | \(290\) | \(1737\) |
\(\chi(n)\) | \(-1\) | \(\zeta_{16}^{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
468.1 |
|
−0.707107 | + | 0.707107i | 0 | 0 | 0 | 0 | −0.382683 | − | 0.923880i | −0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | 0 | ||||||||||||||||||||||||||||||||||
468.2 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | 0 | 0.382683 | + | 0.923880i | −0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||
1266.1 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | 0 | −0.923880 | + | 0.382683i | 0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||
1266.2 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | 0 | 0.923880 | − | 0.382683i | 0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||
1868.1 | 0.707107 | + | 0.707107i | 0 | 0 | 0 | 0 | −0.923880 | − | 0.382683i | 0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||
1868.2 | 0.707107 | + | 0.707107i | 0 | 0 | 0 | 0 | 0.923880 | + | 0.382683i | 0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||
1889.1 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | 0 | −0.382683 | + | 0.923880i | −0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||
1889.2 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | 0 | 0.382683 | − | 0.923880i | −0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
17.b | even | 2 | 1 | inner |
17.c | even | 4 | 2 | inner |
17.d | even | 8 | 4 | inner |
119.d | odd | 2 | 1 | inner |
119.f | odd | 4 | 2 | inner |
119.l | odd | 8 | 4 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2023.1.l.a | 8 | |
7.b | odd | 2 | 1 | CM | 2023.1.l.a | 8 | |
17.b | even | 2 | 1 | inner | 2023.1.l.a | 8 | |
17.c | even | 4 | 2 | inner | 2023.1.l.a | 8 | |
17.d | even | 8 | 4 | inner | 2023.1.l.a | 8 | |
17.e | odd | 16 | 1 | 2023.1.c.a | ✓ | 1 | |
17.e | odd | 16 | 1 | 2023.1.c.b | yes | 1 | |
17.e | odd | 16 | 2 | 2023.1.d.a | 2 | ||
17.e | odd | 16 | 4 | 2023.1.f.a | 4 | ||
119.d | odd | 2 | 1 | inner | 2023.1.l.a | 8 | |
119.f | odd | 4 | 2 | inner | 2023.1.l.a | 8 | |
119.l | odd | 8 | 4 | inner | 2023.1.l.a | 8 | |
119.p | even | 16 | 1 | 2023.1.c.a | ✓ | 1 | |
119.p | even | 16 | 1 | 2023.1.c.b | yes | 1 | |
119.p | even | 16 | 2 | 2023.1.d.a | 2 | ||
119.p | even | 16 | 4 | 2023.1.f.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2023.1.c.a | ✓ | 1 | 17.e | odd | 16 | 1 | |
2023.1.c.a | ✓ | 1 | 119.p | even | 16 | 1 | |
2023.1.c.b | yes | 1 | 17.e | odd | 16 | 1 | |
2023.1.c.b | yes | 1 | 119.p | even | 16 | 1 | |
2023.1.d.a | 2 | 17.e | odd | 16 | 2 | ||
2023.1.d.a | 2 | 119.p | even | 16 | 2 | ||
2023.1.f.a | 4 | 17.e | odd | 16 | 4 | ||
2023.1.f.a | 4 | 119.p | even | 16 | 4 | ||
2023.1.l.a | 8 | 1.a | even | 1 | 1 | trivial | |
2023.1.l.a | 8 | 7.b | odd | 2 | 1 | CM | |
2023.1.l.a | 8 | 17.b | even | 2 | 1 | inner | |
2023.1.l.a | 8 | 17.c | even | 4 | 2 | inner | |
2023.1.l.a | 8 | 17.d | even | 8 | 4 | inner | |
2023.1.l.a | 8 | 119.d | odd | 2 | 1 | inner | |
2023.1.l.a | 8 | 119.f | odd | 4 | 2 | inner | |
2023.1.l.a | 8 | 119.l | odd | 8 | 4 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2023, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + 1)^{2} \)
$3$
\( T^{8} \)
$5$
\( T^{8} \)
$7$
\( T^{8} + 1 \)
$11$
\( T^{8} + 256 \)
$13$
\( T^{8} \)
$17$
\( T^{8} \)
$19$
\( T^{8} \)
$23$
\( T^{8} + 1 \)
$29$
\( T^{8} + 1 \)
$31$
\( T^{8} \)
$37$
\( T^{8} + 1 \)
$41$
\( T^{8} \)
$43$
\( (T^{4} + 1)^{2} \)
$47$
\( T^{8} \)
$53$
\( (T^{4} + 1)^{2} \)
$59$
\( T^{8} \)
$61$
\( T^{8} \)
$67$
\( (T + 2)^{8} \)
$71$
\( T^{8} + 1 \)
$73$
\( T^{8} \)
$79$
\( T^{8} + 1 \)
$83$
\( T^{8} \)
$89$
\( T^{8} \)
$97$
\( T^{8} \)
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