Newspace parameters
Level: | \( N \) | \(=\) | \( 2601 = 3^{2} \cdot 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2601.p (of order \(16\), degree \(8\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.29806809786\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
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: | \(\Z[a_1, \ldots, a_{25}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{2}\) |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{17})\) |
Artin image: | $\OD_{64}$ |
Artin field: | Galois closure of \(\mathbb{Q}[x]/(x^{32} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2601\mathbb{Z}\right)^\times\).
\(n\) | \(290\) | \(2026\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{16}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
802.1 |
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0 | 0 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
1081.1 | 0 | 0 | 0.707107 | + | 0.707107i | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
1405.1 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
1576.1 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
1603.1 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
1774.1 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
2098.1 | 0 | 0 | 0.707107 | + | 0.707107i | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
2377.1 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
17.b | even | 2 | 1 | RM by \(\Q(\sqrt{17}) \) |
51.c | odd | 2 | 1 | CM by \(\Q(\sqrt{-51}) \) |
17.c | even | 4 | 2 | inner |
17.d | even | 8 | 4 | inner |
17.e | odd | 16 | 8 | inner |
51.f | odd | 4 | 2 | inner |
51.g | odd | 8 | 4 | inner |
51.i | even | 16 | 8 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2601.1.p.a | ✓ | 8 |
3.b | odd | 2 | 1 | CM | 2601.1.p.a | ✓ | 8 |
17.b | even | 2 | 1 | RM | 2601.1.p.a | ✓ | 8 |
17.c | even | 4 | 2 | inner | 2601.1.p.a | ✓ | 8 |
17.d | even | 8 | 4 | inner | 2601.1.p.a | ✓ | 8 |
17.e | odd | 16 | 8 | inner | 2601.1.p.a | ✓ | 8 |
51.c | odd | 2 | 1 | CM | 2601.1.p.a | ✓ | 8 |
51.f | odd | 4 | 2 | inner | 2601.1.p.a | ✓ | 8 |
51.g | odd | 8 | 4 | inner | 2601.1.p.a | ✓ | 8 |
51.i | even | 16 | 8 | inner | 2601.1.p.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2601.1.p.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
2601.1.p.a | ✓ | 8 | 3.b | odd | 2 | 1 | CM |
2601.1.p.a | ✓ | 8 | 17.b | even | 2 | 1 | RM |
2601.1.p.a | ✓ | 8 | 17.c | even | 4 | 2 | inner |
2601.1.p.a | ✓ | 8 | 17.d | even | 8 | 4 | inner |
2601.1.p.a | ✓ | 8 | 17.e | odd | 16 | 8 | inner |
2601.1.p.a | ✓ | 8 | 51.c | odd | 2 | 1 | CM |
2601.1.p.a | ✓ | 8 | 51.f | odd | 4 | 2 | inner |
2601.1.p.a | ✓ | 8 | 51.g | odd | 8 | 4 | inner |
2601.1.p.a | ✓ | 8 | 51.i | even | 16 | 8 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7} \)
acting on \(S_{1}^{\mathrm{new}}(2601, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( T^{8} \)
$7$
\( T^{8} \)
$11$
\( T^{8} \)
$13$
\( (T^{4} + 16)^{2} \)
$17$
\( T^{8} \)
$19$
\( T^{8} + 256 \)
$23$
\( T^{8} \)
$29$
\( T^{8} \)
$31$
\( T^{8} \)
$37$
\( T^{8} \)
$41$
\( T^{8} \)
$43$
\( T^{8} + 256 \)
$47$
\( T^{8} \)
$53$
\( T^{8} \)
$59$
\( T^{8} \)
$61$
\( T^{8} \)
$67$
\( (T^{2} + 4)^{4} \)
$71$
\( T^{8} \)
$73$
\( T^{8} \)
$79$
\( T^{8} \)
$83$
\( T^{8} \)
$89$
\( T^{8} \)
$97$
\( T^{8} \)
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