Newspace parameters
Level: | \( N \) | \(=\) | \( 4000 = 2^{5} \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 4000.bo (of order \(20\), degree \(8\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.99626005053\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\Q(\zeta_{20})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 800) |
Projective image: | \(D_{20}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).
\(n\) | \(1377\) | \(2501\) | \(2751\) |
\(\chi(n)\) | \(\zeta_{20}^{7}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
257.1 |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.951057 | + | 0.309017i | 0 | ||||||||||||||||||||||||||||||||||||||||
993.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.951057 | − | 0.309017i | 0 | |||||||||||||||||||||||||||||||||||||||||
1793.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.587785 | + | 0.809017i | 0 | |||||||||||||||||||||||||||||||||||||||||
1857.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.951057 | + | 0.309017i | 0 | |||||||||||||||||||||||||||||||||||||||||
2593.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.587785 | + | 0.809017i | 0 | |||||||||||||||||||||||||||||||||||||||||
2657.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.587785 | − | 0.809017i | 0 | |||||||||||||||||||||||||||||||||||||||||
3393.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.951057 | − | 0.309017i | 0 | |||||||||||||||||||||||||||||||||||||||||
3457.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.587785 | − | 0.809017i | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
25.f | odd | 20 | 1 | inner |
100.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4000.1.bo.b | 8 | |
4.b | odd | 2 | 1 | CM | 4000.1.bo.b | 8 | |
5.b | even | 2 | 1 | 800.1.bo.a | ✓ | 8 | |
5.c | odd | 4 | 1 | 4000.1.bo.a | 8 | ||
5.c | odd | 4 | 1 | 4000.1.bo.c | 8 | ||
20.d | odd | 2 | 1 | 800.1.bo.a | ✓ | 8 | |
20.e | even | 4 | 1 | 4000.1.bo.a | 8 | ||
20.e | even | 4 | 1 | 4000.1.bo.c | 8 | ||
25.d | even | 5 | 1 | 4000.1.bo.a | 8 | ||
25.e | even | 10 | 1 | 4000.1.bo.c | 8 | ||
25.f | odd | 20 | 1 | 800.1.bo.a | ✓ | 8 | |
25.f | odd | 20 | 1 | inner | 4000.1.bo.b | 8 | |
40.e | odd | 2 | 1 | 1600.1.bw.a | 8 | ||
40.f | even | 2 | 1 | 1600.1.bw.a | 8 | ||
100.h | odd | 10 | 1 | 4000.1.bo.c | 8 | ||
100.j | odd | 10 | 1 | 4000.1.bo.a | 8 | ||
100.l | even | 20 | 1 | 800.1.bo.a | ✓ | 8 | |
100.l | even | 20 | 1 | inner | 4000.1.bo.b | 8 | |
200.v | even | 20 | 1 | 1600.1.bw.a | 8 | ||
200.x | odd | 20 | 1 | 1600.1.bw.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
800.1.bo.a | ✓ | 8 | 5.b | even | 2 | 1 | |
800.1.bo.a | ✓ | 8 | 20.d | odd | 2 | 1 | |
800.1.bo.a | ✓ | 8 | 25.f | odd | 20 | 1 | |
800.1.bo.a | ✓ | 8 | 100.l | even | 20 | 1 | |
1600.1.bw.a | 8 | 40.e | odd | 2 | 1 | ||
1600.1.bw.a | 8 | 40.f | even | 2 | 1 | ||
1600.1.bw.a | 8 | 200.v | even | 20 | 1 | ||
1600.1.bw.a | 8 | 200.x | odd | 20 | 1 | ||
4000.1.bo.a | 8 | 5.c | odd | 4 | 1 | ||
4000.1.bo.a | 8 | 20.e | even | 4 | 1 | ||
4000.1.bo.a | 8 | 25.d | even | 5 | 1 | ||
4000.1.bo.a | 8 | 100.j | odd | 10 | 1 | ||
4000.1.bo.b | 8 | 1.a | even | 1 | 1 | trivial | |
4000.1.bo.b | 8 | 4.b | odd | 2 | 1 | CM | |
4000.1.bo.b | 8 | 25.f | odd | 20 | 1 | inner | |
4000.1.bo.b | 8 | 100.l | even | 20 | 1 | inner | |
4000.1.bo.c | 8 | 5.c | odd | 4 | 1 | ||
4000.1.bo.c | 8 | 20.e | even | 4 | 1 | ||
4000.1.bo.c | 8 | 25.e | even | 10 | 1 | ||
4000.1.bo.c | 8 | 100.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{13}^{8} + 2T_{13}^{7} + 2T_{13}^{6} - 4T_{13}^{4} - 10T_{13}^{3} + 13T_{13}^{2} - 4T_{13} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(4000, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( T^{8} \)
$7$
\( T^{8} \)
$11$
\( T^{8} \)
$13$
\( T^{8} + 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 1 \)
$17$
\( T^{8} + 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 1 \)
$19$
\( T^{8} \)
$23$
\( T^{8} \)
$29$
\( (T^{4} + 5 T + 5)^{2} \)
$31$
\( T^{8} \)
$37$
\( T^{8} - 2 T^{7} + 7 T^{6} - 10 T^{5} + \cdots + 1 \)
$41$
\( T^{8} + 10 T^{4} + 25 T^{2} + 25 \)
$43$
\( T^{8} \)
$47$
\( T^{8} \)
$53$
\( T^{8} - 2 T^{7} + 7 T^{6} - 10 T^{5} + \cdots + 1 \)
$59$
\( T^{8} \)
$61$
\( T^{8} + 10 T^{4} + 25 T^{2} + 25 \)
$67$
\( T^{8} \)
$71$
\( T^{8} \)
$73$
\( T^{8} - 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 1 \)
$79$
\( T^{8} \)
$83$
\( T^{8} \)
$89$
\( (T^{4} + 5 T^{3} + 10 T^{2} + 10 T + 5)^{2} \)
$97$
\( T^{8} - 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 1 \)
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