Newspace parameters
Level: | \( N \) | \(=\) | \( 4000 = 2^{5} \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 4000.p (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.99626005053\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
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: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Projective image: | \(D_{20}\) |
Projective field: | Galois closure of 20.2.400000000000000000000000000.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).
\(n\) | \(1377\) | \(2501\) | \(2751\) |
\(\chi(n)\) | \(\zeta_{20}^{5}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
193.1 |
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0 | −1.39680 | − | 1.39680i | 0 | 0 | 0 | 0.642040 | − | 0.642040i | 0 | 2.90211i | 0 | ||||||||||||||||||||||||||||||||||||||
193.2 | 0 | −0.642040 | − | 0.642040i | 0 | 0 | 0 | 0.221232 | − | 0.221232i | 0 | − | 0.175571i | 0 | ||||||||||||||||||||||||||||||||||||||
193.3 | 0 | −0.221232 | − | 0.221232i | 0 | 0 | 0 | −1.26007 | + | 1.26007i | 0 | − | 0.902113i | 0 | ||||||||||||||||||||||||||||||||||||||
193.4 | 0 | 1.26007 | + | 1.26007i | 0 | 0 | 0 | 1.39680 | − | 1.39680i | 0 | 2.17557i | 0 | |||||||||||||||||||||||||||||||||||||||
1057.1 | 0 | −1.39680 | + | 1.39680i | 0 | 0 | 0 | 0.642040 | + | 0.642040i | 0 | − | 2.90211i | 0 | ||||||||||||||||||||||||||||||||||||||
1057.2 | 0 | −0.642040 | + | 0.642040i | 0 | 0 | 0 | 0.221232 | + | 0.221232i | 0 | 0.175571i | 0 | |||||||||||||||||||||||||||||||||||||||
1057.3 | 0 | −0.221232 | + | 0.221232i | 0 | 0 | 0 | −1.26007 | − | 1.26007i | 0 | 0.902113i | 0 | |||||||||||||||||||||||||||||||||||||||
1057.4 | 0 | 1.26007 | − | 1.26007i | 0 | 0 | 0 | 1.39680 | + | 1.39680i | 0 | − | 2.17557i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
20.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-5}) \) |
5.c | odd | 4 | 1 | inner |
20.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4000.1.p.a | ✓ | 8 |
4.b | odd | 2 | 1 | 4000.1.p.b | yes | 8 | |
5.b | even | 2 | 1 | 4000.1.p.b | yes | 8 | |
5.c | odd | 4 | 1 | inner | 4000.1.p.a | ✓ | 8 |
5.c | odd | 4 | 1 | 4000.1.p.b | yes | 8 | |
20.d | odd | 2 | 1 | CM | 4000.1.p.a | ✓ | 8 |
20.e | even | 4 | 1 | inner | 4000.1.p.a | ✓ | 8 |
20.e | even | 4 | 1 | 4000.1.p.b | yes | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4000.1.p.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
4000.1.p.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
4000.1.p.a | ✓ | 8 | 20.d | odd | 2 | 1 | CM |
4000.1.p.a | ✓ | 8 | 20.e | even | 4 | 1 | inner |
4000.1.p.b | yes | 8 | 4.b | odd | 2 | 1 | |
4000.1.p.b | yes | 8 | 5.b | even | 2 | 1 | |
4000.1.p.b | yes | 8 | 5.c | odd | 4 | 1 | |
4000.1.p.b | yes | 8 | 20.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 2T_{3}^{7} + 2T_{3}^{6} + 11T_{3}^{4} + 20T_{3}^{3} + 18T_{3}^{2} + 6T_{3} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(4000, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} + 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \)
$5$
\( T^{8} \)
$7$
\( T^{8} - 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \)
$11$
\( T^{8} \)
$13$
\( T^{8} \)
$17$
\( T^{8} \)
$19$
\( T^{8} \)
$23$
\( T^{8} + 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \)
$29$
\( (T^{4} + 3 T^{2} + 1)^{2} \)
$31$
\( T^{8} \)
$37$
\( T^{8} \)
$41$
\( (T^{4} - 5 T^{2} + 5)^{2} \)
$43$
\( T^{8} - 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \)
$47$
\( T^{8} + 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \)
$53$
\( T^{8} \)
$59$
\( T^{8} \)
$61$
\( (T^{4} - 5 T^{2} + 5)^{2} \)
$67$
\( (T^{2} + 2 T + 2)^{4} \)
$71$
\( T^{8} \)
$73$
\( T^{8} \)
$79$
\( T^{8} \)
$83$
\( T^{8} - 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \)
$89$
\( (T^{4} + 3 T^{2} + 1)^{2} \)
$97$
\( T^{8} \)
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