Newspace parameters
Level: | \( N \) | \(=\) | \( 1600 = 2^{6} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1600.bh (of order \(10\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.798504020213\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
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: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 800) |
Projective image: | \(A_{5}\) |
Projective field: | Galois closure of 5.1.25000000.2 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).
\(n\) | \(577\) | \(901\) | \(1151\) |
\(\chi(n)\) | \(\zeta_{20}^{4}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
191.1 |
|
0 | −0.951057 | + | 0.309017i | 0 | 0.809017 | − | 0.587785i | 0 | 1.61803i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||
191.2 | 0 | 0.951057 | − | 0.309017i | 0 | 0.809017 | − | 0.587785i | 0 | − | 1.61803i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||
511.1 | 0 | −0.951057 | − | 0.309017i | 0 | 0.809017 | + | 0.587785i | 0 | − | 1.61803i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||
511.2 | 0 | 0.951057 | + | 0.309017i | 0 | 0.809017 | + | 0.587785i | 0 | 1.61803i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
831.1 | 0 | −0.587785 | − | 0.809017i | 0 | −0.309017 | + | 0.951057i | 0 | − | 0.618034i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||
831.2 | 0 | 0.587785 | + | 0.809017i | 0 | −0.309017 | + | 0.951057i | 0 | 0.618034i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
1471.1 | 0 | −0.587785 | + | 0.809017i | 0 | −0.309017 | − | 0.951057i | 0 | 0.618034i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
1471.2 | 0 | 0.587785 | − | 0.809017i | 0 | −0.309017 | − | 0.951057i | 0 | − | 0.618034i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
100.j | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1600.1.bh.b | 8 | |
4.b | odd | 2 | 1 | inner | 1600.1.bh.b | 8 | |
8.b | even | 2 | 1 | 800.1.bh.a | ✓ | 8 | |
8.d | odd | 2 | 1 | 800.1.bh.a | ✓ | 8 | |
25.d | even | 5 | 1 | inner | 1600.1.bh.b | 8 | |
40.e | odd | 2 | 1 | 4000.1.bh.a | 8 | ||
40.f | even | 2 | 1 | 4000.1.bh.a | 8 | ||
40.i | odd | 4 | 1 | 4000.1.bf.a | 8 | ||
40.i | odd | 4 | 1 | 4000.1.bf.b | 8 | ||
40.k | even | 4 | 1 | 4000.1.bf.a | 8 | ||
40.k | even | 4 | 1 | 4000.1.bf.b | 8 | ||
100.j | odd | 10 | 1 | inner | 1600.1.bh.b | 8 | |
200.n | odd | 10 | 1 | 800.1.bh.a | ✓ | 8 | |
200.o | even | 10 | 1 | 4000.1.bh.a | 8 | ||
200.s | odd | 10 | 1 | 4000.1.bh.a | 8 | ||
200.t | even | 10 | 1 | 800.1.bh.a | ✓ | 8 | |
200.v | even | 20 | 1 | 4000.1.bf.a | 8 | ||
200.v | even | 20 | 1 | 4000.1.bf.b | 8 | ||
200.x | odd | 20 | 1 | 4000.1.bf.a | 8 | ||
200.x | odd | 20 | 1 | 4000.1.bf.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
800.1.bh.a | ✓ | 8 | 8.b | even | 2 | 1 | |
800.1.bh.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
800.1.bh.a | ✓ | 8 | 200.n | odd | 10 | 1 | |
800.1.bh.a | ✓ | 8 | 200.t | even | 10 | 1 | |
1600.1.bh.b | 8 | 1.a | even | 1 | 1 | trivial | |
1600.1.bh.b | 8 | 4.b | odd | 2 | 1 | inner | |
1600.1.bh.b | 8 | 25.d | even | 5 | 1 | inner | |
1600.1.bh.b | 8 | 100.j | odd | 10 | 1 | inner | |
4000.1.bf.a | 8 | 40.i | odd | 4 | 1 | ||
4000.1.bf.a | 8 | 40.k | even | 4 | 1 | ||
4000.1.bf.a | 8 | 200.v | even | 20 | 1 | ||
4000.1.bf.a | 8 | 200.x | odd | 20 | 1 | ||
4000.1.bf.b | 8 | 40.i | odd | 4 | 1 | ||
4000.1.bf.b | 8 | 40.k | even | 4 | 1 | ||
4000.1.bf.b | 8 | 200.v | even | 20 | 1 | ||
4000.1.bf.b | 8 | 200.x | odd | 20 | 1 | ||
4000.1.bh.a | 8 | 40.e | odd | 2 | 1 | ||
4000.1.bh.a | 8 | 40.f | even | 2 | 1 | ||
4000.1.bh.a | 8 | 200.o | even | 10 | 1 | ||
4000.1.bh.a | 8 | 200.s | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - T_{3}^{6} + T_{3}^{4} - T_{3}^{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(1600, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} - T^{6} + T^{4} - T^{2} + 1 \)
$5$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \)
$7$
\( (T^{4} + 3 T^{2} + 1)^{2} \)
$11$
\( T^{8} \)
$13$
\( (T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1)^{2} \)
$17$
\( T^{8} \)
$19$
\( T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1 \)
$23$
\( T^{8} - T^{6} + T^{4} - T^{2} + 1 \)
$29$
\( (T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1)^{2} \)
$31$
\( T^{8} - T^{6} + T^{4} - T^{2} + 1 \)
$37$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$41$
\( T^{8} \)
$43$
\( (T^{4} + 3 T^{2} + 1)^{2} \)
$47$
\( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \)
$53$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$59$
\( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \)
$61$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \)
$67$
\( T^{8} \)
$71$
\( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \)
$73$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$79$
\( T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1 \)
$83$
\( T^{8} - T^{6} + T^{4} - T^{2} + 1 \)
$89$
\( T^{8} \)
$97$
\( (T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2} \)
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