Newspace parameters
Level: | \( N \) | \(=\) | \( 975 = 3 \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 975.bd (of order \(10\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.486588387317\) |
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: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{10}\) |
Projective field: | Galois closure of 10.0.1059009246826171875.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).
\(n\) | \(301\) | \(326\) | \(352\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-\zeta_{20}^{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
116.1 |
|
−0.951057 | − | 0.690983i | −0.309017 | − | 0.951057i | 0.118034 | + | 0.363271i | 0.951057 | + | 0.309017i | −0.363271 | + | 1.11803i | 0 | −0.224514 | + | 0.690983i | −0.809017 | + | 0.587785i | −0.690983 | − | 0.951057i | ||||||||||||||||||||||||||
116.2 | 0.951057 | + | 0.690983i | −0.309017 | − | 0.951057i | 0.118034 | + | 0.363271i | −0.951057 | − | 0.309017i | 0.363271 | − | 1.11803i | 0 | 0.224514 | − | 0.690983i | −0.809017 | + | 0.587785i | −0.690983 | − | 0.951057i | |||||||||||||||||||||||||||
311.1 | −0.951057 | + | 0.690983i | −0.309017 | + | 0.951057i | 0.118034 | − | 0.363271i | 0.951057 | − | 0.309017i | −0.363271 | − | 1.11803i | 0 | −0.224514 | − | 0.690983i | −0.809017 | − | 0.587785i | −0.690983 | + | 0.951057i | |||||||||||||||||||||||||||
311.2 | 0.951057 | − | 0.690983i | −0.309017 | + | 0.951057i | 0.118034 | − | 0.363271i | −0.951057 | + | 0.309017i | 0.363271 | + | 1.11803i | 0 | 0.224514 | + | 0.690983i | −0.809017 | − | 0.587785i | −0.690983 | + | 0.951057i | |||||||||||||||||||||||||||
506.1 | −0.587785 | − | 1.80902i | 0.809017 | − | 0.587785i | −2.11803 | + | 1.53884i | 0.587785 | − | 0.809017i | −1.53884 | − | 1.11803i | 0 | 2.48990 | + | 1.80902i | 0.309017 | − | 0.951057i | −1.80902 | − | 0.587785i | |||||||||||||||||||||||||||
506.2 | 0.587785 | + | 1.80902i | 0.809017 | − | 0.587785i | −2.11803 | + | 1.53884i | −0.587785 | + | 0.809017i | 1.53884 | + | 1.11803i | 0 | −2.48990 | − | 1.80902i | 0.309017 | − | 0.951057i | −1.80902 | − | 0.587785i | |||||||||||||||||||||||||||
896.1 | −0.587785 | + | 1.80902i | 0.809017 | + | 0.587785i | −2.11803 | − | 1.53884i | 0.587785 | + | 0.809017i | −1.53884 | + | 1.11803i | 0 | 2.48990 | − | 1.80902i | 0.309017 | + | 0.951057i | −1.80902 | + | 0.587785i | |||||||||||||||||||||||||||
896.2 | 0.587785 | − | 1.80902i | 0.809017 | + | 0.587785i | −2.11803 | − | 1.53884i | −0.587785 | − | 0.809017i | 1.53884 | − | 1.11803i | 0 | −2.48990 | + | 1.80902i | 0.309017 | + | 0.951057i | −1.80902 | + | 0.587785i | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
39.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-39}) \) |
3.b | odd | 2 | 1 | inner |
13.b | even | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
75.j | odd | 10 | 1 | inner |
325.q | even | 10 | 1 | inner |
975.bd | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 975.1.bd.c | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 975.1.bd.c | ✓ | 8 |
13.b | even | 2 | 1 | inner | 975.1.bd.c | ✓ | 8 |
25.d | even | 5 | 1 | inner | 975.1.bd.c | ✓ | 8 |
39.d | odd | 2 | 1 | CM | 975.1.bd.c | ✓ | 8 |
75.j | odd | 10 | 1 | inner | 975.1.bd.c | ✓ | 8 |
325.q | even | 10 | 1 | inner | 975.1.bd.c | ✓ | 8 |
975.bd | odd | 10 | 1 | inner | 975.1.bd.c | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
975.1.bd.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
975.1.bd.c | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
975.1.bd.c | ✓ | 8 | 13.b | even | 2 | 1 | inner |
975.1.bd.c | ✓ | 8 | 25.d | even | 5 | 1 | inner |
975.1.bd.c | ✓ | 8 | 39.d | odd | 2 | 1 | CM |
975.1.bd.c | ✓ | 8 | 75.j | odd | 10 | 1 | inner |
975.1.bd.c | ✓ | 8 | 325.q | even | 10 | 1 | inner |
975.1.bd.c | ✓ | 8 | 975.bd | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 5T_{2}^{6} + 10T_{2}^{4} + 25 \)
acting on \(S_{1}^{\mathrm{new}}(975, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 5 T^{6} + 10 T^{4} + 25 \)
$3$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \)
$5$
\( T^{8} - T^{6} + T^{4} - T^{2} + 1 \)
$7$
\( T^{8} \)
$11$
\( T^{8} + 10 T^{4} + 25 T^{2} + 25 \)
$13$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \)
$17$
\( T^{8} \)
$19$
\( T^{8} \)
$23$
\( T^{8} \)
$29$
\( T^{8} \)
$31$
\( T^{8} \)
$37$
\( T^{8} \)
$41$
\( T^{8} + 5 T^{6} + 10 T^{4} + 25 \)
$43$
\( (T^{2} + T - 1)^{4} \)
$47$
\( T^{8} + 10 T^{4} + 25 T^{2} + 25 \)
$53$
\( T^{8} \)
$59$
\( T^{8} + 10 T^{4} + 25 T^{2} + 25 \)
$61$
\( (T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2} \)
$67$
\( T^{8} \)
$71$
\( T^{8} + 5 T^{6} + 10 T^{4} + 25 \)
$73$
\( T^{8} \)
$79$
\( (T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2} \)
$83$
\( T^{8} + 10 T^{4} + 25 T^{2} + 25 \)
$89$
\( T^{8} \)
$97$
\( T^{8} \)
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