Newspace parameters
Level: | \( N \) | \(=\) | \( 1375 = 5^{3} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1375.v (of order \(10\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.686214392370\) |
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: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 5 \) |
Twist minimal: | no (minimal twist has level 275) |
Projective image: | \(D_{10}\) |
Projective field: | Galois closure of 10.2.11170196533203125.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times\).
\(n\) | \(376\) | \(1002\) |
\(\chi(n)\) | \(-1\) | \(\zeta_{20}^{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
76.1 |
|
0 | −0.363271 | − | 1.11803i | 0.309017 | + | 0.951057i | 0 | 0 | 0 | 0 | −0.309017 | + | 0.224514i | 0 | ||||||||||||||||||||||||||||||||||||
76.2 | 0 | 0.363271 | + | 1.11803i | 0.309017 | + | 0.951057i | 0 | 0 | 0 | 0 | −0.309017 | + | 0.224514i | 0 | |||||||||||||||||||||||||||||||||||||
351.1 | 0 | −1.53884 | − | 1.11803i | −0.809017 | − | 0.587785i | 0 | 0 | 0 | 0 | 0.809017 | + | 2.48990i | 0 | |||||||||||||||||||||||||||||||||||||
351.2 | 0 | 1.53884 | + | 1.11803i | −0.809017 | − | 0.587785i | 0 | 0 | 0 | 0 | 0.809017 | + | 2.48990i | 0 | |||||||||||||||||||||||||||||||||||||
901.1 | 0 | −1.53884 | + | 1.11803i | −0.809017 | + | 0.587785i | 0 | 0 | 0 | 0 | 0.809017 | − | 2.48990i | 0 | |||||||||||||||||||||||||||||||||||||
901.2 | 0 | 1.53884 | − | 1.11803i | −0.809017 | + | 0.587785i | 0 | 0 | 0 | 0 | 0.809017 | − | 2.48990i | 0 | |||||||||||||||||||||||||||||||||||||
1176.1 | 0 | −0.363271 | + | 1.11803i | 0.309017 | − | 0.951057i | 0 | 0 | 0 | 0 | −0.309017 | − | 0.224514i | 0 | |||||||||||||||||||||||||||||||||||||
1176.2 | 0 | 0.363271 | − | 1.11803i | 0.309017 | − | 0.951057i | 0 | 0 | 0 | 0 | −0.309017 | − | 0.224514i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
5.b | even | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
25.e | even | 10 | 1 | inner |
55.d | odd | 2 | 1 | inner |
275.s | odd | 10 | 1 | inner |
275.v | odd | 10 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 10T_{3}^{4} + 25T_{3}^{2} + 25 \)
acting on \(S_{1}^{\mathrm{new}}(1375, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} + 10 T^{4} + 25 T^{2} + 25 \)
$5$
\( T^{8} \)
$7$
\( T^{8} \)
$11$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \)
$13$
\( T^{8} \)
$17$
\( T^{8} \)
$19$
\( T^{8} \)
$23$
\( T^{8} + 10 T^{4} + 25 T^{2} + 25 \)
$29$
\( T^{8} \)
$31$
\( (T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2} \)
$37$
\( T^{8} \)
$41$
\( T^{8} \)
$43$
\( T^{8} \)
$47$
\( T^{8} \)
$53$
\( T^{8} + 10 T^{4} + 25 T^{2} + 25 \)
$59$
\( (T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1)^{2} \)
$61$
\( T^{8} \)
$67$
\( T^{8} + 5 T^{6} + 10 T^{4} + 25 \)
$71$
\( (T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1)^{2} \)
$73$
\( T^{8} \)
$79$
\( T^{8} \)
$83$
\( T^{8} \)
$89$
\( (T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1)^{2} \)
$97$
\( T^{8} + 10 T^{4} + 25 T^{2} + 25 \)
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