Newspace parameters
Level: | \( N \) | = | \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | = | \( 1 \) |
Character orbit: | \([\chi]\) | = | 1800.r (of order \(4\) and degree \(2\)) |
Newform invariants
Self dual: | No |
Analytic conductor: | \(0.898317022739\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Projective image | \(D_{4}\) |
Projective field | Galois closure of 4.0.5400.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character Values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).
\(n\) | \(577\) | \(901\) | \(1001\) | \(1351\) |
\(\chi(n)\) | \(-\zeta_{8}^{2}\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
107.1 |
|
−0.707107 | − | 0.707107i | 0 | 1.00000i | 0 | 0 | 1.00000 | − | 1.00000i | 0.707107 | − | 0.707107i | 0 | 0 | ||||||||||||||||||||||||
107.2 | 0.707107 | + | 0.707107i | 0 | 1.00000i | 0 | 0 | 1.00000 | − | 1.00000i | −0.707107 | + | 0.707107i | 0 | 0 | |||||||||||||||||||||||||
1043.1 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 0 | 0 | 1.00000 | + | 1.00000i | 0.707107 | + | 0.707107i | 0 | 0 | ||||||||||||||||||||||||
1043.2 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | 0 | 0 | 1.00000 | + | 1.00000i | −0.707107 | − | 0.707107i | 0 | 0 |
Inner twists
Char. orbit | Parity | Mult. | Self Twist | Proved |
---|---|---|---|---|
1.a | Even | 1 | trivial | yes |
40.e | Odd | 1 | CM by \(\Q(\sqrt{-10}) \) | yes |
3.b | Odd | 1 | yes | |
5.c | Odd | 1 | yes | |
15.e | Even | 1 | yes | |
40.k | Even | 1 | yes | |
120.m | Even | 1 | yes | |
120.q | Odd | 1 | yes |
Hecke kernels
This newform can be constructed as the kernel of the linear operator \( T_{7}^{2} - 2 T_{7} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\).