Properties

Label 1800.1.cj.a
Level 1800
Weight 1
Character orbit 1800.cj
Analytic conductor 0.898
Analytic rank 0
Dimension 8
Projective image \(D_{6}\)
CM disc. -8
Inner twists 16

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Newspace parameters

Level: \( N \) = \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 1800.cj (of order \(12\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.3936600000.3

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{24}^{11} q^{2} -\zeta_{24}^{9} q^{3} -\zeta_{24}^{10} q^{4} + \zeta_{24}^{8} q^{6} + \zeta_{24}^{9} q^{8} -\zeta_{24}^{6} q^{9} +O(q^{10})\) \( q + \zeta_{24}^{11} q^{2} -\zeta_{24}^{9} q^{3} -\zeta_{24}^{10} q^{4} + \zeta_{24}^{8} q^{6} + \zeta_{24}^{9} q^{8} -\zeta_{24}^{6} q^{9} -\zeta_{24}^{7} q^{12} -\zeta_{24}^{8} q^{16} -2 \zeta_{24}^{3} q^{17} + \zeta_{24}^{5} q^{18} -\zeta_{24}^{6} q^{19} + \zeta_{24}^{6} q^{24} -\zeta_{24}^{3} q^{27} + \zeta_{24}^{7} q^{32} + 2 \zeta_{24}^{2} q^{34} -\zeta_{24}^{4} q^{36} + \zeta_{24}^{5} q^{38} + ( 1 - \zeta_{24}^{8} ) q^{41} + ( \zeta_{24} - \zeta_{24}^{9} ) q^{43} -\zeta_{24}^{5} q^{48} + \zeta_{24}^{10} q^{49} -2 q^{51} + \zeta_{24}^{2} q^{54} -\zeta_{24}^{3} q^{57} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{59} -\zeta_{24}^{6} q^{64} + ( \zeta_{24}^{3} - \zeta_{24}^{11} ) q^{67} -2 \zeta_{24} q^{68} + \zeta_{24}^{3} q^{72} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{73} -\zeta_{24}^{4} q^{76} - q^{81} + ( \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{82} + \zeta_{24}^{5} q^{83} + ( -1 + \zeta_{24}^{8} ) q^{86} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{89} + \zeta_{24}^{4} q^{96} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{97} -\zeta_{24}^{9} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{6} + O(q^{10}) \) \( 8q - 4q^{6} + 4q^{16} - 4q^{36} + 12q^{41} - 16q^{51} - 4q^{76} - 8q^{81} - 12q^{86} + 4q^{96} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-\zeta_{24}^{6}\) \(-1\) \(\zeta_{24}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
443.1
0.965926 + 0.258819i
−0.965926 0.258819i
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 + 0.258819i 0.707107 0.707107i 0.866025 0.500000i 0 −0.500000 + 0.866025i 0 −0.707107 + 0.707107i 1.00000i 0
443.2 0.965926 0.258819i −0.707107 + 0.707107i 0.866025 0.500000i 0 −0.500000 + 0.866025i 0 0.707107 0.707107i 1.00000i 0
707.1 −0.965926 0.258819i 0.707107 + 0.707107i 0.866025 + 0.500000i 0 −0.500000 0.866025i 0 −0.707107 0.707107i 1.00000i 0
707.2 0.965926 + 0.258819i −0.707107 0.707107i 0.866025 + 0.500000i 0 −0.500000 0.866025i 0 0.707107 + 0.707107i 1.00000i 0
1307.1 −0.258819 0.965926i −0.707107 0.707107i −0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 0.707107 + 0.707107i 1.00000i 0
1307.2 0.258819 + 0.965926i 0.707107 + 0.707107i −0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 −0.707107 0.707107i 1.00000i 0
1643.1 −0.258819 + 0.965926i −0.707107 + 0.707107i −0.866025 0.500000i 0 −0.500000 0.866025i 0 0.707107 0.707107i 1.00000i 0
1643.2 0.258819 0.965926i 0.707107 0.707107i −0.866025 0.500000i 0 −0.500000 0.866025i 0 −0.707107 + 0.707107i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1643.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes
5.b Even 1 yes
5.c Odd 2 yes
9.d Odd 1 yes
40.e Odd 1 yes
40.k Even 2 yes
45.h Odd 1 yes
45.l Even 2 yes
72.l Even 1 yes
360.bd Even 1 yes
360.bt Odd 2 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{11} \) acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\).