Properties

Label 1800.1.bk.c
Level 1800
Weight 1
Character orbit 1800.bk
Analytic conductor 0.898
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM disc. -8
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 1800.bk (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.16200.2

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{2} - q^{3} -\zeta_{6} q^{4} + \zeta_{6}^{2} q^{6} - q^{8} + q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{2} - q^{3} -\zeta_{6} q^{4} + \zeta_{6}^{2} q^{6} - q^{8} + q^{9} + 2 \zeta_{6}^{2} q^{11} + \zeta_{6} q^{12} + \zeta_{6}^{2} q^{16} -2 q^{17} -\zeta_{6}^{2} q^{18} - q^{19} + 2 \zeta_{6} q^{22} + q^{24} - q^{27} + \zeta_{6} q^{32} -2 \zeta_{6}^{2} q^{33} + 2 \zeta_{6}^{2} q^{34} -\zeta_{6} q^{36} + \zeta_{6}^{2} q^{38} + \zeta_{6} q^{41} + \zeta_{6}^{2} q^{43} + 2 q^{44} -\zeta_{6}^{2} q^{48} -\zeta_{6} q^{49} + 2 q^{51} + \zeta_{6}^{2} q^{54} + q^{57} + \zeta_{6} q^{59} + q^{64} -2 \zeta_{6} q^{66} -\zeta_{6} q^{67} + 2 \zeta_{6} q^{68} - q^{72} + q^{73} + \zeta_{6} q^{76} + q^{81} + q^{82} + \zeta_{6}^{2} q^{83} + \zeta_{6} q^{86} -2 \zeta_{6}^{2} q^{88} - q^{89} -\zeta_{6} q^{96} + \zeta_{6}^{2} q^{97} - q^{98} + 2 \zeta_{6}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - 2q^{3} - q^{4} - q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + q^{2} - 2q^{3} - q^{4} - q^{6} - 2q^{8} + 2q^{9} - 2q^{11} + q^{12} - q^{16} - 4q^{17} + q^{18} - 2q^{19} + 2q^{22} + 2q^{24} - 2q^{27} + q^{32} + 2q^{33} - 2q^{34} - q^{36} - q^{38} + q^{41} - q^{43} + 4q^{44} + q^{48} - q^{49} + 4q^{51} - q^{54} + 2q^{57} + q^{59} + 2q^{64} - 2q^{66} - q^{67} + 2q^{68} - 2q^{72} + 2q^{73} + q^{76} + 2q^{81} + 2q^{82} - q^{83} + q^{86} + 2q^{88} - 2q^{89} - q^{96} - q^{97} - 2q^{98} - 2q^{99} + O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1051.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i −1.00000 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −1.00000 1.00000 0
1651.1 0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
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
9.c Even 1 yes
72.p Odd 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\):

\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{17} + 2 \)
\( T_{19} + 1 \)