Properties

Label 1800.1.bk.a
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 3.1.16200.1
Artin image size \(18\)
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.25920000.1

$q$-expansion

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

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

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 −0.500000 0.866025i −0.500000 0.866025i 0 1.00000 0 1.00000 −0.500000 + 0.866025i 0
1651.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 0 1.00000 −0.500000 0.866025i 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 \(S_{1}^{\mathrm{new}}(1800, [\chi])\):

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