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

Related objects

Downloads

Learn more about

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.

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

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 \(S_{1}^{\mathrm{new}}(1800, [\chi])\):

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