Properties

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

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
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.

\(f(q)\) \(=\) \( q\) \( -\zeta_{12}^{5} q^{2} \) \( -\zeta_{12}^{3} q^{3} \) \( -\zeta_{12}^{4} q^{4} \) \( -\zeta_{12}^{2} q^{6} \) \( -\zeta_{12}^{3} q^{8} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( -\zeta_{12}^{5} q^{2} \) \( -\zeta_{12}^{3} q^{3} \) \( -\zeta_{12}^{4} q^{4} \) \( -\zeta_{12}^{2} q^{6} \) \( -\zeta_{12}^{3} q^{8} \) \(- q^{9}\) \( -2 \zeta_{12}^{2} q^{11} \) \( -\zeta_{12} q^{12} \) \( -\zeta_{12}^{2} q^{16} \) \( + 2 \zeta_{12}^{3} q^{17} \) \( + \zeta_{12}^{5} q^{18} \) \(+ q^{19}\) \( -2 \zeta_{12} q^{22} \) \(- q^{24}\) \( + \zeta_{12}^{3} q^{27} \) \( -\zeta_{12} q^{32} \) \( + 2 \zeta_{12}^{5} q^{33} \) \( + 2 \zeta_{12}^{2} q^{34} \) \( + \zeta_{12}^{4} q^{36} \) \( -\zeta_{12}^{5} q^{38} \) \( -\zeta_{12}^{4} q^{41} \) \( -\zeta_{12}^{5} q^{43} \) \( -2 q^{44} \) \( + \zeta_{12}^{5} q^{48} \) \( -\zeta_{12}^{4} q^{49} \) \( + 2 q^{51} \) \( + \zeta_{12}^{2} q^{54} \) \( -\zeta_{12}^{3} q^{57} \) \( + \zeta_{12}^{4} q^{59} \) \(- q^{64}\) \( + 2 \zeta_{12}^{4} q^{66} \) \( + \zeta_{12} q^{67} \) \( + 2 \zeta_{12} q^{68} \) \( + \zeta_{12}^{3} q^{72} \) \( + \zeta_{12}^{3} q^{73} \) \( -\zeta_{12}^{4} q^{76} \) \(+ q^{81}\) \( -\zeta_{12}^{3} q^{82} \) \( -\zeta_{12}^{5} q^{83} \) \( -\zeta_{12}^{4} q^{86} \) \( + 2 \zeta_{12}^{5} q^{88} \) \(+ q^{89}\) \( + \zeta_{12}^{4} q^{96} \) \( + \zeta_{12}^{5} q^{97} \) \( -\zeta_{12}^{3} q^{98} \) \( + 2 \zeta_{12}^{2} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut -\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 4q^{24} \) \(\mathstrut +\mathstrut 4q^{34} \) \(\mathstrut -\mathstrut 2q^{36} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 8q^{44} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 8q^{51} \) \(\mathstrut +\mathstrut 2q^{54} \) \(\mathstrut -\mathstrut 2q^{59} \) \(\mathstrut -\mathstrut 4q^{64} \) \(\mathstrut -\mathstrut 4q^{66} \) \(\mathstrut +\mathstrut 2q^{76} \) \(\mathstrut +\mathstrut 4q^{81} \) \(\mathstrut +\mathstrut 2q^{86} \) \(\mathstrut +\mathstrut 4q^{89} \) \(\mathstrut -\mathstrut 2q^{96} \) \(\mathstrut +\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{12}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i 1.00000i 0.500000 0.866025i 0 −0.500000 0.866025i 0 1.00000i −1.00000 0
499.2 0.866025 0.500000i 1.00000i 0.500000 0.866025i 0 −0.500000 0.866025i 0 1.00000i −1.00000 0
1699.1 −0.866025 0.500000i 1.00000i 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 1.00000i −1.00000 0
1699.2 0.866025 + 0.500000i 1.00000i 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 1.00000i −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
5.b Even 1 yes
9.c Even 1 yes
40.e Odd 1 yes
45.j Even 1 yes
72.p Odd 1 yes
360.z Odd 1 yes

Hecke kernels

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

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