Properties

Label 1800.1.g.b
Level 1800
Weight 1
Character orbit 1800.g
Self dual Yes
Analytic conductor 0.898
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM disc. -8
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.200.1
Artin image size \(12\)
Artin image $D_6$
Artin field Galois closure of 6.0.5400000.2

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{8} + q^{11} + q^{16} - q^{17} - q^{19} + q^{22} + q^{32} - q^{34} - q^{38} + q^{41} - 2q^{43} + q^{44} + q^{49} - 2q^{59} + q^{64} + q^{67} - q^{68} + q^{73} - q^{76} + q^{82} - q^{83} - 2q^{86} + q^{88} + q^{89} - 2q^{97} + q^{98} + O(q^{100}) \)

Character Values

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

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0
1.00000 0 1.00000 0 0 0 1.00000 0 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

Hecke kernels

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

\( T_{11} - 1 \)
\( T_{17} + 1 \)