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Properties

Label 1800.1.g.a
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 6.0.1080000.2

$q$-expansion

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

Display 100 coefficients 10 coefficients

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

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