Properties

Label 1800.1.g.c
Level 1800
Weight 1
Character orbit 1800.g
Analytic conductor 0.898
Analytic rank 0
Dimension 2
Projective image \(D_{2}\)
CM/RM disc. -15, -40, 24
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.g (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{6}, \sqrt{-10})\)
Artin image size \(16\)
Artin image $D_4:C_2$
Artin field Galois closure of 8.0.729000000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -i q^{2} \) \(- q^{4}\) \( + i q^{8} \) \(+O(q^{10})\) \( q\) \( -i q^{2} \) \(- q^{4}\) \( + i q^{8} \) \(+ q^{16}\) \( + 2 q^{19} \) \( -2 i q^{23} \) \( -i q^{32} \) \( -2 i q^{38} \) \( -2 q^{46} \) \( -2 i q^{47} \) \(+ q^{49}\) \( + 2 i q^{53} \) \(- q^{64}\) \( -2 q^{76} \) \( + 2 i q^{92} \) \( -2 q^{94} \) \( -i q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 4q^{76} \) \(\mathstrut -\mathstrut 4q^{94} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

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
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 0 0
451.2 1.00000i 0 −1.00000 0 0 0 1.00000i 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
15.d Odd 1 CM by \(\Q(\sqrt{-15}) \) yes
24.f Even 1 RM by \(\Q(\sqrt{6}) \) yes
40.e Odd 1 CM by \(\Q(\sqrt{-10}) \) yes
3.b Odd 1 yes
5.b Even 1 yes
8.d Odd 1 yes
120.m Even 1 yes

Hecke kernels

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

\(T_{11} \)
\(T_{17} \)