Properties

Label 1800.1.u.a
Level 1800
Weight 1
Character orbit 1800.u
Analytic conductor 0.898
Analytic rank 0
Dimension 4
Projective image \(D_{2}\)
CM/RM disc. -15, -120, 8
Inner twists 16

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Newspace parameters

Level: \( N \) = \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 1800.u (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{2}, \sqrt{-15})\)
Artin image size \(32\)
Artin image $OD_{16}:C_2$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{8} +O(q^{10})\) \( q -\zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{8} - q^{16} -2 \zeta_{8} q^{17} -2 \zeta_{8}^{3} q^{23} + 2 q^{31} + \zeta_{8} q^{32} + 2 \zeta_{8}^{2} q^{34} -2 q^{46} + 2 \zeta_{8} q^{47} -\zeta_{8}^{2} q^{49} -2 \zeta_{8} q^{62} -\zeta_{8}^{2} q^{64} -2 \zeta_{8}^{3} q^{68} -2 \zeta_{8}^{2} q^{79} + 2 \zeta_{8} q^{92} -2 \zeta_{8}^{2} q^{94} + \zeta_{8}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{16} + 8q^{31} - 8q^{46} + O(q^{100}) \)

Character Values

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

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i 0 1.00000i 0 0 0 0.707107 0.707107i 0 0
757.2 0.707107 + 0.707107i 0 1.00000i 0 0 0 −0.707107 + 0.707107i 0 0
1693.1 −0.707107 + 0.707107i 0 1.00000i 0 0 0 0.707107 + 0.707107i 0 0
1693.2 0.707107 0.707107i 0 1.00000i 0 0 0 −0.707107 0.707107i 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.b Even 1 RM by \(\Q(\sqrt{2}) \) yes
15.d Odd 1 CM by \(\Q(\sqrt{-15}) \) yes
120.i Odd 1 CM by \(\Q(\sqrt{-30}) \) yes
3.b Odd 1 yes
5.b Even 1 yes
5.c Odd 2 yes
15.e Even 2 yes
24.h Odd 1 yes
40.f Even 1 yes
40.i Odd 2 yes
120.w Even 2 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\).