Properties

Label 1800.1.r.b
Level 1800
Weight 1
Character orbit 1800.r
Analytic conductor 0.898
Analytic rank 0
Dimension 4
Projective image \(D_{4}\)
CM disc. -40
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.r (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 \(D_{4}\)
Projective field Galois closure of 4.0.5400.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} + ( 1 + \zeta_{8}^{2} ) q^{7} + \zeta_{8} q^{8} +O(q^{10})\) \( q + \zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} + ( 1 + \zeta_{8}^{2} ) q^{7} + \zeta_{8} q^{8} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{11} + ( 1 - \zeta_{8}^{2} ) q^{13} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{14} - q^{16} + ( 1 + \zeta_{8}^{2} ) q^{22} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{26} + ( 1 - \zeta_{8}^{2} ) q^{28} -\zeta_{8}^{3} q^{32} + ( -1 - \zeta_{8}^{2} ) q^{37} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{44} + 2 \zeta_{8}^{3} q^{47} + \zeta_{8}^{2} q^{49} + ( -1 - \zeta_{8}^{2} ) q^{52} + 2 \zeta_{8} q^{53} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{56} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{59} + \zeta_{8}^{2} q^{64} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{74} -2 \zeta_{8}^{3} q^{77} + ( 1 + \zeta_{8}^{2} ) q^{82} + ( 1 - \zeta_{8}^{2} ) q^{88} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{89} + 2 q^{91} -2 \zeta_{8}^{2} q^{94} -\zeta_{8} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} + 4q^{13} - 4q^{16} + 4q^{22} + 4q^{28} - 4q^{37} - 4q^{52} + 4q^{82} + 4q^{88} + 8q^{91} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
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 1.00000 1.00000i 0.707107 0.707107i 0 0
107.2 0.707107 + 0.707107i 0 1.00000i 0 0 1.00000 1.00000i −0.707107 + 0.707107i 0 0
1043.1 −0.707107 + 0.707107i 0 1.00000i 0 0 1.00000 + 1.00000i 0.707107 + 0.707107i 0 0
1043.2 0.707107 0.707107i 0 1.00000i 0 0 1.00000 + 1.00000i −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
40.e Odd 1 CM by \(\Q(\sqrt{-10}) \) yes
3.b Odd 1 yes
5.c Odd 1 yes
15.e Even 1 yes
40.k Even 1 yes
120.m Even 1 yes
120.q Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{7}^{2} - 2 T_{7} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\).