# Properties

 Label 1800.1.r.a 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{1}^{\mathrm{new}}(1800, [\chi])$$.