# Properties

 Label 1800.1.cr.a Level 1800 Weight 1 Character orbit 1800.cr Analytic conductor 0.898 Analytic rank 0 Dimension 16 Projective image $$D_{20}$$ CM disc. -24 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.cr (of order $$20$$ and degree $$8$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.898317022739$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{20})$$ Coefficient field: $$\Q(\zeta_{40})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Projective image $$D_{20}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{20} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{40}^{19} q^{2} -\zeta_{40}^{18} q^{4} + \zeta_{40}^{9} q^{5} + ( -\zeta_{40}^{2} - \zeta_{40}^{8} ) q^{7} -\zeta_{40}^{17} q^{8} +O(q^{10})$$ $$q -\zeta_{40}^{19} q^{2} -\zeta_{40}^{18} q^{4} + \zeta_{40}^{9} q^{5} + ( -\zeta_{40}^{2} - \zeta_{40}^{8} ) q^{7} -\zeta_{40}^{17} q^{8} + \zeta_{40}^{8} q^{10} + ( -\zeta_{40}^{13} - \zeta_{40}^{15} ) q^{11} + ( -\zeta_{40} - \zeta_{40}^{7} ) q^{14} -\zeta_{40}^{16} q^{16} + \zeta_{40}^{7} q^{20} + ( -\zeta_{40}^{12} - \zeta_{40}^{14} ) q^{22} + \zeta_{40}^{18} q^{25} + ( -1 - \zeta_{40}^{6} ) q^{28} + ( \zeta_{40}^{3} + \zeta_{40}^{13} ) q^{29} + ( \zeta_{40}^{10} + \zeta_{40}^{14} ) q^{31} -\zeta_{40}^{15} q^{32} + ( -\zeta_{40}^{11} - \zeta_{40}^{17} ) q^{35} + \zeta_{40}^{6} q^{40} + ( -\zeta_{40}^{11} - \zeta_{40}^{13} ) q^{44} + ( \zeta_{40}^{4} + \zeta_{40}^{10} + \zeta_{40}^{16} ) q^{49} + \zeta_{40}^{17} q^{50} + ( -\zeta_{40}^{7} - \zeta_{40}^{19} ) q^{53} + ( \zeta_{40}^{2} + \zeta_{40}^{4} ) q^{55} + ( -\zeta_{40}^{5} + \zeta_{40}^{19} ) q^{56} + ( \zeta_{40}^{2} + \zeta_{40}^{12} ) q^{58} + ( -\zeta_{40}^{5} + \zeta_{40}^{7} ) q^{59} + ( \zeta_{40}^{9} + \zeta_{40}^{13} ) q^{62} -\zeta_{40}^{14} q^{64} + ( -\zeta_{40}^{10} - \zeta_{40}^{16} ) q^{70} + ( -\zeta_{40}^{4} + \zeta_{40}^{14} ) q^{73} + ( -\zeta_{40} - \zeta_{40}^{3} + \zeta_{40}^{15} + \zeta_{40}^{17} ) q^{77} + ( -\zeta_{40}^{4} + \zeta_{40}^{12} ) q^{79} + \zeta_{40}^{5} q^{80} + ( \zeta_{40}^{5} - \zeta_{40}^{9} ) q^{83} + ( -\zeta_{40}^{10} - \zeta_{40}^{12} ) q^{88} + ( -1 + \zeta_{40}^{6} ) q^{97} + ( \zeta_{40}^{3} + \zeta_{40}^{9} + \zeta_{40}^{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{7} + O(q^{10})$$ $$16q + 4q^{7} - 4q^{10} + 4q^{16} - 4q^{22} - 16q^{28} + 4q^{55} + 4q^{58} + 4q^{70} - 4q^{73} - 4q^{88} - 16q^{97} + O(q^{100})$$

## Character Values

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

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$\zeta_{40}^{18}$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −0.987688 − 0.156434i 0.987688 + 0.156434i −0.891007 − 0.453990i 0.891007 + 0.453990i −0.453990 + 0.891007i 0.453990 − 0.891007i −0.156434 + 0.987688i 0.156434 − 0.987688i −0.987688 + 0.156434i 0.987688 − 0.156434i −0.891007 + 0.453990i 0.891007 − 0.453990i −0.453990 − 0.891007i 0.453990 + 0.891007i −0.156434 − 0.987688i 0.156434 + 0.987688i
−0.987688 + 0.156434i 0 0.951057 0.309017i −0.156434 0.987688i 0 −1.26007 1.26007i −0.891007 + 0.453990i 0 0.309017 + 0.951057i
37.2 0.987688 0.156434i 0 0.951057 0.309017i 0.156434 + 0.987688i 0 −1.26007 1.26007i 0.891007 0.453990i 0 0.309017 + 0.951057i
253.1 −0.891007 + 0.453990i 0 0.587785 0.809017i 0.453990 + 0.891007i 0 0.221232 0.221232i −0.156434 + 0.987688i 0 −0.809017 0.587785i
253.2 0.891007 0.453990i 0 0.587785 0.809017i −0.453990 0.891007i 0 0.221232 0.221232i 0.156434 0.987688i 0 −0.809017 0.587785i
397.1 −0.453990 0.891007i 0 −0.587785 + 0.809017i 0.891007 0.453990i 0 1.39680 + 1.39680i 0.987688 + 0.156434i 0 −0.809017 0.587785i
397.2 0.453990 + 0.891007i 0 −0.587785 + 0.809017i −0.891007 + 0.453990i 0 1.39680 + 1.39680i −0.987688 0.156434i 0 −0.809017 0.587785i
613.1 −0.156434 0.987688i 0 −0.951057 + 0.309017i −0.987688 + 0.156434i 0 0.642040 0.642040i 0.453990 + 0.891007i 0 0.309017 + 0.951057i
613.2 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 0.987688 0.156434i 0 0.642040 0.642040i −0.453990 0.891007i 0 0.309017 + 0.951057i
973.1 −0.987688 0.156434i 0 0.951057 + 0.309017i −0.156434 + 0.987688i 0 −1.26007 + 1.26007i −0.891007 0.453990i 0 0.309017 0.951057i
973.2 0.987688 + 0.156434i 0 0.951057 + 0.309017i 0.156434 0.987688i 0 −1.26007 + 1.26007i 0.891007 + 0.453990i 0 0.309017 0.951057i
1117.1 −0.891007 0.453990i 0 0.587785 + 0.809017i 0.453990 0.891007i 0 0.221232 + 0.221232i −0.156434 0.987688i 0 −0.809017 + 0.587785i
1117.2 0.891007 + 0.453990i 0 0.587785 + 0.809017i −0.453990 + 0.891007i 0 0.221232 + 0.221232i 0.156434 + 0.987688i 0 −0.809017 + 0.587785i
1333.1 −0.453990 + 0.891007i 0 −0.587785 0.809017i 0.891007 + 0.453990i 0 1.39680 1.39680i 0.987688 0.156434i 0 −0.809017 + 0.587785i
1333.2 0.453990 0.891007i 0 −0.587785 0.809017i −0.891007 0.453990i 0 1.39680 1.39680i −0.987688 + 0.156434i 0 −0.809017 + 0.587785i
1477.1 −0.156434 + 0.987688i 0 −0.951057 0.309017i −0.987688 0.156434i 0 0.642040 + 0.642040i 0.453990 0.891007i 0 0.309017 0.951057i
1477.2 0.156434 0.987688i 0 −0.951057 0.309017i 0.987688 + 0.156434i 0 0.642040 + 0.642040i −0.453990 + 0.891007i 0 0.309017 0.951057i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1477.2 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
24.h Odd 1 CM by $$\Q(\sqrt{-6})$$ yes
3.b Odd 1 yes
8.b Even 1 yes
25.f Odd 1 yes
75.l Even 1 yes
200.x Odd 1 yes
600.bp Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{1}^{\mathrm{new}}(1800, [\chi])$$.