Properties

 Label 1800.1.bk.c Level 1800 Weight 1 Character orbit 1800.bk Analytic conductor 0.898 Analytic rank 0 Dimension 2 Projective image $$D_{3}$$ CM disc. -8 Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 1800.bk (of order $$6$$ and degree $$2$$)

Newform invariants

 Self dual: No Analytic conductor: $$0.898317022739$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Projective image $$D_{3}$$ Projective field Galois closure of 3.1.16200.2

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$-\zeta_{6}^{2} q^{2}$$ $$- q^{3}$$ $$-\zeta_{6} q^{4}$$ $$+ \zeta_{6}^{2} q^{6}$$ $$- q^{8}$$ $$+ q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-\zeta_{6}^{2} q^{2}$$ $$- q^{3}$$ $$-\zeta_{6} q^{4}$$ $$+ \zeta_{6}^{2} q^{6}$$ $$- q^{8}$$ $$+ q^{9}$$ $$+ 2 \zeta_{6}^{2} q^{11}$$ $$+ \zeta_{6} q^{12}$$ $$+ \zeta_{6}^{2} q^{16}$$ $$-2 q^{17}$$ $$-\zeta_{6}^{2} q^{18}$$ $$- q^{19}$$ $$+ 2 \zeta_{6} q^{22}$$ $$+ q^{24}$$ $$- q^{27}$$ $$+ \zeta_{6} q^{32}$$ $$-2 \zeta_{6}^{2} q^{33}$$ $$+ 2 \zeta_{6}^{2} q^{34}$$ $$-\zeta_{6} q^{36}$$ $$+ \zeta_{6}^{2} q^{38}$$ $$+ \zeta_{6} q^{41}$$ $$+ \zeta_{6}^{2} q^{43}$$ $$+ 2 q^{44}$$ $$-\zeta_{6}^{2} q^{48}$$ $$-\zeta_{6} q^{49}$$ $$+ 2 q^{51}$$ $$+ \zeta_{6}^{2} q^{54}$$ $$+ q^{57}$$ $$+ \zeta_{6} q^{59}$$ $$+ q^{64}$$ $$-2 \zeta_{6} q^{66}$$ $$-\zeta_{6} q^{67}$$ $$+ 2 \zeta_{6} q^{68}$$ $$- q^{72}$$ $$+ q^{73}$$ $$+ \zeta_{6} q^{76}$$ $$+ q^{81}$$ $$+ q^{82}$$ $$+ \zeta_{6}^{2} q^{83}$$ $$+ \zeta_{6} q^{86}$$ $$-2 \zeta_{6}^{2} q^{88}$$ $$- q^{89}$$ $$-\zeta_{6} q^{96}$$ $$+ \zeta_{6}^{2} q^{97}$$ $$- q^{98}$$ $$+ 2 \zeta_{6}^{2} q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut +\mathstrut q^{2}$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut -\mathstrut q^{6}$$ $$\mathstrut -\mathstrut 2q^{8}$$ $$\mathstrut +\mathstrut 2q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut +\mathstrut q^{2}$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut -\mathstrut q^{6}$$ $$\mathstrut -\mathstrut 2q^{8}$$ $$\mathstrut +\mathstrut 2q^{9}$$ $$\mathstrut -\mathstrut 2q^{11}$$ $$\mathstrut +\mathstrut q^{12}$$ $$\mathstrut -\mathstrut q^{16}$$ $$\mathstrut -\mathstrut 4q^{17}$$ $$\mathstrut +\mathstrut q^{18}$$ $$\mathstrut -\mathstrut 2q^{19}$$ $$\mathstrut +\mathstrut 2q^{22}$$ $$\mathstrut +\mathstrut 2q^{24}$$ $$\mathstrut -\mathstrut 2q^{27}$$ $$\mathstrut +\mathstrut q^{32}$$ $$\mathstrut +\mathstrut 2q^{33}$$ $$\mathstrut -\mathstrut 2q^{34}$$ $$\mathstrut -\mathstrut q^{36}$$ $$\mathstrut -\mathstrut q^{38}$$ $$\mathstrut +\mathstrut q^{41}$$ $$\mathstrut -\mathstrut q^{43}$$ $$\mathstrut +\mathstrut 4q^{44}$$ $$\mathstrut +\mathstrut q^{48}$$ $$\mathstrut -\mathstrut q^{49}$$ $$\mathstrut +\mathstrut 4q^{51}$$ $$\mathstrut -\mathstrut q^{54}$$ $$\mathstrut +\mathstrut 2q^{57}$$ $$\mathstrut +\mathstrut q^{59}$$ $$\mathstrut +\mathstrut 2q^{64}$$ $$\mathstrut -\mathstrut 2q^{66}$$ $$\mathstrut -\mathstrut q^{67}$$ $$\mathstrut +\mathstrut 2q^{68}$$ $$\mathstrut -\mathstrut 2q^{72}$$ $$\mathstrut +\mathstrut 2q^{73}$$ $$\mathstrut +\mathstrut q^{76}$$ $$\mathstrut +\mathstrut 2q^{81}$$ $$\mathstrut +\mathstrut 2q^{82}$$ $$\mathstrut -\mathstrut q^{83}$$ $$\mathstrut +\mathstrut q^{86}$$ $$\mathstrut +\mathstrut 2q^{88}$$ $$\mathstrut -\mathstrut 2q^{89}$$ $$\mathstrut -\mathstrut q^{96}$$ $$\mathstrut -\mathstrut q^{97}$$ $$\mathstrut -\mathstrut 2q^{98}$$ $$\mathstrut -\mathstrut 2q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Character Values

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

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{6}$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1051.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i −1.00000 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −1.00000 1.00000 0
1651.1 0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −1.00000 1.00000 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
8.d Odd 1 CM by $$\Q(\sqrt{-2})$$ yes
9.c Even 1 yes
72.p Odd 1 yes

Hecke kernels

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

 $$T_{11}^{2}$$ $$\mathstrut +\mathstrut 2 T_{11}$$ $$\mathstrut +\mathstrut 4$$ $$T_{17}$$ $$\mathstrut +\mathstrut 2$$ $$T_{19}$$ $$\mathstrut +\mathstrut 1$$