# Properties

 Label 1800.1.ba.c Level $1800$ Weight $1$ Character orbit 1800.ba Analytic conductor $0.898$ Analytic rank $0$ Dimension $4$ Projective image $D_{3}$ CM discriminant -8 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.ba (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.898317022739$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.16200.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{2} -\zeta_{12} q^{3} -\zeta_{12}^{4} q^{4} + q^{6} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{5} q^{2} -\zeta_{12} q^{3} -\zeta_{12}^{4} q^{4} + q^{6} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + \zeta_{12}^{2} q^{11} + \zeta_{12}^{5} q^{12} -\zeta_{12}^{2} q^{16} + \zeta_{12}^{3} q^{17} -\zeta_{12} q^{18} -2 q^{19} -\zeta_{12} q^{22} -\zeta_{12}^{4} q^{24} -\zeta_{12}^{3} q^{27} + \zeta_{12} q^{32} -\zeta_{12}^{3} q^{33} -\zeta_{12}^{2} q^{34} + q^{36} -2 \zeta_{12}^{5} q^{38} + 2 \zeta_{12}^{4} q^{41} + \zeta_{12}^{5} q^{43} + q^{44} + \zeta_{12}^{3} q^{48} -\zeta_{12}^{4} q^{49} -\zeta_{12}^{4} q^{51} + \zeta_{12}^{2} q^{54} + 2 \zeta_{12} q^{57} + \zeta_{12}^{4} q^{59} - q^{64} + \zeta_{12}^{2} q^{66} + 2 \zeta_{12} q^{67} + \zeta_{12} q^{68} + \zeta_{12}^{5} q^{72} + 2 \zeta_{12}^{3} q^{73} + 2 \zeta_{12}^{4} q^{76} + \zeta_{12}^{4} q^{81} -2 \zeta_{12}^{3} q^{82} + \zeta_{12}^{5} q^{83} -\zeta_{12}^{4} q^{86} + \zeta_{12}^{5} q^{88} + q^{89} -\zeta_{12}^{2} q^{96} -\zeta_{12}^{5} q^{97} + \zeta_{12}^{3} q^{98} + \zeta_{12}^{4} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 4q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{4} + 4q^{6} + 2q^{9} + 2q^{11} - 2q^{16} - 8q^{19} + 2q^{24} - 2q^{34} + 4q^{36} - 4q^{41} + 4q^{44} + 2q^{49} + 2q^{51} + 2q^{54} - 2q^{59} - 4q^{64} + 2q^{66} - 4q^{76} - 2q^{81} + 2q^{86} + 4q^{89} - 2q^{96} - 2q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{12}^{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
499.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0 1.00000 0 1.00000i 0.500000 + 0.866025i 0
499.2 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 0 1.00000 0 1.00000i 0.500000 + 0.866025i 0
1699.1 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 1.00000 0 1.00000i 0.500000 0.866025i 0
1699.2 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 0 1.00000 0 1.00000i 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
5.b even 2 1 inner
9.c even 3 1 inner
40.e odd 2 1 inner
45.j even 6 1 inner
72.p odd 6 1 inner
360.z odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.1.ba.c 4
5.b even 2 1 inner 1800.1.ba.c 4
5.c odd 4 1 1800.1.bk.a 2
5.c odd 4 1 1800.1.bk.e yes 2
8.d odd 2 1 CM 1800.1.ba.c 4
9.c even 3 1 inner 1800.1.ba.c 4
40.e odd 2 1 inner 1800.1.ba.c 4
40.k even 4 1 1800.1.bk.a 2
40.k even 4 1 1800.1.bk.e yes 2
45.j even 6 1 inner 1800.1.ba.c 4
45.k odd 12 1 1800.1.bk.a 2
45.k odd 12 1 1800.1.bk.e yes 2
72.p odd 6 1 inner 1800.1.ba.c 4
360.z odd 6 1 inner 1800.1.ba.c 4
360.bo even 12 1 1800.1.bk.a 2
360.bo even 12 1 1800.1.bk.e yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.1.ba.c 4 1.a even 1 1 trivial
1800.1.ba.c 4 5.b even 2 1 inner
1800.1.ba.c 4 8.d odd 2 1 CM
1800.1.ba.c 4 9.c even 3 1 inner
1800.1.ba.c 4 40.e odd 2 1 inner
1800.1.ba.c 4 45.j even 6 1 inner
1800.1.ba.c 4 72.p odd 6 1 inner
1800.1.ba.c 4 360.z odd 6 1 inner
1800.1.bk.a 2 5.c odd 4 1
1800.1.bk.a 2 40.k even 4 1
1800.1.bk.a 2 45.k odd 12 1
1800.1.bk.a 2 360.bo even 12 1
1800.1.bk.e yes 2 5.c odd 4 1
1800.1.bk.e yes 2 40.k even 4 1
1800.1.bk.e yes 2 45.k odd 12 1
1800.1.bk.e yes 2 360.bo even 12 1

## Hecke kernels

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

 $$T_{11}^{2} - T_{11} + 1$$ $$T_{19} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 1 - T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 1 + T^{2} )^{2}$$
$19$ $$( 2 + T )^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 4 + 2 T + T^{2} )^{2}$$
$43$ $$1 - T^{2} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$( 1 + T + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$16 - 4 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$( 4 + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$1 - T^{2} + T^{4}$$
$89$ $$( -1 + T )^{4}$$
$97$ $$1 - T^{2} + T^{4}$$