# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,1,Mod(1051,1800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 4, 0]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1800.1051");

S:= CuspForms(chi, 1);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.898317022739$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.648.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 0.500000 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i 0 −1.00000 −0.500000 0.866025i 0
1651.1 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 −1.00000 −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 $$\Q(\sqrt{-2})$$
9.c even 3 1 inner
72.p 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.bk.d 2
5.b even 2 1 72.1.p.a 2
5.c odd 4 2 1800.1.ba.b 4
8.d odd 2 1 CM 1800.1.bk.d 2
9.c even 3 1 inner 1800.1.bk.d 2
15.d odd 2 1 216.1.p.a 2
20.d odd 2 1 288.1.t.a 2
35.c odd 2 1 3528.1.cg.a 2
35.i odd 6 1 3528.1.ba.a 2
35.i odd 6 1 3528.1.ce.b 2
35.j even 6 1 3528.1.ba.b 2
35.j even 6 1 3528.1.ce.a 2
40.e odd 2 1 72.1.p.a 2
40.f even 2 1 288.1.t.a 2
40.k even 4 2 1800.1.ba.b 4
45.h odd 6 1 216.1.p.a 2
45.h odd 6 1 648.1.b.a 1
45.j even 6 1 72.1.p.a 2
45.j even 6 1 648.1.b.b 1
45.k odd 12 2 1800.1.ba.b 4
60.h even 2 1 864.1.t.a 2
72.p odd 6 1 inner 1800.1.bk.d 2
80.k odd 4 2 2304.1.o.c 4
80.q even 4 2 2304.1.o.c 4
120.i odd 2 1 864.1.t.a 2
120.m even 2 1 216.1.p.a 2
180.n even 6 1 864.1.t.a 2
180.n even 6 1 2592.1.b.a 1
180.p odd 6 1 288.1.t.a 2
180.p odd 6 1 2592.1.b.b 1
280.n even 2 1 3528.1.cg.a 2
280.ba even 6 1 3528.1.ba.a 2
280.ba even 6 1 3528.1.ce.b 2
280.bi odd 6 1 3528.1.ba.b 2
280.bi odd 6 1 3528.1.ce.a 2
315.q odd 6 1 3528.1.ba.a 2
315.r even 6 1 3528.1.ba.b 2
315.bg odd 6 1 3528.1.cg.a 2
315.bn odd 6 1 3528.1.ce.b 2
315.bo even 6 1 3528.1.ce.a 2
360.z odd 6 1 72.1.p.a 2
360.z odd 6 1 648.1.b.b 1
360.bd even 6 1 216.1.p.a 2
360.bd even 6 1 648.1.b.a 1
360.bh odd 6 1 864.1.t.a 2
360.bh odd 6 1 2592.1.b.a 1
360.bk even 6 1 288.1.t.a 2
360.bk even 6 1 2592.1.b.b 1
360.bo even 12 2 1800.1.ba.b 4
720.ce even 12 2 2304.1.o.c 4
720.cz odd 12 2 2304.1.o.c 4
2520.df odd 6 1 3528.1.ba.b 2
2520.ds even 6 1 3528.1.ba.a 2
2520.dy even 6 1 3528.1.cg.a 2
2520.gj odd 6 1 3528.1.ce.a 2
2520.gp even 6 1 3528.1.ce.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 5.b even 2 1
72.1.p.a 2 40.e odd 2 1
72.1.p.a 2 45.j even 6 1
72.1.p.a 2 360.z odd 6 1
216.1.p.a 2 15.d odd 2 1
216.1.p.a 2 45.h odd 6 1
216.1.p.a 2 120.m even 2 1
216.1.p.a 2 360.bd even 6 1
288.1.t.a 2 20.d odd 2 1
288.1.t.a 2 40.f even 2 1
288.1.t.a 2 180.p odd 6 1
288.1.t.a 2 360.bk even 6 1
648.1.b.a 1 45.h odd 6 1
648.1.b.a 1 360.bd even 6 1
648.1.b.b 1 45.j even 6 1
648.1.b.b 1 360.z odd 6 1
864.1.t.a 2 60.h even 2 1
864.1.t.a 2 120.i odd 2 1
864.1.t.a 2 180.n even 6 1
864.1.t.a 2 360.bh odd 6 1
1800.1.ba.b 4 5.c odd 4 2
1800.1.ba.b 4 40.k even 4 2
1800.1.ba.b 4 45.k odd 12 2
1800.1.ba.b 4 360.bo even 12 2
1800.1.bk.d 2 1.a even 1 1 trivial
1800.1.bk.d 2 8.d odd 2 1 CM
1800.1.bk.d 2 9.c even 3 1 inner
1800.1.bk.d 2 72.p odd 6 1 inner
2304.1.o.c 4 80.k odd 4 2
2304.1.o.c 4 80.q even 4 2
2304.1.o.c 4 720.ce even 12 2
2304.1.o.c 4 720.cz odd 12 2
2592.1.b.a 1 180.n even 6 1
2592.1.b.a 1 360.bh odd 6 1
2592.1.b.b 1 180.p odd 6 1
2592.1.b.b 1 360.bk even 6 1
3528.1.ba.a 2 35.i odd 6 1
3528.1.ba.a 2 280.ba even 6 1
3528.1.ba.a 2 315.q odd 6 1
3528.1.ba.a 2 2520.ds even 6 1
3528.1.ba.b 2 35.j even 6 1
3528.1.ba.b 2 280.bi odd 6 1
3528.1.ba.b 2 315.r even 6 1
3528.1.ba.b 2 2520.df odd 6 1
3528.1.ce.a 2 35.j even 6 1
3528.1.ce.a 2 280.bi odd 6 1
3528.1.ce.a 2 315.bo even 6 1
3528.1.ce.a 2 2520.gj odd 6 1
3528.1.ce.b 2 35.i odd 6 1
3528.1.ce.b 2 280.ba even 6 1
3528.1.ce.b 2 315.bn odd 6 1
3528.1.ce.b 2 2520.gp even 6 1
3528.1.cg.a 2 35.c odd 2 1
3528.1.cg.a 2 280.n even 2 1
3528.1.cg.a 2 315.bg odd 6 1
3528.1.cg.a 2 2520.dy even 6 1

## Hecke kernels

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

 $$T_{11}^{2} - T_{11} + 1$$ T11^2 - T11 + 1 $$T_{17} - 1$$ T17 - 1 $$T_{19} + 1$$ T19 + 1

## Hecke characteristic polynomials

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