# Properties

 Label 1368.1.bz.a Level $1368$ Weight $1$ Character orbit 1368.bz Analytic conductor $0.683$ 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] = [1368,1,Mod(163,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1368.163");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1368.bz (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: $$0.682720937282$$ 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 152) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.2888.1 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{8}+O(q^{10})$$ q + z * q^2 + z^2 * q^4 - q^8 $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{8} + q^{11} - \zeta_{6} q^{16} + \zeta_{6} q^{17} + \zeta_{6}^{2} q^{19} + \zeta_{6} q^{22} + \zeta_{6}^{2} q^{25} - \zeta_{6}^{2} q^{32} + 2 \zeta_{6}^{2} q^{34} - q^{38} - \zeta_{6} q^{41} - \zeta_{6} q^{43} + \zeta_{6}^{2} q^{44} + q^{49} - q^{50} - \zeta_{6} q^{59} + q^{64} - \zeta_{6}^{2} q^{67} - 2 q^{68} + \zeta_{6} q^{73} - \zeta_{6} q^{76} - \zeta_{6}^{2} q^{82} + q^{83} - 2 \zeta_{6}^{2} q^{86} - q^{88} - \zeta_{6}^{2} q^{89} + \zeta_{6} q^{97} + \zeta_{6} q^{98} +O(q^{100})$$ q + z * q^2 + z^2 * q^4 - q^8 + q^11 - z * q^16 + z * q^17 + z^2 * q^19 + z * q^22 + z^2 * q^25 - z^2 * q^32 + 2*z^2 * q^34 - q^38 - z * q^41 - z * q^43 + z^2 * q^44 + q^49 - q^50 - z * q^59 + q^64 - z^2 * q^67 - 2 * q^68 + z * q^73 - z * q^76 - z^2 * q^82 + q^83 - 2*z^2 * q^86 - q^88 - z^2 * q^89 + z * q^97 + z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - 2 q^{8}+O(q^{10})$$ 2 * q + q^2 - q^4 - 2 * q^8 $$2 q + q^{2} - q^{4} - 2 q^{8} + 2 q^{11} - q^{16} + 2 q^{17} - q^{19} + q^{22} - q^{25} + q^{32} - 2 q^{34} - 2 q^{38} - q^{41} - 2 q^{43} - q^{44} + 2 q^{49} - 2 q^{50} - q^{59} + 2 q^{64} + q^{67} - 4 q^{68} + q^{73} - q^{76} + q^{82} + 2 q^{83} + 2 q^{86} - 2 q^{88} + 2 q^{89} + q^{97} + q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 - 2 * q^8 + 2 * q^11 - q^16 + 2 * q^17 - q^19 + q^22 - q^25 + q^32 - 2 * q^34 - 2 * q^38 - q^41 - 2 * q^43 - q^44 + 2 * q^49 - 2 * q^50 - q^59 + 2 * q^64 + q^67 - 4 * q^68 + q^73 - q^76 + q^82 + 2 * q^83 + 2 * q^86 - 2 * q^88 + 2 * q^89 + q^97 + q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$$.

 $$n$$ $$343$$ $$685$$ $$1009$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{6}^{2}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 −1.00000 0 0
235.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 −1.00000 0 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 $$\Q(\sqrt{-2})$$
19.c even 3 1 inner
152.k odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.1.bz.a 2
3.b odd 2 1 152.1.k.a 2
8.d odd 2 1 CM 1368.1.bz.a 2
12.b even 2 1 608.1.o.a 2
15.d odd 2 1 3800.1.bd.c 2
15.e even 4 2 3800.1.bn.b 4
19.c even 3 1 inner 1368.1.bz.a 2
24.f even 2 1 152.1.k.a 2
24.h odd 2 1 608.1.o.a 2
57.d even 2 1 2888.1.k.a 2
57.f even 6 1 2888.1.f.a 1
57.f even 6 1 2888.1.k.a 2
57.h odd 6 1 152.1.k.a 2
57.h odd 6 1 2888.1.f.b 1
57.j even 18 6 2888.1.u.d 6
57.l odd 18 6 2888.1.u.c 6
120.m even 2 1 3800.1.bd.c 2
120.q odd 4 2 3800.1.bn.b 4
152.k odd 6 1 inner 1368.1.bz.a 2
228.m even 6 1 608.1.o.a 2
285.n odd 6 1 3800.1.bd.c 2
285.v even 12 2 3800.1.bn.b 4
456.l odd 2 1 2888.1.k.a 2
456.s odd 6 1 2888.1.f.a 1
456.s odd 6 1 2888.1.k.a 2
456.u even 6 1 152.1.k.a 2
456.u even 6 1 2888.1.f.b 1
456.x odd 6 1 608.1.o.a 2
456.bt odd 18 6 2888.1.u.d 6
456.bu even 18 6 2888.1.u.c 6
2280.co even 6 1 3800.1.bd.c 2
2280.dj odd 12 2 3800.1.bn.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.k.a 2 3.b odd 2 1
152.1.k.a 2 24.f even 2 1
152.1.k.a 2 57.h odd 6 1
152.1.k.a 2 456.u even 6 1
608.1.o.a 2 12.b even 2 1
608.1.o.a 2 24.h odd 2 1
608.1.o.a 2 228.m even 6 1
608.1.o.a 2 456.x odd 6 1
1368.1.bz.a 2 1.a even 1 1 trivial
1368.1.bz.a 2 8.d odd 2 1 CM
1368.1.bz.a 2 19.c even 3 1 inner
1368.1.bz.a 2 152.k odd 6 1 inner
2888.1.f.a 1 57.f even 6 1
2888.1.f.a 1 456.s odd 6 1
2888.1.f.b 1 57.h odd 6 1
2888.1.f.b 1 456.u even 6 1
2888.1.k.a 2 57.d even 2 1
2888.1.k.a 2 57.f even 6 1
2888.1.k.a 2 456.l odd 2 1
2888.1.k.a 2 456.s odd 6 1
2888.1.u.c 6 57.l odd 18 6
2888.1.u.c 6 456.bu even 18 6
2888.1.u.d 6 57.j even 18 6
2888.1.u.d 6 456.bt odd 18 6
3800.1.bd.c 2 15.d odd 2 1
3800.1.bd.c 2 120.m even 2 1
3800.1.bd.c 2 285.n odd 6 1
3800.1.bd.c 2 2280.co even 6 1
3800.1.bn.b 4 15.e even 4 2
3800.1.bn.b 4 120.q odd 4 2
3800.1.bn.b 4 285.v even 12 2
3800.1.bn.b 4 2280.dj odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1368, [\chi])$$.

## Hecke characteristic polynomials

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