# Properties

 Label 1368.2.s.f Level $1368$ Weight $2$ Character orbit 1368.s Analytic conductor $10.924$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(505,1368)]

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([0, 0, 0, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1368.505");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1368.s (of order $$3$$, 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: $$10.9235349965$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{5} + 3 q^{7}+O(q^{10})$$ q + (-2*z + 2) * q^5 + 3 * q^7 $$q + ( - 2 \zeta_{6} + 2) q^{5} + 3 q^{7} + 6 q^{11} + \zeta_{6} q^{13} + ( - 2 \zeta_{6} + 2) q^{17} + (2 \zeta_{6} - 5) q^{19} + \zeta_{6} q^{25} + 2 \zeta_{6} q^{29} - q^{31} + ( - 6 \zeta_{6} + 6) q^{35} - 7 q^{37} + ( - \zeta_{6} + 1) q^{43} + 2 q^{49} + 4 \zeta_{6} q^{53} + ( - 12 \zeta_{6} + 12) q^{55} + ( - 8 \zeta_{6} + 8) q^{59} + 11 \zeta_{6} q^{61} + 2 q^{65} - 15 \zeta_{6} q^{67} + ( - 6 \zeta_{6} + 6) q^{71} + (9 \zeta_{6} - 9) q^{73} + 18 q^{77} + ( - 13 \zeta_{6} + 13) q^{79} + 14 q^{83} - 4 \zeta_{6} q^{85} - 12 \zeta_{6} q^{89} + 3 \zeta_{6} q^{91} + (10 \zeta_{6} - 6) q^{95} + (10 \zeta_{6} - 10) q^{97} +O(q^{100})$$ q + (-2*z + 2) * q^5 + 3 * q^7 + 6 * q^11 + z * q^13 + (-2*z + 2) * q^17 + (2*z - 5) * q^19 + z * q^25 + 2*z * q^29 - q^31 + (-6*z + 6) * q^35 - 7 * q^37 + (-z + 1) * q^43 + 2 * q^49 + 4*z * q^53 + (-12*z + 12) * q^55 + (-8*z + 8) * q^59 + 11*z * q^61 + 2 * q^65 - 15*z * q^67 + (-6*z + 6) * q^71 + (9*z - 9) * q^73 + 18 * q^77 + (-13*z + 13) * q^79 + 14 * q^83 - 4*z * q^85 - 12*z * q^89 + 3*z * q^91 + (10*z - 6) * q^95 + (10*z - 10) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 6 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + 6 * q^7 $$2 q + 2 q^{5} + 6 q^{7} + 12 q^{11} + q^{13} + 2 q^{17} - 8 q^{19} + q^{25} + 2 q^{29} - 2 q^{31} + 6 q^{35} - 14 q^{37} + q^{43} + 4 q^{49} + 4 q^{53} + 12 q^{55} + 8 q^{59} + 11 q^{61} + 4 q^{65} - 15 q^{67} + 6 q^{71} - 9 q^{73} + 36 q^{77} + 13 q^{79} + 28 q^{83} - 4 q^{85} - 12 q^{89} + 3 q^{91} - 2 q^{95} - 10 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + 6 * q^7 + 12 * q^11 + q^13 + 2 * q^17 - 8 * q^19 + q^25 + 2 * q^29 - 2 * q^31 + 6 * q^35 - 14 * q^37 + q^43 + 4 * q^49 + 4 * q^53 + 12 * q^55 + 8 * q^59 + 11 * q^61 + 4 * q^65 - 15 * q^67 + 6 * q^71 - 9 * q^73 + 36 * q^77 + 13 * q^79 + 28 * q^83 - 4 * q^85 - 12 * q^89 + 3 * q^91 - 2 * q^95 - 10 * q^97

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
505.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 1.00000 1.73205i 0 3.00000 0 0 0
577.1 0 0 0 1.00000 + 1.73205i 0 3.00000 0 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.2.s.f yes 2
3.b odd 2 1 1368.2.s.c 2
4.b odd 2 1 2736.2.s.n 2
12.b even 2 1 2736.2.s.e 2
19.c even 3 1 inner 1368.2.s.f yes 2
57.h odd 6 1 1368.2.s.c 2
76.g odd 6 1 2736.2.s.n 2
228.m even 6 1 2736.2.s.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.s.c 2 3.b odd 2 1
1368.2.s.c 2 57.h odd 6 1
1368.2.s.f yes 2 1.a even 1 1 trivial
1368.2.s.f yes 2 19.c even 3 1 inner
2736.2.s.e 2 12.b even 2 1
2736.2.s.e 2 228.m even 6 1
2736.2.s.n 2 4.b odd 2 1
2736.2.s.n 2 76.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1368, [\chi])$$:

 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4 $$T_{7} - 3$$ T7 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$(T - 3)^{2}$$
$11$ $$(T - 6)^{2}$$
$13$ $$T^{2} - T + 1$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 2T + 4$$
$31$ $$(T + 1)^{2}$$
$37$ $$(T + 7)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 4T + 16$$
$59$ $$T^{2} - 8T + 64$$
$61$ $$T^{2} - 11T + 121$$
$67$ $$T^{2} + 15T + 225$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$T^{2} + 9T + 81$$
$79$ $$T^{2} - 13T + 169$$
$83$ $$(T - 14)^{2}$$
$89$ $$T^{2} + 12T + 144$$
$97$ $$T^{2} + 10T + 100$$