# Properties

 Label 768.2.a.k Level $768$ Weight $2$ Character orbit 768.a Self dual yes Analytic conductor $6.133$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.a (trivial)

## 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: yes Analytic conductor: $$6.13251087523$$ 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: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 192) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta q^{5} - \beta q^{7} + q^{9} +O(q^{10})$$ q + q^3 + b * q^5 - b * q^7 + q^9 $$q + q^{3} + \beta q^{5} - \beta q^{7} + q^{9} + \beta q^{15} + 6 q^{17} + 4 q^{19} - \beta q^{21} + 2 \beta q^{23} + 7 q^{25} + q^{27} - \beta q^{29} - \beta q^{31} - 12 q^{35} - 2 \beta q^{37} + 6 q^{41} + 4 q^{43} + \beta q^{45} - 2 \beta q^{47} + 5 q^{49} + 6 q^{51} - \beta q^{53} + 4 q^{57} - 12 q^{59} + 2 \beta q^{61} - \beta q^{63} - 4 q^{67} + 2 \beta q^{69} - 2 \beta q^{71} - 2 q^{73} + 7 q^{75} + 3 \beta q^{79} + q^{81} + 6 \beta q^{85} - \beta q^{87} - 6 q^{89} - \beta q^{93} + 4 \beta q^{95} - 2 q^{97} +O(q^{100})$$ q + q^3 + b * q^5 - b * q^7 + q^9 + b * q^15 + 6 * q^17 + 4 * q^19 - b * q^21 + 2*b * q^23 + 7 * q^25 + q^27 - b * q^29 - b * q^31 - 12 * q^35 - 2*b * q^37 + 6 * q^41 + 4 * q^43 + b * q^45 - 2*b * q^47 + 5 * q^49 + 6 * q^51 - b * q^53 + 4 * q^57 - 12 * q^59 + 2*b * q^61 - b * q^63 - 4 * q^67 + 2*b * q^69 - 2*b * q^71 - 2 * q^73 + 7 * q^75 + 3*b * q^79 + q^81 + 6*b * q^85 - b * q^87 - 6 * q^89 - b * q^93 + 4*b * q^95 - 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{9} + 12 q^{17} + 8 q^{19} + 14 q^{25} + 2 q^{27} - 24 q^{35} + 12 q^{41} + 8 q^{43} + 10 q^{49} + 12 q^{51} + 8 q^{57} - 24 q^{59} - 8 q^{67} - 4 q^{73} + 14 q^{75} + 2 q^{81} - 12 q^{89} - 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^9 + 12 * q^17 + 8 * q^19 + 14 * q^25 + 2 * q^27 - 24 * q^35 + 12 * q^41 + 8 * q^43 + 10 * q^49 + 12 * q^51 + 8 * q^57 - 24 * q^59 - 8 * q^67 - 4 * q^73 + 14 * q^75 + 2 * q^81 - 12 * q^89 - 4 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
0 1.00000 0 −3.46410 0 3.46410 0 1.00000 0
1.2 0 1.00000 0 3.46410 0 −3.46410 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.a.k 2
3.b odd 2 1 2304.2.a.u 2
4.b odd 2 1 768.2.a.j 2
8.b even 2 1 768.2.a.j 2
8.d odd 2 1 inner 768.2.a.k 2
12.b even 2 1 2304.2.a.s 2
16.e even 4 2 192.2.d.a 4
16.f odd 4 2 192.2.d.a 4
24.f even 2 1 2304.2.a.u 2
24.h odd 2 1 2304.2.a.s 2
48.i odd 4 2 576.2.d.b 4
48.k even 4 2 576.2.d.b 4
80.i odd 4 2 4800.2.d.j 4
80.j even 4 2 4800.2.d.j 4
80.k odd 4 2 4800.2.k.j 4
80.q even 4 2 4800.2.k.j 4
80.s even 4 2 4800.2.d.o 4
80.t odd 4 2 4800.2.d.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.2.d.a 4 16.e even 4 2
192.2.d.a 4 16.f odd 4 2
576.2.d.b 4 48.i odd 4 2
576.2.d.b 4 48.k even 4 2
768.2.a.j 2 4.b odd 2 1
768.2.a.j 2 8.b even 2 1
768.2.a.k 2 1.a even 1 1 trivial
768.2.a.k 2 8.d odd 2 1 inner
2304.2.a.s 2 12.b even 2 1
2304.2.a.s 2 24.h odd 2 1
2304.2.a.u 2 3.b odd 2 1
2304.2.a.u 2 24.f even 2 1
4800.2.d.j 4 80.i odd 4 2
4800.2.d.j 4 80.j even 4 2
4800.2.d.o 4 80.s even 4 2
4800.2.d.o 4 80.t odd 4 2
4800.2.k.j 4 80.k odd 4 2
4800.2.k.j 4 80.q even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(768))$$:

 $$T_{5}^{2} - 12$$ T5^2 - 12 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{11}$$ T11 $$T_{19} - 4$$ T19 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - 12$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$(T - 6)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} - 48$$
$29$ $$T^{2} - 12$$
$31$ $$T^{2} - 12$$
$37$ $$T^{2} - 48$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} - 48$$
$53$ $$T^{2} - 12$$
$59$ $$(T + 12)^{2}$$
$61$ $$T^{2} - 48$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} - 48$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2} - 108$$
$83$ $$T^{2}$$
$89$ $$(T + 6)^{2}$$
$97$ $$(T + 2)^{2}$$