# Properties

 Label 768.2.c.j Level $768$ Weight $2$ Character orbit 768.c Analytic conductor $6.133$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(767,768)]

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([1, 0, 1]))

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

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

chi := DirichletCharacter("768.767");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.c (of order $$2$$, degree $$1$$, not 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: $$6.13251087523$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 384) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} - \beta_1 q^{5} - \beta_{3} q^{7} + 3 q^{9}+O(q^{10})$$ q - b2 * q^3 - b1 * q^5 - b3 * q^7 + 3 * q^9 $$q - \beta_{2} q^{3} - \beta_1 q^{5} - \beta_{3} q^{7} + 3 q^{9} - 2 \beta_{2} q^{11} + \beta_{3} q^{15} + 3 \beta_1 q^{21} - 3 q^{25} - 3 \beta_{2} q^{27} + \beta_1 q^{29} + \beta_{3} q^{31} + 6 q^{33} - 8 \beta_{2} q^{35} - 3 \beta_1 q^{45} - 17 q^{49} - 5 \beta_1 q^{53} + 2 \beta_{3} q^{55} - 6 \beta_{2} q^{59} - 3 \beta_{3} q^{63} - 14 q^{73} + 3 \beta_{2} q^{75} + 6 \beta_1 q^{77} - 3 \beta_{3} q^{79} + 9 q^{81} + 10 \beta_{2} q^{83} - \beta_{3} q^{87} - 3 \beta_1 q^{93} + 2 q^{97} - 6 \beta_{2} q^{99}+O(q^{100})$$ q - b2 * q^3 - b1 * q^5 - b3 * q^7 + 3 * q^9 - 2*b2 * q^11 + b3 * q^15 + 3*b1 * q^21 - 3 * q^25 - 3*b2 * q^27 + b1 * q^29 + b3 * q^31 + 6 * q^33 - 8*b2 * q^35 - 3*b1 * q^45 - 17 * q^49 - 5*b1 * q^53 + 2*b3 * q^55 - 6*b2 * q^59 - 3*b3 * q^63 - 14 * q^73 + 3*b2 * q^75 + 6*b1 * q^77 - 3*b3 * q^79 + 9 * q^81 + 10*b2 * q^83 - b3 * q^87 - 3*b1 * q^93 + 2 * q^97 - 6*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{9}+O(q^{10})$$ 4 * q + 12 * q^9 $$4 q + 12 q^{9} - 12 q^{25} + 24 q^{33} - 68 q^{49} - 56 q^{73} + 36 q^{81} + 8 q^{97}+O(q^{100})$$ 4 * q + 12 * q^9 - 12 * q^25 + 24 * q^33 - 68 * q^49 - 56 * q^73 + 36 * q^81 + 8 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu^{3} + 6\nu$$ 2*v^3 + 6*v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$2\nu^{3} + 10\nu$$ 2*v^3 + 10*v
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_1 ) / 4$$ (b3 - b1) / 4 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$( -3\beta_{3} + 5\beta_1 ) / 4$$ (-3*b3 + 5*b1) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
767.1
 0.517638i − 0.517638i − 1.93185i 1.93185i
0 −1.73205 0 2.82843i 0 4.89898i 0 3.00000 0
767.2 0 −1.73205 0 2.82843i 0 4.89898i 0 3.00000 0
767.3 0 1.73205 0 2.82843i 0 4.89898i 0 3.00000 0
767.4 0 1.73205 0 2.82843i 0 4.89898i 0 3.00000 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.c.j 4
3.b odd 2 1 inner 768.2.c.j 4
4.b odd 2 1 inner 768.2.c.j 4
8.b even 2 1 inner 768.2.c.j 4
8.d odd 2 1 inner 768.2.c.j 4
12.b even 2 1 inner 768.2.c.j 4
16.e even 4 2 384.2.f.a 4
16.f odd 4 2 384.2.f.a 4
24.f even 2 1 inner 768.2.c.j 4
24.h odd 2 1 CM 768.2.c.j 4
48.i odd 4 2 384.2.f.a 4
48.k even 4 2 384.2.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.f.a 4 16.e even 4 2
384.2.f.a 4 16.f odd 4 2
384.2.f.a 4 48.i odd 4 2
384.2.f.a 4 48.k even 4 2
768.2.c.j 4 1.a even 1 1 trivial
768.2.c.j 4 3.b odd 2 1 inner
768.2.c.j 4 4.b odd 2 1 inner
768.2.c.j 4 8.b even 2 1 inner
768.2.c.j 4 8.d odd 2 1 inner
768.2.c.j 4 12.b even 2 1 inner
768.2.c.j 4 24.f even 2 1 inner
768.2.c.j 4 24.h odd 2 1 CM

## Hecke kernels

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

 $$T_{5}^{2} + 8$$ T5^2 + 8 $$T_{7}^{2} + 24$$ T7^2 + 24 $$T_{11}^{2} - 12$$ T11^2 - 12 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$(T^{2} + 8)^{2}$$
$7$ $$(T^{2} + 24)^{2}$$
$11$ $$(T^{2} - 12)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$(T^{2} + 8)^{2}$$
$31$ $$(T^{2} + 24)^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 200)^{2}$$
$59$ $$(T^{2} - 108)^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$(T + 14)^{4}$$
$79$ $$(T^{2} + 216)^{2}$$
$83$ $$(T^{2} - 300)^{2}$$
$89$ $$T^{4}$$
$97$ $$(T - 2)^{4}$$