# Properties

 Label 768.1.i.a Level $768$ Weight $1$ Character orbit 768.i Analytic conductor $0.383$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -3 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,1,Mod(65,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("768.65");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 768.i (of order $$4$$, 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: $$0.383281929702$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.18432.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{3} + (\zeta_{8}^{3} + \zeta_{8}) q^{7} + \zeta_{8}^{2} q^{9}+O(q^{10})$$ q + z * q^3 + (z^3 + z) * q^7 + z^2 * q^9 $$q + \zeta_{8} q^{3} + (\zeta_{8}^{3} + \zeta_{8}) q^{7} + \zeta_{8}^{2} q^{9} + ( - \zeta_{8}^{2} - 1) q^{13} + (\zeta_{8}^{2} - 1) q^{21} - \zeta_{8}^{2} q^{25} + \zeta_{8}^{3} q^{27} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{31} + ( - \zeta_{8}^{2} + 1) q^{37} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{39} - q^{49} + (\zeta_{8}^{2} + 1) q^{61} + (\zeta_{8}^{3} - \zeta_{8}) q^{63} - \zeta_{8} q^{67} - \zeta_{8}^{3} q^{75} + (\zeta_{8}^{3} - \zeta_{8}) q^{79} - q^{81} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{91} + (\zeta_{8}^{2} + 1) q^{93} +O(q^{100})$$ q + z * q^3 + (z^3 + z) * q^7 + z^2 * q^9 + (-z^2 - 1) * q^13 + (z^2 - 1) * q^21 - z^2 * q^25 + z^3 * q^27 + (-z^3 + z) * q^31 + (-z^2 + 1) * q^37 + (-z^3 - z) * q^39 - q^49 + (z^2 + 1) * q^61 + (z^3 - z) * q^63 - z * q^67 - z^3 * q^75 + (z^3 - z) * q^79 - q^81 + (-2*z^3 + z) * q^91 + (z^2 + 1) * q^93 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 4 q^{13} - 4 q^{21} + 4 q^{37} - 4 q^{49} + 4 q^{61} - 4 q^{81} + 4 q^{93}+O(q^{100})$$ 4 * q - 4 * q^13 - 4 * q^21 + 4 * q^37 - 4 * q^49 + 4 * q^61 - 4 * q^81 + 4 * q^93

## Character values

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

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 −0.707107 0.707107i 0 0 0 1.41421i 0 1.00000i 0
65.2 0 0.707107 + 0.707107i 0 0 0 1.41421i 0 1.00000i 0
449.1 0 −0.707107 + 0.707107i 0 0 0 1.41421i 0 1.00000i 0
449.2 0 0.707107 0.707107i 0 0 0 1.41421i 0 1.00000i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
12.b even 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner
48.i odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.1.i.a 4
3.b odd 2 1 CM 768.1.i.a 4
4.b odd 2 1 inner 768.1.i.a 4
8.b even 2 1 768.1.i.b yes 4
8.d odd 2 1 768.1.i.b yes 4
12.b even 2 1 inner 768.1.i.a 4
16.e even 4 1 inner 768.1.i.a 4
16.e even 4 1 768.1.i.b yes 4
16.f odd 4 1 inner 768.1.i.a 4
16.f odd 4 1 768.1.i.b yes 4
24.f even 2 1 768.1.i.b yes 4
24.h odd 2 1 768.1.i.b yes 4
32.g even 8 1 3072.1.e.a 2
32.g even 8 1 3072.1.e.b 2
32.g even 8 2 3072.1.h.a 4
32.h odd 8 1 3072.1.e.a 2
32.h odd 8 1 3072.1.e.b 2
32.h odd 8 2 3072.1.h.a 4
48.i odd 4 1 inner 768.1.i.a 4
48.i odd 4 1 768.1.i.b yes 4
48.k even 4 1 inner 768.1.i.a 4
48.k even 4 1 768.1.i.b yes 4
96.o even 8 1 3072.1.e.a 2
96.o even 8 1 3072.1.e.b 2
96.o even 8 2 3072.1.h.a 4
96.p odd 8 1 3072.1.e.a 2
96.p odd 8 1 3072.1.e.b 2
96.p odd 8 2 3072.1.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.1.i.a 4 1.a even 1 1 trivial
768.1.i.a 4 3.b odd 2 1 CM
768.1.i.a 4 4.b odd 2 1 inner
768.1.i.a 4 12.b even 2 1 inner
768.1.i.a 4 16.e even 4 1 inner
768.1.i.a 4 16.f odd 4 1 inner
768.1.i.a 4 48.i odd 4 1 inner
768.1.i.a 4 48.k even 4 1 inner
768.1.i.b yes 4 8.b even 2 1
768.1.i.b yes 4 8.d odd 2 1
768.1.i.b yes 4 16.e even 4 1
768.1.i.b yes 4 16.f odd 4 1
768.1.i.b yes 4 24.f even 2 1
768.1.i.b yes 4 24.h odd 2 1
768.1.i.b yes 4 48.i odd 4 1
768.1.i.b yes 4 48.k even 4 1
3072.1.e.a 2 32.g even 8 1
3072.1.e.a 2 32.h odd 8 1
3072.1.e.a 2 96.o even 8 1
3072.1.e.a 2 96.p odd 8 1
3072.1.e.b 2 32.g even 8 1
3072.1.e.b 2 32.h odd 8 1
3072.1.e.b 2 96.o even 8 1
3072.1.e.b 2 96.p odd 8 1
3072.1.h.a 4 32.g even 8 2
3072.1.h.a 4 32.h odd 8 2
3072.1.h.a 4 96.o even 8 2
3072.1.h.a 4 96.p odd 8 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{2} + 2T_{13} + 2$$ acting on $$S_{1}^{\mathrm{new}}(768, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 1$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 2)^{2}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 2 T + 2)^{2}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} - 2)^{2}$$
$37$ $$(T^{2} - 2 T + 2)^{2}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 2 T + 2)^{2}$$
$67$ $$T^{4} + 16$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} - 2)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$