# Properties

 Label 768.2.c.h Level $768$ Weight $2$ Character orbit 768.c Analytic conductor $6.133$ Analytic rank $0$ Dimension $4$ CM discriminant -8 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: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 24) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + (\beta_{3} + 1) q^{9}+O(q^{10})$$ q + b2 * q^3 + (b3 + 1) * q^9 $$q + \beta_{2} q^{3} + (\beta_{3} + 1) q^{9} + (\beta_{2} + \beta_1) q^{11} + 2 \beta_{3} q^{17} + ( - \beta_{2} + \beta_1) q^{19} + 5 q^{25} + (2 \beta_{2} - 3 \beta_1) q^{27} + (\beta_{3} + 4) q^{33} + 4 \beta_{3} q^{41} + ( - 5 \beta_{2} + 5 \beta_1) q^{43} + 7 q^{49} + (2 \beta_{2} - 6 \beta_1) q^{51} + ( - \beta_{3} + 2) q^{57} + ( - 5 \beta_{2} - 5 \beta_1) q^{59} + ( - 7 \beta_{2} + 7 \beta_1) q^{67} - 2 q^{73} + 5 \beta_{2} q^{75} + (2 \beta_{3} - 7) q^{81} + ( - \beta_{2} - \beta_1) q^{83} - 2 \beta_{3} q^{89} - 10 q^{97} + (5 \beta_{2} - 3 \beta_1) q^{99}+O(q^{100})$$ q + b2 * q^3 + (b3 + 1) * q^9 + (b2 + b1) * q^11 + 2*b3 * q^17 + (-b2 + b1) * q^19 + 5 * q^25 + (2*b2 - 3*b1) * q^27 + (b3 + 4) * q^33 + 4*b3 * q^41 + (-5*b2 + 5*b1) * q^43 + 7 * q^49 + (2*b2 - 6*b1) * q^51 + (-b3 + 2) * q^57 + (-5*b2 - 5*b1) * q^59 + (-7*b2 + 7*b1) * q^67 - 2 * q^73 + 5*b2 * q^75 + (2*b3 - 7) * q^81 + (-b2 - b1) * q^83 - 2*b3 * q^89 - 10 * q^97 + (5*b2 - 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^9 $$4 q + 4 q^{9} + 20 q^{25} + 16 q^{33} + 28 q^{49} + 8 q^{57} - 8 q^{73} - 28 q^{81} - 40 q^{97}+O(q^{100})$$ 4 * q + 4 * q^9 + 20 * q^25 + 16 * q^33 + 28 * q^49 + 8 * q^57 - 8 * q^73 - 28 * q^81 - 40 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8}$$ -v^3 - v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ -v^3 + v^2 + v $$\beta_{3}$$ $$=$$ $$2\zeta_{8}^{3} + 2\zeta_{8}$$ 2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta_1 ) / 4$$ (b3 + b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( \beta_{3} - \beta_{2} - \beta_1 ) / 4$$ (b3 - b2 - 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
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
0 −1.41421 1.00000i 0 0 0 0 0 1.00000 + 2.82843i 0
767.2 0 −1.41421 + 1.00000i 0 0 0 0 0 1.00000 2.82843i 0
767.3 0 1.41421 1.00000i 0 0 0 0 0 1.00000 2.82843i 0
767.4 0 1.41421 + 1.00000i 0 0 0 0 0 1.00000 + 2.82843i 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 $$\Q(\sqrt{-2})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h 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.c.h 4
3.b odd 2 1 inner 768.2.c.h 4
4.b odd 2 1 inner 768.2.c.h 4
8.b even 2 1 inner 768.2.c.h 4
8.d odd 2 1 CM 768.2.c.h 4
12.b even 2 1 inner 768.2.c.h 4
16.e even 4 1 24.2.f.a 2
16.e even 4 1 96.2.f.a 2
16.f odd 4 1 24.2.f.a 2
16.f odd 4 1 96.2.f.a 2
24.f even 2 1 inner 768.2.c.h 4
24.h odd 2 1 inner 768.2.c.h 4
48.i odd 4 1 24.2.f.a 2
48.i odd 4 1 96.2.f.a 2
48.k even 4 1 24.2.f.a 2
48.k even 4 1 96.2.f.a 2
80.i odd 4 1 600.2.m.a 4
80.i odd 4 1 2400.2.m.a 4
80.j even 4 1 600.2.m.a 4
80.j even 4 1 2400.2.m.a 4
80.k odd 4 1 600.2.b.a 2
80.k odd 4 1 2400.2.b.a 2
80.q even 4 1 600.2.b.a 2
80.q even 4 1 2400.2.b.a 2
80.s even 4 1 600.2.m.a 4
80.s even 4 1 2400.2.m.a 4
80.t odd 4 1 600.2.m.a 4
80.t odd 4 1 2400.2.m.a 4
144.u even 12 2 648.2.l.b 4
144.u even 12 2 2592.2.p.b 4
144.v odd 12 2 648.2.l.b 4
144.v odd 12 2 2592.2.p.b 4
144.w odd 12 2 648.2.l.b 4
144.w odd 12 2 2592.2.p.b 4
144.x even 12 2 648.2.l.b 4
144.x even 12 2 2592.2.p.b 4
240.t even 4 1 600.2.b.a 2
240.t even 4 1 2400.2.b.a 2
240.z odd 4 1 600.2.m.a 4
240.z odd 4 1 2400.2.m.a 4
240.bb even 4 1 600.2.m.a 4
240.bb even 4 1 2400.2.m.a 4
240.bd odd 4 1 600.2.m.a 4
240.bd odd 4 1 2400.2.m.a 4
240.bf even 4 1 600.2.m.a 4
240.bf even 4 1 2400.2.m.a 4
240.bm odd 4 1 600.2.b.a 2
240.bm odd 4 1 2400.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.f.a 2 16.e even 4 1
24.2.f.a 2 16.f odd 4 1
24.2.f.a 2 48.i odd 4 1
24.2.f.a 2 48.k even 4 1
96.2.f.a 2 16.e even 4 1
96.2.f.a 2 16.f odd 4 1
96.2.f.a 2 48.i odd 4 1
96.2.f.a 2 48.k even 4 1
600.2.b.a 2 80.k odd 4 1
600.2.b.a 2 80.q even 4 1
600.2.b.a 2 240.t even 4 1
600.2.b.a 2 240.bm odd 4 1
600.2.m.a 4 80.i odd 4 1
600.2.m.a 4 80.j even 4 1
600.2.m.a 4 80.s even 4 1
600.2.m.a 4 80.t odd 4 1
600.2.m.a 4 240.z odd 4 1
600.2.m.a 4 240.bb even 4 1
600.2.m.a 4 240.bd odd 4 1
600.2.m.a 4 240.bf even 4 1
648.2.l.b 4 144.u even 12 2
648.2.l.b 4 144.v odd 12 2
648.2.l.b 4 144.w odd 12 2
648.2.l.b 4 144.x even 12 2
768.2.c.h 4 1.a even 1 1 trivial
768.2.c.h 4 3.b odd 2 1 inner
768.2.c.h 4 4.b odd 2 1 inner
768.2.c.h 4 8.b even 2 1 inner
768.2.c.h 4 8.d odd 2 1 CM
768.2.c.h 4 12.b even 2 1 inner
768.2.c.h 4 24.f even 2 1 inner
768.2.c.h 4 24.h odd 2 1 inner
2400.2.b.a 2 80.k odd 4 1
2400.2.b.a 2 80.q even 4 1
2400.2.b.a 2 240.t even 4 1
2400.2.b.a 2 240.bm odd 4 1
2400.2.m.a 4 80.i odd 4 1
2400.2.m.a 4 80.j even 4 1
2400.2.m.a 4 80.s even 4 1
2400.2.m.a 4 80.t odd 4 1
2400.2.m.a 4 240.z odd 4 1
2400.2.m.a 4 240.bb even 4 1
2400.2.m.a 4 240.bd odd 4 1
2400.2.m.a 4 240.bf even 4 1
2592.2.p.b 4 144.u even 12 2
2592.2.p.b 4 144.v odd 12 2
2592.2.p.b 4 144.w odd 12 2
2592.2.p.b 4 144.x even 12 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 8)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 32)^{2}$$
$19$ $$(T^{2} + 4)^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 128)^{2}$$
$43$ $$(T^{2} + 100)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} - 200)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T^{2} + 196)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T + 2)^{4}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} - 8)^{2}$$
$89$ $$(T^{2} + 32)^{2}$$
$97$ $$(T + 10)^{4}$$