# Properties

 Label 768.3.h.a Level $768$ Weight $3$ Character orbit 768.h Analytic conductor $20.926$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(641,768)]

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, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("768.641");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 768.h (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: $$20.9264843029$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} - 2 q^{7} - 9 q^{9}+O(q^{10})$$ q + 3*i * q^3 - 2 * q^7 - 9 * q^9 $$q + 3 i q^{3} - 2 q^{7} - 9 q^{9} + 22 i q^{13} - 26 i q^{19} - 6 i q^{21} - 25 q^{25} - 27 i q^{27} - 46 q^{31} + 26 i q^{37} - 66 q^{39} - 22 i q^{43} - 45 q^{49} + 78 q^{57} - 74 i q^{61} + 18 q^{63} - 122 i q^{67} + 46 q^{73} - 75 i q^{75} - 142 q^{79} + 81 q^{81} - 44 i q^{91} - 138 i q^{93} + 2 q^{97} +O(q^{100})$$ q + 3*i * q^3 - 2 * q^7 - 9 * q^9 + 22*i * q^13 - 26*i * q^19 - 6*i * q^21 - 25 * q^25 - 27*i * q^27 - 46 * q^31 + 26*i * q^37 - 66 * q^39 - 22*i * q^43 - 45 * q^49 + 78 * q^57 - 74*i * q^61 + 18 * q^63 - 122*i * q^67 + 46 * q^73 - 75*i * q^75 - 142 * q^79 + 81 * q^81 - 44*i * q^91 - 138*i * q^93 + 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7} - 18 q^{9}+O(q^{10})$$ 2 * q - 4 * q^7 - 18 * q^9 $$2 q - 4 q^{7} - 18 q^{9} - 50 q^{25} - 92 q^{31} - 132 q^{39} - 90 q^{49} + 156 q^{57} + 36 q^{63} + 92 q^{73} - 284 q^{79} + 162 q^{81} + 4 q^{97}+O(q^{100})$$ 2 * q - 4 * q^7 - 18 * q^9 - 50 * q^25 - 92 * q^31 - 132 * q^39 - 90 * q^49 + 156 * q^57 + 36 * q^63 + 92 * q^73 - 284 * q^79 + 162 * q^81 + 4 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
641.1
 − 1.00000i 1.00000i
0 3.00000i 0 0 0 −2.00000 0 −9.00000 0
641.2 0 3.00000i 0 0 0 −2.00000 0 −9.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
8.b 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 768.3.h.a 2
3.b odd 2 1 CM 768.3.h.a 2
4.b odd 2 1 768.3.h.b 2
8.b even 2 1 inner 768.3.h.a 2
8.d odd 2 1 768.3.h.b 2
12.b even 2 1 768.3.h.b 2
16.e even 4 1 12.3.c.a 1
16.e even 4 1 192.3.e.b 1
16.f odd 4 1 48.3.e.a 1
16.f odd 4 1 192.3.e.a 1
24.f even 2 1 768.3.h.b 2
24.h odd 2 1 inner 768.3.h.a 2
48.i odd 4 1 12.3.c.a 1
48.i odd 4 1 192.3.e.b 1
48.k even 4 1 48.3.e.a 1
48.k even 4 1 192.3.e.a 1
80.i odd 4 1 300.3.b.a 2
80.j even 4 1 1200.3.c.c 2
80.k odd 4 1 1200.3.l.b 1
80.q even 4 1 300.3.g.b 1
80.s even 4 1 1200.3.c.c 2
80.t odd 4 1 300.3.b.a 2
112.l odd 4 1 588.3.c.c 1
112.w even 12 2 588.3.p.c 2
112.x odd 12 2 588.3.p.b 2
144.u even 12 2 1296.3.q.b 2
144.v odd 12 2 1296.3.q.b 2
144.w odd 12 2 324.3.g.b 2
144.x even 12 2 324.3.g.b 2
176.l odd 4 1 1452.3.e.b 1
240.t even 4 1 1200.3.l.b 1
240.z odd 4 1 1200.3.c.c 2
240.bb even 4 1 300.3.b.a 2
240.bd odd 4 1 1200.3.c.c 2
240.bf even 4 1 300.3.b.a 2
240.bm odd 4 1 300.3.g.b 1
336.y even 4 1 588.3.c.c 1
336.bo even 12 2 588.3.p.b 2
336.bt odd 12 2 588.3.p.c 2
528.x even 4 1 1452.3.e.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.c.a 1 16.e even 4 1
12.3.c.a 1 48.i odd 4 1
48.3.e.a 1 16.f odd 4 1
48.3.e.a 1 48.k even 4 1
192.3.e.a 1 16.f odd 4 1
192.3.e.a 1 48.k even 4 1
192.3.e.b 1 16.e even 4 1
192.3.e.b 1 48.i odd 4 1
300.3.b.a 2 80.i odd 4 1
300.3.b.a 2 80.t odd 4 1
300.3.b.a 2 240.bb even 4 1
300.3.b.a 2 240.bf even 4 1
300.3.g.b 1 80.q even 4 1
300.3.g.b 1 240.bm odd 4 1
324.3.g.b 2 144.w odd 12 2
324.3.g.b 2 144.x even 12 2
588.3.c.c 1 112.l odd 4 1
588.3.c.c 1 336.y even 4 1
588.3.p.b 2 112.x odd 12 2
588.3.p.b 2 336.bo even 12 2
588.3.p.c 2 112.w even 12 2
588.3.p.c 2 336.bt odd 12 2
768.3.h.a 2 1.a even 1 1 trivial
768.3.h.a 2 3.b odd 2 1 CM
768.3.h.a 2 8.b even 2 1 inner
768.3.h.a 2 24.h odd 2 1 inner
768.3.h.b 2 4.b odd 2 1
768.3.h.b 2 8.d odd 2 1
768.3.h.b 2 12.b even 2 1
768.3.h.b 2 24.f even 2 1
1200.3.c.c 2 80.j even 4 1
1200.3.c.c 2 80.s even 4 1
1200.3.c.c 2 240.z odd 4 1
1200.3.c.c 2 240.bd odd 4 1
1200.3.l.b 1 80.k odd 4 1
1200.3.l.b 1 240.t even 4 1
1296.3.q.b 2 144.u even 12 2
1296.3.q.b 2 144.v odd 12 2
1452.3.e.b 1 176.l odd 4 1
1452.3.e.b 1 528.x even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 484$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 676$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 46)^{2}$$
$37$ $$T^{2} + 676$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 484$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 5476$$
$67$ $$T^{2} + 14884$$
$71$ $$T^{2}$$
$73$ $$(T - 46)^{2}$$
$79$ $$(T + 142)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 2)^{2}$$