# Properties

 Label 192.2.d.a Level $192$ Weight $2$ Character orbit 192.d Analytic conductor $1.533$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,2,Mod(97,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("192.97");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 192.d (of order $$2$$, degree $$1$$, 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: $$1.53312771881$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{3} q^{7} - q^{9}+O(q^{10})$$ q + b1 * q^3 - b2 * q^5 + b3 * q^7 - q^9 $$q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{3} q^{7} - q^{9} + \beta_{3} q^{15} + 6 q^{17} + 4 \beta_1 q^{19} + \beta_{2} q^{21} - 2 \beta_{3} q^{23} - 7 q^{25} - \beta_1 q^{27} - \beta_{2} q^{29} - \beta_{3} q^{31} - 12 \beta_1 q^{35} + 2 \beta_{2} q^{37} - 6 q^{41} - 4 \beta_1 q^{43} + \beta_{2} q^{45} - 2 \beta_{3} q^{47} + 5 q^{49} + 6 \beta_1 q^{51} + \beta_{2} q^{53} - 4 q^{57} + 12 \beta_1 q^{59} + 2 \beta_{2} q^{61} - \beta_{3} q^{63} - 4 \beta_1 q^{67} - 2 \beta_{2} q^{69} + 2 \beta_{3} q^{71} + 2 q^{73} - 7 \beta_1 q^{75} + 3 \beta_{3} q^{79} + q^{81} - 6 \beta_{2} q^{85} + \beta_{3} q^{87} + 6 q^{89} - \beta_{2} q^{93} + 4 \beta_{3} q^{95} - 2 q^{97}+O(q^{100})$$ q + b1 * q^3 - b2 * q^5 + b3 * q^7 - q^9 + b3 * q^15 + 6 * q^17 + 4*b1 * q^19 + b2 * q^21 - 2*b3 * q^23 - 7 * q^25 - b1 * q^27 - b2 * q^29 - b3 * q^31 - 12*b1 * q^35 + 2*b2 * q^37 - 6 * q^41 - 4*b1 * q^43 + b2 * q^45 - 2*b3 * q^47 + 5 * q^49 + 6*b1 * q^51 + b2 * q^53 - 4 * q^57 + 12*b1 * q^59 + 2*b2 * q^61 - b3 * q^63 - 4*b1 * q^67 - 2*b2 * q^69 + 2*b3 * q^71 + 2 * q^73 - 7*b1 * q^75 + 3*b3 * q^79 + q^81 - 6*b2 * q^85 + b3 * q^87 + 6 * q^89 - b2 * q^93 + 4*b3 * q^95 - 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} + 24 q^{17} - 28 q^{25} - 24 q^{41} + 20 q^{49} - 16 q^{57} + 8 q^{73} + 4 q^{81} + 24 q^{89} - 8 q^{97}+O(q^{100})$$ 4 * q - 4 * q^9 + 24 * q^17 - 28 * q^25 - 24 * q^41 + 20 * q^49 - 16 * q^57 + 8 * q^73 + 4 * q^81 + 24 * q^89 - 8 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$4\zeta_{12}^{2} - 2$$ 4*v^2 - 2 $$\beta_{3}$$ $$=$$ $$-2\zeta_{12}^{3} + 4\zeta_{12}$$ -2*v^3 + 4*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + 2\beta_1 ) / 4$$ (b3 + 2*b1) / 4 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 2 ) / 4$$ (b2 + 2) / 4 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
0 1.00000i 0 3.46410i 0 −3.46410 0 −1.00000 0
97.2 0 1.00000i 0 3.46410i 0 3.46410 0 −1.00000 0
97.3 0 1.00000i 0 3.46410i 0 3.46410 0 −1.00000 0
97.4 0 1.00000i 0 3.46410i 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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
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 192.2.d.a 4
3.b odd 2 1 576.2.d.b 4
4.b odd 2 1 inner 192.2.d.a 4
5.b even 2 1 4800.2.k.j 4
5.c odd 4 1 4800.2.d.j 4
5.c odd 4 1 4800.2.d.o 4
8.b even 2 1 inner 192.2.d.a 4
8.d odd 2 1 inner 192.2.d.a 4
12.b even 2 1 576.2.d.b 4
16.e even 4 1 768.2.a.j 2
16.e even 4 1 768.2.a.k 2
16.f odd 4 1 768.2.a.j 2
16.f odd 4 1 768.2.a.k 2
20.d odd 2 1 4800.2.k.j 4
20.e even 4 1 4800.2.d.j 4
20.e even 4 1 4800.2.d.o 4
24.f even 2 1 576.2.d.b 4
24.h odd 2 1 576.2.d.b 4
40.e odd 2 1 4800.2.k.j 4
40.f even 2 1 4800.2.k.j 4
40.i odd 4 1 4800.2.d.j 4
40.i odd 4 1 4800.2.d.o 4
40.k even 4 1 4800.2.d.j 4
40.k even 4 1 4800.2.d.o 4
48.i odd 4 1 2304.2.a.s 2
48.i odd 4 1 2304.2.a.u 2
48.k even 4 1 2304.2.a.s 2
48.k even 4 1 2304.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.2.d.a 4 1.a even 1 1 trivial
192.2.d.a 4 4.b odd 2 1 inner
192.2.d.a 4 8.b even 2 1 inner
192.2.d.a 4 8.d odd 2 1 inner
576.2.d.b 4 3.b odd 2 1
576.2.d.b 4 12.b even 2 1
576.2.d.b 4 24.f even 2 1
576.2.d.b 4 24.h odd 2 1
768.2.a.j 2 16.e even 4 1
768.2.a.j 2 16.f odd 4 1
768.2.a.k 2 16.e even 4 1
768.2.a.k 2 16.f odd 4 1
2304.2.a.s 2 48.i odd 4 1
2304.2.a.s 2 48.k even 4 1
2304.2.a.u 2 48.i odd 4 1
2304.2.a.u 2 48.k even 4 1
4800.2.d.j 4 5.c odd 4 1
4800.2.d.j 4 20.e even 4 1
4800.2.d.j 4 40.i odd 4 1
4800.2.d.j 4 40.k even 4 1
4800.2.d.o 4 5.c odd 4 1
4800.2.d.o 4 20.e even 4 1
4800.2.d.o 4 40.i odd 4 1
4800.2.d.o 4 40.k even 4 1
4800.2.k.j 4 5.b even 2 1
4800.2.k.j 4 20.d odd 2 1
4800.2.k.j 4 40.e odd 2 1
4800.2.k.j 4 40.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(192, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$(T^{2} + 12)^{2}$$
$7$ $$(T^{2} - 12)^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T - 6)^{4}$$
$19$ $$(T^{2} + 16)^{2}$$
$23$ $$(T^{2} - 48)^{2}$$
$29$ $$(T^{2} + 12)^{2}$$
$31$ $$(T^{2} - 12)^{2}$$
$37$ $$(T^{2} + 48)^{2}$$
$41$ $$(T + 6)^{4}$$
$43$ $$(T^{2} + 16)^{2}$$
$47$ $$(T^{2} - 48)^{2}$$
$53$ $$(T^{2} + 12)^{2}$$
$59$ $$(T^{2} + 144)^{2}$$
$61$ $$(T^{2} + 48)^{2}$$
$67$ $$(T^{2} + 16)^{2}$$
$71$ $$(T^{2} - 48)^{2}$$
$73$ $$(T - 2)^{4}$$
$79$ $$(T^{2} - 108)^{2}$$
$83$ $$T^{4}$$
$89$ $$(T - 6)^{4}$$
$97$ $$(T + 2)^{4}$$