Properties

 Label 192.3.g.b Level $192$ Weight $3$ Character orbit 192.g Analytic conductor $5.232$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,3,Mod(127,192)]

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([1, 0, 0]))

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

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

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 192.g (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: $$5.23162107572$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 12) 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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + 2 q^{5} - 4 \beta q^{7} - 3 q^{9} +O(q^{10})$$ q - b * q^3 + 2 * q^5 - 4*b * q^7 - 3 * q^9 $$q - \beta q^{3} + 2 q^{5} - 4 \beta q^{7} - 3 q^{9} - 4 \beta q^{11} - 2 q^{13} - 2 \beta q^{15} + 10 q^{17} - 12 \beta q^{19} - 12 q^{21} - 16 \beta q^{23} - 21 q^{25} + 3 \beta q^{27} + 26 q^{29} + 4 \beta q^{31} - 12 q^{33} - 8 \beta q^{35} - 26 q^{37} + 2 \beta q^{39} + 58 q^{41} + 28 \beta q^{43} - 6 q^{45} + 40 \beta q^{47} + q^{49} - 10 \beta q^{51} + 74 q^{53} - 8 \beta q^{55} - 36 q^{57} + 52 \beta q^{59} - 26 q^{61} + 12 \beta q^{63} - 4 q^{65} - 4 \beta q^{67} - 48 q^{69} - 46 q^{73} + 21 \beta q^{75} - 48 q^{77} + 68 \beta q^{79} + 9 q^{81} - 28 \beta q^{83} + 20 q^{85} - 26 \beta q^{87} + 82 q^{89} + 8 \beta q^{91} + 12 q^{93} - 24 \beta q^{95} + 2 q^{97} + 12 \beta q^{99} +O(q^{100})$$ q - b * q^3 + 2 * q^5 - 4*b * q^7 - 3 * q^9 - 4*b * q^11 - 2 * q^13 - 2*b * q^15 + 10 * q^17 - 12*b * q^19 - 12 * q^21 - 16*b * q^23 - 21 * q^25 + 3*b * q^27 + 26 * q^29 + 4*b * q^31 - 12 * q^33 - 8*b * q^35 - 26 * q^37 + 2*b * q^39 + 58 * q^41 + 28*b * q^43 - 6 * q^45 + 40*b * q^47 + q^49 - 10*b * q^51 + 74 * q^53 - 8*b * q^55 - 36 * q^57 + 52*b * q^59 - 26 * q^61 + 12*b * q^63 - 4 * q^65 - 4*b * q^67 - 48 * q^69 - 46 * q^73 + 21*b * q^75 - 48 * q^77 + 68*b * q^79 + 9 * q^81 - 28*b * q^83 + 20 * q^85 - 26*b * q^87 + 82 * q^89 + 8*b * q^91 + 12 * q^93 - 24*b * q^95 + 2 * q^97 + 12*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} - 6 q^{9}+O(q^{10})$$ 2 * q + 4 * q^5 - 6 * q^9 $$2 q + 4 q^{5} - 6 q^{9} - 4 q^{13} + 20 q^{17} - 24 q^{21} - 42 q^{25} + 52 q^{29} - 24 q^{33} - 52 q^{37} + 116 q^{41} - 12 q^{45} + 2 q^{49} + 148 q^{53} - 72 q^{57} - 52 q^{61} - 8 q^{65} - 96 q^{69} - 92 q^{73} - 96 q^{77} + 18 q^{81} + 40 q^{85} + 164 q^{89} + 24 q^{93} + 4 q^{97}+O(q^{100})$$ 2 * q + 4 * q^5 - 6 * q^9 - 4 * q^13 + 20 * q^17 - 24 * q^21 - 42 * q^25 + 52 * q^29 - 24 * q^33 - 52 * q^37 + 116 * q^41 - 12 * q^45 + 2 * q^49 + 148 * q^53 - 72 * q^57 - 52 * q^61 - 8 * q^65 - 96 * q^69 - 92 * q^73 - 96 * q^77 + 18 * q^81 + 40 * q^85 + 164 * q^89 + 24 * q^93 + 4 * q^97

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 2.00000 0 6.92820i 0 −3.00000 0
127.2 0 1.73205i 0 2.00000 0 6.92820i 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
4.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.3.g.b 2
3.b odd 2 1 576.3.g.e 2
4.b odd 2 1 inner 192.3.g.b 2
8.b even 2 1 12.3.d.a 2
8.d odd 2 1 12.3.d.a 2
12.b even 2 1 576.3.g.e 2
16.e even 4 2 768.3.b.c 4
16.f odd 4 2 768.3.b.c 4
24.f even 2 1 36.3.d.c 2
24.h odd 2 1 36.3.d.c 2
40.e odd 2 1 300.3.c.b 2
40.f even 2 1 300.3.c.b 2
40.i odd 4 2 300.3.f.a 4
40.k even 4 2 300.3.f.a 4
48.i odd 4 2 2304.3.b.l 4
48.k even 4 2 2304.3.b.l 4
56.e even 2 1 588.3.g.b 2
56.h odd 2 1 588.3.g.b 2
72.j odd 6 1 324.3.f.a 2
72.j odd 6 1 324.3.f.g 2
72.l even 6 1 324.3.f.a 2
72.l even 6 1 324.3.f.g 2
72.n even 6 1 324.3.f.d 2
72.n even 6 1 324.3.f.j 2
72.p odd 6 1 324.3.f.d 2
72.p odd 6 1 324.3.f.j 2
120.i odd 2 1 900.3.c.e 2
120.m even 2 1 900.3.c.e 2
120.q odd 4 2 900.3.f.c 4
120.w even 4 2 900.3.f.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 8.b even 2 1
12.3.d.a 2 8.d odd 2 1
36.3.d.c 2 24.f even 2 1
36.3.d.c 2 24.h odd 2 1
192.3.g.b 2 1.a even 1 1 trivial
192.3.g.b 2 4.b odd 2 1 inner
300.3.c.b 2 40.e odd 2 1
300.3.c.b 2 40.f even 2 1
300.3.f.a 4 40.i odd 4 2
300.3.f.a 4 40.k even 4 2
324.3.f.a 2 72.j odd 6 1
324.3.f.a 2 72.l even 6 1
324.3.f.d 2 72.n even 6 1
324.3.f.d 2 72.p odd 6 1
324.3.f.g 2 72.j odd 6 1
324.3.f.g 2 72.l even 6 1
324.3.f.j 2 72.n even 6 1
324.3.f.j 2 72.p odd 6 1
576.3.g.e 2 3.b odd 2 1
576.3.g.e 2 12.b even 2 1
588.3.g.b 2 56.e even 2 1
588.3.g.b 2 56.h odd 2 1
768.3.b.c 4 16.e even 4 2
768.3.b.c 4 16.f odd 4 2
900.3.c.e 2 120.i odd 2 1
900.3.c.e 2 120.m even 2 1
900.3.f.c 4 120.q odd 4 2
900.3.f.c 4 120.w even 4 2
2304.3.b.l 4 48.i odd 4 2
2304.3.b.l 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 2$$ acting on $$S_{3}^{\mathrm{new}}(192, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$(T - 2)^{2}$$
$7$ $$T^{2} + 48$$
$11$ $$T^{2} + 48$$
$13$ $$(T + 2)^{2}$$
$17$ $$(T - 10)^{2}$$
$19$ $$T^{2} + 432$$
$23$ $$T^{2} + 768$$
$29$ $$(T - 26)^{2}$$
$31$ $$T^{2} + 48$$
$37$ $$(T + 26)^{2}$$
$41$ $$(T - 58)^{2}$$
$43$ $$T^{2} + 2352$$
$47$ $$T^{2} + 4800$$
$53$ $$(T - 74)^{2}$$
$59$ $$T^{2} + 8112$$
$61$ $$(T + 26)^{2}$$
$67$ $$T^{2} + 48$$
$71$ $$T^{2}$$
$73$ $$(T + 46)^{2}$$
$79$ $$T^{2} + 13872$$
$83$ $$T^{2} + 2352$$
$89$ $$(T - 82)^{2}$$
$97$ $$(T - 2)^{2}$$