# Properties

 Label 192.7.g.b Level $192$ Weight $7$ Character orbit 192.g Analytic conductor $44.170$ 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] = [192,7,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, 7, names="a")

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

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$7$$ 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: $$44.1703840550$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 48) 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.

 $$f(q)$$ $$=$$ $$q - 9 \beta q^{3} - 6 q^{5} + 116 \beta q^{7} - 243 q^{9} +O(q^{10})$$ q - 9*b * q^3 - 6 * q^5 + 116*b * q^7 - 243 * q^9 $$q - 9 \beta q^{3} - 6 q^{5} + 116 \beta q^{7} - 243 q^{9} - 372 \beta q^{11} + 2654 q^{13} + 54 \beta q^{15} - 7206 q^{17} - 2044 \beta q^{19} + 3132 q^{21} + 10080 \beta q^{23} - 15589 q^{25} + 2187 \beta q^{27} - 11550 q^{29} - 4932 \beta q^{31} - 10044 q^{33} - 696 \beta q^{35} - 22346 q^{37} - 23886 \beta q^{39} + 103626 q^{41} + 73548 \beta q^{43} + 1458 q^{45} + 92856 \beta q^{47} + 77281 q^{49} + 64854 \beta q^{51} - 168462 q^{53} + 2232 \beta q^{55} - 55188 q^{57} - 64428 \beta q^{59} + 260470 q^{61} - 28188 \beta q^{63} - 15924 q^{65} + 183068 \beta q^{67} + 272160 q^{69} + 409008 \beta q^{71} - 395918 q^{73} + 140301 \beta q^{75} + 129456 q^{77} + 321436 \beta q^{79} + 59049 q^{81} - 51468 \beta q^{83} + 43236 q^{85} + 103950 \beta q^{87} - 251886 q^{89} + 307864 \beta q^{91} - 133164 q^{93} + 12264 \beta q^{95} + 517474 q^{97} + 90396 \beta q^{99} +O(q^{100})$$ q - 9*b * q^3 - 6 * q^5 + 116*b * q^7 - 243 * q^9 - 372*b * q^11 + 2654 * q^13 + 54*b * q^15 - 7206 * q^17 - 2044*b * q^19 + 3132 * q^21 + 10080*b * q^23 - 15589 * q^25 + 2187*b * q^27 - 11550 * q^29 - 4932*b * q^31 - 10044 * q^33 - 696*b * q^35 - 22346 * q^37 - 23886*b * q^39 + 103626 * q^41 + 73548*b * q^43 + 1458 * q^45 + 92856*b * q^47 + 77281 * q^49 + 64854*b * q^51 - 168462 * q^53 + 2232*b * q^55 - 55188 * q^57 - 64428*b * q^59 + 260470 * q^61 - 28188*b * q^63 - 15924 * q^65 + 183068*b * q^67 + 272160 * q^69 + 409008*b * q^71 - 395918 * q^73 + 140301*b * q^75 + 129456 * q^77 + 321436*b * q^79 + 59049 * q^81 - 51468*b * q^83 + 43236 * q^85 + 103950*b * q^87 - 251886 * q^89 + 307864*b * q^91 - 133164 * q^93 + 12264*b * q^95 + 517474 * q^97 + 90396*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{5} - 486 q^{9}+O(q^{10})$$ 2 * q - 12 * q^5 - 486 * q^9 $$2 q - 12 q^{5} - 486 q^{9} + 5308 q^{13} - 14412 q^{17} + 6264 q^{21} - 31178 q^{25} - 23100 q^{29} - 20088 q^{33} - 44692 q^{37} + 207252 q^{41} + 2916 q^{45} + 154562 q^{49} - 336924 q^{53} - 110376 q^{57} + 520940 q^{61} - 31848 q^{65} + 544320 q^{69} - 791836 q^{73} + 258912 q^{77} + 118098 q^{81} + 86472 q^{85} - 503772 q^{89} - 266328 q^{93} + 1034948 q^{97}+O(q^{100})$$ 2 * q - 12 * q^5 - 486 * q^9 + 5308 * q^13 - 14412 * q^17 + 6264 * q^21 - 31178 * q^25 - 23100 * q^29 - 20088 * q^33 - 44692 * q^37 + 207252 * q^41 + 2916 * q^45 + 154562 * q^49 - 336924 * q^53 - 110376 * q^57 + 520940 * q^61 - 31848 * q^65 + 544320 * q^69 - 791836 * q^73 + 258912 * q^77 + 118098 * q^81 + 86472 * q^85 - 503772 * q^89 - 266328 * q^93 + 1034948 * 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 15.5885i 0 −6.00000 0 200.918i 0 −243.000 0
127.2 0 15.5885i 0 −6.00000 0 200.918i 0 −243.000 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 192.7.g.b 2
3.b odd 2 1 576.7.g.g 2
4.b odd 2 1 inner 192.7.g.b 2
8.b even 2 1 48.7.g.b 2
8.d odd 2 1 48.7.g.b 2
12.b even 2 1 576.7.g.g 2
16.e even 4 2 768.7.b.b 4
16.f odd 4 2 768.7.b.b 4
24.f even 2 1 144.7.g.c 2
24.h odd 2 1 144.7.g.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.7.g.b 2 8.b even 2 1
48.7.g.b 2 8.d odd 2 1
144.7.g.c 2 24.f even 2 1
144.7.g.c 2 24.h odd 2 1
192.7.g.b 2 1.a even 1 1 trivial
192.7.g.b 2 4.b odd 2 1 inner
576.7.g.g 2 3.b odd 2 1
576.7.g.g 2 12.b even 2 1
768.7.b.b 4 16.e even 4 2
768.7.b.b 4 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 6$$ acting on $$S_{7}^{\mathrm{new}}(192, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 243$$
$5$ $$(T + 6)^{2}$$
$7$ $$T^{2} + 40368$$
$11$ $$T^{2} + 415152$$
$13$ $$(T - 2654)^{2}$$
$17$ $$(T + 7206)^{2}$$
$19$ $$T^{2} + 12533808$$
$23$ $$T^{2} + 304819200$$
$29$ $$(T + 11550)^{2}$$
$31$ $$T^{2} + 72973872$$
$37$ $$(T + 22346)^{2}$$
$41$ $$(T - 103626)^{2}$$
$43$ $$T^{2} + 16227924912$$
$47$ $$T^{2} + 25866710208$$
$53$ $$(T + 168462)^{2}$$
$59$ $$T^{2} + 12452901552$$
$61$ $$(T - 260470)^{2}$$
$67$ $$T^{2} + 100541677872$$
$71$ $$T^{2} + 501862632192$$
$73$ $$(T + 395918)^{2}$$
$79$ $$T^{2} + 309963306288$$
$83$ $$T^{2} + 7946865072$$
$89$ $$(T + 251886)^{2}$$
$97$ $$(T - 517474)^{2}$$