# Properties

 Label 192.7.g.a 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.

 $$f(q)$$ $$=$$ $$q - 9 \beta q^{3} - 150 q^{5} - 188 \beta q^{7} - 243 q^{9} +O(q^{10})$$ q - 9*b * q^3 - 150 * q^5 - 188*b * q^7 - 243 * q^9 $$q - 9 \beta q^{3} - 150 q^{5} - 188 \beta q^{7} - 243 q^{9} - 852 \beta q^{11} - 3394 q^{13} + 1350 \beta q^{15} + 5178 q^{17} - 3932 \beta q^{19} - 5076 q^{21} - 2304 \beta q^{23} + 6875 q^{25} + 2187 \beta q^{27} - 32142 q^{29} + 18828 \beta q^{31} - 23004 q^{33} + 28200 \beta q^{35} + 76150 q^{37} + 30546 \beta q^{39} - 70038 q^{41} + 58284 \beta q^{43} + 36450 q^{45} - 87528 \beta q^{47} + 11617 q^{49} - 46602 \beta q^{51} - 66942 q^{53} + 127800 \beta q^{55} - 106164 q^{57} + 225492 \beta q^{59} + 257014 q^{61} + 45684 \beta q^{63} + 509100 q^{65} - 185444 \beta q^{67} - 62208 q^{69} + 198096 \beta q^{71} + 243442 q^{73} - 61875 \beta q^{75} - 480528 q^{77} + 274220 \beta q^{79} + 59049 q^{81} + 597012 \beta q^{83} - 776700 q^{85} + 289278 \beta q^{87} - 686766 q^{89} + 638072 \beta q^{91} + 508356 q^{93} + 589800 \beta q^{95} - 942686 q^{97} + 207036 \beta q^{99} +O(q^{100})$$ q - 9*b * q^3 - 150 * q^5 - 188*b * q^7 - 243 * q^9 - 852*b * q^11 - 3394 * q^13 + 1350*b * q^15 + 5178 * q^17 - 3932*b * q^19 - 5076 * q^21 - 2304*b * q^23 + 6875 * q^25 + 2187*b * q^27 - 32142 * q^29 + 18828*b * q^31 - 23004 * q^33 + 28200*b * q^35 + 76150 * q^37 + 30546*b * q^39 - 70038 * q^41 + 58284*b * q^43 + 36450 * q^45 - 87528*b * q^47 + 11617 * q^49 - 46602*b * q^51 - 66942 * q^53 + 127800*b * q^55 - 106164 * q^57 + 225492*b * q^59 + 257014 * q^61 + 45684*b * q^63 + 509100 * q^65 - 185444*b * q^67 - 62208 * q^69 + 198096*b * q^71 + 243442 * q^73 - 61875*b * q^75 - 480528 * q^77 + 274220*b * q^79 + 59049 * q^81 + 597012*b * q^83 - 776700 * q^85 + 289278*b * q^87 - 686766 * q^89 + 638072*b * q^91 + 508356 * q^93 + 589800*b * q^95 - 942686 * q^97 + 207036*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 300 q^{5} - 486 q^{9}+O(q^{10})$$ 2 * q - 300 * q^5 - 486 * q^9 $$2 q - 300 q^{5} - 486 q^{9} - 6788 q^{13} + 10356 q^{17} - 10152 q^{21} + 13750 q^{25} - 64284 q^{29} - 46008 q^{33} + 152300 q^{37} - 140076 q^{41} + 72900 q^{45} + 23234 q^{49} - 133884 q^{53} - 212328 q^{57} + 514028 q^{61} + 1018200 q^{65} - 124416 q^{69} + 486884 q^{73} - 961056 q^{77} + 118098 q^{81} - 1553400 q^{85} - 1373532 q^{89} + 1016712 q^{93} - 1885372 q^{97}+O(q^{100})$$ 2 * q - 300 * q^5 - 486 * q^9 - 6788 * q^13 + 10356 * q^17 - 10152 * q^21 + 13750 * q^25 - 64284 * q^29 - 46008 * q^33 + 152300 * q^37 - 140076 * q^41 + 72900 * q^45 + 23234 * q^49 - 133884 * q^53 - 212328 * q^57 + 514028 * q^61 + 1018200 * q^65 - 124416 * q^69 + 486884 * q^73 - 961056 * q^77 + 118098 * q^81 - 1553400 * q^85 - 1373532 * q^89 + 1016712 * q^93 - 1885372 * 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 −150.000 0 325.626i 0 −243.000 0
127.2 0 15.5885i 0 −150.000 0 325.626i 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.a 2
3.b odd 2 1 576.7.g.j 2
4.b odd 2 1 inner 192.7.g.a 2
8.b even 2 1 48.7.g.c 2
8.d odd 2 1 48.7.g.c 2
12.b even 2 1 576.7.g.j 2
16.e even 4 2 768.7.b.d 4
16.f odd 4 2 768.7.b.d 4
24.f even 2 1 144.7.g.b 2
24.h odd 2 1 144.7.g.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 150$$ acting on $$S_{7}^{\mathrm{new}}(192, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 243$$
$5$ $$(T + 150)^{2}$$
$7$ $$T^{2} + 106032$$
$11$ $$T^{2} + 2177712$$
$13$ $$(T + 3394)^{2}$$
$17$ $$(T - 5178)^{2}$$
$19$ $$T^{2} + 46381872$$
$23$ $$T^{2} + 15925248$$
$29$ $$(T + 32142)^{2}$$
$31$ $$T^{2} + 1063480752$$
$37$ $$(T - 76150)^{2}$$
$41$ $$(T + 70038)^{2}$$
$43$ $$T^{2} + 10191073968$$
$47$ $$T^{2} + 22983452352$$
$53$ $$(T + 66942)^{2}$$
$59$ $$T^{2} + 152539926192$$
$61$ $$(T - 257014)^{2}$$
$67$ $$T^{2} + 103168431408$$
$71$ $$T^{2} + 117726075648$$
$73$ $$(T - 243442)^{2}$$
$79$ $$T^{2} + 225589825200$$
$83$ $$T^{2} + 1069269984432$$
$89$ $$(T + 686766)^{2}$$
$97$ $$(T + 942686)^{2}$$