# Properties

 Label 192.5.e.d Level $192$ Weight $5$ Character orbit 192.e Analytic conductor $19.847$ 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] = [192,5,Mod(65,192)]

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, 0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("192.65");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 192.e (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: $$19.8470329121$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 6) 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 $$\beta = 6\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 3) q^{3} + 2 \beta q^{5} + 26 q^{7} + ( - 6 \beta - 63) q^{9}+O(q^{10})$$ q + (-b + 3) * q^3 + 2*b * q^5 + 26 * q^7 + (-6*b - 63) * q^9 $$q + ( - \beta + 3) q^{3} + 2 \beta q^{5} + 26 q^{7} + ( - 6 \beta - 63) q^{9} - 14 \beta q^{11} - 50 q^{13} + (6 \beta + 144) q^{15} - 24 \beta q^{17} + 358 q^{19} + ( - 26 \beta + 78) q^{21} - 44 \beta q^{23} + 337 q^{25} + (45 \beta - 621) q^{27} - 170 \beta q^{29} - 742 q^{31} + ( - 42 \beta - 1008) q^{33} + 52 \beta q^{35} - 1874 q^{37} + (50 \beta - 150) q^{39} - 284 \beta q^{41} + 262 q^{43} + ( - 126 \beta + 864) q^{45} + 200 \beta q^{47} - 1725 q^{49} + ( - 72 \beta - 1728) q^{51} - 54 \beta q^{53} + 2016 q^{55} + ( - 358 \beta + 1074) q^{57} - 214 \beta q^{59} + 1486 q^{61} + ( - 156 \beta - 1638) q^{63} - 100 \beta q^{65} + 4486 q^{67} + ( - 132 \beta - 3168) q^{69} - 420 \beta q^{71} + 290 q^{73} + ( - 337 \beta + 1011) q^{75} - 364 \beta q^{77} + 9818 q^{79} + (756 \beta + 1377) q^{81} + 838 \beta q^{83} + 3456 q^{85} + ( - 510 \beta - 12240) q^{87} + 924 \beta q^{89} - 1300 q^{91} + (742 \beta - 2226) q^{93} + 716 \beta q^{95} - 478 q^{97} + (882 \beta - 6048) q^{99} +O(q^{100})$$ q + (-b + 3) * q^3 + 2*b * q^5 + 26 * q^7 + (-6*b - 63) * q^9 - 14*b * q^11 - 50 * q^13 + (6*b + 144) * q^15 - 24*b * q^17 + 358 * q^19 + (-26*b + 78) * q^21 - 44*b * q^23 + 337 * q^25 + (45*b - 621) * q^27 - 170*b * q^29 - 742 * q^31 + (-42*b - 1008) * q^33 + 52*b * q^35 - 1874 * q^37 + (50*b - 150) * q^39 - 284*b * q^41 + 262 * q^43 + (-126*b + 864) * q^45 + 200*b * q^47 - 1725 * q^49 + (-72*b - 1728) * q^51 - 54*b * q^53 + 2016 * q^55 + (-358*b + 1074) * q^57 - 214*b * q^59 + 1486 * q^61 + (-156*b - 1638) * q^63 - 100*b * q^65 + 4486 * q^67 + (-132*b - 3168) * q^69 - 420*b * q^71 + 290 * q^73 + (-337*b + 1011) * q^75 - 364*b * q^77 + 9818 * q^79 + (756*b + 1377) * q^81 + 838*b * q^83 + 3456 * q^85 + (-510*b - 12240) * q^87 + 924*b * q^89 - 1300 * q^91 + (742*b - 2226) * q^93 + 716*b * q^95 - 478 * q^97 + (882*b - 6048) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 52 q^{7} - 126 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 52 * q^7 - 126 * q^9 $$2 q + 6 q^{3} + 52 q^{7} - 126 q^{9} - 100 q^{13} + 288 q^{15} + 716 q^{19} + 156 q^{21} + 674 q^{25} - 1242 q^{27} - 1484 q^{31} - 2016 q^{33} - 3748 q^{37} - 300 q^{39} + 524 q^{43} + 1728 q^{45} - 3450 q^{49} - 3456 q^{51} + 4032 q^{55} + 2148 q^{57} + 2972 q^{61} - 3276 q^{63} + 8972 q^{67} - 6336 q^{69} + 580 q^{73} + 2022 q^{75} + 19636 q^{79} + 2754 q^{81} + 6912 q^{85} - 24480 q^{87} - 2600 q^{91} - 4452 q^{93} - 956 q^{97} - 12096 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 52 * q^7 - 126 * q^9 - 100 * q^13 + 288 * q^15 + 716 * q^19 + 156 * q^21 + 674 * q^25 - 1242 * q^27 - 1484 * q^31 - 2016 * q^33 - 3748 * q^37 - 300 * q^39 + 524 * q^43 + 1728 * q^45 - 3450 * q^49 - 3456 * q^51 + 4032 * q^55 + 2148 * q^57 + 2972 * q^61 - 3276 * q^63 + 8972 * q^67 - 6336 * q^69 + 580 * q^73 + 2022 * q^75 + 19636 * q^79 + 2754 * q^81 + 6912 * q^85 - 24480 * q^87 - 2600 * q^91 - 4452 * q^93 - 956 * q^97 - 12096 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 1.41421i − 1.41421i
0 3.00000 8.48528i 0 16.9706i 0 26.0000 0 −63.0000 50.9117i 0
65.2 0 3.00000 + 8.48528i 0 16.9706i 0 26.0000 0 −63.0000 + 50.9117i 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.5.e.d 2
3.b odd 2 1 inner 192.5.e.d 2
4.b odd 2 1 192.5.e.c 2
8.b even 2 1 6.5.b.a 2
8.d odd 2 1 48.5.e.b 2
12.b even 2 1 192.5.e.c 2
24.f even 2 1 48.5.e.b 2
24.h odd 2 1 6.5.b.a 2
40.f even 2 1 150.5.d.a 2
40.i odd 4 2 150.5.b.a 4
56.h odd 2 1 294.5.b.a 2
72.j odd 6 2 162.5.d.a 4
72.n even 6 2 162.5.d.a 4
120.i odd 2 1 150.5.d.a 2
120.w even 4 2 150.5.b.a 4
168.i even 2 1 294.5.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.5.b.a 2 8.b even 2 1
6.5.b.a 2 24.h odd 2 1
48.5.e.b 2 8.d odd 2 1
48.5.e.b 2 24.f even 2 1
150.5.b.a 4 40.i odd 4 2
150.5.b.a 4 120.w even 4 2
150.5.d.a 2 40.f even 2 1
150.5.d.a 2 120.i odd 2 1
162.5.d.a 4 72.j odd 6 2
162.5.d.a 4 72.n even 6 2
192.5.e.c 2 4.b odd 2 1
192.5.e.c 2 12.b even 2 1
192.5.e.d 2 1.a even 1 1 trivial
192.5.e.d 2 3.b odd 2 1 inner
294.5.b.a 2 56.h odd 2 1
294.5.b.a 2 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(192, [\chi])$$:

 $$T_{5}^{2} + 288$$ T5^2 + 288 $$T_{7} - 26$$ T7 - 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 6T + 81$$
$5$ $$T^{2} + 288$$
$7$ $$(T - 26)^{2}$$
$11$ $$T^{2} + 14112$$
$13$ $$(T + 50)^{2}$$
$17$ $$T^{2} + 41472$$
$19$ $$(T - 358)^{2}$$
$23$ $$T^{2} + 139392$$
$29$ $$T^{2} + 2080800$$
$31$ $$(T + 742)^{2}$$
$37$ $$(T + 1874)^{2}$$
$41$ $$T^{2} + 5807232$$
$43$ $$(T - 262)^{2}$$
$47$ $$T^{2} + 2880000$$
$53$ $$T^{2} + 209952$$
$59$ $$T^{2} + 3297312$$
$61$ $$(T - 1486)^{2}$$
$67$ $$(T - 4486)^{2}$$
$71$ $$T^{2} + 12700800$$
$73$ $$(T - 290)^{2}$$
$79$ $$(T - 9818)^{2}$$
$83$ $$T^{2} + 50561568$$
$89$ $$T^{2} + 61471872$$
$97$ $$(T + 478)^{2}$$