# Properties

 Label 192.9.e.h Level $192$ Weight $9$ Character orbit 192.e Analytic conductor $78.217$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,9,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, 9, names="a")

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

chi := DirichletCharacter("192.65");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ 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: $$78.2166931317$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ 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 $$\beta = 12\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (3 \beta + 63) q^{3} + 34 \beta q^{5} + 2786 q^{7} + (378 \beta + 1377) q^{9}+O(q^{10})$$ q + (3*b + 63) * q^3 + 34*b * q^5 + 2786 * q^7 + (378*b + 1377) * q^9 $$q + (3 \beta + 63) q^{3} + 34 \beta q^{5} + 2786 q^{7} + (378 \beta + 1377) q^{9} + 1322 \beta q^{11} + 13150 q^{13} + (2142 \beta - 29376) q^{15} + 3912 \beta q^{17} - 144002 q^{19} + (8358 \beta + 175518) q^{21} - 2908 \beta q^{23} + 57697 q^{25} + (27945 \beta - 239841) q^{27} - 36970 \beta q^{29} + 728738 q^{31} + (83286 \beta - 1142208) q^{33} + 94724 \beta q^{35} + 1964446 q^{37} + (39450 \beta + 828450) q^{39} - 58108 \beta q^{41} + 78142 q^{43} + (46818 \beta - 3701376) q^{45} + 207400 \beta q^{47} + 1996995 q^{49} + (246456 \beta - 3379968) q^{51} + 30762 \beta q^{53} - 12945024 q^{55} + ( - 432006 \beta - 9072126) q^{57} - 294878 \beta q^{59} - 17578274 q^{61} + (1053108 \beta + 3836322) q^{63} + 447100 \beta q^{65} + 17136766 q^{67} + ( - 183204 \beta + 2512512) q^{69} - 1525620 \beta q^{71} + 28139330 q^{73} + (173091 \beta + 3634911) q^{75} + 3683092 \beta q^{77} + 9182498 q^{79} + (1041012 \beta - 39254463) q^{81} - 5132914 \beta q^{83} - 38306304 q^{85} + ( - 2329110 \beta + 31942080) q^{87} + 4787868 \beta q^{89} + 36635900 q^{91} + (2186214 \beta + 45910494) q^{93} - 4896068 \beta q^{95} - 128722558 q^{97} + (1820394 \beta - 143918208) q^{99} +O(q^{100})$$ q + (3*b + 63) * q^3 + 34*b * q^5 + 2786 * q^7 + (378*b + 1377) * q^9 + 1322*b * q^11 + 13150 * q^13 + (2142*b - 29376) * q^15 + 3912*b * q^17 - 144002 * q^19 + (8358*b + 175518) * q^21 - 2908*b * q^23 + 57697 * q^25 + (27945*b - 239841) * q^27 - 36970*b * q^29 + 728738 * q^31 + (83286*b - 1142208) * q^33 + 94724*b * q^35 + 1964446 * q^37 + (39450*b + 828450) * q^39 - 58108*b * q^41 + 78142 * q^43 + (46818*b - 3701376) * q^45 + 207400*b * q^47 + 1996995 * q^49 + (246456*b - 3379968) * q^51 + 30762*b * q^53 - 12945024 * q^55 + (-432006*b - 9072126) * q^57 - 294878*b * q^59 - 17578274 * q^61 + (1053108*b + 3836322) * q^63 + 447100*b * q^65 + 17136766 * q^67 + (-183204*b + 2512512) * q^69 - 1525620*b * q^71 + 28139330 * q^73 + (173091*b + 3634911) * q^75 + 3683092*b * q^77 + 9182498 * q^79 + (1041012*b - 39254463) * q^81 - 5132914*b * q^83 - 38306304 * q^85 + (-2329110*b + 31942080) * q^87 + 4787868*b * q^89 + 36635900 * q^91 + (2186214*b + 45910494) * q^93 - 4896068*b * q^95 - 128722558 * q^97 + (1820394*b - 143918208) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 126 q^{3} + 5572 q^{7} + 2754 q^{9}+O(q^{10})$$ 2 * q + 126 * q^3 + 5572 * q^7 + 2754 * q^9 $$2 q + 126 q^{3} + 5572 q^{7} + 2754 q^{9} + 26300 q^{13} - 58752 q^{15} - 288004 q^{19} + 351036 q^{21} + 115394 q^{25} - 479682 q^{27} + 1457476 q^{31} - 2284416 q^{33} + 3928892 q^{37} + 1656900 q^{39} + 156284 q^{43} - 7402752 q^{45} + 3993990 q^{49} - 6759936 q^{51} - 25890048 q^{55} - 18144252 q^{57} - 35156548 q^{61} + 7672644 q^{63} + 34273532 q^{67} + 5025024 q^{69} + 56278660 q^{73} + 7269822 q^{75} + 18364996 q^{79} - 78508926 q^{81} - 76612608 q^{85} + 63884160 q^{87} + 73271800 q^{91} + 91820988 q^{93} - 257445116 q^{97} - 287836416 q^{99}+O(q^{100})$$ 2 * q + 126 * q^3 + 5572 * q^7 + 2754 * q^9 + 26300 * q^13 - 58752 * q^15 - 288004 * q^19 + 351036 * q^21 + 115394 * q^25 - 479682 * q^27 + 1457476 * q^31 - 2284416 * q^33 + 3928892 * q^37 + 1656900 * q^39 + 156284 * q^43 - 7402752 * q^45 + 3993990 * q^49 - 6759936 * q^51 - 25890048 * q^55 - 18144252 * q^57 - 35156548 * q^61 + 7672644 * q^63 + 34273532 * q^67 + 5025024 * q^69 + 56278660 * q^73 + 7269822 * q^75 + 18364996 * q^79 - 78508926 * q^81 - 76612608 * q^85 + 63884160 * q^87 + 73271800 * q^91 + 91820988 * q^93 - 257445116 * q^97 - 287836416 * 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 63.0000 50.9117i 0 576.999i 0 2786.00 0 1377.00 6414.87i 0
65.2 0 63.0000 + 50.9117i 0 576.999i 0 2786.00 0 1377.00 + 6414.87i 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 192.9.e.h 2
3.b odd 2 1 inner 192.9.e.h 2
4.b odd 2 1 192.9.e.c 2
8.b even 2 1 6.9.b.a 2
8.d odd 2 1 48.9.e.d 2
12.b even 2 1 192.9.e.c 2
24.f even 2 1 48.9.e.d 2
24.h odd 2 1 6.9.b.a 2
40.f even 2 1 150.9.d.a 2
40.i odd 4 2 150.9.b.a 4
72.j odd 6 2 162.9.d.a 4
72.n even 6 2 162.9.d.a 4
120.i odd 2 1 150.9.d.a 2
120.w even 4 2 150.9.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.9.b.a 2 8.b even 2 1
6.9.b.a 2 24.h odd 2 1
48.9.e.d 2 8.d odd 2 1
48.9.e.d 2 24.f even 2 1
150.9.b.a 4 40.i odd 4 2
150.9.b.a 4 120.w even 4 2
150.9.d.a 2 40.f even 2 1
150.9.d.a 2 120.i odd 2 1
162.9.d.a 4 72.j odd 6 2
162.9.d.a 4 72.n even 6 2
192.9.e.c 2 4.b odd 2 1
192.9.e.c 2 12.b even 2 1
192.9.e.h 2 1.a even 1 1 trivial
192.9.e.h 2 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(192, [\chi])$$:

 $$T_{5}^{2} + 332928$$ T5^2 + 332928 $$T_{7} - 2786$$ T7 - 2786

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 126T + 6561$$
$5$ $$T^{2} + 332928$$
$7$ $$(T - 2786)^{2}$$
$11$ $$T^{2} + 503332992$$
$13$ $$(T - 13150)^{2}$$
$17$ $$T^{2} + 4407478272$$
$19$ $$(T + 144002)^{2}$$
$23$ $$T^{2} + 2435461632$$
$29$ $$T^{2} + 393632899200$$
$31$ $$(T - 728738)^{2}$$
$37$ $$(T - 1964446)^{2}$$
$41$ $$T^{2} + 972443423232$$
$43$ $$(T - 78142)^{2}$$
$47$ $$T^{2} + 12388250880000$$
$53$ $$T^{2} + 272534585472$$
$59$ $$T^{2} + 25042474046592$$
$61$ $$(T + 17578274)^{2}$$
$67$ $$(T - 17136766)^{2}$$
$71$ $$T^{2} + 670324718707200$$
$73$ $$(T - 28139330)^{2}$$
$79$ $$(T - 9182498)^{2}$$
$83$ $$T^{2} + 75\!\cdots\!48$$
$89$ $$T^{2} + 66\!\cdots\!12$$
$97$ $$(T + 128722558)^{2}$$