LMFDB - Newform orbit 192.9.e.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [192,9,Mod(65,192)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [2,0,126,0,0,0,5572,0,2754] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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.

$f(q)$	$=$	$q + (3 \beta + 63) q^{3} + 34 \beta q^{5} + 2786 q^{7} + (378 \beta + 1377) q^{9} + 1322 \beta q^{11} + 13150 q^{13} + (2142 \beta - 29376) q^{15} + 3912 \beta q^{17} - 144002 q^{19} + (8358 \beta + 175518) q^{21}+ \cdots + (1820394 \beta - 143918208) q^{99}+O(q^{100})$ q + (3b + 63) q^3 + 34b q^5 + 2786 * q^7 + (378b + 1377) q^9 + 1322b q^11 + 13150 * q^13 + (2142b - 29376) q^15 + 3912b q^17 - 144002 * q^19 + (8358b + 175518) q^21 - 2908b q^23 + 57697 * q^25 + (27945b - 239841) q^27 - 36970b q^29 + 728738 * q^31 + (83286b - 1142208) q^33 + 94724b q^35 + 1964446 * q^37 + (39450b + 828450) q^39 - 58108b q^41 + 78142 * q^43 + (46818b - 3701376) q^45 + 207400b q^47 + 1996995 * q^49 + (246456b - 3379968) q^51 + 30762b q^53 - 12945024 * q^55 + (-432006b - 9072126) q^57 - 294878b q^59 - 17578274 * q^61 + (1053108b + 3836322) q^63 + 447100b q^65 + 17136766 * q^67 + (-183204b + 2512512) q^69 - 1525620b q^71 + 28139330 * q^73 + (173091b + 3634911) q^75 + 3683092b q^77 + 9182498 * q^79 + (1041012b - 39254463) q^81 - 5132914b q^83 - 38306304 * q^85 + (-2329110b + 31942080) q^87 + 4787868b q^89 + 36635900 * q^91 + (2186214b + 45910494) q^93 - 4896068b q^95 - 128722558 * q^97 + (1820394b - 143918208) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 126 q^{3} + 5572 q^{7} + 2754 q^{9} + 26300 q^{13} - 58752 q^{15} - 288004 q^{19} + 351036 q^{21} + 115394 q^{25} - 479682 q^{27} + 1457476 q^{31} - 2284416 q^{33} + 3928892 q^{37} + 1656900 q^{39}+ \cdots - 287836416 q^{99}+O(q^{100})$ 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

63.0000

−

50.9117i

−

576.999i

2786.00

1377.00

−

6414.87i

65.2

63.0000

50.9117i

576.999i

2786.00

1377.00

6414.87i

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

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 126T + 6561$ T^2 - 126*T + 6561
$5$	$T^{2} + 332928$ T^2 + 332928
$7$	$(T - 2786)^{2}$ (T - 2786)^2
$11$	$T^{2} + 503332992$ T^2 + 503332992
$13$	$(T - 13150)^{2}$ (T - 13150)^2
$17$	$T^{2} + 4407478272$ T^2 + 4407478272
$19$	$(T + 144002)^{2}$ (T + 144002)^2
$23$	$T^{2} + 2435461632$ T^2 + 2435461632
$29$	$T^{2} + 393632899200$ T^2 + 393632899200
$31$	$(T - 728738)^{2}$ (T - 728738)^2
$37$	$(T - 1964446)^{2}$ (T - 1964446)^2
$41$	$T^{2} + 972443423232$ T^2 + 972443423232
$43$	$(T - 78142)^{2}$ (T - 78142)^2
$47$	$T^{2} + 12388250880000$ T^2 + 12388250880000
$53$	$T^{2} + 272534585472$ T^2 + 272534585472
$59$	$T^{2} + 25042474046592$ T^2 + 25042474046592
$61$	$(T + 17578274)^{2}$ (T + 17578274)^2
$67$	$(T - 17136766)^{2}$ (T - 17136766)^2
$71$	$T^{2} + 670324718707200$ T^2 + 670324718707200
$73$	$(T - 28139330)^{2}$ (T - 28139330)^2
$79$	$(T - 9182498)^{2}$ (T - 9182498)^2
$83$	$T^{2} + 75\!\cdots\!48$ T^2 + 7587880165842048
$89$	$T^{2} + 66\!\cdots\!12$ T^2 + 6602019835802112
$97$	$(T + 128722558)^{2}$ (T + 128722558)^2
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