LMFDB - Newform orbit 192.4.c.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [192,4,Mod(191,192)] 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([1, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("192.191"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$192 = 2^{6} \cdot 3$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	192.c (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-54] 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:	$11.3283667211$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2\cdot 3$
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$ -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 $\beta = 3\sqrt{-3}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta q^{3} - 6 \beta q^{7} - 27 q^{9} - 70 q^{13} - 30 \beta q^{19} + 162 q^{21} + 125 q^{25} - 27 \beta q^{27} - 30 \beta q^{31} - 110 q^{37} - 70 \beta q^{39} + 42 \beta q^{43} - 629 q^{49} + 810 q^{57} + \cdots + 1330 q^{97} +O(q^{100})$ q + b * q^3 - 6b q^7 - 27 * q^9 - 70 * q^13 - 30b q^19 + 162 * q^21 + 125 * q^25 - 27b q^27 - 30b q^31 - 110 * q^37 - 70b q^39 + 42b q^43 - 629 * q^49 + 810 * q^57 - 182 * q^61 + 162b q^63 - 126b q^67 - 1190 * q^73 + 125b q^75 + 210b q^79 + 729 * q^81 + 420b q^91 + 810 * q^93 + 1330 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 54 q^{9} - 140 q^{13} + 324 q^{21} + 250 q^{25} - 220 q^{37} - 1258 q^{49} + 1620 q^{57} - 364 q^{61} - 2380 q^{73} + 1458 q^{81} + 1620 q^{93} + 2660 q^{97}+O(q^{100})$ 2 * q - 54 * q^9 - 140 * q^13 + 324 * q^21 + 250 * q^25 - 220 * q^37 - 1258 * q^49 + 1620 * q^57 - 364 * q^61 - 2380 * q^73 + 1458 * q^81 + 1620 * q^93 + 2660 * 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}

191.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−

5.19615i

31.1769i

−27.0000

191.2

5.19615i

−

31.1769i

−27.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	192.4.c.a		2
3.b	odd	2	1	CM	192.4.c.a		2
4.b	odd	2	1	inner	192.4.c.a		2
8.b	even	2	1		48.4.c.a	✓	2
8.d	odd	2	1		48.4.c.a	✓	2
12.b	even	2	1	inner	192.4.c.a		2
16.e	even	4	2		768.4.f.a		4
16.f	odd	4	2		768.4.f.a		4
24.f	even	2	1		48.4.c.a	✓	2
24.h	odd	2	1		48.4.c.a	✓	2
48.i	odd	4	2		768.4.f.a		4
48.k	even	4	2		768.4.f.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
48.4.c.a	✓	2	8.b	even	2	1
48.4.c.a	✓	2	8.d	odd	2	1
48.4.c.a	✓	2	24.f	even	2	1
48.4.c.a	✓	2	24.h	odd	2	1
192.4.c.a		2	1.a	even	1	1	trivial
192.4.c.a		2	3.b	odd	2	1	CM
192.4.c.a		2	4.b	odd	2	1	inner
192.4.c.a		2	12.b	even	2	1	inner
768.4.f.a		4	16.e	even	4	2
768.4.f.a		4	16.f	odd	4	2
768.4.f.a		4	48.i	odd	4	2
768.4.f.a		4	48.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}$ T5 acting on $S_{4}^{\mathrm{new}}(192, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 27$ T^2 + 27
$5$	$T^{2}$ T^2
$7$	$T^{2} + 972$ T^2 + 972
$11$	$T^{2}$ T^2
$13$	$(T + 70)^{2}$ (T + 70)^2
$17$	$T^{2}$ T^2
$19$	$T^{2} + 24300$ T^2 + 24300
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$T^{2} + 24300$ T^2 + 24300
$37$	$(T + 110)^{2}$ (T + 110)^2
$41$	$T^{2}$ T^2
$43$	$T^{2} + 47628$ T^2 + 47628
$47$	$T^{2}$ T^2
$53$	$T^{2}$ T^2
$59$	$T^{2}$ T^2
$61$	$(T + 182)^{2}$ (T + 182)^2
$67$	$T^{2} + 428652$ T^2 + 428652
$71$	$T^{2}$ T^2
$73$	$(T + 1190)^{2}$ (T + 1190)^2
$79$	$T^{2} + 1190700$ T^2 + 1190700
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$(T - 1330)^{2}$ (T - 1330)^2
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