LMFDB - Newform orbit 882.4.g.p

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [882,4,Mod(361,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("882.361"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$882 = 2 \cdot 3^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	882.g (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,2,0,-4,-12,0,0,-16,0,24,48,0,112] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$52.0396846251$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 a primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 12 \zeta_{6} q^{5} - 8 q^{8} + ( - 24 \zeta_{6} + 24) q^{10} + ( - 48 \zeta_{6} + 48) q^{11} + 56 q^{13} - 16 \zeta_{6} q^{16} + (114 \zeta_{6} - 114) q^{17} + \cdots - 1330 q^{97} +O(q^{100})$ q + 2z q^2 + (4z - 4) q^4 - 12z q^5 - 8 * q^8 + (-24z + 24) q^10 + (-48z + 48) q^11 + 56 * q^13 - 16z q^16 + (114z - 114) q^17 - 2z q^19 + 48 * q^20 + 96 * q^22 - 120z q^23 + (19z - 19) q^25 + 112z q^26 + 54 * q^29 + (236z - 236) q^31 + (-32z + 32) q^32 - 228 * q^34 - 146z q^37 + (-4z + 4) q^38 + 96z q^40 - 126 * q^41 - 376 * q^43 + 192z q^44 + (-240z + 240) q^46 - 12z q^47 - 38 * q^50 + (224z - 224) q^52 + (-174z + 174) q^53 - 576 * q^55 + 108z q^58 + (-138z + 138) q^59 - 380z q^61 - 472 * q^62 + 64 * q^64 - 672z q^65 + (-484z + 484) q^67 - 456z q^68 - 576 * q^71 + (-1150z + 1150) q^73 + (-292z + 292) q^74 + 8 * q^76 - 776z q^79 + (192z - 192) q^80 - 252z q^82 - 378 * q^83 + 1368 * q^85 - 752z q^86 + (384z - 384) q^88 - 390z q^89 + 480 * q^92 + (-24z + 24) q^94 + (24z - 24) q^95 - 1330 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{2} - 4 q^{4} - 12 q^{5} - 16 q^{8} + 24 q^{10} + 48 q^{11} + 112 q^{13} - 16 q^{16} - 114 q^{17} - 2 q^{19} + 96 q^{20} + 192 q^{22} - 120 q^{23} - 19 q^{25} + 112 q^{26} + 108 q^{29} - 236 q^{31}+ \cdots - 2660 q^{97}+O(q^{100})$ 2 * q + 2 * q^2 - 4 * q^4 - 12 * q^5 - 16 * q^8 + 24 * q^10 + 48 * q^11 + 112 * q^13 - 16 * q^16 - 114 * q^17 - 2 * q^19 + 96 * q^20 + 192 * q^22 - 120 * q^23 - 19 * q^25 + 112 * q^26 + 108 * q^29 - 236 * q^31 + 32 * q^32 - 456 * q^34 - 146 * q^37 + 4 * q^38 + 96 * q^40 - 252 * q^41 - 752 * q^43 + 192 * q^44 + 240 * q^46 - 12 * q^47 - 76 * q^50 - 224 * q^52 + 174 * q^53 - 1152 * q^55 + 108 * q^58 + 138 * q^59 - 380 * q^61 - 944 * q^62 + 128 * q^64 - 672 * q^65 + 484 * q^67 - 456 * q^68 - 1152 * q^71 + 1150 * q^73 + 292 * q^74 + 16 * q^76 - 776 * q^79 - 192 * q^80 - 252 * q^82 - 756 * q^83 + 2736 * q^85 - 752 * q^86 - 384 * q^88 - 390 * q^89 + 960 * q^92 + 24 * q^94 - 24 * q^95 - 2660 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$ .

$n$	$199$	$785$
$\chi(n)$	$-\zeta_{6}$	$1$

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}

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.00000

1.73205i

−2.00000

3.46410i

−6.00000

−

10.3923i

−8.00000

12.0000

−

20.7846i

667.1

1.00000

−

1.73205i

−2.00000

−

3.46410i

−6.00000

10.3923i

−8.00000

12.0000

20.7846i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	882.4.g.p		2
3.b	odd	2	1		98.4.c.c		2
7.b	odd	2	1		882.4.g.v		2
7.c	even	3	1		126.4.a.d		1
7.c	even	3	1	inner	882.4.g.p		2
7.d	odd	6	1		882.4.a.b		1
7.d	odd	6	1		882.4.g.v		2
21.c	even	2	1		98.4.c.b		2
21.g	even	6	1		98.4.a.e		1
21.g	even	6	1		98.4.c.b		2
21.h	odd	6	1		14.4.a.b	✓	1
21.h	odd	6	1		98.4.c.c		2
28.g	odd	6	1		1008.4.a.r		1
84.j	odd	6	1		784.4.a.h		1
84.n	even	6	1		112.4.a.e		1
105.o	odd	6	1		350.4.a.f		1
105.p	even	6	1		2450.4.a.i		1
105.x	even	12	2		350.4.c.g		2
168.s	odd	6	1		448.4.a.k		1
168.v	even	6	1		448.4.a.g		1
231.l	even	6	1		1694.4.a.b		1
273.w	odd	6	1		2366.4.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.4.a.b	✓	1	21.h	odd	6	1
98.4.a.e		1	21.g	even	6	1
98.4.c.b		2	21.c	even	2	1
98.4.c.b		2	21.g	even	6	1
98.4.c.c		2	3.b	odd	2	1
98.4.c.c		2	21.h	odd	6	1
112.4.a.e		1	84.n	even	6	1
126.4.a.d		1	7.c	even	3	1
350.4.a.f		1	105.o	odd	6	1
350.4.c.g		2	105.x	even	12	2
448.4.a.g		1	168.v	even	6	1
448.4.a.k		1	168.s	odd	6	1
784.4.a.h		1	84.j	odd	6	1
882.4.a.b		1	7.d	odd	6	1
882.4.g.p		2	1.a	even	1	1	trivial
882.4.g.p		2	7.c	even	3	1	inner
882.4.g.v		2	7.b	odd	2	1
882.4.g.v		2	7.d	odd	6	1
1008.4.a.r		1	28.g	odd	6	1
1694.4.a.b		1	231.l	even	6	1
2366.4.a.c		1	273.w	odd	6	1
2450.4.a.i		1	105.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(882, [\chi])$ :

T_{5}^{2} + 12T_{5} + 144

T5^2 + 12*T5 + 144

T_{11}^{2} - 48T_{11} + 2304

T11^2 - 48*T11 + 2304

T_{13} - 56

T13 - 56

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$3$	$T^{2}$ T^2
$5$	$T^{2} + 12T + 144$ T^2 + 12*T + 144
$7$	$T^{2}$ T^2
$11$	$T^{2} - 48T + 2304$ T^2 - 48*T + 2304
$13$	$(T - 56)^{2}$ (T - 56)^2
$17$	$T^{2} + 114T + 12996$ T^2 + 114*T + 12996
$19$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$23$	$T^{2} + 120T + 14400$ T^2 + 120*T + 14400
$29$	$(T - 54)^{2}$ (T - 54)^2
$31$	$T^{2} + 236T + 55696$ T^2 + 236*T + 55696
$37$	$T^{2} + 146T + 21316$ T^2 + 146*T + 21316
$41$	$(T + 126)^{2}$ (T + 126)^2
$43$	$(T + 376)^{2}$ (T + 376)^2
$47$	$T^{2} + 12T + 144$ T^2 + 12*T + 144
$53$	$T^{2} - 174T + 30276$ T^2 - 174*T + 30276
$59$	$T^{2} - 138T + 19044$ T^2 - 138*T + 19044
$61$	$T^{2} + 380T + 144400$ T^2 + 380*T + 144400
$67$	$T^{2} - 484T + 234256$ T^2 - 484*T + 234256
$71$	$(T + 576)^{2}$ (T + 576)^2
$73$	$T^{2} - 1150 T + 1322500$ T^2 - 1150*T + 1322500
$79$	$T^{2} + 776T + 602176$ T^2 + 776*T + 602176
$83$	$(T + 378)^{2}$ (T + 378)^2
$89$	$T^{2} + 390T + 152100$ T^2 + 390*T + 152100
$97$	$(T + 1330)^{2}$ (T + 1330)^2
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