LMFDB - Newform orbit 882.4.g.w

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,-4,18,0,0,-16,0,-36,-72,0,-68] 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 42)
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.

$f(q)$	$=$	$q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 18 \zeta_{6} q^{5} - 8 q^{8} + (36 \zeta_{6} - 36) q^{10} + (72 \zeta_{6} - 72) q^{11} - 34 q^{13} - 16 \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + \cdots - 70 q^{97} +O(q^{100})$ q + 2z q^2 + (4z - 4) q^4 + 18z q^5 - 8 * q^8 + (36z - 36) q^10 + (72z - 72) q^11 - 34 * q^13 - 16z q^16 + (-6z + 6) q^17 - 92z q^19 - 72 * q^20 - 144 * q^22 - 180z q^23 + (199z - 199) q^25 - 68z q^26 + 114 * q^29 + (56z - 56) q^31 + (-32z + 32) q^32 + 12 * q^34 + 34z q^37 + (-184z + 184) q^38 - 144z q^40 - 6 * q^41 + 164 * q^43 - 288z q^44 + (-360z + 360) q^46 + 168z q^47 - 398 * q^50 + (-136z + 136) q^52 + (-654z + 654) q^53 - 1296 * q^55 + 228z q^58 + (492z - 492) q^59 + 250z q^61 - 112 * q^62 + 64 * q^64 - 612z q^65 + (-124z + 124) q^67 + 24z q^68 - 36 * q^71 + (1010z - 1010) q^73 + (68z - 68) q^74 + 368 * q^76 - 56z q^79 + (-288z + 288) q^80 - 12z q^82 - 228 * q^83 + 108 * q^85 + 328z q^86 + (-576z + 576) q^88 + 390z q^89 + 720 * q^92 + (336z - 336) q^94 + (-1656z + 1656) q^95 - 70 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{2} - 4 q^{4} + 18 q^{5} - 16 q^{8} - 36 q^{10} - 72 q^{11} - 68 q^{13} - 16 q^{16} + 6 q^{17} - 92 q^{19} - 144 q^{20} - 288 q^{22} - 180 q^{23} - 199 q^{25} - 68 q^{26} + 228 q^{29} - 56 q^{31}+ \cdots - 140 q^{97}+O(q^{100})$ 2 * q + 2 * q^2 - 4 * q^4 + 18 * q^5 - 16 * q^8 - 36 * q^10 - 72 * q^11 - 68 * q^13 - 16 * q^16 + 6 * q^17 - 92 * q^19 - 144 * q^20 - 288 * q^22 - 180 * q^23 - 199 * q^25 - 68 * q^26 + 228 * q^29 - 56 * q^31 + 32 * q^32 + 24 * q^34 + 34 * q^37 + 184 * q^38 - 144 * q^40 - 12 * q^41 + 328 * q^43 - 288 * q^44 + 360 * q^46 + 168 * q^47 - 796 * q^50 + 136 * q^52 + 654 * q^53 - 2592 * q^55 + 228 * q^58 - 492 * q^59 + 250 * q^61 - 224 * q^62 + 128 * q^64 - 612 * q^65 + 124 * q^67 + 24 * q^68 - 72 * q^71 - 1010 * q^73 - 68 * q^74 + 736 * q^76 - 56 * q^79 + 288 * q^80 - 12 * q^82 - 456 * q^83 + 216 * q^85 + 328 * q^86 + 576 * q^88 + 390 * q^89 + 1440 * q^92 - 336 * q^94 + 1656 * q^95 - 140 * 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

9.00000

15.5885i

−8.00000

−18.0000

31.1769i

667.1

1.00000

−

1.73205i

−2.00000

−

3.46410i

9.00000

−

15.5885i

−8.00000

−18.0000

−

31.1769i

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.w		2
3.b	odd	2	1		294.4.e.c		2
7.b	odd	2	1		882.4.g.o		2
7.c	even	3	1		126.4.a.a		1
7.c	even	3	1	inner	882.4.g.w		2
7.d	odd	6	1		882.4.a.g		1
7.d	odd	6	1		882.4.g.o		2
21.c	even	2	1		294.4.e.b		2
21.g	even	6	1		294.4.a.i		1
21.g	even	6	1		294.4.e.b		2
21.h	odd	6	1		42.4.a.a	✓	1
21.h	odd	6	1		294.4.e.c		2
28.g	odd	6	1		1008.4.a.b		1
84.j	odd	6	1		2352.4.a.a		1
84.n	even	6	1		336.4.a.l		1
105.o	odd	6	1		1050.4.a.g		1
105.x	even	12	2		1050.4.g.a		2
168.s	odd	6	1		1344.4.a.o		1
168.v	even	6	1		1344.4.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.4.a.a	✓	1	21.h	odd	6	1
126.4.a.a		1	7.c	even	3	1
294.4.a.i		1	21.g	even	6	1
294.4.e.b		2	21.c	even	2	1
294.4.e.b		2	21.g	even	6	1
294.4.e.c		2	3.b	odd	2	1
294.4.e.c		2	21.h	odd	6	1
336.4.a.l		1	84.n	even	6	1
882.4.a.g		1	7.d	odd	6	1
882.4.g.o		2	7.b	odd	2	1
882.4.g.o		2	7.d	odd	6	1
882.4.g.w		2	1.a	even	1	1	trivial
882.4.g.w		2	7.c	even	3	1	inner
1008.4.a.b		1	28.g	odd	6	1
1050.4.a.g		1	105.o	odd	6	1
1050.4.g.a		2	105.x	even	12	2
1344.4.a.a		1	168.v	even	6	1
1344.4.a.o		1	168.s	odd	6	1
2352.4.a.a		1	84.j	odd	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} - 18T_{5} + 324

T5^2 - 18*T5 + 324

T_{11}^{2} + 72T_{11} + 5184

T11^2 + 72*T11 + 5184

T_{13} + 34

T13 + 34

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$3$	$T^{2}$ T^2
$5$	$T^{2} - 18T + 324$ T^2 - 18*T + 324
$7$	$T^{2}$ T^2
$11$	$T^{2} + 72T + 5184$ T^2 + 72*T + 5184
$13$	$(T + 34)^{2}$ (T + 34)^2
$17$	$T^{2} - 6T + 36$ T^2 - 6*T + 36
$19$	$T^{2} + 92T + 8464$ T^2 + 92*T + 8464
$23$	$T^{2} + 180T + 32400$ T^2 + 180*T + 32400
$29$	$(T - 114)^{2}$ (T - 114)^2
$31$	$T^{2} + 56T + 3136$ T^2 + 56*T + 3136
$37$	$T^{2} - 34T + 1156$ T^2 - 34*T + 1156
$41$	$(T + 6)^{2}$ (T + 6)^2
$43$	$(T - 164)^{2}$ (T - 164)^2
$47$	$T^{2} - 168T + 28224$ T^2 - 168*T + 28224
$53$	$T^{2} - 654T + 427716$ T^2 - 654*T + 427716
$59$	$T^{2} + 492T + 242064$ T^2 + 492*T + 242064
$61$	$T^{2} - 250T + 62500$ T^2 - 250*T + 62500
$67$	$T^{2} - 124T + 15376$ T^2 - 124*T + 15376
$71$	$(T + 36)^{2}$ (T + 36)^2
$73$	$T^{2} + 1010 T + 1020100$ T^2 + 1010*T + 1020100
$79$	$T^{2} + 56T + 3136$ T^2 + 56*T + 3136
$83$	$(T + 228)^{2}$ (T + 228)^2
$89$	$T^{2} - 390T + 152100$ T^2 - 390*T + 152100
$97$	$(T + 70)^{2}$ (T + 70)^2
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