LMFDB - Newform orbit 162.4.c.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,4,Mod(55,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([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("162.55"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$162 = 2 \cdot 3^{4}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	162.c (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,-4,21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$9.55830942093$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
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.

$f(q)$	$=$	$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 21 \zeta_{6} q^{5} + (8 \zeta_{6} - 8) q^{7} + 8 q^{8} - 42 q^{10} + ( - 36 \zeta_{6} + 36) q^{11} + 49 \zeta_{6} q^{13} - 16 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} + \cdots - 558 q^{98} +O(q^{100})$ q + (2z - 2) q^2 - 4z q^4 + 21z q^5 + (8z - 8) q^7 + 8 * q^8 - 42 * q^10 + (-36z + 36) q^11 + 49z q^13 - 16z q^14 + (16z - 16) q^16 - 21 * q^17 - 112 * q^19 + (-84z + 84) q^20 + 72z q^22 + 180z q^23 + (316z - 316) q^25 - 98 * q^26 + 32 * q^28 + (135z - 135) q^29 - 308z q^31 - 32z q^32 + (-42z + 42) q^34 - 168 * q^35 - q^37 + (-224z + 224) q^38 + 168z q^40 - 42z q^41 + (20z - 20) q^43 - 144 * q^44 - 360 * q^46 + (-84z + 84) q^47 + 279z q^49 - 632z q^50 + (-196z + 196) q^52 + 174 * q^53 + 756 * q^55 + (64z - 64) q^56 - 270z q^58 + 504z q^59 + (-385z + 385) q^61 + 616 * q^62 + 64 * q^64 + (1029z - 1029) q^65 - 272z q^67 + 84z q^68 + (-336z + 336) q^70 + 888 * q^71 + 371 * q^73 + (-2z + 2) q^74 + 448z q^76 + 288z q^77 + (-652z + 652) q^79 - 336 * q^80 + 84 * q^82 + (-84z + 84) q^83 - 441z q^85 - 40z q^86 + (-288z + 288) q^88 - 21 * q^89 - 392 * q^91 + (-720z + 720) q^92 + 168z q^94 - 2352z q^95 + (-1246z + 1246) q^97 - 558 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} - 4 q^{4} + 21 q^{5} - 8 q^{7} + 16 q^{8} - 84 q^{10} + 36 q^{11} + 49 q^{13} - 16 q^{14} - 16 q^{16} - 42 q^{17} - 224 q^{19} + 84 q^{20} + 72 q^{22} + 180 q^{23} - 316 q^{25} - 196 q^{26}+ \cdots - 1116 q^{98}+O(q^{100})$ 2 * q - 2 * q^2 - 4 * q^4 + 21 * q^5 - 8 * q^7 + 16 * q^8 - 84 * q^10 + 36 * q^11 + 49 * q^13 - 16 * q^14 - 16 * q^16 - 42 * q^17 - 224 * q^19 + 84 * q^20 + 72 * q^22 + 180 * q^23 - 316 * q^25 - 196 * q^26 + 64 * q^28 - 135 * q^29 - 308 * q^31 - 32 * q^32 + 42 * q^34 - 336 * q^35 - 2 * q^37 + 224 * q^38 + 168 * q^40 - 42 * q^41 - 20 * q^43 - 288 * q^44 - 720 * q^46 + 84 * q^47 + 279 * q^49 - 632 * q^50 + 196 * q^52 + 348 * q^53 + 1512 * q^55 - 64 * q^56 - 270 * q^58 + 504 * q^59 + 385 * q^61 + 1232 * q^62 + 128 * q^64 - 1029 * q^65 - 272 * q^67 + 84 * q^68 + 336 * q^70 + 1776 * q^71 + 742 * q^73 + 2 * q^74 + 448 * q^76 + 288 * q^77 + 652 * q^79 - 672 * q^80 + 168 * q^82 + 84 * q^83 - 441 * q^85 - 40 * q^86 + 288 * q^88 - 42 * q^89 - 784 * q^91 + 720 * q^92 + 168 * q^94 - 2352 * q^95 + 1246 * q^97 - 1116 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$ .

$n$	$83$
$\chi(n)$	$-\zeta_{6}$

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}

55.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.00000

1.73205i

−2.00000

−

3.46410i

10.5000

18.1865i

−4.00000

6.92820i

8.00000

−42.0000

109.1

−1.00000

−

1.73205i

−2.00000

3.46410i

10.5000

−

18.1865i

−4.00000

−

6.92820i

8.00000

−42.0000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	162.4.c.d		2
3.b	odd	2	1		162.4.c.e		2
9.c	even	3	1		162.4.a.c	yes	1
9.c	even	3	1	inner	162.4.c.d		2
9.d	odd	6	1		162.4.a.b	✓	1
9.d	odd	6	1		162.4.c.e		2
36.f	odd	6	1		1296.4.a.a		1
36.h	even	6	1		1296.4.a.h		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
162.4.a.b	✓	1	9.d	odd	6	1
162.4.a.c	yes	1	9.c	even	3	1
162.4.c.d		2	1.a	even	1	1	trivial
162.4.c.d		2	9.c	even	3	1	inner
162.4.c.e		2	3.b	odd	2	1
162.4.c.e		2	9.d	odd	6	1
1296.4.a.a		1	36.f	odd	6	1
1296.4.a.h		1	36.h	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} - 21T_{5} + 441$ T5^2 - 21*T5 + 441 acting on $S_{4}^{\mathrm{new}}(162, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$3$	$T^{2}$ T^2
$5$	$T^{2} - 21T + 441$ T^2 - 21*T + 441
$7$	$T^{2} + 8T + 64$ T^2 + 8*T + 64
$11$	$T^{2} - 36T + 1296$ T^2 - 36*T + 1296
$13$	$T^{2} - 49T + 2401$ T^2 - 49*T + 2401
$17$	$(T + 21)^{2}$ (T + 21)^2
$19$	$(T + 112)^{2}$ (T + 112)^2
$23$	$T^{2} - 180T + 32400$ T^2 - 180*T + 32400
$29$	$T^{2} + 135T + 18225$ T^2 + 135*T + 18225
$31$	$T^{2} + 308T + 94864$ T^2 + 308*T + 94864
$37$	$(T + 1)^{2}$ (T + 1)^2
$41$	$T^{2} + 42T + 1764$ T^2 + 42*T + 1764
$43$	$T^{2} + 20T + 400$ T^2 + 20*T + 400
$47$	$T^{2} - 84T + 7056$ T^2 - 84*T + 7056
$53$	$(T - 174)^{2}$ (T - 174)^2
$59$	$T^{2} - 504T + 254016$ T^2 - 504*T + 254016
$61$	$T^{2} - 385T + 148225$ T^2 - 385*T + 148225
$67$	$T^{2} + 272T + 73984$ T^2 + 272*T + 73984
$71$	$(T - 888)^{2}$ (T - 888)^2
$73$	$(T - 371)^{2}$ (T - 371)^2
$79$	$T^{2} - 652T + 425104$ T^2 - 652*T + 425104
$83$	$T^{2} - 84T + 7056$ T^2 - 84*T + 7056
$89$	$(T + 21)^{2}$ (T + 21)^2
$97$	$T^{2} - 1246 T + 1552516$ T^2 - 1246*T + 1552516
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