LMFDB - Newform orbit 162.6.a.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,6,Mod(1,162)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$162 = 2 \cdot 3^{4}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$25.9821788097$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 6$ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2\cdot 3$
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -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 $\beta = 6\sqrt{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - 4 q^{2} + 16 q^{4} + (2 \beta + 27) q^{5} + (\beta - 37) q^{7} - 64 q^{8} + ( - 8 \beta - 108) q^{10} + ( - 43 \beta + 39) q^{11} + (20 \beta - 553) q^{13} + ( - 4 \beta + 148) q^{14} + 256 q^{16}+ \cdots + (296 \beta + 60888) q^{98}+O(q^{100})$ q - 4 * q^2 + 16 * q^4 + (2b + 27) q^5 + (b - 37) * q^7 - 64 * q^8 + (-8b - 108) q^10 + (-43b + 39) q^11 + (20b - 553) q^13 + (-4b + 148) q^14 + 256 * q^16 + (94b + 246) q^17 + (-124b - 820) q^19 + (32b + 432) q^20 + (172b - 156) q^22 + (25b + 2769) q^23 + (108b - 1532) q^25 + (-80b + 2212) q^26 + (16b - 592) q^28 + (-124b + 1947) q^29 + (-435b - 2359) q^31 - 1024 * q^32 + (-376b - 984) q^34 + (-47b - 567) q^35 + (826b - 2398) q^37 + (496b + 3280) q^38 + (-128b - 1728) q^40 + (14b - 7677) q^41 + (267b - 16429) q^43 + (-688b + 624) q^44 + (-100b - 11076) q^46 + (249b - 12477) q^47 + (-74b - 15222) q^49 + (-432b + 6128) q^50 + (320b - 8848) q^52 + (-178b - 8166) q^53 + (-1083b - 17523) q^55 + (-64b + 2368) q^56 + (496b - 7788) q^58 + (-311b - 10983) q^59 + (-708b - 1525) q^61 + (1740b + 9436) q^62 + 4096 * q^64 + (-566b - 6291) q^65 + (1671b - 18379) q^67 + (1504b + 3936) q^68 + (188b + 2268) q^70 + (1798b - 36924) q^71 + (-870b - 25594) q^73 + (-3304b + 9592) q^74 + (-1984b - 13120) q^76 + (1630b - 10731) q^77 + (707b + 7463) q^79 + (512b + 6912) q^80 + (-56b + 30708) q^82 + (147b - 45381) q^83 + (3030b + 47250) q^85 + (-1068b + 65716) q^86 + (2752b - 2496) q^88 + (-6086b - 4650) q^89 + (-1293b + 24781) q^91 + (400b + 44304) q^92 + (-996b + 49908) q^94 + (-4988b - 75708) q^95 + (-6574b + 15131) q^97 + (296b + 60888) q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{2} + 32 q^{4} + 54 q^{5} - 74 q^{7} - 128 q^{8} - 216 q^{10} + 78 q^{11} - 1106 q^{13} + 296 q^{14} + 512 q^{16} + 492 q^{17} - 1640 q^{19} + 864 q^{20} - 312 q^{22} + 5538 q^{23} - 3064 q^{25}+ \cdots + 121776 q^{98}+O(q^{100})$ 2 * q - 8 * q^2 + 32 * q^4 + 54 * q^5 - 74 * q^7 - 128 * q^8 - 216 * q^10 + 78 * q^11 - 1106 * q^13 + 296 * q^14 + 512 * q^16 + 492 * q^17 - 1640 * q^19 + 864 * q^20 - 312 * q^22 + 5538 * q^23 - 3064 * q^25 + 4424 * q^26 - 1184 * q^28 + 3894 * q^29 - 4718 * q^31 - 2048 * q^32 - 1968 * q^34 - 1134 * q^35 - 4796 * q^37 + 6560 * q^38 - 3456 * q^40 - 15354 * q^41 - 32858 * q^43 + 1248 * q^44 - 22152 * q^46 - 24954 * q^47 - 30444 * q^49 + 12256 * q^50 - 17696 * q^52 - 16332 * q^53 - 35046 * q^55 + 4736 * q^56 - 15576 * q^58 - 21966 * q^59 - 3050 * q^61 + 18872 * q^62 + 8192 * q^64 - 12582 * q^65 - 36758 * q^67 + 7872 * q^68 + 4536 * q^70 - 73848 * q^71 - 51188 * q^73 + 19184 * q^74 - 26240 * q^76 - 21462 * q^77 + 14926 * q^79 + 13824 * q^80 + 61416 * q^82 - 90762 * q^83 + 94500 * q^85 + 131432 * q^86 - 4992 * q^88 - 9300 * q^89 + 49562 * q^91 + 88608 * q^92 + 99816 * q^94 - 151416 * q^95 + 30262 * q^97 + 121776 * q^98

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}

1.1

−2.44949
2.44949

−4.00000

16.0000

−2.39388

−51.6969

−64.0000

9.57551

1.2

−4.00000

16.0000

56.3939

−22.3031

−64.0000

−225.576

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	162.6.a.e		2
3.b	odd	2	1		162.6.a.f		2
9.c	even	3	2		18.6.c.a	✓	4
9.d	odd	6	2		54.6.c.a		4
36.f	odd	6	2		144.6.i.a		4
36.h	even	6	2		432.6.i.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
18.6.c.a	✓	4	9.c	even	3	2
54.6.c.a		4	9.d	odd	6	2
144.6.i.a		4	36.f	odd	6	2
162.6.a.e		2	1.a	even	1	1	trivial
162.6.a.f		2	3.b	odd	2	1
432.6.i.a		4	36.h	even	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} - 54T_{5} - 135$ T5^2 - 54*T5 - 135 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(162))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T + 4)^{2}$ (T + 4)^2
$3$	$T^{2}$ T^2
$5$	$T^{2} - 54T - 135$ T^2 - 54*T - 135
$7$	$T^{2} + 74T + 1153$ T^2 + 74*T + 1153
$11$	$T^{2} - 78T - 397863$ T^2 - 78*T - 397863
$13$	$T^{2} + 1106 T + 219409$ T^2 + 1106*T + 219409
$17$	$T^{2} - 492 T - 1848060$ T^2 - 492*T - 1848060
$19$	$T^{2} + 1640 T - 2648816$ T^2 + 1640*T - 2648816
$23$	$T^{2} - 5538 T + 7532361$ T^2 - 5538*T + 7532361
$29$	$T^{2} - 3894 T + 469593$ T^2 - 3894*T + 469593
$31$	$T^{2} + 4718 T - 35307719$ T^2 + 4718*T - 35307719
$37$	$T^{2} + 4796 T - 141621212$ T^2 + 4796*T - 141621212
$41$	$T^{2} + 15354 T + 58893993$ T^2 + 15354*T + 58893993
$43$	$T^{2} + 32858 T + 254513617$ T^2 + 32858*T + 254513617
$47$	$T^{2} + 24954 T + 142283313$ T^2 + 24954*T + 142283313
$53$	$T^{2} + 16332 T + 59839812$ T^2 + 16332*T + 59839812
$59$	$T^{2} + 21966 T + 99734553$ T^2 + 21966*T + 99734553
$61$	$T^{2} + 3050 T - 105947399$ T^2 + 3050*T - 105947399
$67$	$T^{2} + 36758 T - 265336415$ T^2 + 36758*T - 265336415
$71$	$T^{2} + 73848 T + 665096112$ T^2 + 73848*T + 665096112
$73$	$T^{2} + 51188 T + 491562436$ T^2 + 51188*T + 491562436
$79$	$T^{2} - 14926 T - 52271015$ T^2 - 14926*T - 52271015
$83$	$T^{2} + \cdots + 2054767617$ T^2 + 90762*T + 2054767617
$89$	$T^{2} + \cdots - 7978887036$ T^2 + 9300*T - 7978887036
$97$	$T^{2} + \cdots - 9106027655$ T^2 - 30262*T - 9106027655
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