LMFDB - Newform orbit 84.8.k.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [84,8,Mod(5,84)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(84, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 5])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("84.5"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$N$	$=$	$84 = 2^{2} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	84.k (of order $6$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$26.2403421407$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$ -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.

$f(q)$	$=$	$q + ( - 27 \zeta_{6} - 27) q^{3} + ( - 757 \zeta_{6} + 1006) q^{7} + 2187 \zeta_{6} q^{9} + (11082 \zeta_{6} - 5541) q^{13} + ( - 23935 \zeta_{6} + 47870) q^{19} + (13716 \zeta_{6} - 47601) q^{21} + ( - 78125 \zeta_{6} + 78125) q^{25}+ \cdots + (15198608 \zeta_{6} - 7599304) q^{97}+O(q^{100})$ q + (-27z - 27) q^3 + (-757z + 1006) q^7 + 2187z q^9 + (11082z - 5541) q^13 + (-23935z + 47870) q^19 + (13716z - 47601) q^21 + (-78125z + 78125) q^25 + (-118098z + 59049) q^27 + (-170101z - 170101) q^31 - 615373z q^37 + (-448821z + 448821) q^39 + 625729 * q^43 + (-950035z + 438987) q^49 - 1938735 * q^57 + (-153716z + 307432) q^61 + (544563z + 1655559) q^63 + (-4058455z + 4058455) q^67 + (-2503873z - 2503873) q^73 + (2109375z - 4218750) q^75 + 4245427z q^79 + (4782969z - 4782969) q^81 + (6953955z + 2814828) q^91 + 13778181z q^93 + (15198608z - 7599304) q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 81 q^{3} + 1255 q^{7} + 2187 q^{9} + 71805 q^{19} - 81486 q^{21} + 78125 q^{25} - 510303 q^{31} - 615373 q^{37} + 448821 q^{39} + 1251458 q^{43} - 72061 q^{49} - 3877470 q^{57} + 461148 q^{61} + 3855681 q^{63}+ \cdots + 13778181 q^{93}+O(q^{100})$ 2 * q - 81 * q^3 + 1255 * q^7 + 2187 * q^9 + 71805 * q^19 - 81486 * q^21 + 78125 * q^25 - 510303 * q^31 - 615373 * q^37 + 448821 * q^39 + 1251458 * q^43 - 72061 * q^49 - 3877470 * q^57 + 461148 * q^61 + 3855681 * q^63 + 4058455 * q^67 - 7511619 * q^73 - 6328125 * q^75 + 4245427 * q^79 - 4782969 * q^81 + 12583611 * q^91 + 13778181 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$ .

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$\zeta_{6}$

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}

5.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−40.5000

23.3827i

627.500

655.581i

1093.50

−

1894.00i

17.1

−40.5000

−

23.3827i

627.500

−

655.581i

1093.50

1894.00i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
7.d	odd	6	1	inner
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	84.8.k.a	✓	2
3.b	odd	2	1	CM	84.8.k.a	✓	2
7.c	even	3	1		588.8.f.a		2
7.d	odd	6	1	inner	84.8.k.a	✓	2
7.d	odd	6	1		588.8.f.a		2
21.g	even	6	1	inner	84.8.k.a	✓	2
21.g	even	6	1		588.8.f.a		2
21.h	odd	6	1		588.8.f.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.8.k.a	✓	2	1.a	even	1	1	trivial
84.8.k.a	✓	2	3.b	odd	2	1	CM
84.8.k.a	✓	2	7.d	odd	6	1	inner
84.8.k.a	✓	2	21.g	even	6	1	inner
588.8.f.a		2	7.c	even	3	1
588.8.f.a		2	7.d	odd	6	1
588.8.f.a		2	21.g	even	6	1
588.8.f.a		2	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}$ T5 acting on $S_{8}^{\mathrm{new}}(84, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 81T + 2187$ T^2 + 81*T + 2187
$5$	$T^{2}$ T^2
$7$	$T^{2} - 1255 T + 823543$ T^2 - 1255*T + 823543
$11$	$T^{2}$ T^2
$13$	$T^{2} + 92108043$ T^2 + 92108043
$17$	$T^{2}$ T^2
$19$	$T^{2} + \cdots + 1718652675$ T^2 - 71805*T + 1718652675
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$T^{2} + \cdots + 86803050603$ T^2 + 510303*T + 86803050603
$37$	$T^{2} + \cdots + 378683929129$ T^2 + 615373*T + 378683929129
$41$	$T^{2}$ T^2
$43$	$(T - 625729)^{2}$ (T - 625729)^2
$47$	$T^{2}$ T^2
$53$	$T^{2}$ T^2
$59$	$T^{2}$ T^2
$61$	$T^{2} + \cdots + 70885825968$ T^2 - 461148*T + 70885825968
$67$	$T^{2} + \cdots + 16471056987025$ T^2 - 4058455*T + 16471056987025
$71$	$T^{2}$ T^2
$73$	$T^{2} + \cdots + 18808140000387$ T^2 + 7511619*T + 18808140000387
$79$	$T^{2} + \cdots + 18023650412329$ T^2 - 4245427*T + 18023650412329
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$T^{2} + 173248263853248$ T^2 + 173248263853248
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