LMFDB - Newform orbit 196.4.e.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [196,4,Mod(165,196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(196, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("196.165");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$196 = 2^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	196.e (of order $3$ , degree $2$ , not minimal)

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:	no
Analytic conductor:	$11.5643743611$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 28)
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.

$f(q)$	$=$	$q + ( - 10 \zeta_{6} + 10) q^{3} + 8 \zeta_{6} q^{5} - 73 \zeta_{6} q^{9} + ( - 40 \zeta_{6} + 40) q^{11} - 12 q^{13} + 80 q^{15} + ( - 58 \zeta_{6} + 58) q^{17} - 26 \zeta_{6} q^{19} + 64 \zeta_{6} q^{23} + \cdots - 2920 q^{99} +O(q^{100})$ q + (-10z + 10) q^3 + 8z q^5 - 73z q^9 + (-40z + 40) q^11 - 12 * q^13 + 80 * q^15 + (-58z + 58) q^17 - 26z q^19 + 64z q^23 + (-61z + 61) q^25 - 460 * q^27 - 62 * q^29 + (252z - 252) q^31 - 400z q^33 - 26z q^37 + (120z - 120) q^39 + 6 * q^41 + 416 * q^43 + (-584z + 584) q^45 + 396z q^47 - 580z q^51 + (-450z + 450) q^53 + 320 * q^55 - 260 * q^57 + (274z - 274) q^59 + 576z q^61 - 96z q^65 + (-476z + 476) q^67 + 640 * q^69 - 448 * q^71 + (-158z + 158) q^73 - 610z q^75 + 936z q^79 + (2629z - 2629) q^81 + 530 * q^83 + 464 * q^85 + (620z - 620) q^87 + 390z q^89 + 2520z q^93 + (-208z + 208) q^95 + 214 * q^97 - 2920 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 10 q^{3} + 8 q^{5} - 73 q^{9} + 40 q^{11} - 24 q^{13} + 160 q^{15} + 58 q^{17} - 26 q^{19} + 64 q^{23} + 61 q^{25} - 920 q^{27} - 124 q^{29} - 252 q^{31} - 400 q^{33} - 26 q^{37} - 120 q^{39} + 12 q^{41}+ \cdots - 5840 q^{99}+O(q^{100})$ 2 * q + 10 * q^3 + 8 * q^5 - 73 * q^9 + 40 * q^11 - 24 * q^13 + 160 * q^15 + 58 * q^17 - 26 * q^19 + 64 * q^23 + 61 * q^25 - 920 * q^27 - 124 * q^29 - 252 * q^31 - 400 * q^33 - 26 * q^37 - 120 * q^39 + 12 * q^41 + 832 * q^43 + 584 * q^45 + 396 * q^47 - 580 * q^51 + 450 * q^53 + 640 * q^55 - 520 * q^57 - 274 * q^59 + 576 * q^61 - 96 * q^65 + 476 * q^67 + 1280 * q^69 - 896 * q^71 + 158 * q^73 - 610 * q^75 + 936 * q^79 - 2629 * q^81 + 1060 * q^83 + 928 * q^85 - 620 * q^87 + 390 * q^89 + 2520 * q^93 + 208 * q^95 + 428 * q^97 - 5840 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/196\mathbb{Z}\right)^\times$ .

$n$	$99$	$101$
$\chi(n)$	$1$	$-\zeta_{6}$

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}

165.1

0.500000	+	0.866025i
0.500000	−	0.866025i

5.00000

−

8.66025i

4.00000

6.92820i

−36.5000

−

63.2199i

177.1

5.00000

8.66025i

4.00000

−

6.92820i

−36.5000

63.2199i

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	196.4.e.f		2
3.b	odd	2	1		1764.4.k.d		2
7.b	odd	2	1		196.4.e.a		2
7.c	even	3	1		28.4.a.a	✓	1
7.c	even	3	1	inner	196.4.e.f		2
7.d	odd	6	1		196.4.a.d		1
7.d	odd	6	1		196.4.e.a		2
21.c	even	2	1		1764.4.k.m		2
21.g	even	6	1		1764.4.a.c		1
21.g	even	6	1		1764.4.k.m		2
21.h	odd	6	1		252.4.a.d		1
21.h	odd	6	1		1764.4.k.d		2
28.f	even	6	1		784.4.a.a		1
28.g	odd	6	1		112.4.a.g		1
35.j	even	6	1		700.4.a.n		1
35.l	odd	12	2		700.4.e.a		2
56.k	odd	6	1		448.4.a.a		1
56.p	even	6	1		448.4.a.p		1
84.n	even	6	1		1008.4.a.o		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.4.a.a	✓	1	7.c	even	3	1
112.4.a.g		1	28.g	odd	6	1
196.4.a.d		1	7.d	odd	6	1
196.4.e.a		2	7.b	odd	2	1
196.4.e.a		2	7.d	odd	6	1
196.4.e.f		2	1.a	even	1	1	trivial
196.4.e.f		2	7.c	even	3	1	inner
252.4.a.d		1	21.h	odd	6	1
448.4.a.a		1	56.k	odd	6	1
448.4.a.p		1	56.p	even	6	1
700.4.a.n		1	35.j	even	6	1
700.4.e.a		2	35.l	odd	12	2
784.4.a.a		1	28.f	even	6	1
1008.4.a.o		1	84.n	even	6	1
1764.4.a.c		1	21.g	even	6	1
1764.4.k.d		2	3.b	odd	2	1
1764.4.k.d		2	21.h	odd	6	1
1764.4.k.m		2	21.c	even	2	1
1764.4.k.m		2	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(196, [\chi])$ :

T_{3}^{2} - 10T_{3} + 100

T_{5}^{2} - 8T_{5} + 64

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 10T + 100$ T^2 - 10*T + 100
$5$	$T^{2} - 8T + 64$ T^2 - 8*T + 64
$7$	$T^{2}$ T^2
$11$	$T^{2} - 40T + 1600$ T^2 - 40*T + 1600
$13$	$(T + 12)^{2}$ (T + 12)^2
$17$	$T^{2} - 58T + 3364$ T^2 - 58*T + 3364
$19$	$T^{2} + 26T + 676$ T^2 + 26*T + 676
$23$	$T^{2} - 64T + 4096$ T^2 - 64*T + 4096
$29$	$(T + 62)^{2}$ (T + 62)^2
$31$	$T^{2} + 252T + 63504$ T^2 + 252*T + 63504
$37$	$T^{2} + 26T + 676$ T^2 + 26*T + 676
$41$	$(T - 6)^{2}$ (T - 6)^2
$43$	$(T - 416)^{2}$ (T - 416)^2
$47$	$T^{2} - 396T + 156816$ T^2 - 396*T + 156816
$53$	$T^{2} - 450T + 202500$ T^2 - 450*T + 202500
$59$	$T^{2} + 274T + 75076$ T^2 + 274*T + 75076
$61$	$T^{2} - 576T + 331776$ T^2 - 576*T + 331776
$67$	$T^{2} - 476T + 226576$ T^2 - 476*T + 226576
$71$	$(T + 448)^{2}$ (T + 448)^2
$73$	$T^{2} - 158T + 24964$ T^2 - 158*T + 24964
$79$	$T^{2} - 936T + 876096$ T^2 - 936*T + 876096
$83$	$(T - 530)^{2}$ (T - 530)^2
$89$	$T^{2} - 390T + 152100$ T^2 - 390*T + 152100
$97$	$(T - 214)^{2}$ (T - 214)^2
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