LMFDB - Newform orbit 700.3.s.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,3,Mod(101,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("700.101");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$700 = 2^{2} \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	700.s (of order $6$ , degree $2$ , 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:	$19.0736185052$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$ -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.

$f(q)$	$=$	$q + ( - \zeta_{6} - 1) q^{3} + 7 q^{7} - 6 \zeta_{6} q^{9} + (15 \zeta_{6} - 15) q^{11} + (16 \zeta_{6} - 8) q^{13} + ( - 17 \zeta_{6} - 17) q^{17} + ( - 9 \zeta_{6} + 18) q^{19} + ( - 7 \zeta_{6} - 7) q^{21} + \cdots + 90 q^{99} +O(q^{100})$ q + (-z - 1) * q^3 + 7 * q^7 - 6z q^9 + (15z - 15) q^11 + (16z - 8) q^13 + (-17z - 17) q^17 + (-9z + 18) q^19 + (-7z - 7) q^21 - 9z q^23 + (30z - 15) q^27 - 6 * q^29 + (-7z - 7) q^31 + (-15z + 30) q^33 + 31z q^37 + (-24z + 24) q^39 + (64z - 32) q^41 - 10 * q^43 + (25z - 50) q^47 + 49 * q^49 + 51z q^51 + (57z - 57) q^53 - 27 * q^57 + (-47z - 47) q^59 + (47z - 94) q^61 - 42z q^63 + (49z - 49) q^67 + (18z - 9) q^69 - 126 * q^71 + (15z + 15) q^73 + (105z - 105) q^77 + 73z q^79 + (9z - 9) q^81 + (-16z + 8) q^83 + (6z + 6) q^87 + (-33z + 66) q^89 + (112z - 56) q^91 + 21z q^93 + (32z - 16) q^97 + 90 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 3 q^{3} + 14 q^{7} - 6 q^{9} - 15 q^{11} - 51 q^{17} + 27 q^{19} - 21 q^{21} - 9 q^{23} - 12 q^{29} - 21 q^{31} + 45 q^{33} + 31 q^{37} + 24 q^{39} - 20 q^{43} - 75 q^{47} + 98 q^{49} + 51 q^{51}+ \cdots + 180 q^{99}+O(q^{100})$ 2 * q - 3 * q^3 + 14 * q^7 - 6 * q^9 - 15 * q^11 - 51 * q^17 + 27 * q^19 - 21 * q^21 - 9 * q^23 - 12 * q^29 - 21 * q^31 + 45 * q^33 + 31 * q^37 + 24 * q^39 - 20 * q^43 - 75 * q^47 + 98 * q^49 + 51 * q^51 - 57 * q^53 - 54 * q^57 - 141 * q^59 - 141 * q^61 - 42 * q^63 - 49 * q^67 - 252 * q^71 + 45 * q^73 - 105 * q^77 + 73 * q^79 - 9 * q^81 + 18 * q^87 + 99 * q^89 + 21 * q^93 + 180 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$ .

$n$	$101$	$351$	$477$
$\chi(n)$	$\zeta_{6}$	$1$	$1$

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}

101.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.50000

−

0.866025i

7.00000

−3.00000

−

5.19615i

201.1

−1.50000

0.866025i

7.00000

−3.00000

5.19615i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	700.3.s.a		2
5.b	even	2	1		28.3.h.a	✓	2
5.c	odd	4	2		700.3.o.a		4
7.d	odd	6	1	inner	700.3.s.a		2
15.d	odd	2	1		252.3.z.a		2
20.d	odd	2	1		112.3.s.a		2
35.c	odd	2	1		196.3.h.a		2
35.i	odd	6	1		28.3.h.a	✓	2
35.i	odd	6	1		196.3.b.a		2
35.j	even	6	1		196.3.b.a		2
35.j	even	6	1		196.3.h.a		2
35.k	even	12	2		700.3.o.a		4
40.e	odd	2	1		448.3.s.b		2
40.f	even	2	1		448.3.s.a		2
60.h	even	2	1		1008.3.cg.c		2
105.g	even	2	1		1764.3.z.f		2
105.o	odd	6	1		1764.3.d.a		2
105.o	odd	6	1		1764.3.z.f		2
105.p	even	6	1		252.3.z.a		2
105.p	even	6	1		1764.3.d.a		2
140.c	even	2	1		784.3.s.b		2
140.p	odd	6	1		784.3.c.a		2
140.p	odd	6	1		784.3.s.b		2
140.s	even	6	1		112.3.s.a		2
140.s	even	6	1		784.3.c.a		2
280.ba	even	6	1		448.3.s.b		2
280.bk	odd	6	1		448.3.s.a		2
420.be	odd	6	1		1008.3.cg.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.3.h.a	✓	2	5.b	even	2	1
28.3.h.a	✓	2	35.i	odd	6	1
112.3.s.a		2	20.d	odd	2	1
112.3.s.a		2	140.s	even	6	1
196.3.b.a		2	35.i	odd	6	1
196.3.b.a		2	35.j	even	6	1
196.3.h.a		2	35.c	odd	2	1
196.3.h.a		2	35.j	even	6	1
252.3.z.a		2	15.d	odd	2	1
252.3.z.a		2	105.p	even	6	1
448.3.s.a		2	40.f	even	2	1
448.3.s.a		2	280.bk	odd	6	1
448.3.s.b		2	40.e	odd	2	1
448.3.s.b		2	280.ba	even	6	1
700.3.o.a		4	5.c	odd	4	2
700.3.o.a		4	35.k	even	12	2
700.3.s.a		2	1.a	even	1	1	trivial
700.3.s.a		2	7.d	odd	6	1	inner
784.3.c.a		2	140.p	odd	6	1
784.3.c.a		2	140.s	even	6	1
784.3.s.b		2	140.c	even	2	1
784.3.s.b		2	140.p	odd	6	1
1008.3.cg.c		2	60.h	even	2	1
1008.3.cg.c		2	420.be	odd	6	1
1764.3.d.a		2	105.o	odd	6	1
1764.3.d.a		2	105.p	even	6	1
1764.3.z.f		2	105.g	even	2	1
1764.3.z.f		2	105.o	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} + 3T_{3} + 3$ T3^2 + 3*T3 + 3 acting on $S_{3}^{\mathrm{new}}(700, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 3T + 3$ T^2 + 3*T + 3
$5$	$T^{2}$ T^2
$7$	$(T - 7)^{2}$ (T - 7)^2
$11$	$T^{2} + 15T + 225$ T^2 + 15*T + 225
$13$	$T^{2} + 192$ T^2 + 192
$17$	$T^{2} + 51T + 867$ T^2 + 51*T + 867
$19$	$T^{2} - 27T + 243$ T^2 - 27*T + 243
$23$	$T^{2} + 9T + 81$ T^2 + 9*T + 81
$29$	$(T + 6)^{2}$ (T + 6)^2
$31$	$T^{2} + 21T + 147$ T^2 + 21*T + 147
$37$	$T^{2} - 31T + 961$ T^2 - 31*T + 961
$41$	$T^{2} + 3072$ T^2 + 3072
$43$	$(T + 10)^{2}$ (T + 10)^2
$47$	$T^{2} + 75T + 1875$ T^2 + 75*T + 1875
$53$	$T^{2} + 57T + 3249$ T^2 + 57*T + 3249
$59$	$T^{2} + 141T + 6627$ T^2 + 141*T + 6627
$61$	$T^{2} + 141T + 6627$ T^2 + 141*T + 6627
$67$	$T^{2} + 49T + 2401$ T^2 + 49*T + 2401
$71$	$(T + 126)^{2}$ (T + 126)^2
$73$	$T^{2} - 45T + 675$ T^2 - 45*T + 675
$79$	$T^{2} - 73T + 5329$ T^2 - 73*T + 5329
$83$	$T^{2} + 192$ T^2 + 192
$89$	$T^{2} - 99T + 3267$ T^2 - 99*T + 3267
$97$	$T^{2} + 768$ T^2 + 768
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