LMFDB - Newform orbit 2352.2.k.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2352,2,Mod(881,2352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2352.881"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$2352 = 2^{4} \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2352.k (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-3,0,-6,0,0,0,3,0,0,0,0,0,9,0,-6,0,0,0,0,0,0,0,8,0,0,0,0, 0,0,0,-9,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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

Self dual:	no
Analytic conductor:	$18.7808145554$
Analytic rank:	$1$
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 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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 a primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\zeta_{6} - 2) q^{3} - 3 q^{5} + ( - 3 \zeta_{6} + 3) q^{9} + (6 \zeta_{6} - 3) q^{11} + ( - 3 \zeta_{6} + 6) q^{15} - 3 q^{17} + (2 \zeta_{6} - 1) q^{19} + (6 \zeta_{6} - 3) q^{23} + 4 q^{25} + (6 \zeta_{6} - 3) q^{27}+ \cdots + (9 \zeta_{6} + 9) q^{99}+O(q^{100})$ q + (z - 2) * q^3 - 3 * q^5 + (-3z + 3) q^9 + (6z - 3) q^11 + (-3z + 6) q^15 - 3 * q^17 + (2z - 1) q^19 + (6z - 3) q^23 + 4 * q^25 + (6z - 3) q^27 + (2z - 1) q^31 - 9z q^33 + 7 * q^37 - 6 * q^41 - 4 * q^43 + (9z - 9) q^45 + 3 * q^47 + (-3z + 6) q^51 + (6z - 3) q^53 + (-18z + 9) q^55 - 3z q^57 - 3 * q^59 + (14z - 7) q^61 - 5 * q^67 - 9z q^69 + (-12z + 6) q^71 + (-14z + 7) q^73 + (4z - 8) q^75 + q^79 - 9z q^81 - 12 * q^83 + 9 * q^85 + 9 * q^89 - 3z q^93 + (-6z + 3) q^95 + (-8z + 4) q^97 + (9z + 9) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 3 q^{3} - 6 q^{5} + 3 q^{9} + 9 q^{15} - 6 q^{17} + 8 q^{25} - 9 q^{33} + 14 q^{37} - 12 q^{41} - 8 q^{43} - 9 q^{45} + 6 q^{47} + 9 q^{51} - 3 q^{57} - 6 q^{59} - 10 q^{67} - 9 q^{69} - 12 q^{75}+ \cdots + 27 q^{99}+O(q^{100})$ 2 * q - 3 * q^3 - 6 * q^5 + 3 * q^9 + 9 * q^15 - 6 * q^17 + 8 * q^25 - 9 * q^33 + 14 * q^37 - 12 * q^41 - 8 * q^43 - 9 * q^45 + 6 * q^47 + 9 * q^51 - 3 * q^57 - 6 * q^59 - 10 * q^67 - 9 * q^69 - 12 * q^75 + 2 * q^79 - 9 * q^81 - 24 * q^83 + 18 * q^85 + 18 * q^89 - 3 * q^93 + 27 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$ .

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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}

881.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.50000

−

0.866025i

−3.00000

1.50000

2.59808i

881.2

−1.50000

0.866025i

−3.00000

1.50000

−

2.59808i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2352.2.k.a		2
3.b	odd	2	1		2352.2.k.d		2
4.b	odd	2	1		588.2.f.c		2
7.b	odd	2	1		2352.2.k.d		2
7.c	even	3	1		336.2.bc.d		2
7.d	odd	6	1		336.2.bc.b		2
12.b	even	2	1		588.2.f.a		2
21.c	even	2	1	inner	2352.2.k.a		2
21.g	even	6	1		336.2.bc.d		2
21.h	odd	6	1		336.2.bc.b		2
28.d	even	2	1		588.2.f.a		2
28.f	even	6	1		84.2.k.b	yes	2
28.f	even	6	1		588.2.k.d		2
28.g	odd	6	1		84.2.k.a	✓	2
28.g	odd	6	1		588.2.k.c		2
84.h	odd	2	1		588.2.f.c		2
84.j	odd	6	1		84.2.k.a	✓	2
84.j	odd	6	1		588.2.k.c		2
84.n	even	6	1		84.2.k.b	yes	2
84.n	even	6	1		588.2.k.d		2
140.p	odd	6	1		2100.2.bi.f		2
140.s	even	6	1		2100.2.bi.e		2
140.w	even	12	2		2100.2.bo.a		4
140.x	odd	12	2		2100.2.bo.f		4
252.n	even	6	1		2268.2.bm.f		2
252.o	even	6	1		2268.2.bm.f		2
252.r	odd	6	1		2268.2.w.f		2
252.u	odd	6	1		2268.2.w.f		2
252.bb	even	6	1		2268.2.w.a		2
252.bj	even	6	1		2268.2.w.a		2
252.bl	odd	6	1		2268.2.bm.a		2
252.bn	odd	6	1		2268.2.bm.a		2
420.ba	even	6	1		2100.2.bi.e		2
420.be	odd	6	1		2100.2.bi.f		2
420.bp	odd	12	2		2100.2.bo.f		4
420.br	even	12	2		2100.2.bo.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.2.k.a	✓	2	28.g	odd	6	1
84.2.k.a	✓	2	84.j	odd	6	1
84.2.k.b	yes	2	28.f	even	6	1
84.2.k.b	yes	2	84.n	even	6	1
336.2.bc.b		2	7.d	odd	6	1
336.2.bc.b		2	21.h	odd	6	1
336.2.bc.d		2	7.c	even	3	1
336.2.bc.d		2	21.g	even	6	1
588.2.f.a		2	12.b	even	2	1
588.2.f.a		2	28.d	even	2	1
588.2.f.c		2	4.b	odd	2	1
588.2.f.c		2	84.h	odd	2	1
588.2.k.c		2	28.g	odd	6	1
588.2.k.c		2	84.j	odd	6	1
588.2.k.d		2	28.f	even	6	1
588.2.k.d		2	84.n	even	6	1
2100.2.bi.e		2	140.s	even	6	1
2100.2.bi.e		2	420.ba	even	6	1
2100.2.bi.f		2	140.p	odd	6	1
2100.2.bi.f		2	420.be	odd	6	1
2100.2.bo.a		4	140.w	even	12	2
2100.2.bo.a		4	420.br	even	12	2
2100.2.bo.f		4	140.x	odd	12	2
2100.2.bo.f		4	420.bp	odd	12	2
2268.2.w.a		2	252.bb	even	6	1
2268.2.w.a		2	252.bj	even	6	1
2268.2.w.f		2	252.r	odd	6	1
2268.2.w.f		2	252.u	odd	6	1
2268.2.bm.a		2	252.bl	odd	6	1
2268.2.bm.a		2	252.bn	odd	6	1
2268.2.bm.f		2	252.n	even	6	1
2268.2.bm.f		2	252.o	even	6	1
2352.2.k.a		2	1.a	even	1	1	trivial
2352.2.k.a		2	21.c	even	2	1	inner
2352.2.k.d		2	3.b	odd	2	1
2352.2.k.d		2	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2352, [\chi])$ :

T_{5} + 3

T5 + 3

T_{13}

T13

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 3T + 3$ T^2 + 3*T + 3
$5$	$(T + 3)^{2}$ (T + 3)^2
$7$	$T^{2}$ T^2
$11$	$T^{2} + 27$ T^2 + 27
$13$	$T^{2}$ T^2
$17$	$(T + 3)^{2}$ (T + 3)^2
$19$	$T^{2} + 3$ T^2 + 3
$23$	$T^{2} + 27$ T^2 + 27
$29$	$T^{2}$ T^2
$31$	$T^{2} + 3$ T^2 + 3
$37$	$(T - 7)^{2}$ (T - 7)^2
$41$	$(T + 6)^{2}$ (T + 6)^2
$43$	$(T + 4)^{2}$ (T + 4)^2
$47$	$(T - 3)^{2}$ (T - 3)^2
$53$	$T^{2} + 27$ T^2 + 27
$59$	$(T + 3)^{2}$ (T + 3)^2
$61$	$T^{2} + 147$ T^2 + 147
$67$	$(T + 5)^{2}$ (T + 5)^2
$71$	$T^{2} + 108$ T^2 + 108
$73$	$T^{2} + 147$ T^2 + 147
$79$	$(T - 1)^{2}$ (T - 1)^2
$83$	$(T + 12)^{2}$ (T + 12)^2
$89$	$(T - 9)^{2}$ (T - 9)^2
$97$	$T^{2} + 48$ T^2 + 48
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