LMFDB - Newform orbit 768.2.d.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(385,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("768.385");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$768 = 2^{8} \cdot 3$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	768.d (of order $2$ , degree $1$ , 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:	$6.13251087523$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 96)
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 $i = \sqrt{-1}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - i q^{3} + 2 i q^{5} + 4 q^{7} - q^{9} + 4 i q^{11} + 2 i q^{13} + 2 q^{15} - 6 q^{17} + 4 i q^{19} - 4 i q^{21} + q^{25} + i q^{27} - 2 i q^{29} + 4 q^{31} + 4 q^{33} + 8 i q^{35} - 2 i q^{37} + 2 q^{39} + \cdots - 4 i q^{99} +O(q^{100})$ q - i * q^3 + 2i q^5 + 4 * q^7 - q^9 + 4i q^11 + 2i q^13 + 2 * q^15 - 6 * q^17 + 4i q^19 - 4i q^21 + q^25 + i * q^27 - 2i q^29 + 4 * q^31 + 4 * q^33 + 8i q^35 - 2i q^37 + 2 * q^39 - 2 * q^41 + 4i q^43 - 2i q^45 + 8 * q^47 + 9 * q^49 + 6i q^51 + 10i q^53 - 8 * q^55 + 4 * q^57 - 4i q^59 - 6i q^61 - 4 * q^63 - 4 * q^65 - 4i q^67 + 16 * q^71 + 6 * q^73 - i * q^75 + 16i q^77 + 4 * q^79 + q^81 - 12i q^83 - 12i q^85 - 2 * q^87 - 10 * q^89 + 8i q^91 - 4i q^93 - 8 * q^95 - 14 * q^97 - 4i q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 8 q^{7} - 2 q^{9} + 4 q^{15} - 12 q^{17} + 2 q^{25} + 8 q^{31} + 8 q^{33} + 4 q^{39} - 4 q^{41} + 16 q^{47} + 18 q^{49} - 16 q^{55} + 8 q^{57} - 8 q^{63} - 8 q^{65} + 32 q^{71} + 12 q^{73} + 8 q^{79}+ \cdots - 28 q^{97}+O(q^{100})$ 2 * q + 8 * q^7 - 2 * q^9 + 4 * q^15 - 12 * q^17 + 2 * q^25 + 8 * q^31 + 8 * q^33 + 4 * q^39 - 4 * q^41 + 16 * q^47 + 18 * q^49 - 16 * q^55 + 8 * q^57 - 8 * q^63 - 8 * q^65 + 32 * q^71 + 12 * q^73 + 8 * q^79 + 2 * q^81 - 4 * q^87 - 20 * q^89 - 16 * q^95 - 28 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$ .

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$1$	$-1$

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}

385.1

		1.00000i
	−	1.00000i

−

1.00000i

2.00000i

4.00000

−1.00000

385.2

1.00000i

−

2.00000i

4.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	768.2.d.h		2
3.b	odd	2	1		2304.2.d.s		2
4.b	odd	2	1		768.2.d.a		2
8.b	even	2	1	inner	768.2.d.h		2
8.d	odd	2	1		768.2.d.a		2
12.b	even	2	1		2304.2.d.c		2
16.e	even	4	1		96.2.a.b	yes	1
16.e	even	4	1		192.2.a.a		1
16.f	odd	4	1		96.2.a.a	✓	1
16.f	odd	4	1		192.2.a.c		1
24.f	even	2	1		2304.2.d.c		2
24.h	odd	2	1		2304.2.d.s		2
48.i	odd	4	1		288.2.a.b		1
48.i	odd	4	1		576.2.a.g		1
48.k	even	4	1		288.2.a.c		1
48.k	even	4	1		576.2.a.h		1
80.i	odd	4	1		2400.2.f.r		2
80.i	odd	4	1		4800.2.f.e		2
80.j	even	4	1		2400.2.f.a		2
80.j	even	4	1		4800.2.f.bh		2
80.k	odd	4	1		2400.2.a.r		1
80.k	odd	4	1		4800.2.a.f		1
80.q	even	4	1		2400.2.a.q		1
80.q	even	4	1		4800.2.a.co		1
80.s	even	4	1		2400.2.f.a		2
80.s	even	4	1		4800.2.f.bh		2
80.t	odd	4	1		2400.2.f.r		2
80.t	odd	4	1		4800.2.f.e		2
112.j	even	4	1		4704.2.a.t		1
112.j	even	4	1		9408.2.a.bj		1
112.l	odd	4	1		4704.2.a.e		1
112.l	odd	4	1		9408.2.a.ct		1
144.u	even	12	2		2592.2.i.q		2
144.v	odd	12	2		2592.2.i.b		2
144.w	odd	12	2		2592.2.i.w		2
144.x	even	12	2		2592.2.i.h		2
240.t	even	4	1		7200.2.a.e		1
240.z	odd	4	1		7200.2.f.x		2
240.bb	even	4	1		7200.2.f.f		2
240.bd	odd	4	1		7200.2.f.x		2
240.bf	even	4	1		7200.2.f.f		2
240.bm	odd	4	1		7200.2.a.bx		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.2.a.a	✓	1	16.f	odd	4	1
96.2.a.b	yes	1	16.e	even	4	1
192.2.a.a		1	16.e	even	4	1
192.2.a.c		1	16.f	odd	4	1
288.2.a.b		1	48.i	odd	4	1
288.2.a.c		1	48.k	even	4	1
576.2.a.g		1	48.i	odd	4	1
576.2.a.h		1	48.k	even	4	1
768.2.d.a		2	4.b	odd	2	1
768.2.d.a		2	8.d	odd	2	1
768.2.d.h		2	1.a	even	1	1	trivial
768.2.d.h		2	8.b	even	2	1	inner
2304.2.d.c		2	12.b	even	2	1
2304.2.d.c		2	24.f	even	2	1
2304.2.d.s		2	3.b	odd	2	1
2304.2.d.s		2	24.h	odd	2	1
2400.2.a.q		1	80.q	even	4	1
2400.2.a.r		1	80.k	odd	4	1
2400.2.f.a		2	80.j	even	4	1
2400.2.f.a		2	80.s	even	4	1
2400.2.f.r		2	80.i	odd	4	1
2400.2.f.r		2	80.t	odd	4	1
2592.2.i.b		2	144.v	odd	12	2
2592.2.i.h		2	144.x	even	12	2
2592.2.i.q		2	144.u	even	12	2
2592.2.i.w		2	144.w	odd	12	2
4704.2.a.e		1	112.l	odd	4	1
4704.2.a.t		1	112.j	even	4	1
4800.2.a.f		1	80.k	odd	4	1
4800.2.a.co		1	80.q	even	4	1
4800.2.f.e		2	80.i	odd	4	1
4800.2.f.e		2	80.t	odd	4	1
4800.2.f.bh		2	80.j	even	4	1
4800.2.f.bh		2	80.s	even	4	1
7200.2.a.e		1	240.t	even	4	1
7200.2.a.bx		1	240.bm	odd	4	1
7200.2.f.f		2	240.bb	even	4	1
7200.2.f.f		2	240.bf	even	4	1
7200.2.f.x		2	240.z	odd	4	1
7200.2.f.x		2	240.bd	odd	4	1
9408.2.a.bj		1	112.j	even	4	1
9408.2.a.ct		1	112.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(768, [\chi])$ :

T_{5}^{2} + 4

T_{7} - 4

T_{23}

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 1$ T^2 + 1
$5$	$T^{2} + 4$ T^2 + 4
$7$	$(T - 4)^{2}$ (T - 4)^2
$11$	$T^{2} + 16$ T^2 + 16
$13$	$T^{2} + 4$ T^2 + 4
$17$	$(T + 6)^{2}$ (T + 6)^2
$19$	$T^{2} + 16$ T^2 + 16
$23$	$T^{2}$ T^2
$29$	$T^{2} + 4$ T^2 + 4
$31$	$(T - 4)^{2}$ (T - 4)^2
$37$	$T^{2} + 4$ T^2 + 4
$41$	$(T + 2)^{2}$ (T + 2)^2
$43$	$T^{2} + 16$ T^2 + 16
$47$	$(T - 8)^{2}$ (T - 8)^2
$53$	$T^{2} + 100$ T^2 + 100
$59$	$T^{2} + 16$ T^2 + 16
$61$	$T^{2} + 36$ T^2 + 36
$67$	$T^{2} + 16$ T^2 + 16
$71$	$(T - 16)^{2}$ (T - 16)^2
$73$	$(T - 6)^{2}$ (T - 6)^2
$79$	$(T - 4)^{2}$ (T - 4)^2
$83$	$T^{2} + 144$ T^2 + 144
$89$	$(T + 10)^{2}$ (T + 10)^2
$97$	$(T + 14)^{2}$ (T + 14)^2
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