LMFDB - Newform orbit 1216.2.c.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(609,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1216 = 2^{6} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1216.c (of order $2$ , degree $1$ , 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:	$9.70980888579$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
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.

$f(q)$	$=$	$q + 3 i q^{3} - 4 i q^{5} - q^{7} - 6 q^{9} + 5 i q^{13} + 12 q^{15} - 5 q^{17} + i q^{19} - 3 i q^{21} + 3 q^{23} - 11 q^{25} - 9 i q^{27} - 7 i q^{29} - 10 q^{31} + 4 i q^{35} - 2 i q^{37} - 15 q^{39} + \cdots - 12 q^{97} +O(q^{100})$ q + 3i q^3 - 4i q^5 - q^7 - 6 * q^9 + 5i q^13 + 12 * q^15 - 5 * q^17 + i * q^19 - 3i q^21 + 3 * q^23 - 11 * q^25 - 9i q^27 - 7i q^29 - 10 * q^31 + 4i q^35 - 2i q^37 - 15 * q^39 - 6 * q^41 - 4i q^43 + 24i q^45 - 8 * q^47 - 6 * q^49 - 15i q^51 - 9i q^53 - 3 * q^57 + i * q^59 + 2i q^61 + 6 * q^63 + 20 * q^65 + 7i q^67 + 9i q^69 + 12 * q^71 + 11 * q^73 - 33i q^75 - 16 * q^79 + 9 * q^81 + 14i q^83 + 20i q^85 + 21 * q^87 - 4 * q^89 - 5i q^91 - 30i q^93 + 4 * q^95 - 12 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{7} - 12 q^{9} + 24 q^{15} - 10 q^{17} + 6 q^{23} - 22 q^{25} - 20 q^{31} - 30 q^{39} - 12 q^{41} - 16 q^{47} - 12 q^{49} - 6 q^{57} + 12 q^{63} + 40 q^{65} + 24 q^{71} + 22 q^{73} - 32 q^{79}+ \cdots - 24 q^{97}+O(q^{100})$ 2 * q - 2 * q^7 - 12 * q^9 + 24 * q^15 - 10 * q^17 + 6 * q^23 - 22 * q^25 - 20 * q^31 - 30 * q^39 - 12 * q^41 - 16 * q^47 - 12 * q^49 - 6 * q^57 + 12 * q^63 + 40 * q^65 + 24 * q^71 + 22 * q^73 - 32 * q^79 + 18 * q^81 + 42 * q^87 - 8 * q^89 + 8 * q^95 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$ .

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$1$	$-1$

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}

609.1

	−	1.00000i
		1.00000i

−

3.00000i

4.00000i

−1.00000

−6.00000

609.2

3.00000i

−

4.00000i

−1.00000

−6.00000

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	1216.2.c.b	✓	2
4.b	odd	2	1		1216.2.c.c	yes	2
8.b	even	2	1	inner	1216.2.c.b	✓	2
8.d	odd	2	1		1216.2.c.c	yes	2
16.e	even	4	1		4864.2.a.a		1
16.e	even	4	1		4864.2.a.p		1
16.f	odd	4	1		4864.2.a.b		1
16.f	odd	4	1		4864.2.a.o		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1216.2.c.b	✓	2	1.a	even	1	1	trivial
1216.2.c.b	✓	2	8.b	even	2	1	inner
1216.2.c.c	yes	2	4.b	odd	2	1
1216.2.c.c	yes	2	8.d	odd	2	1
4864.2.a.a		1	16.e	even	4	1
4864.2.a.b		1	16.f	odd	4	1
4864.2.a.o		1	16.f	odd	4	1
4864.2.a.p		1	16.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1216, [\chi])$ :

T_{3}^{2} + 9

T_{5}^{2} + 16

T_{7} + 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 9$ T^2 + 9
$5$	$T^{2} + 16$ T^2 + 16
$7$	$(T + 1)^{2}$ (T + 1)^2
$11$	$T^{2}$ T^2
$13$	$T^{2} + 25$ T^2 + 25
$17$	$(T + 5)^{2}$ (T + 5)^2
$19$	$T^{2} + 1$ T^2 + 1
$23$	$(T - 3)^{2}$ (T - 3)^2
$29$	$T^{2} + 49$ T^2 + 49
$31$	$(T + 10)^{2}$ (T + 10)^2
$37$	$T^{2} + 4$ T^2 + 4
$41$	$(T + 6)^{2}$ (T + 6)^2
$43$	$T^{2} + 16$ T^2 + 16
$47$	$(T + 8)^{2}$ (T + 8)^2
$53$	$T^{2} + 81$ T^2 + 81
$59$	$T^{2} + 1$ T^2 + 1
$61$	$T^{2} + 4$ T^2 + 4
$67$	$T^{2} + 49$ T^2 + 49
$71$	$(T - 12)^{2}$ (T - 12)^2
$73$	$(T - 11)^{2}$ (T - 11)^2
$79$	$(T + 16)^{2}$ (T + 16)^2
$83$	$T^{2} + 196$ T^2 + 196
$89$	$(T + 4)^{2}$ (T + 4)^2
$97$	$(T + 12)^{2}$ (T + 12)^2
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