LMFDB - Newform orbit 1600.2.s.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(207,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 1]))

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

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

chi := DirichletCharacter("1600.207");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1600 = 2^{6} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1600.s (of order $4$ , 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:	$12.7760643234$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$ -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.

$f(q)$	$=$	$q - 2 q^{3} + ( - 3 i - 3) q^{7} + q^{9} + ( - i + 1) q^{11} - 2 i q^{13} + ( - i - 1) q^{17} + ( - 3 i + 3) q^{19} + (6 i + 6) q^{21} + (i - 1) q^{23} + 4 q^{27} + ( - 7 i - 7) q^{29} + 2 i q^{31} + (2 i - 2) q^{33} + \cdots + ( - i + 1) q^{99} +O(q^{100})$ q - 2 * q^3 + (-3i - 3) q^7 + q^9 + (-i + 1) * q^11 - 2i q^13 + (-i - 1) * q^17 + (-3i + 3) q^19 + (6i + 6) q^21 + (i - 1) * q^23 + 4 * q^27 + (-7i - 7) q^29 + 2i q^31 + (2i - 2) q^33 + 6i q^37 + 4i q^39 + 4i q^41 - 4i q^43 + (7i - 7) q^47 + 11i q^49 + (2i + 2) q^51 + 8 * q^53 + (6i - 6) q^57 + (-3i - 3) q^59 + (i - 1) * q^61 + (-3i - 3) q^63 + 4i q^67 + (-2i + 2) q^69 + (-3i - 3) q^73 - 6 * q^77 - 8 * q^79 - 11 * q^81 - 2 * q^83 + (14i + 14) q^87 - 6 * q^89 + (6i - 6) q^91 - 4i q^93 + (11i + 11) q^97 + (-i + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 4 q^{3} - 6 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{17} + 6 q^{19} + 12 q^{21} - 2 q^{23} + 8 q^{27} - 14 q^{29} - 4 q^{33} - 14 q^{47} + 4 q^{51} + 16 q^{53} - 12 q^{57} - 6 q^{59} - 2 q^{61} - 6 q^{63}+ \cdots + 2 q^{99}+O(q^{100})$ 2 * q - 4 * q^3 - 6 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^17 + 6 * q^19 + 12 * q^21 - 2 * q^23 + 8 * q^27 - 14 * q^29 - 4 * q^33 - 14 * q^47 + 4 * q^51 + 16 * q^53 - 12 * q^57 - 6 * q^59 - 2 * q^61 - 6 * q^63 + 4 * q^69 - 6 * q^73 - 12 * q^77 - 16 * q^79 - 22 * q^81 - 4 * q^83 + 28 * q^87 - 12 * q^89 - 12 * q^91 + 22 * q^97 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$ .

$n$	$577$	$901$	$1151$
$\chi(n)$	$i$	$i$	$-1$

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}

207.1

		1.00000i
	−	1.00000i

−2.00000

−3.00000

−

3.00000i

1.00000

943.1

−2.00000

−3.00000

3.00000i

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
80.s	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1600.2.s.a		2
4.b	odd	2	1		400.2.s.a		2
5.b	even	2	1		320.2.s.a		2
5.c	odd	4	1		320.2.j.a		2
5.c	odd	4	1		1600.2.j.a		2
16.e	even	4	1		400.2.j.a		2
16.f	odd	4	1		1600.2.j.a		2
20.d	odd	2	1		80.2.s.a	yes	2
20.e	even	4	1		80.2.j.a	✓	2
20.e	even	4	1		400.2.j.a		2
40.e	odd	2	1		640.2.s.b		2
40.f	even	2	1		640.2.s.a		2
40.i	odd	4	1		640.2.j.b		2
40.k	even	4	1		640.2.j.a		2
60.h	even	2	1		720.2.z.d		2
60.l	odd	4	1		720.2.bd.a		2
80.i	odd	4	1		400.2.s.a		2
80.i	odd	4	1		640.2.s.b		2
80.j	even	4	1		320.2.s.a		2
80.k	odd	4	1		320.2.j.a		2
80.k	odd	4	1		640.2.j.b		2
80.q	even	4	1		80.2.j.a	✓	2
80.q	even	4	1		640.2.j.a		2
80.s	even	4	1		640.2.s.a		2
80.s	even	4	1	inner	1600.2.s.a		2
80.t	odd	4	1		80.2.s.a	yes	2
240.bf	even	4	1		720.2.z.d		2
240.bm	odd	4	1		720.2.bd.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
80.2.j.a	✓	2	20.e	even	4	1
80.2.j.a	✓	2	80.q	even	4	1
80.2.s.a	yes	2	20.d	odd	2	1
80.2.s.a	yes	2	80.t	odd	4	1
320.2.j.a		2	5.c	odd	4	1
320.2.j.a		2	80.k	odd	4	1
320.2.s.a		2	5.b	even	2	1
320.2.s.a		2	80.j	even	4	1
400.2.j.a		2	16.e	even	4	1
400.2.j.a		2	20.e	even	4	1
400.2.s.a		2	4.b	odd	2	1
400.2.s.a		2	80.i	odd	4	1
640.2.j.a		2	40.k	even	4	1
640.2.j.a		2	80.q	even	4	1
640.2.j.b		2	40.i	odd	4	1
640.2.j.b		2	80.k	odd	4	1
640.2.s.a		2	40.f	even	2	1
640.2.s.a		2	80.s	even	4	1
640.2.s.b		2	40.e	odd	2	1
640.2.s.b		2	80.i	odd	4	1
720.2.z.d		2	60.h	even	2	1
720.2.z.d		2	240.bf	even	4	1
720.2.bd.a		2	60.l	odd	4	1
720.2.bd.a		2	240.bm	odd	4	1
1600.2.j.a		2	5.c	odd	4	1
1600.2.j.a		2	16.f	odd	4	1
1600.2.s.a		2	1.a	even	1	1	trivial
1600.2.s.a		2	80.s	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} + 2$ T3 + 2 acting on $S_{2}^{\mathrm{new}}(1600, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$(T + 2)^{2}$ (T + 2)^2
$5$	$T^{2}$ T^2
$7$	$T^{2} + 6T + 18$ T^2 + 6*T + 18
$11$	$T^{2} - 2T + 2$ T^2 - 2*T + 2
$13$	$T^{2} + 4$ T^2 + 4
$17$	$T^{2} + 2T + 2$ T^2 + 2*T + 2
$19$	$T^{2} - 6T + 18$ T^2 - 6*T + 18
$23$	$T^{2} + 2T + 2$ T^2 + 2*T + 2
$29$	$T^{2} + 14T + 98$ T^2 + 14*T + 98
$31$	$T^{2} + 4$ T^2 + 4
$37$	$T^{2} + 36$ T^2 + 36
$41$	$T^{2} + 16$ T^2 + 16
$43$	$T^{2} + 16$ T^2 + 16
$47$	$T^{2} + 14T + 98$ T^2 + 14*T + 98
$53$	$(T - 8)^{2}$ (T - 8)^2
$59$	$T^{2} + 6T + 18$ T^2 + 6*T + 18
$61$	$T^{2} + 2T + 2$ T^2 + 2*T + 2
$67$	$T^{2} + 16$ T^2 + 16
$71$	$T^{2}$ T^2
$73$	$T^{2} + 6T + 18$ T^2 + 6*T + 18
$79$	$(T + 8)^{2}$ (T + 8)^2
$83$	$(T + 2)^{2}$ (T + 2)^2
$89$	$(T + 6)^{2}$ (T + 6)^2
$97$	$T^{2} - 22T + 242$ T^2 - 22*T + 242
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