LMFDB - Newform orbit 4600.2.e.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4600,2,Mod(4049,4600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4600.4049"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$4600 = 2^{3} \cdot 5^{2} \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$36.7311849298$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 184)
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.

$f(q)$	$=$	$q + 3 i q^{3} + 2 i q^{7} - 6 q^{9} - 5 i q^{13} + 6 i q^{17} - 6 q^{19} - 6 q^{21} + i q^{23} - 9 i q^{27} - 9 q^{29} + 3 q^{31} + 8 i q^{37} + 15 q^{39} + 3 q^{41} - 8 i q^{43} - 7 i q^{47} + 3 q^{49} + \cdots - 6 i q^{97} +O(q^{100})$ q + 3i q^3 + 2i q^7 - 6 * q^9 - 5i q^13 + 6i q^17 - 6 * q^19 - 6 * q^21 + i * q^23 - 9i q^27 - 9 * q^29 + 3 * q^31 + 8i q^37 + 15 * q^39 + 3 * q^41 - 8i q^43 - 7i q^47 + 3 * q^49 - 18 * q^51 - 2i q^53 - 18i q^57 - 4 * q^59 - 10 * q^61 - 12i q^63 - 8i q^67 - 3 * q^69 + 7 * q^71 + 9i q^73 + 6 * q^79 + 9 * q^81 - 14i q^83 - 27i q^87 - 16 * q^89 + 10 * q^91 + 9i q^93 - 6i q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 12 q^{9} - 12 q^{19} - 12 q^{21} - 18 q^{29} + 6 q^{31} + 30 q^{39} + 6 q^{41} + 6 q^{49} - 36 q^{51} - 8 q^{59} - 20 q^{61} - 6 q^{69} + 14 q^{71} + 12 q^{79} + 18 q^{81} - 32 q^{89} + 20 q^{91}+O(q^{100})$ 2 * q - 12 * q^9 - 12 * q^19 - 12 * q^21 - 18 * q^29 + 6 * q^31 + 30 * q^39 + 6 * q^41 + 6 * q^49 - 36 * q^51 - 8 * q^59 - 20 * q^61 - 6 * q^69 + 14 * q^71 + 12 * q^79 + 18 * q^81 - 32 * q^89 + 20 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$ .

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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}

4049.1

	−	1.00000i
		1.00000i

−

3.00000i

−

2.00000i

−6.00000

4049.2

3.00000i

2.00000i

−6.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4600.2.e.a		2
5.b	even	2	1	inner	4600.2.e.a		2
5.c	odd	4	1		184.2.a.d	✓	1
5.c	odd	4	1		4600.2.a.a		1
15.e	even	4	1		1656.2.a.c		1
20.e	even	4	1		368.2.a.a		1
20.e	even	4	1		9200.2.a.bj		1
35.f	even	4	1		9016.2.a.b		1
40.i	odd	4	1		1472.2.a.a		1
40.k	even	4	1		1472.2.a.m		1
60.l	odd	4	1		3312.2.a.i		1
115.e	even	4	1		4232.2.a.j		1
460.k	odd	4	1		8464.2.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
184.2.a.d	✓	1	5.c	odd	4	1
368.2.a.a		1	20.e	even	4	1
1472.2.a.a		1	40.i	odd	4	1
1472.2.a.m		1	40.k	even	4	1
1656.2.a.c		1	15.e	even	4	1
3312.2.a.i		1	60.l	odd	4	1
4232.2.a.j		1	115.e	even	4	1
4600.2.a.a		1	5.c	odd	4	1
4600.2.e.a		2	1.a	even	1	1	trivial
4600.2.e.a		2	5.b	even	2	1	inner
8464.2.a.b		1	460.k	odd	4	1
9016.2.a.b		1	35.f	even	4	1
9200.2.a.bj		1	20.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 $S_{2}^{\mathrm{new}}(4600, [\chi])$ :

T_{3}^{2} + 9

T3^2 + 9

T_{7}^{2} + 4

T7^2 + 4

T_{11}

T11

T_{13}^{2} + 25

T13^2 + 25

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 9$ T^2 + 9
$5$	$T^{2}$ T^2
$7$	$T^{2} + 4$ T^2 + 4
$11$	$T^{2}$ T^2
$13$	$T^{2} + 25$ T^2 + 25
$17$	$T^{2} + 36$ T^2 + 36
$19$	$(T + 6)^{2}$ (T + 6)^2
$23$	$T^{2} + 1$ T^2 + 1
$29$	$(T + 9)^{2}$ (T + 9)^2
$31$	$(T - 3)^{2}$ (T - 3)^2
$37$	$T^{2} + 64$ T^2 + 64
$41$	$(T - 3)^{2}$ (T - 3)^2
$43$	$T^{2} + 64$ T^2 + 64
$47$	$T^{2} + 49$ T^2 + 49
$53$	$T^{2} + 4$ T^2 + 4
$59$	$(T + 4)^{2}$ (T + 4)^2
$61$	$(T + 10)^{2}$ (T + 10)^2
$67$	$T^{2} + 64$ T^2 + 64
$71$	$(T - 7)^{2}$ (T - 7)^2
$73$	$T^{2} + 81$ T^2 + 81
$79$	$(T - 6)^{2}$ (T - 6)^2
$83$	$T^{2} + 196$ T^2 + 196
$89$	$(T + 16)^{2}$ (T + 16)^2
$97$	$T^{2} + 36$ T^2 + 36
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