LMFDB - Newform orbit 2100.4.k.g

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2100,4,Mod(1849,2100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2100.1849"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2100.k (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-18,0,8,0,0,0,0,0,0,0,-184] 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:	$123.904011012$
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:	$\Z[a_1, \ldots, a_{7}]$
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 $i = \sqrt{-1}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 3 i q^{3} + 7 i q^{7} - 9 q^{9} + 4 q^{11} + 54 i q^{13} + 14 i q^{17} - 92 q^{19} - 21 q^{21} - 152 i q^{23} - 27 i q^{27} + 106 q^{29} - 144 q^{31} + 12 i q^{33} - 158 i q^{37} - 162 q^{39} - 390 q^{41} + \cdots - 36 q^{99} +O(q^{100})$ q + 3i q^3 + 7i q^7 - 9 * q^9 + 4 * q^11 + 54i q^13 + 14i q^17 - 92 * q^19 - 21 * q^21 - 152i q^23 - 27i q^27 + 106 * q^29 - 144 * q^31 + 12i q^33 - 158i q^37 - 162 * q^39 - 390 * q^41 - 508i q^43 + 528i q^47 - 49 * q^49 - 42 * q^51 + 606i q^53 - 276i q^57 + 364 * q^59 + 678 * q^61 - 63i q^63 - 844i q^67 + 456 * q^69 - 8 * q^71 - 422i q^73 + 28i q^77 - 384 * q^79 + 81 * q^81 - 548i q^83 + 318i q^87 - 1194 * q^89 - 378 * q^91 - 432i q^93 + 1502i q^97 - 36 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 18 q^{9} + 8 q^{11} - 184 q^{19} - 42 q^{21} + 212 q^{29} - 288 q^{31} - 324 q^{39} - 780 q^{41} - 98 q^{49} - 84 q^{51} + 728 q^{59} + 1356 q^{61} + 912 q^{69} - 16 q^{71} - 768 q^{79} + 162 q^{81}+ \cdots - 72 q^{99}+O(q^{100})$ 2 * q - 18 * q^9 + 8 * q^11 - 184 * q^19 - 42 * q^21 + 212 * q^29 - 288 * q^31 - 324 * q^39 - 780 * q^41 - 98 * q^49 - 84 * q^51 + 728 * q^59 + 1356 * q^61 + 912 * q^69 - 16 * q^71 - 768 * q^79 + 162 * q^81 - 2388 * q^89 - 756 * q^91 - 72 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$ .

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$-1$	$1$

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}

1849.1

	−	1.00000i
		1.00000i

−

3.00000i

−

7.00000i

−9.00000

1849.2

3.00000i

7.00000i

−9.00000

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	2100.4.k.g		2
5.b	even	2	1	inner	2100.4.k.g		2
5.c	odd	4	1		84.4.a.b	✓	1
5.c	odd	4	1		2100.4.a.g		1
15.e	even	4	1		252.4.a.a		1
20.e	even	4	1		336.4.a.e		1
35.f	even	4	1		588.4.a.a		1
35.k	even	12	2		588.4.i.h		2
35.l	odd	12	2		588.4.i.a		2
40.i	odd	4	1		1344.4.a.b		1
40.k	even	4	1		1344.4.a.p		1
60.l	odd	4	1		1008.4.a.d		1
105.k	odd	4	1		1764.4.a.l		1
105.w	odd	12	2		1764.4.k.c		2
105.x	even	12	2		1764.4.k.n		2
140.j	odd	4	1		2352.4.a.v		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.4.a.b	✓	1	5.c	odd	4	1
252.4.a.a		1	15.e	even	4	1
336.4.a.e		1	20.e	even	4	1
588.4.a.a		1	35.f	even	4	1
588.4.i.a		2	35.l	odd	12	2
588.4.i.h		2	35.k	even	12	2
1008.4.a.d		1	60.l	odd	4	1
1344.4.a.b		1	40.i	odd	4	1
1344.4.a.p		1	40.k	even	4	1
1764.4.a.l		1	105.k	odd	4	1
1764.4.k.c		2	105.w	odd	12	2
1764.4.k.n		2	105.x	even	12	2
2100.4.a.g		1	5.c	odd	4	1
2100.4.k.g		2	1.a	even	1	1	trivial
2100.4.k.g		2	5.b	even	2	1	inner
2352.4.a.v		1	140.j	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(2100, [\chi])$ :

T_{11} - 4

T11 - 4

T_{13}^{2} + 2916

T13^2 + 2916

T_{17}^{2} + 196

T17^2 + 196

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 9$ T^2 + 9
$5$	$T^{2}$ T^2
$7$	$T^{2} + 49$ T^2 + 49
$11$	$(T - 4)^{2}$ (T - 4)^2
$13$	$T^{2} + 2916$ T^2 + 2916
$17$	$T^{2} + 196$ T^2 + 196
$19$	$(T + 92)^{2}$ (T + 92)^2
$23$	$T^{2} + 23104$ T^2 + 23104
$29$	$(T - 106)^{2}$ (T - 106)^2
$31$	$(T + 144)^{2}$ (T + 144)^2
$37$	$T^{2} + 24964$ T^2 + 24964
$41$	$(T + 390)^{2}$ (T + 390)^2
$43$	$T^{2} + 258064$ T^2 + 258064
$47$	$T^{2} + 278784$ T^2 + 278784
$53$	$T^{2} + 367236$ T^2 + 367236
$59$	$(T - 364)^{2}$ (T - 364)^2
$61$	$(T - 678)^{2}$ (T - 678)^2
$67$	$T^{2} + 712336$ T^2 + 712336
$71$	$(T + 8)^{2}$ (T + 8)^2
$73$	$T^{2} + 178084$ T^2 + 178084
$79$	$(T + 384)^{2}$ (T + 384)^2
$83$	$T^{2} + 300304$ T^2 + 300304
$89$	$(T + 1194)^{2}$ (T + 1194)^2
$97$	$T^{2} + 2256004$ T^2 + 2256004
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