LMFDB - Newform orbit 1050.4.g.n

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1050,4,Mod(799,1050)] 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("1050.799"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1050.g (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-8,0,12,0,0,-18,0,-16,0,0,28,0,32,0,0,248] 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:	$61.9520055060$
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 42)
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.

$f(q)$	$=$	$q + 2 i q^{2} - 3 i q^{3} - 4 q^{4} + 6 q^{6} - 7 i q^{7} - 8 i q^{8} - 9 q^{9} - 8 q^{11} + 12 i q^{12} + 42 i q^{13} + 14 q^{14} + 16 q^{16} - 2 i q^{17} - 18 i q^{18} + 124 q^{19} - 21 q^{21} - 16 i q^{22} + \cdots + 72 q^{99} +O(q^{100})$ q + 2i q^2 - 3i q^3 - 4 * q^4 + 6 * q^6 - 7i q^7 - 8i q^8 - 9 * q^9 - 8 * q^11 + 12i q^12 + 42i q^13 + 14 * q^14 + 16 * q^16 - 2i q^17 - 18i q^18 + 124 * q^19 - 21 * q^21 - 16i q^22 - 76i q^23 - 24 * q^24 - 84 * q^26 + 27i q^27 + 28i q^28 - 254 * q^29 - 72 * q^31 + 32i q^32 + 24i q^33 + 4 * q^34 + 36 * q^36 + 398i q^37 + 248i q^38 + 126 * q^39 + 462 * q^41 - 42i q^42 - 212i q^43 + 32 * q^44 + 152 * q^46 - 264i q^47 - 48i q^48 - 49 * q^49 - 6 * q^51 - 168i q^52 + 162i q^53 - 54 * q^54 - 56 * q^56 - 372i q^57 - 508i q^58 + 772 * q^59 + 30 * q^61 - 144i q^62 + 63i q^63 - 64 * q^64 - 48 * q^66 - 764i q^67 + 8i q^68 - 228 * q^69 - 236 * q^71 + 72i q^72 - 418i q^73 - 796 * q^74 - 496 * q^76 + 56i q^77 + 252i q^78 - 552 * q^79 + 81 * q^81 + 924i q^82 - 1036i q^83 + 84 * q^84 + 424 * q^86 + 762i q^87 + 64i q^88 - 30 * q^89 + 294 * q^91 + 304i q^92 + 216i q^93 + 528 * q^94 + 96 * q^96 - 1190i q^97 - 98i q^98 + 72 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{4} + 12 q^{6} - 18 q^{9} - 16 q^{11} + 28 q^{14} + 32 q^{16} + 248 q^{19} - 42 q^{21} - 48 q^{24} - 168 q^{26} - 508 q^{29} - 144 q^{31} + 8 q^{34} + 72 q^{36} + 252 q^{39} + 924 q^{41} + 64 q^{44}+ \cdots + 144 q^{99}+O(q^{100})$ 2 * q - 8 * q^4 + 12 * q^6 - 18 * q^9 - 16 * q^11 + 28 * q^14 + 32 * q^16 + 248 * q^19 - 42 * q^21 - 48 * q^24 - 168 * q^26 - 508 * q^29 - 144 * q^31 + 8 * q^34 + 72 * q^36 + 252 * q^39 + 924 * q^41 + 64 * q^44 + 304 * q^46 - 98 * q^49 - 12 * q^51 - 108 * q^54 - 112 * q^56 + 1544 * q^59 + 60 * q^61 - 128 * q^64 - 96 * q^66 - 456 * q^69 - 472 * q^71 - 1592 * q^74 - 992 * q^76 - 1104 * q^79 + 162 * q^81 + 168 * q^84 + 848 * q^86 - 60 * q^89 + 588 * q^91 + 1056 * q^94 + 192 * q^96 + 144 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$ .

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$1$	$1$

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}

799.1

	−	1.00000i
		1.00000i

−

2.00000i

3.00000i

−4.00000

6.00000

7.00000i

8.00000i

−9.00000

799.2

2.00000i

−

3.00000i

−4.00000

6.00000

−

7.00000i

−

8.00000i

−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	1050.4.g.n		2
5.b	even	2	1	inner	1050.4.g.n		2
5.c	odd	4	1		42.4.a.b	✓	1
5.c	odd	4	1		1050.4.a.d		1
15.e	even	4	1		126.4.a.c		1
20.e	even	4	1		336.4.a.d		1
35.f	even	4	1		294.4.a.h		1
35.k	even	12	2		294.4.e.d		2
35.l	odd	12	2		294.4.e.a		2
40.i	odd	4	1		1344.4.a.f		1
40.k	even	4	1		1344.4.a.t		1
60.l	odd	4	1		1008.4.a.j		1
105.k	odd	4	1		882.4.a.d		1
105.w	odd	12	2		882.4.g.r		2
105.x	even	12	2		882.4.g.s		2
140.j	odd	4	1		2352.4.a.ba		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.4.a.b	✓	1	5.c	odd	4	1
126.4.a.c		1	15.e	even	4	1
294.4.a.h		1	35.f	even	4	1
294.4.e.a		2	35.l	odd	12	2
294.4.e.d		2	35.k	even	12	2
336.4.a.d		1	20.e	even	4	1
882.4.a.d		1	105.k	odd	4	1
882.4.g.r		2	105.w	odd	12	2
882.4.g.s		2	105.x	even	12	2
1008.4.a.j		1	60.l	odd	4	1
1050.4.a.d		1	5.c	odd	4	1
1050.4.g.n		2	1.a	even	1	1	trivial
1050.4.g.n		2	5.b	even	2	1	inner
1344.4.a.f		1	40.i	odd	4	1
1344.4.a.t		1	40.k	even	4	1
2352.4.a.ba		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 $S_{4}^{\mathrm{new}}(1050, [\chi])$ :

T_{11} + 8

T11 + 8

T_{13}^{2} + 1764

T13^2 + 1764

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 4$ T^2 + 4
$3$	$T^{2} + 9$ T^2 + 9
$5$	$T^{2}$ T^2
$7$	$T^{2} + 49$ T^2 + 49
$11$	$(T + 8)^{2}$ (T + 8)^2
$13$	$T^{2} + 1764$ T^2 + 1764
$17$	$T^{2} + 4$ T^2 + 4
$19$	$(T - 124)^{2}$ (T - 124)^2
$23$	$T^{2} + 5776$ T^2 + 5776
$29$	$(T + 254)^{2}$ (T + 254)^2
$31$	$(T + 72)^{2}$ (T + 72)^2
$37$	$T^{2} + 158404$ T^2 + 158404
$41$	$(T - 462)^{2}$ (T - 462)^2
$43$	$T^{2} + 44944$ T^2 + 44944
$47$	$T^{2} + 69696$ T^2 + 69696
$53$	$T^{2} + 26244$ T^2 + 26244
$59$	$(T - 772)^{2}$ (T - 772)^2
$61$	$(T - 30)^{2}$ (T - 30)^2
$67$	$T^{2} + 583696$ T^2 + 583696
$71$	$(T + 236)^{2}$ (T + 236)^2
$73$	$T^{2} + 174724$ T^2 + 174724
$79$	$(T + 552)^{2}$ (T + 552)^2
$83$	$T^{2} + 1073296$ T^2 + 1073296
$89$	$(T + 30)^{2}$ (T + 30)^2
$97$	$T^{2} + 1416100$ T^2 + 1416100
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