LMFDB - Newform orbit 1650.4.c.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1650,4,Mod(199,1650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1650.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1650.c (of order $2$ , degree $1$ , 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:	$97.3531515095$
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 66)
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.

$f(q)$	$=$	$q - 2 i q^{2} - 3 i q^{3} - 4 q^{4} - 6 q^{6} + 14 i q^{7} + 8 i q^{8} - 9 q^{9} + 11 q^{11} + 12 i q^{12} - 80 i q^{13} + 28 q^{14} + 16 q^{16} + 30 i q^{17} + 18 i q^{18} - 56 q^{19} + 42 q^{21} - 22 i q^{22} + \cdots - 99 q^{99} +O(q^{100})$ q - 2i q^2 - 3i q^3 - 4 * q^4 - 6 * q^6 + 14i q^7 + 8i q^8 - 9 * q^9 + 11 * q^11 + 12i q^12 - 80i q^13 + 28 * q^14 + 16 * q^16 + 30i q^17 + 18i q^18 - 56 * q^19 + 42 * q^21 - 22i q^22 + 126i q^23 + 24 * q^24 - 160 * q^26 + 27i q^27 - 56i q^28 + 222 * q^29 - 16 * q^31 - 32i q^32 - 33i q^33 + 60 * q^34 + 36 * q^36 - 106i q^37 + 112i q^38 - 240 * q^39 + 114 * q^41 - 84i q^42 + 52i q^43 - 44 * q^44 + 252 * q^46 + 246i q^47 - 48i q^48 + 147 * q^49 + 90 * q^51 + 320i q^52 + 264i q^53 + 54 * q^54 - 112 * q^56 + 168i q^57 - 444i q^58 - 264 * q^59 + 92 * q^61 + 32i q^62 - 126i q^63 - 64 * q^64 - 66 * q^66 - 796i q^67 - 120i q^68 + 378 * q^69 + 426 * q^71 - 72i q^72 + 1174i q^73 - 212 * q^74 + 224 * q^76 + 154i q^77 + 480i q^78 - 842 * q^79 + 81 * q^81 - 228i q^82 - 852i q^83 - 168 * q^84 + 104 * q^86 - 666i q^87 + 88i q^88 + 1062 * q^89 + 1120 * q^91 - 504i q^92 + 48i q^93 + 492 * q^94 - 96 * q^96 - 1282i q^97 - 294i q^98 - 99 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{4} - 12 q^{6} - 18 q^{9} + 22 q^{11} + 56 q^{14} + 32 q^{16} - 112 q^{19} + 84 q^{21} + 48 q^{24} - 320 q^{26} + 444 q^{29} - 32 q^{31} + 120 q^{34} + 72 q^{36} - 480 q^{39} + 228 q^{41} - 88 q^{44}+ \cdots - 198 q^{99}+O(q^{100})$ 2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9 + 22 * q^11 + 56 * q^14 + 32 * q^16 - 112 * q^19 + 84 * q^21 + 48 * q^24 - 320 * q^26 + 444 * q^29 - 32 * q^31 + 120 * q^34 + 72 * q^36 - 480 * q^39 + 228 * q^41 - 88 * q^44 + 504 * q^46 + 294 * q^49 + 180 * q^51 + 108 * q^54 - 224 * q^56 - 528 * q^59 + 184 * q^61 - 128 * q^64 - 132 * q^66 + 756 * q^69 + 852 * q^71 - 424 * q^74 + 448 * q^76 - 1684 * q^79 + 162 * q^81 - 336 * q^84 + 208 * q^86 + 2124 * q^89 + 2240 * q^91 + 984 * q^94 - 192 * q^96 - 198 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times$ .

$n$	$551$	$727$	$1201$
$\chi(n)$	$1$	$-1$	$1$

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}

199.1

		1.00000i
	−	1.00000i

−

2.00000i

−

3.00000i

−4.00000

−6.00000

14.0000i

8.00000i

−9.00000

199.2

2.00000i

3.00000i

−4.00000

−6.00000

−

14.0000i

−

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	1650.4.c.g		2
5.b	even	2	1	inner	1650.4.c.g		2
5.c	odd	4	1		66.4.a.a	✓	1
5.c	odd	4	1		1650.4.a.h		1
15.e	even	4	1		198.4.a.f		1
20.e	even	4	1		528.4.a.d		1
40.i	odd	4	1		2112.4.a.g		1
40.k	even	4	1		2112.4.a.s		1
55.e	even	4	1		726.4.a.h		1
60.l	odd	4	1		1584.4.a.i		1
165.l	odd	4	1		2178.4.a.g		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
66.4.a.a	✓	1	5.c	odd	4	1
198.4.a.f		1	15.e	even	4	1
528.4.a.d		1	20.e	even	4	1
726.4.a.h		1	55.e	even	4	1
1584.4.a.i		1	60.l	odd	4	1
1650.4.a.h		1	5.c	odd	4	1
1650.4.c.g		2	1.a	even	1	1	trivial
1650.4.c.g		2	5.b	even	2	1	inner
2112.4.a.g		1	40.i	odd	4	1
2112.4.a.s		1	40.k	even	4	1
2178.4.a.g		1	165.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(1650, [\chi])$ :

T_{7}^{2} + 196

T_{13}^{2} + 6400

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
$7$	$T^{2} + 196$ T^2 + 196
$11$	$(T - 11)^{2}$ (T - 11)^2
$13$	$T^{2} + 6400$ T^2 + 6400
$17$	$T^{2} + 900$ T^2 + 900
$19$	$(T + 56)^{2}$ (T + 56)^2
$23$	$T^{2} + 15876$ T^2 + 15876
$29$	$(T - 222)^{2}$ (T - 222)^2
$31$	$(T + 16)^{2}$ (T + 16)^2
$37$	$T^{2} + 11236$ T^2 + 11236
$41$	$(T - 114)^{2}$ (T - 114)^2
$43$	$T^{2} + 2704$ T^2 + 2704
$47$	$T^{2} + 60516$ T^2 + 60516
$53$	$T^{2} + 69696$ T^2 + 69696
$59$	$(T + 264)^{2}$ (T + 264)^2
$61$	$(T - 92)^{2}$ (T - 92)^2
$67$	$T^{2} + 633616$ T^2 + 633616
$71$	$(T - 426)^{2}$ (T - 426)^2
$73$	$T^{2} + 1378276$ T^2 + 1378276
$79$	$(T + 842)^{2}$ (T + 842)^2
$83$	$T^{2} + 725904$ T^2 + 725904
$89$	$(T - 1062)^{2}$ (T - 1062)^2
$97$	$T^{2} + 1643524$ T^2 + 1643524
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