LMFDB - Newform orbit 150.4.c.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,4,Mod(49,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("150.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$150 = 2 \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	150.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:	$8.85028650086$
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 30)
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.

$f(q)$	$=$	$q + 2 i q^{2} + 3 i q^{3} - 4 q^{4} - 6 q^{6} - 32 i q^{7} - 8 i q^{8} - 9 q^{9} - 60 q^{11} - 12 i q^{12} - 34 i q^{13} + 64 q^{14} + 16 q^{16} - 42 i q^{17} - 18 i q^{18} + 76 q^{19} + 96 q^{21} - 120 i q^{22} + \cdots + 540 q^{99} +O(q^{100})$ q + 2i q^2 + 3i q^3 - 4 * q^4 - 6 * q^6 - 32i q^7 - 8i q^8 - 9 * q^9 - 60 * q^11 - 12i q^12 - 34i q^13 + 64 * q^14 + 16 * q^16 - 42i q^17 - 18i q^18 + 76 * q^19 + 96 * q^21 - 120i q^22 + 24 * q^24 + 68 * q^26 - 27i q^27 + 128i q^28 - 6 * q^29 - 232 * q^31 + 32i q^32 - 180i q^33 + 84 * q^34 + 36 * q^36 - 134i q^37 + 152i q^38 + 102 * q^39 + 234 * q^41 + 192i q^42 - 412i q^43 + 240 * q^44 + 360i q^47 + 48i q^48 - 681 * q^49 + 126 * q^51 + 136i q^52 + 222i q^53 + 54 * q^54 - 256 * q^56 + 228i q^57 - 12i q^58 - 660 * q^59 - 490 * q^61 - 464i q^62 + 288i q^63 - 64 * q^64 + 360 * q^66 - 812i q^67 + 168i q^68 + 120 * q^71 + 72i q^72 + 746i q^73 + 268 * q^74 - 304 * q^76 + 1920i q^77 + 204i q^78 - 152 * q^79 + 81 * q^81 + 468i q^82 - 804i q^83 - 384 * q^84 + 824 * q^86 - 18i q^87 + 480i q^88 + 678 * q^89 - 1088 * q^91 - 696i q^93 - 720 * q^94 - 96 * q^96 - 194i q^97 - 1362i q^98 + 540 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{4} - 12 q^{6} - 18 q^{9} - 120 q^{11} + 128 q^{14} + 32 q^{16} + 152 q^{19} + 192 q^{21} + 48 q^{24} + 136 q^{26} - 12 q^{29} - 464 q^{31} + 168 q^{34} + 72 q^{36} + 204 q^{39} + 468 q^{41} + 480 q^{44}+ \cdots + 1080 q^{99}+O(q^{100})$ 2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9 - 120 * q^11 + 128 * q^14 + 32 * q^16 + 152 * q^19 + 192 * q^21 + 48 * q^24 + 136 * q^26 - 12 * q^29 - 464 * q^31 + 168 * q^34 + 72 * q^36 + 204 * q^39 + 468 * q^41 + 480 * q^44 - 1362 * q^49 + 252 * q^51 + 108 * q^54 - 512 * q^56 - 1320 * q^59 - 980 * q^61 - 128 * q^64 + 720 * q^66 + 240 * q^71 + 536 * q^74 - 608 * q^76 - 304 * q^79 + 162 * q^81 - 768 * q^84 + 1648 * q^86 + 1356 * q^89 - 2176 * q^91 - 1440 * q^94 - 192 * q^96 + 1080 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$ .

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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}

49.1

	−	1.00000i
		1.00000i

−

2.00000i

−

3.00000i

−4.00000

−6.00000

32.0000i

8.00000i

−9.00000

49.2

2.00000i

3.00000i

−4.00000

−6.00000

−

32.0000i

−

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	150.4.c.a		2
3.b	odd	2	1		450.4.c.k		2
4.b	odd	2	1		1200.4.f.u		2
5.b	even	2	1	inner	150.4.c.a		2
5.c	odd	4	1		30.4.a.a	✓	1
5.c	odd	4	1		150.4.a.e		1
15.d	odd	2	1		450.4.c.k		2
15.e	even	4	1		90.4.a.d		1
15.e	even	4	1		450.4.a.b		1
20.d	odd	2	1		1200.4.f.u		2
20.e	even	4	1		240.4.a.c		1
20.e	even	4	1		1200.4.a.bk		1
35.f	even	4	1		1470.4.a.a		1
40.i	odd	4	1		960.4.a.j		1
40.k	even	4	1		960.4.a.s		1
45.k	odd	12	2		810.4.e.m		2
45.l	even	12	2		810.4.e.e		2
60.l	odd	4	1		720.4.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.4.a.a	✓	1	5.c	odd	4	1
90.4.a.d		1	15.e	even	4	1
150.4.a.e		1	5.c	odd	4	1
150.4.c.a		2	1.a	even	1	1	trivial
150.4.c.a		2	5.b	even	2	1	inner
240.4.a.c		1	20.e	even	4	1
450.4.a.b		1	15.e	even	4	1
450.4.c.k		2	3.b	odd	2	1
450.4.c.k		2	15.d	odd	2	1
720.4.a.b		1	60.l	odd	4	1
810.4.e.e		2	45.l	even	12	2
810.4.e.m		2	45.k	odd	12	2
960.4.a.j		1	40.i	odd	4	1
960.4.a.s		1	40.k	even	4	1
1200.4.a.bk		1	20.e	even	4	1
1200.4.f.u		2	4.b	odd	2	1
1200.4.f.u		2	20.d	odd	2	1
1470.4.a.a		1	35.f	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{2} + 1024$ T7^2 + 1024 acting on $S_{4}^{\mathrm{new}}(150, [\chi])$ .

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
$7$	$T^{2} + 1024$ T^2 + 1024
$11$	$(T + 60)^{2}$ (T + 60)^2
$13$	$T^{2} + 1156$ T^2 + 1156
$17$	$T^{2} + 1764$ T^2 + 1764
$19$	$(T - 76)^{2}$ (T - 76)^2
$23$	$T^{2}$ T^2
$29$	$(T + 6)^{2}$ (T + 6)^2
$31$	$(T + 232)^{2}$ (T + 232)^2
$37$	$T^{2} + 17956$ T^2 + 17956
$41$	$(T - 234)^{2}$ (T - 234)^2
$43$	$T^{2} + 169744$ T^2 + 169744
$47$	$T^{2} + 129600$ T^2 + 129600
$53$	$T^{2} + 49284$ T^2 + 49284
$59$	$(T + 660)^{2}$ (T + 660)^2
$61$	$(T + 490)^{2}$ (T + 490)^2
$67$	$T^{2} + 659344$ T^2 + 659344
$71$	$(T - 120)^{2}$ (T - 120)^2
$73$	$T^{2} + 556516$ T^2 + 556516
$79$	$(T + 152)^{2}$ (T + 152)^2
$83$	$T^{2} + 646416$ T^2 + 646416
$89$	$(T - 678)^{2}$ (T - 678)^2
$97$	$T^{2} + 37636$ T^2 + 37636
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