LMFDB - Newform orbit 150.4.c.c

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.

$f(q)$	$=$	$q + 2 i q^{2} - 3 i q^{3} - 4 q^{4} + 6 q^{6} - 4 i q^{7} - 8 i q^{8} - 9 q^{9} - 48 q^{11} + 12 i q^{12} - 2 i q^{13} + 8 q^{14} + 16 q^{16} - 114 i q^{17} - 18 i q^{18} - 140 q^{19} - 12 q^{21} - 96 i q^{22} + \cdots + 432 q^{99} +O(q^{100})$ q + 2i q^2 - 3i q^3 - 4 * q^4 + 6 * q^6 - 4i q^7 - 8i q^8 - 9 * q^9 - 48 * q^11 + 12i q^12 - 2i q^13 + 8 * q^14 + 16 * q^16 - 114i q^17 - 18i q^18 - 140 * q^19 - 12 * q^21 - 96i q^22 - 72i q^23 - 24 * q^24 + 4 * q^26 + 27i q^27 + 16i q^28 - 210 * q^29 + 272 * q^31 + 32i q^32 + 144i q^33 + 228 * q^34 + 36 * q^36 - 334i q^37 - 280i q^38 - 6 * q^39 - 198 * q^41 - 24i q^42 + 268i q^43 + 192 * q^44 + 144 * q^46 + 216i q^47 - 48i q^48 + 327 * q^49 - 342 * q^51 + 8i q^52 + 78i q^53 - 54 * q^54 - 32 * q^56 + 420i q^57 - 420i q^58 - 240 * q^59 + 302 * q^61 + 544i q^62 + 36i q^63 - 64 * q^64 - 288 * q^66 + 596i q^67 + 456i q^68 - 216 * q^69 - 768 * q^71 + 72i q^72 + 478i q^73 + 668 * q^74 + 560 * q^76 + 192i q^77 - 12i q^78 + 640 * q^79 + 81 * q^81 - 396i q^82 + 348i q^83 + 48 * q^84 - 536 * q^86 + 630i q^87 + 384i q^88 - 210 * q^89 - 8 * q^91 + 288i q^92 - 816i q^93 - 432 * q^94 + 96 * q^96 - 1534i q^97 + 654i q^98 + 432 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{4} + 12 q^{6} - 18 q^{9} - 96 q^{11} + 16 q^{14} + 32 q^{16} - 280 q^{19} - 24 q^{21} - 48 q^{24} + 8 q^{26} - 420 q^{29} + 544 q^{31} + 456 q^{34} + 72 q^{36} - 12 q^{39} - 396 q^{41} + 384 q^{44}+ \cdots + 864 q^{99}+O(q^{100})$ 2 * q - 8 * q^4 + 12 * q^6 - 18 * q^9 - 96 * q^11 + 16 * q^14 + 32 * q^16 - 280 * q^19 - 24 * q^21 - 48 * q^24 + 8 * q^26 - 420 * q^29 + 544 * q^31 + 456 * q^34 + 72 * q^36 - 12 * q^39 - 396 * q^41 + 384 * q^44 + 288 * q^46 + 654 * q^49 - 684 * q^51 - 108 * q^54 - 64 * q^56 - 480 * q^59 + 604 * q^61 - 128 * q^64 - 576 * q^66 - 432 * q^69 - 1536 * q^71 + 1336 * q^74 + 1120 * q^76 + 1280 * q^79 + 162 * q^81 + 96 * q^84 - 1072 * q^86 - 420 * q^89 - 16 * q^91 - 864 * q^94 + 192 * q^96 + 864 * 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

4.00000i

8.00000i

−9.00000

49.2

2.00000i

−

3.00000i

−4.00000

6.00000

−

4.00000i

−

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	150.4.c.c		2
3.b	odd	2	1		450.4.c.j		2
4.b	odd	2	1		1200.4.f.r		2
5.b	even	2	1	inner	150.4.c.c		2
5.c	odd	4	1		30.4.a.b	✓	1
5.c	odd	4	1		150.4.a.b		1
15.d	odd	2	1		450.4.c.j		2
15.e	even	4	1		90.4.a.c		1
15.e	even	4	1		450.4.a.r		1
20.d	odd	2	1		1200.4.f.r		2
20.e	even	4	1		240.4.a.b		1
20.e	even	4	1		1200.4.a.ba		1
35.f	even	4	1		1470.4.a.r		1
40.i	odd	4	1		960.4.a.n		1
40.k	even	4	1		960.4.a.bg		1
45.k	odd	12	2		810.4.e.i		2
45.l	even	12	2		810.4.e.p		2
60.l	odd	4	1		720.4.a.y		1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{2} + 16$ T7^2 + 16 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
$7$	$T^{2} + 16$ T^2 + 16
$11$	$(T + 48)^{2}$ (T + 48)^2
$13$	$T^{2} + 4$ T^2 + 4
$17$	$T^{2} + 12996$ T^2 + 12996
$19$	$(T + 140)^{2}$ (T + 140)^2
$23$	$T^{2} + 5184$ T^2 + 5184
$29$	$(T + 210)^{2}$ (T + 210)^2
$31$	$(T - 272)^{2}$ (T - 272)^2
$37$	$T^{2} + 111556$ T^2 + 111556
$41$	$(T + 198)^{2}$ (T + 198)^2
$43$	$T^{2} + 71824$ T^2 + 71824
$47$	$T^{2} + 46656$ T^2 + 46656
$53$	$T^{2} + 6084$ T^2 + 6084
$59$	$(T + 240)^{2}$ (T + 240)^2
$61$	$(T - 302)^{2}$ (T - 302)^2
$67$	$T^{2} + 355216$ T^2 + 355216
$71$	$(T + 768)^{2}$ (T + 768)^2
$73$	$T^{2} + 228484$ T^2 + 228484
$79$	$(T - 640)^{2}$ (T - 640)^2
$83$	$T^{2} + 121104$ T^2 + 121104
$89$	$(T + 210)^{2}$ (T + 210)^2
$97$	$T^{2} + 2353156$ T^2 + 2353156
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