LMFDB - Newform orbit 150.9.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,9,Mod(101,150)]

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([1, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("150.101");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$N$	$=$	$150 = 2 \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$9$
Character orbit:	$[\chi]$	$=$	150.d (of order $2$ , degree $1$ , 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:	$61.1067915092$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 2$ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 6)
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 $\beta = 4\sqrt{-2}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - 2 \beta q^{2} + ( - 9 \beta + 63) q^{3} - 128 q^{4} + ( - 126 \beta - 576) q^{6} - 2786 q^{7} + 256 \beta q^{8} + ( - 1134 \beta + 1377) q^{9} + 3966 \beta q^{11} + (1152 \beta - 8064) q^{12} + 13150 q^{13} + \cdots + (5461182 \beta + 143918208) q^{99} +O(q^{100})$ q - 2b q^2 + (-9b + 63) q^3 - 128 * q^4 + (-126b - 576) q^6 - 2786 * q^7 + 256b q^8 + (-1134b + 1377) q^9 + 3966b q^11 + (1152b - 8064) q^12 + 13150 * q^13 + 5572b q^14 + 16384 * q^16 + 11736b q^17 + (-2754b - 72576) q^18 + 144002 * q^19 + (25074b - 175518) q^21 + 253824 * q^22 - 8724b q^23 + (16128b + 73728) q^24 - 26300b q^26 + (-83835b - 239841) q^27 + 356608 * q^28 - 110910b q^29 + 728738 * q^31 - 32768b q^32 + (249858b + 1142208) q^33 + 751104 * q^34 + (145152b - 176256) q^36 + 1964446 * q^37 - 288004b q^38 + (-118350b + 828450) q^39 + 174324b q^41 + (351036b + 1604736) q^42 + 78142 * q^43 - 507648b q^44 - 558336 * q^46 + 622200b q^47 + (-147456b + 1032192) q^48 + 1996995 * q^49 + (739368b + 3379968) q^51 - 1683200 * q^52 - 92286b q^53 + (479682b - 5365440) q^54 - 713216b q^56 + (-1296018b + 9072126) q^57 - 7098240 * q^58 - 884634b q^59 + 17578274 * q^61 - 1457476b q^62 + (3159324b - 3836322) q^63 - 2097152 * q^64 + (-2284416b + 15990912) q^66 + 17136766 * q^67 - 1502208b q^68 + (-549612b - 2512512) q^69 + 4576860b q^71 + (352512b + 9289728) q^72 - 28139330 * q^73 - 3928892b q^74 - 18432256 * q^76 - 11049276b q^77 + (-1656900b - 7574400) q^78 + 9182498 * q^79 + (-3123036b - 39254463) q^81 + 11156736 * q^82 + 15398742b q^83 + (-3209472b + 22466304) q^84 - 156284b q^86 + (-6987330b - 31942080) q^87 - 32489472 * q^88 - 14363604b q^89 - 36635900 * q^91 + 1116672b q^92 + (-6558642b + 45910494) q^93 + 39820800 * q^94 + (-2064384b - 9437184) q^96 + 128722558 * q^97 - 3993990b q^98 + (5461182b + 143918208) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 126 q^{3} - 256 q^{4} - 1152 q^{6} - 5572 q^{7} + 2754 q^{9} - 16128 q^{12} + 26300 q^{13} + 32768 q^{16} - 145152 q^{18} + 288004 q^{19} - 351036 q^{21} + 507648 q^{22} + 147456 q^{24} - 479682 q^{27}+ \cdots + 287836416 q^{99}+O(q^{100})$ 2 * q + 126 * q^3 - 256 * q^4 - 1152 * q^6 - 5572 * q^7 + 2754 * q^9 - 16128 * q^12 + 26300 * q^13 + 32768 * q^16 - 145152 * q^18 + 288004 * q^19 - 351036 * q^21 + 507648 * q^22 + 147456 * q^24 - 479682 * q^27 + 713216 * q^28 + 1457476 * q^31 + 2284416 * q^33 + 1502208 * q^34 - 352512 * q^36 + 3928892 * q^37 + 1656900 * q^39 + 3209472 * q^42 + 156284 * q^43 - 1116672 * q^46 + 2064384 * q^48 + 3993990 * q^49 + 6759936 * q^51 - 3366400 * q^52 - 10730880 * q^54 + 18144252 * q^57 - 14196480 * q^58 + 35156548 * q^61 - 7672644 * q^63 - 4194304 * q^64 + 31981824 * q^66 + 34273532 * q^67 - 5025024 * q^69 + 18579456 * q^72 - 56278660 * q^73 - 36864512 * q^76 - 15148800 * q^78 + 18364996 * q^79 - 78508926 * q^81 + 22313472 * q^82 + 44932608 * q^84 - 63884160 * q^87 - 64978944 * q^88 - 73271800 * q^91 + 91820988 * q^93 + 79641600 * q^94 - 18874368 * q^96 + 257445116 * q^97 + 287836416 * 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}

101.1

		1.41421i
	−	1.41421i

−

11.3137i

63.0000

−

50.9117i

−128.000

−576.000

−

712.764i

−2786.00

1448.15i

1377.00

−

6414.87i

101.2

11.3137i

63.0000

50.9117i

−128.000

−576.000

712.764i

−2786.00

−

1448.15i

1377.00

6414.87i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	150.9.d.a		2
3.b	odd	2	1	inner	150.9.d.a		2
5.b	even	2	1		6.9.b.a	✓	2
5.c	odd	4	2		150.9.b.a		4
15.d	odd	2	1		6.9.b.a	✓	2
15.e	even	4	2		150.9.b.a		4
20.d	odd	2	1		48.9.e.d		2
40.e	odd	2	1		192.9.e.c		2
40.f	even	2	1		192.9.e.h		2
45.h	odd	6	2		162.9.d.a		4
45.j	even	6	2		162.9.d.a		4
60.h	even	2	1		48.9.e.d		2
120.i	odd	2	1		192.9.e.h		2
120.m	even	2	1		192.9.e.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.9.b.a	✓	2	5.b	even	2	1
6.9.b.a	✓	2	15.d	odd	2	1
48.9.e.d		2	20.d	odd	2	1
48.9.e.d		2	60.h	even	2	1
150.9.b.a		4	5.c	odd	4	2
150.9.b.a		4	15.e	even	4	2
150.9.d.a		2	1.a	even	1	1	trivial
150.9.d.a		2	3.b	odd	2	1	inner
162.9.d.a		4	45.h	odd	6	2
162.9.d.a		4	45.j	even	6	2
192.9.e.c		2	40.e	odd	2	1
192.9.e.c		2	120.m	even	2	1
192.9.e.h		2	40.f	even	2	1
192.9.e.h		2	120.i	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7} + 2786$ T7 + 2786 acting on $S_{9}^{\mathrm{new}}(150, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 128$ T^2 + 128
$3$	$T^{2} - 126T + 6561$ T^2 - 126*T + 6561
$5$	$T^{2}$ T^2
$7$	$(T + 2786)^{2}$ (T + 2786)^2
$11$	$T^{2} + 503332992$ T^2 + 503332992
$13$	$(T - 13150)^{2}$ (T - 13150)^2
$17$	$T^{2} + 4407478272$ T^2 + 4407478272
$19$	$(T - 144002)^{2}$ (T - 144002)^2
$23$	$T^{2} + 2435461632$ T^2 + 2435461632
$29$	$T^{2} + 393632899200$ T^2 + 393632899200
$31$	$(T - 728738)^{2}$ (T - 728738)^2
$37$	$(T - 1964446)^{2}$ (T - 1964446)^2
$41$	$T^{2} + 972443423232$ T^2 + 972443423232
$43$	$(T - 78142)^{2}$ (T - 78142)^2
$47$	$T^{2} + 12388250880000$ T^2 + 12388250880000
$53$	$T^{2} + 272534585472$ T^2 + 272534585472
$59$	$T^{2} + 25042474046592$ T^2 + 25042474046592
$61$	$(T - 17578274)^{2}$ (T - 17578274)^2
$67$	$(T - 17136766)^{2}$ (T - 17136766)^2
$71$	$T^{2} + 670324718707200$ T^2 + 670324718707200
$73$	$(T + 28139330)^{2}$ (T + 28139330)^2
$79$	$(T - 9182498)^{2}$ (T - 9182498)^2
$83$	$T^{2} + 75\!\cdots\!48$ T^2 + 7587880165842048
$89$	$T^{2} + 66\!\cdots\!12$ T^2 + 6602019835802112
$97$	$(T - 128722558)^{2}$ (T - 128722558)^2
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