LMFDB - Newform orbit 150.8.c.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,8,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, 8, names="a")

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

chi := DirichletCharacter("150.49");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$150 = 2 \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$8$
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:	$46.8577538226$
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.

$f(q)$	$=$	$q - 8 i q^{2} - 27 i q^{3} - 64 q^{4} - 216 q^{6} + 988 i q^{7} + 512 i q^{8} - 729 q^{9} - 8040 q^{11} + 1728 i q^{12} - 3334 i q^{13} + 7904 q^{14} + 4096 q^{16} - 6582 i q^{17} + 5832 i q^{18} + 27436 q^{19} + \cdots + 5861160 q^{99} +O(q^{100})$ q - 8i q^2 - 27i q^3 - 64 * q^4 - 216 * q^6 + 988i q^7 + 512i q^8 - 729 * q^9 - 8040 * q^11 + 1728i q^12 - 3334i q^13 + 7904 * q^14 + 4096 * q^16 - 6582i q^17 + 5832i q^18 + 27436 * q^19 + 26676 * q^21 + 64320i q^22 + 48600i q^23 + 13824 * q^24 - 26672 * q^26 + 19683i q^27 - 63232i q^28 + 132414 * q^29 + 254408 * q^31 - 32768i q^32 + 217080i q^33 - 52656 * q^34 + 46656 * q^36 - 519434i q^37 - 219488i q^38 - 90018 * q^39 + 92394 * q^41 - 213408i q^42 - 234532i q^43 + 514560 * q^44 + 388800 * q^46 + 1277640i q^47 - 110592i q^48 - 152601 * q^49 - 177714 * q^51 + 213376i q^52 - 835278i q^53 + 157464 * q^54 - 505856 * q^56 - 740772i q^57 - 1059312i q^58 + 3068760 * q^59 - 1009330 * q^61 - 2035264i q^62 - 720252i q^63 - 262144 * q^64 + 1736640 * q^66 - 3082172i q^67 + 421248i q^68 + 1312200 * q^69 - 3666720 * q^71 - 373248i q^72 + 1122866i q^73 - 4155472 * q^74 - 1755904 * q^76 - 7943520i q^77 + 720144i q^78 + 4128808 * q^79 + 531441 * q^81 - 739152i q^82 + 4586556i q^83 - 1707264 * q^84 - 1876256 * q^86 - 3575178i q^87 - 4116480i q^88 + 5763678 * q^89 + 3293992 * q^91 - 3110400i q^92 - 6869016i q^93 + 10221120 * q^94 - 884736 * q^96 - 6747554i q^97 + 1220808i q^98 + 5861160 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 128 q^{4} - 432 q^{6} - 1458 q^{9} - 16080 q^{11} + 15808 q^{14} + 8192 q^{16} + 54872 q^{19} + 53352 q^{21} + 27648 q^{24} - 53344 q^{26} + 264828 q^{29} + 508816 q^{31} - 105312 q^{34} + 93312 q^{36}+ \cdots + 11722320 q^{99}+O(q^{100})$ 2 * q - 128 * q^4 - 432 * q^6 - 1458 * q^9 - 16080 * q^11 + 15808 * q^14 + 8192 * q^16 + 54872 * q^19 + 53352 * q^21 + 27648 * q^24 - 53344 * q^26 + 264828 * q^29 + 508816 * q^31 - 105312 * q^34 + 93312 * q^36 - 180036 * q^39 + 184788 * q^41 + 1029120 * q^44 + 777600 * q^46 - 305202 * q^49 - 355428 * q^51 + 314928 * q^54 - 1011712 * q^56 + 6137520 * q^59 - 2018660 * q^61 - 524288 * q^64 + 3473280 * q^66 + 2624400 * q^69 - 7333440 * q^71 - 8310944 * q^74 - 3511808 * q^76 + 8257616 * q^79 + 1062882 * q^81 - 3414528 * q^84 - 3752512 * q^86 + 11527356 * q^89 + 6587984 * q^91 + 20442240 * q^94 - 1769472 * q^96 + 11722320 * 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

−

8.00000i

−

27.0000i

−64.0000

−216.000

988.000i

512.000i

−729.000

49.2

8.00000i

27.0000i

−64.0000

−216.000

−

988.000i

−

512.000i

−729.000

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	150.8.c.a		2
3.b	odd	2	1		450.8.c.r		2
5.b	even	2	1	inner	150.8.c.a		2
5.c	odd	4	1		30.8.a.d	✓	1
5.c	odd	4	1		150.8.a.i		1
15.d	odd	2	1		450.8.c.r		2
15.e	even	4	1		90.8.a.c		1
15.e	even	4	1		450.8.a.x		1
20.e	even	4	1		240.8.a.j		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.8.a.d	✓	1	5.c	odd	4	1
90.8.a.c		1	15.e	even	4	1
150.8.a.i		1	5.c	odd	4	1
150.8.c.a		2	1.a	even	1	1	trivial
150.8.c.a		2	5.b	even	2	1	inner
240.8.a.j		1	20.e	even	4	1
450.8.a.x		1	15.e	even	4	1
450.8.c.r		2	3.b	odd	2	1
450.8.c.r		2	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{2} + 976144$ T7^2 + 976144 acting on $S_{8}^{\mathrm{new}}(150, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 64$ T^2 + 64
$3$	$T^{2} + 729$ T^2 + 729
$5$	$T^{2}$ T^2
$7$	$T^{2} + 976144$ T^2 + 976144
$11$	$(T + 8040)^{2}$ (T + 8040)^2
$13$	$T^{2} + 11115556$ T^2 + 11115556
$17$	$T^{2} + 43322724$ T^2 + 43322724
$19$	$(T - 27436)^{2}$ (T - 27436)^2
$23$	$T^{2} + 2361960000$ T^2 + 2361960000
$29$	$(T - 132414)^{2}$ (T - 132414)^2
$31$	$(T - 254408)^{2}$ (T - 254408)^2
$37$	$T^{2} + 269811680356$ T^2 + 269811680356
$41$	$(T - 92394)^{2}$ (T - 92394)^2
$43$	$T^{2} + 55005259024$ T^2 + 55005259024
$47$	$T^{2} + 1632363969600$ T^2 + 1632363969600
$53$	$T^{2} + 697689337284$ T^2 + 697689337284
$59$	$(T - 3068760)^{2}$ (T - 3068760)^2
$61$	$(T + 1009330)^{2}$ (T + 1009330)^2
$67$	$T^{2} + 9499784237584$ T^2 + 9499784237584
$71$	$(T + 3666720)^{2}$ (T + 3666720)^2
$73$	$T^{2} + 1260828053956$ T^2 + 1260828053956
$79$	$(T - 4128808)^{2}$ (T - 4128808)^2
$83$	$T^{2} + 21036495941136$ T^2 + 21036495941136
$89$	$(T - 5763678)^{2}$ (T - 5763678)^2
$97$	$T^{2} + 45529484982916$ T^2 + 45529484982916
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