LMFDB - Newform orbit 150.8.c.f

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:	yes
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.

$f(q)$	$=$	$q - 8 i q^{2} + 27 i q^{3} - 64 q^{4} + 216 q^{6} + 391 i q^{7} + 512 i q^{8} - 729 q^{9} - 4398 q^{11} - 1728 i q^{12} - 13447 i q^{13} + 3128 q^{14} + 4096 q^{16} + 7686 i q^{17} + 5832 i q^{18} + 13705 q^{19} + \cdots + 3206142 q^{99} +O(q^{100})$ q - 8i q^2 + 27i q^3 - 64 * q^4 + 216 * q^6 + 391i q^7 + 512i q^8 - 729 * q^9 - 4398 * q^11 - 1728i q^12 - 13447i q^13 + 3128 * q^14 + 4096 * q^16 + 7686i q^17 + 5832i q^18 + 13705 * q^19 - 10557 * q^21 + 35184i q^22 + 35478i q^23 - 13824 * q^24 - 107576 * q^26 - 19683i q^27 - 25024i q^28 + 157470 * q^29 - 99343 * q^31 - 32768i q^32 - 118746i q^33 + 61488 * q^34 + 46656 * q^36 + 161926i q^37 - 109640i q^38 + 363069 * q^39 + 521952 * q^41 + 84456i q^42 + 340973i q^43 + 281472 * q^44 + 283824 * q^46 + 50886i q^47 + 110592i q^48 + 670662 * q^49 - 207522 * q^51 + 860608i q^52 - 891132i q^53 - 157464 * q^54 - 200192 * q^56 + 370035i q^57 - 1259760i q^58 + 1344210 * q^59 + 3394127 * q^61 + 794744i q^62 - 285039i q^63 - 262144 * q^64 - 949968 * q^66 + 2248951i q^67 - 491904i q^68 - 957906 * q^69 + 2731872 * q^71 - 373248i q^72 - 5028622i q^73 + 1295408 * q^74 - 877120 * q^76 - 1719618i q^77 - 2904552i q^78 - 1571480 * q^79 + 531441 * q^81 - 4175616i q^82 - 7792962i q^83 + 675648 * q^84 + 2727784 * q^86 + 4251690i q^87 - 2251776i q^88 + 5802240 * q^89 + 5257777 * q^91 - 2270592i q^92 - 2682261i q^93 + 407088 * q^94 + 884736 * q^96 + 2498311i q^97 - 5365296i q^98 + 3206142 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 128 q^{4} + 432 q^{6} - 1458 q^{9} - 8796 q^{11} + 6256 q^{14} + 8192 q^{16} + 27410 q^{19} - 21114 q^{21} - 27648 q^{24} - 215152 q^{26} + 314940 q^{29} - 198686 q^{31} + 122976 q^{34} + 93312 q^{36}+ \cdots + 6412284 q^{99}+O(q^{100})$ 2 * q - 128 * q^4 + 432 * q^6 - 1458 * q^9 - 8796 * q^11 + 6256 * q^14 + 8192 * q^16 + 27410 * q^19 - 21114 * q^21 - 27648 * q^24 - 215152 * q^26 + 314940 * q^29 - 198686 * q^31 + 122976 * q^34 + 93312 * q^36 + 726138 * q^39 + 1043904 * q^41 + 562944 * q^44 + 567648 * q^46 + 1341324 * q^49 - 415044 * q^51 - 314928 * q^54 - 400384 * q^56 + 2688420 * q^59 + 6788254 * q^61 - 524288 * q^64 - 1899936 * q^66 - 1915812 * q^69 + 5463744 * q^71 + 2590816 * q^74 - 1754240 * q^76 - 3142960 * q^79 + 1062882 * q^81 + 1351296 * q^84 + 5455568 * q^86 + 11604480 * q^89 + 10515554 * q^91 + 814176 * q^94 + 1769472 * q^96 + 6412284 * 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

391.000i

512.000i

−729.000

49.2

8.00000i

−

27.0000i

−64.0000

216.000

−

391.000i

−

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.f		2
3.b	odd	2	1		450.8.c.o		2
5.b	even	2	1	inner	150.8.c.f		2
5.c	odd	4	1		150.8.a.c	✓	1
5.c	odd	4	1		150.8.a.o	yes	1
15.d	odd	2	1		450.8.c.o		2
15.e	even	4	1		450.8.a.f		1
15.e	even	4	1		450.8.a.v		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
150.8.a.c	✓	1	5.c	odd	4	1
150.8.a.o	yes	1	5.c	odd	4	1
150.8.c.f		2	1.a	even	1	1	trivial
150.8.c.f		2	5.b	even	2	1	inner
450.8.a.f		1	15.e	even	4	1
450.8.a.v		1	15.e	even	4	1
450.8.c.o		2	3.b	odd	2	1
450.8.c.o		2	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{2} + 152881$ T7^2 + 152881 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
$7$	$T^{2} + 152881$ T^2 + 152881
$11$	$(T + 4398)^{2}$ (T + 4398)^2
$13$	$T^{2} + 180821809$ T^2 + 180821809
$17$	$T^{2} + 59074596$ T^2 + 59074596
$19$	$(T - 13705)^{2}$ (T - 13705)^2
$23$	$T^{2} + 1258688484$ T^2 + 1258688484
$29$	$(T - 157470)^{2}$ (T - 157470)^2
$31$	$(T + 99343)^{2}$ (T + 99343)^2
$37$	$T^{2} + 26220029476$ T^2 + 26220029476
$41$	$(T - 521952)^{2}$ (T - 521952)^2
$43$	$T^{2} + 116262586729$ T^2 + 116262586729
$47$	$T^{2} + 2589384996$ T^2 + 2589384996
$53$	$T^{2} + 794116241424$ T^2 + 794116241424
$59$	$(T - 1344210)^{2}$ (T - 1344210)^2
$61$	$(T - 3394127)^{2}$ (T - 3394127)^2
$67$	$T^{2} + 5057780600401$ T^2 + 5057780600401
$71$	$(T - 2731872)^{2}$ (T - 2731872)^2
$73$	$T^{2} + 25287039218884$ T^2 + 25287039218884
$79$	$(T + 1571480)^{2}$ (T + 1571480)^2
$83$	$T^{2} + 60730256733444$ T^2 + 60730256733444
$89$	$(T - 5802240)^{2}$ (T - 5802240)^2
$97$	$T^{2} + 6241557852721$ T^2 + 6241557852721
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