LMFDB - Newform orbit 300.6.d.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,6,Mod(49,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("300.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$300 = 2^{2} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	300.d (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:	$48.1151459439$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 60)
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.

$f(q)$	$=$	$q + 9 i q^{3} + 44 i q^{7} - 81 q^{9} + 216 q^{11} - 770 i q^{13} + 534 i q^{17} - 1580 q^{19} - 396 q^{21} - 2904 i q^{23} - 729 i q^{27} + 4566 q^{29} + 2744 q^{31} + 1944 i q^{33} + 1442 i q^{37} + 6930 q^{39} + \cdots - 17496 q^{99} +O(q^{100})$ q + 9i q^3 + 44i q^7 - 81 * q^9 + 216 * q^11 - 770i q^13 + 534i q^17 - 1580 * q^19 - 396 * q^21 - 2904i q^23 - 729i q^27 + 4566 * q^29 + 2744 * q^31 + 1944i q^33 + 1442i q^37 + 6930 * q^39 - 13350 * q^41 - 17204i q^43 - 10824i q^47 + 14871 * q^49 - 4806 * q^51 + 9942i q^53 - 14220i q^57 + 15576 * q^59 + 39302 * q^61 - 3564i q^63 + 55796i q^67 + 26136 * q^69 + 57120 * q^71 - 50402i q^73 + 9504i q^77 + 10552 * q^79 + 6561 * q^81 - 108564i q^83 + 41094i q^87 + 116430 * q^89 + 33880 * q^91 + 24696i q^93 - 2782i q^97 - 17496 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 162 q^{9} + 432 q^{11} - 3160 q^{19} - 792 q^{21} + 9132 q^{29} + 5488 q^{31} + 13860 q^{39} - 26700 q^{41} + 29742 q^{49} - 9612 q^{51} + 31152 q^{59} + 78604 q^{61} + 52272 q^{69} + 114240 q^{71}+ \cdots - 34992 q^{99}+O(q^{100})$ 2 * q - 162 * q^9 + 432 * q^11 - 3160 * q^19 - 792 * q^21 + 9132 * q^29 + 5488 * q^31 + 13860 * q^39 - 26700 * q^41 + 29742 * q^49 - 9612 * q^51 + 31152 * q^59 + 78604 * q^61 + 52272 * q^69 + 114240 * q^71 + 21104 * q^79 + 13122 * q^81 + 232860 * q^89 + 67760 * q^91 - 34992 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$ .

$n$	$101$	$151$	$277$
$\chi(n)$	$1$	$1$	$-1$

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

−

9.00000i

−

44.0000i

−81.0000

49.2

9.00000i

44.0000i

−81.0000

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	300.6.d.e		2
3.b	odd	2	1		900.6.d.b		2
5.b	even	2	1	inner	300.6.d.e		2
5.c	odd	4	1		60.6.a.a	✓	1
5.c	odd	4	1		300.6.a.d		1
15.d	odd	2	1		900.6.d.b		2
15.e	even	4	1		180.6.a.d		1
15.e	even	4	1		900.6.a.f		1
20.e	even	4	1		240.6.a.i		1
40.i	odd	4	1		960.6.a.z		1
40.k	even	4	1		960.6.a.i		1
60.l	odd	4	1		720.6.a.p		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
60.6.a.a	✓	1	5.c	odd	4	1
180.6.a.d		1	15.e	even	4	1
240.6.a.i		1	20.e	even	4	1
300.6.a.d		1	5.c	odd	4	1
300.6.d.e		2	1.a	even	1	1	trivial
300.6.d.e		2	5.b	even	2	1	inner
720.6.a.p		1	60.l	odd	4	1
900.6.a.f		1	15.e	even	4	1
900.6.d.b		2	3.b	odd	2	1
900.6.d.b		2	15.d	odd	2	1
960.6.a.i		1	40.k	even	4	1
960.6.a.z		1	40.i	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{2} + 1936$ T7^2 + 1936 acting on $S_{6}^{\mathrm{new}}(300, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 81$ T^2 + 81
$5$	$T^{2}$ T^2
$7$	$T^{2} + 1936$ T^2 + 1936
$11$	$(T - 216)^{2}$ (T - 216)^2
$13$	$T^{2} + 592900$ T^2 + 592900
$17$	$T^{2} + 285156$ T^2 + 285156
$19$	$(T + 1580)^{2}$ (T + 1580)^2
$23$	$T^{2} + 8433216$ T^2 + 8433216
$29$	$(T - 4566)^{2}$ (T - 4566)^2
$31$	$(T - 2744)^{2}$ (T - 2744)^2
$37$	$T^{2} + 2079364$ T^2 + 2079364
$41$	$(T + 13350)^{2}$ (T + 13350)^2
$43$	$T^{2} + 295977616$ T^2 + 295977616
$47$	$T^{2} + 117158976$ T^2 + 117158976
$53$	$T^{2} + 98843364$ T^2 + 98843364
$59$	$(T - 15576)^{2}$ (T - 15576)^2
$61$	$(T - 39302)^{2}$ (T - 39302)^2
$67$	$T^{2} + 3113193616$ T^2 + 3113193616
$71$	$(T - 57120)^{2}$ (T - 57120)^2
$73$	$T^{2} + 2540361604$ T^2 + 2540361604
$79$	$(T - 10552)^{2}$ (T - 10552)^2
$83$	$T^{2} + 11786142096$ T^2 + 11786142096
$89$	$(T - 116430)^{2}$ (T - 116430)^2
$97$	$T^{2} + 7739524$ T^2 + 7739524
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