LMFDB - Newform orbit 336.4.k.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(209,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.209"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.k (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2\cdot 3$
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$ -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 $\beta = 3\sqrt{-3}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta q^{3} + ( - 3 \beta + 10) q^{7} - 27 q^{9} + 12 \beta q^{13} + 30 \beta q^{19} + (10 \beta + 81) q^{21} - 125 q^{25} - 27 \beta q^{27} + 30 \beta q^{31} - 110 q^{37} - 324 q^{39} - 520 q^{43} + \cdots + 264 \beta q^{97} +O(q^{100})$ q + b * q^3 + (-3b + 10) q^7 - 27 * q^9 + 12b q^13 + 30b q^19 + (10b + 81) q^21 - 125 * q^25 - 27b q^27 + 30b q^31 - 110 * q^37 - 324 * q^39 - 520 * q^43 + (-60b - 143) q^49 - 810 * q^57 + 180b q^61 + (81b - 270) q^63 + 880 * q^67 - 72b q^73 - 125b q^75 - 884 * q^79 + 729 * q^81 + (120b + 972) q^91 - 810 * q^93 + 264b q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 20 q^{7} - 54 q^{9} + 162 q^{21} - 250 q^{25} - 220 q^{37} - 648 q^{39} - 1040 q^{43} - 286 q^{49} - 1620 q^{57} - 540 q^{63} + 1760 q^{67} - 1768 q^{79} + 1458 q^{81} + 1944 q^{91} - 1620 q^{93}+O(q^{100})$ 2 * q + 20 * q^7 - 54 * q^9 + 162 * q^21 - 250 * q^25 - 220 * q^37 - 648 * q^39 - 1040 * q^43 - 286 * q^49 - 1620 * q^57 - 540 * q^63 + 1760 * q^67 - 1768 * q^79 + 1458 * q^81 + 1944 * q^91 - 1620 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$ .

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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}

209.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−

5.19615i

10.0000

15.5885i

−27.0000

209.2

5.19615i

10.0000

−

15.5885i

−27.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
7.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	336.4.k.a		2
3.b	odd	2	1	CM	336.4.k.a		2
4.b	odd	2	1		21.4.c.a	✓	2
7.b	odd	2	1	inner	336.4.k.a		2
12.b	even	2	1		21.4.c.a	✓	2
21.c	even	2	1	inner	336.4.k.a		2
28.d	even	2	1		21.4.c.a	✓	2
28.f	even	6	1		147.4.g.a		2
28.f	even	6	1		147.4.g.b		2
28.g	odd	6	1		147.4.g.a		2
28.g	odd	6	1		147.4.g.b		2
84.h	odd	2	1		21.4.c.a	✓	2
84.j	odd	6	1		147.4.g.a		2
84.j	odd	6	1		147.4.g.b		2
84.n	even	6	1		147.4.g.a		2
84.n	even	6	1		147.4.g.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.4.c.a	✓	2	4.b	odd	2	1
21.4.c.a	✓	2	12.b	even	2	1
21.4.c.a	✓	2	28.d	even	2	1
21.4.c.a	✓	2	84.h	odd	2	1
147.4.g.a		2	28.f	even	6	1
147.4.g.a		2	28.g	odd	6	1
147.4.g.a		2	84.j	odd	6	1
147.4.g.a		2	84.n	even	6	1
147.4.g.b		2	28.f	even	6	1
147.4.g.b		2	28.g	odd	6	1
147.4.g.b		2	84.j	odd	6	1
147.4.g.b		2	84.n	even	6	1
336.4.k.a		2	1.a	even	1	1	trivial
336.4.k.a		2	3.b	odd	2	1	CM
336.4.k.a		2	7.b	odd	2	1	inner
336.4.k.a		2	21.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}$ T5 acting on $S_{4}^{\mathrm{new}}(336, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 27$ T^2 + 27
$5$	$T^{2}$ T^2
$7$	$T^{2} - 20T + 343$ T^2 - 20*T + 343
$11$	$T^{2}$ T^2
$13$	$T^{2} + 3888$ T^2 + 3888
$17$	$T^{2}$ T^2
$19$	$T^{2} + 24300$ T^2 + 24300
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$T^{2} + 24300$ T^2 + 24300
$37$	$(T + 110)^{2}$ (T + 110)^2
$41$	$T^{2}$ T^2
$43$	$(T + 520)^{2}$ (T + 520)^2
$47$	$T^{2}$ T^2
$53$	$T^{2}$ T^2
$59$	$T^{2}$ T^2
$61$	$T^{2} + 874800$ T^2 + 874800
$67$	$(T - 880)^{2}$ (T - 880)^2
$71$	$T^{2}$ T^2
$73$	$T^{2} + 139968$ T^2 + 139968
$79$	$(T + 884)^{2}$ (T + 884)^2
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$T^{2} + 1881792$ T^2 + 1881792
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