LMFDB - Newform orbit 336.4.q.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(193,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.193"); S:= CuspForms(chi, 4); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [2,0,3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 a primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - 3 \zeta_{6} + 3) q^{3} + 6 \zeta_{6} q^{5} + ( - 21 \zeta_{6} + 14) q^{7} - 9 \zeta_{6} q^{9} + (30 \zeta_{6} - 30) q^{11} + 53 q^{13} + 18 q^{15} + ( - 84 \zeta_{6} + 84) q^{17} - 97 \zeta_{6} q^{19} + \cdots + 270 q^{99} +O(q^{100})$ q + (-3z + 3) q^3 + 6z q^5 + (-21z + 14) q^7 - 9z q^9 + (30z - 30) q^11 + 53 * q^13 + 18 * q^15 + (-84z + 84) q^17 - 97z q^19 + (-42z - 21) q^21 + 84z q^23 + (-89z + 89) q^25 - 27 * q^27 - 180 * q^29 + (-179z + 179) q^31 + 90z q^33 + (-42z + 126) q^35 + 145z q^37 + (-159z + 159) q^39 + 126 * q^41 + 325 * q^43 + (-54z + 54) q^45 - 366z q^47 + (-147z - 245) q^49 - 252z q^51 + (-768z + 768) q^53 - 180 * q^55 - 291 * q^57 + (264z - 264) q^59 - 818z q^61 + (63z - 189) q^63 + 318z q^65 + (523z - 523) q^67 + 252 * q^69 + 342 * q^71 + (-43z + 43) q^73 - 267z q^75 + (420z + 210) q^77 - 1171z q^79 + (81z - 81) q^81 + 810 * q^83 + 504 * q^85 + (540z - 540) q^87 + 600z q^89 + (-1113z + 742) q^91 - 537z q^93 + (-582z + 582) q^95 + 386 * q^97 + 270 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 3 q^{3} + 6 q^{5} + 7 q^{7} - 9 q^{9} - 30 q^{11} + 106 q^{13} + 36 q^{15} + 84 q^{17} - 97 q^{19} - 84 q^{21} + 84 q^{23} + 89 q^{25} - 54 q^{27} - 360 q^{29} + 179 q^{31} + 90 q^{33} + 210 q^{35}+ \cdots + 540 q^{99}+O(q^{100})$ 2 * q + 3 * q^3 + 6 * q^5 + 7 * q^7 - 9 * q^9 - 30 * q^11 + 106 * q^13 + 36 * q^15 + 84 * q^17 - 97 * q^19 - 84 * q^21 + 84 * q^23 + 89 * q^25 - 54 * q^27 - 360 * q^29 + 179 * q^31 + 90 * q^33 + 210 * q^35 + 145 * q^37 + 159 * q^39 + 252 * q^41 + 650 * q^43 + 54 * q^45 - 366 * q^47 - 637 * q^49 - 252 * q^51 + 768 * q^53 - 360 * q^55 - 582 * q^57 - 264 * q^59 - 818 * q^61 - 315 * q^63 + 318 * q^65 - 523 * q^67 + 504 * q^69 + 684 * q^71 + 43 * q^73 - 267 * q^75 + 840 * q^77 - 1171 * q^79 - 81 * q^81 + 1620 * q^83 + 1008 * q^85 - 540 * q^87 + 600 * q^89 + 371 * q^91 - 537 * q^93 + 582 * q^95 + 772 * q^97 + 540 * q^99

Character values

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

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$-\zeta_{6}$

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}

193.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.50000

−

2.59808i

3.00000

5.19615i

3.50000

−

18.1865i

−4.50000

−

7.79423i

289.1

1.50000

2.59808i

3.00000

−

5.19615i

3.50000

18.1865i

−4.50000

7.79423i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	336.4.q.f		2
4.b	odd	2	1		42.4.e.a	✓	2
7.c	even	3	1	inner	336.4.q.f		2
7.c	even	3	1		2352.4.a.f		1
7.d	odd	6	1		2352.4.a.bf		1
12.b	even	2	1		126.4.g.b		2
28.d	even	2	1		294.4.e.i		2
28.f	even	6	1		294.4.a.c		1
28.f	even	6	1		294.4.e.i		2
28.g	odd	6	1		42.4.e.a	✓	2
28.g	odd	6	1		294.4.a.d		1
84.h	odd	2	1		882.4.g.g		2
84.j	odd	6	1		882.4.a.l		1
84.j	odd	6	1		882.4.g.g		2
84.n	even	6	1		126.4.g.b		2
84.n	even	6	1		882.4.a.o		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.4.e.a	✓	2	4.b	odd	2	1
42.4.e.a	✓	2	28.g	odd	6	1
126.4.g.b		2	12.b	even	2	1
126.4.g.b		2	84.n	even	6	1
294.4.a.c		1	28.f	even	6	1
294.4.a.d		1	28.g	odd	6	1
294.4.e.i		2	28.d	even	2	1
294.4.e.i		2	28.f	even	6	1
336.4.q.f		2	1.a	even	1	1	trivial
336.4.q.f		2	7.c	even	3	1	inner
882.4.a.l		1	84.j	odd	6	1
882.4.a.o		1	84.n	even	6	1
882.4.g.g		2	84.h	odd	2	1
882.4.g.g		2	84.j	odd	6	1
2352.4.a.f		1	7.c	even	3	1
2352.4.a.bf		1	7.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} - 6T_{5} + 36$ T5^2 - 6*T5 + 36 acting on $S_{4}^{\mathrm{new}}(336, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$5$	$T^{2} - 6T + 36$ T^2 - 6*T + 36
$7$	$T^{2} - 7T + 343$ T^2 - 7*T + 343
$11$	$T^{2} + 30T + 900$ T^2 + 30*T + 900
$13$	$(T - 53)^{2}$ (T - 53)^2
$17$	$T^{2} - 84T + 7056$ T^2 - 84*T + 7056
$19$	$T^{2} + 97T + 9409$ T^2 + 97*T + 9409
$23$	$T^{2} - 84T + 7056$ T^2 - 84*T + 7056
$29$	$(T + 180)^{2}$ (T + 180)^2
$31$	$T^{2} - 179T + 32041$ T^2 - 179*T + 32041
$37$	$T^{2} - 145T + 21025$ T^2 - 145*T + 21025
$41$	$(T - 126)^{2}$ (T - 126)^2
$43$	$(T - 325)^{2}$ (T - 325)^2
$47$	$T^{2} + 366T + 133956$ T^2 + 366*T + 133956
$53$	$T^{2} - 768T + 589824$ T^2 - 768*T + 589824
$59$	$T^{2} + 264T + 69696$ T^2 + 264*T + 69696
$61$	$T^{2} + 818T + 669124$ T^2 + 818*T + 669124
$67$	$T^{2} + 523T + 273529$ T^2 + 523*T + 273529
$71$	$(T - 342)^{2}$ (T - 342)^2
$73$	$T^{2} - 43T + 1849$ T^2 - 43*T + 1849
$79$	$T^{2} + 1171 T + 1371241$ T^2 + 1171*T + 1371241
$83$	$(T - 810)^{2}$ (T - 810)^2
$89$	$T^{2} - 600T + 360000$ T^2 - 600*T + 360000
$97$	$(T - 386)^{2}$ (T - 386)^2
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