LMFDB - Newform orbit 336.4.bj.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(95,336)] 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([3, 0, 3, 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("336.95"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.bj (of order $6$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,9,0,0,0,-37] 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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$ -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.

$f(q)$	$=$	$q + ( - 3 \zeta_{6} + 6) q^{3} + (\zeta_{6} - 19) q^{7} + ( - 27 \zeta_{6} + 27) q^{9} - 89 q^{13} + ( - 17 \zeta_{6} - 17) q^{19} + (60 \zeta_{6} - 111) q^{21} - 125 \zeta_{6} q^{25} + ( - 162 \zeta_{6} + 81) q^{27} + \cdots + 1330 q^{97} +O(q^{100})$ q + (-3z + 6) q^3 + (z - 19) * q^7 + (-27z + 27) q^9 - 89 * q^13 + (-17z - 17) q^19 + (60z - 111) q^21 - 125z q^25 + (-162z + 81) q^27 + (199z - 398) q^31 + (-433z + 433) q^37 + (267z - 534) q^39 + (394z - 197) q^43 + (-37z + 360) q^49 - 153 * q^57 + (182z - 182) q^61 + (513z - 486) q^63 + (251z - 502) q^67 - 919z q^73 + (-375z - 375) q^75 + (-757z - 757) q^79 - 729z q^81 + (-89z + 1691) q^91 + (1791z - 1791) q^93 + 1330 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 9 q^{3} - 37 q^{7} + 27 q^{9} - 178 q^{13} - 51 q^{19} - 162 q^{21} - 125 q^{25} - 597 q^{31} + 433 q^{37} - 801 q^{39} + 683 q^{49} - 306 q^{57} - 182 q^{61} - 459 q^{63} - 753 q^{67} - 919 q^{73}+ \cdots + 2660 q^{97}+O(q^{100})$ 2 * q + 9 * q^3 - 37 * q^7 + 27 * q^9 - 178 * q^13 - 51 * q^19 - 162 * q^21 - 125 * q^25 - 597 * q^31 + 433 * q^37 - 801 * q^39 + 683 * q^49 - 306 * q^57 - 182 * q^61 - 459 * q^63 - 753 * q^67 - 919 * q^73 - 1125 * q^75 - 2271 * q^79 - 729 * q^81 + 3293 * q^91 - 1791 * q^93 + 2660 * q^97

Character values

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

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

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}

95.1

0.500000	−	0.866025i
0.500000	+	0.866025i

4.50000

2.59808i

−18.5000

−

0.866025i

13.5000

23.3827i

191.1

4.50000

−

2.59808i

−18.5000

0.866025i

13.5000

−

23.3827i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
28.g	odd	6	1	inner
84.n	even	6	1	inner

Twists

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

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(336, [\chi])$ :

T_{5}

T5

T_{13} + 89

T13 + 89

T_{19}^{2} + 51T_{19} + 867

T19^2 + 51*T19 + 867

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 9T + 27$ T^2 - 9*T + 27
$5$	$T^{2}$ T^2
$7$	$T^{2} + 37T + 343$ T^2 + 37*T + 343
$11$	$T^{2}$ T^2
$13$	$(T + 89)^{2}$ (T + 89)^2
$17$	$T^{2}$ T^2
$19$	$T^{2} + 51T + 867$ T^2 + 51*T + 867
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$T^{2} + 597T + 118803$ T^2 + 597*T + 118803
$37$	$T^{2} - 433T + 187489$ T^2 - 433*T + 187489
$41$	$T^{2}$ T^2
$43$	$T^{2} + 116427$ T^2 + 116427
$47$	$T^{2}$ T^2
$53$	$T^{2}$ T^2
$59$	$T^{2}$ T^2
$61$	$T^{2} + 182T + 33124$ T^2 + 182*T + 33124
$67$	$T^{2} + 753T + 189003$ T^2 + 753*T + 189003
$71$	$T^{2}$ T^2
$73$	$T^{2} + 919T + 844561$ T^2 + 919*T + 844561
$79$	$T^{2} + 2271 T + 1719147$ T^2 + 2271*T + 1719147
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$(T - 1330)^{2}$ (T - 1330)^2
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