LMFDB - Newform orbit 336.4.bj.d

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,-17] 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.

$f(q)$	$=$	$q + ( - 3 \zeta_{6} + 6) q^{3} + ( - 19 \zeta_{6} + 1) q^{7} + ( - 27 \zeta_{6} + 27) q^{9} + 19 q^{13} + ( - 73 \zeta_{6} - 73) q^{19} + ( - 60 \zeta_{6} - 51) q^{21} - 125 \zeta_{6} q^{25} + ( - 162 \zeta_{6} + 81) q^{27} + \cdots + 1330 q^{97} +O(q^{100})$ q + (-3z + 6) q^3 + (-19z + 1) q^7 + (-27z + 27) q^9 + 19 * q^13 + (-73z - 73) q^19 + (-60z - 51) q^21 - 125z q^25 + (-162z + 81) q^27 + (-109z + 218) q^31 + (323z - 323) q^37 + (-57z + 114) q^39 + (-646z + 323) q^43 + (323z - 360) q^49 - 657 * q^57 + (182z - 182) q^61 + (-27z - 486) q^63 + (-629z + 1258) q^67 - 271z q^73 + (-375z - 375) q^75 + (127z + 127) q^79 - 729z q^81 + (-361z + 19) q^91 + (-981z + 981) q^93 + 1330 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 9 q^{3} - 17 q^{7} + 27 q^{9} + 38 q^{13} - 219 q^{19} - 162 q^{21} - 125 q^{25} + 327 q^{31} - 323 q^{37} + 171 q^{39} - 397 q^{49} - 1314 q^{57} - 182 q^{61} - 999 q^{63} + 1887 q^{67} - 271 q^{73}+ \cdots + 2660 q^{97}+O(q^{100})$ 2 * q + 9 * q^3 - 17 * q^7 + 27 * q^9 + 38 * q^13 - 219 * q^19 - 162 * q^21 - 125 * q^25 + 327 * q^31 - 323 * q^37 + 171 * q^39 - 397 * q^49 - 1314 * q^57 - 182 * q^61 - 999 * q^63 + 1887 * q^67 - 271 * q^73 - 1125 * q^75 + 381 * q^79 - 729 * q^81 - 323 * q^91 + 981 * 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

−8.50000

16.4545i

13.5000

23.3827i

191.1

4.50000

−

2.59808i

−8.50000

−

16.4545i

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.d	yes	2
3.b	odd	2	1	CM	336.4.bj.d	yes	2
4.b	odd	2	1		336.4.bj.a	✓	2
7.c	even	3	1		336.4.bj.a	✓	2
12.b	even	2	1		336.4.bj.a	✓	2
21.h	odd	6	1		336.4.bj.a	✓	2
28.g	odd	6	1	inner	336.4.bj.d	yes	2
84.n	even	6	1	inner	336.4.bj.d	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.4.bj.a	✓	2	4.b	odd	2	1
336.4.bj.a	✓	2	7.c	even	3	1
336.4.bj.a	✓	2	12.b	even	2	1
336.4.bj.a	✓	2	21.h	odd	6	1
336.4.bj.d	yes	2	1.a	even	1	1	trivial
336.4.bj.d	yes	2	3.b	odd	2	1	CM
336.4.bj.d	yes	2	28.g	odd	6	1	inner
336.4.bj.d	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} - 19

T13 - 19

T_{19}^{2} + 219T_{19} + 15987

T19^2 + 219*T19 + 15987

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
$7$	$T^{2} + 17T + 343$ T^2 + 17*T + 343
$11$	$T^{2}$ T^2
$13$	$(T - 19)^{2}$ (T - 19)^2
$17$	$T^{2}$ T^2
$19$	$T^{2} + 219T + 15987$ T^2 + 219*T + 15987
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$T^{2} - 327T + 35643$ T^2 - 327*T + 35643
$37$	$T^{2} + 323T + 104329$ T^2 + 323*T + 104329
$41$	$T^{2}$ T^2
$43$	$T^{2} + 312987$ T^2 + 312987
$47$	$T^{2}$ T^2
$53$	$T^{2}$ T^2
$59$	$T^{2}$ T^2
$61$	$T^{2} + 182T + 33124$ T^2 + 182*T + 33124
$67$	$T^{2} - 1887 T + 1186923$ T^2 - 1887*T + 1186923
$71$	$T^{2}$ T^2
$73$	$T^{2} + 271T + 73441$ T^2 + 271*T + 73441
$79$	$T^{2} - 381T + 48387$ T^2 - 381*T + 48387
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$(T - 1330)^{2}$ (T - 1330)^2
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