LMFDB - Newform orbit 336.4.bl.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(31,336)] 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, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.31"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.bl (of order $6$ , degree $2$ , 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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$ -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.

$f(q)$	$=$	$q + ( - 3 \zeta_{6} + 3) q^{3} + (2 \zeta_{6} - 4) q^{5} + ( - 7 \zeta_{6} - 14) q^{7} - 9 \zeta_{6} q^{9} + (38 \zeta_{6} + 38) q^{11} + ( - 42 \zeta_{6} + 21) q^{13} + (12 \zeta_{6} - 6) q^{15} + ( - 36 \zeta_{6} - 36) q^{17} + \cdots + ( - 684 \zeta_{6} + 342) q^{99} +O(q^{100})$ q + (-3z + 3) q^3 + (2z - 4) q^5 + (-7z - 14) q^7 - 9z q^9 + (38z + 38) q^11 + (-42z + 21) q^13 + (12z - 6) q^15 + (-36z - 36) q^17 - 115z q^19 + (42z - 63) q^21 + (112z - 224) q^23 + (113z - 113) q^25 - 27 * q^27 - 96 * q^29 + (209z - 209) q^31 + (-114z + 228) q^33 + (-14z + 70) q^35 - 265z q^37 + (-63z - 63) q^39 + (-404z + 202) q^41 + (634z - 317) q^43 + (18z + 18) q^45 - 66z q^47 + (245z + 147) q^49 + (108z - 216) q^51 + (-288z + 288) q^53 - 228 * q^55 - 345 * q^57 + (720z - 720) q^59 + (280z - 560) q^61 + (189z - 63) q^63 + 126z q^65 + (-339z - 339) q^67 + (672z - 336) q^69 + (-628z + 314) q^71 + (37z + 37) q^73 + 339z q^75 + (-1064z - 266) q^77 + (-99z + 198) q^79 + (81z - 81) q^81 + 66 * q^83 + 216 * q^85 + (288z - 288) q^87 + (-452z + 904) q^89 + (735z - 588) q^91 + 627z q^93 + (230z + 230) q^95 + (-2024z + 1012) q^97 + (-684z + 342) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 3 q^{3} - 6 q^{5} - 35 q^{7} - 9 q^{9} + 114 q^{11} - 108 q^{17} - 115 q^{19} - 84 q^{21} - 336 q^{23} - 113 q^{25} - 54 q^{27} - 192 q^{29} - 209 q^{31} + 342 q^{33} + 126 q^{35} - 265 q^{37} - 189 q^{39}+ \cdots + 690 q^{95}+O(q^{100})$ 2 * q + 3 * q^3 - 6 * q^5 - 35 * q^7 - 9 * q^9 + 114 * q^11 - 108 * q^17 - 115 * q^19 - 84 * q^21 - 336 * q^23 - 113 * q^25 - 54 * q^27 - 192 * q^29 - 209 * q^31 + 342 * q^33 + 126 * q^35 - 265 * q^37 - 189 * q^39 + 54 * q^45 - 66 * q^47 + 539 * q^49 - 324 * q^51 + 288 * q^53 - 456 * q^55 - 690 * q^57 - 720 * q^59 - 840 * q^61 + 63 * q^63 + 126 * q^65 - 1017 * q^67 + 111 * q^73 + 339 * q^75 - 1596 * q^77 + 297 * q^79 - 81 * q^81 + 132 * q^83 + 432 * q^85 - 288 * q^87 + 1356 * q^89 - 441 * q^91 + 627 * q^93 + 690 * q^95

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}

31.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.50000

−

2.59808i

−3.00000

1.73205i

−17.5000

−

6.06218i

−4.50000

−

7.79423i

271.1

1.50000

2.59808i

−3.00000

−

1.73205i

−17.5000

6.06218i

−4.50000

7.79423i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.f	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.bl.c	yes	2
4.b	odd	2	1		336.4.bl.a	✓	2
7.d	odd	6	1		336.4.bl.a	✓	2
28.f	even	6	1	inner	336.4.bl.c	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.4.bl.a	✓	2	4.b	odd	2	1
336.4.bl.a	✓	2	7.d	odd	6	1
336.4.bl.c	yes	2	1.a	even	1	1	trivial
336.4.bl.c	yes	2	28.f	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}^{2} + 6T_{5} + 12

T5^2 + 6*T5 + 12

T_{11}^{2} - 114T_{11} + 4332

T11^2 - 114*T11 + 4332

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$5$	$T^{2} + 6T + 12$ T^2 + 6*T + 12
$7$	$T^{2} + 35T + 343$ T^2 + 35*T + 343
$11$	$T^{2} - 114T + 4332$ T^2 - 114*T + 4332
$13$	$T^{2} + 1323$ T^2 + 1323
$17$	$T^{2} + 108T + 3888$ T^2 + 108*T + 3888
$19$	$T^{2} + 115T + 13225$ T^2 + 115*T + 13225
$23$	$T^{2} + 336T + 37632$ T^2 + 336*T + 37632
$29$	$(T + 96)^{2}$ (T + 96)^2
$31$	$T^{2} + 209T + 43681$ T^2 + 209*T + 43681
$37$	$T^{2} + 265T + 70225$ T^2 + 265*T + 70225
$41$	$T^{2} + 122412$ T^2 + 122412
$43$	$T^{2} + 301467$ T^2 + 301467
$47$	$T^{2} + 66T + 4356$ T^2 + 66*T + 4356
$53$	$T^{2} - 288T + 82944$ T^2 - 288*T + 82944
$59$	$T^{2} + 720T + 518400$ T^2 + 720*T + 518400
$61$	$T^{2} + 840T + 235200$ T^2 + 840*T + 235200
$67$	$T^{2} + 1017 T + 344763$ T^2 + 1017*T + 344763
$71$	$T^{2} + 295788$ T^2 + 295788
$73$	$T^{2} - 111T + 4107$ T^2 - 111*T + 4107
$79$	$T^{2} - 297T + 29403$ T^2 - 297*T + 29403
$83$	$(T - 66)^{2}$ (T - 66)^2
$89$	$T^{2} - 1356 T + 612912$ T^2 - 1356*T + 612912
$97$	$T^{2} + 3072432$ T^2 + 3072432
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