LMFDB - Newform orbit 336.4.b.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(223,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 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("336.223"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.b (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,0,0,34] 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:	$\Q(\sqrt{-6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 6$ x^2 + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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 $\beta = \sqrt{-6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 3 q^{3} + \beta q^{5} + (3 \beta + 17) q^{7} + 9 q^{9} - 23 \beta q^{11} - 30 \beta q^{13} + 3 \beta q^{15} + 21 \beta q^{17} + 10 q^{19} + (9 \beta + 51) q^{21} + 47 \beta q^{23} + 119 q^{25} + 27 q^{27} + \cdots - 207 \beta q^{99} +O(q^{100})$ q + 3 * q^3 + b * q^5 + (3b + 17) q^7 + 9 * q^9 - 23b q^11 - 30b q^13 + 3b q^15 + 21b q^17 + 10 * q^19 + (9b + 51) q^21 + 47b q^23 + 119 * q^25 + 27 * q^27 + 126 * q^29 + 8 * q^31 - 69b q^33 + (17b - 18) q^35 + 244 * q^37 - 90b q^39 - 151b q^41 - 66b q^43 + 9b q^45 - 180 * q^47 + (102b + 235) q^49 + 63b q^51 + 594 * q^53 + 138 * q^55 + 30 * q^57 - 540 * q^59 - 144b q^61 + (27b + 153) q^63 + 180 * q^65 + 432b q^67 + 141b q^69 + 73b q^71 - 270b q^73 + 357 * q^75 + (-391b + 414) q^77 - 420b q^79 + 81 * q^81 - 864 * q^83 - 126 * q^85 + 378 * q^87 + 527b q^89 + (-510b + 540) q^91 + 24 * q^93 + 10b q^95 + 330b q^97 - 207b q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 6 q^{3} + 34 q^{7} + 18 q^{9} + 20 q^{19} + 102 q^{21} + 238 q^{25} + 54 q^{27} + 252 q^{29} + 16 q^{31} - 36 q^{35} + 488 q^{37} - 360 q^{47} + 470 q^{49} + 1188 q^{53} + 276 q^{55} + 60 q^{57} - 1080 q^{59}+ \cdots + 48 q^{93}+O(q^{100})$ 2 * q + 6 * q^3 + 34 * q^7 + 18 * q^9 + 20 * q^19 + 102 * q^21 + 238 * q^25 + 54 * q^27 + 252 * q^29 + 16 * q^31 - 36 * q^35 + 488 * q^37 - 360 * q^47 + 470 * q^49 + 1188 * q^53 + 276 * q^55 + 60 * q^57 - 1080 * q^59 + 306 * q^63 + 360 * q^65 + 714 * q^75 + 828 * q^77 + 162 * q^81 - 1728 * q^83 - 252 * q^85 + 756 * q^87 + 1080 * q^91 + 48 * 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}

223.1

	−	2.44949i
		2.44949i

3.00000

−

2.44949i

17.0000

−

7.34847i

9.00000

223.2

3.00000

2.44949i

17.0000

7.34847i

9.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.d	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.b.d	yes	2
3.b	odd	2	1		1008.4.b.f		2
4.b	odd	2	1		336.4.b.a	✓	2
7.b	odd	2	1		336.4.b.a	✓	2
8.b	even	2	1		1344.4.b.b		2
8.d	odd	2	1		1344.4.b.c		2
12.b	even	2	1		1008.4.b.a		2
21.c	even	2	1		1008.4.b.a		2
28.d	even	2	1	inner	336.4.b.d	yes	2
56.e	even	2	1		1344.4.b.b		2
56.h	odd	2	1		1344.4.b.c		2
84.h	odd	2	1		1008.4.b.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.4.b.a	✓	2	4.b	odd	2	1
336.4.b.a	✓	2	7.b	odd	2	1
336.4.b.d	yes	2	1.a	even	1	1	trivial
336.4.b.d	yes	2	28.d	even	2	1	inner
1008.4.b.a		2	12.b	even	2	1
1008.4.b.a		2	21.c	even	2	1
1008.4.b.f		2	3.b	odd	2	1
1008.4.b.f		2	84.h	odd	2	1
1344.4.b.b		2	8.b	even	2	1
1344.4.b.b		2	56.e	even	2	1
1344.4.b.c		2	8.d	odd	2	1
1344.4.b.c		2	56.h	odd	2	1

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} + 6

T5^2 + 6

T_{19} - 10

T19 - 10

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$(T - 3)^{2}$ (T - 3)^2
$5$	$T^{2} + 6$ T^2 + 6
$7$	$T^{2} - 34T + 343$ T^2 - 34*T + 343
$11$	$T^{2} + 3174$ T^2 + 3174
$13$	$T^{2} + 5400$ T^2 + 5400
$17$	$T^{2} + 2646$ T^2 + 2646
$19$	$(T - 10)^{2}$ (T - 10)^2
$23$	$T^{2} + 13254$ T^2 + 13254
$29$	$(T - 126)^{2}$ (T - 126)^2
$31$	$(T - 8)^{2}$ (T - 8)^2
$37$	$(T - 244)^{2}$ (T - 244)^2
$41$	$T^{2} + 136806$ T^2 + 136806
$43$	$T^{2} + 26136$ T^2 + 26136
$47$	$(T + 180)^{2}$ (T + 180)^2
$53$	$(T - 594)^{2}$ (T - 594)^2
$59$	$(T + 540)^{2}$ (T + 540)^2
$61$	$T^{2} + 124416$ T^2 + 124416
$67$	$T^{2} + 1119744$ T^2 + 1119744
$71$	$T^{2} + 31974$ T^2 + 31974
$73$	$T^{2} + 437400$ T^2 + 437400
$79$	$T^{2} + 1058400$ T^2 + 1058400
$83$	$(T + 864)^{2}$ (T + 864)^2
$89$	$T^{2} + 1666374$ T^2 + 1666374
$97$	$T^{2} + 653400$ T^2 + 653400
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