LMFDB - Newform orbit 336.4.b.c

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,-28] 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:	$2$
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 $\beta = \sqrt{-3}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 3 q^{3} - 8 \beta q^{5} + (7 \beta - 14) q^{7} + 9 q^{9} - 2 \beta q^{11} - 8 \beta q^{13} - 24 \beta q^{15} - 44 \beta q^{17} - 52 q^{19} + (21 \beta - 42) q^{21} - 66 \beta q^{23} - 67 q^{25} + 27 q^{27} + \cdots - 18 \beta q^{99} +O(q^{100})$ q + 3 * q^3 - 8b q^5 + (7b - 14) q^7 + 9 * q^9 - 2b q^11 - 8b q^13 - 24b q^15 - 44b q^17 - 52 * q^19 + (21b - 42) q^21 - 66b q^23 - 67 * q^25 + 27 * q^27 - 246 * q^29 - 116 * q^31 - 6b q^33 + (112b + 168) q^35 - 314 * q^37 - 24b q^39 - 156b q^41 + 218b q^43 - 72b q^45 + 192 * q^47 + (-196b + 49) q^49 - 132b q^51 - 150 * q^53 - 48 * q^55 - 156 * q^57 + 204 * q^59 - 336b q^61 + (63b - 126) q^63 - 192 * q^65 - 294b q^67 - 198b q^69 + 470b q^71 - 72b q^73 - 201 * q^75 + (28b + 42) q^77 - 794b q^79 + 81 * q^81 + 252 * q^83 - 1056 * q^85 - 738 * q^87 + 124b q^89 + (112b + 168) q^91 - 348 * q^93 + 416b q^95 + 832b q^97 - 18b q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 6 q^{3} - 28 q^{7} + 18 q^{9} - 104 q^{19} - 84 q^{21} - 134 q^{25} + 54 q^{27} - 492 q^{29} - 232 q^{31} + 336 q^{35} - 628 q^{37} + 384 q^{47} + 98 q^{49} - 300 q^{53} - 96 q^{55} - 312 q^{57} + 408 q^{59}+ \cdots - 696 q^{93}+O(q^{100})$ 2 * q + 6 * q^3 - 28 * q^7 + 18 * q^9 - 104 * q^19 - 84 * q^21 - 134 * q^25 + 54 * q^27 - 492 * q^29 - 232 * q^31 + 336 * q^35 - 628 * q^37 + 384 * q^47 + 98 * q^49 - 300 * q^53 - 96 * q^55 - 312 * q^57 + 408 * q^59 - 252 * q^63 - 384 * q^65 - 402 * q^75 + 84 * q^77 + 162 * q^81 + 504 * q^83 - 2112 * q^85 - 1476 * q^87 + 336 * q^91 - 696 * 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

0.500000	+	0.866025i
0.500000	−	0.866025i

3.00000

−

13.8564i

−14.0000

12.1244i

9.00000

223.2

3.00000

13.8564i

−14.0000

−

12.1244i

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.c	yes	2
3.b	odd	2	1		1008.4.b.b		2
4.b	odd	2	1		336.4.b.b	✓	2
7.b	odd	2	1		336.4.b.b	✓	2
8.b	even	2	1		1344.4.b.a		2
8.d	odd	2	1		1344.4.b.d		2
12.b	even	2	1		1008.4.b.e		2
21.c	even	2	1		1008.4.b.e		2
28.d	even	2	1	inner	336.4.b.c	yes	2
56.e	even	2	1		1344.4.b.a		2
56.h	odd	2	1		1344.4.b.d		2
84.h	odd	2	1		1008.4.b.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.4.b.b	✓	2	4.b	odd	2	1
336.4.b.b	✓	2	7.b	odd	2	1
336.4.b.c	yes	2	1.a	even	1	1	trivial
336.4.b.c	yes	2	28.d	even	2	1	inner
1008.4.b.b		2	3.b	odd	2	1
1008.4.b.b		2	84.h	odd	2	1
1008.4.b.e		2	12.b	even	2	1
1008.4.b.e		2	21.c	even	2	1
1344.4.b.a		2	8.b	even	2	1
1344.4.b.a		2	56.e	even	2	1
1344.4.b.d		2	8.d	odd	2	1
1344.4.b.d		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} + 192

T5^2 + 192

T_{19} + 52

T19 + 52

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$(T - 3)^{2}$ (T - 3)^2
$5$	$T^{2} + 192$ T^2 + 192
$7$	$T^{2} + 28T + 343$ T^2 + 28*T + 343
$11$	$T^{2} + 12$ T^2 + 12
$13$	$T^{2} + 192$ T^2 + 192
$17$	$T^{2} + 5808$ T^2 + 5808
$19$	$(T + 52)^{2}$ (T + 52)^2
$23$	$T^{2} + 13068$ T^2 + 13068
$29$	$(T + 246)^{2}$ (T + 246)^2
$31$	$(T + 116)^{2}$ (T + 116)^2
$37$	$(T + 314)^{2}$ (T + 314)^2
$41$	$T^{2} + 73008$ T^2 + 73008
$43$	$T^{2} + 142572$ T^2 + 142572
$47$	$(T - 192)^{2}$ (T - 192)^2
$53$	$(T + 150)^{2}$ (T + 150)^2
$59$	$(T - 204)^{2}$ (T - 204)^2
$61$	$T^{2} + 338688$ T^2 + 338688
$67$	$T^{2} + 259308$ T^2 + 259308
$71$	$T^{2} + 662700$ T^2 + 662700
$73$	$T^{2} + 15552$ T^2 + 15552
$79$	$T^{2} + 1891308$ T^2 + 1891308
$83$	$(T - 252)^{2}$ (T - 252)^2
$89$	$T^{2} + 46128$ T^2 + 46128
$97$	$T^{2} + 2076672$ T^2 + 2076672
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