Newform orbit 336.4.bc.f

• Label: 336.4.bc.f
• Level: $336$
• Weight: $4$
• Character orbit: 336.bc
• Analytic conductor: $19.825$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.bc (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) =$ 48 * q - 12 * q^7 + 14 * q^9 + 88 * q^15 + 270 * q^19 + 50 * q^21 - 438 * q^25 - 216 * q^31 - 372 * q^33 + 66 * q^37 - 242 * q^39 - 900 * q^43 - 294 * q^45 + 60 * q^49 + 138 * q^51 + 1384 * q^57 + 108 * q^61 - 1096 * q^63 - 6 * q^67 - 1206 * q^73 + 594 * q^75 + 588 * q^79 - 54 * q^81 - 240 * q^85 + 3522 * q^87 - 234 * q^91 - 608 * q^93 - 1988 * q^99

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.

Label	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
17.1		0		−5.18895	−	0.273513i		0		2.97936	+	5.16040i		0		17.8140	−	5.06584i		0		26.8504	+	2.83849i		0
17.2		0		−5.12717	−	0.843898i		0		−8.29692	−	14.3707i		0		−9.36640	−	15.9772i		0		25.5757	+	8.65361i		0
17.3		0		−4.99832	−	1.42013i		0		−6.11124	−	10.5850i		0		−4.39164	+	17.9920i		0		22.9665	+	14.1965i		0
17.4		0		−4.88874	+	1.76075i		0		6.04693	+	10.4736i		0		−15.9632	−	9.39028i		0		20.7995	−	17.2157i		0
17.5		0		−3.72903	−	3.61861i		0		6.11124	+	10.5850i		0		−4.39164	+	17.9920i		0		0.811307	+	26.9878i		0
17.6		0		−3.62731	+	3.72057i		0		0.263312	+	0.456070i		0		16.6747	+	8.05944i		0		−0.685212	−	26.9913i		0
17.7		0		−3.29442	−	4.01831i		0		8.29692	+	14.3707i		0		−9.36640	−	15.9772i		0		−5.29359	+	26.4760i		0
17.8		0		−3.24856	+	4.05547i		0		3.20701	+	5.55470i		0		−14.8103	+	11.1200i		0		−5.89369	−	26.3489i		0
17.9		0		−2.83134	−	4.35701i		0		−2.97936	−	5.16040i		0		17.8140	−	5.06584i		0		−10.9670	+	24.6724i		0
17.10		0		−2.36622	+	4.62612i		0		−9.22876	−	15.9847i		0		6.70316	−	17.2646i		0		−15.8020	−	21.8928i		0
17.11		0		−0.919514	−	5.11415i		0		−6.04693	−	10.4736i		0		−15.9632	−	9.39028i		0		−25.3090	+	9.40505i		0
17.12		0		0.328966	+	5.18573i		0		−7.91382	−	13.7071i		0		−13.3812	+	12.8040i		0		−26.7836	+	3.41186i		0
17.13		0		0.726622	+	5.14510i		0		9.08545	+	15.7365i		0		18.4348	−	1.77707i		0		−25.9440	+	7.47709i		0
17.14		0		1.35065	+	5.01754i		0		2.40532	+	4.16613i		0		3.40309	−	18.2049i		0		−23.3515	+	13.5539i		0
17.15		0		1.40845	−	5.00163i		0		−0.263312	−	0.456070i		0		16.6747	+	8.05944i		0		−23.0325	−	14.0891i		0
17.16		0		1.88786	−	4.84107i		0		−3.20701	−	5.55470i		0		−14.8103	+	11.1200i		0		−19.8720	−	18.2785i		0
17.17		0		2.82323	−	4.36227i		0		9.22876	+	15.9847i		0		6.70316	−	17.2646i		0		−11.0587	−	24.6314i		0
17.18		0		3.06041	+	4.19928i		0		0.746155	+	1.29238i		0		−15.9029	−	9.49205i		0		−8.26784	+	25.7030i		0
17.19		0		4.27838	+	2.94880i		0		−5.57507	−	9.65631i		0		7.78587	+	16.8042i		0		9.60915	+	25.2322i		0
17.20		0		4.65546	−	2.30797i		0		7.91382	+	13.7071i		0		−13.3812	+	12.8040i		0		16.3465	−	21.4893i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
21.g	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.bc.f		48
3.b	odd	2	1	inner	336.4.bc.f		48
4.b	odd	2	1		168.4.u.a	✓	48
7.d	odd	6	1	inner	336.4.bc.f		48
12.b	even	2	1		168.4.u.a	✓	48
21.g	even	6	1	inner	336.4.bc.f		48
28.f	even	6	1		168.4.u.a	✓	48
84.j	odd	6	1		168.4.u.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.u.a	✓	48	4.b	odd	2	1
168.4.u.a	✓	48	12.b	even	2	1
168.4.u.a	✓	48	28.f	even	6	1
168.4.u.a	✓	48	84.j	odd	6	1
336.4.bc.f		48	1.a	even	1	1	trivial
336.4.bc.f		48	3.b	odd	2	1	inner
336.4.bc.f		48	7.d	odd	6	1	inner
336.4.bc.f		48	21.g	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 $S_{4}^{\mathrm{new}}(336, [\chi])$:

T5^48 + 1719*T5^46 + 1690641*T5^44 + 1132587728*T5^42 + 572282844510*T5^40 + 226442605756098*T5^38 + 72238249800180398*T5^36 + 18797737470498141192*T5^34 + 4029920768129097713463*T5^32 + 712455228165025196764001*T5^30 + 103966839722736959853362895*T5^28 + 12439356023065853409889329288*T5^26 + 1212880901919692449827800991070*T5^24 + 94876826342067626153546462575074*T5^22 + 5885994927108887847752378854388046*T5^20 + 282397564208519908937138598717123568*T5^18 + 10478159830152079472384715616984215657*T5^16 + 290119978884612693133720242244156418775*T5^14 + 5940037636311385940576335365663087916169*T5^12 + 81018546692326059463714541881623179634048*T5^10 + 704008465153785415414301340917434798632960*T5^8 + 1655228075939403326041681975377396129439744*T5^6 + 2975299746147961418698751213220225370030080*T5^4 + 810609805075993705564805442977784643190784*T5^2 + 193663546944588761028507484241951683772416

T13^24 + 27165*T13^22 + 309247695*T13^20 + 1920428280007*T13^18 + 7094435841003132*T13^16 + 15970388380769961168*T13^14 + 21678105215865114133568*T13^12 + 17222393041759569905504256*T13^10 + 7817867758178158337771667456*T13^8 + 1953512426959511638062888648704*T13^6 + 246454401663863447065105784635392*T13^4 + 12618317071117283026475474390876160*T13^2 + 137777953286248718812158641846616064