Newform orbit 336.4.bj.g

• Label: 336.4.bj.g
• Level: $336$
• Weight: $4$
• Character orbit: 336.bj
• Analytic conductor: $19.825$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $8$

Newspace parameters

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

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
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.

$\operatorname{Tr}(f)(q) =$ $32 q + 32 q^{9} - 40 q^{13} - 148 q^{21} + 316 q^{25} - 128 q^{33} - 644 q^{37} + 316 q^{45} + 632 q^{49} + 1136 q^{57} + 328 q^{61} - 1424 q^{69} + 1124 q^{73} + 1564 q^{81} - 912 q^{85} + 24 q^{93} - 3304 q^{97}+O(q^{100})$

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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
95.1		0		−5.19375	−	0.158148i		0		17.5520	−	10.1337i		0		18.5198	+	0.135878i		0		26.9500	+	1.64276i		0
95.2		0		−5.16981	−	0.522553i		0		−5.91262	+	3.41365i		0		−13.7402	−	12.4180i		0		26.4539	+	5.40300i		0
95.3		0		−5.06455	+	1.16202i		0		−6.80588	+	3.92938i		0		0.395544	+	18.5160i		0		24.2994	−	11.7702i		0
95.4		0		−3.03745	−	4.21591i		0		5.91262	−	3.41365i		0		−13.7402	−	12.4180i		0		−8.54780	+	25.6112i		0
95.5		0		−2.89222	+	4.31684i		0		−6.70049	+	3.86853i		0		13.9129	−	12.2242i		0		−10.2702	−	24.9705i		0
95.6		0		−2.73383	−	4.41884i		0		−17.5520	+	10.1337i		0		18.5198	+	0.135878i		0		−12.0523	+	24.1607i		0
95.7		0		−2.29238	+	4.66315i		0		6.70049	−	3.86853i		0		−13.9129	+	12.2242i		0		−16.4900	−	21.3795i		0
95.8		0		−1.52594	−	4.96704i		0		6.80588	−	3.92938i		0		0.395544	+	18.5160i		0		−22.3430	+	15.1588i		0
95.9		0		1.52594	+	4.96704i		0		6.80588	−	3.92938i		0		−0.395544	−	18.5160i		0		−22.3430	+	15.1588i		0
95.10		0		2.29238	−	4.66315i		0		6.70049	−	3.86853i		0		13.9129	−	12.2242i		0		−16.4900	−	21.3795i		0
95.11		0		2.73383	+	4.41884i		0		−17.5520	+	10.1337i		0		−18.5198	−	0.135878i		0		−12.0523	+	24.1607i		0
95.12		0		2.89222	−	4.31684i		0		−6.70049	+	3.86853i		0		−13.9129	+	12.2242i		0		−10.2702	−	24.9705i		0
95.13		0		3.03745	+	4.21591i		0		5.91262	−	3.41365i		0		13.7402	+	12.4180i		0		−8.54780	+	25.6112i		0
95.14		0		5.06455	−	1.16202i		0		−6.80588	+	3.92938i		0		−0.395544	−	18.5160i		0		24.2994	−	11.7702i		0
95.15		0		5.16981	+	0.522553i		0		−5.91262	+	3.41365i		0		13.7402	+	12.4180i		0		26.4539	+	5.40300i		0
95.16		0		5.19375	+	0.158148i		0		17.5520	−	10.1337i		0		−18.5198	−	0.135878i		0		26.9500	+	1.64276i		0
191.1		0		−5.19375	+	0.158148i		0		17.5520	+	10.1337i		0		18.5198	−	0.135878i		0		26.9500	−	1.64276i		0
191.2		0		−5.16981	+	0.522553i		0		−5.91262	−	3.41365i		0		−13.7402	+	12.4180i		0		26.4539	−	5.40300i		0
191.3		0		−5.06455	−	1.16202i		0		−6.80588	−	3.92938i		0		0.395544	−	18.5160i		0		24.2994	+	11.7702i		0
191.4		0		−3.03745	+	4.21591i		0		5.91262	+	3.41365i		0		−13.7402	+	12.4180i		0		−8.54780	−	25.6112i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
7.c	even	3	1	inner
12.b	even	2	1	inner
21.h	odd	6	1	inner
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.g	✓	32
3.b	odd	2	1	inner	336.4.bj.g	✓	32
4.b	odd	2	1	inner	336.4.bj.g	✓	32
7.c	even	3	1	inner	336.4.bj.g	✓	32
12.b	even	2	1	inner	336.4.bj.g	✓	32
21.h	odd	6	1	inner	336.4.bj.g	✓	32
28.g	odd	6	1	inner	336.4.bj.g	✓	32
84.n	even	6	1	inner	336.4.bj.g	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.4.bj.g	✓	32	1.a	even	1	1	trivial
336.4.bj.g	✓	32	3.b	odd	2	1	inner
336.4.bj.g	✓	32	4.b	odd	2	1	inner
336.4.bj.g	✓	32	7.c	even	3	1	inner
336.4.bj.g	✓	32	12.b	even	2	1	inner
336.4.bj.g	✓	32	21.h	odd	6	1	inner
336.4.bj.g	✓	32	28.g	odd	6	1	inner
336.4.bj.g	✓	32	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 $S_{4}^{\mathrm{new}}(336, [\chi])$:

T5^16 - 579*T5^14 + 256770*T5^12 - 37395467*T5^10 + 3759550482*T5^8 - 233452565091*T5^6 + 10602642000841*T5^4 - 284536107802800*T5^2 + 5010771054240000

$T_{13}^{4} + 5T_{13}^{3} - 3684T_{13}^{2} + 85684T_{13} - 180880$

T19^16 - 19023*T19^14 + 243345570*T19^12 - 1748942749615*T19^10 + 9176937791252898*T19^8 - 27659455872985546239*T19^6 + 56743799430348782765041*T19^4 - 15412671982647108235105200*T19^2 + 3713645554182899116417440000