Newform orbit 684.2.j.a

• Label: 684.2.j.a
• Level: $684$
• Weight: $2$
• Character orbit: 684.j
• Analytic conductor: $5.462$
• Analytic rank: $0$
• Dimension: $40$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$684 = 2^{2} \cdot 3^{2} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	684.j (of order $3$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$5.46176749826$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) =$ 40 * q - 2 * q^3 - q^7 - 2 * q^9 - q^11 + 2 * q^13 + q^15 + 5 * q^17 + q^19 + 6 * q^21 + 8 * q^23 - 20 * q^25 + 7 * q^27 - 9 * q^29 + 2 * q^31 - q^33 - 6 * q^35 + 2 * q^37 + 3 * q^39 - 25 * q^41 - 16 * q^43 + 44 * q^45 - 10 * q^47 - 21 * q^49 + 19 * q^51 - 8 * q^53 - 22 * q^59 - 7 * q^61 + 10 * q^63 - 6 * q^65 + 20 * q^67 + 38 * q^69 - 5 * q^71 - 10 * q^73 - 10 * q^77 + 2 * q^79 + 22 * q^81 + q^83 - 52 * q^87 - 34 * q^89 + 16 * q^91 - 8 * q^93 + 36 * q^95 + 2 * q^97 + 28 * 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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
49.1		0		−1.73202	+	0.0101310i		0		−1.94402	−	3.36714i		0		−0.0132784	−	0.0229989i		0		2.99979	−	0.0350943i		0
49.2		0		−1.73108	−	0.0580258i		0		0.748116	+	1.29577i		0		−1.75655	−	3.04244i		0		2.99327	+	0.200895i		0
49.3		0		−1.65259	+	0.518616i		0		1.53869	+	2.66510i		0		2.34238	+	4.05712i		0		2.46208	−	1.71411i		0
49.4		0		−1.55715	−	0.758482i		0		−0.482969	−	0.836526i		0		0.302844	+	0.524541i		0		1.84941	+	2.36214i		0
49.5		0		−1.16446	+	1.28220i		0		−0.145159	−	0.251423i		0		−0.636745	−	1.10287i		0		−0.288081	−	2.98614i		0
49.6		0		−0.961962	−	1.44036i		0		1.35066	+	2.33941i		0		−0.793446	−	1.37429i		0		−1.14926	+	2.77114i		0
49.7		0		−0.833161	+	1.51850i		0		−1.50323	−	2.60367i		0		0.886284	+	1.53509i		0		−1.61168	−	2.53031i		0
49.8		0		−0.709119	−	1.58024i		0		−0.702434	−	1.21665i		0		1.81436	+	3.14256i		0		−1.99430	+	2.24115i		0
49.9		0		−0.672276	+	1.59626i		0		1.20282	+	2.08335i		0		−1.53172	−	2.65301i		0		−2.09609	−	2.14625i		0
49.10		0		−0.404917	−	1.68406i		0		−1.80780	−	3.13120i		0		−2.29625	−	3.97722i		0		−2.67209	+	1.36380i		0
49.11		0		0.0878220	+	1.72982i		0		1.41926	+	2.45824i		0		0.648230	+	1.12277i		0		−2.98457	+	0.303833i		0
49.12		0		0.143302	−	1.72611i		0		−0.479354	−	0.830266i		0		2.11741	+	3.66746i		0		−2.95893	−	0.494710i		0
49.13		0		0.597890	+	1.62559i		0		−1.25752	−	2.17808i		0		−0.950930	−	1.64706i		0		−2.28505	+	1.94384i		0
49.14		0		0.747847	−	1.56228i		0		0.136750	+	0.236858i		0		−0.546712	−	0.946933i		0		−1.88145	−	2.33670i		0
49.15		0		1.23208	−	1.21737i		0		2.12356	+	3.67811i		0		1.38218	+	2.39401i		0		0.0360230	−	2.99978i		0
49.16		0		1.24864	+	1.20038i		0		0.745287	+	1.29087i		0		0.903772	+	1.56538i		0		0.118185	+	2.99767i		0
49.17		0		1.30529	+	1.13852i		0		−1.34080	−	2.32233i		0		−0.125175	−	0.216810i		0		0.407555	+	2.97219i		0
49.18		0		1.61687	−	0.621068i		0		−0.400523	−	0.693726i		0		−1.41378	−	2.44874i		0		2.22855	−	2.00838i		0
49.19		0		1.71780	+	0.221694i		0		1.53086	+	2.65152i		0		−2.37338	−	4.11082i		0		2.90170	+	0.761653i		0
49.20		0		1.72118	−	0.193715i		0		−0.732210	−	1.26823i		0		1.54051	+	2.66824i		0		2.92495	−	0.666840i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
171.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	684.2.j.a	✓	40
3.b	odd	2	1		2052.2.j.a		40
9.c	even	3	1		684.2.l.a	yes	40
9.d	odd	6	1		2052.2.l.a		40
19.c	even	3	1		684.2.l.a	yes	40
57.h	odd	6	1		2052.2.l.a		40
171.h	even	3	1	inner	684.2.j.a	✓	40
171.j	odd	6	1		2052.2.j.a		40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
684.2.j.a	✓	40	1.a	even	1	1	trivial
684.2.j.a	✓	40	171.h	even	3	1	inner
684.2.l.a	yes	40	9.c	even	3	1
684.2.l.a	yes	40	19.c	even	3	1
2052.2.j.a		40	3.b	odd	2	1
2052.2.j.a		40	171.j	odd	6	1
2052.2.l.a		40	9.d	odd	6	1
2052.2.l.a		40	57.h	odd	6	1

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(684, [\chi])$.