Newform orbit 171.2.g.c

• Label: 171.2.g.c
• Level: $171$
• Weight: $2$
• Character orbit: 171.g
• Analytic conductor: $1.365$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.36544187456\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
106.1	−1.23404	−	2.13742i	1.62292	+	0.605076i	−2.04570	+	3.54325i	1.91524			−0.709450	−	4.21555i	−0.324708	+	0.562412i	5.16173			2.26777	+	1.96399i	−2.36348	−	4.09366i
106.2	−1.04884	−	1.81665i	0.154304	+	1.72516i	−1.20014	+	2.07870i	−2.89593			2.97217	−	2.08974i	0.116480	−	0.201749i	0.839660			−2.95238	+	0.532400i	3.03737	+	5.26088i
106.3	−0.978515	−	1.69484i	−1.73022	−	0.0795352i	−0.914982	+	1.58480i	−0.196235			1.55825	+	3.01027i	−2.23368	+	3.86885i	−0.332766			2.98735	+	0.275227i	0.192018	+	0.332586i
106.4	−0.888985	−	1.53977i	1.15573	−	1.29008i	−0.580589	+	1.00561i	−1.27957			−3.01384	−	0.632688i	0.657761	−	1.13928i	−1.49140			−0.328599	−	2.98195i	1.13752	+	1.97024i
106.5	−0.803309	−	1.39137i	−1.24014	−	1.20915i	−0.290611	+	0.503353i	3.75880			−0.686165	+	2.69682i	2.27973	−	3.94861i	−2.27943			0.0758986	+	2.99904i	−3.01948	−	5.22989i
106.6	−0.395929	−	0.685769i	0.659141	+	1.60173i	0.686481	−	1.18902i	2.59093			0.837442	−	1.08619i	−0.373088	+	0.646207i	−2.67091			−2.13107	+	2.11153i	−1.02582	−	1.77678i
106.7	−0.269545	−	0.466866i	−1.40907	+	1.00723i	0.854691	−	1.48037i	−0.947325			0.850050	+	0.386354i	1.18430	−	2.05126i	−1.99969			0.970970	−	2.83852i	0.255347	+	0.442274i
106.8	−0.185445	−	0.321199i	−0.894876	−	1.48297i	0.931221	−	1.61292i	−3.55521			−0.310379	+	0.562442i	−0.124876	+	0.216291i	−1.43254			−1.39839	+	2.65415i	0.659294	+	1.14193i
106.9	−0.0732670	−	0.126902i	0.157437	−	1.72488i	0.989264	−	1.71346i	2.57004			−0.230426	+	0.106398i	−1.73898	+	3.01201i	−0.582990			−2.95043	−	0.543121i	−0.188299	−	0.326143i
106.10	0.0973467	+	0.168609i	1.55267	+	0.767609i	0.981047	−	1.69922i	−1.90563			0.0217210	+	0.336519i	1.69446	−	2.93489i	0.771394			1.82155	+	2.38368i	−0.185507	−	0.321308i
106.11	0.616796	+	1.06832i	0.506944	+	1.65620i	0.239126	−	0.414178i	−0.551543			−1.45668	+	1.56312i	−1.62156	+	2.80862i	3.05715			−2.48602	+	1.67920i	−0.340189	−	0.589225i
106.12	0.847114	+	1.46725i	0.0521113	−	1.73127i	−0.435206	+	0.753799i	0.0882176			2.58434	−	1.39012i	1.84695	−	3.19901i	1.91378			−2.99457	−	0.180437i	0.0747304	+	0.129437i
106.13	1.01016	+	1.74964i	−1.70377	+	0.311691i	−1.04083	+	1.80277i	−4.18739			−2.26643	−	2.66614i	−0.976107	+	1.69067i	−0.164982			2.80570	−	1.06210i	−4.22991	−	7.32643i
106.14	1.19971	+	2.07797i	−0.340918	+	1.69817i	−1.87863	+	3.25388i	0.719678			−3.93774	+	1.32890i	1.65862	−	2.87282i	−4.21641			−2.76755	−	1.15787i	0.863408	+	1.49547i
106.15	1.30160	+	2.25443i	−1.17382	−	1.27363i	−2.38830	+	4.13666i	2.87794			1.34348	−	4.30405i	−1.80240	+	3.12185i	−7.22804			−0.244287	+	2.99004i	3.74592	+	6.48812i
106.16	1.30515	+	2.26059i	1.63157	−	0.581354i	−2.40684	+	4.16877i	−2.00201			3.44365	+	2.92956i	0.257107	−	0.445323i	−7.34455			2.32405	−	1.89704i	−2.61292	−	4.52572i
121.1	−1.23404	+	2.13742i	1.62292	−	0.605076i	−2.04570	−	3.54325i	1.91524			−0.709450	+	4.21555i	−0.324708	−	0.562412i	5.16173			2.26777	−	1.96399i	−2.36348	+	4.09366i
121.2	−1.04884	+	1.81665i	0.154304	−	1.72516i	−1.20014	−	2.07870i	−2.89593			2.97217	+	2.08974i	0.116480	+	0.201749i	0.839660			−2.95238	−	0.532400i	3.03737	−	5.26088i
121.3	−0.978515	+	1.69484i	−1.73022	+	0.0795352i	−0.914982	−	1.58480i	−0.196235			1.55825	−	3.01027i	−2.23368	−	3.86885i	−0.332766			2.98735	−	0.275227i	0.192018	−	0.332586i
121.4	−0.888985	+	1.53977i	1.15573	+	1.29008i	−0.580589	−	1.00561i	−1.27957			−3.01384	+	0.632688i	0.657761	+	1.13928i	−1.49140			−0.328599	+	2.98195i	1.13752	−	1.97024i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	171.2.g.c	✓	32
3.b	odd	2	1		513.2.g.c		32
9.c	even	3	1		171.2.h.c	yes	32
9.d	odd	6	1		513.2.h.c		32
19.c	even	3	1		171.2.h.c	yes	32
57.h	odd	6	1		513.2.h.c		32
171.g	even	3	1	inner	171.2.g.c	✓	32
171.n	odd	6	1		513.2.g.c		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
171.2.g.c	✓	32	1.a	even	1	1	trivial
171.2.g.c	✓	32	171.g	even	3	1	inner
171.2.h.c	yes	32	9.c	even	3	1
171.2.h.c	yes	32	19.c	even	3	1
513.2.g.c		32	3.b	odd	2	1
513.2.g.c		32	171.n	odd	6	1
513.2.h.c		32	9.d	odd	6	1
513.2.h.c		32	57.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\):

T2^32 - T2^31 + 25*T2^30 - 10*T2^29 + 358*T2^28 - 34*T2^27 + 3447*T2^26 + 717*T2^25 + 24681*T2^24 + 10382*T2^23 + 135298*T2^22 + 82109*T2^21 + 580960*T2^20 + 420887*T2^19 + 1945837*T2^18 + 1581978*T2^17 + 5070162*T2^16 + 4256568*T2^15 + 9953691*T2^14 + 8450307*T2^13 + 14418975*T2^12 + 11385480*T2^11 + 14005971*T2^10 + 10297053*T2^9 + 8939259*T2^8 + 4637646*T2^7 + 2245509*T2^6 + 590571*T2^5 + 154773*T2^4 + 18333*T2^3 + 5562*T2^2 + 567*T2 + 81

T5^16 + 3*T5^15 - 40*T5^14 - 115*T5^13 + 600*T5^12 + 1690*T5^11 - 4152*T5^10 - 12111*T5^9 + 12405*T5^8 + 43371*T5^7 - 6663*T5^6 - 67191*T5^5 - 25815*T5^4 + 23535*T5^3 + 13887*T5^2 + 756*T5 - 189