Newform orbit 171.3.n.a

• Label: 171.3.n.a
• Level: $171$
• Weight: $3$
• Character orbit: 171.n
• Analytic conductor: $4.659$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	171.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65941252056\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 3 q^{2} + q^{3} + 73 q^{4} - 13 q^{6} - q^{7} + 5 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−3.43451	+	1.98292i	2.13391	−	2.10865i	5.86391	−	10.1566i			0.416904i	−3.14766	+	11.4736i	−3.19072	+	5.52649i			30.6472i	0.107158	−	8.99936i	−0.826686	−	1.43186i
11.2	−3.29397	+	1.90178i	0.857338	+	2.87489i	5.23350	−	9.06469i			4.15749i	−8.29143	−	7.83933i	2.97063	−	5.14528i			24.5976i	−7.52994	+	4.92950i	−7.90662	−	13.6947i
11.3	−2.98070	+	1.72091i	−2.95378	+	0.524601i	3.92303	−	6.79489i			3.58462i	7.90152	−	6.64684i	1.78162	−	3.08586i			13.2374i	8.44959	−	3.09911i	−6.16879	−	10.6847i
11.4	−2.97755	+	1.71909i	−2.30348	−	1.92197i	3.91053	−	6.77323i		−	8.37705i	10.1628	+	1.76288i	2.22581	−	3.85521i			13.1374i	1.61204	+	8.85445i	14.4009	+	24.9431i
11.5	−2.63101	+	1.51902i	2.35103	+	1.86351i	2.61482	−	4.52901i		−	6.84159i	−9.01629	−	1.33168i	−3.34863	+	5.80000i			3.73570i	2.05464	+	8.76233i	10.3925	+	18.0003i
11.6	−2.57535	+	1.48688i	−1.89614	+	2.32479i	2.42163	−	4.19438i		−	0.856662i	1.42655	−	8.80649i	−5.41085	+	9.37188i			2.50764i	−1.80929	−	8.81626i	1.27375	+	2.20621i
11.7	−2.51271	+	1.45071i	−0.407391	−	2.97221i	2.20914	−	3.82634i			7.56102i	5.33548	+	6.87729i	3.82876	−	6.63160i			1.21360i	−8.66807	+	2.42170i	−10.9689	−	18.9987i
11.8	−2.46085	+	1.42077i	2.98129	+	0.334500i	2.03719	−	3.52852i		−	0.513997i	−7.81177	+	3.41259i	3.84223	−	6.65493i			0.211380i	8.77622	+	1.99449i	0.730273	+	1.26487i
11.9	−2.18680	+	1.26255i	0.700979	−	2.91696i	1.18805	−	2.05776i		−	3.32218i	2.14989	+	7.26380i	−0.588043	+	1.01852i		−	4.10051i	−8.01726	−	4.08945i	4.19441	+	7.26493i
11.10	−1.87868	+	1.08466i	0.803747	+	2.89033i	0.352955	−	0.611336i			8.45042i	−4.64499	−	4.55821i	−2.96980	+	5.14384i		−	7.14591i	−7.70798	+	4.64619i	−9.16580	−	15.8756i
11.11	−1.79858	+	1.03841i	2.81797	−	1.02909i	0.156597	−	0.271233i			5.76968i	−3.99973	+	4.77712i	−5.23061	+	9.05969i		−	7.65684i	6.88194	−	5.79990i	−5.99130	−	10.3772i
11.12	−1.67797	+	0.968778i	−0.241130	+	2.99029i	−0.122940	+	0.212939i		−	3.56553i	−2.49232	−	5.25123i	5.75629	−	9.97018i		−	8.22663i	−8.88371	−	1.44210i	3.45421	+	5.98286i
11.13	−1.52713	+	0.881691i	−2.86614	+	0.886146i	−0.445243	+	0.771183i		−	4.20095i	3.59567	−	3.88031i	1.07447	−	1.86104i		−	8.62379i	7.42949	−	5.07963i	3.70394	+	6.41541i
11.14	−1.17515	+	0.678474i	−2.77157	−	1.14822i	−1.07935	+	1.86948i			5.28553i	4.03605	−	0.531103i	−1.20457	+	2.08637i		−	8.35703i	6.36318	+	6.36474i	−3.58609	−	6.21130i
11.15	−1.09394	+	0.631586i	−1.73513	−	2.44731i	−1.20220	+	2.08227i		−	3.47204i	3.44381	+	1.58132i	−3.61953	+	6.26920i		−	8.08985i	−2.97864	+	8.49280i	2.19289	+	3.79820i
11.16	−0.783509	+	0.452359i	2.48319	−	1.68338i	−1.59074	+	2.75525i			1.12143i	−1.18411	+	2.44224i	4.59151	−	7.95273i		−	6.49722i	3.33246	−	8.36031i	−0.507290	−	0.878652i
11.17	−0.552143	+	0.318780i	2.67057	−	1.36676i	−1.79676	+	3.11208i		−	9.82496i	−1.03884	+	1.60597i	−1.72131	+	2.98139i		−	4.84132i	5.26392	−	7.30008i	3.13200	+	5.42478i
11.18	−0.276176	+	0.159450i	2.41106	+	1.78516i	−1.94915	+	3.37603i			3.48138i	−0.950522	−	0.108573i	1.52603	−	2.64316i		−	2.51878i	2.62643	+	8.60824i	−0.555107	−	0.961474i
11.19	−0.190755	+	0.110133i	−0.267723	+	2.98803i	−1.97574	+	3.42208i		−	5.73814i	−0.278010	−	0.599468i	−3.79245	+	6.56872i		−	1.75144i	−8.85665	−	1.59993i	0.631957	+	1.09458i
11.20	−0.0367265	+	0.0212040i	−2.01808	+	2.21976i	−1.99910	+	3.46254i			6.93179i	0.0270491	−	0.124316i	1.72622	−	2.98990i		−	0.339188i	−0.854685	−	8.95933i	−0.146982	−	0.254580i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
171.n	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	171.3.n.a	yes	76
3.b	odd	2	1		513.3.n.a		76
9.c	even	3	1		513.3.j.a		76
9.d	odd	6	1		171.3.j.a	✓	76
19.c	even	3	1		171.3.j.a	✓	76
57.h	odd	6	1		513.3.j.a		76
171.g	even	3	1		513.3.n.a		76
171.n	odd	6	1	inner	171.3.n.a	yes	76

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
171.3.j.a	✓	76	9.d	odd	6	1
171.3.j.a	✓	76	19.c	even	3	1
171.3.n.a	yes	76	1.a	even	1	1	trivial
171.3.n.a	yes	76	171.n	odd	6	1	inner
513.3.j.a		76	9.c	even	3	1
513.3.j.a		76	57.h	odd	6	1
513.3.n.a		76	3.b	odd	2	1
513.3.n.a		76	171.g	even	3	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(171, [\chi])\).