Newform orbit 171.3.q.a

• Label: 171.3.q.a
• Level: $171$
• Weight: $3$
• Character orbit: 171.q
• Analytic conductor: $4.659$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65941252056\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 2 q^{3} + 72 q^{4} - 18 q^{5} - 22 q^{6} + 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
20.1	−3.35010	−	1.93418i	−1.25076	−	2.72683i	5.48210	+	9.49527i	−7.85315	+	4.53402i	−1.08401	+	11.5543i	0.551401	−	0.955054i		−	26.9400i	−5.87120	+	6.82122i	35.0784
20.2	−3.26875	−	1.88722i	−2.62151	+	1.45865i	5.12317	+	8.87359i	3.99506	−	2.30655i	11.3219	+	0.179376i	−3.48382	+	6.03415i		−	23.5764i	4.74465	−	7.64776i	−17.4118
20.3	−3.09705	−	1.78808i	2.66600	−	1.37566i	4.39447	+	7.61144i	4.10114	−	2.36780i	−10.7165	−	0.506538i	−1.24181	+	2.15087i		−	17.1260i	5.21511	−	7.33502i	−16.9353
20.4	−2.99030	−	1.72645i	0.0547441	+	2.99950i	3.96126	+	6.86110i	−1.97273	+	1.13895i	5.01478	−	9.06391i	5.96281	−	10.3279i		−	13.5440i	−8.99401	+	0.328410i	7.86538
20.5	−2.96952	−	1.71445i	2.47292	+	1.69843i	3.87870	+	6.71811i	−2.50796	+	1.44797i	−4.43150	−	9.28323i	−3.77541	+	6.53920i		−	12.8838i	3.23066	+	8.40017i	9.92990
20.6	−2.40334	−	1.38757i	1.68015	−	2.48538i	1.85070	+	3.20551i	−1.08225	+	0.624835i	−7.48661	+	3.64189i	3.10197	−	5.37276i			0.828638i	−3.35421	−	8.35160i	3.46801
20.7	−2.32987	−	1.34515i	−2.46014	−	1.71689i	1.61888	+	2.80398i	1.31696	−	0.760345i	3.42235	+	7.30940i	−2.02468	+	3.50685i			2.05068i	3.10461	+	8.44757i	−4.09112
20.8	−2.31996	−	1.33943i	−2.92815	−	0.652642i	1.58813	+	2.75073i	2.50859	−	1.44833i	5.91901	+	5.43615i	6.06415	−	10.5034i			2.20667i	8.14812	+	3.82207i	−7.75975
20.9	−2.19209	−	1.26560i	2.05203	+	2.18841i	1.20350	+	2.08453i	8.14302	−	4.70138i	−1.72858	−	7.39426i	1.59509	−	2.76277i			4.03219i	−0.578305	+	8.98140i	−23.8003
20.10	−2.05717	−	1.18771i	−2.58832	+	1.51678i	0.821296	+	1.42253i	−7.91441	+	4.56938i	7.12609	−	0.0461048i	−1.26225	+	2.18629i			5.59982i	4.39877	−	7.85180i	21.7084
20.11	−1.84414	−	1.06471i	−0.0845080	−	2.99881i	0.267231	+	0.462857i	0.590019	−	0.340648i	−3.03703	+	5.62020i	−6.43659	+	11.1485i			7.37961i	−8.98572	+	0.506847i	−1.45077
20.12	−1.77501	−	1.02480i	2.96735	+	0.441388i	0.100449	+	0.173983i	−5.79916	+	3.34815i	−4.81475	−	3.82443i	0.488361	−	0.845867i			7.78667i	8.61035	+	2.61951i	13.7248
20.13	−0.692274	−	0.399685i	0.0689180	−	2.99921i	−1.68050	−	2.91072i	8.02032	−	4.63053i	−1.24645	+	2.04873i	3.10019	−	5.36969i			5.88416i	−8.99050	−	0.413399i	−7.40301
20.14	−0.658324	−	0.380083i	1.47417	+	2.61282i	−1.71107	−	2.96367i	−0.162364	+	0.0937407i	0.0226040	−	2.28039i	−4.08229	+	7.07074i			5.64207i	−4.65362	+	7.70349i	0.142517
20.15	−0.511116	−	0.295093i	−1.58134	−	2.54938i	−1.82584	−	3.16245i	−4.25975	+	2.45937i	0.0559453	+	1.76967i	3.55508	−	6.15757i			4.51592i	−3.99870	+	8.06290i	2.90297
20.16	−0.506424	−	0.292384i	2.83118	−	0.992190i	−1.82902	−	3.16796i	2.36501	−	1.36544i	−1.72388	−	0.325322i	0.857721	−	1.48562i			4.47818i	7.03112	−	5.61813i	−1.59693
20.17	−0.406613	−	0.234758i	0.510975	+	2.95616i	−1.88978	−	3.27319i	−5.03210	+	2.90528i	0.486214	−	1.32197i	5.32202	−	9.21801i			3.65262i	−8.47781	+	3.02105i	2.72815
20.18	−0.00940539	−	0.00543021i	−2.99982	+	0.0330393i	−1.99994	−	3.46400i	−2.13068	+	1.23015i	0.0283939	+	0.0159789i	−1.68412	+	2.91698i			0.0868820i	8.99782	−	0.198224i	0.0267199
20.19	0.0217615	+	0.0125640i	−2.91865	−	0.693896i	−1.99968	−	3.46355i	7.22268	−	4.17002i	−0.0547961	−	0.0517702i	−3.62212	+	6.27369i		−	0.201008i	8.03702	+	4.05048i	0.209569
20.20	0.133214	+	0.0769109i	1.82791	−	2.37881i	−1.98817	−	3.44361i	−7.01226	+	4.04853i	0.426459	−	0.176304i	−5.54783	+	9.60912i		−	1.22693i	−2.31749	−	8.69651i	−1.24550

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	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.q.a	✓	72
3.b	odd	2	1		513.3.q.a		72
9.c	even	3	1		513.3.q.a		72
9.d	odd	6	1	inner	171.3.q.a	✓	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
171.3.q.a	✓	72	1.a	even	1	1	trivial
171.3.q.a	✓	72	9.d	odd	6	1	inner
513.3.q.a		72	3.b	odd	2	1
513.3.q.a		72	9.c	even	3	1

Hecke kernels

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