Newform orbit 171.2.x.a

• Label: 171.2.x.a
• Level: $171$
• Weight: $2$
• Character orbit: 171.x
• Analytic conductor: $1.365$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.36544187456\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 9 q^{2} - 3 q^{4} - 9 q^{5} + 3 q^{7} - 24 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
14.1	−2.02450	−	1.69876i	1.43995	+	0.962572i	0.865531	+	4.90867i	−0.259455	−	0.712848i	−1.28000	−	4.39486i	−1.63029	+	2.82374i	3.94358	−	6.83047i	1.14691	+	2.77211i	−0.685689	+	1.88391i
14.2	−1.86688	−	1.56649i	−1.40345	+	1.01505i	0.684023	+	3.87929i	0.0603078	+	0.165694i	4.21014	+	0.303523i	0.340059	−	0.588999i	2.36286	−	4.09260i	0.939343	−	2.84915i	0.146972	−	0.403802i
14.3	−1.80901	−	1.51794i	−0.419122	−	1.68058i	0.621081	+	3.52233i	−1.22473	−	3.36492i	−1.79282	+	3.67638i	0.850912	−	1.47382i	1.86164	−	3.22446i	−2.64867	+	1.40873i	−2.89220	+	7.94624i
14.4	−1.50803	−	1.26539i	−1.01330	−	1.40472i	0.325651	+	1.84686i	1.48603	+	4.08283i	−0.249426	+	3.40056i	−0.0709257	+	0.122847i	−0.122694	+	0.212511i	−0.946454	+	2.84679i	2.92538	−	8.03742i
14.5	−1.28119	−	1.07504i	1.71187	−	0.263611i	0.138425	+	0.785049i	0.287058	+	0.788686i	−2.47662	−	1.50260i	1.60000	−	2.77129i	−1.00586	+	1.74220i	2.86102	−	0.902536i	0.480097	−	1.31905i
14.6	−0.907216	−	0.761244i	0.509726	+	1.65535i	−0.103749	−	0.588390i	−1.20969	−	3.32360i	0.797694	−	1.88978i	0.600840	−	1.04068i	−1.53807	+	2.66402i	−2.48036	+	1.68755i	−1.43262	+	3.93610i
14.7	−0.788568	−	0.661687i	−0.897794	+	1.48120i	−0.163287	−	0.926044i	0.446483	+	1.22670i	1.68807	−	0.573972i	−0.947545	+	1.64120i	−1.51339	+	2.62127i	−1.38793	−	2.65963i	0.459610	−	1.26277i
14.8	−0.507087	−	0.425497i	−1.51869	−	0.832819i	−0.271206	−	1.53809i	−0.735283	−	2.02017i	0.415745	+	1.06851i	−2.31844	+	4.01566i	−1.17888	+	2.04188i	1.61282	+	2.52958i	−0.486725	+	1.33727i
14.9	−0.247109	−	0.207349i	0.668060	−	1.59803i	−0.329227	−	1.86714i	−0.0294890	−	0.0810203i	−0.496432	+	0.256365i	−0.0754752	+	0.130727i	−0.628371	+	1.08837i	−2.10739	−	2.13516i	−0.00951247	+	0.0261353i
14.10	0.103187	+	0.0865845i	−1.73169	−	0.0351497i	−0.344146	−	1.95175i	0.862577	+	2.36991i	−0.175646	−	0.153565i	1.62680	−	2.81770i	0.268181	−	0.464503i	2.99753	+	0.121737i	−0.116191	+	0.319231i
14.11	0.353154	+	0.296331i	0.726580	+	1.57229i	−0.310391	−	1.76031i	0.919654	+	2.52673i	−0.209323	+	0.770567i	2.00770	−	3.47744i	0.873030	−	1.51213i	−1.94416	+	2.28478i	−0.423969	+	1.16485i
14.12	0.469442	+	0.393909i	1.72885	+	0.105209i	−0.282084	−	1.59978i	−0.821139	−	2.25606i	0.770154	+	0.730400i	−0.676894	+	1.17241i	1.11056	−	1.92354i	2.97786	+	0.363781i	0.503205	−	1.38254i
14.13	1.12318	+	0.942457i	−1.13434	−	1.30892i	0.0260041	+	0.147477i	−0.668800	−	1.83751i	−0.0404630	−	2.53922i	1.13552	−	1.96677i	1.35642	−	2.34939i	−0.426545	+	2.96952i	0.980596	−	2.69417i
14.14	1.28121	+	1.07506i	0.658438	+	1.60202i	0.138440	+	0.785130i	−0.199661	−	0.548565i	−0.878670	+	2.76037i	−1.25763	+	2.17828i	1.00580	−	1.74210i	−2.13292	+	2.10966i	0.333933	−	0.917472i
14.15	1.30761	+	1.09721i	0.387860	−	1.68807i	0.158664	+	0.899827i	1.05557	+	2.90015i	2.35933	−	1.78176i	−0.421554	+	0.730152i	0.927128	−	1.60583i	−2.69913	−	1.30947i	−1.80181	+	4.95044i
14.16	1.42288	+	1.19394i	−1.62073	+	0.610926i	0.251804	+	1.42805i	0.629748	+	1.73022i	−3.03552	−	1.06578i	−1.66649	+	2.88644i	0.510719	−	0.884592i	2.25354	−	1.98029i	−1.16972	+	3.21378i
14.17	1.98542	+	1.66596i	0.915994	−	1.47002i	0.819153	+	4.64565i	−1.06492	−	2.92585i	4.26763	−	1.39259i	−1.52472	+	2.64090i	−3.52134	+	6.09914i	−1.32191	−	2.69306i	2.76004	−	7.58315i
14.18	1.98591	+	1.66638i	−0.713953	+	1.57806i	0.819730	+	4.64892i	−0.441857	−	1.21399i	−4.04748	+	1.94417i	2.16209	−	3.74485i	−3.52652	+	6.10811i	−1.98054	−	2.25332i	1.14548	−	3.14718i
29.1	−2.53216	−	0.921632i	0.224069	+	1.71750i	4.03035	+	3.38187i	0.619688	−	0.109268i	1.01552	−	4.55549i	0.508133	+	0.880112i	−4.39400	−	7.61064i	−2.89959	+	0.769674i	−1.66986	−	0.294441i
29.2	−2.34386	−	0.853094i	1.04482	−	1.38143i	3.23381	+	2.71349i	−1.83975	+	0.324398i	−3.62740	+	2.34655i	−1.22269	−	2.11775i	−2.77045	−	4.79855i	−0.816708	−	2.88669i	4.58886	+	0.809141i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
171.x	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	171.2.x.a	✓	108
3.b	odd	2	1		513.2.bo.a		108
9.c	even	3	1		513.2.cd.a		108
9.d	odd	6	1		171.2.bd.a	yes	108
19.f	odd	18	1		171.2.bd.a	yes	108
57.j	even	18	1		513.2.cd.a		108
171.x	even	18	1	inner	171.2.x.a	✓	108
171.bc	odd	18	1		513.2.bo.a		108

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
171.2.x.a	✓	108	1.a	even	1	1	trivial
171.2.x.a	✓	108	171.x	even	18	1	inner
171.2.bd.a	yes	108	9.d	odd	6	1
171.2.bd.a	yes	108	19.f	odd	18	1
513.2.bo.a		108	3.b	odd	2	1
513.2.bo.a		108	171.bc	odd	18	1
513.2.cd.a		108	9.c	even	3	1
513.2.cd.a		108	57.j	even	18	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(171, [\chi])\).