Newform orbit 171.2.bd.a

• Label: 171.2.bd.a
• Level: $171$
• Weight: $2$
• Character orbit: 171.bd
• 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.bd (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.

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 9 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} - 9 q^{6} - 6 q^{7} + 3 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−2.06424	−	1.73210i	1.65249	−	0.518905i	0.913607	+	5.18132i	0.404473	−	0.482032i	−4.30994	−	1.79115i	−1.01627			4.39400	−	7.61064i	2.46148	−	1.71497i	−1.66986	+	0.294441i
2.2	−1.91073	−	1.60329i	−1.54188	−	0.789062i	0.733045	+	4.15731i	−1.20081	+	1.43107i	1.68101	+	3.97976i	2.44537			2.77045	−	4.79855i	1.75476	+	2.43327i	4.58886	−	0.809141i
2.3	−1.73314	−	1.45428i	−1.29784	+	1.14700i	0.541559	+	3.07134i	1.93467	−	2.30565i	3.91740	−	0.100500i	−4.02280			1.26552	−	2.19195i	0.368770	−	2.97725i	−6.70611	+	1.18247i
2.4	−1.49018	−	1.25041i	0.939095	+	1.45537i	0.309818	+	1.75707i	−1.38825	+	1.65445i	0.420388	−	3.34302i	−0.126291			−0.209926	+	0.363602i	−1.23620	+	2.73346i	4.13748	−	0.729550i
2.5	−1.20013	−	1.00703i	0.499246	−	1.65854i	0.0789109	+	0.447526i	1.56758	−	1.86817i	−2.26936	+	1.48771i	4.63240			−1.21069	+	2.09698i	−2.50151	−	1.65604i	−3.76261	+	0.663450i
2.6	−1.09154	−	0.915910i	0.396744	−	1.68600i	0.00526991	+	0.0298872i	−2.61333	+	3.11445i	−1.97729	+	1.47695i	−3.25194			−1.40328	+	2.43055i	−2.68519	−	1.33782i	5.70511	−	1.00597i
2.7	−0.965067	−	0.809788i	−1.15795	+	1.28808i	−0.0716975	−	0.406617i	−0.180362	+	0.214947i	2.16057	−	0.305396i	3.15185			−1.51989	+	2.63252i	−0.318319	−	2.98306i	0.348122	−	0.0613834i
2.8	−0.586300	−	0.491964i	−1.39355	−	1.02859i	−0.245577	−	1.39274i	1.29387	−	1.54197i	0.311010	+	1.28864i	−1.87387			−1.30656	+	2.26302i	0.883991	+	2.86680i	−1.51719	+	0.267522i
2.9	−0.373949	−	0.313780i	1.65202	−	0.520407i	−0.305917	−	1.73494i	1.23004	−	1.46591i	−0.781065	−	0.323766i	−3.29328			−0.918148	+	1.59028i	2.45835	−	1.71945i	−0.919944	+	0.162211i
2.10	0.0307038	+	0.0257635i	1.68465	+	0.402423i	−0.347017	−	1.96803i	−1.41680	+	1.68848i	0.0413574	+	0.0557585i	2.99977			0.0801297	−	0.138789i	2.67611	+	1.35589i	−0.0870023	+	0.0153409i
2.11	0.388217	+	0.325752i	−1.51148	+	0.845833i	−0.302699	−	1.71669i	−2.48025	+	2.95585i	−0.862313	−	0.164001i	−2.21821			0.948484	−	1.64282i	1.56913	−	2.55692i	−1.92575	+	0.339561i
2.12	0.400522	+	0.336078i	0.388639	+	1.68789i	−0.299827	−	1.70040i	2.05581	−	2.45002i	−0.411603	+	0.806648i	2.01117			0.974224	−	1.68741i	−2.69792	+	1.31196i	1.64680	−	0.290374i
2.13	0.704973	+	0.591543i	−1.72842	+	0.112107i	−0.200232	−	1.13557i	0.899185	−	1.07161i	−1.28480	−	0.943401i	1.63939			1.45086	−	2.51296i	2.97486	−	0.387535i	1.26780	−	0.223548i
2.14	0.742125	+	0.622716i	−0.329326	−	1.70045i	−0.184323	−	1.04535i	0.194712	−	0.232049i	0.814500	−	1.46703i	−1.08691			1.48294	−	2.56853i	−2.78309	+	1.12001i	0.289001	−	0.0509587i
2.15	1.37859	+	1.15678i	1.50503	+	0.857254i	0.215090	+	1.21983i	0.265173	−	0.316021i	1.08317	+	2.92279i	−4.97178			0.685071	−	1.18658i	1.53023	+	2.58039i	0.731132	−	0.128918i
2.16	1.46425	+	1.22865i	1.16023	−	1.28602i	0.287148	+	1.62850i	−1.02278	+	1.21890i	3.27895	−	0.457536i	0.784390			0.331042	−	0.573381i	−0.307711	−	2.98418i	−2.99522	+	0.528138i
2.17	1.78700	+	1.49947i	−0.951155	+	1.44752i	0.597656	+	3.38948i	0.797436	−	0.950347i	−3.87022	+	1.16048i	−0.314886			−1.68164	+	2.91269i	−1.19061	−	2.75363i	2.85003	−	0.502538i
2.18	1.90556	+	1.59895i	−1.46656	−	0.921515i	0.727205	+	4.12418i	−2.43276	+	2.89925i	−1.32116	−	4.10097i	1.63251			−2.72112	+	4.71311i	1.30162	+	2.70292i	−9.27152	+	1.63482i
32.1	−2.57424	−	0.936948i	−1.03372	+	1.38976i	4.21676	+	3.53829i	−1.12978	+	3.10405i	3.96317	−	2.60903i	−0.922102			−4.80033	−	8.31442i	−0.862851	−	2.87324i	5.81667	−	6.93203i
32.2	−2.26067	−	0.822816i	1.53353	+	0.805164i	2.90150	+	2.43465i	0.773557	−	2.12533i	−2.80430	−	3.08202i	−0.302529			−2.15031	−	3.72445i	1.70342	+	2.46949i	−3.49751	+	4.16817i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
171.bd	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.bd.a	yes	108
3.b	odd	2	1		513.2.cd.a		108
9.c	even	3	1		513.2.bo.a		108
9.d	odd	6	1		171.2.x.a	✓	108
19.f	odd	18	1		171.2.x.a	✓	108
57.j	even	18	1		513.2.bo.a		108
171.bd	even	18	1	inner	171.2.bd.a	yes	108
171.be	odd	18	1		513.2.cd.a		108

    

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

Hecke kernels

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