Newform orbit 603.2.k.a

• Label: 603.2.k.a
• Level: $603$
• Weight: $2$
• Character orbit: 603.k
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 132 q - 3 q^{2} - 3 q^{3} - 63 q^{4} + 3 q^{5} + 7 q^{6} + 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
38.1	−1.38849	−	2.40493i	−1.72141	+	0.191707i	−2.85580	+	4.94640i	−1.25488	−	2.17352i	2.85120	+	3.87369i			0.988984i	10.3071			2.92650	−	0.660014i	−3.48478	+	6.03582i
38.2	−1.37071	−	2.37413i	0.181503	−	1.72251i	−2.75766	+	4.77642i	1.15839	+	2.00639i	−4.33826	+	1.93015i			0.0829299i	9.63696			−2.93411	−	0.625282i	3.17562	−	5.50034i
38.3	−1.28266	−	2.22163i	−0.385568	+	1.68859i	−2.29042	+	3.96712i	0.714646	+	1.23780i	4.24597	−	1.30929i		−	2.23337i	6.62064			−2.70268	−	1.30213i	1.83329	−	3.17535i
38.4	−1.26076	−	2.18370i	1.57859	−	0.712778i	−2.17904	+	3.77421i	−0.296268	−	0.513151i	−3.54672	−	2.54853i			4.87307i	5.94595			1.98389	−	2.25037i	−0.747047	+	1.29392i
38.5	−1.24202	−	2.15125i	1.30166	+	1.14266i	−2.08524	+	3.61175i	−0.106943	−	0.185231i	0.841451	−	4.21941i		−	2.10526i	5.39159			0.388652	+	2.97472i	−0.265651	+	0.460121i
38.6	−1.20278	−	2.08327i	−1.17583	−	1.27178i	−1.89335	+	3.27938i	−0.366678	−	0.635105i	−1.23520	+	3.97925i		−	4.48576i	4.29802			−0.234838	+	2.99079i	−0.882065	+	1.52778i
38.7	−1.17719	−	2.03895i	1.35469	+	1.07927i	−1.77155	+	3.06842i	−1.13402	−	1.96417i	0.605848	−	4.03265i			0.964702i	3.63306			0.670366	+	2.92414i	−2.66990	+	4.62441i
38.8	−1.17138	−	2.02889i	−1.66573	+	0.474716i	−1.74427	+	3.02116i	1.98526	+	3.43857i	2.91435	+	2.82350i			2.23626i	3.48727			2.54929	−	1.58149i	4.65099	−	8.05576i
38.9	−1.16723	−	2.02170i	1.72007	−	0.203369i	−1.72484	+	2.98751i	1.79437	+	3.10794i	−2.41886	−	3.24008i		−	2.98859i	3.38421			2.91728	−	0.699617i	4.18888	−	7.25535i
38.10	−1.12708	−	1.95215i	−0.276897	−	1.70977i	−1.54060	+	2.66839i	−1.58205	−	2.74019i	−3.02565	+	2.46759i			1.46181i	2.43718			−2.84666	+	0.946863i	−3.56617	+	6.17679i
38.11	−1.11390	−	1.92933i	−0.523970	+	1.65090i	−1.48155	+	2.56612i	−0.551755	−	0.955668i	3.76878	−	0.828021i			5.15352i	2.14560			−2.45091	−	1.73004i	−1.22920	+	2.12904i
38.12	−0.992895	−	1.71975i	−1.16560	−	1.28117i	−0.971682	+	1.68300i	1.04440	+	1.80895i	−1.04596	+	3.27659i			0.822941i	−0.112467			−0.282772	+	2.98664i	2.07395	−	3.59219i
38.13	−0.932560	−	1.61524i	−0.793584	+	1.53955i	−0.739338	+	1.28057i	−2.22779	−	3.85864i	3.22682	−	0.153896i		−	3.82176i	−0.972332			−1.74045	−	2.44353i	−4.15509	+	7.19683i
38.14	−0.912112	−	1.57982i	1.09608	−	1.34112i	−0.663895	+	1.14990i	−0.255149	−	0.441931i	−3.11848	−	0.508369i		−	3.55097i	−1.22626			−0.597201	−	2.93996i	−0.465449	+	0.806182i
38.15	−0.889016	−	1.53982i	−1.72990	−	0.0863251i	−0.580697	+	1.00580i	0.315907	+	0.547167i	1.40498	+	2.74048i		−	2.72071i	−1.49107			2.98510	+	0.298667i	0.561692	−	0.972879i
38.16	−0.880451	−	1.52499i	1.37106	−	1.05839i	−0.550389	+	0.953302i	0.675144	+	1.16938i	−2.82118	−	1.15899i			1.51810i	−1.58344			0.759619	−	2.90224i	1.18886	−	2.05917i
38.17	−0.795430	−	1.37773i	1.42944	+	0.978118i	−0.265418	+	0.459718i	1.73659	+	3.00787i	0.210563	−	2.74739i			3.47454i	−2.33723			1.08657	+	2.79631i	2.76268	−	4.78509i
38.18	−0.791284	−	1.37054i	0.843344	+	1.51287i	−0.252260	+	0.436927i	1.28815	+	2.23114i	1.40613	−	2.35295i		−	0.0941343i	−2.36670			−1.57754	+	2.55174i	2.03858	−	3.53093i
38.19	−0.714188	−	1.23701i	1.73068	−	0.0688806i	−0.0201279	+	0.0348626i	−1.51065	−	2.61653i	−1.32124	−	2.09167i		−	0.535303i	−2.79925			2.99051	−	0.238421i	−2.15778	+	3.73738i
38.20	−0.688777	−	1.19300i	−1.60949	−	0.639965i	0.0511733	−	0.0886348i	−1.73441	−	3.00409i	0.345100	+	2.36090i			2.96153i	−2.89609			2.18089	+	2.06003i	−2.38925	+	4.13830i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
603.k	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	603.2.k.a	✓	132
9.d	odd	6	1		603.2.t.a	yes	132
67.d	odd	6	1		603.2.t.a	yes	132
603.k	even	6	1	inner	603.2.k.a	✓	132

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
603.2.k.a	✓	132	1.a	even	1	1	trivial
603.2.k.a	✓	132	603.k	even	6	1	inner
603.2.t.a	yes	132	9.d	odd	6	1
603.2.t.a	yes	132	67.d	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(603, [\chi])\).