Newform orbit 603.2.t.a

• Label: 603.2.t.a
• Level: $603$
• Weight: $2$
• Character orbit: 603.t
• 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.t (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.

\(\operatorname{Tr}(f)(q) = \) \( 132 q - 6 q^{2} + 3 q^{3} + 126 q^{4} - 3 q^{5} - 2 q^{6} - 6 q^{7} - 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
164.1	−2.77698			1.72141	−	0.191707i	5.71161			1.25488	+	2.17352i	−4.78031	+	0.532367i	−0.856486	+	0.494492i	−10.3071			2.92650	−	0.660014i	−3.48478	−	6.03582i
164.2	−2.74141			−0.181503	+	1.72251i	5.51533			−1.15839	−	2.00639i	0.497573	−	4.72212i	−0.0718194	+	0.0414650i	−9.63696			−2.93411	−	0.625282i	3.17562	+	5.50034i
164.3	−2.56531			0.385568	−	1.68859i	4.58083			−0.714646	−	1.23780i	−0.989101	+	4.33176i	1.93415	−	1.11668i	−6.62064			−2.70268	−	1.30213i	1.83329	+	3.17535i
164.4	−2.52152			−1.57859	+	0.712778i	4.35808			0.296268	+	0.513151i	3.98045	−	1.79729i	−4.22020	+	2.43653i	−5.94595			1.98389	−	2.25037i	−0.747047	−	1.29392i
164.5	−2.48405			−1.30166	−	1.14266i	4.17049			0.106943	+	0.185231i	3.23339	+	2.83842i	1.82321	−	1.05263i	−5.39159			0.388652	+	2.97472i	−0.265651	−	0.460121i
164.6	−2.40556			1.17583	+	1.27178i	3.78670			0.366678	+	0.635105i	−2.82853	−	3.05933i	3.88478	−	2.24288i	−4.29802			−0.234838	+	2.99079i	−0.882065	−	1.52778i
164.7	−2.35438			−1.35469	−	1.07927i	3.54311			1.13402	+	1.96417i	3.18945	+	2.54100i	−0.835456	+	0.482351i	−3.63306			0.670366	+	2.92414i	−2.66990	−	4.62441i
164.8	−2.34276			1.66573	−	0.474716i	3.48853			−1.98526	−	3.43857i	−3.90240	+	1.11215i	−1.93666	+	1.11813i	−3.48727			2.54929	−	1.58149i	4.65099	+	8.05576i
164.9	−2.33446			−1.72007	+	0.203369i	3.44968			−1.79437	−	3.10794i	4.01543	−	0.474755i	2.58819	−	1.49429i	−3.38421			2.91728	−	0.699617i	4.18888	+	7.25535i
164.10	−2.25415			0.276897	+	1.70977i	3.08119			1.58205	+	2.74019i	−0.624168	−	3.85409i	−1.26597	+	0.730907i	−2.43718			−2.84666	+	0.946863i	−3.56617	−	6.17679i
164.11	−2.22780			0.523970	−	1.65090i	2.96310			0.551755	+	0.955668i	−1.16730	+	3.67787i	−4.46308	+	2.57676i	−2.14560			−2.45091	−	1.73004i	−1.22920	−	2.12904i
164.12	−1.98579			1.16560	+	1.28117i	1.94336			−1.04440	−	1.80895i	−2.31463	−	2.54413i	−0.712688	+	0.411470i	0.112467			−0.282772	+	2.98664i	2.07395	+	3.59219i
164.13	−1.86512			0.793584	−	1.53955i	1.47868			2.22779	+	3.85864i	−1.48013	+	2.87145i	3.30974	−	1.91088i	0.972332			−1.74045	−	2.44353i	−4.15509	−	7.19683i
164.14	−1.82422			−1.09608	+	1.34112i	1.32779			0.255149	+	0.441931i	1.99950	−	2.44650i	3.07523	−	1.77549i	1.22626			−0.597201	−	2.93996i	−0.465449	−	0.806182i
164.15	−1.77803			1.72990	+	0.0863251i	1.16139			−0.315907	−	0.547167i	−3.07581	−	0.153489i	2.35620	−	1.36035i	1.49107			2.98510	+	0.298667i	0.561692	+	0.972879i
164.16	−1.76090			−1.37106	+	1.05839i	1.10078			−0.675144	−	1.16938i	2.41431	−	1.86372i	−1.31471	+	0.759049i	1.58344			0.759619	−	2.90224i	1.18886	+	2.05917i
164.17	−1.59086			−1.42944	−	0.978118i	0.530836			−1.73659	−	3.00787i	2.27403	+	1.55605i	−3.00904	+	1.73727i	2.33723			1.08657	+	2.79631i	2.76268	+	4.78509i
164.18	−1.58257			−0.843344	−	1.51287i	0.504520			−1.28815	−	2.23114i	1.33465	+	2.39422i	0.0815227	−	0.0470672i	2.36670			−1.57754	+	2.55174i	2.03858	+	3.53093i
164.19	−1.42838			−1.73068	+	0.0688806i	0.0402558			1.51065	+	2.61653i	2.47206	−	0.0983874i	0.463586	−	0.267651i	2.79925			2.99051	−	0.238421i	−2.15778	−	3.73738i
164.20	−1.37755			1.60949	+	0.639965i	−0.102347			1.73441	+	3.00409i	−2.21715	−	0.881586i	−2.56476	+	1.48077i	2.89609			2.18089	+	2.06003i	−2.38925	−	4.13830i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
603.t	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.t.a	yes	132
9.d	odd	6	1		603.2.k.a	✓	132
67.d	odd	6	1		603.2.k.a	✓	132
603.t	even	6	1	inner	603.2.t.a	yes	132

    

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

Hecke kernels

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