Newform orbit 368.3.l.a

• Label: 368.3.l.a
• Level: $368$
• Weight: $3$
• Character orbit: 368.l
• Analytic conductor: $10.027$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	368.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.0272737285\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(88\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176 q + 6 q^{6} - 12 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
139.1	−2.00000	+	0.00218807i	−0.0285067	−	0.0285067i	3.99999	−	0.00875226i	−5.21813	−	5.21813i	0.0570756	+	0.0569509i	−10.6343			−7.99996	+	0.0262567i		−	8.99837i	10.4477	+	10.4248i
139.2	−1.99927	−	0.0541245i	−2.61280	−	2.61280i	3.99414	+	0.216419i	0.543869	+	0.543869i	5.08227	+	5.36510i	1.21227			−7.97364	−	0.648859i			4.65343i	−1.05790	−	1.11678i
139.3	−1.99519	+	0.138574i	2.85816	+	2.85816i	3.96159	−	0.552963i	−4.66000	−	4.66000i	−6.09864	−	5.30651i	−0.714696			−7.82752	+	1.65224i			7.33812i	9.94335	+	8.65184i
139.4	−1.97089	−	0.339992i	2.68098	+	2.68098i	3.76881	+	1.34017i	−0.683891	−	0.683891i	−4.37241	−	6.19544i	−1.47597			−6.97226	−	3.92270i			5.37535i	1.11536	+	1.58039i
139.5	−1.97049	+	0.342296i	3.44479	+	3.44479i	3.76567	−	1.34898i	4.32371	+	4.32371i	−7.96706	−	5.60879i	7.38910			−6.95846	+	3.94713i			14.7331i	−9.99983	−	7.03985i
139.6	−1.96969	+	0.346850i	−3.37928	−	3.37928i	3.75939	−	1.36638i	3.19897	+	3.19897i	7.82825	+	5.48404i	1.98675			−6.93092	+	3.99529i			13.8391i	−7.41055	−	5.19142i
139.7	−1.95258	+	0.432951i	−0.363044	−	0.363044i	3.62511	−	1.69074i	−2.26139	−	2.26139i	0.866052	+	0.551691i	13.7813			−6.34629	+	4.87079i		−	8.73640i	5.39460	+	3.43646i
139.8	−1.92318	+	0.548981i	0.226855	+	0.226855i	3.39724	−	2.11158i	3.94577	+	3.94577i	−0.560821	−	0.311743i	−11.0564			−5.37428	+	5.92597i		−	8.89707i	−9.75458	−	5.42227i
139.9	−1.91831	−	0.565765i	0.255986	+	0.255986i	3.35982	+	2.17063i	1.03745	+	1.03745i	−0.346232	−	0.635888i	−1.20555			−5.21711	−	6.06480i		−	8.86894i	−1.40319	−	2.57710i
139.10	−1.91659	−	0.571548i	−3.27740	−	3.27740i	3.34666	+	2.19085i	−6.04143	−	6.04143i	4.40825	+	8.15463i	5.20480			−5.16202	−	6.11176i			12.4827i	8.12600	+	15.0319i
139.11	−1.88876	−	0.657722i	−0.624710	−	0.624710i	3.13480	+	2.48455i	6.53098	+	6.53098i	0.769040	+	1.59081i	3.30040			−4.28673	−	6.75455i		−	8.21947i	−8.03986	−	16.6310i
139.12	−1.84149	+	0.780331i	0.856994	+	0.856994i	2.78217	−	2.87394i	3.39872	+	3.39872i	−2.24688	−	0.909406i	−1.99678			−2.88070	+	7.46335i		−	7.53112i	−8.91082	−	3.60657i
139.13	−1.78827	−	0.895592i	1.19232	+	1.19232i	2.39583	+	3.20312i	−1.11167	−	1.11167i	−1.06436	−	3.20003i	13.2568			−1.41571	−	7.87374i		−	6.15674i	0.992365	+	2.98357i
139.14	−1.75879	+	0.952188i	−1.40543	−	1.40543i	2.18668	−	3.34939i	−4.58241	−	4.58241i	3.81010	+	1.13363i	0.0883802			−0.656650	+	7.97301i		−	5.04951i	12.4228	+	3.69618i
139.15	−1.75863	−	0.952488i	3.41173	+	3.41173i	2.18553	+	3.35014i	5.19725	+	5.19725i	−2.75033	−	9.24959i	−12.1330			−0.652563	−	7.97334i			14.2798i	−4.18970	−	14.0903i
139.16	−1.72434	−	1.01324i	−1.83742	−	1.83742i	1.94670	+	3.49433i	−2.51330	−	2.51330i	1.30659	+	5.03008i	−3.07621			0.183823	−	7.99789i		−	2.24778i	1.78721	+	6.88036i
139.17	−1.65959	+	1.11613i	−3.13372	−	3.13372i	1.50850	−	3.70465i	−1.67188	−	1.67188i	8.69836	+	1.70306i	−9.46824			1.63139	+	7.83189i			10.6405i	4.64068	+	0.908601i
139.18	−1.63730	−	1.14858i	−3.49836	−	3.49836i	1.36151	+	3.76116i	2.67647	+	2.67647i	1.70971	+	9.74602i	−12.1235			2.09080	−	7.72195i			15.4770i	−1.30804	−	7.45634i
139.19	−1.57334	−	1.23475i	4.09182	+	4.09182i	0.950778	+	3.88536i	−4.38523	−	4.38523i	−1.38543	−	11.4902i	5.72212			3.30156	−	7.28695i			24.4860i	1.48478	+	12.3141i
139.20	−1.52951	+	1.28864i	3.91358	+	3.91358i	0.678811	−	3.94198i	−0.822460	−	0.822460i	−11.0291	−	0.942668i	−13.5678			4.04155	+	6.90405i			21.6323i	2.31782	+	0.198107i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.f	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	368.3.l.a	✓	176
16.f	odd	4	1	inner	368.3.l.a	✓	176

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
368.3.l.a	✓	176	1.a	even	1	1	trivial
368.3.l.a	✓	176	16.f	odd	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(368, [\chi])\).