Newform orbit 308.3.m.a

• Label: 308.3.m.a
• Level: $308$
• Weight: $3$
• Character orbit: 308.m
• Analytic conductor: $8.392$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	308.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.39239214230\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(92\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 184 q - 8 q^{4} - 12 q^{5} - 256 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
87.1	−1.99954	+	0.0430829i	0.572268	−	0.991198i	3.99629	−	0.172291i	−1.53373	+	0.885497i	−1.10157	+	2.00659i	−4.34069	−	5.49166i	−7.98330	+	0.516674i	3.84502	+	6.65977i	3.02859	−	1.83666i
87.2	−1.99721	−	0.105517i	−1.65669	+	2.86947i	3.97773	+	0.421479i	4.44645	−	2.56716i	3.61154	−	5.55613i	−0.136950	−	6.99866i	−7.89991	−	1.26150i	−0.989223	−	1.71339i	−9.15140	+	4.65799i
87.3	−1.99594	−	0.127333i	−2.13824	+	3.70354i	3.96757	+	0.508297i	4.61015	−	2.66167i	4.73939	−	7.11979i	−5.17399	+	4.71486i	−7.85432	−	1.51973i	−4.64415	−	8.04391i	−9.54052	+	4.72553i
87.4	−1.97995	−	0.282471i	2.70187	−	4.67977i	3.84042	+	1.11856i	0.997477	−	0.575894i	−6.67147	+	8.50253i	6.56338	−	2.43353i	−7.28789	−	3.29950i	−10.1002	−	17.4940i	−2.13763	+	0.858483i
87.5	−1.96647	−	0.364674i	1.86885	−	3.23694i	3.73403	+	1.43424i	3.54290	−	2.04549i	−4.85547	+	5.68384i	−2.10136	+	6.67715i	−6.81983	−	4.18210i	−2.48520	−	4.30450i	−7.71295	+	2.73040i
87.6	−1.95634	+	0.415613i	−1.88145	+	3.25877i	3.65453	−	1.62616i	−2.10212	+	1.21366i	2.32637	−	7.15722i	6.81166	−	1.61285i	−6.47365	+	4.70019i	−2.57973	−	4.46823i	3.60805	−	3.24800i
87.7	−1.95276	+	0.432105i	−0.404714	+	0.700986i	3.62657	−	1.68760i	−6.57122	+	3.79389i	0.487412	−	1.54374i	−5.00346	+	4.89544i	−6.35261	+	4.86254i	4.17241	+	7.22683i	11.1927	−	10.2480i
87.8	−1.94582	+	0.462387i	0.816848	−	1.41482i	3.57240	−	1.79944i	−0.284953	+	0.164518i	−0.935240	+	3.13068i	1.74590	+	6.77878i	−6.11918	+	5.15321i	3.16552	+	5.48284i	0.478396	−	0.451880i
87.9	−1.88955	−	0.655435i	1.38231	−	2.39423i	3.14081	+	2.47696i	−7.47927	+	4.31816i	−4.18121	+	3.61801i	2.34199	−	6.59660i	−4.31124	−	6.73893i	0.678436	+	1.17509i	16.9627	−	3.25721i
87.10	−1.85625	−	0.744526i	−1.08449	+	1.87839i	2.89136	+	2.76406i	−5.27883	+	3.04773i	3.41160	−	2.67934i	6.17008	+	3.30608i	−3.30919	−	7.28349i	2.14777	+	3.72004i	12.0680	−	1.72714i
87.11	−1.84963	−	0.760825i	−0.00372851	+	0.00645797i	2.84229	+	2.81450i	4.17026	−	2.40770i	0.0118098	−	0.00910814i	6.98174	+	0.505253i	−3.11585	−	7.36827i	4.49997	+	7.79418i	−9.54530	+	1.28053i
87.12	−1.82515	+	0.817816i	2.35144	−	4.07281i	2.66235	−	2.98528i	1.51478	−	0.874558i	−0.960921	+	9.35653i	−4.31417	−	5.51253i	−2.41779	+	7.62589i	−6.55851	−	11.3597i	−2.04947	+	2.83501i
87.13	−1.80218	+	0.867272i	0.619480	−	1.07297i	2.49568	−	3.12595i	6.76288	−	3.90455i	−0.185855	+	2.47094i	6.66914	−	2.12662i	−1.78660	+	7.79795i	3.73249	+	6.46486i	−8.80160	+	12.9019i
87.14	−1.74320	−	0.980428i	−0.553191	+	0.958155i	2.07752	+	3.41817i	−3.12355	+	1.80338i	1.90373	−	1.12790i	−6.94563	+	0.870733i	−0.270273	−	7.99543i	3.88796	+	6.73414i	7.21307	−	0.0812480i
87.15	−1.74161	+	0.983256i	−2.37235	+	4.10904i	2.06641	−	3.42490i	−6.99471	+	4.03840i	0.0914799	−	9.48898i	−6.09373	−	3.44477i	−0.231336	+	7.99665i	−6.75613	−	11.7020i	8.21128	−	13.9109i
87.16	−1.72233	+	1.01665i	2.37235	−	4.10904i	1.93284	−	3.50202i	−6.99471	+	4.03840i	0.0914799	+	9.48898i	6.09373	+	3.44477i	0.231336	+	7.99665i	−6.75613	−	11.7020i	7.94156	−	14.0666i
87.17	−1.69188	−	1.06655i	−2.83243	+	4.90592i	1.72493	+	3.60896i	−4.83498	+	2.79148i	10.0246	−	5.27929i	−0.542966	−	6.97891i	0.930779	−	7.94567i	−11.5454	−	19.9972i	11.1575	+	0.433915i
87.18	−1.65217	+	1.12709i	−0.619480	+	1.07297i	1.45932	−	3.72430i	6.76288	−	3.90455i	−0.185855	−	2.47094i	−6.66914	+	2.12662i	1.78660	+	7.79795i	3.73249	+	6.46486i	−6.77262	+	14.0734i
87.19	−1.62082	+	1.17172i	−2.35144	+	4.07281i	1.25415	−	3.79830i	1.51478	−	0.874558i	−0.960921	−	9.35653i	4.31417	+	5.51253i	2.41779	+	7.62589i	−6.55851	−	11.3597i	−1.43045	+	3.19240i
87.20	−1.61766	−	1.17609i	2.38737	−	4.13504i	1.23362	+	3.80502i	−2.95021	+	1.70331i	−8.72512	+	3.88131i	−5.97183	+	3.65203i	2.47948	−	7.60606i	−6.89903	−	11.9495i	6.77567	+	0.714355i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.d	odd	6	1	inner
11.b	odd	2	1	inner
28.f	even	6	1	inner
44.c	even	2	1	inner
77.i	even	6	1	inner
308.m	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	308.3.m.a	✓	184
4.b	odd	2	1	inner	308.3.m.a	✓	184
7.d	odd	6	1	inner	308.3.m.a	✓	184
11.b	odd	2	1	inner	308.3.m.a	✓	184
28.f	even	6	1	inner	308.3.m.a	✓	184
44.c	even	2	1	inner	308.3.m.a	✓	184
77.i	even	6	1	inner	308.3.m.a	✓	184
308.m	odd	6	1	inner	308.3.m.a	✓	184

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
308.3.m.a	✓	184	1.a	even	1	1	trivial
308.3.m.a	✓	184	4.b	odd	2	1	inner
308.3.m.a	✓	184	7.d	odd	6	1	inner
308.3.m.a	✓	184	11.b	odd	2	1	inner
308.3.m.a	✓	184	28.f	even	6	1	inner
308.3.m.a	✓	184	44.c	even	2	1	inner
308.3.m.a	✓	184	77.i	even	6	1	inner
308.3.m.a	✓	184	308.m	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(308, [\chi])\).