Newform orbit 147.3.j.a

• Label: 147.3.j.a
• Level: $147$
• Weight: $3$
• Character orbit: 147.j
• Analytic conductor: $4.005$
• Analytic rank: $0$
• Dimension: $108$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	147.j (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00545988610\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 32 q^{4} - 2 q^{7} + 12 q^{8} + 54 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−0.828971	−	3.63196i	1.35417	−	1.07992i	−8.90007	+	4.28605i	−5.61004	+	4.47386i	−5.04478	−	4.02308i	−3.05350	−	6.29890i	13.6538	+	17.1213i	0.667563	−	2.92478i	20.8995	+	16.6668i
13.2	−0.773997	−	3.39110i	−1.35417	+	1.07992i	−7.29663	+	3.51387i	−4.11712	+	3.28330i	4.71023	+	3.75628i	6.78745	+	1.71188i	8.88871	+	11.1461i	0.667563	−	2.92478i	14.3206	+	11.4203i
13.3	−0.706945	−	3.09733i	−1.35417	+	1.07992i	−5.48980	+	2.64375i	6.01934	−	4.80027i	4.30218	+	3.43087i	0.865390	−	6.94630i	4.14628	+	5.19927i	0.667563	−	2.92478i	−19.1234	−	15.2504i
13.4	−0.603450	−	2.64389i	1.35417	−	1.07992i	−3.02212	+	1.45538i	2.36014	−	1.88215i	−3.67235	−	2.92860i	6.86496	−	1.36831i	−1.09177	−	1.36903i	0.667563	−	2.92478i	−6.40041	−	5.10416i
13.5	−0.505025	−	2.21266i	1.35417	−	1.07992i	−1.03694	+	0.499363i	−0.315841	+	0.251875i	−3.07338	−	2.45094i	−6.99310	+	0.310829i	−4.03160	−	5.05546i	0.667563	−	2.92478i	0.716821	+	0.571645i
13.6	−0.319991	−	1.40197i	−1.35417	+	1.07992i	1.74074	−	0.838296i	1.59667	−	1.27330i	1.94734	+	1.55295i	5.60091	+	4.19878i	−5.31867	−	6.66941i	0.667563	−	2.92478i	−2.29606	−	1.83104i
13.7	−0.261314	−	1.14489i	−1.35417	+	1.07992i	2.36138	−	1.13718i	−5.42860	+	4.32916i	1.59025	+	1.26818i	−1.30136	−	6.87797i	−4.84776	−	6.07890i	0.667563	−	2.92478i	6.37500	+	5.08389i
13.8	−0.193320	−	0.846990i	1.35417	−	1.07992i	2.92386	−	1.40805i	7.20531	−	5.74604i	−1.17647	−	0.938201i	−4.78185	+	5.11213i	−3.92453	−	4.92121i	0.667563	−	2.92478i	−6.25977	−	4.99200i
13.9	0.0266310	+	0.116678i	−1.35417	+	1.07992i	3.59097	−	1.72932i	4.71829	−	3.76271i	−0.162066	−	0.129243i	−6.37412	−	2.89320i	0.595879	+	0.747209i	0.667563	−	2.92478i	0.564680	+	0.450317i
13.10	0.0269077	+	0.117890i	1.35417	−	1.07992i	3.59070	−	1.72919i	−1.53269	+	1.22228i	0.163749	+	0.130586i	3.03105	−	6.30973i	0.602047	+	0.754943i	0.667563	−	2.92478i	−0.185336	−	0.147801i
13.11	0.138992	+	0.608965i	−1.35417	+	1.07992i	3.25236	−	1.56625i	−3.41184	+	2.72085i	−0.845850	−	0.674543i	−4.25494	+	5.55837i	2.96363	+	3.71628i	0.667563	−	2.92478i	−2.13112	−	1.69951i
13.12	0.225717	+	0.988930i	1.35417	−	1.07992i	2.67684	−	1.28910i	−0.139490	+	0.111239i	1.37362	+	1.09543i	0.360741	+	6.99070i	4.40882	+	5.52848i	0.667563	−	2.92478i	−0.141493	−	0.112837i
13.13	0.408782	+	1.79099i	−1.35417	+	1.07992i	0.563331	−	0.271286i	3.73365	−	2.97749i	−2.48768	−	1.98386i	6.36620	−	2.91058i	5.29768	+	6.64308i	0.667563	−	2.92478i	6.85890	+	5.46979i
13.14	0.578194	+	2.53323i	1.35417	−	1.07992i	−2.47908	+	1.19386i	5.94954	−	4.74460i	3.51865	+	2.80603i	−1.51760	−	6.83351i	2.02253	+	2.53617i	0.667563	−	2.92478i	15.4592	+	12.3283i
13.15	0.642006	+	2.81281i	1.35417	−	1.07992i	−3.89586	+	1.87615i	−6.93256	+	5.52853i	3.90699	+	3.11572i	−6.97777	−	0.557365i	−0.582974	−	0.731026i	0.667563	−	2.92478i	−20.0015	−	15.9506i
13.16	0.657943	+	2.88264i	1.35417	−	1.07992i	−4.27282	+	2.05768i	0.899409	−	0.717255i	4.00397	+	3.19306i	5.24404	+	4.63682i	−1.36877	−	1.71638i	0.667563	−	2.92478i	2.65934	+	2.12076i
13.17	0.668697	+	2.92975i	−1.35417	+	1.07992i	−4.53242	+	2.18270i	−3.04415	+	2.42763i	−4.06942	−	3.24525i	1.44694	+	6.84882i	−1.93099	−	2.42138i	0.667563	−	2.92478i	−9.14798	−	7.29527i
13.18	0.819146	+	3.58891i	−1.35417	+	1.07992i	−8.60543	+	4.14416i	−0.593123	+	0.473000i	−4.98499	−	3.97540i	−3.45051	−	6.09048i	−12.7413	−	15.9771i	0.667563	−	2.92478i	−2.18341	−	1.74121i
34.1	−0.828971	+	3.63196i	1.35417	+	1.07992i	−8.90007	−	4.28605i	−5.61004	−	4.47386i	−5.04478	+	4.02308i	−3.05350	+	6.29890i	13.6538	−	17.1213i	0.667563	+	2.92478i	20.8995	−	16.6668i
34.2	−0.773997	+	3.39110i	−1.35417	−	1.07992i	−7.29663	−	3.51387i	−4.11712	−	3.28330i	4.71023	−	3.75628i	6.78745	−	1.71188i	8.88871	−	11.1461i	0.667563	+	2.92478i	14.3206	−	11.4203i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
49.f	odd	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	147.3.j.a	✓	108
3.b	odd	2	1		441.3.v.c		108
49.f	odd	14	1	inner	147.3.j.a	✓	108
147.k	even	14	1		441.3.v.c		108

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
147.3.j.a	✓	108	1.a	even	1	1	trivial
147.3.j.a	✓	108	49.f	odd	14	1	inner
441.3.v.c		108	3.b	odd	2	1
441.3.v.c		108	147.k	even	14	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(147, [\chi])\).