Newform orbit 49.3.f.a

• Label: 49.3.f.a
• Level: $49$
• Weight: $3$
• Character orbit: 49.f
• Analytic conductor: $1.335$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	49.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.33515329537\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(9\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 5 q^{2} - 7 q^{3} - 25 q^{4} - 7 q^{5} - 35 q^{6} - 3 q^{8} + 62 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
6.1	−2.18762	+	2.74319i	1.78809	−	3.71301i	−1.84933	−	8.10243i	3.43314	−	7.12898i	6.27383	+	13.0277i	−5.30895	+	4.56235i	13.6273	+	6.56257i	−4.97777	−	6.24192i	12.0458	+	25.0133i
6.2	−1.96408	+	2.46288i	−2.23248	+	4.63580i	−1.31807	−	5.77485i	1.13327	−	2.35326i	−7.03263	−	14.6034i	6.18203	+	3.28367i	5.45883	+	2.62883i	−10.8953	−	13.6622i	3.56996	+	7.41309i
6.3	−1.68012	+	2.10681i	0.149221	−	0.309861i	−0.725737	−	3.17966i	−3.73262	+	7.75087i	0.402107	+	0.834985i	−4.15473	−	5.63367i	−1.79312	−	0.863523i	5.53766	+	6.94401i	−10.0583	−	20.8863i
6.4	−0.523443	+	0.656377i	−0.418220	+	0.868443i	0.733246	+	3.21256i	0.0740213	−	0.153707i	−0.351112	−	0.729091i	−0.110195	+	6.99913i	−5.51805	−	2.65735i	5.03212	+	6.31008i	0.0621436	+	0.129043i
6.5	−0.303238	+	0.380248i	1.38579	−	2.87762i	0.837448	+	3.66910i	1.33228	−	2.76651i	0.673986	+	1.39955i	5.25745	−	4.62161i	−3.40188	−	1.63826i	−0.748878	−	0.939064i	0.647963	+	1.34551i
6.6	0.712240	−	0.893121i	−2.03455	+	4.22478i	0.599705	+	2.62748i	−0.756289	+	1.57045i	2.32415	+	4.82616i	1.39870	−	6.85884i	6.89066	+	3.31837i	−8.09799	−	10.1546i	0.863943	+	1.79400i
6.7	1.21271	−	1.52069i	1.94575	−	4.04040i	0.0482519	+	0.211405i	−2.79684	+	5.80770i	−3.78456	−	7.85872i	−6.81297	+	1.60731i	7.38966	+	3.55867i	−6.92747	−	8.68677i	5.43995	+	11.2962i
6.8	1.58614	−	1.98896i	−0.520045	+	1.07988i	−0.550031	−	2.40984i	3.04025	−	6.31315i	1.32298	+	2.74720i	−6.63215	+	2.23934i	3.50266	+	1.68679i	4.71571	+	5.91331i	−7.73433	−	16.0605i
6.9	2.36993	−	2.97180i	−0.440069	+	0.913813i	−2.32493	−	10.1862i	−2.94973	+	6.12518i	1.67273	+	3.47347i	6.96434	−	0.705711i	−22.0827	−	10.6345i	4.97002	+	6.23220i	11.2122	+	23.2823i
13.1	−0.855647	−	3.74883i	2.14296	−	1.70895i	−9.71775	+	4.67982i	4.77045	−	3.80430i	−8.24019	−	6.57133i	−1.28845	+	6.88040i	16.2690	+	20.4006i	−0.330939	+	1.44994i	−18.3435	−	14.6285i
13.2	−0.636208	−	2.78741i	−4.36946	+	3.48453i	−3.76102	+	1.81121i	−2.14189	+	1.70810i	12.4927	+	9.96260i	−5.93707	+	3.70826i	0.310918	+	0.389879i	4.94757	−	21.6767i	6.12387	+	4.88362i
13.3	−0.412846	−	1.80879i	0.312940	−	0.249561i	0.502578	−	0.242029i	1.52116	−	1.21309i	−0.580601	−	0.463014i	−2.64109	−	6.48264i	−5.27234	−	6.61130i	−1.96704	+	8.61816i	−2.82223	−	2.25065i
13.4	−0.335140	−	1.46834i	3.79786	−	3.02869i	1.56016	−	0.751333i	−7.21066	+	5.75031i	−5.71998	−	4.56153i	3.24558	+	6.20211i	−5.38225	−	6.74913i	3.24807	−	14.2307i	10.8600	+	8.66057i
13.5	−0.0996049	−	0.436397i	−1.48486	+	1.18414i	3.42335	−	1.64860i	3.42347	−	2.73012i	0.664654	+	0.530044i	5.89216	+	3.77921i	−2.17677	−	2.72959i	−1.20006	+	5.25779i	−1.53241	−	1.22206i
13.6	0.339789	+	1.48871i	1.79939	−	1.43497i	1.50306	−	0.723836i	−0.186747	+	0.148926i	2.74767	+	2.19119i	−6.85227	−	1.43051i	5.39658	+	6.76710i	−0.824008	+	3.61022i	−0.285162	−	0.227409i
13.7	0.372065	+	1.63012i	−2.94160	+	2.34585i	1.08500	−	0.522509i	−6.25969	+	4.99193i	−4.91849	−	3.92237i	5.44207	−	4.40271i	5.42546	+	6.80331i	1.14732	−	5.02673i	−10.4665	−	8.34674i
13.8	0.692224	+	3.03283i	−3.02407	+	2.41162i	−5.11501	+	2.46326i	5.95062	−	4.74546i	−9.40735	−	7.50211i	−4.61128	+	5.26651i	−3.25311	−	4.07927i	1.32642	−	5.81144i	18.5114	+	14.7623i
13.9	0.836336	+	3.66423i	2.54433	−	2.02904i	−9.12324	+	4.39352i	−1.76769	+	1.40968i	9.56276	+	7.62605i	6.23947	−	3.17318i	−14.3555	−	18.0012i	0.353941	−	1.55072i	−6.64379	−	5.29824i
20.1	−3.50149	−	1.68623i	3.90069	−	0.890306i	6.92313	+	8.68133i	0.503685	−	0.114963i	−15.1595	−	3.46006i	5.67726	−	4.09496i	−6.14338	−	26.9159i	6.31399	−	3.04066i	−1.95750	−	0.446788i
20.2	−2.79066	−	1.34391i	−4.46528	+	1.01917i	3.48771	+	4.37345i	6.99590	−	1.59677i	13.8307	+	3.15678i	−0.883869	+	6.94397i	−1.09854	−	4.81301i	10.7913	−	5.19683i	−21.6691	−	4.94582i

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	49.3.f.a	✓	54
3.b	odd	2	1		441.3.v.a		54
7.b	odd	2	1		343.3.f.a		54
7.c	even	3	2		343.3.h.e		108
7.d	odd	6	2		343.3.h.d		108
49.e	even	7	1		343.3.f.a		54
49.f	odd	14	1	inner	49.3.f.a	✓	54
49.g	even	21	2		343.3.h.d		108
49.h	odd	42	2		343.3.h.e		108
147.k	even	14	1		441.3.v.a		54

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
49.3.f.a	✓	54	1.a	even	1	1	trivial
49.3.f.a	✓	54	49.f	odd	14	1	inner
343.3.f.a		54	7.b	odd	2	1
343.3.f.a		54	49.e	even	7	1
343.3.h.d		108	7.d	odd	6	2
343.3.h.d		108	49.g	even	21	2
343.3.h.e		108	7.c	even	3	2
343.3.h.e		108	49.h	odd	42	2
441.3.v.a		54	3.b	odd	2	1
441.3.v.a		54	147.k	even	14	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(49, [\chi])\).