Newform orbit 49.2.g.a

• Label: 49.2.g.a
• Level: $49$
• Weight: $2$
• Character orbit: 49.g
• Analytic conductor: $0.391$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.391266969904\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(4\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 13 q^{2} - 14 q^{3} - 9 q^{4} - 14 q^{5} - 14 q^{7} - 20 q^{8} + 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−2.50546	−	0.772833i	−0.539916	+	1.37568i	4.02760	+	2.74597i	2.04183	−	0.307757i	2.41591	−	3.02946i	0.189789	+	2.63894i	−4.69930	−	5.89274i	0.598158	+	0.555009i	−5.35358	−	0.806922i
2.2	−1.16438	−	0.359164i	0.671120	−	1.70999i	−0.425690	−	0.290231i	−0.478452	+	0.0721150i	−1.39561	+	1.75004i	1.90196	−	1.83918i	1.91089	+	2.39618i	−0.274497	−	0.254696i	0.583002	+	0.0878735i
2.3	0.681025	+	0.210069i	−0.953694	+	2.42997i	−1.23281	−	0.840516i	2.65874	−	0.400741i	−1.15995	+	1.45453i	−0.518691	−	2.59441i	−1.55172	−	1.94579i	−2.79608	−	2.59438i	1.89485	+	0.285603i
2.4	1.16258	+	0.358609i	0.190201	−	0.484624i	−0.429482	−	0.292816i	−2.56601	+	0.386764i	0.394914	−	0.495207i	−0.339612	+	2.62386i	−1.91142	−	2.39684i	2.00047	+	1.85617i	−3.12189	−	0.470550i
4.1	−2.09732	−	1.42993i	−1.40598	−	1.30456i	1.62338	+	4.13631i	−2.38971	+	0.737129i	1.08336	+	4.74653i	−1.40100	−	2.24437i	1.38019	−	6.04701i	0.0507148	+	0.676742i	6.06605	+	1.87113i
4.2	−0.968018	−	0.659983i	0.630335	+	0.584866i	−0.229202	−	0.583997i	3.03447	−	0.936009i	−0.224174	−	0.982171i	−2.50722	+	0.844891i	−0.684966	+	3.00103i	−0.168936	−	2.25429i	−3.55517	−	1.09662i
4.3	0.174575	+	0.119023i	1.58372	+	1.46948i	−0.714372	−	1.82019i	−3.75050	+	1.15688i	0.101576	+	0.445032i	1.85388	−	1.88763i	0.185965	−	0.814768i	0.124614	+	1.66286i	−0.792437	−	0.244435i
4.4	1.52543	+	1.04002i	−1.14975	−	1.06682i	0.514606	+	1.31120i	−1.32986	+	0.410207i	−0.644358	−	2.82312i	−1.73764	+	1.99515i	0.242977	−	1.06455i	−0.0403518	−	0.538458i	−2.45523	−	0.757337i
9.1	−1.73170	−	1.60678i	2.78061	+	0.419109i	0.267571	+	3.57049i	−0.567937	−	1.44708i	−4.14175	−	5.19359i	−2.62161	+	0.356599i	2.32789	−	2.91908i	4.68940	+	1.44649i	−1.34164	+	3.41845i
9.2	−1.02480	−	0.950878i	−1.96135	−	0.295625i	−0.00340854	−	0.0454838i	−1.11243	−	2.83442i	1.72889	+	2.16796i	2.44418	+	1.01290i	−1.78303	+	2.23585i	0.892763	+	0.275381i	−1.55517	+	3.96251i
9.3	0.168237	+	0.156102i	0.223157	+	0.0336355i	−0.145524	−	1.94188i	0.711168	+	1.81203i	0.0322928	+	0.0404939i	−2.04196	+	1.68237i	0.564834	−	0.708279i	−2.81805	−	0.869254i	−0.163215	+	0.415865i
9.4	1.51353	+	1.40435i	−2.00916	−	0.302832i	0.169112	+	2.25665i	−0.0830372	−	0.211575i	−2.61564	−	3.27991i	1.07894	−	2.41576i	−0.338532	+	0.424506i	1.07830	+	0.332612i	0.171447	−	0.436839i
11.1	−1.73170	+	1.60678i	2.78061	−	0.419109i	0.267571	−	3.57049i	−0.567937	+	1.44708i	−4.14175	+	5.19359i	−2.62161	−	0.356599i	2.32789	+	2.91908i	4.68940	−	1.44649i	−1.34164	−	3.41845i
11.2	−1.02480	+	0.950878i	−1.96135	+	0.295625i	−0.00340854	+	0.0454838i	−1.11243	+	2.83442i	1.72889	−	2.16796i	2.44418	−	1.01290i	−1.78303	−	2.23585i	0.892763	−	0.275381i	−1.55517	−	3.96251i
11.3	0.168237	−	0.156102i	0.223157	−	0.0336355i	−0.145524	+	1.94188i	0.711168	−	1.81203i	0.0322928	−	0.0404939i	−2.04196	−	1.68237i	0.564834	+	0.708279i	−2.81805	+	0.869254i	−0.163215	−	0.415865i
11.4	1.51353	−	1.40435i	−2.00916	+	0.302832i	0.169112	−	2.25665i	−0.0830372	+	0.211575i	−2.61564	+	3.27991i	1.07894	+	2.41576i	−0.338532	−	0.424506i	1.07830	−	0.332612i	0.171447	+	0.436839i
16.1	−0.887059	−	2.26019i	−0.0483362	−	0.645002i	−2.85548	+	2.64950i	1.29033	−	0.879733i	−1.41495	+	0.681404i	−2.25101	+	1.39030i	4.14619	+	1.99670i	2.55280	−	0.384773i	−3.13296	−	2.13602i
16.2	−0.354207	−	0.902506i	0.218016	+	2.90922i	0.777050	−	0.720997i	−0.605902	+	0.413097i	2.54837	−	1.22723i	−0.310399	−	2.62748i	−2.67297	−	1.28723i	−5.44954	+	0.821385i	0.587437	+	0.400508i
16.3	0.0341744	+	0.0870750i	−0.0823774	−	1.09925i	1.45969	−	1.35439i	−2.09852	+	1.43075i	0.0929020	−	0.0447392i	−0.301609	+	2.62850i	0.336373	+	0.161989i	1.76493	−	0.266020i	−0.196298	−	0.133834i
16.4	0.940144	+	2.39545i	−0.173202	−	2.31121i	−3.38820	+	3.14379i	−0.392378	+	0.267519i	5.37356	−	2.58777i	−1.60453	−	2.10369i	−6.07919	−	2.92758i	−2.34522	+	0.353485i	−1.00972	−	0.688415i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
49.g	even	21	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	49.2.g.a	✓	48
3.b	odd	2	1		441.2.bb.d		48
4.b	odd	2	1		784.2.bg.c		48
7.b	odd	2	1		343.2.g.g		48
7.c	even	3	1		343.2.e.d		48
7.c	even	3	1		343.2.g.i		48
7.d	odd	6	1		343.2.e.c		48
7.d	odd	6	1		343.2.g.h		48
49.e	even	7	1		343.2.g.i		48
49.f	odd	14	1		343.2.g.h		48
49.g	even	21	1	inner	49.2.g.a	✓	48
49.g	even	21	1		343.2.e.d		48
49.g	even	21	1		2401.2.a.h		24
49.h	odd	42	1		343.2.e.c		48
49.h	odd	42	1		343.2.g.g		48
49.h	odd	42	1		2401.2.a.i		24
147.n	odd	42	1		441.2.bb.d		48
196.o	odd	42	1		784.2.bg.c		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
49.2.g.a	✓	48	1.a	even	1	1	trivial
49.2.g.a	✓	48	49.g	even	21	1	inner
343.2.e.c		48	7.d	odd	6	1
343.2.e.c		48	49.h	odd	42	1
343.2.e.d		48	7.c	even	3	1
343.2.e.d		48	49.g	even	21	1
343.2.g.g		48	7.b	odd	2	1
343.2.g.g		48	49.h	odd	42	1
343.2.g.h		48	7.d	odd	6	1
343.2.g.h		48	49.f	odd	14	1
343.2.g.i		48	7.c	even	3	1
343.2.g.i		48	49.e	even	7	1
441.2.bb.d		48	3.b	odd	2	1
441.2.bb.d		48	147.n	odd	42	1
784.2.bg.c		48	4.b	odd	2	1
784.2.bg.c		48	196.o	odd	42	1
2401.2.a.h		24	49.g	even	21	1
2401.2.a.i		24	49.h	odd	42	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(49, [\chi])\).