Newform orbit 51.2.i.a

• Label: 51.2.i.a
• Level: $51$
• Weight: $2$
• Character orbit: 51.i
• Analytic conductor: $0.407$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	51.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.407237050309\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1.88095	+	0.779115i	1.10321	−	1.33527i	1.51674	−	1.51674i	2.03727	+	0.405238i	−1.03475	+	3.37109i	0.388855	+	1.95490i	−0.112963	+	0.272718i	−0.565877	−	2.94615i	−4.14772	+	0.825033i
5.2	−0.419490	+	0.173759i	−0.706330	+	1.58149i	−1.26843	+	1.26843i	3.00087	+	0.596910i	0.0215020	−	0.786149i	−0.464975	−	2.33759i	0.659211	−	1.59148i	−2.00219	−	2.23410i	−1.36255	+	0.271029i
5.3	0.419490	−	0.173759i	1.73140	−	0.0473558i	−1.26843	+	1.26843i	−3.00087	−	0.596910i	0.718078	−	0.320711i	−0.464975	−	2.33759i	−0.659211	+	1.59148i	2.99551	−	0.163984i	−1.36255	+	0.271029i
5.4	1.88095	−	0.779115i	−1.65580	+	0.508244i	1.51674	−	1.51674i	−2.03727	−	0.405238i	−2.71850	+	2.24604i	0.388855	+	1.95490i	0.112963	−	0.272718i	2.48338	−	1.68311i	−4.14772	+	0.825033i
11.1	−0.978223	+	2.36164i	0.605951	+	1.62260i	−3.20621	−	3.20621i	1.28940	−	1.92973i	−4.42475	−	0.156224i	0.0883372	−	0.0590250i	5.98501	−	2.47907i	−2.26565	+	1.96643i	3.29600	+	4.93281i
11.2	−0.503008	+	1.21437i	0.802099	−	1.53513i	0.192538	+	0.192538i	−0.0781694	+	0.116989i	1.46076	+	1.74623i	−1.47102	+	0.982905i	−2.75940	+	1.14298i	−1.71327	−	2.46266i	−0.102748	−	0.153773i
11.3	0.503008	−	1.21437i	−1.32851	−	1.11133i	0.192538	+	0.192538i	0.0781694	−	0.116989i	−2.01782	+	1.05430i	−1.47102	+	0.982905i	2.75940	−	1.14298i	0.529896	+	2.95283i	−0.102748	−	0.153773i
11.4	0.978223	−	2.36164i	0.0611151	+	1.73097i	−3.20621	−	3.20621i	−1.28940	+	1.92973i	4.14772	+	1.54895i	0.0883372	−	0.0590250i	−5.98501	+	2.47907i	−2.99253	+	0.211577i	3.29600	+	4.93281i
14.1	−0.978223	−	2.36164i	0.605951	−	1.62260i	−3.20621	+	3.20621i	1.28940	+	1.92973i	−4.42475	+	0.156224i	0.0883372	+	0.0590250i	5.98501	+	2.47907i	−2.26565	−	1.96643i	3.29600	−	4.93281i
14.2	−0.503008	−	1.21437i	0.802099	+	1.53513i	0.192538	−	0.192538i	−0.0781694	−	0.116989i	1.46076	−	1.74623i	−1.47102	−	0.982905i	−2.75940	−	1.14298i	−1.71327	+	2.46266i	−0.102748	+	0.153773i
14.3	0.503008	+	1.21437i	−1.32851	+	1.11133i	0.192538	−	0.192538i	0.0781694	+	0.116989i	−2.01782	−	1.05430i	−1.47102	−	0.982905i	2.75940	+	1.14298i	0.529896	−	2.95283i	−0.102748	+	0.153773i
14.4	0.978223	+	2.36164i	0.0611151	−	1.73097i	−3.20621	+	3.20621i	−1.28940	−	1.92973i	4.14772	−	1.54895i	0.0883372	+	0.0590250i	−5.98501	−	2.47907i	−2.99253	−	0.211577i	3.29600	−	4.93281i
20.1	−0.762502	−	1.84084i	−1.47444	−	0.908865i	−1.39308	+	1.39308i	−0.360100	+	0.240611i	−0.548817	+	3.40722i	1.52969	−	2.28935i	−0.0550167	−	0.0227887i	1.34793	+	2.68013i	0.717504	+	0.479421i
20.2	−0.295520	−	0.713449i	−0.521102	+	1.65180i	0.992537	−	0.992537i	2.16012	−	1.44335i	1.33247	−	0.116362i	−2.14701	+	3.21323i	−2.42834	−	1.00585i	−2.45691	−	1.72151i	−1.66811	−	1.11460i
20.3	0.295520	+	0.713449i	0.150682	−	1.72548i	0.992537	−	0.992537i	−2.16012	+	1.44335i	1.27557	−	0.402411i	−2.14701	+	3.21323i	2.42834	+	1.00585i	−2.95459	−	0.520000i	−1.66811	−	1.11460i
20.4	0.762502	+	1.84084i	−1.71001	+	0.275439i	−1.39308	+	1.39308i	0.360100	−	0.240611i	−1.81093	−	2.93784i	1.52969	−	2.28935i	0.0550167	+	0.0227887i	2.84827	−	0.942008i	0.717504	+	0.479421i
23.1	−0.762502	+	1.84084i	−1.47444	+	0.908865i	−1.39308	−	1.39308i	−0.360100	−	0.240611i	−0.548817	−	3.40722i	1.52969	+	2.28935i	−0.0550167	+	0.0227887i	1.34793	−	2.68013i	0.717504	−	0.479421i
23.2	−0.295520	+	0.713449i	−0.521102	−	1.65180i	0.992537	+	0.992537i	2.16012	+	1.44335i	1.33247	+	0.116362i	−2.14701	−	3.21323i	−2.42834	+	1.00585i	−2.45691	+	1.72151i	−1.66811	+	1.11460i
23.3	0.295520	−	0.713449i	0.150682	+	1.72548i	0.992537	+	0.992537i	−2.16012	−	1.44335i	1.27557	+	0.402411i	−2.14701	−	3.21323i	2.42834	−	1.00585i	−2.95459	+	0.520000i	−1.66811	+	1.11460i
23.4	0.762502	−	1.84084i	−1.71001	−	0.275439i	−1.39308	−	1.39308i	0.360100	+	0.240611i	−1.81093	+	2.93784i	1.52969	+	2.28935i	0.0550167	−	0.0227887i	2.84827	+	0.942008i	0.717504	−	0.479421i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
17.e	odd	16	1	inner
51.i	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	51.2.i.a	✓	32
3.b	odd	2	1	inner	51.2.i.a	✓	32
4.b	odd	2	1		816.2.cj.c		32
12.b	even	2	1		816.2.cj.c		32
17.b	even	2	1		867.2.i.h		32
17.c	even	4	1		867.2.i.c		32
17.c	even	4	1		867.2.i.d		32
17.d	even	8	1		867.2.i.b		32
17.d	even	8	1		867.2.i.f		32
17.d	even	8	1		867.2.i.g		32
17.d	even	8	1		867.2.i.i		32
17.e	odd	16	1	inner	51.2.i.a	✓	32
17.e	odd	16	1		867.2.i.b		32
17.e	odd	16	1		867.2.i.c		32
17.e	odd	16	1		867.2.i.d		32
17.e	odd	16	1		867.2.i.f		32
17.e	odd	16	1		867.2.i.g		32
17.e	odd	16	1		867.2.i.h		32
17.e	odd	16	1		867.2.i.i		32
51.c	odd	2	1		867.2.i.h		32
51.f	odd	4	1		867.2.i.c		32
51.f	odd	4	1		867.2.i.d		32
51.g	odd	8	1		867.2.i.b		32
51.g	odd	8	1		867.2.i.f		32
51.g	odd	8	1		867.2.i.g		32
51.g	odd	8	1		867.2.i.i		32
51.i	even	16	1	inner	51.2.i.a	✓	32
51.i	even	16	1		867.2.i.b		32
51.i	even	16	1		867.2.i.c		32
51.i	even	16	1		867.2.i.d		32
51.i	even	16	1		867.2.i.f		32
51.i	even	16	1		867.2.i.g		32
51.i	even	16	1		867.2.i.h		32
51.i	even	16	1		867.2.i.i		32
68.i	even	16	1		816.2.cj.c		32
204.t	odd	16	1		816.2.cj.c		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
51.2.i.a	✓	32	1.a	even	1	1	trivial
51.2.i.a	✓	32	3.b	odd	2	1	inner
51.2.i.a	✓	32	17.e	odd	16	1	inner
51.2.i.a	✓	32	51.i	even	16	1	inner
816.2.cj.c		32	4.b	odd	2	1
816.2.cj.c		32	12.b	even	2	1
816.2.cj.c		32	68.i	even	16	1
816.2.cj.c		32	204.t	odd	16	1
867.2.i.b		32	17.d	even	8	1
867.2.i.b		32	17.e	odd	16	1
867.2.i.b		32	51.g	odd	8	1
867.2.i.b		32	51.i	even	16	1
867.2.i.c		32	17.c	even	4	1
867.2.i.c		32	17.e	odd	16	1
867.2.i.c		32	51.f	odd	4	1
867.2.i.c		32	51.i	even	16	1
867.2.i.d		32	17.c	even	4	1
867.2.i.d		32	17.e	odd	16	1
867.2.i.d		32	51.f	odd	4	1
867.2.i.d		32	51.i	even	16	1
867.2.i.f		32	17.d	even	8	1
867.2.i.f		32	17.e	odd	16	1
867.2.i.f		32	51.g	odd	8	1
867.2.i.f		32	51.i	even	16	1
867.2.i.g		32	17.d	even	8	1
867.2.i.g		32	17.e	odd	16	1
867.2.i.g		32	51.g	odd	8	1
867.2.i.g		32	51.i	even	16	1
867.2.i.h		32	17.b	even	2	1
867.2.i.h		32	17.e	odd	16	1
867.2.i.h		32	51.c	odd	2	1
867.2.i.h		32	51.i	even	16	1
867.2.i.i		32	17.d	even	8	1
867.2.i.i		32	17.e	odd	16	1
867.2.i.i		32	51.g	odd	8	1
867.2.i.i		32	51.i	even	16	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(51, [\chi])\).