Newform orbit 867.2.i.h

• Label: 867.2.i.h
• Level: $867$
• Weight: $2$
• Character orbit: 867.i
• Analytic conductor: $6.923$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.92302985525\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{16})\)
Twist minimal:	no (minimal twist has level 51)
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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
65.1	−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
65.2	−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
65.3	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
65.4	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
131.1	−1.34436	+	0.556851i	1.69899	−	0.336782i	0.0830021	−	0.0830021i	0.595296	−	2.99276i	−2.09652	+	1.39884i	2.11533	−	0.420765i	1.04834	−	2.53091i	2.77316	−	1.14438i	0.866229	+	4.35483i
131.2	−0.774648	+	0.320870i	−1.60072	−	0.661591i	−0.917091	+	0.917091i	−0.159652	+	0.802626i	1.45228	−	0.00112159i	−0.191449	+	0.0380817i	1.05790	−	2.55399i	2.12459	+	2.11804i	−0.133864	−	0.672980i
131.3	0.774648	−	0.320870i	−0.00133765	+	1.73205i	−0.917091	+	0.917091i	0.159652	−	0.802626i	0.554726	+	1.34216i	−0.191449	+	0.0380817i	−1.05790	+	2.55399i	−3.00000	−	0.00463376i	−0.133864	−	0.672980i
131.4	1.34436	−	0.556851i	0.961322	−	1.44078i	0.0830021	−	0.0830021i	−0.595296	+	2.99276i	0.490059	−	2.47224i	2.11533	−	0.420765i	−1.04834	+	2.53091i	−1.15172	−	2.77012i	0.866229	+	4.35483i
158.1	−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
158.2	−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
158.3	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
158.4	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
224.1	−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
224.2	−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
224.3	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
224.4	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
329.1	−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
329.2	−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
329.3	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
329.4	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	867.2.i.h		32
3.b	odd	2	1	inner	867.2.i.h		32
17.b	even	2	1		51.2.i.a	✓	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		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	inner	867.2.i.h		32
17.e	odd	16	1		867.2.i.i		32
51.c	odd	2	1		51.2.i.a	✓	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		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	inner	867.2.i.h		32
51.i	even	16	1		867.2.i.i		32
68.d	odd	2	1		816.2.cj.c		32
68.i	even	16	1		816.2.cj.c		32
204.h	even	2	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	17.b	even	2	1
51.2.i.a	✓	32	17.e	odd	16	1
51.2.i.a	✓	32	51.c	odd	2	1
51.2.i.a	✓	32	51.i	even	16	1
816.2.cj.c		32	68.d	odd	2	1
816.2.cj.c		32	68.i	even	16	1
816.2.cj.c		32	204.h	even	2	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	1.a	even	1	1	trivial
867.2.i.h		32	3.b	odd	2	1	inner
867.2.i.h		32	17.e	odd	16	1	inner
867.2.i.h		32	51.i	even	16	1	inner
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 can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\):

T2^32 + 8*T2^30 + 32*T2^28 - 40*T2^26 + 366*T2^24 + 2552*T2^22 + 9504*T2^20 - 9864*T2^18 + 14941*T2^16 + 54576*T2^14 + 130944*T2^12 - 63424*T2^10 + 20436*T2^8 + 17024*T2^6 + 21632*T2^4 - 7072*T2^2 + 1156

T5^32 - 8*T5^30 - 52*T5^28 + 656*T5^26 + 456*T5^24 - 27912*T5^22 + 352752*T5^20 - 1443704*T5^18 + 10227589*T5^16 - 61607408*T5^14 + 108290404*T5^12 + 183041688*T5^10 + 61746760*T5^8 - 4108752*T5^6 + 2895864*T5^4 + 42704*T5^2 + 1156

T7^16 - 8*T7^15 + 44*T7^14 - 160*T7^13 + 534*T7^12 - 1632*T7^11 + 4252*T7^10 - 9944*T7^9 + 19298*T7^8 - 29696*T7^7 + 38424*T7^6 - 34752*T7^5 + 8280*T7^4 + 14048*T7^3 + 3680*T7^2 + 384*T7 + 16