Newform orbit 867.2.i.g

• Label: 867.2.i.g
• 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} + 8 q^{6} + 24 q^{9}+O(q^{10}) \)

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	−0.779115	−	1.88095i	−1.53021	−	0.811448i	−1.51674	+	1.51674i	1.15402	+	1.72711i	−0.334080	+	3.51046i	−1.65729	−	1.10736i	0.272718	+	0.112963i	1.68311	+	2.48338i	2.34950	−	3.51627i
65.2	−0.173759	−	0.419490i	1.25777	+	1.19080i	1.26843	−	1.26843i	1.69986	+	2.54401i	0.280981	−	0.734535i	1.98171	+	1.32414i	−1.59148	−	0.659211i	0.163984	+	2.99551i	0.771824	−	1.15512i
65.3	0.173759	+	0.419490i	−1.61773	+	0.618828i	1.26843	−	1.26843i	−1.69986	−	2.54401i	−0.540687	−	0.571095i	1.98171	+	1.32414i	1.59148	+	0.659211i	2.23410	−	2.00219i	0.771824	−	1.15512i
65.4	0.779115	+	1.88095i	1.72426	−	0.164093i	−1.51674	+	1.51674i	−1.15402	−	1.72711i	1.65205	+	3.11540i	−1.65729	−	1.10736i	−0.272718	−	0.112963i	2.94615	−	0.565877i	2.34950	−	3.51627i
131.1	−2.36164	+	0.978223i	−1.57582	+	0.718877i	3.20621	−	3.20621i	−0.452778	+	2.27627i	3.01830	−	3.23923i	0.104201	−	0.0207268i	−2.47907	+	5.98501i	1.96643	−	2.26565i	−1.15740	−	5.81865i
131.2	−1.21437	+	0.503008i	0.518334	−	1.65267i	−0.192538	+	0.192538i	0.0274494	−	0.137998i	0.201860	+	2.26768i	−1.73519	+	0.345150i	1.14298	−	2.75940i	−2.46266	−	1.71327i	0.0360802	+	0.181387i
131.3	1.21437	−	0.503008i	1.72523	+	0.153573i	−0.192538	+	0.192538i	−0.0274494	+	0.137998i	2.17231	−	0.681311i	−1.73519	+	0.345150i	−1.14298	+	2.75940i	2.95283	+	0.529896i	0.0360802	+	0.181387i
131.4	2.36164	−	0.978223i	−1.26720	+	1.18077i	3.20621	−	3.20621i	0.452778	−	2.27627i	−1.83761	+	4.02815i	0.104201	−	0.0207268i	2.47907	−	5.98501i	0.211577	−	2.99253i	−1.15740	−	5.81865i
158.1	−1.84084	+	0.762502i	0.399920	+	1.68525i	1.39308	−	1.39308i	−0.424767	−	0.0844913i	−2.02120	−	2.79734i	−0.537157	−	2.70047i	0.0227887	−	0.0550167i	−2.68013	+	1.34793i	0.846353	−	0.168350i
158.2	−0.713449	+	0.295520i	1.53648	−	0.799527i	−0.992537	+	0.992537i	2.54804	+	0.506836i	−0.859920	+	1.02448i	0.753930	+	3.79026i	1.00585	−	2.42834i	1.72151	−	2.45691i	−1.96767	+	0.391395i
158.3	0.713449	−	0.295520i	−1.32665	+	1.11355i	−0.992537	+	0.992537i	−2.54804	−	0.506836i	−0.617419	+	1.18651i	0.753930	+	3.79026i	−1.00585	+	2.42834i	0.520000	−	2.95459i	−1.96767	+	0.391395i
158.4	1.84084	−	0.762502i	1.40392	+	1.01439i	1.39308	−	1.39308i	0.424767	+	0.0844913i	3.35788	+	0.796845i	−0.537157	−	2.70047i	−0.0227887	+	0.0550167i	0.942008	+	2.84827i	0.846353	−	0.168350i
224.1	−0.556851	−	1.34436i	−1.69855	+	0.339031i	−0.0830021	+	0.0830021i	−2.53714	+	1.69526i	1.40162	+	2.09466i	1.19824	−	1.79329i	−2.53091	−	1.04834i	2.77012	−	1.15172i	3.69185	+	2.46681i
224.2	−0.320870	−	0.774648i	1.22569	−	1.22380i	0.917091	−	0.917091i	0.680434	−	0.454651i	−1.34130	−	0.556799i	−0.108447	+	0.162303i	−2.55399	−	1.05790i	0.00463376	−	3.00000i	−0.570525	−	0.381213i
224.3	0.320870	+	0.774648i	0.664063	+	1.59969i	0.917091	−	0.917091i	−0.680434	+	0.454651i	−1.02612	+	1.02771i	−0.108447	+	0.162303i	2.55399	+	1.05790i	−2.11804	+	2.12459i	−0.570525	−	0.381213i
224.4	0.556851	+	1.34436i	−1.43951	−	0.963229i	−0.0830021	+	0.0830021i	2.53714	−	1.69526i	0.493332	−	2.47159i	1.19824	−	1.79329i	2.53091	+	1.04834i	1.14438	+	2.77316i	3.69185	+	2.46681i
329.1	−0.556851	+	1.34436i	−1.69855	−	0.339031i	−0.0830021	−	0.0830021i	−2.53714	−	1.69526i	1.40162	−	2.09466i	1.19824	+	1.79329i	−2.53091	+	1.04834i	2.77012	+	1.15172i	3.69185	−	2.46681i
329.2	−0.320870	+	0.774648i	1.22569	+	1.22380i	0.917091	+	0.917091i	0.680434	+	0.454651i	−1.34130	+	0.556799i	−0.108447	−	0.162303i	−2.55399	+	1.05790i	0.00463376	+	3.00000i	−0.570525	+	0.381213i
329.3	0.320870	−	0.774648i	0.664063	−	1.59969i	0.917091	+	0.917091i	−0.680434	−	0.454651i	−1.02612	−	1.02771i	−0.108447	−	0.162303i	2.55399	−	1.05790i	−2.11804	−	2.12459i	−0.570525	+	0.381213i
329.4	0.556851	−	1.34436i	−1.43951	+	0.963229i	−0.0830021	−	0.0830021i	2.53714	+	1.69526i	0.493332	+	2.47159i	1.19824	+	1.79329i	2.53091	−	1.04834i	1.14438	−	2.77316i	3.69185	−	2.46681i

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.g		32
3.b	odd	2	1	inner	867.2.i.g		32
17.b	even	2	1		867.2.i.f		32
17.c	even	4	1		867.2.i.b		32
17.c	even	4	1		867.2.i.i		32
17.d	even	8	1		51.2.i.a	✓	32
17.d	even	8	1		867.2.i.c		32
17.d	even	8	1		867.2.i.d		32
17.d	even	8	1		867.2.i.h		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	inner	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.f		32
51.f	odd	4	1		867.2.i.b		32
51.f	odd	4	1		867.2.i.i		32
51.g	odd	8	1		51.2.i.a	✓	32
51.g	odd	8	1		867.2.i.c		32
51.g	odd	8	1		867.2.i.d		32
51.g	odd	8	1		867.2.i.h		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	inner	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.g	odd	8	1		816.2.cj.c		32
68.i	even	16	1		816.2.cj.c		32
204.p	even	8	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.d	even	8	1
51.2.i.a	✓	32	17.e	odd	16	1
51.2.i.a	✓	32	51.g	odd	8	1
51.2.i.a	✓	32	51.i	even	16	1
816.2.cj.c		32	68.g	odd	8	1
816.2.cj.c		32	68.i	even	16	1
816.2.cj.c		32	204.p	even	8	1
816.2.cj.c		32	204.t	odd	16	1
867.2.i.b		32	17.c	even	4	1
867.2.i.b		32	17.e	odd	16	1
867.2.i.b		32	51.f	odd	4	1
867.2.i.b		32	51.i	even	16	1
867.2.i.c		32	17.d	even	8	1
867.2.i.c		32	17.e	odd	16	1
867.2.i.c		32	51.g	odd	8	1
867.2.i.c		32	51.i	even	16	1
867.2.i.d		32	17.d	even	8	1
867.2.i.d		32	17.e	odd	16	1
867.2.i.d		32	51.g	odd	8	1
867.2.i.d		32	51.i	even	16	1
867.2.i.f		32	17.b	even	2	1
867.2.i.f		32	17.e	odd	16	1
867.2.i.f		32	51.c	odd	2	1
867.2.i.f		32	51.i	even	16	1
867.2.i.g		32	1.a	even	1	1	trivial
867.2.i.g		32	3.b	odd	2	1	inner
867.2.i.g		32	17.e	odd	16	1	inner
867.2.i.g		32	51.i	even	16	1	inner
867.2.i.h		32	17.d	even	8	1
867.2.i.h		32	17.e	odd	16	1
867.2.i.h		32	51.g	odd	8	1
867.2.i.h		32	51.i	even	16	1
867.2.i.i		32	17.c	even	4	1
867.2.i.i		32	17.e	odd	16	1
867.2.i.i		32	51.f	odd	4	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 + 84*T5^28 - 96*T5^26 + 3528*T5^24 - 5352*T5^22 - 182320*T5^20 - 5848*T5^18 + 8354373*T5^16 + 39094712*T5^14 + 154359612*T5^12 - 126009128*T5^10 + 111019784*T5^8 - 27518912*T5^6 + 1835336*T5^4 + 95472*T5^2 + 1156

T7^16 + 16*T7^14 + 24*T7^13 + 50*T7^12 + 168*T7^11 + 244*T7^10 + 1016*T7^9 + 3554*T7^8 + 2816*T7^7 + 3912*T7^6 + 28704*T7^5 + 37448*T7^4 + 416*T7^3 - 192*T7 + 16