Newform orbit 867.2.i.d

• Label: 867.2.i.d
• 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.

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

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	−0.762502	−	1.84084i	0.275439	+	1.71001i	−1.39308	+	1.39308i	−0.240611	−	0.360100i	2.93784	−	1.81093i	2.28935	+	1.52969i	−0.0550167	−	0.0227887i	−2.84827	+	0.942008i	−0.479421	+	0.717504i
65.2	−0.295520	−	0.713449i	−1.72548	−	0.150682i	0.992537	−	0.992537i	1.44335	+	2.16012i	0.402411	+	1.27557i	−3.21323	−	2.14701i	−2.42834	−	1.00585i	2.95459	+	0.520000i	1.11460	−	1.66811i
65.3	0.295520	+	0.713449i	1.65180	+	0.521102i	0.992537	−	0.992537i	−1.44335	−	2.16012i	0.116362	+	1.33247i	−3.21323	−	2.14701i	2.42834	+	1.00585i	2.45691	+	1.72151i	1.11460	−	1.66811i
65.4	0.762502	+	1.84084i	−0.908865	+	1.47444i	−1.39308	+	1.39308i	0.240611	+	0.360100i	−3.40722	−	0.548817i	2.28935	+	1.52969i	0.0550167	+	0.0227887i	−1.34793	−	2.68013i	−0.479421	+	0.717504i
131.1	−1.88095	+	0.779115i	0.508244	+	1.65580i	1.51674	−	1.51674i	−0.405238	+	2.03727i	−2.24604	−	2.71850i	−1.95490	+	0.388855i	−0.112963	+	0.272718i	−2.48338	+	1.68311i	−0.825033	−	4.14772i
131.2	−0.419490	+	0.173759i	−0.0473558	−	1.73140i	−1.26843	+	1.26843i	−0.596910	+	3.00087i	0.320711	+	0.718078i	2.33759	−	0.464975i	0.659211	−	1.59148i	−2.99551	+	0.163984i	−0.271029	−	1.36255i
131.3	0.419490	−	0.173759i	1.58149	+	0.706330i	−1.26843	+	1.26843i	0.596910	−	3.00087i	0.786149	+	0.0215020i	2.33759	−	0.464975i	−0.659211	+	1.59148i	2.00219	+	2.23410i	−0.271029	−	1.36255i
131.4	1.88095	−	0.779115i	−1.33527	−	1.10321i	1.51674	−	1.51674i	0.405238	−	2.03727i	−3.37109	−	1.03475i	−1.95490	+	0.388855i	0.112963	−	0.272718i	0.565877	+	2.94615i	−0.825033	−	4.14772i
158.1	−1.34436	+	0.556851i	1.44078	+	0.961322i	0.0830021	−	0.0830021i	−2.99276	−	0.595296i	−2.47224	−	0.490059i	−0.420765	−	2.11533i	1.04834	−	2.53091i	1.15172	+	2.77012i	4.35483	−	0.866229i
158.2	−0.774648	+	0.320870i	−1.73205	−	0.00133765i	−0.917091	+	0.917091i	0.802626	+	0.159652i	1.34216	−	0.554726i	0.0380817	+	0.191449i	1.05790	−	2.55399i	3.00000	+	0.00463376i	−0.672980	+	0.133864i
158.3	0.774648	−	0.320870i	0.661591	−	1.60072i	−0.917091	+	0.917091i	−0.802626	−	0.159652i	−0.00112159	−	1.45228i	0.0380817	+	0.191449i	−1.05790	+	2.55399i	−2.12459	−	2.11804i	−0.672980	+	0.133864i
158.4	1.34436	−	0.556851i	0.336782	+	1.69899i	0.0830021	−	0.0830021i	2.99276	+	0.595296i	1.39884	+	2.09652i	−0.420765	−	2.11533i	−1.04834	+	2.53091i	−2.77316	+	1.14438i	4.35483	−	0.866229i
224.1	−0.978223	−	2.36164i	−1.73097	−	0.0611151i	−3.20621	+	3.20621i	−1.92973	+	1.28940i	1.54895	+	4.14772i	−0.0590250	+	0.0883372i	5.98501	+	2.47907i	2.99253	+	0.211577i	4.93281	+	3.29600i
224.2	−0.503008	−	1.21437i	1.11133	+	1.32851i	0.192538	−	0.192538i	0.116989	−	0.0781694i	1.05430	−	2.01782i	0.982905	−	1.47102i	−2.75940	−	1.14298i	−0.529896	+	2.95283i	−0.153773	−	0.102748i
224.3	0.503008	+	1.21437i	1.53513	−	0.802099i	0.192538	−	0.192538i	−0.116989	+	0.0781694i	1.74623	+	1.46076i	0.982905	−	1.47102i	2.75940	+	1.14298i	1.71327	−	2.46266i	−0.153773	−	0.102748i
224.4	0.978223	+	2.36164i	−1.62260	−	0.605951i	−3.20621	+	3.20621i	1.92973	−	1.28940i	−0.156224	−	4.42475i	−0.0590250	+	0.0883372i	−5.98501	−	2.47907i	2.26565	+	1.96643i	4.93281	+	3.29600i
329.1	−0.978223	+	2.36164i	−1.73097	+	0.0611151i	−3.20621	−	3.20621i	−1.92973	−	1.28940i	1.54895	−	4.14772i	−0.0590250	−	0.0883372i	5.98501	−	2.47907i	2.99253	−	0.211577i	4.93281	−	3.29600i
329.2	−0.503008	+	1.21437i	1.11133	−	1.32851i	0.192538	+	0.192538i	0.116989	+	0.0781694i	1.05430	+	2.01782i	0.982905	+	1.47102i	−2.75940	+	1.14298i	−0.529896	−	2.95283i	−0.153773	+	0.102748i
329.3	0.503008	−	1.21437i	1.53513	+	0.802099i	0.192538	+	0.192538i	−0.116989	−	0.0781694i	1.74623	−	1.46076i	0.982905	+	1.47102i	2.75940	−	1.14298i	1.71327	+	2.46266i	−0.153773	+	0.102748i
329.4	0.978223	−	2.36164i	−1.62260	+	0.605951i	−3.20621	−	3.20621i	1.92973	+	1.28940i	−0.156224	+	4.42475i	−0.0590250	−	0.0883372i	−5.98501	+	2.47907i	2.26565	−	1.96643i	4.93281	−	3.29600i

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.d		32
3.b	odd	2	1	inner	867.2.i.d		32
17.b	even	2	1		867.2.i.c		32
17.c	even	4	1		51.2.i.a	✓	32
17.c	even	4	1		867.2.i.h		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	inner	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.c		32
51.f	odd	4	1		51.2.i.a	✓	32
51.f	odd	4	1		867.2.i.h		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	inner	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.f	odd	4	1		816.2.cj.c		32
68.i	even	16	1		816.2.cj.c		32
204.l	even	4	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.c	even	4	1
51.2.i.a	✓	32	17.e	odd	16	1
51.2.i.a	✓	32	51.f	odd	4	1
51.2.i.a	✓	32	51.i	even	16	1
816.2.cj.c		32	68.f	odd	4	1
816.2.cj.c		32	68.i	even	16	1
816.2.cj.c		32	204.l	even	4	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.b	even	2	1
867.2.i.c		32	17.e	odd	16	1
867.2.i.c		32	51.c	odd	2	1
867.2.i.c		32	51.i	even	16	1
867.2.i.d		32	1.a	even	1	1	trivial
867.2.i.d		32	3.b	odd	2	1	inner
867.2.i.d		32	17.e	odd	16	1	inner
867.2.i.d		32	51.i	even	16	1	inner
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.c	even	4	1
867.2.i.h		32	17.e	odd	16	1
867.2.i.h		32	51.f	odd	4	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 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 - 12*T7^14 - 8*T7^13 + 150*T7^12 - 176*T7^11 - 60*T7^10 + 120*T7^9 - 2270*T7^8 + 6128*T7^7 - 1496*T7^6 - 16928*T7^5 + 36440*T7^4 + 864*T7^3 + 1440*T7^2 + 128*T7 + 16