Newform orbit 867.2.h.m

• Label: 867.2.h.m
• Level: $867$
• Weight: $2$
• Character orbit: 867.h
• Analytic conductor: $6.923$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.h (of order \(8\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.92302985525\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
688.1	−1.94326	+	1.94326i	−0.382683	+	0.923880i		−	5.55252i	0.899799	+	0.372709i	−1.05168	−	2.53899i	1.48128	−	0.613566i	6.90347	+	6.90347i	−0.707107	−	0.707107i	−2.47281	+	1.02427i
688.2	−1.94326	+	1.94326i	0.382683	−	0.923880i		−	5.55252i	−0.899799	−	0.372709i	1.05168	+	2.53899i	−1.48128	+	0.613566i	6.90347	+	6.90347i	−0.707107	−	0.707107i	2.47281	−	1.02427i
688.3	−1.72814	+	1.72814i	−0.382683	+	0.923880i		−	3.97290i	−2.65475	−	1.09964i	−0.935260	−	2.25792i	1.44358	−	0.597950i	3.40944	+	3.40944i	−0.707107	−	0.707107i	6.48809	−	2.68746i
688.4	−1.72814	+	1.72814i	0.382683	−	0.923880i		−	3.97290i	2.65475	+	1.09964i	0.935260	+	2.25792i	−1.44358	+	0.597950i	3.40944	+	3.40944i	−0.707107	−	0.707107i	−6.48809	+	2.68746i
688.5	−0.641392	+	0.641392i	−0.382683	+	0.923880i			1.17723i	−2.95025	−	1.22203i	−0.347119	−	0.838019i	−3.29117	+	1.36325i	−2.03785	−	2.03785i	−0.707107	−	0.707107i	2.67607	−	1.10847i
688.6	−0.641392	+	0.641392i	0.382683	−	0.923880i			1.17723i	2.95025	+	1.22203i	0.347119	+	0.838019i	3.29117	−	1.36325i	−2.03785	−	2.03785i	−0.707107	−	0.707107i	−2.67607	+	1.10847i
688.7	−0.307898	+	0.307898i	−0.382683	+	0.923880i			1.81040i	3.89949	+	1.61522i	−0.166633	−	0.402288i	−2.68832	+	1.11354i	−1.17321	−	1.17321i	−0.707107	−	0.707107i	−1.69797	+	0.703322i
688.8	−0.307898	+	0.307898i	0.382683	−	0.923880i			1.81040i	−3.89949	−	1.61522i	0.166633	+	0.402288i	2.68832	−	1.11354i	−1.17321	−	1.17321i	−0.707107	−	0.707107i	1.69797	−	0.703322i
688.9	1.01764	−	1.01764i	−0.382683	+	0.923880i		−	0.0711653i	2.13781	+	0.885508i	0.550741	+	1.32961i	4.10362	−	1.69978i	1.96285	+	1.96285i	−0.707107	−	0.707107i	3.07663	−	1.27438i
688.10	1.01764	−	1.01764i	0.382683	−	0.923880i		−	0.0711653i	−2.13781	−	0.885508i	−0.550741	−	1.32961i	−4.10362	+	1.69978i	1.96285	+	1.96285i	−0.707107	−	0.707107i	−3.07663	+	1.27438i
688.11	1.48173	−	1.48173i	−0.382683	+	0.923880i		−	2.39104i	1.43955	+	0.596279i	0.801906	+	1.93597i	−3.82062	+	1.58255i	−0.579419	−	0.579419i	−0.707107	−	0.707107i	3.01654	−	1.24949i
688.12	1.48173	−	1.48173i	0.382683	−	0.923880i		−	2.39104i	−1.43955	−	0.596279i	−0.801906	−	1.93597i	3.82062	−	1.58255i	−0.579419	−	0.579419i	−0.707107	−	0.707107i	−3.01654	+	1.24949i
712.1	−1.94326	−	1.94326i	−0.382683	−	0.923880i			5.55252i	0.899799	−	0.372709i	−1.05168	+	2.53899i	1.48128	+	0.613566i	6.90347	−	6.90347i	−0.707107	+	0.707107i	−2.47281	−	1.02427i
712.2	−1.94326	−	1.94326i	0.382683	+	0.923880i			5.55252i	−0.899799	+	0.372709i	1.05168	−	2.53899i	−1.48128	−	0.613566i	6.90347	−	6.90347i	−0.707107	+	0.707107i	2.47281	+	1.02427i
712.3	−1.72814	−	1.72814i	−0.382683	−	0.923880i			3.97290i	−2.65475	+	1.09964i	−0.935260	+	2.25792i	1.44358	+	0.597950i	3.40944	−	3.40944i	−0.707107	+	0.707107i	6.48809	+	2.68746i
712.4	−1.72814	−	1.72814i	0.382683	+	0.923880i			3.97290i	2.65475	−	1.09964i	0.935260	−	2.25792i	−1.44358	−	0.597950i	3.40944	−	3.40944i	−0.707107	+	0.707107i	−6.48809	−	2.68746i
712.5	−0.641392	−	0.641392i	−0.382683	−	0.923880i		−	1.17723i	−2.95025	+	1.22203i	−0.347119	+	0.838019i	−3.29117	−	1.36325i	−2.03785	+	2.03785i	−0.707107	+	0.707107i	2.67607	+	1.10847i
712.6	−0.641392	−	0.641392i	0.382683	+	0.923880i		−	1.17723i	2.95025	−	1.22203i	0.347119	−	0.838019i	3.29117	+	1.36325i	−2.03785	+	2.03785i	−0.707107	+	0.707107i	−2.67607	−	1.10847i
712.7	−0.307898	−	0.307898i	−0.382683	−	0.923880i		−	1.81040i	3.89949	−	1.61522i	−0.166633	+	0.402288i	−2.68832	−	1.11354i	−1.17321	+	1.17321i	−0.707107	+	0.707107i	−1.69797	−	0.703322i
712.8	−0.307898	−	0.307898i	0.382683	+	0.923880i		−	1.81040i	−3.89949	+	1.61522i	0.166633	−	0.402288i	2.68832	+	1.11354i	−1.17321	+	1.17321i	−0.707107	+	0.707107i	1.69797	+	0.703322i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.b	even	2	1	inner
17.c	even	4	2	inner
17.d	even	8	4	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	867.2.h.m		48
17.b	even	2	1	inner	867.2.h.m		48
17.c	even	4	2	inner	867.2.h.m		48
17.d	even	8	4	inner	867.2.h.m		48
17.e	odd	16	1		867.2.a.o	✓	6
17.e	odd	16	1		867.2.a.p	yes	6
17.e	odd	16	2		867.2.d.g		12
17.e	odd	16	4		867.2.e.k		24
51.i	even	16	1		2601.2.a.bh		6
51.i	even	16	1		2601.2.a.bi		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
867.2.a.o	✓	6	17.e	odd	16	1
867.2.a.p	yes	6	17.e	odd	16	1
867.2.d.g		12	17.e	odd	16	2
867.2.e.k		24	17.e	odd	16	4
867.2.h.m		48	1.a	even	1	1	trivial
867.2.h.m		48	17.b	even	2	1	inner
867.2.h.m		48	17.c	even	4	2	inner
867.2.h.m		48	17.d	even	8	4	inner
2601.2.a.bh		6	51.i	even	16	1
2601.2.a.bi		6	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 \(S_{2}^{\mathrm{new}}(867, [\chi])\):

\( T_{2}^{24} + 117T_{2}^{20} + 4386T_{2}^{16} + 58705T_{2}^{12} + 208080T_{2}^{8} + 121344T_{2}^{4} + 4096 \)

T5^48 + 117045*T5^40 + 1707261258*T5^32 + 6444357521537*T5^24 + 4387921274829312*T5^16 + 148115041150107648*T5^8 + 117033789351264256