Newform orbit 164.5.h.a

• Label: 164.5.h.a
• Level: $164$
• Weight: $5$
• Character orbit: 164.h
• Analytic conductor: $16.953$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	164.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.9526739458\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 16 q^{3} + 48 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
85.1		0		−6.50619	+	15.7073i		0		11.9441	+	11.9441i		0		27.5778	−	66.5788i		0		−147.114	−	147.114i		0
85.2		0		−5.06683	+	12.2324i		0		−5.49066	−	5.49066i		0		−23.2374	+	56.1000i		0		−66.6837	−	66.6837i		0
85.3		0		−4.23537	+	10.2251i		0		11.7033	+	11.7033i		0		−10.2485	+	24.7421i		0		−29.3384	−	29.3384i		0
85.4		0		−3.88138	+	9.37049i		0		−31.2234	−	31.2234i		0		13.0527	−	31.5121i		0		−15.4653	−	15.4653i		0
85.5		0		−2.16982	+	5.23842i		0		14.6042	+	14.6042i		0		12.5724	−	30.3524i		0		34.5428	+	34.5428i		0
85.6		0		−0.881831	+	2.12893i		0		32.7552	+	32.7552i		0		−22.6182	+	54.6051i		0		53.5209	+	53.5209i		0
85.7		0		−0.314706	+	0.759767i		0		−2.58279	−	2.58279i		0		32.4835	−	78.4222i		0		56.7974	+	56.7974i		0
85.8		0		0.0880302	−	0.212524i		0		−16.3525	−	16.3525i		0		−14.6034	+	35.2556i		0		57.2382	+	57.2382i		0
85.9		0		1.51555	−	3.65887i		0		−15.4582	−	15.4582i		0		−8.96870	+	21.6523i		0		46.1852	+	46.1852i		0
85.10		0		3.11888	−	7.52964i		0		22.8730	+	22.8730i		0		5.53820	−	13.3704i		0		10.3075	+	10.3075i		0
85.11		0		3.55815	−	8.59014i		0		12.9792	+	12.9792i		0		12.2211	−	29.5042i		0		−3.85441	−	3.85441i		0
85.12		0		4.11580	−	9.93641i		0		−8.65776	−	8.65776i		0		−32.5373	+	78.5519i		0		−24.5168	−	24.5168i		0
85.13		0		5.31219	−	12.8248i		0		−24.5239	−	24.5239i		0		21.1406	−	51.0379i		0		−78.9797	−	78.9797i		0
85.14		0		6.51910	−	15.7385i		0		14.4008	+	14.4008i		0		−12.3730	+	29.8710i		0		−147.926	−	147.926i		0
109.1		0		−14.8296	+	6.14263i		0		18.3281	−	18.3281i		0		−25.9038	+	10.7297i		0		124.910	−	124.910i		0
109.2		0		−12.4576	+	5.16010i		0		−4.22194	+	4.22194i		0		43.5193	−	18.0263i		0		71.2893	−	71.2893i		0
109.3		0		−10.6578	+	4.41461i		0		−4.00655	+	4.00655i		0		−46.1462	+	19.1144i		0		36.8245	−	36.8245i		0
109.4		0		−6.27730	+	2.60014i		0		−33.2574	+	33.2574i		0		68.7068	−	28.4593i		0		−24.6319	+	24.6319i		0
109.5		0		−5.81579	+	2.40898i		0		−24.2556	+	24.2556i		0		−62.5434	+	25.9063i		0		−29.2554	+	29.2554i		0
109.6		0		−4.61947	+	1.91345i		0		26.2443	−	26.2443i		0		43.5243	−	18.0284i		0		−39.5974	+	39.5974i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
41.e	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	164.5.h.a	✓	56
41.e	odd	8	1	inner	164.5.h.a	✓	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
164.5.h.a	✓	56	1.a	even	1	1	trivial
164.5.h.a	✓	56	41.e	odd	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(164, [\chi])\).