Newform orbit 164.3.p.a

• Label: 164.3.p.a
• Level: $164$
• Weight: $3$
• Character orbit: 164.p
• Analytic conductor: $4.469$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	164.p (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.46867633551\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(7\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 8 q^{3} - 24 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1		0		−1.82897	+	4.41553i		0		4.72709	+	0.748697i		0		6.86243	+	5.86107i		0		−9.78777	−	9.78777i		0
13.2		0		−1.54809	+	3.73742i		0		−3.32602	−	0.526789i		0		−6.11952	−	5.22656i		0		−5.20775	−	5.20775i		0
13.3		0		−0.155806	+	0.376149i		0		−3.98726	−	0.631520i		0		3.14094	+	2.68261i		0		6.24675	+	6.24675i		0
13.4		0		0.116511	−	0.281281i		0		6.74584	+	1.06844i		0		−3.28559	−	2.80616i		0		6.29842	+	6.29842i		0
13.5		0		1.01026	−	2.43897i		0		−7.34954	−	1.16405i		0		−0.253886	−	0.216839i		0		1.43598	+	1.43598i		0
13.6		0		1.56061	−	3.76765i		0		4.24816	+	0.672843i		0		9.38440	+	8.01503i		0		−5.39573	−	5.39573i		0
13.7		0		2.00947	−	4.85128i		0		−0.400423	−	0.0634208i		0		−9.72877	−	8.30915i		0		−13.1330	−	13.1330i		0
17.1		0		−1.74685	−	4.21727i		0		−4.27795	−	2.17973i		0		−1.85942	−	1.13945i		0		−8.36992	+	8.36992i		0
17.2		0		−1.50568	−	3.63502i		0		7.12193	+	3.62880i		0		5.35696	+	3.28275i		0		−4.58236	+	4.58236i		0
17.3		0		−0.378915	−	0.914782i		0		0.598740	+	0.305073i		0		−8.88688	−	5.44589i		0		5.67071	−	5.67071i		0
17.4		0		0.131950	+	0.318555i		0		−7.59062	−	3.86762i		0		9.54444	+	5.84884i		0		6.27989	−	6.27989i		0
17.5		0		0.716369	+	1.72947i		0		1.44524	+	0.736388i		0		2.96256	+	1.81546i		0		3.88609	−	3.88609i		0
17.6		0		2.03675	+	4.91716i		0		7.12792	+	3.63186i		0		0.954557	+	0.584953i		0		−13.6661	+	13.6661i		0
17.7		0		2.05153	+	4.95284i		0		−8.35605	−	4.25762i		0		−8.07223	−	4.94667i		0		−13.9579	+	13.9579i		0
29.1		0		−1.74685	+	4.21727i		0		−4.27795	+	2.17973i		0		−1.85942	+	1.13945i		0		−8.36992	−	8.36992i		0
29.2		0		−1.50568	+	3.63502i		0		7.12193	−	3.62880i		0		5.35696	−	3.28275i		0		−4.58236	−	4.58236i		0
29.3		0		−0.378915	+	0.914782i		0		0.598740	−	0.305073i		0		−8.88688	+	5.44589i		0		5.67071	+	5.67071i		0
29.4		0		0.131950	−	0.318555i		0		−7.59062	+	3.86762i		0		9.54444	−	5.84884i		0		6.27989	+	6.27989i		0
29.5		0		0.716369	−	1.72947i		0		1.44524	−	0.736388i		0		2.96256	−	1.81546i		0		3.88609	+	3.88609i		0
29.6		0		2.03675	−	4.91716i		0		7.12792	−	3.63186i		0		0.954557	−	0.584953i		0		−13.6661	−	13.6661i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
41.h	odd	40	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	164.3.p.a	✓	112
41.h	odd	40	1	inner	164.3.p.a	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
164.3.p.a	✓	112	1.a	even	1	1	trivial
164.3.p.a	✓	112	41.h	odd	40	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(164, [\chi])\).