Newform orbit 162.5.f.a

• Label: 162.5.f.a
• Level: $162$
• Weight: $5$
• Character orbit: 162.f
• Analytic conductor: $16.746$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	162.f (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.7459340196\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{18})\)
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 18 q^{5}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1	−0.967379	+	2.65785i		0		−6.12836	−	5.14230i	45.7302	+	8.06347i		0		−14.1135	+	11.8427i	19.5959	−	11.3137i		0		−65.6700	+	113.744i
17.2	−0.967379	+	2.65785i		0		−6.12836	−	5.14230i	5.14300	+	0.906850i		0		−21.7252	+	18.2296i	19.5959	−	11.3137i		0		−7.38550	+	12.7921i
17.3	−0.967379	+	2.65785i		0		−6.12836	−	5.14230i	3.66245	+	0.645790i		0		67.2236	−	56.4073i	19.5959	−	11.3137i		0		−5.25940	+	9.10954i
17.4	−0.967379	+	2.65785i		0		−6.12836	−	5.14230i	20.4117	+	3.59913i		0		−31.5466	+	26.4708i	19.5959	−	11.3137i		0		−29.3118	+	50.7695i
17.5	−0.967379	+	2.65785i		0		−6.12836	−	5.14230i	−31.9962	−	5.64180i		0		29.5375	−	24.7849i	19.5959	−	11.3137i		0		45.9475	−	79.5835i
17.6	−0.967379	+	2.65785i		0		−6.12836	−	5.14230i	−39.1530	−	6.90373i		0		−3.77039	+	3.16373i	19.5959	−	11.3137i		0		56.2249	−	97.3844i
17.7	0.967379	−	2.65785i		0		−6.12836	−	5.14230i	−41.7224	−	7.35678i		0		19.0320	−	15.9698i	−19.5959	+	11.3137i		0		−59.9146	+	103.775i
17.8	0.967379	−	2.65785i		0		−6.12836	−	5.14230i	33.5989	+	5.92440i		0		−63.3453	+	53.1530i	−19.5959	+	11.3137i		0		48.2491	−	83.5699i
17.9	0.967379	−	2.65785i		0		−6.12836	−	5.14230i	−5.05113	−	0.890650i		0		14.8841	−	12.4892i	−19.5959	+	11.3137i		0		−7.25357	+	12.5636i
17.10	0.967379	−	2.65785i		0		−6.12836	−	5.14230i	−6.60957	−	1.16545i		0		−57.7034	+	48.4189i	−19.5959	+	11.3137i		0		−9.49154	+	16.4398i
17.11	0.967379	−	2.65785i		0		−6.12836	−	5.14230i	−0.622849	−	0.109825i		0		42.1999	−	35.4099i	−19.5959	+	11.3137i		0		−0.894430	+	1.54920i
17.12	0.967379	−	2.65785i		0		−6.12836	−	5.14230i	24.2051	+	4.26801i		0		19.3273	−	16.2175i	−19.5959	+	11.3137i		0		34.7593	−	60.2048i
35.1	−1.81808	+	2.16670i		0		−1.38919	−	7.87846i	−4.26691	+	11.7232i		0		3.24231	−	18.3880i	19.5959	+	11.3137i		0		−17.6432	−	30.5589i
35.2	−1.81808	+	2.16670i		0		−1.38919	−	7.87846i	5.13581	−	14.1105i		0		13.6904	−	77.6422i	19.5959	+	11.3137i		0		21.2360	+	36.7818i
35.3	−1.81808	+	2.16670i		0		−1.38919	−	7.87846i	−6.13288	+	16.8500i		0		8.35982	−	47.4109i	19.5959	+	11.3137i		0		−25.3588	−	43.9227i
35.4	−1.81808	+	2.16670i		0		−1.38919	−	7.87846i	7.68730	−	21.1207i		0		−9.39505	+	53.2820i	19.5959	+	11.3137i		0		31.7861	+	55.0551i
35.5	−1.81808	+	2.16670i		0		−1.38919	−	7.87846i	−12.8057	+	35.1833i		0		−14.9416	+	84.7379i	19.5959	+	11.3137i		0		−52.9500	−	91.7121i
35.6	−1.81808	+	2.16670i		0		−1.38919	−	7.87846i	12.4033	−	34.0778i		0		−2.97171	+	16.8534i	19.5959	+	11.3137i		0		51.2862	+	88.8303i
35.7	1.81808	−	2.16670i		0		−1.38919	−	7.87846i	−14.9830	+	41.1655i		0		12.4750	−	70.7490i	−19.5959	−	11.3137i		0		61.9530	+	107.306i
35.8	1.81808	−	2.16670i		0		−1.38919	−	7.87846i	2.28016	−	6.26470i		0		−10.8277	+	61.4069i	−19.5959	−	11.3137i		0		−9.42821	−	16.3301i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
27.f	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	162.5.f.a		72
3.b	odd	2	1		54.5.f.a	✓	72
27.e	even	9	1		54.5.f.a	✓	72
27.f	odd	18	1	inner	162.5.f.a		72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
54.5.f.a	✓	72	3.b	odd	2	1
54.5.f.a	✓	72	27.e	even	9	1
162.5.f.a		72	1.a	even	1	1	trivial
162.5.f.a		72	27.f	odd	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(162, [\chi])\).