Newform orbit 160.2.z.a

• Label: 160.2.z.a
• Level: $160$
• Weight: $2$
• Character orbit: 160.z
• Analytic conductor: $1.278$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	160.z (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.27760643234\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 8 q^{4} - 4 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1	−1.41382	+	0.0334026i	1.02536	−	2.47544i	1.99777	−	0.0944503i	1.80031	+	1.32623i	−1.36699	+	3.53408i	2.81608	+	2.81608i	−2.82133	+	0.200266i	−2.95514	−	2.95514i	−2.58962	−	1.81491i
29.2	−1.41308	−	0.0566552i	−0.529266	+	1.27776i	1.99358	+	0.160116i	1.47606	−	1.67965i	0.820286	−	1.77559i	−0.242153	−	0.242153i	−2.80801	−	0.339204i	0.768771	+	0.768771i	−2.18095	+	2.28986i
29.3	−1.31115	+	0.529979i	0.763124	−	1.84234i	1.43825	−	1.38977i	−1.81451	−	1.30674i	−0.0241693	+	2.82004i	−2.19624	−	2.19624i	−1.14921	+	2.58444i	−0.690555	−	0.690555i	3.07164	+	0.751682i
29.4	−1.20513	−	0.740042i	0.0462717	−	0.111710i	0.904676	+	1.78369i	−2.14654	+	0.626377i	−0.138433	+	0.100382i	2.44906	+	2.44906i	0.229754	−	2.81908i	2.11098	+	2.11098i	3.05041	+	0.833667i
29.5	−1.08863	+	0.902710i	−0.624514	+	1.50771i	0.370229	−	1.96543i	1.41683	+	1.72991i	−0.681161	−	2.20509i	0.611546	+	0.611546i	1.37117	+	2.47384i	0.238149	+	0.238149i	−3.10401	−	0.604247i
29.6	−0.771562	−	1.18520i	−1.12821	+	2.72375i	−0.809383	+	1.82891i	−1.18407	−	1.89683i	4.09867	−	0.764387i	−1.51399	−	1.51399i	2.79210	−	0.451838i	−4.02463	−	4.02463i	−1.33454	+	2.86688i
29.7	−0.685946	+	1.23672i	0.0724046	−	0.174800i	−1.05896	−	1.69665i	−0.320762	−	2.21294i	0.166513	+	0.209448i	2.49757	+	2.49757i	2.82467	−	0.145826i	2.09601	+	2.09601i	2.95682	+	1.12127i
29.8	−0.578672	−	1.29040i	−0.245584	+	0.592892i	−1.33028	+	1.49344i	1.89351	+	1.18938i	0.907181	−	0.0261881i	−0.502274	−	0.502274i	2.69693	+	0.852380i	1.83011	+	1.83011i	0.439066	−	3.13165i
29.9	−0.425344	+	1.34873i	0.872653	−	2.10677i	−1.63817	−	1.14735i	2.19410	+	0.431197i	2.47030	+	2.07308i	−3.02408	−	3.02408i	2.24425	−	1.72143i	−1.55564	−	1.55564i	−1.51482	+	2.77585i
29.10	−0.381706	−	1.36173i	1.23949	−	2.99239i	−1.70860	+	1.03956i	−1.80901	+	1.31434i	−4.54794	−	0.545632i	−1.10874	−	1.10874i	2.06778	+	1.92984i	−5.29676	−	5.29676i	2.48028	+	1.96169i
29.11	−0.258628	+	1.39036i	−0.644653	+	1.55633i	−1.86622	−	0.719173i	−2.20768	+	0.355198i	−1.99714	−	1.29881i	−1.63067	−	1.63067i	1.48257	−	2.40873i	0.114733	+	0.114733i	0.0771112	−	3.16134i
29.12	0.258628	−	1.39036i	0.644653	−	1.55633i	−1.86622	−	0.719173i	1.81223	−	1.30990i	−1.99714	−	1.29881i	1.63067	+	1.63067i	−1.48257	+	2.40873i	0.114733	+	0.114733i	−1.35255	−	2.85843i
29.13	0.381706	+	1.36173i	−1.23949	+	2.99239i	−1.70860	+	1.03956i	2.20854	−	0.349785i	−4.54794	−	0.545632i	1.10874	+	1.10874i	−2.06778	−	1.92984i	−5.29676	−	5.29676i	1.31932	+	2.87391i
29.14	0.425344	−	1.34873i	−0.872653	+	2.10677i	−1.63817	−	1.14735i	−1.24656	+	1.85636i	2.47030	+	2.07308i	3.02408	+	3.02408i	−2.24425	+	1.72143i	−1.55564	−	1.55564i	1.97353	+	2.47087i
29.15	0.578672	+	1.29040i	0.245584	−	0.592892i	−1.33028	+	1.49344i	−0.497888	+	2.17993i	0.907181	−	0.0261881i	0.502274	+	0.502274i	−2.69693	−	0.852380i	1.83011	+	1.83011i	−3.10110	+	0.618990i
29.16	0.685946	−	1.23672i	−0.0724046	+	0.174800i	−1.05896	−	1.69665i	−1.33797	−	1.79160i	0.166513	+	0.209448i	−2.49757	−	2.49757i	−2.82467	+	0.145826i	2.09601	+	2.09601i	−3.13349	+	0.425759i
29.17	0.771562	+	1.18520i	1.12821	−	2.72375i	−0.809383	+	1.82891i	−0.504002	−	2.17853i	4.09867	−	0.764387i	1.51399	+	1.51399i	−2.79210	+	0.451838i	−4.02463	−	4.02463i	2.19312	−	2.27821i
29.18	1.08863	−	0.902710i	0.624514	−	1.50771i	0.370229	−	1.96543i	0.221383	+	2.22508i	−0.681161	−	2.20509i	−0.611546	−	0.611546i	−1.37117	−	2.47384i	0.238149	+	0.238149i	2.24961	+	2.22245i
29.19	1.20513	+	0.740042i	−0.0462717	+	0.111710i	0.904676	+	1.78369i	1.96075	−	1.07492i	−0.138433	+	0.100382i	−2.44906	−	2.44906i	−0.229754	+	2.81908i	2.11098	+	2.11098i	3.15845	+	0.155618i
29.20	1.31115	−	0.529979i	−0.763124	+	1.84234i	1.43825	−	1.38977i	0.359048	−	2.20705i	−0.0241693	+	2.82004i	2.19624	+	2.19624i	1.14921	−	2.58444i	−0.690555	−	0.690555i	−0.698924	−	3.08407i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
32.g	even	8	1	inner
160.z	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	160.2.z.a	✓	88
4.b	odd	2	1		640.2.z.a		88
5.b	even	2	1	inner	160.2.z.a	✓	88
5.c	odd	4	2		800.2.y.f		88
20.d	odd	2	1		640.2.z.a		88
32.g	even	8	1	inner	160.2.z.a	✓	88
32.h	odd	8	1		640.2.z.a		88
160.v	odd	8	1		800.2.y.f		88
160.y	odd	8	1		640.2.z.a		88
160.z	even	8	1	inner	160.2.z.a	✓	88
160.bb	odd	8	1		800.2.y.f		88

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.2.z.a	✓	88	1.a	even	1	1	trivial
160.2.z.a	✓	88	5.b	even	2	1	inner
160.2.z.a	✓	88	32.g	even	8	1	inner
160.2.z.a	✓	88	160.z	even	8	1	inner
640.2.z.a		88	4.b	odd	2	1
640.2.z.a		88	20.d	odd	2	1
640.2.z.a		88	32.h	odd	8	1
640.2.z.a		88	160.y	odd	8	1
800.2.y.f		88	5.c	odd	4	2
800.2.y.f		88	160.v	odd	8	1
800.2.y.f		88	160.bb	odd	8	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(160, [\chi])\).