Newform orbit 160.2.ba.a

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

Newspace parameters

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	160.ba (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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−1.36334	−	0.375898i	1.22690	−	0.508197i	1.71740	+	1.02496i	0.868914	−	2.06034i	−1.86371	+	0.231658i	0.810621			−1.95613	−	2.04293i	−0.874309	+	0.874309i	−1.95910	+	2.48232i
3.2	−1.33787	+	0.458371i	−0.528116	+	0.218753i	1.57979	−	1.22648i	−2.13281	−	0.671666i	0.606280	−	0.534736i	−0.814088			−1.55137	+	2.36500i	−1.89027	+	1.89027i	3.16129	−	0.0790144i
3.3	−1.32621	+	0.491080i	−2.16430	+	0.896482i	1.51768	−	1.30255i	2.08728	−	0.802025i	2.43008	−	2.25177i	0.225996			−1.37311	+	2.47277i	1.75919	−	1.75919i	−2.37432	+	2.08868i
3.4	−1.26065	−	0.640905i	−0.237464	+	0.0983610i	1.17848	+	1.61591i	0.189205	+	2.22805i	0.362400	+	0.0281932i	−4.12414			−0.450008	−	2.79240i	−2.07461	+	2.07461i	1.18944	−	2.93005i
3.5	−1.15485	−	0.816290i	−3.09930	+	1.28377i	0.667342	+	1.88538i	−2.14296	−	0.638517i	4.62714	+	1.04737i	0.906290			0.768338	−	2.72207i	5.83625	−	5.83625i	1.95358	+	2.48667i
3.6	−0.971453	−	1.02775i	2.54348	−	1.05354i	−0.112559	+	1.99683i	−1.68821	+	1.46627i	−3.55365	−	1.59060i	4.43630			2.16160	−	1.82414i	3.23800	−	3.23800i	3.14698	+	0.310658i
3.7	−0.949552	+	1.04802i	0.616647	−	0.255424i	−0.196703	−	1.99030i	−0.915235	+	2.04018i	−0.317849	+	0.888798i	2.27809			2.27266	+	1.68375i	−1.80631	+	1.80631i	−1.26909	−	2.89645i
3.8	−0.762560	+	1.19101i	2.68192	−	1.11089i	−0.837005	−	1.81643i	−0.603584	−	2.15306i	−0.722048	+	4.04131i	−0.874514			2.80165	+	0.388258i	3.83732	−	3.83732i	3.02459	+	0.922966i
3.9	−0.535863	−	1.30876i	−1.11473	+	0.461737i	−1.42570	+	1.40263i	2.18433	+	0.478219i	1.20165	+	1.21149i	2.85280			2.59969	+	1.11428i	−1.09189	+	1.09189i	−0.544629	−	3.11502i
3.10	−0.343520	+	1.37186i	−2.69126	+	1.11476i	−1.76399	−	0.942521i	0.100236	+	2.23382i	−0.604785	−	4.07496i	−0.518179			1.89897	−	2.09617i	3.87887	−	3.87887i	−3.09892	−	0.629853i
3.11	−0.178125	−	1.40295i	2.54622	−	1.05468i	−1.93654	+	0.499802i	2.21873	−	0.277906i	−1.93321	−	3.38436i	−3.70855			1.04614	+	2.62785i	3.24959	−	3.24959i	−0.785101	−	3.06327i
3.12	0.0556644	+	1.41312i	0.532554	−	0.220591i	−1.99380	+	0.157321i	2.20626	−	0.363921i	0.341365	+	0.740282i	3.48272			−0.333297	−	2.80872i	−1.88637	+	1.88637i	0.637074	+	3.09744i
3.13	0.184269	+	1.40216i	−1.10776	+	0.458849i	−1.93209	+	0.516749i	−1.41630	−	1.73035i	−0.847505	−	1.46870i	−4.27741			−1.08059	−	2.61387i	−1.10473	+	1.10473i	2.16524	−	2.30472i
3.14	0.500260	+	1.32278i	2.50226	−	1.03647i	−1.49948	+	1.32347i	0.0688146	+	2.23501i	2.62280	+	2.79143i	−2.65674			−2.50078	−	1.32140i	3.06572	−	3.06572i	−2.92199	+	1.20911i
3.15	0.521383	−	1.31460i	−2.23011	+	0.923741i	−1.45632	−	1.37081i	0.881968	−	2.05478i	0.0516058	+	3.41331i	−3.63945			−2.56137	+	1.19975i	1.99876	−	1.99876i	−2.24136	−	2.23076i
3.16	0.551886	−	1.30208i	1.39485	−	0.577765i	−1.39084	−	1.43720i	−1.78859	−	1.34199i	0.0174990	−	2.13507i	1.62907			−2.63895	+	1.01782i	−0.509534	+	0.509534i	−2.73448	+	1.58827i
3.17	1.00177	+	0.998231i	−1.22899	+	0.509063i	0.00706901	+	1.99999i	−1.94326	+	1.10623i	−1.73932	−	0.716851i	2.73471			−1.98937	+	2.01058i	−0.870059	+	0.870059i	−3.05096	−	0.831636i
3.18	1.00742	−	0.992527i	0.673021	−	0.278775i	0.0297801	−	1.99978i	1.32500	+	1.80122i	0.401322	−	0.948834i	0.467309			−1.95483	−	2.04417i	−1.74608	+	1.74608i	3.12258	+	0.499480i
3.19	1.19311	+	0.759267i	0.608697	−	0.252131i	0.847028	+	1.81178i	1.26769	−	1.84200i	0.917677	+	0.161344i	−1.49067			−0.365026	+	2.80477i	−1.81438	+	1.81438i	2.91106	−	1.23519i
3.20	1.35631	−	0.400521i	−1.51557	+	0.627770i	1.67917	−	1.08646i	−0.661993	−	2.13583i	−1.80415	+	1.45847i	4.80429			1.84232	−	2.14613i	−0.218458	+	0.218458i	−1.75331	−	2.63171i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
160.ba	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.ba.a	yes	88
4.b	odd	2	1		640.2.ba.a		88
5.b	even	2	1		800.2.bb.b		88
5.c	odd	4	1		160.2.u.a	✓	88
5.c	odd	4	1		800.2.v.b		88
20.e	even	4	1		640.2.u.a		88
32.g	even	8	1		640.2.u.a		88
32.h	odd	8	1		160.2.u.a	✓	88
160.u	even	8	1		800.2.bb.b		88
160.v	odd	8	1		640.2.ba.a		88
160.y	odd	8	1		800.2.v.b		88
160.ba	even	8	1	inner	160.2.ba.a	yes	88

    

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

Hecke kernels

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