Embedded newform 160.4.ba.a.3.4

• Label: 160.4.ba.a.3.4
• Level: $160$
• Weight: $4$
• Character: 160.3
• Analytic conductor: $9.440$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.44030560092\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(70\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			3.4
Character	\(\chi\)	\(=\)	160.3
Dual form			160.4.ba.a.107.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.82209 + 0.189157i) q^{2} +(7.87428 - 3.26164i) q^{3} +(7.92844 - 1.06764i) q^{4} +(-5.83539 - 9.53668i) q^{5} +(-21.6050 + 10.6941i) q^{6} +21.2036 q^{7} +(-22.1729 + 4.51268i) q^{8} +(32.2742 - 32.2742i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} - 8 q^{7} - 88 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(97\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{3}{8}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.82209	+	0.189157i	−0.997761	+	0.0668769i
							\(3\)	7.87428	−	3.26164i	1.51541	−	0.627702i	0.538742	−	0.842471i	\(-0.318900\pi\)
0.976665	+	0.214769i	\(0.0688999\pi\)
\(5\)	−5.83539	−	9.53668i	−0.521934	−	0.852986i
							\(7\)	21.2036			1.14489			0.572444	−	0.819944i	\(-0.305996\pi\)
0.572444	+	0.819944i	\(0.305996\pi\)
\(11\)	−12.2710	−	29.6248i	−0.336349	−	0.812019i	−0.998060	−	0.0622598i	\(-0.980169\pi\)
							0.661711	−	0.749759i	\(-0.269831\pi\)
\(13\)	21.2489	−	8.80158i	0.453337	−	0.187778i	−0.144319	−	0.989531i	\(-0.546099\pi\)
							0.597656	+	0.801753i	\(0.296099\pi\)
\(17\)	0.104581	+	0.104581i	0.00149204	+	0.00149204i	0.707852	−	0.706360i	\(-0.249664\pi\)
							−0.706360	+	0.707852i	\(0.749664\pi\)
\(19\)	−47.5734	+	114.852i	−0.574426	+	1.38679i	0.323327	+	0.946287i	\(0.395199\pi\)
							−0.897753	+	0.440500i	\(0.854801\pi\)
\(23\)	−69.0960			−0.626414			−0.313207	−	0.949685i	\(-0.601403\pi\)
							−0.313207	+	0.949685i	\(0.601403\pi\)
\(29\)	109.313	−	263.904i	0.699960	−	1.68985i	−0.0237208	−	0.999719i	\(-0.507551\pi\)
							0.723681	−	0.690135i	\(-0.242449\pi\)
\(31\)		−	178.805i		−	1.03595i	−0.855397	−	0.517973i	\(-0.826687\pi\)
							0.855397	−	0.517973i	\(-0.173313\pi\)
\(37\)	70.5198	+	29.2102i	0.313335	+	0.129787i	0.533808	−	0.845605i	\(-0.320760\pi\)
							−0.220474	+	0.975393i	\(0.570760\pi\)
\(41\)	294.509	+	294.509i	1.12182	+	1.12182i	0.991468	+	0.130353i	\(0.0416110\pi\)
							0.130353	+	0.991468i	\(0.458389\pi\)
\(43\)	114.746	−	277.022i	0.406945	−	0.982451i	−0.578992	−	0.815333i	\(-0.696554\pi\)
							0.985937	−	0.167118i	\(-0.0534461\pi\)
\(47\)	30.1639	−	30.1639i	0.0936139	−	0.0936139i	−0.658749	−	0.752363i	\(-0.728914\pi\)
							0.752363	+	0.658749i	\(0.228914\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.4.ba.a.3.4	yes	280
5.2	odd	4		160.4.u.a.67.34	yes	280
32.11	odd	8		160.4.u.a.43.34	✓	280
160.107	even	8	inner	160.4.ba.a.107.4	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.u.a.43.34	✓	280	32.11	odd	8
160.4.u.a.67.34	yes	280	5.2	odd	4
160.4.ba.a.3.4	yes	280	1.1	even	1	trivial
160.4.ba.a.107.4	yes	280	160.107	even	8	inner