Embedded newform 160.3.v.a.13.11

• Label: 160.3.v.a.13.11
• Level: $160$
• Weight: $3$
• Character: 160.13
• Analytic conductor: $4.360$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	160.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.35968422976\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(46\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			13.11
Character	\(\chi\)	\(=\)	160.13
Dual form			160.3.v.a.37.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53272 - 1.28482i) q^{2} +(-0.518083 + 0.214597i) q^{3} +(0.698458 + 3.93855i) q^{4} +(-1.24748 + 4.84188i) q^{5} +(1.06980 + 0.336728i) q^{6} -7.03161i q^{7} +(3.98980 - 6.93408i) q^{8} +(-6.14160 + 6.14160i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} + 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(97\)	\(101\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{7}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.53272	−	1.28482i	−0.766360	−	0.642412i
							\(3\)	−0.518083	+	0.214597i	−0.172694	+	0.0715324i	−0.467356	−	0.884069i	\(-0.654793\pi\)
0.294661	+	0.955602i	\(0.404793\pi\)
\(5\)	−1.24748	+	4.84188i	−0.249496	+	0.968376i
							\(7\)		−	7.03161i		−	1.00452i	−0.864718	−	0.502258i	\(-0.832503\pi\)
0.864718	−	0.502258i	\(-0.167497\pi\)
\(11\)	−5.00915	−	12.0932i	−0.455377	−	1.09938i	−0.970249	−	0.242110i	\(-0.922160\pi\)
							0.514872	−	0.857267i	\(-0.327840\pi\)
\(13\)	−5.86338	−	14.1555i	−0.451030	−	1.08888i	−0.971931	−	0.235265i	\(-0.924404\pi\)
							0.520902	−	0.853617i	\(-0.325596\pi\)
\(17\)	−6.97383	−	6.97383i	−0.410225	−	0.410225i	0.471592	−	0.881817i	\(-0.343680\pi\)
							−0.881817	+	0.471592i	\(0.843680\pi\)
\(19\)	1.45405	−	3.51040i	0.0765292	−	0.184758i	−0.880985	−	0.473144i	\(-0.843119\pi\)
							0.957514	+	0.288387i	\(0.0931189\pi\)
\(23\)			30.7853i			1.33849i	0.743041	+	0.669246i	\(0.233383\pi\)
							−0.743041	+	0.669246i	\(0.766617\pi\)
\(29\)	−49.5646	−	20.5303i	−1.70912	−	0.707942i	−0.999996	−	0.00286094i	\(-0.999089\pi\)
							−0.709127	−	0.705081i	\(-0.750911\pi\)
\(31\)	−16.2127			−0.522989			−0.261495	−	0.965205i	\(-0.584215\pi\)
							−0.261495	+	0.965205i	\(0.584215\pi\)
\(37\)	13.9546	−	33.6894i	0.377151	−	0.910524i	−0.615346	−	0.788257i	\(-0.710984\pi\)
							0.992497	−	0.122267i	\(-0.0390163\pi\)
\(41\)	14.1732	+	14.1732i	0.345688	+	0.345688i	0.858500	−	0.512813i	\(-0.171397\pi\)
							−0.512813	+	0.858500i	\(0.671397\pi\)
\(43\)	19.4815	−	47.0325i	0.453058	−	1.09378i	−0.518096	−	0.855323i	\(-0.673359\pi\)
							0.971153	−	0.238456i	\(-0.0766412\pi\)
\(47\)	−2.83122	−	2.83122i	−0.0602388	−	0.0602388i	0.676346	−	0.736584i	\(-0.263563\pi\)
							−0.736584	+	0.676346i	\(0.763563\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.v.a.13.11	✓	184
5.2	odd	4		160.3.bb.a.77.34	yes	184
32.5	even	8		160.3.bb.a.133.34	yes	184
160.37	odd	8	inner	160.3.v.a.37.11	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.v.a.13.11	✓	184	1.1	even	1	trivial
160.3.v.a.37.11	yes	184	160.37	odd	8	inner
160.3.bb.a.77.34	yes	184	5.2	odd	4
160.3.bb.a.133.34	yes	184	32.5	even	8