Embedded newform 105.3.o.a.74.3

• Label: 105.3.o.a.74.3
• Level: $105$
• Weight: $3$
• Character: 105.74
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	105.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 4*x^15 - 4*x^14 + 12*x^13 + 162*x^12 - 524*x^11 - 88*x^10 + 1492*x^9 + 6266*x^8 - 41348*x^7 + 103092*x^6 - 152724*x^5 + 148342*x^4 - 96348*x^3 + 41016*x^2 - 10764*x + 1521
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			74.3
Root			\(-2.89591 - 1.43281i\) of defining polynomial
Character	\(\chi\)	\(=\)	105.74
Dual form			105.3.o.a.44.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.530805 - 0.919382i) q^{2} +(-1.89916 - 2.32232i) q^{3} +(1.43649 - 2.48808i) q^{4} +(-0.901243 - 4.91811i) q^{5} +(-1.12702 + 2.97876i) q^{6} +(3.04726 + 6.30192i) q^{7} -7.29643 q^{8} +(-1.78637 + 8.82093i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} - 80 q^{6} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.530805	−	0.919382i	−0.265403	−	0.459691i	0.702266	−	0.711914i	\(-0.252172\pi\)
							−0.967669	+	0.252223i	\(0.918838\pi\)
\(3\)	−1.89916	−	2.32232i	−0.633054	−	0.774108i
							\(5\)	−0.901243	−	4.91811i	−0.180249	−	0.983621i
\(7\)	3.04726	+	6.30192i	0.435323	+	0.900274i
							\(11\)	−9.99105	−	5.76833i	−0.908277	−	0.524394i	−0.0284008	−	0.999597i	\(-0.509041\pi\)
−0.879876	+	0.475203i	\(0.842375\pi\)
\(13\)		−	5.56650i		−	0.428192i	−0.976813	−	0.214096i	\(-0.931319\pi\)
							0.976813	−	0.214096i	\(-0.0686806\pi\)
\(17\)	7.82724	−	13.5572i	0.460426	−	0.797481i	−0.538556	−	0.842589i	\(-0.681030\pi\)
							0.998982	+	0.0451088i	\(0.0143635\pi\)
\(19\)	1.06351	+	1.84205i	0.0559741	+	0.0969500i	0.892655	−	0.450741i	\(-0.148840\pi\)
							−0.836681	+	0.547691i	\(0.815507\pi\)
\(23\)	13.0004	+	22.5174i	0.565237	+	0.979019i	0.997028	+	0.0770450i	\(0.0245485\pi\)
							−0.431791	+	0.901974i	\(0.642118\pi\)
\(29\)		−	13.3804i		−	0.461392i	−0.973026	−	0.230696i	\(-0.925900\pi\)
							0.973026	−	0.230696i	\(-0.0741003\pi\)
\(31\)	25.3730	−	43.9473i	0.818483	−	1.41765i	−0.0883161	−	0.996092i	\(-0.528149\pi\)
							0.906799	−	0.421562i	\(-0.138518\pi\)
\(37\)	19.8268	−	11.4470i	0.535859	−	0.309378i	−0.207540	−	0.978227i	\(-0.566546\pi\)
							0.743399	+	0.668848i	\(0.233212\pi\)
\(41\)		−	53.9959i		−	1.31697i	−0.752592	−	0.658487i	\(-0.771197\pi\)
							0.752592	−	0.658487i	\(-0.228803\pi\)
\(43\)		−	38.8354i		−	0.903150i	−0.892233	−	0.451575i	\(-0.850862\pi\)
							0.892233	−	0.451575i	\(-0.149138\pi\)
\(47\)	2.72145	+	4.71368i	0.0579031	+	0.100291i	0.893524	−	0.449016i	\(-0.148225\pi\)
							−0.835621	+	0.549307i	\(0.814892\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.o.a.74.3	yes	16
3.2	odd	2	inner	105.3.o.a.74.5	yes	16
5.4	even	2	inner	105.3.o.a.74.6	yes	16
7.2	even	3	inner	105.3.o.a.44.4	yes	16
15.14	odd	2	inner	105.3.o.a.74.4	yes	16
21.2	odd	6	inner	105.3.o.a.44.6	yes	16
35.9	even	6	inner	105.3.o.a.44.5	yes	16
105.44	odd	6	inner	105.3.o.a.44.3	✓	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.o.a.44.3	✓	16	105.44	odd	6	inner
105.3.o.a.44.4	yes	16	7.2	even	3	inner
105.3.o.a.44.5	yes	16	35.9	even	6	inner
105.3.o.a.44.6	yes	16	21.2	odd	6	inner
105.3.o.a.74.3	yes	16	1.1	even	1	trivial
105.3.o.a.74.4	yes	16	15.14	odd	2	inner
105.3.o.a.74.5	yes	16	3.2	odd	2	inner
105.3.o.a.74.6	yes	16	5.4	even	2	inner