Embedded newform 105.3.o.b.74.1

• Label: 105.3.o.b.74.1
• Level: $105$
• Weight: $3$
• Character: 105.74
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $40$
• 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:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			74.1
Character	\(\chi\)	\(=\)	105.74
Dual form			105.3.o.b.44.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85988 - 3.22141i) q^{2} +(-2.47330 - 1.69788i) q^{3} +(-4.91833 + 8.51879i) q^{4} +(4.36624 + 2.43637i) q^{5} +(-0.869509 + 11.1254i) q^{6} +(-4.87678 + 5.02166i) q^{7} +21.7110 q^{8} +(3.23443 + 8.39872i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 44 q^{4} + 80 q^{6} + 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85988	−	3.22141i	−0.929941	−	1.61071i	−0.783415	−	0.621499i	\(-0.786524\pi\)
							−0.146526	−	0.989207i	\(-0.546809\pi\)
\(3\)	−2.47330	−	1.69788i	−0.824434	−	0.565959i
							\(5\)	4.36624	+	2.43637i	0.873249	+	0.487275i
\(7\)	−4.87678	+	5.02166i	−0.696682	+	0.717380i
							\(11\)	−10.0814	−	5.82052i	−0.916494	−	0.529138i	−0.0339793	−	0.999423i	\(-0.510818\pi\)
−0.882515	+	0.470284i	\(0.844151\pi\)
\(13\)			9.22710i			0.709777i	0.934909	+	0.354888i	\(0.115481\pi\)
							−0.934909	+	0.354888i	\(0.884519\pi\)
\(17\)	−1.56346	+	2.70799i	−0.0919682	+	0.159294i	−0.908339	−	0.418234i	\(-0.862649\pi\)
							0.816371	+	0.577528i	\(0.195982\pi\)
\(19\)	5.39398	+	9.34265i	0.283894	+	0.491719i	0.972340	−	0.233569i	\(-0.0750403\pi\)
							−0.688447	+	0.725287i	\(0.741707\pi\)
\(23\)	2.93334	+	5.08070i	0.127537	+	0.220900i	0.922722	−	0.385467i	\(-0.125960\pi\)
							−0.795185	+	0.606367i	\(0.792626\pi\)
\(29\)			38.3541i			1.32256i	0.750141	+	0.661278i	\(0.229986\pi\)
							−0.750141	+	0.661278i	\(0.770014\pi\)
\(31\)	−15.7425	+	27.2669i	−0.507824	+	0.879577i	0.492135	+	0.870519i	\(0.336217\pi\)
							−0.999959	+	0.00905794i	\(0.997117\pi\)
\(37\)	−20.0791	+	11.5927i	−0.542678	+	0.313315i	−0.746164	−	0.665763i	\(-0.768106\pi\)
							0.203486	+	0.979078i	\(0.434773\pi\)
\(41\)		−	22.7035i		−	0.553744i	−0.960907	−	0.276872i	\(-0.910702\pi\)
							0.960907	−	0.276872i	\(-0.0892978\pi\)
\(43\)			29.1447i			0.677784i	0.940825	+	0.338892i	\(0.110052\pi\)
							−0.940825	+	0.338892i	\(0.889948\pi\)
\(47\)	−30.0800	−	52.1000i	−0.639999	−	1.10851i	−0.985432	−	0.170068i	\(-0.945601\pi\)
							0.345433	−	0.938443i	\(-0.387732\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.o.b.74.1	yes	40
3.2	odd	2	inner	105.3.o.b.74.19	yes	40
5.4	even	2	inner	105.3.o.b.74.20	yes	40
7.2	even	3	inner	105.3.o.b.44.2	yes	40
15.14	odd	2	inner	105.3.o.b.74.2	yes	40
21.2	odd	6	inner	105.3.o.b.44.20	yes	40
35.9	even	6	inner	105.3.o.b.44.19	yes	40
105.44	odd	6	inner	105.3.o.b.44.1	✓	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.o.b.44.1	✓	40	105.44	odd	6	inner
105.3.o.b.44.2	yes	40	7.2	even	3	inner
105.3.o.b.44.19	yes	40	35.9	even	6	inner
105.3.o.b.44.20	yes	40	21.2	odd	6	inner
105.3.o.b.74.1	yes	40	1.1	even	1	trivial
105.3.o.b.74.2	yes	40	15.14	odd	2	inner
105.3.o.b.74.19	yes	40	3.2	odd	2	inner
105.3.o.b.74.20	yes	40	5.4	even	2	inner