Embedded newform 105.3.t.b.86.1

• Label: 105.3.t.b.86.1
• Level: $105$
• Weight: $3$
• Character: 105.86
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			86.1
Character	\(\chi\)	\(=\)	105.86
Dual form			105.3.t.b.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.31814 - 1.91573i) q^{2} +(1.84164 - 2.36819i) q^{3} +(5.34002 + 9.24919i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(-10.6476 + 4.32991i) q^{6} +(-5.86414 - 3.82255i) q^{7} -25.5943i q^{8} +(-2.21669 - 8.72274i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 4 q^{3} + 36 q^{4} - 24 q^{6} - 58 q^{7} - 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.31814	−	1.91573i	−1.65907	−	0.957864i	−0.973146	−	0.230189i	\(-0.926066\pi\)
							−0.685923	−	0.727675i	\(-0.740601\pi\)
\(3\)	1.84164	−	2.36819i	0.613881	−	0.789398i
							\(5\)	−1.93649	−	1.11803i	−0.387298	−	0.223607i
\(7\)	−5.86414	−	3.82255i	−0.837734	−	0.546078i
							\(11\)	−9.48205	+	5.47446i	−0.862004	+	0.497679i	−0.864683	−	0.502318i	\(-0.832481\pi\)
0.00267854	+	0.999996i	\(0.499147\pi\)
\(13\)	−3.75260			−0.288662			−0.144331	−	0.989529i	\(-0.546103\pi\)
							−0.144331	+	0.989529i	\(0.546103\pi\)
\(17\)	−12.6515	+	7.30435i	−0.744206	+	0.429668i	−0.823597	−	0.567176i	\(-0.808036\pi\)
							0.0793904	+	0.996844i	\(0.474703\pi\)
\(19\)	5.54612	−	9.60616i	0.291901	−	0.505587i	−0.682358	−	0.731018i	\(-0.739046\pi\)
							0.974259	+	0.225431i	\(0.0723789\pi\)
\(23\)	8.53608	+	4.92831i	0.371134	+	0.214274i	0.673954	−	0.738774i	\(-0.264595\pi\)
							−0.302820	+	0.953048i	\(0.597928\pi\)
\(29\)			10.0771i			0.347485i	0.984791	+	0.173743i	\(0.0555861\pi\)
							−0.984791	+	0.173743i	\(0.944414\pi\)
\(31\)	−12.0674	−	20.9013i	−0.389270	−	0.674236i	0.603081	−	0.797680i	\(-0.293939\pi\)
							−0.992352	+	0.123444i	\(0.960606\pi\)
\(37\)	19.1895	−	33.2372i	0.518636	−	0.898303i	−0.481130	−	0.876649i	\(-0.659773\pi\)
							0.999766	−	0.0216538i	\(-0.00689317\pi\)
\(41\)		−	67.5044i		−	1.64645i	−0.567716	−	0.823224i	\(-0.692173\pi\)
							0.567716	−	0.823224i	\(-0.307827\pi\)
\(43\)	−77.4222			−1.80052			−0.900258	−	0.435356i	\(-0.856623\pi\)
							−0.900258	+	0.435356i	\(0.856623\pi\)
\(47\)	−4.91770	−	2.83924i	−0.104632	−	0.0604093i	0.446771	−	0.894648i	\(-0.352574\pi\)
							−0.551403	+	0.834239i	\(0.685907\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.t.b.86.1	yes	36
3.2	odd	2	inner	105.3.t.b.86.18	yes	36
7.4	even	3	inner	105.3.t.b.11.18	yes	36
21.11	odd	6	inner	105.3.t.b.11.1	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.t.b.11.1	✓	36	21.11	odd	6	inner
105.3.t.b.11.18	yes	36	7.4	even	3	inner
105.3.t.b.86.1	yes	36	1.1	even	1	trivial
105.3.t.b.86.18	yes	36	3.2	odd	2	inner