Embedded newform 105.4.s.b.26.9

• Label: 105.4.s.b.26.9
• Level: $105$
• Weight: $4$
• Character: 105.26
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			26.9
Character	\(\chi\)	\(=\)	105.26
Dual form			105.4.s.b.101.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.133140 + 0.0768685i) q^{2} +(-2.72308 - 4.42548i) q^{3} +(-3.98818 - 6.90773i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-0.0223714 - 0.798528i) q^{6} +(-15.0457 + 10.7995i) q^{7} -2.45616i q^{8} +(-12.1697 + 24.1018i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2 q^{3} + 64 q^{4} + 80 q^{5} + 28 q^{6} + 46 q^{7} - 98 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.133140	+	0.0768685i	0.0470722	+	0.0271771i	0.523351	−	0.852117i	\(-0.324682\pi\)
							−0.476279	+	0.879294i	\(0.658015\pi\)
\(3\)	−2.72308	−	4.42548i	−0.524057	−	0.851683i
							\(5\)	2.50000	−	4.33013i	0.223607	−	0.387298i
\(7\)	−15.0457	+	10.7995i	−0.812389	+	0.583116i
							\(11\)	14.9856	−	8.65197i	0.410758	−	0.237152i	−0.280357	−	0.959896i	\(-0.590453\pi\)
0.691116	+	0.722744i	\(0.257120\pi\)
\(13\)			36.4846i			0.778385i	0.921156	+	0.389192i	\(0.127246\pi\)
							−0.921156	+	0.389192i	\(0.872754\pi\)
\(17\)	14.8621	+	25.7418i	0.212034	+	0.367254i	0.952351	−	0.305004i	\(-0.0986579\pi\)
							−0.740317	+	0.672258i	\(0.765325\pi\)
\(19\)	−112.843	−	65.1498i	−1.36252	−	0.786652i	−0.372562	−	0.928007i	\(-0.621521\pi\)
							−0.989959	+	0.141355i	\(0.954854\pi\)
\(23\)	−134.961	−	77.9197i	−1.22353	−	0.706408i	−0.257865	−	0.966181i	\(-0.583019\pi\)
							−0.965670	+	0.259773i	\(0.916352\pi\)
\(29\)		−	165.529i		−	1.05993i	−0.848021	−	0.529963i	\(-0.822206\pi\)
							0.848021	−	0.529963i	\(-0.177794\pi\)
\(31\)	12.9990	−	7.50496i	0.0753123	−	0.0434816i	−0.461871	−	0.886947i	\(-0.652822\pi\)
							0.537183	+	0.843466i	\(0.319488\pi\)
\(37\)	−16.8835	+	29.2431i	−0.0750170	+	0.129933i	−0.901094	−	0.433625i	\(-0.857234\pi\)
							0.826077	+	0.563558i	\(0.190568\pi\)
\(41\)	−274.227			−1.04456			−0.522282	−	0.852773i	\(-0.674919\pi\)
							−0.522282	+	0.852773i	\(0.674919\pi\)
\(43\)	248.354			0.880784			0.440392	−	0.897806i	\(-0.354840\pi\)
							0.440392	+	0.897806i	\(0.354840\pi\)
\(47\)	−229.515	+	397.532i	−0.712302	+	1.23374i	0.251689	+	0.967808i	\(0.419014\pi\)
							−0.963991	+	0.265936i	\(0.914319\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.b.26.9	yes	32
3.2	odd	2		105.4.s.a.26.8	✓	32
7.3	odd	6		105.4.s.a.101.8	yes	32
21.17	even	6	inner	105.4.s.b.101.9	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.8	✓	32	3.2	odd	2
105.4.s.a.101.8	yes	32	7.3	odd	6
105.4.s.b.26.9	yes	32	1.1	even	1	trivial
105.4.s.b.101.9	yes	32	21.17	even	6	inner