Embedded newform 105.3.t.b.86.9

• Label: 105.3.t.b.86.9
• 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			86.9
Character	\(\chi\)	\(=\)	105.86
Dual form			105.3.t.b.11.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.644768 - 0.372257i) q^{2} +(-2.67627 - 1.35556i) q^{3} +(-1.72285 - 2.98406i) q^{4} +(1.93649 + 1.11803i) q^{5} +(1.22096 + 1.87029i) q^{6} +(-5.98242 + 3.63464i) q^{7} +5.54343i q^{8} +(5.32489 + 7.25573i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 4 q^{3} + 36 q^{4} - 24 q^{6} - 58 q^{7} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.644768	−	0.372257i	−0.322384	−	0.186129i	0.330071	−	0.943956i	\(-0.392927\pi\)
							−0.652455	+	0.757828i	\(0.726261\pi\)
\(3\)	−2.67627	−	1.35556i	−0.892091	−	0.451855i
							\(5\)	1.93649	+	1.11803i	0.387298	+	0.223607i
\(7\)	−5.98242	+	3.63464i	−0.854632	+	0.519234i
							\(11\)	−8.35340	+	4.82284i	−0.759400	+	0.438440i	−0.829080	−	0.559130i	\(-0.811135\pi\)
0.0696804	+	0.997569i	\(0.477802\pi\)
\(13\)	−0.0661184			−0.00508603			−0.00254302	−	0.999997i	\(-0.500809\pi\)
							−0.00254302	+	0.999997i	\(0.500809\pi\)
\(17\)	−28.1197	+	16.2349i	−1.65410	+	0.954995i	−0.678739	+	0.734380i	\(0.737473\pi\)
							−0.975361	+	0.220615i	\(0.929194\pi\)
\(19\)	12.7759	−	22.1286i	0.672418	−	1.16466i	−0.304798	−	0.952417i	\(-0.598589\pi\)
							0.977216	−	0.212246i	\(-0.0680777\pi\)
\(23\)	−8.49789	−	4.90626i	−0.369473	−	0.213316i	0.303755	−	0.952750i	\(-0.401760\pi\)
							−0.673228	+	0.739435i	\(0.735093\pi\)
\(29\)			6.58972i			0.227232i	0.993525	+	0.113616i	\(0.0362433\pi\)
							−0.993525	+	0.113616i	\(0.963757\pi\)
\(31\)	−16.4028	−	28.4105i	−0.529123	−	0.916467i	−0.999423	−	0.0339609i	\(-0.989188\pi\)
							0.470301	−	0.882506i	\(-0.344146\pi\)
\(37\)	−27.3632	+	47.3945i	−0.739546	+	1.28093i	0.213154	+	0.977019i	\(0.431627\pi\)
							−0.952700	+	0.303913i	\(0.901707\pi\)
\(41\)		−	14.8128i		−	0.361287i	−0.983549	−	0.180643i	\(-0.942182\pi\)
							0.983549	−	0.180643i	\(-0.0578180\pi\)
\(43\)	−14.3286			−0.333223			−0.166612	−	0.986023i	\(-0.553283\pi\)
							−0.166612	+	0.986023i	\(0.553283\pi\)
\(47\)	−63.7562	−	36.8097i	−1.35652	−	0.783184i	−0.367363	−	0.930078i	\(-0.619739\pi\)
							−0.989152	+	0.146893i	\(0.953073\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.t.b.86.9	yes	36
3.2	odd	2	inner	105.3.t.b.86.10	yes	36
7.4	even	3	inner	105.3.t.b.11.10	yes	36
21.11	odd	6	inner	105.3.t.b.11.9	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.t.b.11.9	✓	36	21.11	odd	6	inner
105.3.t.b.11.10	yes	36	7.4	even	3	inner
105.3.t.b.86.9	yes	36	1.1	even	1	trivial
105.3.t.b.86.10	yes	36	3.2	odd	2	inner